src/HOL/Analysis/Connected.thy
 author immler Tue Jul 10 09:38:35 2018 +0200 (11 months ago) changeset 68607 67bb59e49834 parent 68527 2f4e2aab190a child 69064 5840724b1d71 permissions -rw-r--r--
make theorem, corollary, and proposition %important for HOL-Analysis manual
     1 (*  Author:     L C Paulson, University of Cambridge

     2     Material split off from Topology_Euclidean_Space

     3 *)

     4

     5 section \<open>Connected Components, Homeomorphisms, Baire property, etc\<close>

     6

     7 theory Connected

     8 imports Topology_Euclidean_Space

     9 begin

    10

    11 subsection%unimportant \<open>More properties of closed balls, spheres, etc\<close>

    12

    13 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

    14   apply (simp add: interior_def, safe)

    15   apply (force simp: open_contains_cball)

    16   apply (rule_tac x="ball x e" in exI)

    17   apply (simp add: subset_trans [OF ball_subset_cball])

    18   done

    19

    20 lemma islimpt_ball:

    21   fixes x y :: "'a::{real_normed_vector,perfect_space}"

    22   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"

    23   (is "?lhs \<longleftrightarrow> ?rhs")

    24 proof

    25   show ?rhs if ?lhs

    26   proof

    27     {

    28       assume "e \<le> 0"

    29       then have *: "ball x e = {}"

    30         using ball_eq_empty[of x e] by auto

    31       have False using \<open>?lhs\<close>

    32         unfolding * using islimpt_EMPTY[of y] by auto

    33     }

    34     then show "e > 0" by (metis not_less)

    35     show "y \<in> cball x e"

    36       using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

    37         ball_subset_cball[of x e] \<open>?lhs\<close>

    38       unfolding closed_limpt by auto

    39   qed

    40   show ?lhs if ?rhs

    41   proof -

    42     from that have "e > 0" by auto

    43     {

    44       fix d :: real

    45       assume "d > 0"

    46       have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

    47       proof (cases "d \<le> dist x y")

    48         case True

    49         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

    50         proof (cases "x = y")

    51           case True

    52           then have False

    53             using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto

    54           then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

    55             by auto

    56         next

    57           case False

    58           have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =

    59             norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

    60             unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]

    61             by auto

    62           also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

    63             using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]

    64             unfolding scaleR_minus_left scaleR_one

    65             by (auto simp: norm_minus_commute)

    66           also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

    67             unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

    68             unfolding distrib_right using \<open>x\<noteq>y\<close>  by auto

    69           also have "\<dots> \<le> e - d/2" using \<open>d \<le> dist x y\<close> and \<open>d>0\<close> and \<open>?rhs\<close>

    70             by (auto simp: dist_norm)

    71           finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using \<open>d>0\<close>

    72             by auto

    73           moreover

    74           have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

    75             using \<open>x\<noteq>y\<close>[unfolded dist_nz] \<open>d>0\<close> unfolding scaleR_eq_0_iff

    76             by (auto simp: dist_commute)

    77           moreover

    78           have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"

    79             unfolding dist_norm

    80             apply simp

    81             unfolding norm_minus_cancel

    82             using \<open>d > 0\<close> \<open>x\<noteq>y\<close>[unfolded dist_nz] dist_commute[of x y]

    83             unfolding dist_norm

    84             apply auto

    85             done

    86           ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

    87             apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)

    88             apply auto

    89             done

    90         qed

    91       next

    92         case False

    93         then have "d > dist x y" by auto

    94         show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"

    95         proof (cases "x = y")

    96           case True

    97           obtain z where **: "z \<noteq> y" "dist z y < min e d"

    98             using perfect_choose_dist[of "min e d" y]

    99             using \<open>d > 0\<close> \<open>e>0\<close> by auto

   100           show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

   101             unfolding \<open>x = y\<close>

   102             using \<open>z \<noteq> y\<close> **

   103             apply (rule_tac x=z in bexI)

   104             apply (auto simp: dist_commute)

   105             done

   106         next

   107           case False

   108           then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

   109             using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close>

   110             apply (rule_tac x=x in bexI, auto)

   111             done

   112         qed

   113       qed

   114     }

   115     then show ?thesis

   116       unfolding mem_cball islimpt_approachable mem_ball by auto

   117   qed

   118 qed

   119

   120 lemma closure_ball_lemma:

   121   fixes x y :: "'a::real_normed_vector"

   122   assumes "x \<noteq> y"

   123   shows "y islimpt ball x (dist x y)"

   124 proof (rule islimptI)

   125   fix T

   126   assume "y \<in> T" "open T"

   127   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

   128     unfolding open_dist by fast

   129   (* choose point between x and y, within distance r of y. *)

   130   define k where "k = min 1 (r / (2 * dist x y))"

   131   define z where "z = y + scaleR k (x - y)"

   132   have z_def2: "z = x + scaleR (1 - k) (y - x)"

   133     unfolding z_def by (simp add: algebra_simps)

   134   have "dist z y < r"

   135     unfolding z_def k_def using \<open>0 < r\<close>

   136     by (simp add: dist_norm min_def)

   137   then have "z \<in> T"

   138     using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp

   139   have "dist x z < dist x y"

   140     unfolding z_def2 dist_norm

   141     apply (simp add: norm_minus_commute)

   142     apply (simp only: dist_norm [symmetric])

   143     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

   144     apply (rule mult_strict_right_mono)

   145     apply (simp add: k_def \<open>0 < r\<close> \<open>x \<noteq> y\<close>)

   146     apply (simp add: \<open>x \<noteq> y\<close>)

   147     done

   148   then have "z \<in> ball x (dist x y)"

   149     by simp

   150   have "z \<noteq> y"

   151     unfolding z_def k_def using \<open>x \<noteq> y\<close> \<open>0 < r\<close>

   152     by (simp add: min_def)

   153   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

   154     using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>

   155     by fast

   156 qed

   157

   158 lemma at_within_ball_bot_iff:

   159   fixes x y :: "'a::{real_normed_vector,perfect_space}"

   160   shows "at x within ball y r = bot \<longleftrightarrow> (r=0 \<or> x \<notin> cball y r)"

   161   unfolding trivial_limit_within

   162 apply (auto simp add:trivial_limit_within islimpt_ball )

   163 by (metis le_less_trans less_eq_real_def zero_le_dist)

   164

   165 lemma closure_ball [simp]:

   166   fixes x :: "'a::real_normed_vector"

   167   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

   168   apply (rule equalityI)

   169   apply (rule closure_minimal)

   170   apply (rule ball_subset_cball)

   171   apply (rule closed_cball)

   172   apply (rule subsetI, rename_tac y)

   173   apply (simp add: le_less [where 'a=real])

   174   apply (erule disjE)

   175   apply (rule subsetD [OF closure_subset], simp)

   176   apply (simp add: closure_def, clarify)

   177   apply (rule closure_ball_lemma)

   178   apply simp

   179   done

   180

   181 (* In a trivial vector space, this fails for e = 0. *)

   182 lemma interior_cball [simp]:

   183   fixes x :: "'a::{real_normed_vector, perfect_space}"

   184   shows "interior (cball x e) = ball x e"

   185 proof (cases "e \<ge> 0")

   186   case False note cs = this

   187   from cs have null: "ball x e = {}"

   188     using ball_empty[of e x] by auto

   189   moreover

   190   {

   191     fix y

   192     assume "y \<in> cball x e"

   193     then have False

   194       by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball subset_antisym subset_ball)

   195   }

   196   then have "cball x e = {}" by auto

   197   then have "interior (cball x e) = {}"

   198     using interior_empty by auto

   199   ultimately show ?thesis by blast

   200 next

   201   case True note cs = this

   202   have "ball x e \<subseteq> cball x e"

   203     using ball_subset_cball by auto

   204   moreover

   205   {

   206     fix S y

   207     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

   208     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"

   209       unfolding open_dist by blast

   210     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

   211       using perfect_choose_dist [of d] by auto

   212     have "xa \<in> S"

   213       using d[THEN spec[where x = xa]]

   214       using xa by (auto simp: dist_commute)

   215     then have xa_cball: "xa \<in> cball x e"

   216       using as(1) by auto

   217     then have "y \<in> ball x e"

   218     proof (cases "x = y")

   219       case True

   220       then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce

   221       then show "y \<in> ball x e"

   222         using \<open>x = y \<close> by simp

   223     next

   224       case False

   225       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"

   226         unfolding dist_norm

   227         using \<open>d>0\<close> norm_ge_zero[of "y - x"] \<open>x \<noteq> y\<close> by auto

   228       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"

   229         using d as(1)[unfolded subset_eq] by blast

   230       have "y - x \<noteq> 0" using \<open>x \<noteq> y\<close> by auto

   231       hence **:"d / (2 * norm (y - x)) > 0"

   232         unfolding zero_less_norm_iff[symmetric] using \<open>d>0\<close> by auto

   233       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =

   234         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

   235         by (auto simp: dist_norm algebra_simps)

   236       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

   237         by (auto simp: algebra_simps)

   238       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

   239         using ** by auto

   240       also have "\<dots> = (dist y x) + d/2"

   241         using ** by (auto simp: distrib_right dist_norm)

   242       finally have "e \<ge> dist x y +d/2"

   243         using *[unfolded mem_cball] by (auto simp: dist_commute)

   244       then show "y \<in> ball x e"

   245         unfolding mem_ball using \<open>d>0\<close> by auto

   246     qed

   247   }

   248   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"

   249     by auto

   250   ultimately show ?thesis

   251     using interior_unique[of "ball x e" "cball x e"]

   252     using open_ball[of x e]

   253     by auto

   254 qed

   255

   256 lemma interior_ball [simp]: "interior (ball x e) = ball x e"

   257   by (simp add: interior_open)

   258

   259 lemma frontier_ball [simp]:

   260   fixes a :: "'a::real_normed_vector"

   261   shows "0 < e \<Longrightarrow> frontier (ball a e) = sphere a e"

   262   by (force simp: frontier_def)

   263

   264 lemma frontier_cball [simp]:

   265   fixes a :: "'a::{real_normed_vector, perfect_space}"

   266   shows "frontier (cball a e) = sphere a e"

   267   by (force simp: frontier_def)

   268

   269 lemma cball_eq_empty [simp]: "cball x e = {} \<longleftrightarrow> e < 0"

   270   apply (simp add: set_eq_iff not_le)

   271   apply (metis zero_le_dist dist_self order_less_le_trans)

   272   done

   273

   274 lemma cball_empty [simp]: "e < 0 \<Longrightarrow> cball x e = {}"

   275   by simp

   276

   277 lemma cball_eq_sing:

   278   fixes x :: "'a::{metric_space,perfect_space}"

   279   shows "cball x e = {x} \<longleftrightarrow> e = 0"

   280 proof (rule linorder_cases)

   281   assume e: "0 < e"

   282   obtain a where "a \<noteq> x" "dist a x < e"

   283     using perfect_choose_dist [OF e] by auto

   284   then have "a \<noteq> x" "dist x a \<le> e"

   285     by (auto simp: dist_commute)

   286   with e show ?thesis by (auto simp: set_eq_iff)

   287 qed auto

   288

   289 lemma cball_sing:

   290   fixes x :: "'a::metric_space"

   291   shows "e = 0 \<Longrightarrow> cball x e = {x}"

   292   by (auto simp: set_eq_iff)

   293

   294 lemma ball_divide_subset: "d \<ge> 1 \<Longrightarrow> ball x (e/d) \<subseteq> ball x e"

   295   apply (cases "e \<le> 0")

   296   apply (simp add: ball_empty divide_simps)

   297   apply (rule subset_ball)

   298   apply (simp add: divide_simps)

   299   done

   300

   301 lemma ball_divide_subset_numeral: "ball x (e / numeral w) \<subseteq> ball x e"

   302   using ball_divide_subset one_le_numeral by blast

   303

   304 lemma cball_divide_subset: "d \<ge> 1 \<Longrightarrow> cball x (e/d) \<subseteq> cball x e"

   305   apply (cases "e < 0")

   306   apply (simp add: divide_simps)

   307   apply (rule subset_cball)

   308   apply (metis div_by_1 frac_le not_le order_refl zero_less_one)

   309   done

   310

   311 lemma cball_divide_subset_numeral: "cball x (e / numeral w) \<subseteq> cball x e"

   312   using cball_divide_subset one_le_numeral by blast

   313

   314 lemma compact_cball[simp]:

   315   fixes x :: "'a::heine_borel"

   316   shows "compact (cball x e)"

   317   using compact_eq_bounded_closed bounded_cball closed_cball

   318   by blast

   319

   320 lemma compact_frontier_bounded[intro]:

   321   fixes S :: "'a::heine_borel set"

   322   shows "bounded S \<Longrightarrow> compact (frontier S)"

   323   unfolding frontier_def

   324   using compact_eq_bounded_closed

   325   by blast

   326

   327 lemma compact_frontier[intro]:

   328   fixes S :: "'a::heine_borel set"

   329   shows "compact S \<Longrightarrow> compact (frontier S)"

   330   using compact_eq_bounded_closed compact_frontier_bounded

   331   by blast

   332

   333 corollary compact_sphere [simp]:

   334   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"

   335   shows "compact (sphere a r)"

   336 using compact_frontier [of "cball a r"] by simp

   337

   338 corollary bounded_sphere [simp]:

   339   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"

   340   shows "bounded (sphere a r)"

   341 by (simp add: compact_imp_bounded)

   342

   343 corollary closed_sphere  [simp]:

   344   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"

   345   shows "closed (sphere a r)"

   346 by (simp add: compact_imp_closed)

   347

   348 subsection%unimportant \<open>Connectedness\<close>

   349

   350 lemma connected_local:

   351  "connected S \<longleftrightarrow>

   352   \<not> (\<exists>e1 e2.

   353       openin (subtopology euclidean S) e1 \<and>

   354       openin (subtopology euclidean S) e2 \<and>

   355       S \<subseteq> e1 \<union> e2 \<and>

   356       e1 \<inter> e2 = {} \<and>

   357       e1 \<noteq> {} \<and>

   358       e2 \<noteq> {})"

   359   unfolding connected_def openin_open

   360   by safe blast+

   361

   362 lemma exists_diff:

   363   fixes P :: "'a set \<Rightarrow> bool"

   364   shows "(\<exists>S. P (- S)) \<longleftrightarrow> (\<exists>S. P S)"

   365     (is "?lhs \<longleftrightarrow> ?rhs")

   366 proof -

   367   have ?rhs if ?lhs

   368     using that by blast

   369   moreover have "P (- (- S))" if "P S" for S

   370   proof -

   371     have "S = - (- S)" by simp

   372     with that show ?thesis by metis

   373   qed

   374   ultimately show ?thesis by metis

   375 qed

   376

   377 lemma connected_clopen: "connected S \<longleftrightarrow>

   378   (\<forall>T. openin (subtopology euclidean S) T \<and>

   379      closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")

   380 proof -

   381   have "\<not> connected S \<longleftrightarrow>

   382     (\<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> {})"

   383     unfolding connected_def openin_open closedin_closed

   384     by (metis double_complement)

   385   then have th0: "connected S \<longleftrightarrow>

   386     \<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> {})"

   387     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")

   388     by (simp add: closed_def) metis

   389   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'))"

   390     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")

   391     unfolding connected_def openin_open closedin_closed by auto

   392   have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" for e2

   393   proof -

   394     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)" for e1

   395       by auto

   396     then show ?thesis

   397       by metis

   398   qed

   399   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"

   400     by blast

   401   then show ?thesis

   402     by (simp add: th0 th1)

   403 qed

   404

   405 lemma connected_linear_image:

   406   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"

   407   assumes "linear f" and "connected s"

   408   shows "connected (f  s)"

   409 using connected_continuous_image assms linear_continuous_on linear_conv_bounded_linear by blast

   410

   411 subsection \<open>Connected components, considered as a connectedness relation or a set\<close>

   412

   413 definition%important "connected_component s x y \<equiv> \<exists>t. connected t \<and> t \<subseteq> s \<and> x \<in> t \<and> y \<in> t"

   414

   415 abbreviation "connected_component_set s x \<equiv> Collect (connected_component s x)"

   416

   417 lemma connected_componentI:

   418   "connected t \<Longrightarrow> t \<subseteq> s \<Longrightarrow> x \<in> t \<Longrightarrow> y \<in> t \<Longrightarrow> connected_component s x y"

   419   by (auto simp: connected_component_def)

   420

   421 lemma connected_component_in: "connected_component s x y \<Longrightarrow> x \<in> s \<and> y \<in> s"

   422   by (auto simp: connected_component_def)

   423

   424 lemma connected_component_refl: "x \<in> s \<Longrightarrow> connected_component s x x"

   425   by (auto simp: connected_component_def) (use connected_sing in blast)

   426

   427 lemma connected_component_refl_eq [simp]: "connected_component s x x \<longleftrightarrow> x \<in> s"

   428   by (auto simp: connected_component_refl) (auto simp: connected_component_def)

   429

   430 lemma connected_component_sym: "connected_component s x y \<Longrightarrow> connected_component s y x"

   431   by (auto simp: connected_component_def)

   432

   433 lemma connected_component_trans:

   434   "connected_component s x y \<Longrightarrow> connected_component s y z \<Longrightarrow> connected_component s x z"

   435   unfolding connected_component_def

   436   by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)

   437

   438 lemma connected_component_of_subset:

   439   "connected_component s x y \<Longrightarrow> s \<subseteq> t \<Longrightarrow> connected_component t x y"

   440   by (auto simp: connected_component_def)

   441

   442 lemma connected_component_Union: "connected_component_set s x = \<Union>{t. connected t \<and> x \<in> t \<and> t \<subseteq> s}"

   443   by (auto simp: connected_component_def)

   444

   445 lemma connected_connected_component [iff]: "connected (connected_component_set s x)"

   446   by (auto simp: connected_component_Union intro: connected_Union)

   447

   448 lemma connected_iff_eq_connected_component_set:

   449   "connected s \<longleftrightarrow> (\<forall>x \<in> s. connected_component_set s x = s)"

   450 proof (cases "s = {}")

   451   case True

   452   then show ?thesis by simp

   453 next

   454   case False

   455   then obtain x where "x \<in> s" by auto

   456   show ?thesis

   457   proof

   458     assume "connected s"

   459     then show "\<forall>x \<in> s. connected_component_set s x = s"

   460       by (force simp: connected_component_def)

   461   next

   462     assume "\<forall>x \<in> s. connected_component_set s x = s"

   463     then show "connected s"

   464       by (metis \<open>x \<in> s\<close> connected_connected_component)

   465   qed

   466 qed

   467

   468 lemma connected_component_subset: "connected_component_set s x \<subseteq> s"

   469   using connected_component_in by blast

   470

   471 lemma connected_component_eq_self: "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> connected_component_set s x = s"

   472   by (simp add: connected_iff_eq_connected_component_set)

   473

   474 lemma connected_iff_connected_component:

   475   "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component s x y)"

   476   using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)

   477

   478 lemma connected_component_maximal:

   479   "x \<in> t \<Longrightarrow> connected t \<Longrightarrow> t \<subseteq> s \<Longrightarrow> t \<subseteq> (connected_component_set s x)"

   480   using connected_component_eq_self connected_component_of_subset by blast

   481

   482 lemma connected_component_mono:

   483   "s \<subseteq> t \<Longrightarrow> connected_component_set s x \<subseteq> connected_component_set t x"

   484   by (simp add: Collect_mono connected_component_of_subset)

   485

   486 lemma connected_component_eq_empty [simp]: "connected_component_set s x = {} \<longleftrightarrow> x \<notin> s"

   487   using connected_component_refl by (fastforce simp: connected_component_in)

   488

   489 lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"

   490   using connected_component_eq_empty by blast

   491

   492 lemma connected_component_eq:

   493   "y \<in> connected_component_set s x \<Longrightarrow> (connected_component_set s y = connected_component_set s x)"

   494   by (metis (no_types, lifting)

   495       Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)

   496

   497 lemma closed_connected_component:

   498   assumes s: "closed s"

   499   shows "closed (connected_component_set s x)"

   500 proof (cases "x \<in> s")

   501   case False

   502   then show ?thesis

   503     by (metis connected_component_eq_empty closed_empty)

   504 next

   505   case True

   506   show ?thesis

   507     unfolding closure_eq [symmetric]

   508   proof

   509     show "closure (connected_component_set s x) \<subseteq> connected_component_set s x"

   510       apply (rule connected_component_maximal)

   511         apply (simp add: closure_def True)

   512        apply (simp add: connected_imp_connected_closure)

   513       apply (simp add: s closure_minimal connected_component_subset)

   514       done

   515   next

   516     show "connected_component_set s x \<subseteq> closure (connected_component_set s x)"

   517       by (simp add: closure_subset)

   518   qed

   519 qed

   520

   521 lemma connected_component_disjoint:

   522   "connected_component_set s a \<inter> connected_component_set s b = {} \<longleftrightarrow>

   523     a \<notin> connected_component_set s b"

   524   apply (auto simp: connected_component_eq)

   525   using connected_component_eq connected_component_sym

   526   apply blast

   527   done

   528

   529 lemma connected_component_nonoverlap:

   530   "connected_component_set s a \<inter> connected_component_set s b = {} \<longleftrightarrow>

   531     a \<notin> s \<or> b \<notin> s \<or> connected_component_set s a \<noteq> connected_component_set s b"

   532   apply (auto simp: connected_component_in)

   533   using connected_component_refl_eq

   534     apply blast

   535    apply (metis connected_component_eq mem_Collect_eq)

   536   apply (metis connected_component_eq mem_Collect_eq)

   537   done

   538

   539 lemma connected_component_overlap:

   540   "connected_component_set s a \<inter> connected_component_set s b \<noteq> {} \<longleftrightarrow>

   541     a \<in> s \<and> b \<in> s \<and> connected_component_set s a = connected_component_set s b"

   542   by (auto simp: connected_component_nonoverlap)

   543

   544 lemma connected_component_sym_eq: "connected_component s x y \<longleftrightarrow> connected_component s y x"

   545   using connected_component_sym by blast

   546

   547 lemma connected_component_eq_eq:

   548   "connected_component_set s x = connected_component_set s y \<longleftrightarrow>

   549     x \<notin> s \<and> y \<notin> s \<or> x \<in> s \<and> y \<in> s \<and> connected_component s x y"

   550   apply (cases "y \<in> s", simp)

   551    apply (metis connected_component_eq connected_component_eq_empty connected_component_refl_eq mem_Collect_eq)

   552   apply (cases "x \<in> s", simp)

   553    apply (metis connected_component_eq_empty)

   554   using connected_component_eq_empty

   555   apply blast

   556   done

   557

   558 lemma connected_iff_connected_component_eq:

   559   "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component_set s x = connected_component_set s y)"

   560   by (simp add: connected_component_eq_eq connected_iff_connected_component)

   561

   562 lemma connected_component_idemp:

   563   "connected_component_set (connected_component_set s x) x = connected_component_set s x"

   564   apply (rule subset_antisym)

   565    apply (simp add: connected_component_subset)

   566   apply (metis connected_component_eq_empty connected_component_maximal

   567       connected_component_refl_eq connected_connected_component mem_Collect_eq set_eq_subset)

   568   done

   569

   570 lemma connected_component_unique:

   571   "\<lbrakk>x \<in> c; c \<subseteq> s; connected c;

   572     \<And>c'. x \<in> c' \<and> c' \<subseteq> s \<and> connected c'

   573               \<Longrightarrow> c' \<subseteq> c\<rbrakk>

   574         \<Longrightarrow> connected_component_set s x = c"

   575 apply (rule subset_antisym)

   576 apply (meson connected_component_maximal connected_component_subset connected_connected_component contra_subsetD)

   577 by (simp add: connected_component_maximal)

   578

   579 lemma joinable_connected_component_eq:

   580   "\<lbrakk>connected t; t \<subseteq> s;

   581     connected_component_set s x \<inter> t \<noteq> {};

   582     connected_component_set s y \<inter> t \<noteq> {}\<rbrakk>

   583     \<Longrightarrow> connected_component_set s x = connected_component_set s y"

   584 apply (simp add: ex_in_conv [symmetric])

   585 apply (rule connected_component_eq)

   586 by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in connected_component_maximal subsetD mem_Collect_eq)

   587

   588

   589 lemma Union_connected_component: "\<Union>(connected_component_set s  s) = s"

   590   apply (rule subset_antisym)

   591   apply (simp add: SUP_least connected_component_subset)

   592   using connected_component_refl_eq

   593   by force

   594

   595

   596 lemma complement_connected_component_unions:

   597     "s - connected_component_set s x =

   598      \<Union>(connected_component_set s  s - {connected_component_set s x})"

   599   apply (subst Union_connected_component [symmetric], auto)

   600   apply (metis connected_component_eq_eq connected_component_in)

   601   by (metis connected_component_eq mem_Collect_eq)

   602

   603 lemma connected_component_intermediate_subset:

   604         "\<lbrakk>connected_component_set u a \<subseteq> t; t \<subseteq> u\<rbrakk>

   605         \<Longrightarrow> connected_component_set t a = connected_component_set u a"

   606   apply (case_tac "a \<in> u")

   607   apply (simp add: connected_component_maximal connected_component_mono subset_antisym)

   608   using connected_component_eq_empty by blast

   609

   610

   611 subsection \<open>The set of connected components of a set\<close>

   612

   613 definition%important components:: "'a::topological_space set \<Rightarrow> 'a set set"

   614   where "components s \<equiv> connected_component_set s  s"

   615

   616 lemma components_iff: "s \<in> components u \<longleftrightarrow> (\<exists>x. x \<in> u \<and> s = connected_component_set u x)"

   617   by (auto simp: components_def)

   618

   619 lemma componentsI: "x \<in> u \<Longrightarrow> connected_component_set u x \<in> components u"

   620   by (auto simp: components_def)

   621

   622 lemma componentsE:

   623   assumes "s \<in> components u"

   624   obtains x where "x \<in> u" "s = connected_component_set u x"

   625   using assms by (auto simp: components_def)

   626

   627 lemma Union_components [simp]: "\<Union>(components u) = u"

   628   apply (rule subset_antisym)

   629   using Union_connected_component components_def apply fastforce

   630   apply (metis Union_connected_component components_def set_eq_subset)

   631   done

   632

   633 lemma pairwise_disjoint_components: "pairwise (\<lambda>X Y. X \<inter> Y = {}) (components u)"

   634   apply (simp add: pairwise_def)

   635   apply (auto simp: components_iff)

   636   apply (metis connected_component_eq_eq connected_component_in)+

   637   done

   638

   639 lemma in_components_nonempty: "c \<in> components s \<Longrightarrow> c \<noteq> {}"

   640     by (metis components_iff connected_component_eq_empty)

   641

   642 lemma in_components_subset: "c \<in> components s \<Longrightarrow> c \<subseteq> s"

   643   using Union_components by blast

   644

   645 lemma in_components_connected: "c \<in> components s \<Longrightarrow> connected c"

   646   by (metis components_iff connected_connected_component)

   647

   648 lemma in_components_maximal:

   649   "c \<in> components s \<longleftrightarrow>

   650     c \<noteq> {} \<and> c \<subseteq> s \<and> connected c \<and> (\<forall>d. d \<noteq> {} \<and> c \<subseteq> d \<and> d \<subseteq> s \<and> connected d \<longrightarrow> d = c)"

   651   apply (rule iffI)

   652    apply (simp add: in_components_nonempty in_components_connected)

   653    apply (metis (full_types) components_iff connected_component_eq_self connected_component_intermediate_subset connected_component_refl in_components_subset mem_Collect_eq rev_subsetD)

   654   apply (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)

   655   done

   656

   657 lemma joinable_components_eq:

   658   "connected t \<and> t \<subseteq> s \<and> c1 \<in> components s \<and> c2 \<in> components s \<and> c1 \<inter> t \<noteq> {} \<and> c2 \<inter> t \<noteq> {} \<Longrightarrow> c1 = c2"

   659   by (metis (full_types) components_iff joinable_connected_component_eq)

   660

   661 lemma closed_components: "\<lbrakk>closed s; c \<in> components s\<rbrakk> \<Longrightarrow> closed c"

   662   by (metis closed_connected_component components_iff)

   663

   664 lemma compact_components:

   665   fixes s :: "'a::heine_borel set"

   666   shows "\<lbrakk>compact s; c \<in> components s\<rbrakk> \<Longrightarrow> compact c"

   667 by (meson bounded_subset closed_components in_components_subset compact_eq_bounded_closed)

   668

   669 lemma components_nonoverlap:

   670     "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c \<inter> c' = {}) \<longleftrightarrow> (c \<noteq> c')"

   671   apply (auto simp: in_components_nonempty components_iff)

   672     using connected_component_refl apply blast

   673    apply (metis connected_component_eq_eq connected_component_in)

   674   by (metis connected_component_eq mem_Collect_eq)

   675

   676 lemma components_eq: "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c = c' \<longleftrightarrow> c \<inter> c' \<noteq> {})"

   677   by (metis components_nonoverlap)

   678

   679 lemma components_eq_empty [simp]: "components s = {} \<longleftrightarrow> s = {}"

   680   by (simp add: components_def)

   681

   682 lemma components_empty [simp]: "components {} = {}"

   683   by simp

   684

   685 lemma connected_eq_connected_components_eq: "connected s \<longleftrightarrow> (\<forall>c \<in> components s. \<forall>c' \<in> components s. c = c')"

   686   by (metis (no_types, hide_lams) components_iff connected_component_eq_eq connected_iff_connected_component)

   687

   688 lemma components_eq_sing_iff: "components s = {s} \<longleftrightarrow> connected s \<and> s \<noteq> {}"

   689   apply (rule iffI)

   690   using in_components_connected apply fastforce

   691   apply safe

   692   using Union_components apply fastforce

   693    apply (metis components_iff connected_component_eq_self)

   694   using in_components_maximal

   695   apply auto

   696   done

   697

   698 lemma components_eq_sing_exists: "(\<exists>a. components s = {a}) \<longleftrightarrow> connected s \<and> s \<noteq> {}"

   699   apply (rule iffI)

   700   using connected_eq_connected_components_eq apply fastforce

   701   apply (metis components_eq_sing_iff)

   702   done

   703

   704 lemma connected_eq_components_subset_sing: "connected s \<longleftrightarrow> components s \<subseteq> {s}"

   705   by (metis Union_components components_empty components_eq_sing_iff connected_empty insert_subset order_refl subset_singletonD)

   706

   707 lemma connected_eq_components_subset_sing_exists: "connected s \<longleftrightarrow> (\<exists>a. components s \<subseteq> {a})"

   708   by (metis components_eq_sing_exists connected_eq_components_subset_sing empty_iff subset_iff subset_singletonD)

   709

   710 lemma in_components_self: "s \<in> components s \<longleftrightarrow> connected s \<and> s \<noteq> {}"

   711   by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)

   712

   713 lemma components_maximal: "\<lbrakk>c \<in> components s; connected t; t \<subseteq> s; c \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> t \<subseteq> c"

   714   apply (simp add: components_def ex_in_conv [symmetric], clarify)

   715   by (meson connected_component_def connected_component_trans)

   716

   717 lemma exists_component_superset: "\<lbrakk>t \<subseteq> s; s \<noteq> {}; connected t\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> components s \<and> t \<subseteq> c"

   718   apply (cases "t = {}", force)

   719   apply (metis components_def ex_in_conv connected_component_maximal contra_subsetD image_eqI)

   720   done

   721

   722 lemma components_intermediate_subset: "\<lbrakk>s \<in> components u; s \<subseteq> t; t \<subseteq> u\<rbrakk> \<Longrightarrow> s \<in> components t"

   723   apply (auto simp: components_iff)

   724   apply (metis connected_component_eq_empty connected_component_intermediate_subset)

   725   done

   726

   727 lemma in_components_unions_complement: "c \<in> components s \<Longrightarrow> s - c = \<Union>(components s - {c})"

   728   by (metis complement_connected_component_unions components_def components_iff)

   729

   730 lemma connected_intermediate_closure:

   731   assumes cs: "connected s" and st: "s \<subseteq> t" and ts: "t \<subseteq> closure s"

   732   shows "connected t"

   733 proof (rule connectedI)

   734   fix A B

   735   assume A: "open A" and B: "open B" and Alap: "A \<inter> t \<noteq> {}" and Blap: "B \<inter> t \<noteq> {}"

   736     and disj: "A \<inter> B \<inter> t = {}" and cover: "t \<subseteq> A \<union> B"

   737   have disjs: "A \<inter> B \<inter> s = {}"

   738     using disj st by auto

   739   have "A \<inter> closure s \<noteq> {}"

   740     using Alap Int_absorb1 ts by blast

   741   then have Alaps: "A \<inter> s \<noteq> {}"

   742     by (simp add: A open_Int_closure_eq_empty)

   743   have "B \<inter> closure s \<noteq> {}"

   744     using Blap Int_absorb1 ts by blast

   745   then have Blaps: "B \<inter> s \<noteq> {}"

   746     by (simp add: B open_Int_closure_eq_empty)

   747   then show False

   748     using cs [unfolded connected_def] A B disjs Alaps Blaps cover st

   749     by blast

   750 qed

   751

   752 lemma closedin_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"

   753 proof (cases "connected_component_set s x = {}")

   754   case True

   755   then show ?thesis

   756     by (metis closedin_empty)

   757 next

   758   case False

   759   then obtain y where y: "connected_component s x y"

   760     by blast

   761   have *: "connected_component_set s x \<subseteq> s \<inter> closure (connected_component_set s x)"

   762     by (auto simp: closure_def connected_component_in)

   763   have "connected_component s x y \<Longrightarrow> s \<inter> closure (connected_component_set s x) \<subseteq> connected_component_set s x"

   764     apply (rule connected_component_maximal, simp)

