src/HOL/Analysis/Topology_Euclidean_Space.thy
 author wenzelm Tue Jan 17 13:59:10 2017 +0100 (2017-01-17) changeset 64911 f0e07600de47 parent 64910 6108dddad9f0 child 65036 ab7e11730ad8 permissions -rw-r--r--
isabelle update_cartouches -c -t;
```     1 (*  Author:     L C Paulson, University of Cambridge
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3     Author:     Robert Himmelmann, TU Muenchen
```
```     4     Author:     Brian Huffman, Portland State University
```
```     5 *)
```
```     6
```
```     7 section \<open>Elementary topology in Euclidean space.\<close>
```
```     8
```
```     9 theory Topology_Euclidean_Space
```
```    10 imports
```
```    11   "~~/src/HOL/Library/Indicator_Function"
```
```    12   "~~/src/HOL/Library/Countable_Set"
```
```    13   "~~/src/HOL/Library/FuncSet"
```
```    14   Linear_Algebra
```
```    15   Norm_Arith
```
```    16 begin
```
```    17
```
```    18 (* FIXME: move elsewhere *)
```
```    19
```
```    20 lemma Times_eq_image_sum:
```
```    21   fixes S :: "'a :: comm_monoid_add set" and T :: "'b :: comm_monoid_add set"
```
```    22   shows "S \<times> T = {u + v |u v. u \<in> (\<lambda>x. (x, 0)) ` S \<and> v \<in> Pair 0 ` T}"
```
```    23   by force
```
```    24
```
```    25 lemma halfspace_Int_eq:
```
```    26      "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
```
```    27      "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
```
```    28   by auto
```
```    29
```
```    30 definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
```
```    31   where "support_on s f = {x\<in>s. f x \<noteq> 0}"
```
```    32
```
```    33 lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
```
```    34   by (simp add: support_on_def)
```
```    35
```
```    36 lemma support_on_simps[simp]:
```
```    37   "support_on {} f = {}"
```
```    38   "support_on (insert x s) f =
```
```    39     (if f x = 0 then support_on s f else insert x (support_on s f))"
```
```    40   "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
```
```    41   "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
```
```    42   "support_on (s - t) f = support_on s f - support_on t f"
```
```    43   "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
```
```    44   unfolding support_on_def by auto
```
```    45
```
```    46 lemma support_on_cong:
```
```    47   "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
```
```    48   by (auto simp: support_on_def)
```
```    49
```
```    50 lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
```
```    51   by (auto simp: support_on_def)
```
```    52
```
```    53 lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
```
```    54   by (auto simp: support_on_def)
```
```    55
```
```    56 lemma finite_support[intro]: "finite s \<Longrightarrow> finite (support_on s f)"
```
```    57   unfolding support_on_def by auto
```
```    58
```
```    59 (* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
```
```    60 definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    61   where "supp_sum f s = (\<Sum>x\<in>support_on s f. f x)"
```
```    62
```
```    63 lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
```
```    64   unfolding supp_sum_def by auto
```
```    65
```
```    66 lemma supp_sum_insert[simp]:
```
```    67   "finite (support_on s f) \<Longrightarrow>
```
```    68     supp_sum f (insert x s) = (if x \<in> s then supp_sum f s else f x + supp_sum f s)"
```
```    69   by (simp add: supp_sum_def in_support_on insert_absorb)
```
```    70
```
```    71 lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
```
```    72   by (cases "r = 0")
```
```    73      (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
```
```    74
```
```    75 (*END OF SUPPORT, ETC.*)
```
```    76
```
```    77 lemma image_affinity_interval:
```
```    78   fixes c :: "'a::ordered_real_vector"
```
```    79   shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) = (if {a..b}={} then {}
```
```    80             else if 0 <= m then {m *\<^sub>R a + c .. m  *\<^sub>R b + c}
```
```    81             else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
```
```    82   apply (case_tac "m=0", force)
```
```    83   apply (auto simp: scaleR_left_mono)
```
```    84   apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI, auto simp: pos_le_divideR_eq le_diff_eq scaleR_left_mono_neg)
```
```    85   apply (metis diff_le_eq inverse_inverse_eq order.not_eq_order_implies_strict pos_le_divideR_eq positive_imp_inverse_positive)
```
```    86   apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI, auto simp: not_le neg_le_divideR_eq diff_le_eq)
```
```    87   using le_diff_eq scaleR_le_cancel_left_neg
```
```    88   apply fastforce
```
```    89   done
```
```    90
```
```    91 lemma countable_PiE:
```
```    92   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
```
```    93   by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
```
```    94
```
```    95 lemma continuous_on_cases:
```
```    96   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>
```
```    97     \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>
```
```    98     continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
```
```    99   by (rule continuous_on_If) auto
```
```   100
```
```   101 lemma open_sums:
```
```   102   fixes T :: "('b::real_normed_vector) set"
```
```   103   assumes "open S \<or> open T"
```
```   104   shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
```
```   105   using assms
```
```   106 proof
```
```   107   assume S: "open S"
```
```   108   show ?thesis
```
```   109   proof (clarsimp simp: open_dist)
```
```   110     fix x y
```
```   111     assume "x \<in> S" "y \<in> T"
```
```   112     with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
```
```   113       by (auto simp: open_dist)
```
```   114     then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
```
```   115       by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
```
```   116     then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
```
```   117       using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
```
```   118   qed
```
```   119 next
```
```   120   assume T: "open T"
```
```   121   show ?thesis
```
```   122   proof (clarsimp simp: open_dist)
```
```   123     fix x y
```
```   124     assume "x \<in> S" "y \<in> T"
```
```   125     with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
```
```   126       by (auto simp: open_dist)
```
```   127     then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
```
```   128       by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
```
```   129     then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
```
```   130       using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
```
```   131   qed
```
```   132 qed
```
```   133
```
```   134
```
```   135 subsection \<open>Topological Basis\<close>
```
```   136
```
```   137 context topological_space
```
```   138 begin
```
```   139
```
```   140 definition "topological_basis B \<longleftrightarrow>
```
```   141   (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
```
```   142
```
```   143 lemma topological_basis:
```
```   144   "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
```
```   145   unfolding topological_basis_def
```
```   146   apply safe
```
```   147      apply fastforce
```
```   148     apply fastforce
```
```   149    apply (erule_tac x="x" in allE)
```
```   150    apply simp
```
```   151    apply (rule_tac x="{x}" in exI)
```
```   152   apply auto
```
```   153   done
```
```   154
```
```   155 lemma topological_basis_iff:
```
```   156   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
```
```   157   shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
```
```   158     (is "_ \<longleftrightarrow> ?rhs")
```
```   159 proof safe
```
```   160   fix O' and x::'a
```
```   161   assume H: "topological_basis B" "open O'" "x \<in> O'"
```
```   162   then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
```
```   163   then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
```
```   164   then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
```
```   165 next
```
```   166   assume H: ?rhs
```
```   167   show "topological_basis B"
```
```   168     using assms unfolding topological_basis_def
```
```   169   proof safe
```
```   170     fix O' :: "'a set"
```
```   171     assume "open O'"
```
```   172     with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
```
```   173       by (force intro: bchoice simp: Bex_def)
```
```   174     then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
```
```   175       by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
```
```   176   qed
```
```   177 qed
```
```   178
```
```   179 lemma topological_basisI:
```
```   180   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
```
```   181     and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
```
```   182   shows "topological_basis B"
```
```   183   using assms by (subst topological_basis_iff) auto
```
```   184
```
```   185 lemma topological_basisE:
```
```   186   fixes O'
```
```   187   assumes "topological_basis B"
```
```   188     and "open O'"
```
```   189     and "x \<in> O'"
```
```   190   obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
```
```   191 proof atomize_elim
```