   765     using closure_subset connected_component_in apply fastforce

   766     using * connected_intermediate_closure apply blast+

   767     done

   768   with y * show ?thesis

   769     by (auto simp: closedin_closed)

   770 qed

   771

   772 lemma closedin_component:

   773    "C \<in> components s \<Longrightarrow> closedin (subtopology euclidean s) C"

   774   using closedin_connected_component componentsE by blast

   775

   776

   777 subsection \<open>Intersecting chains of compact sets and the Baire property\<close>

   778

   779 proposition bounded_closed_chain:

   780   fixes \<F> :: "'a::heine_borel set set"

   781   assumes "B \<in> \<F>" "bounded B" and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" and "{} \<notin> \<F>"

   782       and chain: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"

   783     shows "\<Inter>\<F> \<noteq> {}"

   784 proof -

   785   have "B \<inter> \<Inter>\<F> \<noteq> {}"

   786   proof (rule compact_imp_fip)

   787     show "compact B" "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"

   788       by (simp_all add: assms compact_eq_bounded_closed)

   789     show "\<lbrakk>finite \<G>; \<G> \<subseteq> \<F>\<rbrakk> \<Longrightarrow> B \<inter> \<Inter>\<G> \<noteq> {}" for \<G>

   790     proof (induction \<G> rule: finite_induct)

   791       case empty

   792       with assms show ?case by force

   793     next

   794       case (insert U \<G>)

   795       then have "U \<in> \<F>" and ne: "B \<inter> \<Inter>\<G> \<noteq> {}" by auto

   796       then consider "B \<subseteq> U" | "U \<subseteq> B"

   797           using \<open>B \<in> \<F>\<close> chain by blast

   798         then show ?case

   799         proof cases

   800           case 1

   801           then show ?thesis

   802             using Int_left_commute ne by auto

   803         next

   804           case 2

   805           have "U \<noteq> {}"

   806             using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast

   807           moreover

   808           have False if "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. x \<notin> Y"

   809           proof -

   810             have "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. Y \<subseteq> U"

   811               by (metis chain contra_subsetD insert.prems insert_subset that)

   812             then obtain Y where "Y \<in> \<G>" "Y \<subseteq> U"

   813               by (metis all_not_in_conv \<open>U \<noteq> {}\<close>)

   814             moreover obtain x where "x \<in> \<Inter>\<G>"

   815               by (metis Int_emptyI ne)

   816             ultimately show ?thesis

   817               by (metis Inf_lower subset_eq that)

   818           qed

   819           with 2 show ?thesis

   820             by blast

   821         qed

   822       qed

   823   qed

   824   then show ?thesis by blast

   825 qed

   826

   827 corollary compact_chain:

   828   fixes \<F> :: "'a::heine_borel set set"

   829   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" "{} \<notin> \<F>"

   830           "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"

   831     shows "\<Inter> \<F> \<noteq> {}"

   832 proof (cases "\<F> = {}")

   833   case True

   834   then show ?thesis by auto

   835 next

   836   case False

   837   show ?thesis

   838     by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)

   839 qed

   840

   841 lemma compact_nest:

   842   fixes F :: "'a::linorder \<Rightarrow> 'b::heine_borel set"

   843   assumes F: "\<And>n. compact(F n)" "\<And>n. F n \<noteq> {}" and mono: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

   844   shows "\<Inter>range F \<noteq> {}"

   845 proof -

   846   have *: "\<And>S T. S \<in> range F \<and> T \<in> range F \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"

   847     by (metis mono image_iff le_cases)

   848   show ?thesis

   849     apply (rule compact_chain [OF _ _ *])

   850     using F apply (blast intro: dest: *)+

   851     done

   852 qed

   853

   854 text\<open>The Baire property of dense sets\<close>

   855 theorem Baire:

   856   fixes S::"'a::{real_normed_vector,heine_borel} set"

   857   assumes "closed S" "countable \<G>"

   858       and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (subtopology euclidean S) T \<and> S \<subseteq> closure T"

   859  shows "S \<subseteq> closure(\<Inter>\<G>)"

   860 proof (cases "\<G> = {}")

   861   case True

   862   then show ?thesis

   863     using closure_subset by auto

   864 next

   865   let ?g = "from_nat_into \<G>"

   866   case False

   867   then have gin: "?g n \<in> \<G>" for n

   868     by (simp add: from_nat_into)

   869   show ?thesis

   870   proof (clarsimp simp: closure_approachable)

   871     fix x and e::real

   872     assume "x \<in> S" "0 < e"

   873     obtain TF where opeF: "\<And>n. openin (subtopology euclidean S) (TF n)"

   874                and ne: "\<And>n. TF n \<noteq> {}"

   875                and subg: "\<And>n. S \<inter> closure(TF n) \<subseteq> ?g n"

   876                and subball: "\<And>n. closure(TF n) \<subseteq> ball x e"

   877                and decr: "\<And>n. TF(Suc n) \<subseteq> TF n"

   878     proof -

   879       have *: "\<exists>Y. (openin (subtopology euclidean S) Y \<and> Y \<noteq> {} \<and>

   880                    S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U"

   881         if opeU: "openin (subtopology euclidean S) U" and "U \<noteq> {}" and cloU: "closure U \<subseteq> ball x e" for U n

   882       proof -

   883         obtain T where T: "open T" "U = T \<inter> S"

   884           using \<open>openin (subtopology euclidean S) U\<close> by (auto simp: openin_subtopology)

   885         with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"

   886           using gin ope by fastforce

   887         then have "T \<inter> ?g n \<noteq> {}"

   888           using \<open>open T\<close> open_Int_closure_eq_empty by blast

   889         then obtain y where "y \<in> U" "y \<in> ?g n"

   890           using T ope [of "?g n", OF gin] by (blast dest:  openin_imp_subset)

   891         moreover have "openin (subtopology euclidean S) (U \<inter> ?g n)"

   892           using gin ope opeU by blast

   893         ultimately obtain d where U: "U \<inter> ?g n \<subseteq> S" and "d > 0" and d: "ball y d \<inter> S \<subseteq> U \<inter> ?g n"

   894           by (force simp: openin_contains_ball)

   895         show ?thesis

   896         proof (intro exI conjI)

   897           show "openin (subtopology euclidean S) (S \<inter> ball y (d/2))"

   898             by (simp add: openin_open_Int)

   899           show "S \<inter> ball y (d/2) \<noteq> {}"

   900             using \<open>0 < d\<close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce

   901           have "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> closure (ball y (d/2))"

   902             using closure_mono by blast

   903           also have "... \<subseteq> ?g n"

   904             using \<open>d > 0\<close> d by force

   905           finally show "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> ?g n" .

   906           have "closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> ball y d"

   907           proof -

   908             have "closure (ball y (d/2)) \<subseteq> ball y d"

   909               using \<open>d > 0\<close> by auto

   910             then have "closure (S \<inter> ball y (d/2)) \<subseteq> ball y d"

   911               by (meson closure_mono inf.cobounded2 subset_trans)

   912             then show ?thesis

   913               by (simp add: \<open>closed S\<close> closure_minimal)

   914           qed

   915           also have "...  \<subseteq> ball x e"

   916             using cloU closure_subset d by blast

   917           finally show "closure (S \<inter> ball y (d/2)) \<subseteq> ball x e" .

   918           show "S \<inter> ball y (d/2) \<subseteq> U"

   919             using ball_divide_subset_numeral d by blast

   920         qed

   921       qed

   922       let ?\<Phi> = "\<lambda>n X. openin (subtopology euclidean S) X \<and> X \<noteq> {} \<and>

   923                       S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"

   924       have "closure (S \<inter> ball x (e / 2)) \<subseteq> closure(ball x (e/2))"

   925         by (simp add: closure_mono)

   926       also have "...  \<subseteq> ball x e"

   927         using \<open>e > 0\<close> by auto

   928       finally have "closure (S \<inter> ball x (e / 2)) \<subseteq> ball x e" .

   929       moreover have"openin (subtopology euclidean S) (S \<inter> ball x (e / 2))" "S \<inter> ball x (e / 2) \<noteq> {}"

   930         using \<open>0 < e\<close> \<open>x \<in> S\<close> by auto

   931       ultimately obtain Y where Y: "?\<Phi> 0 Y \<and> Y \<subseteq> S \<inter> ball x (e / 2)"

   932             using * [of "S \<inter> ball x (e/2)" 0] by metis

   933       show thesis

   934       proof (rule exE [OF dependent_nat_choice [of ?\<Phi> "\<lambda>n X Y. Y \<subseteq> X"]])

   935         show "\<exists>x. ?\<Phi> 0 x"

   936           using Y by auto

   937         show "\<exists>Y. ?\<Phi> (Suc n) Y \<and> Y \<subseteq> X" if "?\<Phi> n X" for X n

   938           using that by (blast intro: *)

   939       qed (use that in metis)

   940     qed

   941     have "(\<Inter>n. S \<inter> closure (TF n)) \<noteq> {}"

   942     proof (rule compact_nest)

   943       show "\<And>n. compact (S \<inter> closure (TF n))"

   944         by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF \<open>closed S\<close>])

   945       show "\<And>n. S \<inter> closure (TF n) \<noteq> {}"

   946         by (metis Int_absorb1 opeF \<open>closed S\<close> closure_eq_empty closure_minimal ne openin_imp_subset)

   947       show "\<And>m n. m \<le> n \<Longrightarrow> S \<inter> closure (TF n) \<subseteq> S \<inter> closure (TF m)"

   948         by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)

   949     qed

   950     moreover have "(\<Inter>n. S \<inter> closure (TF n)) \<subseteq> {y \<in> \<Inter>\<G>. dist y x < e}"

   951     proof (clarsimp, intro conjI)

   952       fix y

   953       assume "y \<in> S" and y: "\<forall>n. y \<in> closure (TF n)"

   954       then show "\<forall>T\<in>\<G>. y \<in> T"

   955         by (metis Int_iff from_nat_into_surj [OF \<open>countable \<G>\<close>] set_mp subg)

   956       show "dist y x < e"

   957         by (metis y dist_commute mem_ball subball subsetCE)

   958     qed

   959     ultimately show "\<exists>y \<in> \<Inter>\<G>. dist y x < e"

   960       by auto

   961   qed

   962 qed

   963

   964 subsection%unimportant \<open>Some theorems on sups and infs using the notion "bounded"\<close>

   965

   966 lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"

   967   by (simp add: bounded_iff)

   968

   969 lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"

   970   by (auto simp: bounded_def bdd_above_def dist_real_def)

   971      (metis abs_le_D1 abs_minus_commute diff_le_eq)

   972

   973 lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"

   974   by (auto simp: bounded_def bdd_below_def dist_real_def)

   975      (metis abs_le_D1 add.commute diff_le_eq)

   976

   977 lemma bounded_inner_imp_bdd_above:

   978   assumes "bounded s"

   979     shows "bdd_above ((\<lambda>x. x \<bullet> a)  s)"

   980 by (simp add: assms bounded_imp_bdd_above bounded_linear_image bounded_linear_inner_left)

   981

   982 lemma bounded_inner_imp_bdd_below:

   983   assumes "bounded s"

   984     shows "bdd_below ((\<lambda>x. x \<bullet> a)  s)"

   985 by (simp add: assms bounded_imp_bdd_below bounded_linear_image bounded_linear_inner_left)

   986

   987 lemma bounded_has_Sup:

   988   fixes S :: "real set"

   989   assumes "bounded S"

   990     and "S \<noteq> {}"

   991   shows "\<forall>x\<in>S. x \<le> Sup S"

   992     and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

   993 proof

   994   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

   995     using assms by (metis cSup_least)

   996 qed (metis cSup_upper assms(1) bounded_imp_bdd_above)

   997

   998 lemma Sup_insert:

   999   fixes S :: "real set"

  1000   shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  1001   by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)

  1002

  1003 lemma Sup_insert_finite:

  1004   fixes S :: "'a::conditionally_complete_linorder set"

  1005   shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  1006 by (simp add: cSup_insert sup_max)

  1007

  1008 lemma bounded_has_Inf:

  1009   fixes S :: "real set"

  1010   assumes "bounded S"

  1011     and "S \<noteq> {}"

  1012   shows "\<forall>x\<in>S. x \<ge> Inf S"

  1013     and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  1014 proof

  1015   show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  1016     using assms by (metis cInf_greatest)

  1017 qed (metis cInf_lower assms(1) bounded_imp_bdd_below)

  1018

  1019 lemma Inf_insert:

  1020   fixes S :: "real set"

  1021   shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  1022   by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)

  1023

  1024 lemma Inf_insert_finite:

  1025   fixes S :: "'a::conditionally_complete_linorder set"

  1026   shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  1027 by (simp add: cInf_eq_Min)

  1028

  1029 lemma finite_imp_less_Inf:

  1030   fixes a :: "'a::conditionally_complete_linorder"

  1031   shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a < x\<rbrakk> \<Longrightarrow> a < Inf X"

  1032   by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)

  1033

  1034 lemma finite_less_Inf_iff:

  1035   fixes a :: "'a :: conditionally_complete_linorder"

  1036   shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a < Inf X \<longleftrightarrow> (\<forall>x \<in> X. a < x)"

  1037   by (auto simp: cInf_eq_Min)

  1038

  1039 lemma finite_imp_Sup_less:

  1040   fixes a :: "'a::conditionally_complete_linorder"

  1041   shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a > x\<rbrakk> \<Longrightarrow> a > Sup X"

  1042   by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)

  1043

  1044 lemma finite_Sup_less_iff:

  1045   fixes a :: "'a :: conditionally_complete_linorder"

  1046   shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a > Sup X \<longleftrightarrow> (\<forall>x \<in> X. a > x)"

  1047   by (auto simp: cSup_eq_Max)

  1048

  1049 proposition is_interval_compact:

  1050    "is_interval S \<and> compact S \<longleftrightarrow> (\<exists>a b. S = cbox a b)"   (is "?lhs = ?rhs")

  1051 proof (cases "S = {}")

  1052   case True

  1053   with empty_as_interval show ?thesis by auto

  1054 next

  1055   case False

  1056   show ?thesis

  1057   proof

  1058     assume L: ?lhs

  1059     then have "is_interval S" "compact S" by auto

  1060     define a where "a \<equiv> \<Sum>i\<in>Basis. (INF x:S. x \<bullet> i) *\<^sub>R i"

  1061     define b where "b \<equiv> \<Sum>i\<in>Basis. (SUP x:S. x \<bullet> i) *\<^sub>R i"

  1062     have 1: "\<And>x i. \<lbrakk>x \<in> S; i \<in> Basis\<rbrakk> \<Longrightarrow> (INF x:S. x \<bullet> i) \<le> x \<bullet> i"

  1063       by (simp add: cInf_lower bounded_inner_imp_bdd_below compact_imp_bounded L)

  1064     have 2: "\<And>x i. \<lbrakk>x \<in> S; i \<in> Basis\<rbrakk> \<Longrightarrow> x \<bullet> i \<le> (SUP x:S. x \<bullet> i)"

  1065       by (simp add: cSup_upper bounded_inner_imp_bdd_above compact_imp_bounded L)

  1066     have 3: "x \<in> S" if inf: "\<And>i. i \<in> Basis \<Longrightarrow> (INF x:S. x \<bullet> i) \<le> x \<bullet> i"

  1067                    and sup: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<le> (SUP x:S. x \<bullet> i)" for x

  1068     proof (rule mem_box_componentwiseI [OF \<open>is_interval S\<close>])

  1069       fix i::'a

  1070       assume i: "i \<in> Basis"

  1071       have cont: "continuous_on S (\<lambda>x. x \<bullet> i)"

  1072         by (intro continuous_intros)

  1073       obtain a where "a \<in> S" and a: "\<And>y. y\<in>S \<Longrightarrow> a \<bullet> i \<le> y \<bullet> i"

  1074         using continuous_attains_inf [OF \<open>compact S\<close> False cont] by blast

  1075       obtain b where "b \<in> S" and b: "\<And>y. y\<in>S \<Longrightarrow> y \<bullet> i \<le> b \<bullet> i"

  1076         using continuous_attains_sup [OF \<open>compact S\<close> False cont] by blast

  1077       have "a \<bullet> i \<le> (INF x:S. x \<bullet> i)"

  1078         by (simp add: False a cINF_greatest)

  1079       also have "\<dots> \<le> x \<bullet> i"

  1080         by (simp add: i inf)

  1081       finally have ai: "a \<bullet> i \<le> x \<bullet> i" .

  1082       have "x \<bullet> i \<le> (SUP x:S. x \<bullet> i)"

  1083         by (simp add: i sup)

  1084       also have "(SUP x:S. x \<bullet> i) \<le> b \<bullet> i"

  1085         by (simp add: False b cSUP_least)

  1086       finally have bi: "x \<bullet> i \<le> b \<bullet> i" .

  1087       show "x \<bullet> i \<in> (\<lambda>x. x \<bullet> i)  S"

  1088         apply (rule_tac x="\<Sum>j\<in>Basis. (if j = i then x \<bullet> i else a \<bullet> j) *\<^sub>R j" in image_eqI)

  1089         apply (simp add: i)

  1090         apply (rule mem_is_intervalI [OF \<open>is_interval S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>])

  1091         using i ai bi apply force

  1092         done

  1093     qed

  1094     have "S = cbox a b"

  1095       by (auto simp: a_def b_def mem_box intro: 1 2 3)

  1096     then show ?rhs

  1097       by blast

  1098   next

  1099     assume R: ?rhs

  1100     then show ?lhs

  1101       using compact_cbox is_interval_cbox by blast

  1102   qed

  1103 qed

  1104

  1105 text \<open>Hence some handy theorems on distance, diameter etc. of/from a set.\<close>

  1106

  1107 lemma distance_attains_sup:

  1108   assumes "compact s" "s \<noteq> {}"

  1109   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  1110 proof (rule continuous_attains_sup [OF assms])

  1111   {

  1112     fix x

  1113     assume "x\<in>s"

  1114     have "(dist a \<longlongrightarrow> dist a x) (at x within s)"

  1115       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  1116   }

  1117   then show "continuous_on s (dist a)"

  1118     unfolding continuous_on ..

  1119 qed

  1120

  1121 text \<open>For \emph{minimal} distance, we only need closure, not compactness.\<close>

  1122

  1123 lemma distance_attains_inf:

  1124   fixes a :: "'a::heine_borel"

  1125   assumes "closed s" and "s \<noteq> {}"

  1126   obtains x where "x\<in>s" "\<And>y. y \<in> s \<Longrightarrow> dist a x \<le> dist a y"

  1127 proof -

  1128   from assms obtain b where "b \<in> s" by auto

  1129   let ?B = "s \<inter> cball a (dist b a)"

  1130   have "?B \<noteq> {}" using \<open>b \<in> s\<close>

  1131     by (auto simp: dist_commute)

  1132   moreover have "continuous_on ?B (dist a)"

  1133     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_ident continuous_const)

  1134   moreover have "compact ?B"

  1135     by (intro closed_Int_compact \<open>closed s\<close> compact_cball)

  1136   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  1137     by (metis continuous_attains_inf)

  1138   with that show ?thesis by fastforce

  1139 qed

  1140

  1141

  1142 subsection%unimportant\<open>Relations among convergence and absolute convergence for power series\<close>

  1143

  1144 lemma summable_imp_bounded:

  1145   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"

  1146   shows "summable f \<Longrightarrow> bounded (range f)"

  1147 by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)

  1148

  1149 lemma summable_imp_sums_bounded:

  1150    "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"

  1151 by (auto simp: summable_def sums_def dest: convergent_imp_bounded)

  1152

  1153 lemma power_series_conv_imp_absconv_weak:

  1154   fixes a:: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" and w :: 'a

  1155   assumes sum: "summable (\<lambda>n. a n * z ^ n)" and no: "norm w < norm z"

  1156     shows "summable (\<lambda>n. of_real(norm(a n)) * w ^ n)"

  1157 proof -

  1158   obtain M where M: "\<And>x. norm (a x * z ^ x) \<le> M"

  1159     using summable_imp_bounded [OF sum] by (force simp: bounded_iff)

  1160   then have *: "summable (\<lambda>n. norm (a n) * norm w ^ n)"

  1161     by (rule_tac M=M in Abel_lemma) (auto simp: norm_mult norm_power intro: no)

  1162   show ?thesis

  1163     apply (rule series_comparison_complex [of "(\<lambda>n. of_real(norm(a n) * norm w ^ n))"])

  1164     apply (simp only: summable_complex_of_real *)

  1165     apply (auto simp: norm_mult norm_power)

  1166     done

  1167 qed

  1168

  1169 subsection%unimportant \<open>Bounded closed nest property (proof does not use Heine-Borel)\<close>

  1170

  1171 lemma bounded_closed_nest:

  1172   fixes S :: "nat \<Rightarrow> ('a::heine_borel) set"

  1173   assumes "\<And>n. closed (S n)"

  1174       and "\<And>n. S n \<noteq> {}"

  1175       and "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"

  1176       and "bounded (S 0)"

  1177   obtains a where "\<And>n. a \<in> S n"

  1178 proof -

  1179   from assms(2) obtain x where x: "\<forall>n. x n \<in> S n"

  1180     using choice[of "\<lambda>n x. x \<in> S n"] by auto

  1181   from assms(4,1) have "seq_compact (S 0)"

  1182     by (simp add: bounded_closed_imp_seq_compact)

  1183   then obtain l r where lr: "l \<in> S 0" "strict_mono r" "(x \<circ> r) \<longlonglongrightarrow> l"

  1184     using x and assms(3) unfolding seq_compact_def by blast

  1185   have "\<forall>n. l \<in> S n"

  1186   proof

  1187     fix n :: nat

  1188     have "closed (S n)"

  1189       using assms(1) by simp

  1190     moreover have "\<forall>i. (x \<circ> r) i \<in> S i"

  1191       using x and assms(3) and lr(2) [THEN seq_suble] by auto

  1192     then have "\<forall>i. (x \<circ> r) (i + n) \<in> S n"

  1193       using assms(3) by (fast intro!: le_add2)

  1194     moreover have "(\<lambda>i. (x \<circ> r) (i + n)) \<longlonglongrightarrow> l"

  1195       using lr(3) by (rule LIMSEQ_ignore_initial_segment)

  1196     ultimately show "l \<in> S n"

  1197       by (rule closed_sequentially)

  1198   qed

  1199   then show ?thesis

  1200     using that by blast

  1201 qed

  1202

  1203 text \<open>Decreasing case does not even need compactness, just completeness.\<close>

  1204

  1205 lemma decreasing_closed_nest:

  1206   fixes S :: "nat \<Rightarrow> ('a::complete_space) set"

  1207   assumes "\<And>n. closed (S n)"

  1208           "\<And>n. S n \<noteq> {}"

  1209           "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"

  1210           "\<And>e. e>0 \<Longrightarrow> \<exists>n. \<forall>x\<in>S n. \<forall>y\<in>S n. dist x y < e"

  1211   obtains a where "\<And>n. a \<in> S n"

  1212 proof -

  1213   have "\<forall>n. \<exists>x. x \<in> S n"

  1214     using assms(2) by auto

  1215   then have "\<exists>t. \<forall>n. t n \<in> S n"

  1216     using choice[of "\<lambda>n x. x \<in> S n"] by auto

  1217   then obtain t where t: "\<forall>n. t n \<in> S n" by auto

  1218   {

  1219     fix e :: real

  1220     assume "e > 0"

  1221     then obtain N where N: "\<forall>x\<in>S N. \<forall>y\<in>S N. dist x y < e"

  1222       using assms(4) by blast

  1223     {

  1224       fix m n :: nat

  1225       assume "N \<le> m \<and> N \<le> n"

  1226       then have "t m \<in> S N" "t n \<in> S N"

  1227         using assms(3) t unfolding  subset_eq t by blast+

  1228       then have "dist (t m) (t n) < e"

  1229         using N by auto

  1230     }

  1231     then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"

  1232       by auto

  1233   }

  1234   then have "Cauchy t"

  1235     unfolding cauchy_def by auto

  1236   then obtain l where l:"(t \<longlongrightarrow> l) sequentially"

  1237     using complete_UNIV unfolding complete_def by auto

  1238   { fix n :: nat

  1239     { fix e :: real

  1240       assume "e > 0"

  1241       then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"

  1242         using l[unfolded lim_sequentially] by auto

  1243       have "t (max n N) \<in> S n"

  1244         by (meson assms(3) contra_subsetD max.cobounded1 t)

  1245       then have "\<exists>y\<in>S n. dist y l < e"

  1246         using N max.cobounded2 by blast

  1247     }

  1248     then have "l \<in> S n"

  1249       using closed_approachable[of "S n" l] assms(1) by auto

  1250   }

  1251   then show ?thesis

  1252     using that by blast

  1253 qed

  1254

  1255 text \<open>Strengthen it to the intersection actually being a singleton.\<close>

  1256

  1257 lemma decreasing_closed_nest_sing:

  1258   fixes S :: "nat \<Rightarrow> 'a::complete_space set"

  1259   assumes "\<And>n. closed(S n)"

  1260           "\<And>n. S n \<noteq> {}"

  1261           "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"

  1262           "\<And>e. e>0 \<Longrightarrow> \<exists>n. \<forall>x \<in> (S n). \<forall> y\<in>(S n). dist x y < e"

  1263   shows "\<exists>a. \<Inter>(range S) = {a}"

  1264 proof -

  1265   obtain a where a: "\<forall>n. a \<in> S n"

  1266     using decreasing_closed_nest[of S] using assms by auto

  1267   { fix b

  1268     assume b: "b \<in> \<Inter>(range S)"

  1269     { fix e :: real

  1270       assume "e > 0"

  1271       then have "dist a b < e"

  1272         using assms(4) and b and a by blast

  1273     }

  1274     then have "dist a b = 0"

  1275       by (metis dist_eq_0_iff dist_nz less_le)

  1276   }

  1277   with a have "\<Inter>(range S) = {a}"

  1278     unfolding image_def by auto

  1279   then show ?thesis ..

  1280 qed

  1281

  1282

  1283 subsection \<open>Infimum Distance\<close>

  1284

  1285 definition%important "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"

  1286

  1287 lemma bdd_below_image_dist[intro, simp]: "bdd_below (dist x  A)"

  1288   by (auto intro!: zero_le_dist)

  1289

  1290 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"

  1291   by (simp add: infdist_def)

  1292

  1293 lemma infdist_nonneg: "0 \<le> infdist x A"

  1294   by (auto simp: infdist_def intro: cINF_greatest)

  1295

  1296 lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"

  1297   by (auto intro: cINF_lower simp add: infdist_def)

  1298

  1299 lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"

  1300   by (auto intro!: cINF_lower2 simp add: infdist_def)

  1301

  1302 lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"

  1303   by (auto intro!: antisym infdist_nonneg infdist_le2)

  1304

  1305 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"

  1306 proof (cases "A = {}")

  1307   case True

  1308   then show ?thesis by (simp add: infdist_def)

  1309 next

  1310   case False

  1311   then obtain a where "a \<in> A" by auto

  1312   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  1313   proof (rule cInf_greatest)

  1314     from \<open>A \<noteq> {}\<close> show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"

  1315       by simp

  1316     fix d

  1317     assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  1318     then obtain a where d: "d = dist x y + dist y a" "a \<in> A"

  1319       by auto

  1320     show "infdist x A \<le> d"

  1321       unfolding infdist_notempty[OF \<open>A \<noteq> {}\<close>]

  1322     proof (rule cINF_lower2)

  1323       show "a \<in> A" by fact

  1324       show "dist x a \<le> d"

  1325         unfolding d by (rule dist_triangle)

  1326     qed simp

  1327   qed

  1328   also have "\<dots> = dist x y + infdist y A"

  1329   proof (rule cInf_eq, safe)

  1330     fix a

  1331     assume "a \<in> A"

  1332     then show "dist x y + infdist y A \<le> dist x y + dist y a"

  1333       by (auto intro: infdist_le)

  1334   next

  1335     fix i

  1336     assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  1337     then have "i - dist x y \<le> infdist y A"

  1338       unfolding infdist_notempty[OF \<open>A \<noteq> {}\<close>] using \<open>a \<in> A\<close>

  1339       by (intro cINF_greatest) (auto simp: field_simps)

  1340     then show "i \<le> dist x y + infdist y A"

  1341       by simp

  1342   qed

  1343   finally show ?thesis by simp

  1344 qed

  1345

  1346 lemma in_closure_iff_infdist_zero:

  1347   assumes "A \<noteq> {}"

  1348   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1349 proof

  1350   assume "x \<in> closure A"

  1351   show "infdist x A = 0"

  1352   proof (rule ccontr)

  1353     assume "infdist x A \<noteq> 0"

  1354     with infdist_nonneg[of x A] have "infdist x A > 0"

  1355       by auto

  1356     then have "ball x (infdist x A) \<inter> closure A = {}"

  1357       apply auto

  1358       apply (metis \<open>x \<in> closure A\<close> closure_approachable dist_commute infdist_le not_less)

  1359       done

  1360     then have "x \<notin> closure A"

  1361       by (metis \<open>0 < infdist x A\<close> centre_in_ball disjoint_iff_not_equal)

  1362     then show False using \<open>x \<in> closure A\<close> by simp

  1363   qed

  1364 next

  1365   assume x: "infdist x A = 0"

  1366   then obtain a where "a \<in> A"

  1367     by atomize_elim (metis all_not_in_conv assms)

  1368   show "x \<in> closure A"

  1369     unfolding closure_approachable

  1370     apply safe

  1371   proof (rule ccontr)

  1372     fix e :: real

  1373     assume "e > 0"

  1374     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  1375     then have "infdist x A \<ge> e" using \<open>a \<in> A\<close>

  1376       unfolding infdist_def

  1377       by (force simp: dist_commute intro: cINF_greatest)

  1378     with x \<open>e > 0\<close> show False by auto

  1379   qed

  1380 qed

  1381

  1382 lemma in_closed_iff_infdist_zero:

  1383   assumes "closed A" "A \<noteq> {}"

  1384   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  1385 proof -

  1386   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1387     by (rule in_closure_iff_infdist_zero) fact

  1388   with assms show ?thesis by simp

  1389 qed

  1390

  1391 lemma infdist_pos_not_in_closed:

  1392   assumes "closed S" "S \<noteq> {}" "x \<notin> S"

  1393   shows "infdist x S > 0"

  1394 using in_closed_iff_infdist_zero[OF assms(1) assms(2), of x] assms(3) infdist_nonneg le_less by fastforce

  1395

  1396 lemma

  1397   infdist_attains_inf:

  1398   fixes X::"'a::heine_borel set"

  1399   assumes "closed X"

  1400   assumes "X \<noteq> {}"

  1401   obtains x where "x \<in> X" "infdist y X = dist y x"

  1402 proof -

  1403   have "bdd_below (dist y  X)"

  1404     by auto

  1405   from distance_attains_inf[OF assms, of y]

  1406   obtain x where INF: "x \<in> X" "\<And>z. z \<in> X \<Longrightarrow> dist y x \<le> dist y z" by auto

  1407   have "infdist y X = dist y x"

  1408     by (auto simp: infdist_def assms

  1409       intro!: antisym cINF_lower[OF _ \<open>x \<in> X\<close>] cINF_greatest[OF assms(2) INF(2)])

  1410   with \<open>x \<in> X\<close> show ?thesis ..

  1411 qed

  1412

  1413

  1414 text \<open>Every metric space is a T4 space:\<close>

  1415

  1416 instance metric_space \<subseteq> t4_space

  1417 proof

  1418   fix S T::"'a set" assume H: "closed S" "closed T" "S \<inter> T = {}"

  1419   consider "S = {}" | "T = {}" | "S \<noteq> {} \<and> T \<noteq> {}" by auto

  1420   then show "\<exists>U V. open U \<and> open V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> U \<inter> V = {}"

  1421   proof (cases)

  1422     case 1

  1423     show ?thesis

  1424       apply (rule exI[of _ "{}"], rule exI[of _ UNIV]) using 1 by auto

  1425   next

  1426     case 2

  1427     show ?thesis

  1428       apply (rule exI[of _ UNIV], rule exI[of _ "{}"]) using 2 by auto

  1429   next

  1430     case 3

  1431     define U where "U = (\<Union>x\<in>S. ball x ((infdist x T)/2))"

  1432     have A: "open U" unfolding U_def by auto

  1433     have "infdist x T > 0" if "x \<in> S" for x

  1434       using H that 3 by (auto intro!: infdist_pos_not_in_closed)

  1435     then have B: "S \<subseteq> U" unfolding U_def by auto

  1436     define V where "V = (\<Union>x\<in>T. ball x ((infdist x S)/2))"

  1437     have C: "open V" unfolding V_def by auto

  1438     have "infdist x S > 0" if "x \<in> T" for x

  1439       using H that 3 by (auto intro!: infdist_pos_not_in_closed)

  1440     then have D: "T \<subseteq> V" unfolding V_def by auto

  1441

  1442     have "(ball x ((infdist x T)/2)) \<inter> (ball y ((infdist y S)/2)) = {}" if "x \<in> S" "y \<in> T" for x y

  1443     proof (auto)

  1444       fix z assume H: "dist x z * 2 < infdist x T" "dist y z * 2 < infdist y S"

  1445       have "2 * dist x y \<le> 2 * dist x z + 2 * dist y z"

  1446         using dist_triangle[of x y z] by (auto simp add: dist_commute)

  1447       also have "... < infdist x T + infdist y S"

  1448         using H by auto

  1449       finally have "dist x y < infdist x T \<or> dist x y < infdist y S"

  1450         by auto

  1451       then show False

  1452         using infdist_le[OF \<open>x \<in> S\<close>, of y] infdist_le[OF \<open>y \<in> T\<close>, of x] by (auto simp add: dist_commute)

  1453     qed

  1454     then have E: "U \<inter> V = {}"

  1455       unfolding U_def V_def by auto

  1456     show ?thesis

  1457       apply (rule exI[of _ U], rule exI[of _ V]) using A B C D E by auto

  1458   qed

  1459 qed

  1460

  1461 lemma tendsto_infdist [tendsto_intros]:

  1462   assumes f: "(f \<longlongrightarrow> l) F"

  1463   shows "((\<lambda>x. infdist (f x) A) \<longlongrightarrow> infdist l A) F"

  1464 proof (rule tendstoI)

  1465   fix e ::real

  1466   assume "e > 0"

  1467   from tendstoD[OF f this]

  1468   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  1469   proof (eventually_elim)

  1470     fix x

  1471     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  1472     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  1473       by (simp add: dist_commute dist_real_def)

  1474     also assume "dist (f x) l < e"

  1475     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  1476   qed

  1477 qed

  1478

  1479 lemma continuous_infdist[continuous_intros]:

  1480   assumes "continuous F f"

  1481   shows "continuous F (\<lambda>x. infdist (f x) A)"

  1482   using assms unfolding continuous_def by (rule tendsto_infdist)

  1483

  1484 lemma compact_infdist_le:

  1485   fixes A::"'a::heine_borel set"

  1486   assumes "A \<noteq> {}"

  1487   assumes "compact A"

  1488   assumes "e > 0"

  1489   shows "compact {x. infdist x A \<le> e}"

  1490 proof -

  1491   from continuous_closed_vimage[of "{0..e}" "\<lambda>x. infdist x A"]

  1492     continuous_infdist[OF continuous_ident, of _ UNIV A]

  1493   have "closed {x. infdist x A \<le> e}" by (auto simp: vimage_def infdist_nonneg)

  1494   moreover

  1495   from assms obtain x0 b where b: "\<And>x. x \<in> A \<Longrightarrow> dist x0 x \<le> b" "closed A"

  1496     by (auto simp: compact_eq_bounded_closed bounded_def)

  1497   {

  1498     fix y

  1499     assume le: "infdist y A \<le> e"

  1500     from infdist_attains_inf[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>, of y]

  1501     obtain z where z: "z \<in> A" "infdist y A = dist y z" by blast

  1502     have "dist x0 y \<le> dist y z + dist x0 z"

  1503       by (metis dist_commute dist_triangle)

  1504     also have "dist y z \<le> e" using le z by simp

  1505     also have "dist x0 z \<le> b" using b z by simp

  1506     finally have "dist x0 y \<le> b + e" by arith

  1507   } then

  1508   have "bounded {x. infdist x A \<le> e}"

  1509     by (auto simp: bounded_any_center[where a=x0] intro!: exI[where x="b + e"])

  1510   ultimately show "compact {x. infdist x A \<le> e}"

  1511     by (simp add: compact_eq_bounded_closed)

  1512 qed

  1513

  1514 subsection%unimportant \<open>Equality of continuous functions on closure and related results\<close>

  1515

  1516 lemma continuous_closedin_preimage_constant:

  1517   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  1518   shows "continuous_on S f \<Longrightarrow> closedin (subtopology euclidean S) {x \<in> S. f x = a}"

  1519   using continuous_closedin_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)

  1520

  1521 lemma continuous_closed_preimage_constant:

  1522   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  1523   shows "continuous_on S f \<Longrightarrow> closed S \<Longrightarrow> closed {x \<in> S. f x = a}"

  1524   using continuous_closed_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)

  1525

  1526 lemma continuous_constant_on_closure:

  1527   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  1528   assumes "continuous_on (closure S) f"

  1529       and "\<And>x. x \<in> S \<Longrightarrow> f x = a"

  1530       and "x \<in> closure S"

  1531   shows "f x = a"

  1532     using continuous_closed_preimage_constant[of "closure S" f a]

  1533       assms closure_minimal[of S "{x \<in> closure S. f x = a}"] closure_subset

  1534     unfolding subset_eq

  1535     by auto

  1536

  1537 lemma image_closure_subset:

  1538   assumes contf: "continuous_on (closure S) f"

  1539     and "closed T"

  1540     and "(f  S) \<subseteq> T"

  1541   shows "f  (closure S) \<subseteq> T"

  1542 proof -

  1543   have "S \<subseteq> {x \<in> closure S. f x \<in> T}"

  1544     using assms(3) closure_subset by auto

  1545   moreover have "closed (closure S \<inter> f - T)"

  1546     using continuous_closed_preimage[OF contf] \<open>closed T\<close> by auto

  1547   ultimately have "closure S = (closure S \<inter> f - T)"

  1548     using closure_minimal[of S "(closure S \<inter> f - T)"] by auto

  1549   then show ?thesis by auto

  1550 qed

  1551

  1552 lemma continuous_on_closure_norm_le:

  1553   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  1554   assumes "continuous_on (closure s) f"

  1555     and "\<forall>y \<in> s. norm(f y) \<le> b"

  1556     and "x \<in> (closure s)"

  1557   shows "norm (f x) \<le> b"

  1558 proof -

  1559   have *: "f  s \<subseteq> cball 0 b"

  1560     using assms(2)[unfolded mem_cball_0[symmetric]] by auto

  1561   show ?thesis

  1562     by (meson "*" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)

  1563 qed

  1564

  1565 lemma isCont_indicator:

  1566   fixes x :: "'a::t2_space"

  1567   shows "isCont (indicator A :: 'a \<Rightarrow> real) x = (x \<notin> frontier A)"

  1568 proof auto

  1569   fix x

  1570   assume cts_at: "isCont (indicator A :: 'a \<Rightarrow> real) x" and fr: "x \<in> frontier A"

  1571   with continuous_at_open have 1: "\<forall>V::real set. open V \<and> indicator A x \<in> V \<longrightarrow>

  1572     (\<exists>U::'a set. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> V))" by auto

  1573   show False

  1574   proof (cases "x \<in> A")

  1575     assume x: "x \<in> A"

  1576     hence "indicator A x \<in> ({0<..<2} :: real set)" by simp

  1577     hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({0<..<2} :: real set))"

  1578       using 1 open_greaterThanLessThan by blast

  1579     then guess U .. note U = this

  1580     hence "\<forall>y\<in>U. indicator A y > (0::real)"

  1581       unfolding greaterThanLessThan_def by auto

  1582     hence "U \<subseteq> A" using indicator_eq_0_iff by force

  1583     hence "x \<in> interior A" using U interiorI by auto

  1584     thus ?thesis using fr unfolding frontier_def by simp

  1585   next

  1586     assume x: "x \<notin> A"

  1587     hence "indicator A x \<in> ({-1<..<1} :: real set)" by simp

  1588     hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({-1<..<1} :: real set))"

  1589       using 1 open_greaterThanLessThan by blast

  1590     then guess U .. note U = this

  1591     hence "\<forall>y\<in>U. indicator A y < (1::real)"

  1592       unfolding greaterThanLessThan_def by auto

  1593     hence "U \<subseteq> -A" by auto

  1594     hence "x \<in> interior (-A)" using U interiorI by auto

  1595     thus ?thesis using fr interior_complement unfolding frontier_def by auto

  1596   qed

  1597 next

  1598   assume nfr: "x \<notin> frontier A"

  1599   hence "x \<in> interior A \<or> x \<in> interior (-A)"

  1600     by (auto simp: frontier_def closure_interior)

  1601   thus "isCont ((indicator A)::'a \<Rightarrow> real) x"

  1602   proof

  1603     assume int: "x \<in> interior A"

  1604     then obtain U where U: "open U" "x \<in> U" "U \<subseteq> A" unfolding interior_def by auto

  1605     hence "\<forall>y\<in>U. indicator A y = (1::real)" unfolding indicator_def by auto

  1606     hence "continuous_on U (indicator A)" by (simp add: continuous_on_const indicator_eq_1_iff)

  1607     thus ?thesis using U continuous_on_eq_continuous_at by auto

  1608   next

  1609     assume ext: "x \<in> interior (-A)"

  1610     then obtain U where U: "open U" "x \<in> U" "U \<subseteq> -A" unfolding interior_def by auto

  1611     then have "continuous_on U (indicator A)"

  1612       using continuous_on_topological by (auto simp: subset_iff)

  1613     thus ?thesis using U continuous_on_eq_continuous_at by auto

  1614   qed

  1615 qed

  1616

  1617 subsection%unimportant \<open>A function constant on a set\<close>

  1618

  1619 definition constant_on  (infixl "(constant'_on)" 50)

  1620   where "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"

  1621

  1622 lemma constant_on_subset: "\<lbrakk>f constant_on A; B \<subseteq> A\<rbrakk> \<Longrightarrow> f constant_on B"

  1623   unfolding constant_on_def by blast

  1624

  1625 lemma injective_not_constant:

  1626   fixes S :: "'a::{perfect_space} set"

  1627   shows "\<lbrakk>open S; inj_on f S; f constant_on S\<rbrakk> \<Longrightarrow> S = {}"

  1628 unfolding constant_on_def

  1629 by (metis equals0I inj_on_contraD islimpt_UNIV islimpt_def)

  1630

  1631 lemma constant_on_closureI:

  1632   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  1633   assumes cof: "f constant_on S" and contf: "continuous_on (closure S) f"

  1634     shows "f constant_on (closure S)"

  1635 using continuous_constant_on_closure [OF contf] cof unfolding constant_on_def

  1636 by metis

  1637

  1638 subsection%unimportant\<open>Relating linear images to open/closed/interior/closure\<close>

  1639

  1640 proposition open_surjective_linear_image:

  1641   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"

  1642   assumes "open A" "linear f" "surj f"

  1643     shows "open(f  A)"

  1644 unfolding open_dist

  1645 proof clarify

  1646   fix x

  1647   assume "x \<in> A"

  1648   have "bounded (inv f  Basis)"

  1649     by (simp add: finite_imp_bounded)

  1650   with bounded_pos obtain B where "B > 0" and B: "\<And>x. x \<in> inv f  Basis \<Longrightarrow> norm x \<le> B"

  1651     by metis

  1652   obtain e where "e > 0" and e: "\<And>z. dist z x < e \<Longrightarrow> z \<in> A"

  1653     by (metis open_dist \<open>x \<in> A\<close> \<open>open A\<close>)

  1654   define \<delta> where "\<delta> \<equiv> e / B / DIM('b)"

  1655   show "\<exists>e>0. \<forall>y. dist y (f x) < e \<longrightarrow> y \<in> f  A"

  1656   proof (intro exI conjI)

  1657     show "\<delta> > 0"

  1658       using \<open>e > 0\<close> \<open>B > 0\<close>  by (simp add: \<delta>_def divide_simps)

  1659     have "y \<in> f  A" if "dist y (f x) * (B * real DIM('b)) < e" for y

  1660     proof -

  1661       define u where "u \<equiv> y - f x"

  1662       show ?thesis

  1663       proof (rule image_eqI)

  1664         show "y = f (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i))"

  1665           apply (simp add: linear_add linear_sum linear.scaleR \<open>linear f\<close> surj_f_inv_f \<open>surj f\<close>)

  1666           apply (simp add: euclidean_representation u_def)

  1667           done

  1668         have "dist (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i)) x \<le> (\<Sum>i\<in>Basis. norm ((u \<bullet> i) *\<^sub>R inv f i))"

  1669           by (simp add: dist_norm sum_norm_le)

  1670         also have "... = (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar> * norm (inv f i))"

  1671           by simp

  1672         also have "... \<le> (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar>) * B"

  1673           by (simp add: B sum_distrib_right sum_mono mult_left_mono)

  1674         also have "... \<le> DIM('b) * dist y (f x) * B"

  1675           apply (rule mult_right_mono [OF sum_bounded_above])

  1676           using \<open>0 < B\<close> by (auto simp: Basis_le_norm dist_norm u_def)

  1677         also have "... < e"

  1678           by (metis mult.commute mult.left_commute that)

  1679         finally show "x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i) \<in> A"

  1680           by (rule e)

  1681       qed

  1682     qed

  1683     then show "\<forall>y. dist y (f x) < \<delta> \<longrightarrow> y \<in> f  A"

  1684       using \<open>e > 0\<close> \<open>B > 0\<close>

  1685       by (auto simp: \<delta>_def divide_simps mult_less_0_iff)

  1686   qed

  1687 qed

  1688

  1689 corollary open_bijective_linear_image_eq:

  1690   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"

  1691   assumes "linear f" "bij f"

  1692     shows "open(f  A) \<longleftrightarrow> open A"

  1693 proof

  1694   assume "open(f  A)"

  1695   then have "open(f - (f  A))"

  1696     using assms by (force simp: linear_continuous_at linear_conv_bounded_linear continuous_open_vimage)

  1697   then show "open A"

  1698     by (simp add: assms bij_is_inj inj_vimage_image_eq)

  1699 next

  1700   assume "open A"

  1701   then show "open(f  A)"

  1702     by (simp add: assms bij_is_surj open_surjective_linear_image)

  1703 qed

  1704

  1705 corollary interior_bijective_linear_image:

  1706   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"

  1707   assumes "linear f" "bij f"

  1708   shows "interior (f  S) = f  interior S"  (is "?lhs = ?rhs")

  1709 proof safe

  1710   fix x

  1711   assume x: "x \<in> ?lhs"

  1712   then obtain T where "open T" and "x \<in> T" and "T \<subseteq> f  S"

  1713     by (metis interiorE)

  1714   then show "x \<in> ?rhs"

  1715     by (metis (no_types, hide_lams) assms subsetD interior_maximal open_bijective_linear_image_eq subset_image_iff)

  1716 next

  1717   fix x

  1718   assume x: "x \<in> interior S"

  1719   then show "f x \<in> interior (f  S)"

  1720     by (meson assms imageI image_mono interiorI interior_subset open_bijective_linear_image_eq open_interior)

  1721 qed

  1722

  1723 lemma interior_injective_linear_image:

  1724   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"

  1725   assumes "linear f" "inj f"

  1726    shows "interior(f  S) = f  (interior S)"

  1727   by (simp add: linear_injective_imp_surjective assms bijI interior_bijective_linear_image)

  1728

  1729 lemma interior_surjective_linear_image:

  1730   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"

  1731   assumes "linear f" "surj f"

  1732    shows "interior(f  S) = f  (interior S)"

  1733   by (simp add: assms interior_injective_linear_image linear_surjective_imp_injective)

  1734

  1735 lemma interior_negations:

  1736   fixes S :: "'a::euclidean_space set"

  1737   shows "interior(uminus  S) = image uminus (interior S)"

  1738   by (simp add: bij_uminus interior_bijective_linear_image linear_uminus)

  1739

  1740 text \<open>Preservation of compactness and connectedness under continuous function.\<close>

  1741

  1742 lemma compact_eq_openin_cover:

  1743   "compact S \<longleftrightarrow>

  1744     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  1745       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  1746 proof safe

  1747   fix C

  1748   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  1749   then have "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"

  1750     unfolding openin_open by force+

  1751   with \<open>compact S\<close> obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"

  1752     by (meson compactE)

  1753   then have "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"

  1754     by auto

  1755   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  1756 next

  1757   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  1758         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  1759   show "compact S"

  1760   proof (rule compactI)

  1761     fix C

  1762     let ?C = "image (\<lambda>T. S \<inter> T) C"

  1763     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  1764     then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  1765       unfolding openin_open by auto

  1766     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  1767       by metis

  1768     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  1769     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  1770     proof (intro conjI)

  1771       from \<open>D \<subseteq> ?C\<close> show "?D \<subseteq> C"

  1772         by (fast intro: inv_into_into)

  1773       from \<open>finite D\<close> show "finite ?D"

  1774         by (rule finite_imageI)

  1775       from \<open>S \<subseteq> \<Union>D\<close> show "S \<subseteq> \<Union>?D"

  1776         apply (rule subset_trans, clarsimp)

  1777         apply (frule subsetD [OF \<open>D \<subseteq> ?C\<close>, THEN f_inv_into_f])

  1778         apply (erule rev_bexI, fast)

  1779         done

  1780     qed

  1781     then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  1782   qed

  1783 qed

  1784

  1785 subsection%unimportant\<open> Theorems relating continuity and uniform continuity to closures\<close>

  1786

  1787 lemma continuous_on_closure:

  1788    "continuous_on (closure S) f \<longleftrightarrow>

  1789     (\<forall>x e. x \<in> closure S \<and> 0 < e

  1790            \<longrightarrow> (\<exists>d. 0 < d \<and> (\<forall>y. y \<in> S \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e)))"

  1791    (is "?lhs = ?rhs")

  1792 proof

  1793   assume ?lhs then show ?rhs

  1794     unfolding continuous_on_iff  by (metis Un_iff closure_def)

  1795 next

  1796   assume R [rule_format]: ?rhs

  1797   show ?lhs

  1798   proof

  1799     fix x and e::real

  1800     assume "0 < e" and x: "x \<in> closure S"

  1801     obtain \<delta>::real where "\<delta> > 0"

  1802                    and \<delta>: "\<And>y. \<lbrakk>y \<in> S; dist y x < \<delta>\<rbrakk> \<Longrightarrow> dist (f y) (f x) < e/2"

  1803       using R [of x "e/2"] \<open>0 < e\<close> x by auto

  1804     have "dist (f y) (f x) \<le> e" if y: "y \<in> closure S" and dyx: "dist y x < \<delta>/2" for y

  1805     proof -

  1806       obtain \<delta>'::real where "\<delta>' > 0"

  1807                       and \<delta>': "\<And>z. \<lbrakk>z \<in> S; dist z y < \<delta>'\<rbrakk> \<Longrightarrow> dist (f z) (f y) < e/2"

  1808         using R [of y "e/2"] \<open>0 < e\<close> y by auto

  1809       obtain z where "z \<in> S" and z: "dist z y < min \<delta>' \<delta> / 2"

  1810         using closure_approachable y

  1811         by (metis \<open>0 < \<delta>'\<close> \<open>0 < \<delta>\<close> divide_pos_pos min_less_iff_conj zero_less_numeral)

  1812       have "dist (f z) (f y) < e/2"

  1813         apply (rule \<delta>' [OF \<open>z \<in> S\<close>])

  1814         using z \<open>0 < \<delta>'\<close> by linarith

  1815       moreover have "dist (f z) (f x) < e/2"

  1816         apply (rule \<delta> [OF \<open>z \<in> S\<close>])

  1817         using z \<open>0 < \<delta>\<close>  dist_commute[of y z] dist_triangle_half_r [of y] dyx by auto

  1818       ultimately show ?thesis

  1819         by (metis dist_commute dist_triangle_half_l less_imp_le)

  1820     qed

  1821     then show "\<exists>d>0. \<forall>x'\<in>closure S. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"

  1822       by (rule_tac x="\<delta>/2" in exI) (simp add: \<open>\<delta> > 0\<close>)

  1823   qed

  1824 qed

  1825

  1826 lemma continuous_on_closure_sequentially:

  1827   fixes f :: "'a::metric_space \<Rightarrow> 'b :: metric_space"

  1828   shows

  1829    "continuous_on (closure S) f \<longleftrightarrow>

  1830     (\<forall>x a. a \<in> closure S \<and> (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"

  1831    (is "?lhs = ?rhs")

  1832 proof -

  1833   have "continuous_on (closure S) f \<longleftrightarrow>

  1834            (\<forall>x \<in> closure S. continuous (at x within S) f)"

  1835     by (force simp: continuous_on_closure continuous_within_eps_delta)

  1836   also have "... = ?rhs"

  1837     by (force simp: continuous_within_sequentially)

  1838   finally show ?thesis .

  1839 qed

  1840

  1841 lemma uniformly_continuous_on_closure:

  1842   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  1843   assumes ucont: "uniformly_continuous_on S f"

  1844       and cont: "continuous_on (closure S) f"

  1845     shows "uniformly_continuous_on (closure S) f"

  1846 unfolding uniformly_continuous_on_def

  1847 proof (intro allI impI)

  1848   fix e::real

  1849   assume "0 < e"

  1850   then obtain d::real

  1851     where "d>0"

  1852       and d: "\<And>x x'. \<lbrakk>x\<in>S; x'\<in>S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e/3"

  1853     using ucont [unfolded uniformly_continuous_on_def, rule_format, of "e/3"] by auto

  1854   show "\<exists>d>0. \<forall>x\<in>closure S. \<forall>x'\<in>closure S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  1855   proof (rule exI [where x="d/3"], clarsimp simp: \<open>d > 0\<close>)

  1856     fix x y

  1857     assume x: "x \<in> closure S" and y: "y \<in> closure S" and dyx: "dist y x * 3 < d"

  1858     obtain d1::real where "d1 > 0"

  1859            and d1: "\<And>w. \<lbrakk>w \<in> closure S; dist w x < d1\<rbrakk> \<Longrightarrow> dist (f w) (f x) < e/3"

  1860       using cont [unfolded continuous_on_iff, rule_format, of "x" "e/3"] \<open>0 < e\<close> x by auto

  1861      obtain x' where "x' \<in> S" and x': "dist x' x < min d1 (d / 3)"

  1862         using closure_approachable [of x S]

  1863         by (metis \<open>0 < d1\<close> \<open>0 < d\<close> divide_pos_pos min_less_iff_conj x zero_less_numeral)

  1864     obtain d2::real where "d2 > 0"

  1865            and d2: "\<forall>w \<in> closure S. dist w y < d2 \<longrightarrow> dist (f w) (f y) < e/3"

  1866       using cont [unfolded continuous_on_iff, rule_format, of "y" "e/3"] \<open>0 < e\<close> y by auto

  1867      obtain y' where "y' \<in> S" and y': "dist y' y < min d2 (d / 3)"

  1868         using closure_approachable [of y S]

  1869         by (metis \<open>0 < d2\<close> \<open>0 < d\<close> divide_pos_pos min_less_iff_conj y zero_less_numeral)

  1870      have "dist x' x < d/3" using x' by auto

  1871      moreover have "dist x y < d/3"

  1872        by (metis dist_commute dyx less_divide_eq_numeral1(1))

  1873      moreover have "dist y y' < d/3"

  1874        by (metis (no_types) dist_commute min_less_iff_conj y')

  1875      ultimately have "dist x' y' < d/3 + d/3 + d/3"

  1876        by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)

  1877      then have "dist x' y' < d" by simp

  1878      then have "dist (f x') (f y') < e/3"

  1879        by (rule d [OF \<open>y' \<in> S\<close> \<open>x' \<in> S\<close>])

  1880      moreover have "dist (f x') (f x) < e/3" using \<open>x' \<in> S\<close> closure_subset x' d1

  1881        by (simp add: closure_def)

  1882      moreover have "dist (f y') (f y) < e/3" using \<open>y' \<in> S\<close> closure_subset y' d2

  1883        by (simp add: closure_def)

  1884      ultimately have "dist (f y) (f x) < e/3 + e/3 + e/3"

  1885        by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)

  1886     then show "dist (f y) (f x) < e" by simp

  1887   qed

  1888 qed

  1889

  1890 lemma uniformly_continuous_on_extension_at_closure:

  1891   fixes f::"'a::metric_space \<Rightarrow> 'b::complete_space"

  1892   assumes uc: "uniformly_continuous_on X f"

  1893   assumes "x \<in> closure X"

  1894   obtains l where "(f \<longlongrightarrow> l) (at x within X)"

  1895 proof -

  1896   from assms obtain xs where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X"

  1897     by (auto simp: closure_sequential)

  1898

  1899   from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF xs]

  1900   obtain l where l: "(\<lambda>n. f (xs n)) \<longlonglongrightarrow> l"

  1901     by atomize_elim (simp only: convergent_eq_Cauchy)

  1902

  1903   have "(f \<longlongrightarrow> l) (at x within X)"

  1904   proof (safe intro!: Lim_within_LIMSEQ)

  1905     fix xs'

  1906     assume "\<forall>n. xs' n \<noteq> x \<and> xs' n \<in> X"

  1907       and xs': "xs' \<longlonglongrightarrow> x"

  1908     then have "xs' n \<noteq> x" "xs' n \<in> X" for n by auto

  1909

  1910     from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF \<open>xs' \<longlonglongrightarrow> x\<close> \<open>xs' _ \<in> X\<close>]

  1911     obtain l' where l': "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l'"

  1912       by atomize_elim (simp only: convergent_eq_Cauchy)

  1913

  1914     show "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l"

  1915     proof (rule tendstoI)

  1916       fix e::real assume "e > 0"

  1917       define e' where "e' \<equiv> e / 2"

  1918       have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def)

  1919

  1920       have "\<forall>\<^sub>F n in sequentially. dist (f (xs n)) l < e'"

  1921         by (simp add: \<open>0 < e'\<close> l tendstoD)

  1922       moreover

  1923       from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>e' > 0\<close>]

  1924       obtain d where d: "d > 0" "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x x' < d \<Longrightarrow> dist (f x) (f x') < e'"

  1925         by auto

  1926       have "\<forall>\<^sub>F n in sequentially. dist (xs n) (xs' n) < d"

  1927         by (auto intro!: \<open>0 < d\<close> order_tendstoD tendsto_eq_intros xs xs')

  1928       ultimately

  1929       show "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) l < e"

  1930       proof eventually_elim

  1931         case (elim n)

  1932         have "dist (f (xs' n)) l \<le> dist (f (xs n)) (f (xs' n)) + dist (f (xs n)) l"

  1933           by (metis dist_triangle dist_commute)

  1934         also have "dist (f (xs n)) (f (xs' n)) < e'"

  1935           by (auto intro!: d xs \<open>xs' _ \<in> _\<close> elim)

  1936         also note \<open>dist (f (xs n)) l < e'\<close>

  1937         also have "e' + e' = e" by (simp add: e'_def)

  1938         finally show ?case by simp

  1939       qed

  1940     qed

  1941   qed

  1942   thus ?thesis ..

  1943 qed

  1944

  1945 lemma uniformly_continuous_on_extension_on_closure:

  1946   fixes f::"'a::metric_space \<Rightarrow> 'b::complete_space"

  1947   assumes uc: "uniformly_continuous_on X f"

  1948   obtains g where "uniformly_continuous_on (closure X) g" "\<And>x. x \<in> X \<Longrightarrow> f x = g x"

  1949     "\<And>Y h x. X \<subseteq> Y \<Longrightarrow> Y \<subseteq> closure X \<Longrightarrow> continuous_on Y h \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> f x = h x) \<Longrightarrow> x \<in> Y \<Longrightarrow> h x = g x"

  1950 proof -

  1951   from uc have cont_f: "continuous_on X f"

  1952     by (simp add: uniformly_continuous_imp_continuous)

  1953   obtain y where y: "(f \<longlongrightarrow> y x) (at x within X)" if "x \<in> closure X" for x

  1954     apply atomize_elim

  1955     apply (rule choice)

  1956     using uniformly_continuous_on_extension_at_closure[OF assms]

  1957     by metis

  1958   let ?g = "\<lambda>x. if x \<in> X then f x else y x"

  1959

  1960   have "uniformly_continuous_on (closure X) ?g"

  1961     unfolding uniformly_continuous_on_def

  1962   proof safe

  1963     fix e::real assume "e > 0"

  1964     define e' where "e' \<equiv> e / 3"

  1965     have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def)

  1966     from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>0 < e'\<close>]

  1967     obtain d where "d > 0" and d: "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x' x < d \<Longrightarrow> dist (f x') (f x) < e'"

  1968       by auto

  1969     define d' where "d' = d / 3"

  1970     have "d' > 0" using \<open>d > 0\<close> by (simp add: d'_def)

  1971     show "\<exists>d>0. \<forall>x\<in>closure X. \<forall>x'\<in>closure X. dist x' x < d \<longrightarrow> dist (?g x') (?g x) < e"

  1972     proof (safe intro!: exI[where x=d'] \<open>d' > 0\<close>)

  1973       fix x x' assume x: "x \<in> closure X" and x': "x' \<in> closure X" and dist: "dist x' x < d'"

  1974       then obtain xs xs' where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X"

  1975         and xs': "xs' \<longlonglongrightarrow> x'" "\<And>n. xs' n \<in> X"

  1976         by (auto simp: closure_sequential)

  1977       have "\<forall>\<^sub>F n in sequentially. dist (xs' n) x' < d'"

  1978         and "\<forall>\<^sub>F n in sequentially. dist (xs n) x < d'"

  1979         by (auto intro!: \<open>0 < d'\<close> order_tendstoD tendsto_eq_intros xs xs')

  1980       moreover

  1981       have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x" if "x \<in> closure X" "x \<notin> X" "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" for xs x

  1982         using that not_eventuallyD

  1983         by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at)

  1984       then have "(\<lambda>x. f (xs' x)) \<longlonglongrightarrow> ?g x'" "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> ?g x"

  1985         using x x'

  1986         by (auto intro!: continuous_on_tendsto_compose[OF cont_f] simp: xs' xs)

  1987       then have "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) (?g x') < e'"

  1988         "\<forall>\<^sub>F n in sequentially. dist (f (xs n)) (?g x) < e'"

  1989         by (auto intro!: \<open>0 < e'\<close> order_tendstoD tendsto_eq_intros)

  1990       ultimately

  1991       have "\<forall>\<^sub>F n in sequentially. dist (?g x') (?g x) < e"

  1992       proof eventually_elim

  1993         case (elim n)

  1994         have "dist (?g x') (?g x) \<le>

  1995           dist (f (xs' n)) (?g x') + dist (f (xs' n)) (f (xs n)) + dist (f (xs n)) (?g x)"

  1996           by (metis add.commute add_le_cancel_left dist_commute dist_triangle dist_triangle_le)

  1997         also

  1998         {

  1999           have "dist (xs' n) (xs n) \<le> dist (xs' n) x' + dist x' x + dist (xs n) x"

  2000             by (metis add.commute add_le_cancel_left  dist_triangle dist_triangle_le)

  2001           also note \<open>dist (xs' n) x' < d'\<close>

  2002           also note \<open>dist x' x < d'\<close>

  2003           also note \<open>dist (xs n) x < d'\<close>

  2004           finally have "dist (xs' n) (xs n) < d" by (simp add: d'_def)

  2005         }

  2006         with \<open>xs _ \<in> X\<close> \<open>xs' _ \<in> X\<close> have "dist (f (xs' n)) (f (xs n)) < e'"

  2007           by (rule d)

  2008         also note \<open>dist (f (xs' n)) (?g x') < e'\<close>

  2009         also note \<open>dist (f (xs n)) (?g x) < e'\<close>

  2010         finally show ?case by (simp add: e'_def)

  2011       qed

  2012       then show "dist (?g x') (?g x) < e" by simp

  2013     qed

  2014   qed

  2015   moreover have "f x = ?g x" if "x \<in> X" for x using that by simp

  2016   moreover

  2017   {

  2018     fix Y h x

  2019     assume Y: "x \<in> Y" "X \<subseteq> Y" "Y \<subseteq> closure X" and cont_h: "continuous_on Y h"

  2020       and extension: "(\<And>x. x \<in> X \<Longrightarrow> f x = h x)"

  2021     {

  2022       assume "x \<notin> X"

  2023       have "x \<in> closure X" using Y by auto

  2024       then obtain xs where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X"

  2025         by (auto simp: closure_sequential)

  2026       from continuous_on_tendsto_compose[OF cont_h xs(1)] xs(2) Y

  2027       have hx: "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> h x"

  2028         by (auto simp: set_mp extension)

  2029       then have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x"

  2030         using \<open>x \<notin> X\<close> not_eventuallyD xs(2)

  2031         by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at xs)

  2032       with hx have "h x = y x" by (rule LIMSEQ_unique)

  2033     } then

  2034     have "h x = ?g x"

  2035       using extension by auto

  2036   }

  2037   ultimately show ?thesis ..