```   192   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
```
```   193     by (simp add: topological_basis_def)
```
```   194   with topological_basis_iff assms
```
```   195   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
```
```   196     using assms by (simp add: Bex_def)
```
```   197 qed
```
```   198
```
```   199 lemma topological_basis_open:
```
```   200   assumes "topological_basis B"
```
```   201     and "X \<in> B"
```
```   202   shows "open X"
```
```   203   using assms by (simp add: topological_basis_def)
```
```   204
```
```   205 lemma topological_basis_imp_subbasis:
```
```   206   assumes B: "topological_basis B"
```
```   207   shows "open = generate_topology B"
```
```   208 proof (intro ext iffI)
```
```   209   fix S :: "'a set"
```
```   210   assume "open S"
```
```   211   with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
```
```   212     unfolding topological_basis_def by blast
```
```   213   then show "generate_topology B S"
```
```   214     by (auto intro: generate_topology.intros dest: topological_basis_open)
```
```   215 next
```
```   216   fix S :: "'a set"
```
```   217   assume "generate_topology B S"
```
```   218   then show "open S"
```
```   219     by induct (auto dest: topological_basis_open[OF B])
```
```   220 qed
```
```   221
```
```   222 lemma basis_dense:
```
```   223   fixes B :: "'a set set"
```
```   224     and f :: "'a set \<Rightarrow> 'a"
```
```   225   assumes "topological_basis B"
```
```   226     and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
```
```   227   shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
```
```   228 proof (intro allI impI)
```
```   229   fix X :: "'a set"
```
```   230   assume "open X" and "X \<noteq> {}"
```
```   231   from topological_basisE[OF \<open>topological_basis B\<close> \<open>open X\<close> choosefrom_basis[OF \<open>X \<noteq> {}\<close>]]
```
```   232   obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
```
```   233   then show "\<exists>B'\<in>B. f B' \<in> X"
```
```   234     by (auto intro!: choosefrom_basis)
```
```   235 qed
```
```   236
```
```   237 end
```
```   238
```
```   239 lemma topological_basis_prod:
```
```   240   assumes A: "topological_basis A"
```
```   241     and B: "topological_basis B"
```
```   242   shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
```
```   243   unfolding topological_basis_def
```
```   244 proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
```
```   245   fix S :: "('a \<times> 'b) set"
```
```   246   assume "open S"
```
```   247   then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
```
```   248   proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
```
```   249     fix x y
```
```   250     assume "(x, y) \<in> S"
```
```   251     from open_prod_elim[OF \<open>open S\<close> this]
```
```   252     obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
```
```   253       by (metis mem_Sigma_iff)
```
```   254     moreover
```
```   255     from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
```
```   256       by (rule topological_basisE)
```
```   257     moreover
```
```   258     from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
```
```   259       by (rule topological_basisE)
```
```   260     ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
```
```   261       by (intro UN_I[of "(A0, B0)"]) auto
```
```   262   qed auto
```
```   263 qed (metis A B topological_basis_open open_Times)
```
```   264
```
```   265
```
```   266 subsection \<open>Countable Basis\<close>
```
```   267
```
```   268 locale countable_basis =
```
```   269   fixes B :: "'a::topological_space set set"
```
```   270   assumes is_basis: "topological_basis B"
```
```   271     and countable_basis: "countable B"
```
```   272 begin
```
```   273
```
```   274 lemma open_countable_basis_ex:
```
```   275   assumes "open X"
```
```   276   shows "\<exists>B' \<subseteq> B. X = \<Union>B'"
```
```   277   using assms countable_basis is_basis
```
```   278   unfolding topological_basis_def by blast
```
```   279
```
```   280 lemma open_countable_basisE:
```
```   281   assumes "open X"
```
```   282   obtains B' where "B' \<subseteq> B" "X = \<Union>B'"
```
```   283   using assms open_countable_basis_ex
```
```   284   by (atomize_elim) simp
```
```   285
```
```   286 lemma countable_dense_exists:
```
```   287   "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
```
```   288 proof -
```
```   289   let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
```
```   290   have "countable (?f ` B)" using countable_basis by simp
```
```   291   with basis_dense[OF is_basis, of ?f] show ?thesis
```
```   292     by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
```
```   293 qed
```
```   294
```
```   295 lemma countable_dense_setE:
```
```   296   obtains D :: "'a set"
```
```   297   where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
```
```   298   using countable_dense_exists by blast
```
```   299
```
```   300 end
```
```   301
```
```   302 lemma (in first_countable_topology) first_countable_basisE:
```
```   303   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
```
```   304     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
```
```   305   using first_countable_basis[of x]
```
```   306   apply atomize_elim
```
```   307   apply (elim exE)
```
```   308   apply (rule_tac x="range A" in exI)
```
```   309   apply auto
```
```   310   done
```
```   311
```
```   312 lemma (in first_countable_topology) first_countable_basis_Int_stableE:
```
```   313   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
```
```   314     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
```
```   315     "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"
```
```   316 proof atomize_elim
```
```   317   obtain A' where A':
```
```   318     "countable A'"
```
```   319     "\<And>a. a \<in> A' \<Longrightarrow> x \<in> a"
```
```   320     "\<And>a. a \<in> A' \<Longrightarrow> open a"
```
```   321     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A'. a \<subseteq> S"
```
```   322     by (rule first_countable_basisE) blast
```
```   323   define A where [abs_def]:
```
```   324     "A = (\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"
```
```   325   then show "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>
```
```   326         (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)"
```
```   327   proof (safe intro!: exI[where x=A])
```
```   328     show "countable A"
```
```   329       unfolding A_def by (intro countable_image countable_Collect_finite)
```
```   330     fix a
```
```   331     assume "a \<in> A"
```
```   332     then show "x \<in> a" "open a"
```
```   333       using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
```
```   334   next
```
```   335     let ?int = "\<lambda>N. \<Inter>(from_nat_into A' ` N)"
```
```   336     fix a b
```
```   337     assume "a \<in> A" "b \<in> A"
```
```   338     then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)"
```
```   339       by (auto simp: A_def)
```
```   340     then show "a \<inter> b \<in> A"
```
```   341       by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
```
```   342   next
```
```   343     fix S
```
```   344     assume "open S" "x \<in> S"
```
```   345     then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
```
```   346     then show "\<exists>a\<in>A. a \<subseteq> S" using a A'
```
```   347       by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])
```
```   348   qed
```
```   349 qed
```
```   350
```
```   351 lemma (in topological_space) first_countableI:
```
```   352   assumes "countable A"
```
```   353     and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
```
```   354     and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
```
```   355   shows "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
```
```   356 proof (safe intro!: exI[of _ "from_nat_into A"])
```
```   357   fix i
```
```   358   have "A \<noteq> {}" using 2[of UNIV] by auto
```
```   359   show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
```
```   360     using range_from_nat_into_subset[OF \<open>A \<noteq> {}\<close>] 1 by auto
```
```   361 next
```
```   362   fix S
```
```   363   assume "open S" "x\<in>S" from 2[OF this]
```
```   364   show "\<exists>i. from_nat_into A i \<subseteq> S"
```
```   365     using subset_range_from_nat_into[OF \<open>countable A\<close>] by auto
```
```   366 qed
```
```   367
```
```   368 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
```
```   369 proof
```
```   370   fix x :: "'a \<times> 'b"
```
```   371   obtain A where A:
```
```   372       "countable A"
```
```   373       "\<And>a. a \<in> A \<Longrightarrow> fst x \<in> a"
```
```   374       "\<And>a. a \<in> A \<Longrightarrow> open a"
```
```   375       "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
```
```   376     by (rule first_countable_basisE[of "fst x"]) blast
```
```   377   obtain B where B:
```
```   378       "countable B"
```
```   379       "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
```
```   380       "\<And>a. a \<in> B \<Longrightarrow> open a"
```
```   381       "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
```
```   382     by (rule first_countable_basisE[of "snd x"]) blast
```
```   383   show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.