  2038 qed

  2039

  2040 lemma bounded_uniformly_continuous_image:

  2041   fixes f :: "'a :: heine_borel \<Rightarrow> 'b :: heine_borel"

  2042   assumes "uniformly_continuous_on S f" "bounded S"

  2043   shows "bounded(f  S)"

  2044   by (metis (no_types, lifting) assms bounded_closure_image compact_closure compact_continuous_image compact_eq_bounded_closed image_cong uniformly_continuous_imp_continuous uniformly_continuous_on_extension_on_closure)

  2045

  2046 subsection%unimportant \<open>Making a continuous function avoid some value in a neighbourhood\<close>

  2047

  2048 lemma continuous_within_avoid:

  2049   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  2050   assumes "continuous (at x within s) f"

  2051     and "f x \<noteq> a"

  2052   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  2053 proof -

  2054   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  2055     using t1_space [OF \<open>f x \<noteq> a\<close>] by fast

  2056   have "(f \<longlongrightarrow> f x) (at x within s)"

  2057     using assms(1) by (simp add: continuous_within)

  2058   then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  2059     using \<open>open U\<close> and \<open>f x \<in> U\<close>

  2060     unfolding tendsto_def by fast

  2061   then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  2062     using \<open>a \<notin> U\<close> by (fast elim: eventually_mono)

  2063   then show ?thesis

  2064     using \<open>f x \<noteq> a\<close> by (auto simp: dist_commute eventually_at)

  2065 qed

  2066

  2067 lemma continuous_at_avoid:

  2068   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  2069   assumes "continuous (at x) f"

  2070     and "f x \<noteq> a"

  2071   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  2072   using assms continuous_within_avoid[of x UNIV f a] by simp

  2073

  2074 lemma continuous_on_avoid:

  2075   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  2076   assumes "continuous_on s f"

  2077     and "x \<in> s"

  2078     and "f x \<noteq> a"

  2079   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  2080   using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],

  2081     OF assms(2)] continuous_within_avoid[of x s f a]

  2082   using assms(3)

  2083   by auto

  2084

  2085 lemma continuous_on_open_avoid:

  2086   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  2087   assumes "continuous_on s f"

  2088     and "open s"

  2089     and "x \<in> s"

  2090     and "f x \<noteq> a"

  2091   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  2092   using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]

  2093   using continuous_at_avoid[of x f a] assms(4)

  2094   by auto

  2095

  2096 subsection%unimportant\<open>Quotient maps\<close>

  2097

  2098 lemma quotient_map_imp_continuous_open:

  2099   assumes T: "f  S \<subseteq> T"

  2100       and ope: "\<And>U. U \<subseteq> T

  2101               \<Longrightarrow> (openin (subtopology euclidean S) (S \<inter> f - U) \<longleftrightarrow>

  2102                    openin (subtopology euclidean T) U)"

  2103     shows "continuous_on S f"

  2104 proof -

  2105   have [simp]: "S \<inter> f - f  S = S" by auto

  2106   show ?thesis

  2107     using ope [OF T]

  2108     apply (simp add: continuous_on_open)

  2109     by (meson ope openin_imp_subset openin_trans)

  2110 qed

  2111

  2112 lemma quotient_map_imp_continuous_closed:

  2113   assumes T: "f  S \<subseteq> T"

  2114       and ope: "\<And>U. U \<subseteq> T

  2115                   \<Longrightarrow> (closedin (subtopology euclidean S) (S \<inter> f - U) \<longleftrightarrow>

  2116                        closedin (subtopology euclidean T) U)"

  2117     shows "continuous_on S f"

  2118 proof -

  2119   have [simp]: "S \<inter> f - f  S = S" by auto

  2120   show ?thesis

  2121     using ope [OF T]

  2122     apply (simp add: continuous_on_closed)

  2123     by (metis (no_types, lifting) ope closedin_imp_subset closedin_trans)

  2124 qed

  2125

  2126 lemma open_map_imp_quotient_map:

  2127   assumes contf: "continuous_on S f"

  2128       and T: "T \<subseteq> f  S"

  2129       and ope: "\<And>T. openin (subtopology euclidean S) T

  2130                    \<Longrightarrow> openin (subtopology euclidean (f  S)) (f  T)"

  2131     shows "openin (subtopology euclidean S) (S \<inter> f - T) =

  2132            openin (subtopology euclidean (f  S)) T"

  2133 proof -

  2134   have "T = f  (S \<inter> f - T)"

  2135     using T by blast

  2136   then show ?thesis

  2137     using "ope" contf continuous_on_open by metis

  2138 qed

  2139

  2140 lemma closed_map_imp_quotient_map:

  2141   assumes contf: "continuous_on S f"

  2142       and T: "T \<subseteq> f  S"

  2143       and ope: "\<And>T. closedin (subtopology euclidean S) T

  2144               \<Longrightarrow> closedin (subtopology euclidean (f  S)) (f  T)"

  2145     shows "openin (subtopology euclidean S) (S \<inter> f - T) \<longleftrightarrow>

  2146            openin (subtopology euclidean (f  S)) T"

  2147           (is "?lhs = ?rhs")

  2148 proof

  2149   assume ?lhs

  2150   then have *: "closedin (subtopology euclidean S) (S - (S \<inter> f - T))"

  2151     using closedin_diff by fastforce

  2152   have [simp]: "(f  S - f  (S - (S \<inter> f - T))) = T"

  2153     using T by blast

  2154   show ?rhs

  2155     using ope [OF *, unfolded closedin_def] by auto

  2156 next

  2157   assume ?rhs

  2158   with contf show ?lhs

  2159     by (auto simp: continuous_on_open)

  2160 qed

  2161

  2162 lemma continuous_right_inverse_imp_quotient_map:

  2163   assumes contf: "continuous_on S f" and imf: "f  S \<subseteq> T"

  2164       and contg: "continuous_on T g" and img: "g  T \<subseteq> S"

  2165       and fg [simp]: "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"

  2166       and U: "U \<subseteq> T"

  2167     shows "openin (subtopology euclidean S) (S \<inter> f - U) \<longleftrightarrow>

  2168            openin (subtopology euclidean T) U"

  2169           (is "?lhs = ?rhs")

  2170 proof -

  2171   have f: "\<And>Z. openin (subtopology euclidean (f  S)) Z \<Longrightarrow>

  2172                 openin (subtopology euclidean S) (S \<inter> f - Z)"

  2173   and  g: "\<And>Z. openin (subtopology euclidean (g  T)) Z \<Longrightarrow>

  2174                 openin (subtopology euclidean T) (T \<inter> g - Z)"

  2175     using contf contg by (auto simp: continuous_on_open)

  2176   show ?thesis

  2177   proof

  2178     have "T \<inter> g - (g  T \<inter> (S \<inter> f - U)) = {x \<in> T. f (g x) \<in> U}"

  2179       using imf img by blast

  2180     also have "... = U"

  2181       using U by auto

  2182     finally have eq: "T \<inter> g - (g  T \<inter> (S \<inter> f - U)) = U" .

  2183     assume ?lhs

  2184     then have *: "openin (subtopology euclidean (g  T)) (g  T \<inter> (S \<inter> f - U))"

  2185       by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)

  2186     show ?rhs

  2187       using g [OF *] eq by auto

  2188   next

  2189     assume rhs: ?rhs

  2190     show ?lhs

  2191       by (metis f fg image_eqI image_subset_iff imf img openin_subopen openin_subtopology_self openin_trans rhs)

  2192   qed

  2193 qed

  2194

  2195 lemma continuous_left_inverse_imp_quotient_map:

  2196   assumes "continuous_on S f"

  2197       and "continuous_on (f  S) g"

  2198       and  "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"

  2199       and "U \<subseteq> f  S"

  2200     shows "openin (subtopology euclidean S) (S \<inter> f - U) \<longleftrightarrow>

  2201            openin (subtopology euclidean (f  S)) U"

  2202 apply (rule continuous_right_inverse_imp_quotient_map)

  2203 using assms apply force+

  2204 done

  2205

  2206

  2207 text \<open>Proving a function is constant by proving that a level set is open\<close>

  2208

  2209 lemma continuous_levelset_openin_cases:

  2210   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  2211   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  2212         openin (subtopology euclidean s) {x \<in> s. f x = a}

  2213         \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  2214   unfolding connected_clopen

  2215   using continuous_closedin_preimage_constant by auto

  2216

  2217 lemma continuous_levelset_openin:

  2218   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  2219   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  2220         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  2221         (\<exists>x \<in> s. f x = a)  \<Longrightarrow> (\<forall>x \<in> s. f x = a)"

  2222   using continuous_levelset_openin_cases[of s f ]

  2223   by meson

  2224

  2225 lemma continuous_levelset_open:

  2226   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  2227   assumes "connected s"

  2228     and "continuous_on s f"

  2229     and "open {x \<in> s. f x = a}"

  2230     and "\<exists>x \<in> s.  f x = a"

  2231   shows "\<forall>x \<in> s. f x = a"

  2232   using continuous_levelset_openin[OF assms(1,2), of a, unfolded openin_open]

  2233   using assms (3,4)

  2234   by fast

  2235

  2236 text \<open>Some arithmetical combinations (more to prove).\<close>

  2237

  2238 lemma open_scaling[intro]:

  2239   fixes s :: "'a::real_normed_vector set"

  2240   assumes "c \<noteq> 0"

  2241     and "open s"

  2242   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  2243 proof -

  2244   {

  2245     fix x

  2246     assume "x \<in> s"

  2247     then obtain e where "e>0"

  2248       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]

  2249       by auto

  2250     have "e * \<bar>c\<bar> > 0"

  2251       using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto

  2252     moreover

  2253     {

  2254       fix y

  2255       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  2256       then have "norm ((1 / c) *\<^sub>R y - x) < e"

  2257         unfolding dist_norm

  2258         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  2259           assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)

  2260       then have "y \<in> ( *\<^sub>R) c  s"

  2261         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "( *\<^sub>R) c"]

  2262         using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]

  2263         using assms(1)

  2264         unfolding dist_norm scaleR_scaleR

  2265         by auto

  2266     }

  2267     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> ( *\<^sub>R) c  s"

  2268       apply (rule_tac x="e * \<bar>c\<bar>" in exI, auto)

  2269       done

  2270   }

  2271   then show ?thesis unfolding open_dist by auto

  2272 qed

  2273

  2274 lemma minus_image_eq_vimage:

  2275   fixes A :: "'a::ab_group_add set"

  2276   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  2277   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  2278

  2279 lemma open_negations:

  2280   fixes S :: "'a::real_normed_vector set"

  2281   shows "open S \<Longrightarrow> open ((\<lambda>x. - x)  S)"

  2282   using open_scaling [of "- 1" S] by simp

  2283

  2284 lemma open_translation:

  2285   fixes S :: "'a::real_normed_vector set"

  2286   assumes "open S"

  2287   shows "open((\<lambda>x. a + x)  S)"

  2288 proof -

  2289   {

  2290     fix x

  2291     have "continuous (at x) (\<lambda>x. x - a)"

  2292       by (intro continuous_diff continuous_ident continuous_const)

  2293   }

  2294   moreover have "{x. x - a \<in> S} = (+) a  S"

  2295     by force

  2296   ultimately show ?thesis

  2297     by (metis assms continuous_open_vimage vimage_def)

  2298 qed

  2299

  2300 lemma open_neg_translation:

  2301   fixes s :: "'a::real_normed_vector set"

  2302   assumes "open s"

  2303   shows "open((\<lambda>x. a - x)  s)"

  2304   using open_translation[OF open_negations[OF assms], of a]

  2305   by (auto simp: image_image)

  2306

  2307 lemma open_affinity:

  2308   fixes S :: "'a::real_normed_vector set"

  2309   assumes "open S"  "c \<noteq> 0"

  2310   shows "open ((\<lambda>x. a + c *\<^sub>R x)  S)"

  2311 proof -

  2312   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"

  2313     unfolding o_def ..

  2314   have "(+) a  ( *\<^sub>R) c  S = ((+) a \<circ> ( *\<^sub>R) c)  S"

  2315     by auto

  2316   then show ?thesis

  2317     using assms open_translation[of "( *\<^sub>R) c  S" a]

  2318     unfolding *

  2319     by auto

  2320 qed

  2321

  2322 lemma interior_translation:

  2323   fixes S :: "'a::real_normed_vector set"

  2324   shows "interior ((\<lambda>x. a + x)  S) = (\<lambda>x. a + x)  (interior S)"

  2325 proof (rule set_eqI, rule)

  2326   fix x

  2327   assume "x \<in> interior ((+) a  S)"

  2328   then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) a  S"

  2329     unfolding mem_interior by auto

  2330   then have "ball (x - a) e \<subseteq> S"

  2331     unfolding subset_eq Ball_def mem_ball dist_norm

  2332     by (auto simp: diff_diff_eq)

  2333   then show "x \<in> (+) a  interior S"

  2334     unfolding image_iff

  2335     apply (rule_tac x="x - a" in bexI)

  2336     unfolding mem_interior

  2337     using \<open>e > 0\<close>

  2338     apply auto

  2339     done

  2340 next

  2341   fix x

  2342   assume "x \<in> (+) a  interior S"

  2343   then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"

  2344     unfolding image_iff Bex_def mem_interior by auto

  2345   {

  2346     fix z

  2347     have *: "a + y - z = y + a - z" by auto

  2348     assume "z \<in> ball x e"

  2349     then have "z - a \<in> S"

  2350       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]

  2351       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *

  2352       by auto

  2353     then have "z \<in> (+) a  S"

  2354       unfolding image_iff by (auto intro!: bexI[where x="z - a"])

  2355   }

  2356   then have "ball x e \<subseteq> (+) a  S"

  2357     unfolding subset_eq by auto

  2358   then show "x \<in> interior ((+) a  S)"

  2359     unfolding mem_interior using \<open>e > 0\<close> by auto

  2360 qed

  2361

  2362 subsection \<open>Continuity implies uniform continuity on a compact domain\<close>

  2363

  2364 text\<open>From the proof of the Heine-Borel theorem: Lemma 2 in section 3.7, page 69 of

  2365 J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis (CUP, 2002)\<close>

  2366

  2367 lemma Heine_Borel_lemma:

  2368   assumes "compact S" and Ssub: "S \<subseteq> \<Union>\<G>" and opn: "\<And>G. G \<in> \<G> \<Longrightarrow> open G"

  2369   obtains e where "0 < e" "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> \<G>. ball x e \<subseteq> G"

  2370 proof -

  2371   have False if neg: "\<And>e. 0 < e \<Longrightarrow> \<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x e \<subseteq> G"

  2372   proof -

  2373     have "\<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x (1 / Suc n) \<subseteq> G" for n

  2374       using neg by simp

  2375     then obtain f where "\<And>n. f n \<in> S" and fG: "\<And>G n. G \<in> \<G> \<Longrightarrow> \<not> ball (f n) (1 / Suc n) \<subseteq> G"

  2376       by metis

  2377     then obtain l r where "l \<in> S" "strict_mono r" and to_l: "(f \<circ> r) \<longlonglongrightarrow> l"

  2378       using \<open>compact S\<close> compact_def that by metis

  2379     then obtain G where "l \<in> G" "G \<in> \<G>"

  2380       using Ssub by auto

  2381     then obtain e where "0 < e" and e: "\<And>z. dist z l < e \<Longrightarrow> z \<in> G"

  2382       using opn open_dist by blast

  2383     obtain N1 where N1: "\<And>n. n \<ge> N1 \<Longrightarrow> dist (f (r n)) l < e/2"

  2384       using to_l apply (simp add: lim_sequentially)

  2385       using \<open>0 < e\<close> half_gt_zero that by blast

  2386     obtain N2 where N2: "of_nat N2 > 2/e"

  2387       using reals_Archimedean2 by blast

  2388     obtain x where "x \<in> ball (f (r (max N1 N2))) (1 / real (Suc (r (max N1 N2))))" and "x \<notin> G"

  2389       using fG [OF \<open>G \<in> \<G>\<close>, of "r (max N1 N2)"] by blast

  2390     then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"

  2391       by simp

  2392     also have "... \<le> 1 / real (Suc (max N1 N2))"

  2393       apply (simp add: divide_simps del: max.bounded_iff)

  2394       using \<open>strict_mono r\<close> seq_suble by blast

  2395     also have "... \<le> 1 / real (Suc N2)"

  2396       by (simp add: field_simps)

  2397     also have "... < e/2"

  2398       using N2 \<open>0 < e\<close> by (simp add: field_simps)

  2399     finally have "dist (f (r (max N1 N2))) x < e / 2" .

  2400     moreover have "dist (f (r (max N1 N2))) l < e/2"

  2401       using N1 max.cobounded1 by blast

  2402     ultimately have "dist x l < e"

  2403       using dist_triangle_half_r by blast

  2404     then show ?thesis

  2405       using e \<open>x \<notin> G\<close> by blast

  2406   qed

  2407   then show ?thesis

  2408     by (meson that)

  2409 qed

  2410

  2411 lemma compact_uniformly_equicontinuous:

  2412   assumes "compact S"

  2413       and cont: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>

  2414                         \<Longrightarrow> \<exists>d. 0 < d \<and>

  2415                                 (\<forall>f \<in> \<F>. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  2416       and "0 < e"

  2417   obtains d where "0 < d"

  2418                   "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"

  2419 proof -

  2420   obtain d where d_pos: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> 0 < d x e"

  2421      and d_dist : "\<And>x x' e f. \<lbrakk>dist x' x < d x e; x \<in> S; x' \<in> S; 0 < e; f \<in> \<F>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"

  2422     using cont by metis

  2423   let ?\<G> = "((\<lambda>x. ball x (d x (e / 2)))  S)"

  2424   have Ssub: "S \<subseteq> \<Union> ?\<G>"

  2425     by clarsimp (metis d_pos \<open>0 < e\<close> dist_self half_gt_zero_iff)

  2426   then obtain k where "0 < k" and k: "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> ?\<G>. ball x k \<subseteq> G"

  2427     by (rule Heine_Borel_lemma [OF \<open>compact S\<close>]) auto

  2428   moreover have "dist (f v) (f u) < e" if "f \<in> \<F>" "u \<in> S" "v \<in> S" "dist v u < k" for f u v

  2429   proof -

  2430     obtain G where "G \<in> ?\<G>" "u \<in> G" "v \<in> G"

  2431       using k that

  2432       by (metis \<open>dist v u < k\<close> \<open>u \<in> S\<close> \<open>0 < k\<close> centre_in_ball subsetD dist_commute mem_ball)

  2433     then obtain w where w: "dist w u < d w (e / 2)" "dist w v < d w (e / 2)" "w \<in> S"

  2434       by auto

  2435     with that d_dist have "dist (f w) (f v) < e/2"

  2436       by (metis \<open>0 < e\<close> dist_commute half_gt_zero)

  2437     moreover

  2438     have "dist (f w) (f u) < e/2"

  2439       using that d_dist w by (metis \<open>0 < e\<close> dist_commute divide_pos_pos zero_less_numeral)

  2440     ultimately show ?thesis

  2441       using dist_triangle_half_r by blast

  2442   qed

  2443   ultimately show ?thesis using that by blast

  2444 qed

  2445

  2446 corollary compact_uniformly_continuous:

  2447   fixes f :: "'a :: metric_space \<Rightarrow> 'b :: metric_space"

  2448   assumes f: "continuous_on S f" and S: "compact S"

  2449   shows "uniformly_continuous_on S f"

  2450   using f

  2451     unfolding continuous_on_iff uniformly_continuous_on_def

  2452     by (force intro: compact_uniformly_equicontinuous [OF S, of "{f}"])

  2453

  2454 subsection%unimportant \<open>Topological stuff about the set of Reals\<close>

  2455

  2456 lemma open_real:

  2457   fixes s :: "real set"

  2458   shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. \<bar>x' - x\<bar> < e --> x' \<in> s)"

  2459   unfolding open_dist dist_norm by simp

  2460

  2461 lemma islimpt_approachable_real:

  2462   fixes s :: "real set"

  2463   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e)"

  2464   unfolding islimpt_approachable dist_norm by simp

  2465

  2466 lemma closed_real:

  2467   fixes s :: "real set"

  2468   shows "closed s \<longleftrightarrow> (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e) \<longrightarrow> x \<in> s)"

  2469   unfolding closed_limpt islimpt_approachable dist_norm by simp

  2470

  2471 lemma continuous_at_real_range:

  2472   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  2473   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> \<bar>f x' - f x\<bar> < e)"

  2474   unfolding continuous_at

  2475   unfolding Lim_at

  2476   unfolding dist_norm

  2477   apply auto

  2478   apply (erule_tac x=e in allE, auto)

  2479   apply (rule_tac x=d in exI, auto)

  2480   apply (erule_tac x=x' in allE, auto)

  2481   apply (erule_tac x=e in allE, auto)

  2482   done

  2483

  2484 lemma continuous_on_real_range:

  2485   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  2486   shows "continuous_on s f \<longleftrightarrow>

  2487     (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e))"

  2488   unfolding continuous_on_iff dist_norm by simp

  2489

  2490

  2491 subsection%unimportant \<open>Cartesian products\<close>

  2492

  2493 lemma bounded_Times:

  2494   assumes "bounded s" "bounded t"

  2495   shows "bounded (s \<times> t)"

  2496 proof -

  2497   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  2498     using assms [unfolded bounded_def] by auto

  2499   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"

  2500     by (auto simp: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  2501   then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  2502 qed

  2503

  2504 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  2505   by (induct x) simp

  2506

  2507 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  2508   unfolding seq_compact_def

  2509   apply clarify

  2510   apply (drule_tac x="fst \<circ> f" in spec)

  2511   apply (drule mp, simp add: mem_Times_iff)

  2512   apply (clarify, rename_tac l1 r1)

  2513   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  2514   apply (drule mp, simp add: mem_Times_iff)

  2515   apply (clarify, rename_tac l2 r2)

  2516   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  2517   apply (rule_tac x="r1 \<circ> r2" in exI)

  2518   apply (rule conjI, simp add: strict_mono_def)

  2519   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  2520   apply (drule (1) tendsto_Pair) back

  2521   apply (simp add: o_def)

  2522   done

  2523

  2524 lemma compact_Times:

  2525   assumes "compact s" "compact t"

  2526   shows "compact (s \<times> t)"

  2527 proof (rule compactI)

  2528   fix C

  2529   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  2530   have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  2531   proof

  2532     fix x

  2533     assume "x \<in> s"

  2534     have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")

  2535     proof

  2536       fix y

  2537       assume "y \<in> t"

  2538       with \<open>x \<in> s\<close> C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  2539       then show "?P y" by (auto elim!: open_prod_elim)

  2540     qed

  2541     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  2542       and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"

  2543       by metis

  2544     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  2545     with compactE_image[OF \<open>compact t\<close>] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  2546       by metis

  2547     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  2548       by (fastforce simp: subset_eq)

  2549     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  2550       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  2551   qed

  2552   then obtain a d where a: "\<And>x. x\<in>s \<Longrightarrow> open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  2553     and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>d x"

  2554     unfolding subset_eq UN_iff by metis

  2555   moreover

  2556   from compactE_image[OF \<open>compact s\<close> a]

  2557   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"

  2558     by auto

  2559   moreover

  2560   {

  2561     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"

  2562       by auto

  2563     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"

  2564       using d \<open>e \<subseteq> s\<close> by (intro UN_mono) auto

  2565     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .

  2566   }

  2567   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  2568     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp: subset_eq)

  2569 qed

  2570

  2571 text\<open>Hence some useful properties follow quite easily.\<close>

  2572

  2573 lemma compact_scaling:

  2574   fixes s :: "'a::real_normed_vector set"

  2575   assumes "compact s"

  2576   shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  2577 proof -

  2578   let ?f = "\<lambda>x. scaleR c x"

  2579   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  2580   show ?thesis

  2581     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  2582     using linear_continuous_at[OF *] assms

  2583     by auto

  2584 qed

  2585

  2586 lemma compact_negations:

  2587   fixes s :: "'a::real_normed_vector set"

  2588   assumes "compact s"

  2589   shows "compact ((\<lambda>x. - x)  s)"

  2590   using compact_scaling [OF assms, of "- 1"] by auto

  2591

  2592 lemma compact_sums:

  2593   fixes s t :: "'a::real_normed_vector set"

  2594   assumes "compact s"

  2595     and "compact t"

  2596   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  2597 proof -

  2598   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  2599     apply auto

  2600     unfolding image_iff

  2601     apply (rule_tac x="(xa, y)" in bexI)

  2602     apply auto

  2603     done

  2604   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  2605     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  2606   then show ?thesis

  2607     unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  2608 qed

  2609

  2610 lemma compact_differences:

  2611   fixes s t :: "'a::real_normed_vector set"

  2612   assumes "compact s"

  2613     and "compact t"

  2614   shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  2615 proof-

  2616   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  2617     apply auto

  2618     apply (rule_tac x= xa in exI, auto)

  2619     done

  2620   then show ?thesis

  2621     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  2622 qed

  2623

  2624 lemma compact_translation:

  2625   fixes s :: "'a::real_normed_vector set"

  2626   assumes "compact s"

  2627   shows "compact ((\<lambda>x. a + x)  s)"

  2628 proof -

  2629   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s"

  2630     by auto

  2631   then show ?thesis

  2632     using compact_sums[OF assms compact_sing[of a]] by auto

  2633 qed

  2634

  2635 lemma compact_affinity:

  2636   fixes s :: "'a::real_normed_vector set"

  2637   assumes "compact s"

  2638   shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  2639 proof -

  2640   have "(+) a  ( *\<^sub>R) c  s = (\<lambda>x. a + c *\<^sub>R x)  s"

  2641     by auto

  2642   then show ?thesis

  2643     using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  2644 qed

  2645

  2646 text \<open>Hence we get the following.\<close>

  2647

  2648 lemma compact_sup_maxdistance:

  2649   fixes s :: "'a::metric_space set"

  2650   assumes "compact s"

  2651     and "s \<noteq> {}"

  2652   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  2653 proof -

  2654   have "compact (s \<times> s)"

  2655     using \<open>compact s\<close> by (intro compact_Times)

  2656   moreover have "s \<times> s \<noteq> {}"

  2657     using \<open>s \<noteq> {}\<close> by auto

  2658   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  2659     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  2660   ultimately show ?thesis

  2661     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  2662 qed

  2663

  2664

  2665 subsection \<open>The diameter of a set\<close>

  2666

  2667 definition%important diameter :: "'a::metric_space set \<Rightarrow> real" where

  2668   "diameter S = (if S = {} then 0 else SUP (x,y):S\<times>S. dist x y)"

  2669

  2670 lemma diameter_empty [simp]: "diameter{} = 0"

  2671   by (auto simp: diameter_def)

  2672

  2673 lemma diameter_singleton [simp]: "diameter{x} = 0"

  2674   by (auto simp: diameter_def)

  2675

  2676 lemma diameter_le:

  2677   assumes "S \<noteq> {} \<or> 0 \<le> d"

  2678       and no: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> norm(x - y) \<le> d"

  2679     shows "diameter S \<le> d"

  2680 using assms

  2681   by (auto simp: dist_norm diameter_def intro: cSUP_least)

  2682

  2683 lemma diameter_bounded_bound:

  2684   fixes s :: "'a :: metric_space set"

  2685   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  2686   shows "dist x y \<le> diameter s"

  2687 proof -

  2688   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  2689     unfolding bounded_def by auto

  2690   have "bdd_above (case_prod dist  (s\<times>s))"

  2691   proof (intro bdd_aboveI, safe)

  2692     fix a b

  2693     assume "a \<in> s" "b \<in> s"

  2694     with z[of a] z[of b] dist_triangle[of a b z]

  2695     show "dist a b \<le> 2 * d"

  2696       by (simp add: dist_commute)

  2697   qed

  2698   moreover have "(x,y) \<in> s\<times>s" using s by auto

  2699   ultimately have "dist x y \<le> (SUP (x,y):s\<times>s. dist x y)"

  2700     by (rule cSUP_upper2) simp

  2701   with \<open>x \<in> s\<close> show ?thesis

  2702     by (auto simp: diameter_def)

  2703 qed

  2704

  2705 lemma diameter_lower_bounded:

  2706   fixes s :: "'a :: metric_space set"

  2707   assumes s: "bounded s"

  2708     and d: "0 < d" "d < diameter s"

  2709   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  2710 proof (rule ccontr)

  2711   assume contr: "\<not> ?thesis"

  2712   moreover have "s \<noteq> {}"

  2713     using d by (auto simp: diameter_def)

  2714   ultimately have "diameter s \<le> d"

  2715     by (auto simp: not_less diameter_def intro!: cSUP_least)

  2716   with \<open>d < diameter s\<close> show False by auto

  2717 qed

  2718

  2719 lemma diameter_bounded:

  2720   assumes "bounded s"

  2721   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  2722     and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  2723   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  2724   by auto

  2725

  2726 lemma bounded_two_points:

  2727   "bounded S \<longleftrightarrow> (\<exists>e. \<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> e)"

  2728   apply (rule iffI)

  2729   subgoal using diameter_bounded(1) by auto

  2730   subgoal using bounded_any_center[of S] by meson

  2731   done

  2732

  2733 lemma diameter_compact_attained:

  2734   assumes "compact s"

  2735     and "s \<noteq> {}"

  2736   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  2737 proof -

  2738   have b: "bounded s" using assms(1)

  2739     by (rule compact_imp_bounded)

  2740   then obtain x y where xys: "x\<in>s" "y\<in>s"

  2741     and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  2742     using compact_sup_maxdistance[OF assms] by auto

  2743   then have "diameter s \<le> dist x y"

  2744     unfolding diameter_def

  2745     apply clarsimp

  2746     apply (rule cSUP_least, fast+)

  2747     done

  2748   then show ?thesis

  2749     by (metis b diameter_bounded_bound order_antisym xys)

  2750 qed

  2751

  2752 lemma diameter_ge_0:

  2753   assumes "bounded S"  shows "0 \<le> diameter S"

  2754   by (metis all_not_in_conv assms diameter_bounded_bound diameter_empty dist_self order_refl)

  2755

  2756 lemma diameter_subset:

  2757   assumes "S \<subseteq> T" "bounded T"

  2758   shows "diameter S \<le> diameter T"

  2759 proof (cases "S = {} \<or> T = {}")

  2760   case True

  2761   with assms show ?thesis

  2762     by (force simp: diameter_ge_0)

  2763 next

  2764   case False

  2765   then have "bdd_above ((\<lambda>x. case x of (x, xa) \<Rightarrow> dist x xa)  (T \<times> T))"

  2766     using \<open>bounded T\<close> diameter_bounded_bound by (force simp: bdd_above_def)

  2767   with False \<open>S \<subseteq> T\<close> show ?thesis

  2768     apply (simp add: diameter_def)

  2769     apply (rule cSUP_subset_mono, auto)

  2770     done

  2771 qed

  2772

  2773 lemma diameter_closure:

  2774   assumes "bounded S"

  2775   shows "diameter(closure S) = diameter S"

  2776 proof (rule order_antisym)

  2777   have "False" if "diameter S < diameter (closure S)"

  2778   proof -

  2779     define d where "d = diameter(closure S) - diameter(S)"

  2780     have "d > 0"

  2781       using that by (simp add: d_def)

  2782     then have "diameter(closure(S)) - d / 2 < diameter(closure(S))"

  2783       by simp

  2784     have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"

  2785       by (simp add: d_def divide_simps)

  2786      have bocl: "bounded (closure S)"

  2787       using assms by blast

  2788     moreover have "0 \<le> diameter S"

  2789       using assms diameter_ge_0 by blast

  2790     ultimately obtain x y where "x \<in> closure S" "y \<in> closure S" and xy: "diameter(closure(S)) - d / 2 < dist x y"

  2791       using diameter_bounded(2) [OF bocl, rule_format, of "diameter(closure(S)) - d / 2"] \<open>d > 0\<close> d_def by auto

  2792     then obtain x' y' where x'y': "x' \<in> S" "dist x' x < d/4" "y' \<in> S" "dist y' y < d/4"

  2793       using closure_approachable

  2794       by (metis \<open>0 < d\<close> zero_less_divide_iff zero_less_numeral)

  2795     then have "dist x' y' \<le> diameter S"

  2796       using assms diameter_bounded_bound by blast

  2797     with x'y' have "dist x y \<le> d / 4 + diameter S + d / 4"

  2798       by (meson add_mono_thms_linordered_semiring(1) dist_triangle dist_triangle3 less_eq_real_def order_trans)

  2799     then show ?thesis

  2800       using xy d_def by linarith

  2801   qed

  2802   then show "diameter (closure S) \<le> diameter S"

  2803     by fastforce

  2804   next

  2805     show "diameter S \<le> diameter (closure S)"

  2806       by (simp add: assms bounded_closure closure_subset diameter_subset)

  2807 qed

  2808

  2809 lemma diameter_cball [simp]:

  2810   fixes a :: "'a::euclidean_space"

  2811   shows "diameter(cball a r) = (if r < 0 then 0 else 2*r)"

  2812 proof -

  2813   have "diameter(cball a r) = 2*r" if "r \<ge> 0"

  2814   proof (rule order_antisym)

  2815     show "diameter (cball a r) \<le> 2*r"

  2816     proof (rule diameter_le)

  2817       fix x y assume "x \<in> cball a r" "y \<in> cball a r"

  2818       then have "norm (x - a) \<le> r" "norm (a - y) \<le> r"

  2819         by (auto simp: dist_norm norm_minus_commute)

  2820       then have "norm (x - y) \<le> r+r"

  2821         using norm_diff_triangle_le by blast

  2822       then show "norm (x - y) \<le> 2*r" by simp

  2823     qed (simp add: that)

  2824     have "2*r = dist (a + r *\<^sub>R (SOME i. i \<in> Basis)) (a - r *\<^sub>R (SOME i. i \<in> Basis))"

  2825       apply (simp add: dist_norm)

  2826       by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)

  2827     also have "... \<le> diameter (cball a r)"

  2828       apply (rule diameter_bounded_bound)

  2829       using that by (auto simp: dist_norm)

  2830     finally show "2*r \<le> diameter (cball a r)" .