```
```   384     (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
```
```   385   proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
```
```   386     fix a b
```
```   387     assume x: "a \<in> A" "b \<in> B"
```
```   388     with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)"
```
```   389       unfolding mem_Times_iff
```
```   390       by (auto intro: open_Times)
```
```   391   next
```
```   392     fix S
```
```   393     assume "open S" "x \<in> S"
```
```   394     then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
```
```   395       by (rule open_prod_elim)
```
```   396     moreover
```
```   397     from a'b' A(4)[of a'] B(4)[of b']
```
```   398     obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
```
```   399       by auto
```
```   400     ultimately
```
```   401     show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
```
```   402       by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
```
```   403   qed (simp add: A B)
```
```   404 qed
```
```   405
```
```   406 class second_countable_topology = topological_space +
```
```   407   assumes ex_countable_subbasis:
```
```   408     "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
```
```   409 begin
```
```   410
```
```   411 lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
```
```   412 proof -
```
```   413   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
```
```   414     by blast
```
```   415   let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
```
```   416
```
```   417   show ?thesis
```
```   418   proof (intro exI conjI)
```
```   419     show "countable ?B"
```
```   420       by (intro countable_image countable_Collect_finite_subset B)
```
```   421     {
```
```   422       fix S
```
```   423       assume "open S"
```
```   424       then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
```
```   425         unfolding B
```
```   426       proof induct
```
```   427         case UNIV
```
```   428         show ?case by (intro exI[of _ "{{}}"]) simp
```
```   429       next
```
```   430         case (Int a b)
```
```   431         then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
```
```   432           and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
```
```   433           by blast
```
```   434         show ?case
```
```   435           unfolding x y Int_UN_distrib2
```
```   436           by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
```
```   437       next
```
```   438         case (UN K)
```
```   439         then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
```
```   440         then obtain k where
```
```   441             "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> UNION (k ka) Inter = ka"
```
```   442           unfolding bchoice_iff ..
```
```   443         then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
```
```   444           by (intro exI[of _ "UNION K k"]) auto
```
```   445       next
```
```   446         case (Basis S)
```
```   447         then show ?case
```
```   448           by (intro exI[of _ "{{S}}"]) auto
```
```   449       qed
```
```   450       then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
```
```   451         unfolding subset_image_iff by blast }
```
```   452     then show "topological_basis ?B"
```
```   453       unfolding topological_space_class.topological_basis_def
```
```   454       by (safe intro!: topological_space_class.open_Inter)
```
```   455          (simp_all add: B generate_topology.Basis subset_eq)
```
```   456   qed
```
```   457 qed
```
```   458
```
```   459 end
```
```   460
```
```   461 sublocale second_countable_topology <
```
```   462   countable_basis "SOME B. countable B \<and> topological_basis B"
```
```   463   using someI_ex[OF ex_countable_basis]
```
```   464   by unfold_locales safe
```
```   465
```
```   466 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
```
```   467 proof
```
```   468   obtain A :: "'a set set" where "countable A" "topological_basis A"
```
```   469     using ex_countable_basis by auto
```
```   470   moreover
```
```   471   obtain B :: "'b set set" where "countable B" "topological_basis B"
```
```   472     using ex_countable_basis by auto
```
```   473   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
```
```   474     by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
```
```   475       topological_basis_imp_subbasis)
```
```   476 qed
```
```   477
```
```   478 instance second_countable_topology \<subseteq> first_countable_topology
```
```   479 proof
```
```   480   fix x :: 'a
```
```   481   define B :: "'a set set" where "B = (SOME B. countable B \<and> topological_basis B)"
```
```   482   then have B: "countable B" "topological_basis B"
```
```   483     using countable_basis is_basis
```
```   484     by (auto simp: countable_basis is_basis)
```
```   485   then show "\<exists>A::nat \<Rightarrow> 'a set.