  2831   qed

  2832   then show ?thesis by simp

  2833 qed

  2834

  2835 lemma diameter_ball [simp]:

  2836   fixes a :: "'a::euclidean_space"

  2837   shows "diameter(ball a r) = (if r < 0 then 0 else 2*r)"

  2838 proof -

  2839   have "diameter(ball a r) = 2*r" if "r > 0"

  2840     by (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)

  2841   then show ?thesis

  2842     by (simp add: diameter_def)

  2843 qed

  2844

  2845 lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"

  2846 proof -

  2847   have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"

  2848     by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)

  2849   then show ?thesis

  2850     by simp

  2851 qed

  2852

  2853 lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"

  2854 proof -

  2855   have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"

  2856     by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)

  2857   then show ?thesis

  2858     by simp

  2859 qed

  2860

  2861 proposition Lebesgue_number_lemma:

  2862   assumes "compact S" "\<C> \<noteq> {}" "S \<subseteq> \<Union>\<C>" and ope: "\<And>B. B \<in> \<C> \<Longrightarrow> open B"

  2863   obtains \<delta> where "0 < \<delta>" "\<And>T. \<lbrakk>T \<subseteq> S; diameter T < \<delta>\<rbrakk> \<Longrightarrow> \<exists>B \<in> \<C>. T \<subseteq> B"

  2864 proof (cases "S = {}")

  2865   case True

  2866   then show ?thesis

  2867     by (metis \<open>\<C> \<noteq> {}\<close> zero_less_one empty_subsetI equals0I subset_trans that)

  2868 next

  2869   case False

  2870   { fix x assume "x \<in> S"

  2871     then obtain C where C: "x \<in> C" "C \<in> \<C>"

  2872       using \<open>S \<subseteq> \<Union>\<C>\<close> by blast

  2873     then obtain r where r: "r>0" "ball x (2*r) \<subseteq> C"

  2874       by (metis mult.commute mult_2_right not_le ope openE field_sum_of_halves zero_le_numeral zero_less_mult_iff)

  2875     then have "\<exists>r C. r > 0 \<and> ball x (2*r) \<subseteq> C \<and> C \<in> \<C>"

  2876       using C by blast

  2877   }

  2878   then obtain r where r: "\<And>x. x \<in> S \<Longrightarrow> r x > 0 \<and> (\<exists>C \<in> \<C>. ball x (2*r x) \<subseteq> C)"

  2879     by metis

  2880   then have "S \<subseteq> (\<Union>x \<in> S. ball x (r x))"

  2881     by auto

  2882   then obtain \<T> where "finite \<T>" "S \<subseteq> \<Union>\<T>" and \<T>: "\<T> \<subseteq> (\<lambda>x. ball x (r x))  S"

  2883     by (rule compactE [OF \<open>compact S\<close>]) auto

  2884   then obtain S0 where "S0 \<subseteq> S" "finite S0" and S0: "\<T> = (\<lambda>x. ball x (r x))  S0"

  2885     by (meson finite_subset_image)

  2886   then have "S0 \<noteq> {}"

  2887     using False \<open>S \<subseteq> \<Union>\<T>\<close> by auto

  2888   define \<delta> where "\<delta> = Inf (r  S0)"

  2889   have "\<delta> > 0"

  2890     using \<open>finite S0\<close> \<open>S0 \<subseteq> S\<close> \<open>S0 \<noteq> {}\<close> r by (auto simp: \<delta>_def finite_less_Inf_iff)

  2891   show ?thesis

  2892   proof

  2893     show "0 < \<delta>"

  2894       by (simp add: \<open>0 < \<delta>\<close>)

  2895     show "\<exists>B \<in> \<C>. T \<subseteq> B" if "T \<subseteq> S" and dia: "diameter T < \<delta>" for T

  2896     proof (cases "T = {}")

  2897       case True

  2898       then show ?thesis

  2899         using \<open>\<C> \<noteq> {}\<close> by blast

  2900     next

  2901       case False

  2902       then obtain y where "y \<in> T" by blast

  2903       then have "y \<in> S"

  2904         using \<open>T \<subseteq> S\<close> by auto

  2905       then obtain x where "x \<in> S0" and x: "y \<in> ball x (r x)"

  2906         using \<open>S \<subseteq> \<Union>\<T>\<close> S0 that by blast

  2907       have "ball y \<delta> \<subseteq> ball y (r x)"

  2908         by (metis \<delta>_def \<open>S0 \<noteq> {}\<close> \<open>finite S0\<close> \<open>x \<in> S0\<close> empty_is_image finite_imageI finite_less_Inf_iff imageI less_irrefl not_le subset_ball)

  2909       also have "... \<subseteq> ball x (2*r x)"

  2910         by clarsimp (metis dist_commute dist_triangle_less_add mem_ball mult_2 x)

  2911       finally obtain C where "C \<in> \<C>" "ball y \<delta> \<subseteq> C"

  2912         by (meson r \<open>S0 \<subseteq> S\<close> \<open>x \<in> S0\<close> dual_order.trans subsetCE)

  2913       have "bounded T"

  2914         using \<open>compact S\<close> bounded_subset compact_imp_bounded \<open>T \<subseteq> S\<close> by blast

  2915       then have "T \<subseteq> ball y \<delta>"

  2916         using \<open>y \<in> T\<close> dia diameter_bounded_bound by fastforce

  2917       then show ?thesis

  2918         apply (rule_tac x=C in bexI)

  2919         using \<open>ball y \<delta> \<subseteq> C\<close> \<open>C \<in> \<C>\<close> by auto

  2920     qed

  2921   qed

  2922 qed

  2923

  2924 lemma diameter_cbox:

  2925   fixes a b::"'a::euclidean_space"

  2926   shows "(\<forall>i \<in> Basis. a \<bullet> i \<le> b \<bullet> i) \<Longrightarrow> diameter (cbox a b) = dist a b"

  2927   by (force simp: diameter_def intro!: cSup_eq_maximum L2_set_mono

  2928      simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)

  2929

  2930 subsection \<open>Separation between points and sets\<close>

  2931

  2932 proposition separate_point_closed:

  2933   fixes s :: "'a::heine_borel set"

  2934   assumes "closed s" and "a \<notin> s"

  2935   shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"

  2936 proof (cases "s = {}")

  2937   case True

  2938   then show ?thesis by(auto intro!: exI[where x=1])

  2939 next

  2940   case False

  2941   from assms obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y"

  2942     using \<open>s \<noteq> {}\<close> by (blast intro: distance_attains_inf [of s a])

  2943   with \<open>x\<in>s\<close> show ?thesis using dist_pos_lt[of a x] and\<open>a \<notin> s\<close>

  2944     by blast

  2945 qed

  2946

  2947 proposition separate_compact_closed:

  2948   fixes s t :: "'a::heine_borel set"

  2949   assumes "compact s"

  2950     and t: "closed t" "s \<inter> t = {}"

  2951   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  2952 proof cases

  2953   assume "s \<noteq> {} \<and> t \<noteq> {}"

  2954   then have "s \<noteq> {}" "t \<noteq> {}" by auto

  2955   let ?inf = "\<lambda>x. infdist x t"

  2956   have "continuous_on s ?inf"

  2957     by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_ident)

  2958   then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"

  2959     using continuous_attains_inf[OF \<open>compact s\<close> \<open>s \<noteq> {}\<close>] by auto

  2960   then have "0 < ?inf x"

  2961     using t \<open>t \<noteq> {}\<close> in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)

  2962   moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"

  2963     using x by (auto intro: order_trans infdist_le)

  2964   ultimately show ?thesis by auto

  2965 qed (auto intro!: exI[of _ 1])

  2966

  2967 proposition separate_closed_compact:

  2968   fixes s t :: "'a::heine_borel set"

  2969   assumes "closed s"

  2970     and "compact t"

  2971     and "s \<inter> t = {}"

  2972   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  2973 proof -

  2974   have *: "t \<inter> s = {}"

  2975     using assms(3) by auto

  2976   show ?thesis

  2977     using separate_compact_closed[OF assms(2,1) *] by (force simp: dist_commute)

  2978 qed

  2979

  2980 proposition compact_in_open_separated:

  2981   fixes A::"'a::heine_borel set"

  2982   assumes "A \<noteq> {}"

  2983   assumes "compact A"

  2984   assumes "open B"

  2985   assumes "A \<subseteq> B"

  2986   obtains e where "e > 0" "{x. infdist x A \<le> e} \<subseteq> B"

  2987 proof atomize_elim

  2988   have "closed (- B)" "compact A" "- B \<inter> A = {}"

  2989     using assms by (auto simp: open_Diff compact_eq_bounded_closed)

  2990   from separate_closed_compact[OF this]

  2991   obtain d'::real where d': "d'>0" "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d' \<le> dist x y"

  2992     by auto

  2993   define d where "d = d' / 2"

  2994   hence "d>0" "d < d'" using d' by auto

  2995   with d' have d: "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d < dist x y"

  2996     by force

  2997   show "\<exists>e>0. {x. infdist x A \<le> e} \<subseteq> B"

  2998   proof (rule ccontr)

  2999     assume "\<nexists>e. 0 < e \<and> {x. infdist x A \<le> e} \<subseteq> B"

  3000     with \<open>d > 0\<close> obtain x where x: "infdist x A \<le> d" "x \<notin> B"

  3001       by auto

  3002     from assms have "closed A" "A \<noteq> {}" by (auto simp: compact_eq_bounded_closed)

  3003     from infdist_attains_inf[OF this]

  3004     obtain y where y: "y \<in> A" "infdist x A = dist x y"

  3005       by auto

  3006     have "dist x y \<le> d" using x y by simp

  3007     also have "\<dots> < dist x y" using y d x by auto

  3008     finally show False by simp

  3009   qed

  3010 qed

  3011

  3012

  3013 subsection%unimportant \<open>Compact sets and the closure operation\<close>

  3014

  3015 lemma closed_scaling:

  3016   fixes S :: "'a::real_normed_vector set"

  3017   assumes "closed S"

  3018   shows "closed ((\<lambda>x. c *\<^sub>R x)  S)"

  3019 proof (cases "c = 0")

  3020   case True then show ?thesis

  3021     by (auto simp: image_constant_conv)

  3022 next

  3023   case False

  3024   from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) - S)"

  3025     by (simp add: continuous_closed_vimage)

  3026   also have "(\<lambda>x. inverse c *\<^sub>R x) - S = (\<lambda>x. c *\<^sub>R x)  S"

  3027     using \<open>c \<noteq> 0\<close> by (auto elim: image_eqI [rotated])

  3028   finally show ?thesis .

  3029 qed

  3030

  3031 lemma closed_negations:

  3032   fixes S :: "'a::real_normed_vector set"

  3033   assumes "closed S"

  3034   shows "closed ((\<lambda>x. -x)  S)"

  3035   using closed_scaling[OF assms, of "- 1"] by simp

  3036

  3037 lemma compact_closed_sums:

  3038   fixes S :: "'a::real_normed_vector set"

  3039   assumes "compact S" and "closed T"

  3040   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"

  3041 proof -

  3042   let ?S = "{x + y |x y. x \<in> S \<and> y \<in> T}"

  3043   {

  3044     fix x l

  3045     assume as: "\<forall>n. x n \<in> ?S"  "(x \<longlongrightarrow> l) sequentially"

  3046     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"

  3047       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> S \<and> snd y \<in> T"] by auto

  3048     obtain l' r where "l'\<in>S" and r: "strict_mono r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"

  3049       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  3050     have "((\<lambda>n. snd (f (r n))) \<longlongrightarrow> l - l') sequentially"

  3051       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)

  3052       unfolding o_def

  3053       by auto

  3054     then have "l - l' \<in> T"

  3055       using assms(2)[unfolded closed_sequential_limits,

  3056         THEN spec[where x="\<lambda> n. snd (f (r n))"],

  3057         THEN spec[where x="l - l'"]]

  3058       using f(3)

  3059       by auto

  3060     then have "l \<in> ?S"

  3061       using \<open>l' \<in> S\<close>

  3062       apply auto

  3063       apply (rule_tac x=l' in exI)

  3064       apply (rule_tac x="l - l'" in exI, auto)

  3065       done

  3066   }

  3067   moreover have "?S = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"

  3068     by force

  3069   ultimately show ?thesis

  3070     unfolding closed_sequential_limits

  3071     by (metis (no_types, lifting))

  3072 qed

  3073

  3074 lemma closed_compact_sums:

  3075   fixes S T :: "'a::real_normed_vector set"

  3076   assumes "closed S" "compact T"

  3077   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"

  3078 proof -

  3079   have "(\<Union>x\<in> T. \<Union>y \<in> S. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"

  3080     by auto

  3081   then show ?thesis

  3082     using compact_closed_sums[OF assms(2,1)] by simp

  3083 qed

  3084

  3085 lemma compact_closed_differences:

  3086   fixes S T :: "'a::real_normed_vector set"

  3087   assumes "compact S" "closed T"

  3088   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"

  3089 proof -

  3090   have "(\<Union>x\<in> S. \<Union>y \<in> uminus  T. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"

  3091     by force

  3092   then show ?thesis

  3093     using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  3094 qed

  3095

  3096 lemma closed_compact_differences:

  3097   fixes S T :: "'a::real_normed_vector set"

  3098   assumes "closed S" "compact T"

  3099   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"

  3100 proof -

  3101   have "(\<Union>x\<in> S. \<Union>y \<in> uminus  T. {x + y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"

  3102     by auto

  3103  then show ?thesis

  3104   using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  3105 qed

  3106

  3107 lemma closed_translation:

  3108   fixes a :: "'a::real_normed_vector"

  3109   assumes "closed S"

  3110   shows "closed ((\<lambda>x. a + x)  S)"

  3111 proof -

  3112   have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = ((+) a  S)" by auto

  3113   then show ?thesis

  3114     using compact_closed_sums[OF compact_sing[of a] assms] by auto

  3115 qed

  3116

  3117 lemma closure_translation:

  3118   fixes a :: "'a::real_normed_vector"

  3119   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  3120 proof -

  3121   have *: "(+) a  (- s) = - (+) a  s"

  3122     apply auto

  3123     unfolding image_iff

  3124     apply (rule_tac x="x - a" in bexI, auto)

  3125     done

  3126   show ?thesis

  3127     unfolding closure_interior translation_Compl

  3128     using interior_translation[of a "- s"]

  3129     unfolding *

  3130     by auto

  3131 qed

  3132

  3133 lemma frontier_translation:

  3134   fixes a :: "'a::real_normed_vector"

  3135   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  3136   unfolding frontier_def translation_diff interior_translation closure_translation

  3137   by auto

  3138

  3139 lemma sphere_translation:

  3140   fixes a :: "'n::euclidean_space"

  3141   shows "sphere (a+c) r = (+) a  sphere c r"

  3142 apply safe

  3143 apply (rule_tac x="x-a" in image_eqI)

  3144 apply (auto simp: dist_norm algebra_simps)

  3145 done

  3146

  3147 lemma cball_translation:

  3148   fixes a :: "'n::euclidean_space"

  3149   shows "cball (a+c) r = (+) a  cball c r"

  3150 apply safe

  3151 apply (rule_tac x="x-a" in image_eqI)

  3152 apply (auto simp: dist_norm algebra_simps)

  3153 done

  3154

  3155 lemma ball_translation:

  3156   fixes a :: "'n::euclidean_space"

  3157   shows "ball (a+c) r = (+) a  ball c r"

  3158 apply safe

  3159 apply (rule_tac x="x-a" in image_eqI)

  3160 apply (auto simp: dist_norm algebra_simps)

  3161 done

  3162

  3163

  3164 subsection%unimportant \<open>Closure of halfspaces and hyperplanes\<close>

  3165

  3166 lemma continuous_on_closed_Collect_le:

  3167   fixes f g :: "'a::t2_space \<Rightarrow> real"

  3168   assumes f: "continuous_on s f" and g: "continuous_on s g" and s: "closed s"

  3169   shows "closed {x \<in> s. f x \<le> g x}"

  3170 proof -

  3171   have "closed ((\<lambda>x. g x - f x) - {0..} \<inter> s)"

  3172     using closed_real_atLeast continuous_on_diff [OF g f]

  3173     by (simp add: continuous_on_closed_vimage [OF s])

  3174   also have "((\<lambda>x. g x - f x) - {0..} \<inter> s) = {x\<in>s. f x \<le> g x}"

  3175     by auto

  3176   finally show ?thesis .

  3177 qed

  3178

  3179 lemma continuous_at_inner: "continuous (at x) (inner a)"

  3180   unfolding continuous_at by (intro tendsto_intros)

  3181

  3182 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"

  3183   by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

  3184

  3185 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"

  3186   by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

  3187

  3188 lemma closed_hyperplane: "closed {x. inner a x = b}"

  3189   by (simp add: closed_Collect_eq continuous_on_inner continuous_on_const continuous_on_id)

  3190

  3191 lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"

  3192   by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

  3193

  3194 lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"

  3195   by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

  3196

  3197 lemma closed_interval_left:

  3198   fixes b :: "'a::euclidean_space"

  3199   shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"

  3200   by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

  3201

  3202 lemma closed_interval_right:

  3203   fixes a :: "'a::euclidean_space"

  3204   shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"

  3205   by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

  3206

  3207 lemma continuous_le_on_closure:

  3208   fixes a::real

  3209   assumes f: "continuous_on (closure s) f"

  3210       and x: "x \<in> closure(s)"

  3211       and xlo: "\<And>x. x \<in> s ==> f(x) \<le> a"

  3212     shows "f(x) \<le> a"

  3213     using image_closure_subset [OF f]

  3214   using image_closure_subset [OF f] closed_halfspace_le [of "1::real" a] assms

  3215   by force

  3216

  3217 lemma continuous_ge_on_closure:

  3218   fixes a::real

  3219   assumes f: "continuous_on (closure s) f"

  3220       and x: "x \<in> closure(s)"

  3221       and xlo: "\<And>x. x \<in> s ==> f(x) \<ge> a"

  3222     shows "f(x) \<ge> a"

  3223   using image_closure_subset [OF f] closed_halfspace_ge [of a "1::real"] assms

  3224   by force

  3225

  3226 lemma Lim_component_le:

  3227   fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  3228   assumes "(f \<longlongrightarrow> l) net"

  3229     and "\<not> (trivial_limit net)"

  3230     and "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"

  3231   shows "l\<bullet>i \<le> b"

  3232   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])

  3233

  3234 lemma Lim_component_ge:

  3235   fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  3236   assumes "(f \<longlongrightarrow> l) net"

  3237     and "\<not> (trivial_limit net)"

  3238     and "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"

  3239   shows "b \<le> l\<bullet>i"

  3240   by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])

  3241

  3242 lemma Lim_component_eq:

  3243   fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  3244   assumes net: "(f \<longlongrightarrow> l) net" "\<not> trivial_limit net"

  3245     and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"

  3246   shows "l\<bullet>i = b"

  3247   using ev[unfolded order_eq_iff eventually_conj_iff]

  3248   using Lim_component_ge[OF net, of b i]

  3249   using Lim_component_le[OF net, of i b]

  3250   by auto

  3251

  3252 text \<open>Limits relative to a union.\<close>

  3253

  3254 lemma eventually_within_Un:

  3255   "eventually P (at x within (s \<union> t)) \<longleftrightarrow>

  3256     eventually P (at x within s) \<and> eventually P (at x within t)"

  3257   unfolding eventually_at_filter

  3258   by (auto elim!: eventually_rev_mp)

  3259

  3260 lemma Lim_within_union:

  3261  "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>

  3262   (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"

  3263   unfolding tendsto_def

  3264   by (auto simp: eventually_within_Un)

  3265

  3266 lemma Lim_topological:

  3267   "(f \<longlongrightarrow> l) net \<longleftrightarrow>

  3268     trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  3269   unfolding tendsto_def trivial_limit_eq by auto

  3270

  3271 text \<open>Continuity relative to a union.\<close>

  3272

  3273 lemma continuous_on_Un_local:

  3274     "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;

  3275       continuous_on s f; continuous_on t f\<rbrakk>

  3276      \<Longrightarrow> continuous_on (s \<union> t) f"

  3277   unfolding continuous_on closedin_limpt

  3278   by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)

  3279

  3280 lemma continuous_on_cases_local:

  3281      "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;

  3282        continuous_on s f; continuous_on t g;

  3283        \<And>x. \<lbrakk>x \<in> s \<and> ~P x \<or> x \<in> t \<and> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk>

  3284       \<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

  3285   by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)

  3286

  3287 lemma continuous_on_cases_le:

  3288   fixes h :: "'a :: topological_space \<Rightarrow> real"

  3289   assumes "continuous_on {t \<in> s. h t \<le> a} f"

  3290       and "continuous_on {t \<in> s. a \<le> h t} g"

  3291       and h: "continuous_on s h"

  3292       and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t"

  3293     shows "continuous_on s (\<lambda>t. if h t \<le> a then f(t) else g(t))"

  3294 proof -

  3295   have s: "s = (s \<inter> h - atMost a) \<union> (s \<inter> h - atLeast a)"

  3296     by force

  3297   have 1: "closedin (subtopology euclidean s) (s \<inter> h - atMost a)"

  3298     by (rule continuous_closedin_preimage [OF h closed_atMost])

  3299   have 2: "closedin (subtopology euclidean s) (s \<inter> h - atLeast a)"

  3300     by (rule continuous_closedin_preimage [OF h closed_atLeast])

  3301   have eq: "s \<inter> h - {..a} = {t \<in> s. h t \<le> a}" "s \<inter> h - {a..} = {t \<in> s. a \<le> h t}"

  3302     by auto

  3303   show ?thesis

  3304     apply (rule continuous_on_subset [of s, OF _ order_refl])

  3305     apply (subst s)

  3306     apply (rule continuous_on_cases_local)

  3307     using 1 2 s assms apply (auto simp: eq)

  3308     done

  3309 qed

  3310

  3311 lemma continuous_on_cases_1:

  3312   fixes s :: "real set"

  3313   assumes "continuous_on {t \<in> s. t \<le> a} f"

  3314       and "continuous_on {t \<in> s. a \<le> t} g"

  3315       and "a \<in> s \<Longrightarrow> f a = g a"

  3316     shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"

  3317 using assms

  3318 by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])

  3319

  3320 subsubsection\<open>Some more convenient intermediate-value theorem formulations\<close>

  3321

  3322 lemma connected_ivt_hyperplane:

  3323   assumes "connected S" and xy: "x \<in> S" "y \<in> S" and b: "inner a x \<le> b" "b \<le> inner a y"

  3324   shows "\<exists>z \<in> S. inner a z = b"

  3325 proof (rule ccontr)

  3326   assume as:"\<not> (\<exists>z\<in>S. inner a z = b)"

  3327   let ?A = "{x. inner a x < b}"

  3328   let ?B = "{x. inner a x > b}"

  3329   have "open ?A" "open ?B"

  3330     using open_halfspace_lt and open_halfspace_gt by auto

  3331   moreover have "?A \<inter> ?B = {}" by auto

  3332   moreover have "S \<subseteq> ?A \<union> ?B" using as by auto

  3333   ultimately show False

  3334     using \<open>connected S\<close>[unfolded connected_def not_ex,

  3335       THEN spec[where x="?A"], THEN spec[where x="?B"]]

  3336     using xy b by auto

  3337 qed

  3338

  3339 lemma connected_ivt_component:

  3340   fixes x::"'a::euclidean_space"

  3341   shows "connected S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>S.  z\<bullet>k = a)"

  3342   using connected_ivt_hyperplane[of S x y "k::'a" a]

  3343   by (auto simp: inner_commute)

  3344

  3345 lemma image_affinity_cbox: fixes m::real

  3346   fixes a b c :: "'a::euclidean_space"

  3347   shows "(\<lambda>x. m *\<^sub>R x + c)  cbox a b =

  3348     (if cbox a b = {} then {}

  3349      else (if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)

  3350      else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c)))"

  3351 proof (cases "m = 0")

  3352   case True

  3353   {

  3354     fix x

  3355     assume "\<forall>i\<in>Basis. x \<bullet> i \<le> c \<bullet> i" "\<forall>i\<in>Basis. c \<bullet> i \<le> x \<bullet> i"

  3356     then have "x = c"

  3357       by (simp add: dual_order.antisym euclidean_eqI)

  3358   }

  3359   moreover have "c \<in> cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)"

  3360     unfolding True by (auto simp: cbox_sing)

  3361   ultimately show ?thesis using True by (auto simp: cbox_def)

  3362 next

  3363   case False

  3364   {

  3365     fix y

  3366     assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m > 0"

  3367     then have "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"

  3368       by (auto simp: inner_distrib)

  3369   }

  3370   moreover

  3371   {

  3372     fix y

  3373     assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m < 0"

  3374     then have "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i"

  3375       by (auto simp: mult_left_mono_neg inner_distrib)

  3376   }

  3377   moreover

  3378   {

  3379     fix y

  3380     assume "m > 0" and "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> y \<bullet> i" and "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"

  3381     then have "y \<in> (\<lambda>x. m *\<^sub>R x + c)  cbox a b"

  3382       unfolding image_iff Bex_def mem_box

  3383       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])

  3384       apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)

  3385       done

  3386   }

  3387   moreover

  3388   {

  3389     fix y

  3390     assume "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i" "m < 0"

  3391     then have "y \<in> (\<lambda>x. m *\<^sub>R x + c)  cbox a b"

  3392       unfolding image_iff Bex_def mem_box

  3393       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])

  3394       apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)

  3395       done

  3396   }

  3397   ultimately show ?thesis using False by (auto simp: cbox_def)

  3398 qed

  3399

  3400 lemma image_smult_cbox:"(\<lambda>x. m *\<^sub>R (x::_::euclidean_space))  cbox a b =

  3401   (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"

  3402   using image_affinity_cbox[of m 0 a b] by auto

  3403

  3404 lemma islimpt_greaterThanLessThan1:

  3405   fixes a b::"'a::{linorder_topology, dense_order}"

  3406   assumes "a < b"

  3407   shows  "a islimpt {a<..<b}"

  3408 proof (rule islimptI)

  3409   fix T

  3410   assume "open T" "a \<in> T"

  3411   from open_right[OF this \<open>a < b\<close>]

  3412   obtain c where c: "a < c" "{a..<c} \<subseteq> T" by auto

  3413   with assms dense[of a "min c b"]

  3414   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> a"

  3415     by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj

  3416       not_le order.strict_implies_order subset_eq)

  3417 qed

  3418

  3419 lemma islimpt_greaterThanLessThan2:

  3420   fixes a b::"'a::{linorder_topology, dense_order}"

  3421   assumes "a < b"

  3422   shows  "b islimpt {a<..<b}"

  3423 proof (rule islimptI)

  3424   fix T

  3425   assume "open T" "b \<in> T"

  3426   from open_left[OF this \<open>a < b\<close>]

  3427   obtain c where c: "c < b" "{c<..b} \<subseteq> T" by auto

  3428   with assms dense[of "max a c" b]

  3429   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> b"

  3430     by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj

  3431       not_le order.strict_implies_order subset_eq)

  3432 qed

  3433

  3434 lemma closure_greaterThanLessThan[simp]:

  3435   fixes a b::"'a::{linorder_topology, dense_order}"

  3436   shows "a < b \<Longrightarrow> closure {a <..< b} = {a .. b}" (is "_ \<Longrightarrow> ?l = ?r")

  3437 proof

  3438   have "?l \<subseteq> closure ?r"

  3439     by (rule closure_mono) auto

  3440   thus "closure {a<..<b} \<subseteq> {a..b}" by simp

  3441 qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1

  3442   islimpt_greaterThanLessThan2)

  3443

  3444 lemma closure_greaterThan[simp]:

  3445   fixes a b::"'a::{no_top, linorder_topology, dense_order}"

  3446   shows "closure {a<..} = {a..}"

  3447 proof -

  3448   from gt_ex obtain b where "a < b" by auto

  3449   hence "{a<..} = {a<..<b} \<union> {b..}" by auto

  3450   also have "closure \<dots> = {a..}" using \<open>a < b\<close> unfolding closure_Un

  3451     by auto

  3452   finally show ?thesis .

  3453 qed

  3454

  3455 lemma closure_lessThan[simp]:

  3456   fixes b::"'a::{no_bot, linorder_topology, dense_order}"

  3457   shows "closure {..<b} = {..b}"

  3458 proof -

  3459   from lt_ex obtain a where "a < b" by auto

  3460   hence "{..<b} = {a<..<b} \<union> {..a}" by auto

  3461   also have "closure \<dots> = {..b}" using \<open>a < b\<close> unfolding closure_Un

  3462     by auto

  3463   finally show ?thesis .

  3464 qed

  3465

  3466 lemma closure_atLeastLessThan[simp]:

  3467   fixes a b::"'a::{linorder_topology, dense_order}"

  3468   assumes "a < b"

  3469   shows "closure {a ..< b} = {a .. b}"

  3470 proof -

  3471   from assms have "{a ..< b} = {a} \<union> {a <..< b}" by auto

  3472   also have "closure \<dots> = {a .. b}" unfolding closure_Un

  3473     by (auto simp: assms less_imp_le)

  3474   finally show ?thesis .

  3475 qed

  3476

  3477 lemma closure_greaterThanAtMost[simp]:

  3478   fixes a b::"'a::{linorder_topology, dense_order}"

  3479   assumes "a < b"

  3480   shows "closure {a <.. b} = {a .. b}"

  3481 proof -

  3482   from assms have "{a <.. b} = {b} \<union> {a <..< b}" by auto

  3483   also have "closure \<dots> = {a .. b}" unfolding closure_Un

  3484     by (auto simp: assms less_imp_le)

  3485   finally show ?thesis .

  3486 qed

  3487

  3488

  3489 subsection \<open>Homeomorphisms\<close>

  3490

  3491 definition%important "homeomorphism s t f g \<longleftrightarrow>

  3492   (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>

  3493   (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"

  3494

  3495 lemma homeomorphismI [intro?]:

  3496   assumes "continuous_on S f" "continuous_on T g"

  3497           "f  S \<subseteq> T" "g  T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"

  3498     shows "homeomorphism S T f g"

  3499   using assms by (force simp: homeomorphism_def)

  3500

  3501 lemma homeomorphism_translation:

  3502   fixes a :: "'a :: real_normed_vector"

  3503   shows "homeomorphism ((+) a  S) S ((+) (- a)) ((+) a)"

  3504 unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)

  3505

  3506 lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"

  3507   by (rule homeomorphismI) (auto simp: continuous_on_id)

  3508

  3509 lemma homeomorphism_compose:

  3510   assumes "homeomorphism S T f g" "homeomorphism T U h k"

  3511     shows "homeomorphism S U (h o f) (g o k)"

  3512   using assms

  3513   unfolding homeomorphism_def

  3514   by (intro conjI ballI continuous_on_compose) (auto simp: image_comp [symmetric])

  3515

  3516 lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"

  3517   by (simp add: homeomorphism_def)

  3518

  3519 lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"

  3520   by (force simp: homeomorphism_def)

  3521

  3522 definition homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"

  3523     (infixr "homeomorphic" 60)

  3524   where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"

  3525

  3526 lemma homeomorphic_empty [iff]:

  3527      "S homeomorphic {} \<longleftrightarrow> S = {}" "{} homeomorphic S \<longleftrightarrow> S = {}"

  3528   by (auto simp: homeomorphic_def homeomorphism_def)

  3529

  3530 lemma homeomorphic_refl: "s homeomorphic s"

  3531   unfolding homeomorphic_def homeomorphism_def

  3532   using continuous_on_id

  3533   apply (rule_tac x = "(\<lambda>x. x)" in exI)

  3534   apply (rule_tac x = "(\<lambda>x. x)" in exI)

  3535   apply blast

  3536   done

  3537

  3538 lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"

  3539   unfolding homeomorphic_def homeomorphism_def

  3540   by blast

  3541

  3542 lemma homeomorphic_trans [trans]:

  3543   assumes "S homeomorphic T"

  3544       and "T homeomorphic U"

  3545     shows "S homeomorphic U"

  3546   using assms

  3547   unfolding homeomorphic_def

  3548 by (metis homeomorphism_compose)

  3549

  3550 lemma homeomorphic_minimal:

  3551   "s homeomorphic t \<longleftrightarrow>

  3552     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>

  3553            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>

  3554            continuous_on s f \<and> continuous_on t g)"

  3555    (is "?lhs = ?rhs")

  3556 proof

  3557   assume ?lhs

  3558   then show ?rhs

  3559     by (fastforce simp: homeomorphic_def homeomorphism_def)

  3560 next

  3561   assume ?rhs

  3562   then show ?lhs

  3563     apply clarify

  3564     unfolding homeomorphic_def homeomorphism_def

  3565     by (metis equalityI image_subset_iff subsetI)

  3566  qed

  3567

  3568 lemma homeomorphicI [intro?]:

  3569    "\<lbrakk>f  S = T; g  T = S;

  3570      continuous_on S f; continuous_on T g;

  3571      \<And>x. x \<in> S \<Longrightarrow> g(f(x)) = x;

  3572      \<And>y. y \<in> T \<Longrightarrow> f(g(y)) = y\<rbrakk> \<Longrightarrow> S homeomorphic T"

  3573 unfolding homeomorphic_def homeomorphism_def by metis

  3574

  3575 lemma homeomorphism_of_subsets:

  3576    "\<lbrakk>homeomorphism S T f g; S' \<subseteq> S; T'' \<subseteq> T; f  S' = T'\<rbrakk>

  3577     \<Longrightarrow> homeomorphism S' T' f g"

  3578 apply (auto simp: homeomorphism_def elim!: continuous_on_subset)

  3579 by (metis subsetD imageI)

  3580

  3581 lemma homeomorphism_apply1: "\<lbrakk>homeomorphism S T f g; x \<in> S\<rbrakk> \<Longrightarrow> g(f x) = x"

  3582   by (simp add: homeomorphism_def)

  3583

  3584 lemma homeomorphism_apply2: "\<lbrakk>homeomorphism S T f g; x \<in> T\<rbrakk> \<Longrightarrow> f(g x) = x"

  3585   by (simp add: homeomorphism_def)

  3586

  3587 lemma homeomorphism_image1: "homeomorphism S T f g \<Longrightarrow> f  S = T"

  3588   by (simp add: homeomorphism_def)

  3589

  3590 lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g  T = S"

  3591   by (simp add: homeomorphism_def)

  3592

  3593 lemma homeomorphism_cont1: "homeomorphism S T f g \<Longrightarrow> continuous_on S f"

  3594   by (simp add: homeomorphism_def)

  3595

  3596 lemma homeomorphism_cont2: "homeomorphism S T f g \<Longrightarrow> continuous_on T g"

  3597   by (simp add: homeomorphism_def)

  3598

  3599 lemma continuous_on_no_limpt:

  3600    "(\<And>x. \<not> x islimpt S) \<Longrightarrow> continuous_on S f"

  3601   unfolding continuous_on_def

  3602   by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)

  3603

  3604 lemma continuous_on_finite:

  3605   fixes S :: "'a::t1_space set"

  3606   shows "finite S \<Longrightarrow> continuous_on S f"

  3607 by (metis continuous_on_no_limpt islimpt_finite)

  3608

  3609 lemma homeomorphic_finite:

  3610   fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"

  3611   assumes "finite T"

  3612   shows "S homeomorphic T \<longleftrightarrow> finite S \<and> finite T \<and> card S = card T" (is "?lhs = ?rhs")

  3613 proof

  3614   assume "S homeomorphic T"

  3615   with assms show ?rhs

  3616     apply (auto simp: homeomorphic_def homeomorphism_def)

  3617      apply (metis finite_imageI)

  3618     by (metis card_image_le finite_imageI le_antisym)

  3619 next

  3620   assume R: ?rhs

  3621   with finite_same_card_bij obtain h where "bij_betw h S T"

  3622     by auto

  3623   with R show ?lhs

  3624     apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)

  3625     apply (rule_tac x=h in exI)

  3626     apply (rule_tac x="inv_into S h" in exI)

  3627     apply (auto simp:  bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)

  3628     apply (metis bij_betw_def bij_betw_inv_into)

  3629     done

  3630 qed

  3631

  3632 text \<open>Relatively weak hypotheses if a set is compact.\<close>

  3633

  3634 lemma homeomorphism_compact:

  3635   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  3636   assumes "compact s" "continuous_on s f"  "f  s = t"  "inj_on f s"

  3637   shows "\<exists>g. homeomorphism s t f g"

  3638 proof -

  3639   define g where "g x = (SOME y. y\<in>s \<and> f y = x)" for x

  3640   have g: "\<forall>x\<in>s. g (f x) = x"

  3641     using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto

  3642   {

  3643     fix y

  3644     assume "y \<in> t"

  3645     then obtain x where x:"f x = y" "x\<in>s"

  3646       using assms(3) by auto

  3647     then have "g (f x) = x" using g by auto

  3648     then have "f (g y) = y" unfolding x(1)[symmetric] by auto

  3649   }

  3650   then have g':"\<forall>x\<in>t. f (g x) = x" by auto

  3651   moreover

  3652   {

  3653     fix x

  3654     have "x\<in>s \<Longrightarrow> x \<in> g  t"

  3655       using g[THEN bspec[where x=x]]

  3656       unfolding image_iff

  3657       using assms(3)

  3658       by (auto intro!: bexI[where x="f x"])

  3659     moreover

  3660     {

  3661       assume "x\<in>g  t"

  3662       then obtain y where y:"y\<in>t" "g y = x" by auto

  3663       then obtain x' where x':"x'\<in>s" "f x' = y"

  3664         using assms(3) by auto

  3665       then have "x \<in> s"

  3666         unfolding g_def

  3667         using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]

  3668         unfolding y(2)[symmetric] and g_def

  3669         by auto

  3670     }

  3671     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g  t" ..