```
```   486     (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
```
```   487     by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
```
```   488        (fastforce simp: topological_space_class.topological_basis_def)+
```
```   489 qed
```
```   490
```
```   491 instance nat :: second_countable_topology
```
```   492 proof
```
```   493   show "\<exists>B::nat set set. countable B \<and> open = generate_topology B"
```
```   494     by (intro exI[of _ "range lessThan \<union> range greaterThan"]) (auto simp: open_nat_def)
```
```   495 qed
```
```   496
```
```   497 lemma countable_separating_set_linorder1:
```
```   498   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
```
```   499 proof -
```
```   500   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
```
```   501   define B1 where "B1 = {(LEAST x. x \<in> U)| U. U \<in> A}"
```
```   502   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
```
```   503   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
```
```   504   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
```
```   505   have "\<exists>b \<in> B1 \<union> B2. x < b \<and> b \<le> y" if "x < y" for x y
```
```   506   proof (cases)
```
```   507     assume "\<exists>z. x < z \<and> z < y"
```
```   508     then obtain z where z: "x < z \<and> z < y" by auto
```
```   509     define U where "U = {x<..<y}"
```
```   510     then have "open U" by simp
```
```   511     moreover have "z \<in> U" using z U_def by simp
```
```   512     ultimately obtain V where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
```
```   513     define w where "w = (SOME x. x \<in> V)"
```
```   514     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
```
```   515     then have "x < w \<and> w \<le> y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
```
```   516     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
```
```   517     ultimately show ?thesis by auto
```
```   518   next
```
```   519     assume "\<not>(\<exists>z. x < z \<and> z < y)"
```
```   520     then have *: "\<And>z. z > x \<Longrightarrow> z \<ge> y" by auto
```
```   521     define U where "U = {x<..}"
```
```   522     then have "open U" by simp
```
```   523     moreover have "y \<in> U" using \<open>x < y\<close> U_def by simp
```
```   524     ultimately obtain "V" where "V \<in> A" "y \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
```
```   525     have "U = {y..}" unfolding U_def using * \<open>x < y\<close> by auto
```
```   526     then have "V \<subseteq> {y..}" using \<open>V \<subseteq> U\<close> by simp
```
```   527     then have "(LEAST w. w \<in> V) = y" using \<open>y \<in> V\<close> by (meson Least_equality atLeast_iff subsetCE)
```
```   528     then have "y \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
```
```   529     moreover have "x < y \<and> y \<le> y" using \<open>x < y\<close> by simp
```
```   530     ultimately show ?thesis by auto
```
```   531   qed
```
```   532   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
```
```   533   ultimately show ?thesis by auto
```
```   534 qed
```
```   535
```
```   536 lemma countable_separating_set_linorder2:
```
```   537   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x \<le> b \<and> b < y))"
```
```   538 proof -
```
```   539   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
```
```   540   define B1 where "B1 = {(GREATEST x. x \<in> U) | U. U \<in> A}"
```
```   541   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
```
```   542   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
```
```   543   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
```
```   544   have "\<exists>b \<in> B1 \<union> B2. x \<le> b \<and> b < y" if "x < y" for x y
```
```   545   proof (cases)
```
```   546     assume "\<exists>z. x < z \<and> z < y"
```
```   547     then obtain z where z: "x < z \<and> z < y" by auto
```
```   548     define U where "U = {x<..<y}"
```
```   549     then have "open U" by simp
```
```   550     moreover have "z \<in> U" using z U_def by simp
```
```   551     ultimately obtain "V" where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
```
```   552     define w where "w = (SOME x. x \<in> V)"
```
```   553     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
```
```   554     then have "x \<le> w \<and> w < y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
```
```   555     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
```
```   556     ultimately show ?thesis by auto
```
```   557   next
```
```   558     assume "\<not>(\<exists>z. x < z \<and> z < y)"
```
```   559     then have *: "\<And>z. z < y \<Longrightarrow> z \<le> x" using leI by blast
```
```   560     define U where "U = {..<y}"
```
```   561     then have "open U" by simp
```
```   562     moreover have "x \<in> U" using \<open>x < y\<close> U_def by simp
```
```   563     ultimately obtain "V" where "V \<in> A" "x \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
```
```   564     have "U = {..x}" unfolding U_def using * \<open>x < y\<close> by auto
```
```   565     then have "V \<subseteq> {..x}" using \<open>V \<subseteq> U\<close> by simp
```
```   566     then have "(GREATEST x. x \<in> V) = x" using \<open>x \<in> V\<close> by (meson Greatest_equality atMost_iff subsetCE)
```
```   567     then have "x \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
```
```   568     moreover have "x \<le> x \<and> x < y" using \<open>x < y\<close> by simp
```
```   569     ultimately show ?thesis by auto
```
```   570   qed
```
```   571   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
```
```   572   ultimately show ?thesis by auto
```
```   573 qed
```
```   574
```
```   575 lemma countable_separating_set_dense_linorder:
```
```   576   shows "\<exists>B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b < y))"
```
```   577 proof -
```
```   578   obtain B::"'a set" where B: "countable B" "\<And>x y. x < y \<Longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y)"
```
```   579     using countable_separating_set_linorder1 by auto
```
```   580   have "\<exists>b \<in> B. x < b \<and> b < y" if "x < y" for x y
```
```   581   proof -
```
```   582     obtain z where "x < z" "z < y" using \<open>x < y\<close> dense by blast
```
```   583     then obtain b where "b \<in> B" "x < b \<and> b \<le> z" using B(2) by auto
```
```   584     then have "x < b \<and> b < y" using \<open>z < y\<close> by auto
```
```   585     then show ?thesis using \<open>b \<in> B\<close> by auto
```
```   586   qed
```
```   587   then show ?thesis using B(1) by auto
```
```   588 qed
```
```   589
```
```   590 subsection \<open>Polish spaces\<close>
```
```   591
```
```   592 text \<open>Textbooks define Polish spaces as completely metrizable.
```
```   593   We assume the topology to be complete for a given metric.\<close>
```
```   594
```
```   595 class polish_space = complete_space + second_countable_topology
```
```   596
```
```   597 subsection \<open>General notion of a topology as a value\<close>
```
```   598
```
```   599 definition "istopology L \<longleftrightarrow>
```
```   600   L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union>K))"
```
```   601
```
```   602 typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
```
```   603   morphisms "openin" "topology"
```
```   604   unfolding istopology_def by blast
```
```   605
```
```   606 lemma istopology_openin[intro]: "istopology(openin U)"
```
```   607   using openin[of U] by blast
```
```   608
```
```   609 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
```
```   610   using topology_inverse[unfolded mem_Collect_eq] .
```
```   611
```
```   612 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
```
```   613   using topology_inverse[of U] istopology_openin[of "topology U"] by auto
```
```   614
```
```   615 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
```
```   616 proof
```
```   617   assume "T1 = T2"
```
```   618   then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
```
```   619 next
```
```   620   assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
```
```   621   then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
```
```   622   then have "topology (openin T1) = topology (openin T2)" by simp
```
```   623   then show "T1 = T2" unfolding openin_inverse .
```
```   624 qed
```
```   625
```
```   626 text\<open>Infer the "universe" from union of all sets in the topology.\<close>
```
```   627
```
```   628 definition "topspace T = \<Union>{S. openin T S}"
```
```   629
```
```   630 subsubsection \<open>Main properties of open sets\<close>
```
```   631
```
```   632 lemma openin_clauses:
```
```   633   fixes U :: "'a topology"
```
```   634   shows
```
```   635     "openin U {}"
```
```   636     "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
```
```   637     "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
```
```   638   using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
```
```   639
```
```   640 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
```
```   641   unfolding topspace_def by blast
```
```   642
```
```   643 lemma openin_empty[simp]: "openin U {}"
```
```   644   by (rule openin_clauses)
```
```   645
```
```   646 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
```
```   647   by (rule openin_clauses)
```
```   648
```
```   649 lemma openin_Union[intro]: "(\<And>S. S \<in> K \<Longrightarrow> openin U S) \<Longrightarrow> openin U (\<Union>K)"
```
```   650   using openin_clauses by blast
```
```   651
```
```   652 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
```
```   653   using openin_Union[of "{S,T}" U] by auto
```
```   654
```
```   655 lemma openin_topspace[intro, simp]: "openin U (topspace U)"
```
```   656   by (force simp add: openin_Union topspace_def)
```
```   657
```
```   658 lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
```
```   659   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   660 proof
```
```   661   assume ?lhs
```
```   662   then show ?rhs by auto
```
```   663 next
```
```   664   assume H: ?rhs
```
```   665   let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
```
```   666   have "openin U ?t" by (force simp add: openin_Union)
```
```   667   also have "?t = S" using H by auto
```
```   668   finally show "openin U S" .
```
```   669 qed
```
```   670
```
```   671 lemma openin_INT [intro]:
```
```   672   assumes "finite I"
```
```   673           "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
```
```   674   shows "openin T ((\<Inter>i \<in> I. U i) \<inter> topspace T)"
```
```   675 using assms by (induct, auto simp add: inf_sup_aci(2) openin_Int)
```
```   676
```
```   677 lemma openin_INT2 [intro]:
```
```   678   assumes "finite I" "I \<noteq> {}"
```
```   679           "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
```
```   680   shows "openin T (\<Inter>i \<in> I. U i)"
```
```   681 proof -
```
```   682   have "(\<Inter>i \<in> I. U i) \<subseteq> topspace T"
```
```   683     using \<open>I \<noteq> {}\<close> openin_subset[OF assms(3)] by auto
```
```   684   then show ?thesis
```
```   685     using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
```
```   686 qed
```
```   687
```
```   688
```
```   689 subsubsection \<open>Closed sets\<close>
```
```   690
```
```   691 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
```
```   692
```
```   693 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
```
```   694   by (metis closedin_def)
```
```   695
```
```   696 lemma closedin_empty[simp]: "closedin U {}"
```
```   697   by (simp add: closedin_def)
```
```   698
```
```   699 lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
```
```   700   by (simp add: closedin_def)
```
```   701
```
```   702 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
```
```   703   by (auto simp add: Diff_Un closedin_def)
```
```   704
```
```   705 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
```
```   706   by auto
```
```   707
```
```   708 lemma closedin_Union:
```
```   709   assumes "finite S" "\<And>T. T \<in> S \<Longrightarrow> closedin U T"
```
```   710     shows "closedin U (\<Union>S)"
```
```   711   using assms by induction auto
```
```   712
```
```   713 lemma closedin_Inter[intro]:
```
```   714   assumes Ke: "K \<noteq> {}"
```
```   715     and Kc: "\<And>S. S \<in>K \<Longrightarrow> closedin U S"
```
```   716   shows "closedin U (\<Inter>K)"
```
```   717   using Ke Kc unfolding closedin_def Diff_Inter by auto
```
```   718
```
```   719 lemma closedin_INT[intro]:
```
```   720   assumes "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)"
```
```   721   shows "closedin U (\<Inter>x\<in>A. B x)"
```
```   722   apply (rule closedin_Inter)
```
```   723   using assms
```
```   724   apply auto
```
```   725   done
```
```   726
```
```   727 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
```
```   728   using closedin_Inter[of "{S,T}" U] by auto
```
```   729
```
```   730 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
```
```   731   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
```
```   732   apply (metis openin_subset subset_eq)
```
```   733   done
```
```   734
```
```   735 lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
```
```   736   by (simp add: openin_closedin_eq)
```
```   737
```
```   738 lemma openin_diff[intro]:
```
```   739   assumes oS: "openin U S"
```
```   740     and cT: "closedin U T"
```
```   741   shows "openin U (S - T)"
```
```   742 proof -
```
```   743   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
```
```   744     by (auto simp add: topspace_def openin_subset)
```
```   745   then show ?thesis using oS cT
```
```   746     by (auto simp add: closedin_def)
```
```   747 qed
```
```   748
```
```   749 lemma closedin_diff[intro]:
```
```   750   assumes oS: "closedin U S"
```
```   751     and cT: "openin U T"
```
```   752   shows "closedin U (S - T)"
```
```   753 proof -
```
```   754   have "S - T = S \<inter> (topspace U - T)"
```
```   755     using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)
```
```   756   then show ?thesis
```
```   757     using oS cT by (auto simp add: openin_closedin_eq)
```
```   758 qed
```
```   759
```
```   760
```
```   761 subsubsection \<open>Subspace topology\<close>
```
```   762
```
```   763 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
```
```   764
```
```   765 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
```
```   766   (is "istopology ?L")
```
```   767 proof -
```
```   768   have "?L {}" by blast
```
```   769   {
```
```   770     fix A B
```
```   771     assume A: "?L A" and B: "?L B"
```
```   772     from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V"
```
```   773       by blast
```
```   774     have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
```
```   775       using Sa Sb by blast+
```
```   776     then have "?L (A \<inter> B)" by blast
```
```   777   }
```
```   778   moreover
```
```   779   {
```
```   780     fix K
```
```   781     assume K: "K \<subseteq> Collect ?L"
```
```   782     have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
```
```   783       by blast
```
```   784     from K[unfolded th0 subset_image_iff]
```
```   785     obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
```
```   786       by blast
```
```   787     have "\<Union>K = (\<Union>Sk) \<inter> V"
```
```   788       using Sk by auto
```
```   789     moreover have "openin U (\<Union>Sk)"
```
```   790       using Sk by (auto simp add: subset_eq)
```
```   791     ultimately have "?L (\<Union>K)" by blast
```
```   792   }
```
```   793   ultimately show ?thesis
```
```   794     unfolding subset_eq mem_Collect_eq istopology_def by auto
```
```   795 qed
```
```   796
```
```   797 lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
```
```   798   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
```
```   799   by auto
```
```   800
```
```   801 lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
```
```   802   by (auto simp add: topspace_def openin_subtopology)
```
```   803
```
```   804 lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
```
```   805   unfolding closedin_def topspace_subtopology
```
```   806   by (auto simp add: openin_subtopology)
```
```   807
```
```   808 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
```
```   809   unfolding openin_subtopology
```
```   810   by auto (metis IntD1 in_mono openin_subset)
```
```   811
```
```   812 lemma subtopology_superset:
```
```   813   assumes UV: "topspace U \<subseteq> V"
```
```   814   shows "subtopology U V = U"
```
```   815 proof -
```
```   816   {
```
```   817     fix S
```
```   818     {
```
```   819       fix T
```
```   820       assume T: "openin U T" "S = T \<inter> V"
```
```   821       from T openin_subset[OF T(1)] UV have eq: "S = T"
```
```   822         by blast
```
```   823       have "openin U S"
```
```   824         unfolding eq using T by blast
```
```   825     }
```
```   826     moreover
```
```   827     {
```
```   828       assume S: "openin U S"
```
```   829       then have "\<exists>T. openin U T \<and> S = T \<inter> V"
```
```   830         using openin_subset[OF S] UV by auto
```
```   831     }
```
```   832     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
```
```   833       by blast
```
```   834   }
```
```   835   then show ?thesis
```
```   836     unfolding topology_eq openin_subtopology by blast
```
```   837 qed
```
```   838
```
```   839 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
```
```   840   by (simp add: subtopology_superset)
```
```   841
```
```   842 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
```
```   843   by (simp add: subtopology_superset)
```
```   844
```
```   845 lemma openin_subtopology_empty:
```
```   846    "openin (subtopology U {}) S \<longleftrightarrow> S = {}"
```
```   847 by (metis Int_empty_right openin_empty openin_subtopology)
```
```   848
```
```   849 lemma closedin_subtopology_empty:
```
```   850    "closedin (subtopology U {}) S \<longleftrightarrow> S = {}"
```
```   851 by (metis Int_empty_right closedin_empty closedin_subtopology)
```
```   852
```
```   853 lemma closedin_subtopology_refl [simp]:
```
```   854    "closedin (subtopology U X) X \<longleftrightarrow> X \<subseteq> topspace U"
```
```   855 by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)
```
```   856
```
```   857 lemma openin_imp_subset:
```
```   858    "openin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
```
```   859 by (metis Int_iff openin_subtopology subsetI)
```