  3672   }

  3673   then have "g  t = s" by auto

  3674   ultimately show ?thesis

  3675     unfolding homeomorphism_def homeomorphic_def

  3676     apply (rule_tac x=g in exI)

  3677     using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)

  3678     apply auto

  3679     done

  3680 qed

  3681

  3682 lemma homeomorphic_compact:

  3683   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  3684   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f  s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"

  3685   unfolding homeomorphic_def by (metis homeomorphism_compact)

  3686

  3687 text\<open>Preservation of topological properties.\<close>

  3688

  3689 lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"

  3690   unfolding homeomorphic_def homeomorphism_def

  3691   by (metis compact_continuous_image)

  3692

  3693 text\<open>Results on translation, scaling etc.\<close>

  3694

  3695 lemma homeomorphic_scaling:

  3696   fixes s :: "'a::real_normed_vector set"

  3697   assumes "c \<noteq> 0"

  3698   shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x)  s)"

  3699   unfolding homeomorphic_minimal

  3700   apply (rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)

  3701   apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)

  3702   using assms

  3703   apply (auto simp: continuous_intros)

  3704   done

  3705

  3706 lemma homeomorphic_translation:

  3707   fixes s :: "'a::real_normed_vector set"

  3708   shows "s homeomorphic ((\<lambda>x. a + x)  s)"

  3709   unfolding homeomorphic_minimal

  3710   apply (rule_tac x="\<lambda>x. a + x" in exI)

  3711   apply (rule_tac x="\<lambda>x. -a + x" in exI)

  3712   using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]

  3713     continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a  s" "- a"]

  3714   apply auto

  3715   done

  3716

  3717 lemma homeomorphic_affinity:

  3718   fixes s :: "'a::real_normed_vector set"

  3719   assumes "c \<noteq> 0"

  3720   shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x)  s)"

  3721 proof -

  3722   have *: "(+) a  ( *\<^sub>R) c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  3723   show ?thesis

  3724     using homeomorphic_trans

  3725     using homeomorphic_scaling[OF assms, of s]

  3726     using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x)  s" a]

  3727     unfolding *

  3728     by auto

  3729 qed

  3730

  3731 lemma homeomorphic_balls:

  3732   fixes a b ::"'a::real_normed_vector"

  3733   assumes "0 < d"  "0 < e"

  3734   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)

  3735     and "(cball a d) homeomorphic (cball b e)" (is ?cth)

  3736 proof -

  3737   show ?th unfolding homeomorphic_minimal

  3738     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)

  3739     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)

  3740     using assms

  3741     apply (auto intro!: continuous_intros

  3742       simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)

  3743     done

  3744   show ?cth unfolding homeomorphic_minimal

  3745     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)

  3746     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)

  3747     using assms

  3748     apply (auto intro!: continuous_intros

  3749       simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)

  3750     done

  3751 qed

  3752

  3753 lemma homeomorphic_spheres:

  3754   fixes a b ::"'a::real_normed_vector"

  3755   assumes "0 < d"  "0 < e"

  3756   shows "(sphere a d) homeomorphic (sphere b e)"

  3757 unfolding homeomorphic_minimal

  3758     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)

  3759     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)

  3760     using assms

  3761     apply (auto intro!: continuous_intros

  3762       simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)

  3763     done

  3764

  3765 lemma homeomorphic_ball01_UNIV:

  3766   "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"

  3767   (is "?B homeomorphic ?U")

  3768 proof

  3769   have "x \<in> (\<lambda>z. z /\<^sub>R (1 - norm z))  ball 0 1" for x::'a

  3770     apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)

  3771      apply (auto simp: divide_simps)

  3772     using norm_ge_zero [of x] apply linarith+

  3773     done

  3774   then show "(\<lambda>z::'a. z /\<^sub>R (1 - norm z))  ?B = ?U"

  3775     by blast

  3776   have "x \<in> range (\<lambda>z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a

  3777     apply (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI)

  3778     using that apply (auto simp: divide_simps)

  3779     done

  3780   then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z))  ?U = ?B"

  3781     by (force simp: divide_simps dest: add_less_zeroD)

  3782   show "continuous_on (ball 0 1) (\<lambda>z. z /\<^sub>R (1 - norm z))"

  3783     by (rule continuous_intros | force)+

  3784   show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"

  3785     apply (intro continuous_intros)

  3786     apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)

  3787     done

  3788   show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>

  3789          x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"

  3790     by (auto simp: divide_simps)

  3791   show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"

  3792     apply (auto simp: divide_simps)

  3793     apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)

  3794     done

  3795 qed

  3796

  3797 proposition homeomorphic_ball_UNIV:

  3798   fixes a ::"'a::real_normed_vector"

  3799   assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"

  3800   using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast

  3801

  3802

  3803 text \<open>Connectedness is invariant under homeomorphisms.\<close>

  3804

  3805 lemma homeomorphic_connectedness:

  3806   assumes "s homeomorphic t"

  3807   shows "connected s \<longleftrightarrow> connected t"

  3808 using assms unfolding homeomorphic_def homeomorphism_def by (metis connected_continuous_image)

  3809

  3810

  3811 subsection%unimportant\<open>Inverse function property for open/closed maps\<close>

  3812

  3813 lemma continuous_on_inverse_open_map:

  3814   assumes contf: "continuous_on S f"

  3815     and imf: "f  S = T"

  3816     and injf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"

  3817     and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f  U)"

  3818   shows "continuous_on T g"

  3819 proof -

  3820   from imf injf have gTS: "g  T = S"

  3821     by force

  3822   from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f  U) = T \<inter> g - U" for U

  3823     by force

  3824   show ?thesis

  3825     by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)

  3826 qed

  3827

  3828 lemma continuous_on_inverse_closed_map:

  3829   assumes contf: "continuous_on S f"

  3830     and imf: "f  S = T"

  3831     and injf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"

  3832     and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f  U)"

  3833   shows "continuous_on T g"

  3834 proof -

  3835   from imf injf have gTS: "g  T = S"

  3836     by force

  3837   from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f  U) = T \<inter> g - U" for U

  3838     by force

  3839   show ?thesis

  3840     by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)

  3841 qed

  3842

  3843 lemma homeomorphism_injective_open_map:

  3844   assumes contf: "continuous_on S f"

  3845     and imf: "f  S = T"

  3846     and injf: "inj_on f S"

  3847     and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f  U)"

  3848   obtains g where "homeomorphism S T f g"

  3849 proof

  3850   have "continuous_on T (inv_into S f)"

  3851     by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)

  3852   with imf injf contf show "homeomorphism S T f (inv_into S f)"

  3853     by (auto simp: homeomorphism_def)

  3854 qed

  3855

  3856 lemma homeomorphism_injective_closed_map:

  3857   assumes contf: "continuous_on S f"

  3858     and imf: "f  S = T"

  3859     and injf: "inj_on f S"

  3860     and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f  U)"

  3861   obtains g where "homeomorphism S T f g"

  3862 proof

  3863   have "continuous_on T (inv_into S f)"

  3864     by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)

  3865   with imf injf contf show "homeomorphism S T f (inv_into S f)"

  3866     by (auto simp: homeomorphism_def)

  3867 qed

  3868

  3869 lemma homeomorphism_imp_open_map:

  3870   assumes hom: "homeomorphism S T f g"

  3871     and oo: "openin (subtopology euclidean S) U"

  3872   shows "openin (subtopology euclidean T) (f  U)"

  3873 proof -

  3874   from hom oo have [simp]: "f  U = T \<inter> g - U"

  3875     using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)

  3876   from hom have "continuous_on T g"

  3877     unfolding homeomorphism_def by blast

  3878   moreover have "g  T = S"

  3879     by (metis hom homeomorphism_def)

  3880   ultimately show ?thesis

  3881     by (simp add: continuous_on_open oo)

  3882 qed

  3883

  3884 lemma homeomorphism_imp_closed_map:

  3885   assumes hom: "homeomorphism S T f g"

  3886     and oo: "closedin (subtopology euclidean S) U"

  3887   shows "closedin (subtopology euclidean T) (f  U)"

  3888 proof -

  3889   from hom oo have [simp]: "f  U = T \<inter> g - U"

  3890     using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)

  3891   from hom have "continuous_on T g"

  3892     unfolding homeomorphism_def by blast

  3893   moreover have "g  T = S"

  3894     by (metis hom homeomorphism_def)

  3895   ultimately show ?thesis

  3896     by (simp add: continuous_on_closed oo)

  3897 qed

  3898

  3899

  3900 subsection \<open>"Isometry" (up to constant bounds) of injective linear map etc\<close>

  3901

  3902 lemma cauchy_isometric:

  3903   assumes e: "e > 0"

  3904     and s: "subspace s"

  3905     and f: "bounded_linear f"

  3906     and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"

  3907     and xs: "\<forall>n. x n \<in> s"

  3908     and cf: "Cauchy (f \<circ> x)"

  3909   shows "Cauchy x"

  3910 proof -

  3911   interpret f: bounded_linear f by fact

  3912   have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" if "d > 0" for d :: real

  3913   proof -

  3914     from that obtain N where N: "\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"

  3915       using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e

  3916       by auto

  3917     have "norm (x n - x N) < d" if "n \<ge> N" for n

  3918     proof -

  3919       have "e * norm (x n - x N) \<le> norm (f (x n - x N))"

  3920         using subspace_diff[OF s, of "x n" "x N"]

  3921         using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]

  3922         using normf[THEN bspec[where x="x n - x N"]]

  3923         by auto

  3924       also have "norm (f (x n - x N)) < e * d"

  3925         using \<open>N \<le> n\<close> N unfolding f.diff[symmetric] by auto

  3926       finally show ?thesis

  3927         using \<open>e>0\<close> by simp

  3928     qed

  3929     then show ?thesis by auto

  3930   qed

  3931   then show ?thesis

  3932     by (simp add: Cauchy_altdef2 dist_norm)

  3933 qed

  3934

  3935 lemma complete_isometric_image:

  3936   assumes "0 < e"

  3937     and s: "subspace s"

  3938     and f: "bounded_linear f"

  3939     and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"

  3940     and cs: "complete s"

  3941   shows "complete (f  s)"

  3942 proof -

  3943   have "\<exists>l\<in>f  s. (g \<longlongrightarrow> l) sequentially"

  3944     if as:"\<forall>n::nat. g n \<in> f  s" and cfg:"Cauchy g" for g

  3945   proof -

  3946     from that obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"

  3947       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto

  3948     then have x: "\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto

  3949     then have "f \<circ> x = g" by (simp add: fun_eq_iff)

  3950     then obtain l where "l\<in>s" and l:"(x \<longlongrightarrow> l) sequentially"

  3951       using cs[unfolded complete_def, THEN spec[where x=x]]

  3952       using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)

  3953       by auto

  3954     then show ?thesis

  3955       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]

  3956       by (auto simp: \<open>f \<circ> x = g\<close>)

  3957   qed

  3958   then show ?thesis

  3959     unfolding complete_def by auto

  3960 qed

  3961

  3962 proposition injective_imp_isometric:

  3963   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"

  3964   assumes s: "closed s" "subspace s"

  3965     and f: "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0"

  3966   shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm x"

  3967 proof (cases "s \<subseteq> {0::'a}")

  3968   case True

  3969   have "norm x \<le> norm (f x)" if "x \<in> s" for x

  3970   proof -

  3971     from True that have "x = 0" by auto

  3972     then show ?thesis by simp

  3973   qed

  3974   then show ?thesis

  3975     by (auto intro!: exI[where x=1])

  3976 next

  3977   case False

  3978   interpret f: bounded_linear f by fact

  3979   from False obtain a where a: "a \<noteq> 0" "a \<in> s"

  3980     by auto

  3981   from False have "s \<noteq> {}"

  3982     by auto

  3983   let ?S = "{f x| x. x \<in> s \<and> norm x = norm a}"

  3984   let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"

  3985   let ?S'' = "{x::'a. norm x = norm a}"

  3986

  3987   have "?S'' = frontier (cball 0 (norm a))"

  3988     by (simp add: sphere_def dist_norm)

  3989   then have "compact ?S''" by (metis compact_cball compact_frontier)

  3990   moreover have "?S' = s \<inter> ?S''" by auto

  3991   ultimately have "compact ?S'"

  3992     using closed_Int_compact[of s ?S''] using s(1) by auto

  3993   moreover have *:"f  ?S' = ?S" by auto

  3994   ultimately have "compact ?S"

  3995     using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto

  3996   then have "closed ?S"

  3997     using compact_imp_closed by auto

  3998   moreover from a have "?S \<noteq> {}" by auto

  3999   ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y"

  4000     using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto

  4001   then obtain b where "b\<in>s"

  4002     and ba: "norm b = norm a"

  4003     and b: "\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"

  4004     unfolding *[symmetric] unfolding image_iff by auto

  4005

  4006   let ?e = "norm (f b) / norm b"

  4007   have "norm b > 0"

  4008     using ba and a and norm_ge_zero by auto

  4009   moreover have "norm (f b) > 0"

  4010     using f(2)[THEN bspec[where x=b], OF \<open>b\<in>s\<close>]

  4011     using \<open>norm b >0\<close> by simp

  4012   ultimately have "0 < norm (f b) / norm b" by simp

  4013   moreover

  4014   have "norm (f b) / norm b * norm x \<le> norm (f x)" if "x\<in>s" for x

  4015   proof (cases "x = 0")

  4016     case True

  4017     then show "norm (f b) / norm b * norm x \<le> norm (f x)"

  4018       by auto

  4019   next

  4020     case False

  4021     with \<open>a \<noteq> 0\<close> have *: "0 < norm a / norm x"

  4022       unfolding zero_less_norm_iff[symmetric] by simp

  4023     have "\<forall>x\<in>s. c *\<^sub>R x \<in> s" for c

  4024       using s[unfolded subspace_def] by simp

  4025     with \<open>x \<in> s\<close> \<open>x \<noteq> 0\<close> have "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}"

  4026       by simp

  4027     with \<open>x \<noteq> 0\<close> \<open>a \<noteq> 0\<close> show "norm (f b) / norm b * norm x \<le> norm (f x)"

  4028       using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]

  4029       unfolding f.scaleR and ba

  4030       by (auto simp: mult.commute pos_le_divide_eq pos_divide_le_eq)

  4031   qed

  4032   ultimately show ?thesis by auto

  4033 qed

  4034

  4035 proposition closed_injective_image_subspace:

  4036   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"

  4037   assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0" "closed s"

  4038   shows "closed(f  s)"

  4039 proof -

  4040   obtain e where "e > 0" and e: "\<forall>x\<in>s. e * norm x \<le> norm (f x)"

  4041     using injective_imp_isometric[OF assms(4,1,2,3)] by auto

  4042   show ?thesis

  4043     using complete_isometric_image[OF \<open>e>0\<close> assms(1,2) e] and assms(4)

  4044     unfolding complete_eq_closed[symmetric] by auto

  4045 qed

  4046

  4047

  4048 subsection%unimportant \<open>Some properties of a canonical subspace\<close>

  4049

  4050 lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"

  4051   by (auto simp: subspace_def inner_add_left)

  4052

  4053 lemma closed_substandard: "closed {x::'a::euclidean_space. \<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0}"

  4054   (is "closed ?A")

  4055 proof -

  4056   let ?D = "{i\<in>Basis. P i}"

  4057   have "closed (\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0})"

  4058     by (simp add: closed_INT closed_Collect_eq continuous_on_inner

  4059         continuous_on_const continuous_on_id)

  4060   also have "(\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0}) = ?A"

  4061     by auto

  4062   finally show "closed ?A" .

  4063 qed

  4064

  4065 lemma dim_substandard:

  4066   assumes d: "d \<subseteq> Basis"

  4067   shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")

  4068 proof (rule dim_unique)

  4069   from d show "d \<subseteq> ?A"

  4070     by (auto simp: inner_Basis)

  4071   from d show "independent d"

  4072     by (rule independent_mono [OF independent_Basis])

  4073   have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x

  4074   proof -

  4075     have "finite d"

  4076       by (rule finite_subset [OF d finite_Basis])

  4077     then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"

  4078       by (simp add: span_sum span_clauses)

  4079     also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"

  4080       by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)

  4081     finally show "x \<in> span d"

  4082       by (simp only: euclidean_representation)

  4083   qed

  4084   then show "?A \<subseteq> span d" by auto

  4085 qed simp

  4086

  4087 text \<open>Hence closure and completeness of all subspaces.\<close>

  4088 lemma ex_card:

  4089   assumes "n \<le> card A"

  4090   shows "\<exists>S\<subseteq>A. card S = n"

  4091 proof (cases "finite A")

  4092   case True

  4093   from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..

  4094   moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"

  4095     by (auto simp: bij_betw_def intro: subset_inj_on)

  4096   ultimately have "f  {..< n} \<subseteq> A" "card (f  {..< n}) = n"

  4097     by (auto simp: bij_betw_def card_image)

  4098   then show ?thesis by blast

  4099 next

  4100   case False

  4101   with \<open>n \<le> card A\<close> show ?thesis by force

  4102 qed

  4103

  4104 lemma closed_subspace:

  4105   fixes s :: "'a::euclidean_space set"

  4106   assumes "subspace s"

  4107   shows "closed s"

  4108 proof -

  4109   have "dim s \<le> card (Basis :: 'a set)"

  4110     using dim_subset_UNIV by auto

  4111   with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d \<subseteq> Basis"

  4112     by auto

  4113   let ?t = "{x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"

  4114   have "\<exists>f. linear f \<and> f  {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s \<and>

  4115       inj_on f {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"

  4116     using dim_substandard[of d] t d assms

  4117     by (intro subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]]) (auto simp: inner_Basis)

  4118   then obtain f where f:

  4119       "linear f"

  4120       "f  {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s"

  4121       "inj_on f {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"

  4122     by blast

  4123   interpret f: bounded_linear f

  4124     using f by (simp add: linear_conv_bounded_linear)

  4125   have "x \<in> ?t \<Longrightarrow> f x = 0 \<Longrightarrow> x = 0" for x

  4126     using f.zero d f(3)[THEN inj_onD, of x 0] by auto

  4127   moreover have "closed ?t" by (rule closed_substandard)

  4128   moreover have "subspace ?t" by (rule subspace_substandard)

  4129   ultimately show ?thesis

  4130     using closed_injective_image_subspace[of ?t f]

  4131     unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto

  4132 qed

  4133

  4134 lemma complete_subspace: "subspace s \<Longrightarrow> complete s"

  4135   for s :: "'a::euclidean_space set"

  4136   using complete_eq_closed closed_subspace by auto

  4137

  4138 lemma closed_span [iff]: "closed (span s)"

  4139   for s :: "'a::euclidean_space set"

  4140   by (simp add: closed_subspace subspace_span)

  4141

  4142 lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")

  4143   for s :: "'a::euclidean_space set"

  4144 proof -

  4145   have "?dc \<le> ?d"

  4146     using closure_minimal[OF span_superset, of s]

  4147     using closed_subspace[OF subspace_span, of s]

  4148     using dim_subset[of "closure s" "span s"]

  4149     by simp

  4150   then show ?thesis

  4151     using dim_subset[OF closure_subset, of s]

  4152     by simp

  4153 qed

  4154

  4155

  4156 subsection%unimportant \<open>Affine transformations of intervals\<close>

  4157

  4158 lemma real_affinity_le: "0 < m \<Longrightarrow> m * x + c \<le> y \<longleftrightarrow> x \<le> inverse m * y + - (c / m)"

  4159   for m :: "'a::linordered_field"

  4160   by (simp add: field_simps)

  4161

  4162 lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x"

  4163   for m :: "'a::linordered_field"

  4164   by (simp add: field_simps)

  4165

  4166 lemma real_affinity_lt: "0 < m \<Longrightarrow> m * x + c < y \<longleftrightarrow> x < inverse m * y + - (c / m)"

  4167   for m :: "'a::linordered_field"

  4168   by (simp add: field_simps)

  4169

  4170 lemma real_lt_affinity: "0 < m \<Longrightarrow> y < m * x + c \<longleftrightarrow> inverse m * y + - (c / m) < x"

  4171   for m :: "'a::linordered_field"

  4172   by (simp add: field_simps)

  4173

  4174 lemma real_affinity_eq: "m \<noteq> 0 \<Longrightarrow> m * x + c = y \<longleftrightarrow> x = inverse m * y + - (c / m)"

  4175   for m :: "'a::linordered_field"

  4176   by (simp add: field_simps)

  4177

  4178 lemma real_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m * x + c  \<longleftrightarrow> inverse m * y + - (c / m) = x"

  4179   for m :: "'a::linordered_field"

  4180   by (simp add: field_simps)

  4181

  4182

  4183 subsection \<open>Banach fixed point theorem (not really topological ...)\<close>

  4184

  4185 theorem banach_fix:

  4186   assumes s: "complete s" "s \<noteq> {}"

  4187     and c: "0 \<le> c" "c < 1"

  4188     and f: "f  s \<subseteq> s"

  4189     and lipschitz: "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"

  4190   shows "\<exists>!x\<in>s. f x = x"

  4191 proof -

  4192   from c have "1 - c > 0" by simp

  4193

  4194   from s(2) obtain z0 where z0: "z0 \<in> s" by blast

  4195   define z where "z n = (f ^^ n) z0" for n

  4196   with f z0 have z_in_s: "z n \<in> s" for n :: nat

  4197     by (induct n) auto

  4198   define d where "d = dist (z 0) (z 1)"

  4199

  4200   have fzn: "f (z n) = z (Suc n)" for n

  4201     by (simp add: z_def)

  4202   have cf_z: "dist (z n) (z (Suc n)) \<le> (c ^ n) * d" for n :: nat

  4203   proof (induct n)

  4204     case 0

  4205     then show ?case

  4206       by (simp add: d_def)

  4207   next

  4208     case (Suc m)

  4209     with \<open>0 \<le> c\<close> have "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"

  4210       using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by simp

  4211     then show ?case

  4212       using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]

  4213       by (simp add: fzn mult_le_cancel_left)

  4214   qed

  4215

  4216   have cf_z2: "(1 - c) * dist (z m) (z (m + n)) \<le> (c ^ m) * d * (1 - c ^ n)" for n m :: nat

  4217   proof (induct n)

  4218     case 0

  4219     show ?case by simp

  4220   next

  4221     case (Suc k)

  4222     from c have "(1 - c) * dist (z m) (z (m + Suc k)) \<le>

  4223         (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"

  4224       by (simp add: dist_triangle)

  4225     also from c cf_z[of "m + k"] have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"

  4226       by simp

  4227     also from Suc have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"

  4228       by (simp add: field_simps)

  4229     also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"

  4230       by (simp add: power_add field_simps)

  4231     also from c have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"

  4232       by (simp add: field_simps)

  4233     finally show ?case by simp

  4234   qed

  4235

  4236   have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e" if "e > 0" for e

  4237   proof (cases "d = 0")

  4238     case True

  4239     from \<open>1 - c > 0\<close> have "(1 - c) * x \<le> 0 \<longleftrightarrow> x \<le> 0" for x

  4240       by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)

  4241     with c cf_z2[of 0] True have "z n = z0" for n

  4242       by (simp add: z_def)

  4243     with \<open>e > 0\<close> show ?thesis by simp

  4244   next

  4245     case False

  4246     with zero_le_dist[of "z 0" "z 1"] have "d > 0"

  4247       by (metis d_def less_le)

  4248     with \<open>1 - c > 0\<close> \<open>e > 0\<close> have "0 < e * (1 - c) / d"

  4249       by simp

  4250     with c obtain N where N: "c ^ N < e * (1 - c) / d"

  4251       using real_arch_pow_inv[of "e * (1 - c) / d" c] by auto

  4252     have *: "dist (z m) (z n) < e" if "m > n" and as: "m \<ge> N" "n \<ge> N" for m n :: nat

  4253     proof -

  4254       from c \<open>n \<ge> N\<close> have *: "c ^ n \<le> c ^ N"

  4255         using power_decreasing[OF \<open>n\<ge>N\<close>, of c] by simp

  4256       from c \<open>m > n\<close> have "1 - c ^ (m - n) > 0"

  4257         using power_strict_mono[of c 1 "m - n"] by simp

  4258       with \<open>d > 0\<close> \<open>0 < 1 - c\<close> have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"

  4259         by simp

  4260       from cf_z2[of n "m - n"] \<open>m > n\<close>

  4261       have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"

  4262         by (simp add: pos_le_divide_eq[OF \<open>1 - c > 0\<close>] mult.commute dist_commute)

  4263       also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"

  4264         using mult_right_mono[OF * order_less_imp_le[OF **]]

  4265         by (simp add: mult.assoc)

  4266       also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"

  4267         using mult_strict_right_mono[OF N **] by (auto simp: mult.assoc)

  4268       also from c \<open>d > 0\<close> \<open>1 - c > 0\<close> have "\<dots> = e * (1 - c ^ (m - n))"

  4269         by simp

  4270       also from c \<open>1 - c ^ (m - n) > 0\<close> \<open>e > 0\<close> have "\<dots> \<le> e"

  4271         using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto

  4272       finally show ?thesis by simp

  4273     qed

  4274     have "dist (z n) (z m) < e" if "N \<le> m" "N \<le> n" for m n :: nat

  4275     proof (cases "n = m")

  4276       case True

  4277       with \<open>e > 0\<close> show ?thesis by simp

  4278     next

  4279       case False

  4280       with *[of n m] *[of m n] and that show ?thesis

  4281         by (auto simp: dist_commute nat_neq_iff)

  4282     qed

  4283     then show ?thesis by auto

  4284   qed

  4285   then have "Cauchy z"

  4286     by (simp add: cauchy_def)

  4287   then obtain x where "x\<in>s" and x:"(z \<longlongrightarrow> x) sequentially"

  4288     using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto

  4289

  4290   define e where "e = dist (f x) x"

  4291   have "e = 0"

  4292   proof (rule ccontr)

  4293     assume "e \<noteq> 0"

  4294     then have "e > 0"

  4295       unfolding e_def using zero_le_dist[of "f x" x]

  4296       by (metis dist_eq_0_iff dist_nz e_def)

  4297     then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"

  4298       using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto

  4299     then have N':"dist (z N) x < e / 2" by auto

  4300     have *: "c * dist (z N) x \<le> dist (z N) x"

  4301       unfolding mult_le_cancel_right2

  4302       using zero_le_dist[of "z N" x] and c

  4303       by (metis dist_eq_0_iff dist_nz order_less_asym less_le)

  4304     have "dist (f (z N)) (f x) \<le> c * dist (z N) x"

  4305       using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]

  4306       using z_in_s[of N] \<open>x\<in>s\<close>

  4307       using c

  4308       by auto

  4309     also have "\<dots> < e / 2"

  4310       using N' and c using * by auto

  4311     finally show False

  4312       unfolding fzn

  4313       using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]

  4314       unfolding e_def

  4315       by auto

  4316   qed

  4317   then have "f x = x" by (auto simp: e_def)

  4318   moreover have "y = x" if "f y = y" "y \<in> s" for y

  4319   proof -

  4320     from \<open>x \<in> s\<close> \<open>f x = x\<close> that have "dist x y \<le> c * dist x y"

  4321       using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]] by simp

  4322     with c and zero_le_dist[of x y] have "dist x y = 0"

  4323       by (simp add: mult_le_cancel_right1)

  4324     then show ?thesis by simp

  4325   qed

  4326   ultimately show ?thesis

  4327     using \<open>x\<in>s\<close> by blast

  4328 qed

  4329

  4330 lemma banach_fix_type:

  4331   fixes f::"'a::complete_space\<Rightarrow>'a"

  4332   assumes c:"0 \<le> c" "c < 1"

  4333       and lipschitz:"\<forall>x. \<forall>y. dist (f x) (f y) \<le> c * dist x y"

  4334   shows "\<exists>!x. (f x = x)"

  4335   using assms banach_fix[OF complete_UNIV UNIV_not_empty assms(1,2) subset_UNIV, of f]

  4336   by auto

  4337

  4338

  4339 subsection \<open>Edelstein fixed point theorem\<close>

  4340

  4341 theorem edelstein_fix:

  4342   fixes s :: "'a::metric_space set"

  4343   assumes s: "compact s" "s \<noteq> {}"

  4344     and gs: "(g  s) \<subseteq> s"

  4345     and dist: "\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"

  4346   shows "\<exists>!x\<in>s. g x = x"

  4347 proof -

  4348   let ?D = "(\<lambda>x. (x, x))  s"

  4349   have D: "compact ?D" "?D \<noteq> {}"

  4350     by (rule compact_continuous_image)

  4351        (auto intro!: s continuous_Pair continuous_ident simp: continuous_on_eq_continuous_within)

  4352

  4353   have "\<And>x y e. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 < e \<Longrightarrow> dist y x < e \<Longrightarrow> dist (g y) (g x) < e"

  4354     using dist by fastforce

  4355   then have "continuous_on s g"

  4356     by (auto simp: continuous_on_iff)

  4357   then have cont: "continuous_on ?D (\<lambda>x. dist ((g \<circ> fst) x) (snd x))"

  4358     unfolding continuous_on_eq_continuous_within

  4359     by (intro continuous_dist ballI continuous_within_compose)

  4360        (auto intro!: continuous_fst continuous_snd continuous_ident simp: image_image)

  4361

  4362   obtain a where "a \<in> s" and le: "\<And>x. x \<in> s \<Longrightarrow> dist (g a) a \<le> dist (g x) x"

  4363     using continuous_attains_inf[OF D cont] by auto

  4364

  4365   have "g a = a"

  4366   proof (rule ccontr)

  4367     assume "g a \<noteq> a"

  4368     with \<open>a \<in> s\<close> gs have "dist (g (g a)) (g a) < dist (g a) a"

  4369       by (intro dist[rule_format]) auto

  4370     moreover have "dist (g a) a \<le> dist (g (g a)) (g a)"

  4371       using \<open>a \<in> s\<close> gs by (intro le) auto

  4372     ultimately show False by auto

  4373   qed

  4374   moreover have "\<And>x. x \<in> s \<Longrightarrow> g x = x \<Longrightarrow> x = a"

  4375     using dist[THEN bspec[where x=a]] \<open>g a = a\<close> and \<open>a\<in>s\<close> by auto

  4376   ultimately show "\<exists>!x\<in>s. g x = x"

  4377     using \<open>a \<in> s\<close> by blast

  4378 qed

  4379

  4380

  4381 lemma cball_subset_cball_iff:

  4382   fixes a :: "'a :: euclidean_space"

  4383   shows "cball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r < 0"

  4384     (is "?lhs \<longleftrightarrow> ?rhs")

  4385 proof

  4386   assume ?lhs

  4387   then show ?rhs

  4388   proof (cases "r < 0")

  4389     case True

  4390     then show ?rhs by simp

  4391   next

  4392     case False

  4393     then have [simp]: "r \<ge> 0" by simp

  4394     have "norm (a - a') + r \<le> r'"

  4395     proof (cases "a = a'")

  4396       case True

  4397       then show ?thesis

  4398         using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]

  4399         by (force simp: SOME_Basis dist_norm)

  4400     next

  4401       case False

  4402       have "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = norm (a' - a - (r / norm (a - a')) *\<^sub>R (a - a'))"

  4403         by (simp add: algebra_simps)

  4404       also have "... = norm ((-1 - (r / norm (a - a'))) *\<^sub>R (a - a'))"

  4405         by (simp add: algebra_simps)

  4406       also from \<open>a \<noteq> a'\<close> have "... = \<bar>- norm (a - a') - r\<bar>"