```   860
```
```   861 lemma closedin_imp_subset:
```
```   862    "closedin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
```
```   863 by (simp add: closedin_def topspace_subtopology)
```
```   864
```
```   865 lemma openin_subtopology_Un:
```
```   866     "openin (subtopology U T) S \<and> openin (subtopology U u) S
```
```   867      \<Longrightarrow> openin (subtopology U (T \<union> u)) S"
```
```   868 by (simp add: openin_subtopology) blast
```
```   869
```
```   870
```
```   871 subsubsection \<open>The standard Euclidean topology\<close>
```
```   872
```
```   873 definition euclidean :: "'a::topological_space topology"
```
```   874   where "euclidean = topology open"
```
```   875
```
```   876 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
```
```   877   unfolding euclidean_def
```
```   878   apply (rule cong[where x=S and y=S])
```
```   879   apply (rule topology_inverse[symmetric])
```
```   880   apply (auto simp add: istopology_def)
```
```   881   done
```
```   882
```
```   883 declare open_openin [symmetric, simp]
```
```   884
```
```   885 lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
```
```   886   by (force simp add: topspace_def)
```
```   887
```
```   888 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
```
```   889   by (simp add: topspace_subtopology)
```
```   890
```
```   891 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
```
```   892   by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV)
```
```   893
```
```   894 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
```
```   895   using openI by auto
```
```   896
```
```   897 lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
```
```   898   by (metis openin_topspace topspace_euclidean_subtopology)
```
```   899
```
```   900 text \<open>Basic "localization" results are handy for connectedness.\<close>
```
```   901
```
```   902 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
```
```   903   by (auto simp add: openin_subtopology)
```
```   904
```
```   905 lemma openin_Int_open:
```
```   906    "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
```
```   907         \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
```
```   908 by (metis open_Int Int_assoc openin_open)
```
```   909
```
```   910 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
```
```   911   by (auto simp add: openin_open)
```
```   912
```
```   913 lemma open_openin_trans[trans]:
```
```   914   "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
```
```   915   by (metis Int_absorb1  openin_open_Int)
```
```   916
```
```   917 lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
```
```   918   by (auto simp add: openin_open)
```
```   919
```
```   920 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
```
```   921   by (simp add: closedin_subtopology closed_closedin Int_ac)
```
```   922
```
```   923 lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
```
```   924   by (metis closedin_closed)
```
```   925
```
```   926 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
```
```   927   by (auto simp add: closedin_closed)
```
```   928
```
```   929 lemma closedin_closed_subset:
```
```   930  "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
```
```   931              \<Longrightarrow> closedin (subtopology euclidean T) S"
```
```   932   by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
```
```   933
```
```   934 lemma finite_imp_closedin:
```
```   935   fixes S :: "'a::t1_space set"
```
```   936   shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
```
```   937     by (simp add: finite_imp_closed closed_subset)
```
```   938
```
```   939 lemma closedin_singleton [simp]:
```
```   940   fixes a :: "'a::t1_space"
```
```   941   shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
```
```   942 using closedin_subset  by (force intro: closed_subset)
```
```   943
```
```   944 lemma openin_euclidean_subtopology_iff:
```
```   945   fixes S U :: "'a::metric_space set"
```
```   946   shows "openin (subtopology euclidean U) S \<longleftrightarrow>
```
```   947     S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"
```
```   948   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   949 proof
```
```   950   assume ?lhs
```
```   951   then show ?rhs
```
```   952     unfolding openin_open open_dist by blast
```
```   953 next
```
```   954   define T where "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
```
```   955   have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
```
```   956     unfolding T_def
```
```   957     apply clarsimp
```
```   958     apply (rule_tac x="d - dist x a" in exI)
```
```   959     apply (clarsimp simp add: less_diff_eq)
```
```   960     by (metis dist_commute dist_triangle_lt)
```
```   961   assume ?rhs then have 2: "S = U \<inter> T"
```
```   962     unfolding T_def
```
```   963     by auto (metis dist_self)
```
```   964   from 1 2 show ?lhs
```
```   965     unfolding openin_open open_dist by fast
```
```   966 qed
```
```   967
```
```   968 lemma connected_openin:
```
```   969       "connected s \<longleftrightarrow>
```
```   970        ~(\<exists>e1 e2. openin (subtopology euclidean s) e1 \<and>
```
```   971                  openin (subtopology euclidean s) e2 \<and>
```
```   972                  s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and> e1 \<noteq> {} \<and> e2 \<noteq> {})"
```
```   973   apply (simp add: connected_def openin_open, safe)
```
```   974   apply (simp_all, blast+)  (* SLOW *)
```
```   975   done
```
```   976
```
```   977 lemma connected_openin_eq:
```
```   978       "connected s \<longleftrightarrow>
```
```   979        ~(\<exists>e1 e2. openin (subtopology euclidean s) e1 \<and>
```
```   980                  openin (subtopology euclidean s) e2 \<and>
```
```   981                  e1 \<union> e2 = s \<and> e1 \<inter> e2 = {} \<and>
```
```   982                  e1 \<noteq> {} \<and> e2 \<noteq> {})"
```
```   983   apply (simp add: connected_openin, safe)
```
```   984   apply blast
```
```   985   by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)
```
```   986
```
```   987 lemma connected_closedin:
```
```   988       "connected s \<longleftrightarrow>
```
```   989        ~(\<exists>e1 e2.
```
```   990              closedin (subtopology euclidean s) e1 \<and>
```
```   991              closedin (subtopology euclidean s) e2 \<and>
```
```   992              s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and>
```
```   993              e1 \<noteq> {} \<and> e2 \<noteq> {})"
```
```   994 proof -
```
```   995   { fix A B x x'
```
```   996     assume s_sub: "s \<subseteq> A \<union> B"
```
```   997        and disj: "A \<inter> B \<inter> s = {}"
```
```   998        and x: "x \<in> s" "x \<in> B" and x': "x' \<in> s" "x' \<in> A"
```
```   999        and cl: "closed A" "closed B"
```
```  1000     assume "\<forall>e1. (\<forall>T. closed T \<longrightarrow> e1 \<noteq> s \<inter> T) \<or> (\<forall>e2. e1 \<inter> e2 = {} \<longrightarrow> s \<subseteq> e1 \<union> e2 \<longrightarrow> (\<forall>T. closed T \<longrightarrow> e2 \<noteq> s \<inter> T) \<or> e1 = {} \<or> e2 = {})"
```
```  1001     then have "\<And>C D. s \<inter> C = {} \<or> s \<inter> D = {} \<or> s \<inter> (C \<inter> (s \<inter> D)) \<noteq> {} \<or> \<not> s \<subseteq> s \<inter> (C \<union> D) \<or> \<not> closed C \<or> \<not> closed D"
```
```  1002       by (metis (no_types) Int_Un_distrib Int_assoc)
```
```  1003     moreover have "s \<inter> (A \<inter> B) = {}" "s \<inter> (A \<union> B) = s" "s \<inter> B \<noteq> {}"
```
```  1004       using disj s_sub x by blast+
```
```  1005     ultimately have "s \<inter> A = {}"
```
```  1006       using cl by (metis inf.left_commute inf_bot_right order_refl)
```
```  1007     then have False
```
```  1008       using x' by blast
```
```  1009   } note * = this
```
```  1010   show ?thesis
```
```  1011     apply (simp add: connected_closed closedin_closed)
```
```  1012     apply (safe; simp)
```
```  1013     apply blast
```
```  1014     apply (blast intro: *)
```
```  1015     done
```
```  1016 qed
```
```  1017
```
```  1018 lemma connected_closedin_eq:
```
```  1019       "connected s \<longleftrightarrow>
```
```  1020            ~(\<exists>e1 e2.