  4407         by (simp add: abs_mult_pos field_simps)

  4408       finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \<bar>norm (a - a') + r\<bar>"

  4409         by linarith

  4410       from \<open>a \<noteq> a'\<close> show ?thesis

  4411         using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>]

  4412         by (simp add: dist_norm scaleR_add_left)

  4413     qed

  4414     then show ?rhs

  4415       by (simp add: dist_norm)

  4416   qed

  4417 next

  4418   assume ?rhs

  4419   then show ?lhs

  4420     by (auto simp: ball_def dist_norm)

  4421       (metis add.commute add_le_cancel_right dist_norm dist_triangle3 order_trans)

  4422 qed

  4423

  4424 lemma cball_subset_ball_iff: "cball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r < r' \<or> r < 0"

  4425   (is "?lhs \<longleftrightarrow> ?rhs")

  4426   for a :: "'a::euclidean_space"

  4427 proof

  4428   assume ?lhs

  4429   then show ?rhs

  4430   proof (cases "r < 0")

  4431     case True then

  4432     show ?rhs by simp

  4433   next

  4434     case False

  4435     then have [simp]: "r \<ge> 0" by simp

  4436     have "norm (a - a') + r < r'"

  4437     proof (cases "a = a'")

  4438       case True

  4439       then show ?thesis

  4440         using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]

  4441         by (force simp: SOME_Basis dist_norm)

  4442     next

  4443       case False

  4444       have False if "norm (a - a') + r \<ge> r'"

  4445       proof -

  4446         from that have "\<bar>r' - norm (a - a')\<bar> \<le> r"

  4447           by (simp split: abs_split)

  4448             (metis \<open>0 \<le> r\<close> \<open>?lhs\<close> centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)

  4449         then show ?thesis

  4450           using subsetD [where c = "a + (r' / norm(a - a') - 1) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>

  4451           by (simp add: dist_norm field_simps)

  4452             (simp add: diff_divide_distrib scaleR_left_diff_distrib)

  4453       qed

  4454       then show ?thesis by force

  4455     qed

  4456     then show ?rhs by (simp add: dist_norm)

  4457   qed

  4458 next

  4459   assume ?rhs

  4460   then show ?lhs

  4461     by (auto simp: ball_def dist_norm)

  4462       (metis add.commute add_le_cancel_right dist_norm dist_triangle3 le_less_trans)

  4463 qed

  4464

  4465 lemma ball_subset_cball_iff: "ball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"

  4466   (is "?lhs = ?rhs")

  4467   for a :: "'a::euclidean_space"

  4468 proof (cases "r \<le> 0")

  4469   case True

  4470   then show ?thesis

  4471     using dist_not_less_zero less_le_trans by force

  4472 next

  4473   case False

  4474   show ?thesis

  4475   proof

  4476     assume ?lhs

  4477     then have "(cball a r \<subseteq> cball a' r')"

  4478       by (metis False closed_cball closure_ball closure_closed closure_mono not_less)

  4479     with False show ?rhs

  4480       by (fastforce iff: cball_subset_cball_iff)

  4481   next

  4482     assume ?rhs

  4483     with False show ?lhs

  4484       using ball_subset_cball cball_subset_cball_iff by blast

  4485   qed

  4486 qed

  4487

  4488 lemma ball_subset_ball_iff:

  4489   fixes a :: "'a :: euclidean_space"

  4490   shows "ball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"

  4491         (is "?lhs = ?rhs")

  4492 proof (cases "r \<le> 0")

  4493   case True then show ?thesis

  4494     using dist_not_less_zero less_le_trans by force

  4495 next

  4496   case False show ?thesis

  4497   proof

  4498     assume ?lhs

  4499     then have "0 < r'"

  4500       by (metis (no_types) False \<open>?lhs\<close> centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less set_mp)

  4501     then have "(cball a r \<subseteq> cball a' r')"

  4502       by (metis False\<open>?lhs\<close> closure_ball closure_mono not_less)

  4503     then show ?rhs

  4504       using False cball_subset_cball_iff by fastforce

  4505   next

  4506   assume ?rhs then show ?lhs

  4507     apply (auto simp: ball_def)

  4508     apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)

  4509     using dist_not_less_zero order.strict_trans2 apply blast

  4510     done

  4511   qed

  4512 qed

  4513

  4514

  4515 lemma ball_eq_ball_iff:

  4516   fixes x :: "'a :: euclidean_space"

  4517   shows "ball x d = ball y e \<longleftrightarrow> d \<le> 0 \<and> e \<le> 0 \<or> x=y \<and> d=e"

  4518         (is "?lhs = ?rhs")

  4519 proof

  4520   assume ?lhs

  4521   then show ?rhs

  4522   proof (cases "d \<le> 0 \<or> e \<le> 0")

  4523     case True

  4524       with \<open>?lhs\<close> show ?rhs

  4525         by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])

  4526   next

  4527     case False

  4528     with \<open>?lhs\<close> show ?rhs

  4529       apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)

  4530       apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)

  4531       apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)

  4532       done

  4533   qed

  4534 next

  4535   assume ?rhs then show ?lhs

  4536     by (auto simp: set_eq_subset ball_subset_ball_iff)

  4537 qed

  4538

  4539 lemma cball_eq_cball_iff:

  4540   fixes x :: "'a :: euclidean_space"

  4541   shows "cball x d = cball y e \<longleftrightarrow> d < 0 \<and> e < 0 \<or> x=y \<and> d=e"

  4542         (is "?lhs = ?rhs")

  4543 proof

  4544   assume ?lhs

  4545   then show ?rhs

  4546   proof (cases "d < 0 \<or> e < 0")

  4547     case True

  4548       with \<open>?lhs\<close> show ?rhs

  4549         by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])

  4550   next

  4551     case False

  4552     with \<open>?lhs\<close> show ?rhs

  4553       apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)

  4554       apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)

  4555       apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)

  4556       done

  4557   qed

  4558 next

  4559   assume ?rhs then show ?lhs

  4560     by (auto simp: set_eq_subset cball_subset_cball_iff)

  4561 qed

  4562

  4563 lemma ball_eq_cball_iff:

  4564   fixes x :: "'a :: euclidean_space"

  4565   shows "ball x d = cball y e \<longleftrightarrow> d \<le> 0 \<and> e < 0" (is "?lhs = ?rhs")

  4566 proof

  4567   assume ?lhs

  4568   then show ?rhs

  4569     apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)

  4570     apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)

  4571     apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)

  4572     using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+

  4573     done

  4574 next

  4575   assume ?rhs then show ?lhs by auto

  4576 qed

  4577

  4578 lemma cball_eq_ball_iff:

  4579   fixes x :: "'a :: euclidean_space"

  4580   shows "cball x d = ball y e \<longleftrightarrow> d < 0 \<and> e \<le> 0"

  4581   using ball_eq_cball_iff by blast

  4582

  4583 lemma finite_ball_avoid:

  4584   fixes S :: "'a :: euclidean_space set"

  4585   assumes "open S" "finite X" "p \<in> S"

  4586   shows "\<exists>e>0. \<forall>w\<in>ball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"

  4587 proof -

  4588   obtain e1 where "0 < e1" and e1_b:"ball p e1 \<subseteq> S"

  4589     using open_contains_ball_eq[OF \<open>open S\<close>] assms by auto

  4590   obtain e2 where "0 < e2" and "\<forall>x\<in>X. x \<noteq> p \<longrightarrow> e2 \<le> dist p x"

  4591     using finite_set_avoid[OF \<open>finite X\<close>,of p] by auto

  4592   hence "\<forall>w\<in>ball p (min e1 e2). w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)" using e1_b by auto

  4593   thus "\<exists>e>0. \<forall>w\<in>ball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> \<open>e1>0\<close>

  4594     apply (rule_tac x="min e1 e2" in exI)

  4595     by auto

  4596 qed

  4597

  4598 lemma finite_cball_avoid:

  4599   fixes S :: "'a :: euclidean_space set"

  4600   assumes "open S" "finite X" "p \<in> S"

  4601   shows "\<exists>e>0. \<forall>w\<in>cball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"

  4602 proof -

  4603   obtain e1 where "e1>0" and e1: "\<forall>w\<in>ball p e1. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"

  4604     using finite_ball_avoid[OF assms] by auto

  4605   define e2 where "e2 \<equiv> e1/2"

  4606   have "e2>0" and "e2 < e1" unfolding e2_def using \<open>e1>0\<close> by auto

  4607   then have "cball p e2 \<subseteq> ball p e1" by (subst cball_subset_ball_iff,auto)

  4608   then show "\<exists>e>0. \<forall>w\<in>cball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> e1 by auto

  4609 qed

  4610

  4611 subsection\<open>Various separability-type properties\<close>

  4612

  4613 lemma univ_second_countable:

  4614   obtains \<B> :: "'a::euclidean_space set set"

  4615   where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"

  4616        "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"

  4617 by (metis ex_countable_basis topological_basis_def)

  4618

  4619 lemma subset_second_countable:

  4620   obtains \<B> :: "'a:: euclidean_space set set"

  4621     where "countable \<B>"

  4622           "{} \<notin> \<B>"

  4623           "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"

  4624           "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"

  4625 proof -

  4626   obtain \<B> :: "'a set set"

  4627     where "countable \<B>"

  4628       and opeB: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"

  4629       and \<B>:    "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"

  4630   proof -

  4631     obtain \<C> :: "'a set set"

  4632       where "countable \<C>" and ope: "\<And>C. C \<in> \<C> \<Longrightarrow> open C"

  4633         and \<C>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<C> \<and> S = \<Union>U"

  4634       by (metis univ_second_countable that)

  4635     show ?thesis

  4636     proof

  4637       show "countable ((\<lambda>C. S \<inter> C)  \<C>)"

  4638         by (simp add: \<open>countable \<C>\<close>)

  4639       show "\<And>C. C \<in> (\<inter>) S  \<C> \<Longrightarrow> openin (subtopology euclidean S) C"

  4640         using ope by auto

  4641       show "\<And>T. openin (subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>\<subseteq>(\<inter>) S  \<C>. T = \<Union>\<U>"

  4642         by (metis \<C> image_mono inf_Sup openin_open)

  4643     qed

  4644   qed

  4645   show ?thesis

  4646   proof

  4647     show "countable (\<B> - {{}})"

  4648       using \<open>countable \<B>\<close> by blast

  4649     show "\<And>C. \<lbrakk>C \<in> \<B> - {{}}\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) C"

  4650       by (simp add: \<open>\<And>C. C \<in> \<B> \<Longrightarrow> openin (subtopology euclidean S) C\<close>)

  4651     show "\<exists>\<U>\<subseteq>\<B> - {{}}. T = \<Union>\<U>" if "openin (subtopology euclidean S) T" for T

  4652       using \<B> [OF that]

  4653       apply clarify

  4654       apply (rule_tac x="\<U> - {{}}" in exI, auto)

  4655         done

  4656   qed auto

  4657 qed

  4658

  4659 lemma univ_second_countable_sequence:

  4660   obtains B :: "nat \<Rightarrow> 'a::euclidean_space set"

  4661     where "inj B" "\<And>n. open(B n)" "\<And>S. open S \<Longrightarrow> \<exists>k. S = \<Union>{B n |n. n \<in> k}"

  4662 proof -

  4663   obtain \<B> :: "'a set set"

  4664   where "countable \<B>"

  4665     and opn: "\<And>C. C \<in> \<B> \<Longrightarrow> open C"

  4666     and Un: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"

  4667     using univ_second_countable by blast

  4668   have *: "infinite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"

  4669     apply (rule Infinite_Set.range_inj_infinite)

  4670     apply (simp add: inj_on_def ball_eq_ball_iff)

  4671     done

  4672   have "infinite \<B>"

  4673   proof

  4674     assume "finite \<B>"

  4675     then have "finite (Union  (Pow \<B>))"

  4676       by simp

  4677     then have "finite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"

  4678       apply (rule rev_finite_subset)

  4679       by (metis (no_types, lifting) PowI image_eqI image_subset_iff Un [OF open_ball])

  4680     with * show False by simp

  4681   qed

  4682   obtain f :: "nat \<Rightarrow> 'a set" where "\<B> = range f" "inj f"

  4683     by (blast intro: countable_as_injective_image [OF \<open>countable \<B>\<close> \<open>infinite \<B>\<close>])

  4684   have *: "\<exists>k. S = \<Union>{f n |n. n \<in> k}" if "open S" for S

  4685     using Un [OF that]

  4686     apply clarify

  4687     apply (rule_tac x="f-U" in exI)

  4688     using \<open>inj f\<close> \<open>\<B> = range f\<close> apply force

  4689     done

  4690   show ?thesis

  4691     apply (rule that [OF \<open>inj f\<close> _ *])

  4692     apply (auto simp: \<open>\<B> = range f\<close> opn)

  4693     done

  4694 qed

  4695

  4696 proposition separable:

  4697   fixes S :: "'a:: euclidean_space set"

  4698   obtains T where "countable T" "T \<subseteq> S" "S \<subseteq> closure T"

  4699 proof -

  4700   obtain \<B> :: "'a:: euclidean_space set set"

  4701     where "countable \<B>"

  4702       and "{} \<notin> \<B>"

  4703       and ope: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"

  4704       and if_ope: "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"

  4705     by (meson subset_second_countable)

  4706   then obtain f where f: "\<And>C. C \<in> \<B> \<Longrightarrow> f C \<in> C"

  4707     by (metis equals0I)

  4708   show ?thesis

  4709   proof

  4710     show "countable (f  \<B>)"

  4711       by (simp add: \<open>countable \<B>\<close>)

  4712     show "f  \<B> \<subseteq> S"

  4713       using ope f openin_imp_subset by blast

  4714     show "S \<subseteq> closure (f  \<B>)"

  4715     proof (clarsimp simp: closure_approachable)

  4716       fix x and e::real

  4717       assume "x \<in> S" "0 < e"

  4718       have "openin (subtopology euclidean S) (S \<inter> ball x e)"

  4719         by (simp add: openin_Int_open)

  4720       with if_ope obtain \<U> where  \<U>: "\<U> \<subseteq> \<B>" "S \<inter> ball x e = \<Union>\<U>"

  4721         by meson

  4722       show "\<exists>C \<in> \<B>. dist (f C) x < e"

  4723       proof (cases "\<U> = {}")

  4724         case True

  4725         then show ?thesis

  4726           using \<open>0 < e\<close>  \<U> \<open>x \<in> S\<close> by auto

  4727       next

  4728         case False

  4729         then obtain C where "C \<in> \<U>" by blast

  4730         show ?thesis

  4731         proof

  4732           show "dist (f C) x < e"

  4733             by (metis Int_iff Union_iff \<U> \<open>C \<in> \<U>\<close> dist_commute f mem_ball subsetCE)

  4734           show "C \<in> \<B>"

  4735             using \<open>\<U> \<subseteq> \<B>\<close> \<open>C \<in> \<U>\<close> by blast

  4736         qed

  4737       qed

  4738     qed

  4739   qed

  4740 qed

  4741

  4742 proposition Lindelof:

  4743   fixes \<F> :: "'a::euclidean_space set set"

  4744   assumes \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> open S"

  4745   obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"

  4746 proof -

  4747   obtain \<B> :: "'a set set"

  4748     where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"

  4749       and \<B>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"

  4750     using univ_second_countable by blast

  4751   define \<D> where "\<D> \<equiv> {S. S \<in> \<B> \<and> (\<exists>U. U \<in> \<F> \<and> S \<subseteq> U)}"

  4752   have "countable \<D>"

  4753     apply (rule countable_subset [OF _ \<open>countable \<B>\<close>])

  4754     apply (force simp: \<D>_def)

  4755     done

  4756   have "\<And>S. \<exists>U. S \<in> \<D> \<longrightarrow> U \<in> \<F> \<and> S \<subseteq> U"

  4757     by (simp add: \<D>_def)

  4758   then obtain G where G: "\<And>S. S \<in> \<D> \<longrightarrow> G S \<in> \<F> \<and> S \<subseteq> G S"

  4759     by metis

  4760   have "\<Union>\<F> \<subseteq> \<Union>\<D>"

  4761     unfolding \<D>_def by (blast dest: \<F> \<B>)

  4762   moreover have "\<Union>\<D> \<subseteq> \<Union>\<F>"

  4763     using \<D>_def by blast

  4764   ultimately have eq1: "\<Union>\<F> = \<Union>\<D>" ..

  4765   have eq2: "\<Union>\<D> = UNION \<D> G"

  4766     using G eq1 by auto

  4767   show ?thesis

  4768     apply (rule_tac \<F>' = "G  \<D>" in that)

  4769     using G \<open>countable \<D>\<close>  apply (auto simp: eq1 eq2)

  4770     done

  4771 qed

  4772

  4773 lemma Lindelof_openin:

  4774   fixes \<F> :: "'a::euclidean_space set set"

  4775   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean U) S"

  4776   obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"

  4777 proof -

  4778   have "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>T. open T \<and> S = U \<inter> T"

  4779     using assms by (simp add: openin_open)

  4780   then obtain tf where tf: "\<And>S. S \<in> \<F> \<Longrightarrow> open (tf S) \<and> (S = U \<inter> tf S)"

  4781     by metis

  4782   have [simp]: "\<And>\<F>'. \<F>' \<subseteq> \<F> \<Longrightarrow> \<Union>\<F>' = U \<inter> \<Union>(tf  \<F>')"

  4783     using tf by fastforce

  4784   obtain \<G> where "countable \<G> \<and> \<G> \<subseteq> tf  \<F>" "\<Union>\<G> = UNION \<F> tf"

  4785     using tf by (force intro: Lindelof [of "tf  \<F>"])

  4786   then obtain \<F>' where \<F>': "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"

  4787     by (clarsimp simp add: countable_subset_image)

  4788   then show ?thesis ..

  4789 qed

  4790

  4791 lemma countable_disjoint_open_subsets:

  4792   fixes \<F> :: "'a::euclidean_space set set"

  4793   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S" and pw: "pairwise disjnt \<F>"

  4794     shows "countable \<F>"

  4795 proof -

  4796   obtain \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"

  4797     by (meson assms Lindelof)

  4798   with pw have "\<F> \<subseteq> insert {} \<F>'"

  4799     by (fastforce simp add: pairwise_def disjnt_iff)

  4800   then show ?thesis

  4801     by (simp add: \<open>countable \<F>'\<close> countable_subset)

  4802 qed

  4803

  4804 lemma countable_disjoint_nonempty_interior_subsets:

  4805   fixes \<F> :: "'a::euclidean_space set set"

  4806   assumes pw: "pairwise disjnt \<F>" and int: "\<And>S. \<lbrakk>S \<in> \<F>; interior S = {}\<rbrakk> \<Longrightarrow> S = {}"

  4807   shows "countable \<F>"

  4808 proof (rule countable_image_inj_on)

  4809   have "disjoint (interior  \<F>)"

  4810     using pw by (simp add: disjoint_image_subset interior_subset)

  4811   then show "countable (interior  \<F>)"

  4812     by (auto intro: countable_disjoint_open_subsets)

  4813   show "inj_on interior \<F>"

  4814     using pw apply (clarsimp simp: inj_on_def pairwise_def)

  4815     apply (metis disjnt_def disjnt_subset1 inf.orderE int interior_subset)

  4816     done

  4817 qed

  4818

  4819 lemma closedin_compact:

  4820    "\<lbrakk>compact S; closedin (subtopology euclidean S) T\<rbrakk> \<Longrightarrow> compact T"

  4821 by (metis closedin_closed compact_Int_closed)

  4822

  4823 lemma closedin_compact_eq:

  4824   fixes S :: "'a::t2_space set"

  4825   shows

  4826    "compact S

  4827          \<Longrightarrow> (closedin (subtopology euclidean S) T \<longleftrightarrow>

  4828               compact T \<and> T \<subseteq> S)"

  4829 by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)

  4830

  4831 lemma continuous_imp_closed_map:

  4832   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4833   assumes "closedin (subtopology euclidean S) U"

  4834           "continuous_on S f" "f  S = T" "compact S"

  4835     shows "closedin (subtopology euclidean T) (f  U)"

  4836   by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)

  4837

  4838 lemma continuous_imp_quotient_map:

  4839   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4840   assumes "continuous_on S f" "f  S = T" "compact S" "U \<subseteq> T"

  4841     shows "openin (subtopology euclidean S) (S \<inter> f - U) \<longleftrightarrow>

  4842            openin (subtopology euclidean T) U"

  4843   by (metis (no_types, lifting) assms closed_map_imp_quotient_map continuous_imp_closed_map)

  4844

  4845

  4846 lemma open_map_restrict:

  4847   assumes opeU: "openin (subtopology euclidean (S \<inter> f - T')) U"

  4848     and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f  U)"

  4849     and "T' \<subseteq> T"

  4850   shows "openin (subtopology euclidean T') (f  U)"

  4851 proof -

  4852   obtain V where "open V" "U = S \<inter> f - T' \<inter> V"

  4853     using opeU by (auto simp: openin_open)

  4854   with oo [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis

  4855     by (fastforce simp add: openin_open)

  4856 qed

  4857

  4858 lemma closed_map_restrict:

  4859   assumes cloU: "closedin (subtopology euclidean (S \<inter> f - T')) U"

  4860     and cc: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f  U)"

  4861     and "T' \<subseteq> T"

  4862   shows "closedin (subtopology euclidean T') (f  U)"

  4863 proof -

  4864   obtain V where "closed V" "U = S \<inter> f - T' \<inter> V"

  4865     using cloU by (auto simp: closedin_closed)

  4866   with cc [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis

  4867     by (fastforce simp add: closedin_closed)

  4868 qed

  4869

  4870 lemma connected_monotone_quotient_preimage:

  4871   assumes "connected T"

  4872       and contf: "continuous_on S f" and fim: "f  S = T"

  4873       and opT: "\<And>U. U \<subseteq> T

  4874                  \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f - U) \<longleftrightarrow>

  4875                      openin (subtopology euclidean T) U"

  4876       and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f - {y})"

  4877     shows "connected S"

  4878 proof (rule connectedI)

  4879   fix U V

  4880   assume "open U" and "open V" and "U \<inter> S \<noteq> {}" and "V \<inter> S \<noteq> {}"

  4881     and "U \<inter> V \<inter> S = {}" and "S \<subseteq> U \<union> V"

  4882   moreover

  4883   have disjoint: "f  (S \<inter> U) \<inter> f  (S \<inter> V) = {}"

  4884   proof -

  4885     have False if "y \<in> f  (S \<inter> U) \<inter> f  (S \<inter> V)" for y

  4886     proof -

  4887       have "y \<in> T"

  4888         using fim that by blast

  4889       show ?thesis

  4890         using connectedD [OF connT [OF \<open>y \<in> T\<close>] \<open>open U\<close> \<open>open V\<close>]

  4891               \<open>S \<subseteq> U \<union> V\<close> \<open>U \<inter> V \<inter> S = {}\<close> that by fastforce

  4892     qed

  4893     then show ?thesis by blast

  4894   qed

  4895   ultimately have UU: "(S \<inter> f - f  (S \<inter> U)) = S \<inter> U" and VV: "(S \<inter> f - f  (S \<inter> V)) = S \<inter> V"

  4896     by auto

  4897   have opeU: "openin (subtopology euclidean T) (f  (S \<inter> U))"

  4898     by (metis UU \<open>open U\<close> fim image_Int_subset le_inf_iff opT openin_open_Int)

  4899   have opeV: "openin (subtopology euclidean T) (f  (S \<inter> V))"

  4900     by (metis opT fim VV \<open>open V\<close> openin_open_Int image_Int_subset inf.bounded_iff)

  4901   have "T \<subseteq> f  (S \<inter> U) \<union> f  (S \<inter> V)"

  4902     using \<open>S \<subseteq> U \<union> V\<close> fim by auto

  4903   then show False

  4904     using \<open>connected T\<close> disjoint opeU opeV \<open>U \<inter> S \<noteq> {}\<close> \<open>V \<inter> S \<noteq> {}\<close>

  4905     by (auto simp: connected_openin)

  4906 qed

  4907

  4908 lemma connected_open_monotone_preimage:

  4909   assumes contf: "continuous_on S f" and fim: "f  S = T"

  4910     and ST: "\<And>C. openin (subtopology euclidean S) C \<Longrightarrow> openin (subtopology euclidean T) (f  C)"

  4911     and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f - {y})"

  4912     and "connected C" "C \<subseteq> T"

  4913   shows "connected (S \<inter> f - C)"

  4914 proof -

  4915   have contf': "continuous_on (S \<inter> f - C) f"

  4916     by (meson contf continuous_on_subset inf_le1)

  4917   have eqC: "f  (S \<inter> f - C) = C"

  4918     using \<open>C \<subseteq> T\<close> fim by blast

  4919   show ?thesis

  4920   proof (rule connected_monotone_quotient_preimage [OF \<open>connected C\<close> contf' eqC])

  4921     show "connected (S \<inter> f - C \<inter> f - {y})" if "y \<in> C" for y

  4922     proof -

  4923       have "S \<inter> f - C \<inter> f - {y} = S \<inter> f - {y}"

  4924         using that by blast

  4925       moreover have "connected (S \<inter> f - {y})"

  4926         using \<open>C \<subseteq> T\<close> connT that by blast

  4927       ultimately show ?thesis

  4928         by metis

  4929     qed

  4930     have "\<And>U. openin (subtopology euclidean (S \<inter> f - C)) U

  4931                \<Longrightarrow> openin (subtopology euclidean C) (f  U)"

  4932       using open_map_restrict [OF _ ST \<open>C \<subseteq> T\<close>] by metis

  4933     then show "\<And>D. D \<subseteq> C

  4934           \<Longrightarrow> openin (subtopology euclidean (S \<inter> f - C)) (S \<inter> f - C \<inter> f - D) =

  4935               openin (subtopology euclidean C) D"

  4936       using open_map_imp_quotient_map [of "(S \<inter> f - C)" f] contf' by (simp add: eqC)

  4937   qed

  4938 qed

  4939

  4940

  4941 lemma connected_closed_monotone_preimage:

  4942   assumes contf: "continuous_on S f" and fim: "f  S = T"

  4943     and ST: "\<And>C. closedin (subtopology euclidean S) C \<Longrightarrow> closedin (subtopology euclidean T) (f  C)"

  4944     and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f - {y})"

  4945     and "connected C" "C \<subseteq> T"

  4946   shows "connected (S \<inter> f - C)"

  4947 proof -

  4948   have contf': "continuous_on (S \<inter> f - C) f"

  4949     by (meson contf continuous_on_subset inf_le1)

  4950   have eqC: "f  (S \<inter> f - C) = C"

  4951     using \<open>C \<subseteq> T\<close> fim by blast

  4952   show ?thesis

  4953   proof (rule connected_monotone_quotient_preimage [OF \<open>connected C\<close> contf' eqC])

  4954     show "connected (S \<inter> f - C \<inter> f - {y})" if "y \<in> C" for y

  4955     proof -

  4956       have "S \<inter> f - C \<inter> f - {y} = S \<inter> f - {y}"

  4957         using that by blast

  4958       moreover have "connected (S \<inter> f - {y})"

  4959         using \<open>C \<subseteq> T\<close> connT that by blast

  4960       ultimately show ?thesis

  4961         by metis

  4962     qed

  4963     have "\<And>U. closedin (subtopology euclidean (S \<inter> f - C)) U

  4964                \<Longrightarrow> closedin (subtopology euclidean C) (f  U)"

  4965       using closed_map_restrict [OF _ ST \<open>C \<subseteq> T\<close>] by metis

  4966     then show "\<And>D. D \<subseteq> C

  4967           \<Longrightarrow> openin (subtopology euclidean (S \<inter> f - C)) (S \<inter> f - C \<inter> f - D) =

  4968               openin (subtopology euclidean C) D"

  4969       using closed_map_imp_quotient_map [of "(S \<inter> f - C)" f] contf' by (simp add: eqC)

  4970   qed

  4971 qed

  4972

  4973

  4974

  4975 subsection\<open>A couple of lemmas about components (see Newman IV, 3.3 and 3.4)\<close>

  4976

  4977

  4978 lemma connected_Un_clopen_in_complement:

  4979   fixes S U :: "'a::metric_space set"

  4980   assumes "connected S" "connected U" "S \<subseteq> U"

  4981       and opeT: "openin (subtopology euclidean (U - S)) T"

  4982       and cloT: "closedin (subtopology euclidean (U - S)) T"

  4983     shows "connected (S \<union> T)"

  4984 proof -

  4985   have *: "\<lbrakk>\<And>x y. P x y \<longleftrightarrow> P y x; \<And>x y. P x y \<Longrightarrow> S \<subseteq> x \<or> S \<subseteq> y;

  4986             \<And>x y. \<lbrakk>P x y; S \<subseteq> x\<rbrakk> \<Longrightarrow> False\<rbrakk> \<Longrightarrow> ~(\<exists>x y. (P x y))" for P

  4987     by metis

  4988   show ?thesis

  4989     unfolding connected_closedin_eq

  4990   proof (rule *)

  4991     fix H1 H2

  4992     assume H: "closedin (subtopology euclidean (S \<union> T)) H1 \<and>

  4993                closedin (subtopology euclidean (S \<union> T)) H2 \<and>

  4994                H1 \<union> H2 = S \<union> T \<and> H1 \<inter> H2 = {} \<and> H1 \<noteq> {} \<and> H2 \<noteq> {}"

  4995     then have clo: "closedin (subtopology euclidean S) (S \<inter> H1)"

  4996                    "closedin (subtopology euclidean S) (S \<inter> H2)"

  4997       by (metis Un_upper1 closedin_closed_subset inf_commute)+

  4998     have Seq: "S \<inter> (H1 \<union> H2) = S"

  4999       by (simp add: H)

  5000     have "S \<inter> ((S \<union> T) \<inter> H1) \<union> S \<inter> ((S \<union> T) \<inter> H2) = S"

  5001       using Seq by auto

  5002     moreover have "H1 \<inter> (S \<inter> ((S \<union> T) \<inter> H2)) = {}"

  5003       using H by blast

  5004     ultimately have "S \<inter> H1 = {} \<or> S \<inter> H2 = {}"

  5005       by (metis (no_types) H Int_assoc \<open>S \<inter> (H1 \<union> H2) = S\<close> \<open>connected S\<close>

  5006           clo Seq connected_closedin inf_bot_right inf_le1)

  5007     then show "S \<subseteq> H1 \<or> S \<subseteq> H2"

  5008       using H \<open>connected S\<close> unfolding connected_closedin by blast

  5009   next

  5010     fix H1 H2

  5011     assume H: "closedin (subtopology euclidean (S \<union> T)) H1 \<and>

  5012                closedin (subtopology euclidean (S \<union> T)) H2 \<and>

  5013                H1 \<union> H2 = S \<union> T \<and> H1 \<inter> H2 = {} \<and> H1 \<noteq> {} \<and> H2 \<noteq> {}"

  5014        and "S \<subseteq> H1"

  5015     then have H2T: "H2 \<subseteq> T"

  5016       by auto

  5017     have "T \<subseteq> U"

  5018       using Diff_iff opeT openin_imp_subset by auto

  5019     with \<open>S \<subseteq> U\<close> have Ueq: "U = (U - S) \<union> (S \<union> T)"

  5020       by auto

  5021     have "openin (subtopology euclidean ((U - S) \<union> (S \<union> T))) H2"

  5022     proof (rule openin_subtopology_Un)

  5023       show "openin (subtopology euclidean (S \<union> T)) H2"

  5024         using \<open>H2 \<subseteq> T\<close> apply (auto simp: openin_closedin_eq)

  5025         by (metis Diff_Diff_Int Diff_disjoint Diff_partition Diff_subset H Int_absorb1 Un_Diff)

  5026       then show "openin (subtopology euclidean (U - S)) H2"

  5027         by (meson H2T Un_upper2 opeT openin_subset_trans openin_trans)

  5028     qed

  5029     moreover have "closedin (subtopology euclidean ((U - S) \<union> (S \<union> T))) H2"

  5030     proof (rule closedin_subtopology_Un)

  5031       show "closedin (subtopology euclidean (U - S)) H2"

  5032         using H H2T cloT closedin_subset_trans

  5033         by (blast intro: closedin_subtopology_Un closedin_trans)

  5034     qed (simp add: H)

  5035     ultimately

  5036     have H2: "H2 = {} \<or> H2 = U"

  5037       using Ueq \<open>connected U\<close> unfolding connected_clopen by metis

  5038     then have "H2 \<subseteq> S"

  5039       by (metis Diff_partition H Un_Diff_cancel Un_subset_iff \<open>H2 \<subseteq> T\<close> assms(3) inf.orderE opeT openin_imp_subset)

  5040     moreover have "T \<subseteq> H2 - S"

  5041       by (metis (no_types) H2 H opeT openin_closedin_eq topspace_euclidean_subtopology)

  5042     ultimately show False

  5043       using H \<open>S \<subseteq> H1\<close> by blast

  5044   qed blast

  5045 qed

  5046

  5047

  5048 proposition component_diff_connected:

  5049   fixes S :: "'a::metric_space set"

  5050   assumes "connected S" "connected U" "S \<subseteq> U" and C: "C \<in> components (U - S)"

  5051   shows "connected(U - C)"

  5052   using \<open>connected S\<close> unfolding connected_closedin_eq not_ex de_Morgan_conj

  5053 proof clarify

  5054   fix H3 H4

  5055   assume clo3: "closedin (subtopology euclidean (U - C)) H3"

  5056     and clo4: "closedin (subtopology euclidean (U - C)) H4"

  5057     and "H3 \<union> H4 = U - C" and "H3 \<inter> H4 = {}" and "H3 \<noteq> {}" and "H4 \<noteq> {}"

  5058     and * [rule_format]:

  5059     "\<forall>H1 H2. \<not> closedin (subtopology euclidean S) H1 \<or>

  5060                       \<not> closedin (subtopology euclidean S) H2 \<or>

  5061                       H1 \<union> H2 \<noteq> S \<or> H1 \<inter> H2 \<noteq> {} \<or> \<not> H1 \<noteq> {} \<or> \<not> H2 \<noteq> {}"

  5062   then have "H3 \<subseteq> U-C" and ope3: "openin (subtopology euclidean (U - C)) (U - C - H3)"

  5063     and "H4 \<subseteq> U-C" and ope4: "openin (subtopology euclidean (U - C)) (U - C - H4)"

  5064     by (auto simp: closedin_def)

  5065   have "C \<noteq> {}" "C \<subseteq> U-S" "connected C"

  5066     using C in_components_nonempty in_components_subset in_components_maximal by blast+

  5067   have cCH3: "connected (C \<union> H3)"

  5068   proof (rule connected_Un_clopen_in_complement [OF \<open>connected C\<close> \<open>connected U\<close> _ _ clo3])

  5069     show "openin (subtopology euclidean (U - C)) H3"

  5070       apply (simp add: openin_closedin_eq \<open>H3 \<subseteq> U - C\<close>)

  5071       apply (simp add: closedin_subtopology)

  5072       by (metis Diff_cancel Diff_triv Un_Diff clo4 \<open>H3 \<inter> H4 = {}\<close> \<open>H3 \<union> H4 = U - C\<close> closedin_closed inf_commute sup_bot.left_neutral)

  5073   qed (use clo3 \<open>C \<subseteq> U - S\<close> in auto)

  5074   have cCH4: "connected (C \<union> H4)"

  5075   proof (rule connected_Un_clopen_in_complement [OF \<open>connected C\<close> \<open>connected U\<close> _ _ clo4])

  5076     show "openin (subtopology euclidean (U - C)) H4"

  5077       apply (simp add: openin_closedin_eq \<open>H4 \<subseteq> U - C\<close>)

  5078       apply (simp add: closedin_subtopology)

  5079       by (metis Diff_cancel Int_commute Un_Diff Un_Diff_Int \<open>H3 \<inter> H4 = {}\<close> \<open>H3 \<union> H4 = U - C\<close> clo3 closedin_closed)

  5080   qed (use clo4 \<open>C \<subseteq> U - S\<close> in auto)

  5081   have "closedin (subtopology euclidean S) (S \<inter> H3)" "closedin (subtopology euclidean S) (S \<inter> H4)"

  5082     using clo3 clo4 \<open>S \<subseteq> U\<close> \<open>C \<subseteq> U - S\<close> by (auto simp: closedin_closed)

  5083   moreover have "S \<inter> H3 \<noteq> {}"

  5084     using components_maximal [OF C cCH3] \<open>C \<noteq> {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H3 \<noteq> {}\<close> \<open>H3 \<subseteq> U - C\<close> by auto

  5085   moreover have "S \<inter> H4 \<noteq> {}"

  5086     using components_maximal [OF C cCH4] \<open>C \<noteq> {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H4 \<noteq> {}\<close> \<open>H4 \<subseteq> U - C\<close> by auto

  5087   ultimately show False

  5088     using * [of "S \<inter> H3" "S \<inter> H4"] \<open>H3 \<inter> H4 = {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H3 \<union> H4 = U - C\<close> \<open>S \<subseteq> U\<close>

  5089     by auto

  5090 qed

  5091

  5092 subsection%unimportant\<open> Finite intersection property\<close>

  5093

  5094 text\<open>Also developed in HOL's toplogical spaces theory, but the Heine-Borel type class isn't available there.\<close>

  5095

  5096 lemma closed_imp_fip:

  5097   fixes S :: "'a::heine_borel set"

  5098   assumes "closed S"

  5099       and T: "T \<in> \<F>" "bounded T"

  5100       and clof: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"

  5101       and none: "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter>\<F>' \<noteq> {}"

  5102     shows "S \<inter> \<Inter>\<F> \<noteq> {}"

  5103 proof -

  5104   have "compact (S \<inter> T)"

  5105     using \<open>closed S\<close> clof compact_eq_bounded_closed T by blast

  5106   then have "(S \<inter> T) \<inter> \<Inter>\<F> \<noteq> {}"

  5107     apply (rule compact_imp_fip)

  5108      apply (simp add: clof)

  5109     by (metis Int_assoc complete_lattice_class.Inf_insert finite_insert insert_subset none \<open>T \<in> \<F>\<close>)

  5110   then show ?thesis by blast

  5111 qed

  5112

  5113 lemma closed_imp_fip_compact:

  5114   fixes S :: "'a::heine_borel set"

  5115   shows

  5116    "\<lbrakk>closed S; \<And>T. T \<in> \<F> \<Longrightarrow> compact T;

  5117      \<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter>\<F>' \<noteq> {}\<rbrakk>

  5118         \<Longrightarrow> S \<inter> \<Inter>\<F> \<noteq> {}"

  5119 by (metis Inf_greatest closed_imp_fip compact_eq_bounded_closed empty_subsetI finite.emptyI inf.orderE)

  5120

  5121 lemma closed_fip_heine_borel:

  5122   fixes \<F> :: "'a::heine_borel set set"

  5123   assumes "closed S" "T \<in> \<F>" "bounded T"

  5124       and "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"

  5125       and "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> \<Inter>\<F>' \<noteq> {}"

  5126     shows "\<Inter>\<F> \<noteq> {}"

  5127 proof -

  5128   have "UNIV \<inter> \<Inter>\<F> \<noteq> {}"

  5129     using assms closed_imp_fip [OF closed_UNIV] by auto

  5130   then show ?thesis by simp

  5131 qed

  5132

  5133 lemma compact_fip_heine_borel:

  5134   fixes \<F> :: "'a::heine_borel set set"

  5135   assumes clof: "\<And>T. T \<in> \<F> \<Longrightarrow> compact T"

  5136       and none: "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> \<Inter>\<F>' \<noteq> {}"

  5137     shows "\<Inter>\<F> \<noteq> {}"

  5138 by (metis InterI all_not_in_conv clof closed_fip_heine_borel compact_eq_bounded_closed none)

  5139

  5140 lemma compact_sequence_with_limit:

  5141   fixes f :: "nat \<Rightarrow> 'a::heine_borel"

  5142   shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> compact (insert l (range f))"

  5143 apply (simp add: compact_eq_bounded_closed, auto)

  5144 apply (simp add: convergent_imp_bounded)

  5145 by (simp add: closed_limpt islimpt_insert sequence_unique_limpt)

  5146

  5147

  5148 subsection%unimportant\<open>Componentwise limits and continuity\<close>

  5149

  5150 text\<open>But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}\<close>

  5151 lemma Euclidean_dist_upper: "i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y"

  5152   by (metis (no_types) member_le_L2_set euclidean_dist_l2 finite_Basis)

  5153

  5154 text\<open>But is the premise @{term \<open>i \<in> Basis\<close>} really necessary?\<close>

  5155 lemma open_preimage_inner:

  5156   assumes "open S" "i \<in> Basis"

  5157     shows "open {x. x \<bullet> i \<in> S}"

  5158 proof (rule openI, simp)

  5159   fix x

  5160   assume x: "x \<bullet> i \<in> S"

  5161   with assms obtain e where "0 < e" and e: "ball (x \<bullet> i) e \<subseteq> S"

  5162     by (auto simp: open_contains_ball_eq)

  5163   have "\<exists>e>0. ball (y \<bullet> i) e \<subseteq> S" if dxy: "dist x y < e / 2" for y

  5164   proof (intro exI conjI)

  5165     have "dist (x \<bullet> i) (y \<bullet> i) < e / 2"

  5166       by (meson \<open>i \<in> Basis\<close> dual_order.trans Euclidean_dist_upper not_le that)

  5167     then have "dist (x \<bullet> i) z < e" if "dist (y \<bullet> i) z < e / 2" for z

  5168       by (metis dist_commute dist_triangle_half_l that)

  5169     then have "ball (y \<bullet> i) (e / 2) \<subseteq> ball (x \<bullet> i) e"

  5170       using mem_ball by blast

  5171       with e show "ball (y \<bullet> i) (e / 2) \<subseteq> S"

  5172         by (metis order_trans)

  5173   qed (simp add: \<open>0 < e\<close>)

  5174   then show "\<exists>e>0. ball x e \<subseteq> {s. s \<bullet> i \<in> S}"

  5175     by (metis (no_types, lifting) \<open>0 < e\<close> \<open>open S\<close> half_gt_zero_iff mem_Collect_eq mem_ball open_contains_ball_eq subsetI)

  5176 qed

  5177

  5178 proposition tendsto_componentwise_iff:

  5179   fixes f :: "_ \<Rightarrow> 'b::euclidean_space"

  5180   shows "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>i \<in> Basis. ((\<lambda>x. (f x \<bullet> i)) \<longlongrightarrow> (l \<bullet> i)) F)"

  5181          (is "?lhs = ?rhs")

  5182 proof

  5183   assume ?lhs

  5184   then show ?rhs

  5185     unfolding tendsto_def

  5186     apply clarify

  5187     apply (drule_tac x="{s. s \<bullet> i \<in> S}" in spec)

  5188     apply (auto simp: open_preimage_inner)

  5189     done

  5190 next

  5191   assume R: ?rhs

  5192   then have "\<And>e. e > 0 \<Longrightarrow> \<forall>i\<in>Basis. \<forall>\<^sub>F x in F. dist (f x \<bullet> i) (l \<bullet> i) < e"

  5193     unfolding tendsto_iff by blast

  5194   then have R': "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e"

  5195       by (simp add: eventually_ball_finite_distrib [symmetric])

  5196   show ?lhs

  5197   unfolding tendsto_iff

  5198   proof clarify

  5199     fix e::real

  5200     assume "0 < e"

  5201     have *: "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e"

  5202              if "\<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / real DIM('b)" for x

  5203     proof -

  5204       have "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis \<le> sum (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis"

  5205         by (simp add: L2_set_le_sum)

  5206       also have "... < DIM('b) * (e / real DIM('b))"

  5207         apply (rule sum_bounded_above_strict)

  5208         using that by auto

  5209       also have "... = e"

  5210         by (simp add: field_simps)

  5211       finally show "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e" .

  5212     qed

  5213     have "\<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / DIM('b)"

  5214       apply (rule R')

  5215       using \<open>0 < e\<close> by simp

  5216     then show "\<forall>\<^sub>F x in F. dist (f x) l < e"

  5217       apply (rule eventually_mono)

  5218       apply (subst euclidean_dist_l2)

  5219       using * by blast

  5220   qed

  5221 qed

  5222

  5223

  5224 corollary continuous_componentwise:

  5225    "continuous F f \<longleftrightarrow> (\<forall>i \<in> Basis. continuous F (\<lambda>x. (f x \<bullet> i)))"

  5226 by (simp add: continuous_def tendsto_componentwise_iff [symmetric])

  5227

  5228 corollary continuous_on_componentwise:

  5229   fixes S :: "'a :: t2_space set"

  5230   shows "continuous_on S f \<longleftrightarrow> (\<forall>i \<in> Basis. continuous_on S (\<lambda>x. (f x \<bullet> i)))"

  5231   apply (simp add: continuous_on_eq_continuous_within)

  5232   using continuous_componentwise by blast

  5233

  5234 lemma linear_componentwise_iff:

  5235      "(linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. linear (\<lambda>x. f' x \<bullet> i))"

  5236   apply (auto simp: linear_iff inner_left_distrib)

  5237    apply (metis inner_left_distrib euclidean_eq_iff)

  5238   by (metis euclidean_eqI inner_scaleR_left)

  5239

  5240 lemma bounded_linear_componentwise_iff:

  5241      "(bounded_linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. bounded_linear (\<lambda>x. f' x \<bullet> i))"

  5242      (is "?lhs = ?rhs")

  5243 proof

  5244   assume ?lhs then show ?rhs

  5245     by (simp add: bounded_linear_inner_left_comp)

  5246 next

  5247   assume ?rhs

  5248   then have "(\<forall>i\<in>Basis. \<exists>K. \<forall>x. \<bar>f' x \<bullet> i\<bar> \<le> norm x * K)" "linear f'"

  5249     by (auto simp: bounded_linear_def bounded_linear_axioms_def linear_componentwise_iff [symmetric] ball_conj_distrib)

  5250   then obtain F where F: "\<And>i x. i \<in> Basis \<Longrightarrow> \<bar>f' x \<bullet> i\<bar> \<le> norm x * F i"

  5251     by metis

  5252   have "norm (f' x) \<le> norm x * sum F Basis" for x

  5253   proof -

  5254     have "norm (f' x) \<le> (\<Sum>i\<in>Basis. \<bar>f' x \<bullet> i\<bar>)"

  5255       by (rule norm_le_l1)

  5256     also have "... \<le> (\<Sum>i\<in>Basis. norm x * F i)"

  5257       by (metis F sum_mono)

  5258     also have "... = norm x * sum F Basis"

  5259       by (simp add: sum_distrib_left)

  5260     finally show ?thesis .

  5261   qed

  5262   then show ?lhs

  5263     by (force simp: bounded_linear_def bounded_linear_axioms_def \<open>linear f'\<close>)

  5264 qed

  5265

  5266 subsection%unimportant\<open>Pasting functions together\<close>

  5267

  5268 subsubsection%unimportant\<open>on open sets\<close>

  5269

  5270 lemma pasting_lemma:

  5271   fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"

  5272   assumes clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"

  5273       and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"

  5274       and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"

  5275       and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"

  5276     shows "continuous_on S g"

  5277 proof (clarsimp simp: continuous_openin_preimage_eq)

  5278   fix U :: "'b set"

  5279   assume "open U"

  5280   have S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"

  5281     using clo openin_imp_subset by blast

  5282   have *: "(S \<inter> g - U) = (\<Union>i \<in> I. T i \<inter> f i - U)"

  5283     using S f g by fastforce

  5284   show "openin (subtopology euclidean S) (S \<inter> g - U)"

  5285     apply (subst *)

  5286     apply (rule openin_Union, clarify)

  5287     using \<open>open U\<close> clo cont continuous_openin_preimage_gen openin_trans by blast

  5288 qed

  5289

  5290 lemma pasting_lemma_exists:

  5291   fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"

  5292   assumes S: "S \<subseteq> (\<Union>i \<in> I. T i)"

  5293       and clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"

  5294       and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"

  5295       and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"

  5296     obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"

  5297 proof

  5298   show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"

  5299     apply (rule pasting_lemma [OF clo cont])

  5300      apply (blast intro: f)+

  5301     apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)

  5302     done

  5303 next

  5304   fix x i

  5305   assume "i \<in> I" "x \<in> S \<inter> T i"

  5306   then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"

  5307     by (metis (no_types, lifting) IntD2 IntI f someI_ex)

  5308 qed

  5309

  5310 subsubsection%unimportant\<open>Likewise on closed sets, with a finiteness assumption\<close>

  5311

  5312 lemma pasting_lemma_closed:

  5313   fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"

  5314   assumes "finite I"

  5315       and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"

  5316       and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"

  5317       and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"

  5318       and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"

  5319     shows "continuous_on S g"

  5320 proof (clarsimp simp: continuous_closedin_preimage_eq)

  5321   fix U :: "'b set"

  5322   assume "closed U"

  5323   have S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"

  5324     using clo closedin_imp_subset by blast

  5325   have *: "(S \<inter> g - U) = (\<Union>i \<in> I. T i \<inter> f i - U)"

  5326     using S f g by fastforce

  5327   show "closedin (subtopology euclidean S) (S \<inter> g - U)"

  5328     apply (subst *)

  5329     apply (rule closedin_Union)

  5330     using \<open>finite I\<close> apply simp

  5331     apply (blast intro: \<open>closed U\<close> continuous_closedin_preimage cont clo closedin_trans)

  5332     done

  5333 qed

  5334

  5335 lemma pasting_lemma_exists_closed:

  5336   fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"

  5337   assumes "finite I"

  5338       and S: "S \<subseteq> (\<Union>i \<in> I. T i)"

  5339       and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"

  5340       and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"

  5341       and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"

  5342     obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"

  5343 proof

  5344   show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"

  5345     apply (rule pasting_lemma_closed [OF \<open>finite I\<close> clo cont])

  5346      apply (blast intro: f)+

  5347     apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)

  5348     done

  5349 next

  5350   fix x i

  5351   assume "i \<in> I" "x \<in> S \<inter> T i"

  5352   then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"

  5353     by (metis (no_types, lifting) IntD2 IntI f someI_ex)

  5354 qed

  5355

  5356 lemma tube_lemma:

  5357   assumes "compact K"

  5358   assumes "open W"

  5359   assumes "{x0} \<times> K \<subseteq> W"

  5360   shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"

  5361 proof -

  5362   {

  5363     fix y assume "y \<in> K"

  5364     then have "(x0, y) \<in> W" using assms by auto

  5365     with \<open>open W\<close>

  5366     have "\<exists>X0 Y. open X0 \<and> open Y \<and> x0 \<in> X0 \<and> y \<in> Y \<and> X0 \<times> Y \<subseteq> W"

  5367       by (rule open_prod_elim) blast

  5368   }

  5369   then obtain X0 Y where

  5370     *: "\<forall>y \<in> K. open (X0 y) \<and> open (Y y) \<and> x0 \<in> X0 y \<and> y \<in> Y y \<and> X0 y \<times> Y y \<subseteq> W"

  5371     by metis

  5372   from * have "\<forall>t\<in>Y  K. open t" "K \<subseteq> \<Union>(Y  K)" by auto

  5373   with \<open>compact K\<close> obtain CC where CC: "CC \<subseteq> Y  K" "finite CC" "K \<subseteq> \<Union>CC"

  5374     by (meson compactE)

  5375   then obtain c where c: "\<And>C. C \<in> CC \<Longrightarrow> c C \<in> K \<and> C = Y (c C)"

  5376     by (force intro!: choice)

  5377   with * CC show ?thesis

  5378     by (force intro!: exI[where x="\<Inter>C\<in>CC. X0 (c C)"]) (* SLOW *)

  5379 qed

  5380

  5381 lemma continuous_on_prod_compactE:

  5382   fixes fx::"'a::topological_space \<times> 'b::topological_space \<Rightarrow> 'c::metric_space"

  5383     and e::real

  5384   assumes cont_fx: "continuous_on (U \<times> C) fx"

  5385   assumes "compact C"

  5386   assumes [intro]: "x0 \<in> U"

  5387   notes [continuous_intros] = continuous_on_compose2[OF cont_fx]

  5388   assumes "e > 0"

  5389   obtains X0 where "x0 \<in> X0" "open X0"

  5390     "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"

  5391 proof -

  5392   define psi where "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"

  5393   define W0 where "W0 = {(x, t) \<in> U \<times> C. psi (x, t) < e}"

  5394   have W0_eq: "W0 = psi - {..<e} \<inter> U \<times> C"

  5395     by (auto simp: vimage_def W0_def)

  5396   have "open {..<e}" by simp

  5397   have "continuous_on (U \<times> C) psi"

  5398     by (auto intro!: continuous_intros simp: psi_def split_beta')

  5399   from this[unfolded continuous_on_open_invariant, rule_format, OF \<open>open {..<e}\<close>]

  5400   obtain W where W: "open W" "W \<inter> U \<times> C = W0 \<inter> U \<times> C"

  5401     unfolding W0_eq by blast

  5402   have "{x0} \<times> C \<subseteq> W \<inter> U \<times> C"

  5403     unfolding W

  5404     by (auto simp: W0_def psi_def \<open>0 < e\<close>)

  5405   then have "{x0} \<times> C \<subseteq> W" by blast

  5406   from tube_lemma[OF \<open>compact C\<close> \<open>open W\<close> this]

  5407   obtain X0 where X0: "x0 \<in> X0" "open X0" "X0 \<times> C \<subseteq> W"

  5408     by blast

  5409

  5410   have "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"

  5411   proof safe

  5412     fix x assume x: "x \<in> X0" "x \<in> U"

  5413     fix t assume t: "t \<in> C"

  5414     have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"

  5415       by (auto simp: psi_def)

  5416     also

  5417     {

  5418       have "(x, t) \<in> X0 \<times> C"

  5419         using t x

  5420         by auto

  5421       also note \<open>\<dots> \<subseteq> W\<close>

  5422       finally have "(x, t) \<in> W" .

  5423       with t x have "(x, t) \<in> W \<inter> U \<times> C"

  5424         by blast

  5425       also note \<open>W \<inter> U \<times> C = W0 \<inter> U \<times> C\<close>

  5426       finally  have "psi (x, t) < e"

  5427         by (auto simp: W0_def)

  5428     }

  5429     finally show "dist (fx (x, t)) (fx (x0, t)) \<le> e" by simp

  5430   qed

  5431   from X0(1,2) this show ?thesis ..

  5432 qed

  5433

  5434

  5435 subsection%unimportant\<open>Constancy of a function from a connected set into a finite, disconnected or discrete set\<close>

  5436

  5437 text\<open>Still missing: versions for a set that is smaller than R, or countable.\<close>

  5438

  5439 lemma continuous_disconnected_range_constant:

  5440   assumes S: "connected S"

  5441       and conf: "continuous_on S f"

  5442       and fim: "f  S \<subseteq> t"

  5443       and cct: "\<And>y. y \<in> t \<Longrightarrow> connected_component_set t y = {y}"

  5444     shows "f constant_on S"

  5445 proof (cases "S = {}")

  5446   case True then show ?thesis

  5447     by (simp add: constant_on_def)

  5448 next

  5449   case False

  5450   { fix x assume "x \<in> S"

  5451     then have "f  S \<subseteq> {f x}"

  5452     by (metis connected_continuous_image conf connected_component_maximal fim image_subset_iff rev_image_eqI S cct)

  5453   }

  5454   with False show ?thesis

  5455     unfolding constant_on_def by blast

  5456 qed

  5457

  5458 lemma discrete_subset_disconnected:

  5459   fixes S :: "'a::topological_space set"

  5460   fixes t :: "'b::real_normed_vector set"

  5461   assumes conf: "continuous_on S f"

  5462       and no: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"

  5463    shows "f  S \<subseteq> {y. connected_component_set (f  S) y = {y}}"

  5464 proof -

  5465   { fix x assume x: "x \<in> S"

  5466     then obtain e where "e>0" and ele: "\<And>y. \<lbrakk>y \<in> S; f y \<noteq> f x\<rbrakk> \<Longrightarrow> e \<le> norm (f y - f x)"

  5467       using conf no [OF x] by auto

  5468     then have e2: "0 \<le> e / 2"

  5469       by simp

  5470     have "f y = f x" if "y \<in> S" and ccs: "f y \<in> connected_component_set (f  S) (f x)" for y

  5471       apply (rule ccontr)

  5472       using connected_closed [of "connected_component_set (f  S) (f x)"] \<open>e>0\<close>

  5473       apply (simp add: del: ex_simps)

  5474       apply (drule spec [where x="cball (f x) (e / 2)"])

  5475       apply (drule spec [where x="- ball(f x) e"])

  5476       apply (auto simp: dist_norm open_closed [symmetric] simp del: le_divide_eq_numeral1 dest!: connected_component_in)

  5477         apply (metis diff_self e2 ele norm_minus_commute norm_zero not_less)

  5478        using centre_in_cball connected_component_refl_eq e2 x apply blast

  5479       using ccs

  5480       apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF \<open>y \<in> S\<close>])

  5481       done

  5482     moreover have "connected_component_set (f  S) (f x) \<subseteq> f  S"

  5483       by (auto simp: connected_component_in)

  5484     ultimately have "connected_component_set (f  S) (f x) = {f x}"

  5485       by (auto simp: x)

  5486   }

  5487   with assms show ?thesis

  5488     by blast

  5489 qed

  5490

  5491 lemma finite_implies_discrete:

  5492   fixes S :: "'a::topological_space set"

  5493   assumes "finite (f  S)"

  5494   shows "(\<forall>x \<in> S. \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x))"

  5495 proof -

  5496   have "\<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)" if "x \<in> S" for x

  5497   proof (cases "f  S - {f x} = {}")

  5498     case True

  5499     with zero_less_numeral show ?thesis

  5500       by (fastforce simp add: Set.image_subset_iff cong: conj_cong)

  5501   next

  5502     case False

  5503     then obtain z where z: "z \<in> S" "f z \<noteq> f x"

  5504       by blast

  5505     have finn: "finite {norm (z - f x) |z. z \<in> f  S - {f x}}"

  5506       using assms by simp

  5507     then have *: "0 < Inf{norm(z - f x) | z. z \<in> f  S - {f x}}"

  5508       apply (rule finite_imp_less_Inf)

  5509       using z apply force+

  5510       done

  5511     show ?thesis

  5512       by (force intro!: * cInf_le_finite [OF finn])

  5513   qed

  5514   with assms show ?thesis

  5515     by blast

  5516 qed

  5517

  5518 text\<open>This proof requires the existence of two separate values of the range type.\<close>

  5519 lemma finite_range_constant_imp_connected:

  5520   assumes "\<And>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.

  5521               \<lbrakk>continuous_on S f; finite(f  S)\<rbrakk> \<Longrightarrow> f constant_on S"

  5522     shows "connected S"

  5523 proof -

  5524   { fix t u

  5525     assume clt: "closedin (subtopology euclidean S) t"

  5526        and clu: "closedin (subtopology euclidean S) u"

  5527        and tue: "t \<inter> u = {}" and tus: "t \<union> u = S"

  5528     have conif: "continuous_on S (\<lambda>x. if x \<in> t then 0 else 1)"

  5529       apply (subst tus [symmetric])

  5530       apply (rule continuous_on_cases_local)

  5531       using clt clu tue

  5532       apply (auto simp: tus continuous_on_const)

  5533       done

  5534     have fi: "finite ((\<lambda>x. if x \<in> t then 0 else 1)  S)"

  5535       by (rule finite_subset [of _ "{0,1}"]) auto

  5536     have "t = {} \<or> u = {}"

  5537       using assms [OF conif fi] tus [symmetric]

  5538       by (auto simp: Ball_def constant_on_def) (metis IntI empty_iff one_neq_zero tue)

  5539   }

  5540   then show ?thesis

  5541     by (simp add: connected_closedin_eq)

  5542 qed

  5543

  5544 lemma continuous_disconnected_range_constant_eq:

  5545       "(connected S \<longleftrightarrow>

  5546            (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.

  5547             \<forall>t. continuous_on S f \<and> f  S \<subseteq> t \<and> (\<forall>y \<in> t. connected_component_set t y = {y})

  5548             \<longrightarrow> f constant_on S))" (is ?thesis1)

  5549   and continuous_discrete_range_constant_eq:

  5550       "(connected S \<longleftrightarrow>

  5551          (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.

  5552           continuous_on S f \<and>

  5553           (\<forall>x \<in> S. \<exists>e. 0 < e \<and> (\<forall>y. y \<in> S \<and> (f y \<noteq> f x) \<longrightarrow> e \<le> norm(f y - f x)))

  5554           \<longrightarrow> f constant_on S))" (is ?thesis2)

  5555   and continuous_finite_range_constant_eq:

  5556       "(connected S \<longleftrightarrow>

  5557          (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.

  5558           continuous_on S f \<and> finite (f  S)

  5559           \<longrightarrow> f constant_on S))" (is ?thesis3)

  5560 proof -

  5561   have *: "\<And>s t u v. \<lbrakk>s \<Longrightarrow> t; t \<Longrightarrow> u; u \<Longrightarrow> v; v \<Longrightarrow> s\<rbrakk>

  5562     \<Longrightarrow> (s \<longleftrightarrow> t) \<and> (s \<longleftrightarrow> u) \<and> (s \<longleftrightarrow> v)"

  5563     by blast

  5564   have "?thesis1 \<and> ?thesis2 \<and> ?thesis3"

  5565     apply (rule *)

  5566     using continuous_disconnected_range_constant apply metis

  5567     apply clarify

  5568     apply (frule discrete_subset_disconnected; blast)

  5569     apply (blast dest: finite_implies_discrete)

  5570     apply (blast intro!: finite_range_constant_imp_connected)

  5571     done

  5572   then show ?thesis1 ?thesis2 ?thesis3

  5573     by blast+

  5574 qed

  5575

  5576 lemma continuous_discrete_range_constant:

  5577   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"

  5578   assumes S: "connected S"

  5579       and "continuous_on S f"

  5580       and "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"

  5581     shows "f constant_on S"

  5582   using continuous_discrete_range_constant_eq [THEN iffD1, OF S] assms by blast

  5583

  5584 lemma continuous_finite_range_constant:

  5585   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"

  5586   assumes "connected S"

  5587       and "continuous_on S f"

  5588       and "finite (f  S)"

  5589     shows "f constant_on S"

  5590   using assms continuous_finite_range_constant_eq  by blast

  5591

  5592

  5593

  5594 subsection%unimportant \<open>Continuous Extension\<close>

  5595

  5596 definition clamp :: "'a::euclidean_space \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where

  5597   "clamp a b x = (if (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)

  5598     then (\<Sum>i\<in>Basis. (if x\<bullet>i < a\<bullet>i then a\<bullet>i else if x\<bullet>i \<le> b\<bullet>i then x\<bullet>i else b\<bullet>i) *\<^sub>R i)

  5599     else a)"

  5600

  5601 lemma clamp_in_interval[simp]:

  5602   assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"

  5603   shows "clamp a b x \<in> cbox a b"

  5604   unfolding clamp_def

  5605   using box_ne_empty(1)[of a b] assms by (auto simp: cbox_def)

  5606

  5607 lemma clamp_cancel_cbox[simp]:

  5608   fixes x a b :: "'a::euclidean_space"

  5609   assumes x: "x \<in> cbox a b"

  5610   shows "clamp a b x = x"

  5611   using assms

  5612   by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a])

  5613

  5614 lemma clamp_empty_interval:

  5615   assumes "i \<in> Basis" "a \<bullet> i > b \<bullet> i"

  5616   shows "clamp a b = (\<lambda>_. a)"

  5617   using assms

  5618   by (force simp: clamp_def[abs_def] split: if_splits intro!: ext)

  5619

  5620 lemma dist_clamps_le_dist_args:

  5621   fixes x :: "'a::euclidean_space"

  5622   shows "dist (clamp a b y) (clamp a b x) \<le> dist y x"

  5623 proof cases

  5624   assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"

  5625   then have "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2) \<le>

  5626     (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"

  5627     by (auto intro!: sum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])

  5628   then show ?thesis

  5629     by (auto intro: real_sqrt_le_mono

  5630       simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] L2_set_def)

  5631 qed (auto simp: clamp_def)

  5632

  5633 lemma clamp_continuous_at:

  5634   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"

  5635     and x :: 'a

  5636   assumes f_cont: "continuous_on (cbox a b) f"

  5637   shows "continuous (at x) (\<lambda>x. f (clamp a b x))"

  5638 proof cases

  5639   assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"

  5640   show ?thesis

  5641     unfolding continuous_at_eps_delta

  5642   proof safe

  5643     fix x :: 'a

  5644     fix e :: real

  5645     assume "e > 0"

  5646     moreover have "clamp a b x \<in> cbox a b"

  5647       by (simp add: clamp_in_interval le)

  5648     moreover note f_cont[simplified continuous_on_iff]

  5649     ultimately

  5650     obtain d where d: "0 < d"

  5651       "\<And>x'. x' \<in> cbox a b \<Longrightarrow> dist x' (clamp a b x) < d \<Longrightarrow> dist (f x') (f (clamp a b x)) < e"

  5652       by force

  5653     show "\<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow>

  5654       dist (f (clamp a b x')) (f (clamp a b x)) < e"

  5655       using le

  5656       by (auto intro!: d clamp_in_interval dist_clamps_le_dist_args[THEN le_less_trans])

  5657   qed

  5658 qed (auto simp: clamp_empty_interval)

  5659

  5660 lemma clamp_continuous_on:

  5661   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"

  5662   assumes f_cont: "continuous_on (cbox a b) f"

  5663   shows "continuous_on S (\<lambda>x. f (clamp a b x))"

  5664   using assms

  5665   by (auto intro: continuous_at_imp_continuous_on clamp_continuous_at)

  5666

  5667 lemma clamp_bounded:

  5668   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"

  5669   assumes bounded: "bounded (f  (cbox a b))"

  5670   shows "bounded (range (\<lambda>x. f (clamp a b x)))"

  5671 proof cases

  5672   assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"

  5673   from bounded obtain c where f_bound: "\<forall>x\<in>f  cbox a b. dist undefined x \<le> c"

  5674     by (auto simp: bounded_any_center[where a=undefined])

  5675   then show ?thesis

  5676     by (auto intro!: exI[where x=c] clamp_in_interval[OF le[rule_format]]

  5677         simp: bounded_any_center[where a=undefined])

  5678 qed (auto simp: clamp_empty_interval image_def)

  5679

  5680

  5681 definition ext_cont :: "('a::euclidean_space \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"

  5682   where "ext_cont f a b = (\<lambda>x. f (clamp a b x))"

  5683

  5684 lemma ext_cont_cancel_cbox[simp]:

  5685   fixes x a b :: "'a::euclidean_space"

  5686   assumes x: "x \<in> cbox a b"

  5687   shows "ext_cont f a b x = f x"

  5688   using assms

  5689   unfolding ext_cont_def

  5690   by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a] arg_cong[where f=f])

  5691

  5692 lemma continuous_on_ext_cont[continuous_intros]:

  5693   "continuous_on (cbox a b) f \<Longrightarrow> continuous_on S (ext_cont f a b)"

  5694   by (auto intro!: clamp_continuous_on simp: ext_cont_def)

  5695

  5696 end