```
```  1021                  closedin (subtopology euclidean s) e1 \<and>
```
```  1022                  closedin (subtopology euclidean s) e2 \<and>
```
```  1023                  e1 \<union> e2 = s \<and> e1 \<inter> e2 = {} \<and>
```
```  1024                  e1 \<noteq> {} \<and> e2 \<noteq> {})"
```
```  1025   apply (simp add: connected_closedin, safe)
```
```  1026   apply blast
```
```  1027   by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
```
```  1028
```
```  1029 text \<open>These "transitivity" results are handy too\<close>
```
```  1030
```
```  1031 lemma openin_trans[trans]:
```
```  1032   "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
```
```  1033     openin (subtopology euclidean U) S"
```
```  1034   unfolding open_openin openin_open by blast
```
```  1035
```
```  1036 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
```
```  1037   by (auto simp add: openin_open intro: openin_trans)
```
```  1038
```
```  1039 lemma closedin_trans[trans]:
```
```  1040   "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
```
```  1041     closedin (subtopology euclidean U) S"
```
```  1042   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
```
```  1043
```
```  1044 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
```
```  1045   by (auto simp add: closedin_closed intro: closedin_trans)
```
```  1046
```
```  1047 lemma openin_subtopology_Int_subset:
```
```  1048    "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
```
```  1049   by (auto simp: openin_subtopology)
```
```  1050
```
```  1051 lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
```
```  1052   using open_subset openin_open_trans openin_subset by fastforce
```
```  1053
```
```  1054
```
```  1055 subsection \<open>Open and closed balls\<close>
```
```  1056
```
```  1057 definition ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
```
```  1058   where "ball x e = {y. dist x y < e}"
```
```  1059
```
```  1060 definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
```
```  1061   where "cball x e = {y. dist x y \<le> e}"
```
```  1062
```
```  1063 definition sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
```
```  1064   where "sphere x e = {y. dist x y = e}"
```
```  1065
```
```  1066 lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
```
```  1067   by (simp add: ball_def)
```
```  1068
```
```  1069 lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
```
```  1070   by (simp add: cball_def)
```
```  1071
```
```  1072 lemma mem_sphere [simp]: "y \<in> sphere x e \<longleftrightarrow> dist x y = e"
```
```  1073   by (simp add: sphere_def)
```
```  1074
```
```  1075 lemma ball_trivial [simp]: "ball x 0 = {}"
```
```  1076   by (simp add: ball_def)
```
```  1077
```
```  1078 lemma cball_trivial [simp]: "cball x 0 = {x}"
```
```  1079   by (simp add: cball_def)
```
```  1080
```
```  1081 lemma sphere_trivial [simp]: "sphere x 0 = {x}"
```
```  1082   by (simp add: sphere_def)
```
```  1083
```
```  1084 lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
```
```  1085   for x :: "'a::real_normed_vector"
```
```  1086   by (simp add: dist_norm)
```
```  1087
```
```  1088 lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
```
```  1089   for x :: "'a::real_normed_vector"
```
```  1090   by (simp add: dist_norm)
```
```  1091
```
```  1092 lemma disjoint_ballI: "dist x y \<ge> r+s \<Longrightarrow> ball x r \<inter> ball y s = {}"
```
```  1093   using dist_triangle_less_add not_le by fastforce
```
```  1094
```
```  1095 lemma disjoint_cballI: "dist x y > r + s \<Longrightarrow> cball x r \<inter> cball y s = {}"
```
```  1096   by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball)
```
```  1097
```
```  1098 lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
```
```  1099   for x :: "'a::real_normed_vector"
```
```  1100   by (simp add: dist_norm)
```
```  1101
```
```  1102 lemma sphere_empty [simp]: "r < 0 \<Longrightarrow> sphere a r = {}"
```
```  1103   for a :: "'a::metric_space"
```
```  1104   by auto
```
```  1105
```
```  1106 lemma centre_in_ball [simp]: "x \<in> ball x e \<longleftrightarrow> 0 < e"
```
```  1107   by simp
```
```  1108
```
```  1109 lemma centre_in_cball [simp]: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
```
```  1110   by simp
```
```  1111
```
```  1112 lemma ball_subset_cball [simp, intro]: "ball x e \<subseteq> cball x e"
```
```  1113   by (simp add: subset_eq)
```
```  1114
```
```  1115 lemma sphere_cball [simp,intro]: "sphere z r \<subseteq> cball z r"
```
```  1116   by force
```
```  1117
```
```  1118 lemma cball_diff_sphere: "cball a r - sphere a r = ball a r"
```
```  1119   by auto
```
```  1120
```
```  1121 lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
```
```  1122   by (simp add: subset_eq)
```
```  1123
```
```  1124 lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
```
```  1125   by (simp add: subset_eq)
```
```  1126
```
```  1127 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
```
```  1128   by (simp add: set_eq_iff) arith
```
```  1129
```
```  1130 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
```
```  1131   by (simp add: set_eq_iff)
```
```  1132
```
```  1133 lemma cball_max_Un: "cball a (max r s) = cball a r \<union> cball a s"
```
```  1134   by (simp add: set_eq_iff) arith
```
```  1135
```
```  1136 lemma cball_min_Int: "cball a (min r s) = cball a r \<inter> cball a s"
```
```  1137   by (simp add: set_eq_iff)
```
```  1138
```
```  1139 lemma cball_diff_eq_sphere: "cball a r - ball a r =  sphere a r"
```
```  1140   by (auto simp: cball_def ball_def dist_commute)
```
```  1141
```
```  1142 lemma image_add_ball [simp]:
```
```  1143   fixes a :: "'a::real_normed_vector"
```
```  1144   shows "op + b ` ball a r = ball (a+b) r"
```
```  1145 apply (intro equalityI subsetI)
```
```  1146 apply (force simp: dist_norm)
```
```  1147 apply (rule_tac x="x-b" in image_eqI)
```
```  1148 apply (auto simp: dist_norm algebra_simps)
```
```  1149 done
```
```  1150
```
```  1151 lemma image_add_cball [simp]:
```
```  1152   fixes a :: "'a::real_normed_vector"
```
```  1153   shows "op + b ` cball a r = cball (a+b) r"
```
```  1154 apply (intro equalityI subsetI)
```
```  1155 apply (force simp: dist_norm)
```
```  1156 apply (rule_tac x="x-b" in image_eqI)
```
```  1157 apply (auto simp: dist_norm algebra_simps)
```
```  1158 done
```
```  1159
```
```  1160 lemma open_ball [intro, simp]: "open (ball x e)"
```
```  1161 proof -
```
```  1162   have "open (dist x -` {..<e})"
```
```  1163     by (intro open_vimage open_lessThan continuous_intros)
```
```  1164   also have "dist x -` {..<e} = ball x e"
```
```  1165     by auto
```
```  1166   finally show ?thesis .
```
```  1167 qed
```
```  1168
```
```  1169 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
```
```  1170   by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute)
```
```  1171
```
```  1172 lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S) \<Longrightarrow> open S"
```
```  1173   by (auto simp: open_contains_ball)
```
```  1174
```
```  1175 lemma openE[elim?]:
```
```  1176   assumes "open S" "x\<in>S"
```
```  1177   obtains e where "e>0" "ball x e \<subseteq> S"
```
```  1178   using assms unfolding open_contains_ball by auto
```
```  1179
```
```  1180 lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
```
```  1181   by (metis open_contains_ball subset_eq centre_in_ball)
```
```  1182
```
```  1183 lemma openin_contains_ball:
```
```  1184     "openin (subtopology euclidean t) s \<longleftrightarrow>
```
```  1185      s \<subseteq> t \<and> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> ball x e \<inter> t \<subseteq> s)"
```
```  1186     (is "?lhs = ?rhs")
```
```  1187 proof
```
```  1188   assume ?lhs
```
```  1189   then show ?rhs
```
```  1190     apply (simp add: openin_open)
```
```  1191     apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
```
```  1192     done
```
```  1193 next
```
```  1194   assume ?rhs
```
```  1195   then show ?lhs
```
```  1196     apply (simp add: openin_euclidean_subtopology_iff)
```
```  1197     by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball)
```
```  1198 qed
```
```  1199
```
```  1200 lemma openin_contains_cball:
```
```  1201    "openin (subtopology euclidean t) s \<longleftrightarrow>
```
```  1202         s \<subseteq> t \<and>
```
```  1203         (\<forall>x \<in> s. \<exists>e. 0 < e \<and> cball x e \<inter> t \<subseteq> s)"
```
```  1204 apply (simp add: openin_contains_ball)
```
```  1205 apply (rule iffI)
```
```  1206 apply (auto dest!: bspec)
```
```  1207 apply (rule_tac x="e/2" in exI)
```
```  1208 apply force+
```
```  1209 done
```
```  1210
```
```  1211 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
```
```  1212   unfolding mem_ball set_eq_iff
```
```  1213   apply (simp add: not_less)
```
```  1214   apply (metis zero_le_dist order_trans dist_self)
```
```  1215   done
```
```  1216
```
```  1217 lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
```
```  1218
```
```  1219 lemma euclidean_dist_l2:
```
```  1220   fixes x y :: "'a :: euclidean_space"
```
```  1221   shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
```
```  1222   unfolding dist_norm norm_eq_sqrt_inner setL2_def
```
```  1223   by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
```
```  1224
```
```  1225 lemma eventually_nhds_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z d) (nhds z)"
```
```  1226   by (rule eventually_nhds_in_open) simp_all
```
```  1227
```
```  1228 lemma eventually_at_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<in> A) (at z within A)"
```
```  1229   unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
```
```  1230
```
```  1231 lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
```
```  1232   unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
```
```  1233
```
```  1234
```
```  1235 subsection \<open>Boxes\<close>
```
```  1236
```
```  1237 abbreviation One :: "'a::euclidean_space"
```
```  1238   where "One \<equiv> \<Sum>Basis"
```
```  1239
```
```  1240 lemma One_non_0: assumes "One = (0::'a::euclidean_space)" shows False
```
```  1241 proof -
```
```  1242   have "dependent (Basis :: 'a set)"
```
```  1243     apply (simp add: dependent_finite)
```
```  1244     apply (rule_tac x="\<lambda>i. 1" in exI)
```
```  1245     using SOME_Basis apply (auto simp: assms)
```
```  1246     done
```
```  1247   with independent_Basis show False by force
```
```  1248 qed
```
```  1249
```
```  1250 corollary One_neq_0[iff]: "One \<noteq> 0"
```
```  1251   by (metis One_non_0)
```
```  1252
```
```  1253 corollary Zero_neq_One[iff]: "0 \<noteq> One"
```
```  1254   by (metis One_non_0)
```
```  1255
```
```  1256 definition (in euclidean_space) eucl_less (infix "<e" 50)
```
```  1257   where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
```
```  1258
```
```  1259 definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"
```
```  1260 definition "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
```
```  1261
```
```  1262 lemma box_def: "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
```
```  1263   and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"
```
```  1264   and mem_box: "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"
```
```  1265     "x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
```
```  1266   by (auto simp: box_eucl_less eucl_less_def cbox_def)
```
```  1267
```
```  1268 lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b \<times> cbox c d"
```
```  1269   by (force simp: cbox_def Basis_prod_def)
```
```  1270
```
```  1271 lemma cbox_Pair_iff [iff]: "(x, y) \<in> cbox (a, c) (b, d) \<longleftrightarrow> x \<in> cbox a b \<and> y \<in> cbox c d"
```
```  1272   by (force simp: cbox_Pair_eq)
```
```  1273
```
```  1274 lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} \<longleftrightarrow> cbox a b = {} \<or> cbox c d = {}"
```
```  1275   by (force simp: cbox_Pair_eq)
```
```  1276
```
```  1277 lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)"
```
```  1278   by auto
```
```  1279
```
```  1280 lemma mem_box_real[simp]:
```
```  1281   "(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b"
```
```  1282   "(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b"
```
```  1283   by (auto simp: mem_box)
```
```  1284
```
```  1285 lemma box_real[simp]:
```
```  1286   fixes a b:: real
```
```  1287   shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
```
```  1288   by auto
```
```  1289
```
```  1290 lemma box_Int_box:
```
```  1291   fixes a :: "'a::euclidean_space"
```
```  1292   shows "box a b \<inter> box c d =
```
```  1293     box (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
```
```  1294   unfolding set_eq_iff and Int_iff and mem_box by auto
```
```  1295
```
```  1296 lemma rational_boxes:
```
```  1297   fixes x :: "'a::euclidean_space"
```
```  1298   assumes "e > 0"
```
```  1299   shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
```
```  1300 proof -
```
```  1301   define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
```
```  1302   then have e: "e' > 0"
```
```  1303     using assms by (auto simp: DIM_positive)
```
```  1304   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
```
```  1305   proof
```
```  1306     fix i
```
```  1307     from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
```
```  1308     show "?th i" by auto
```
```  1309   qed
```
```  1310   from choice[OF this] obtain a where
```
```  1311     a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'" ..
```
```  1312   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
```
```  1313   proof
```
```  1314     fix i
```
```  1315     from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
```
```  1316     show "?th i" by auto
```
```  1317   qed
```
```  1318   from choice[OF this] obtain b where
```
```  1319     b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'" ..
```
```  1320   let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
```
```  1321   show ?thesis
```
```  1322   proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
```
```  1323     fix y :: 'a
```
```  1324     assume *: "y \<in> box ?a ?b"
```
```  1325     have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
```
```  1326       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
```
```  1327     also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
```
```  1328     proof (rule real_sqrt_less_mono, rule sum_strict_mono)
```
```  1329       fix i :: "'a"
```
```  1330       assume i: "i \<in> Basis"
```
```  1331       have "a i < y\<bullet>i \<and> y\<bullet>i < b i"
```
```  1332         using * i by (auto simp: box_def)
```
```  1333       moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
```
```  1334         using a by auto
```
```  1335       moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
```
```  1336         using b by auto
```
```  1337       ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
```
```  1338         by auto
```
```  1339       then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
```
```  1340         unfolding e'_def by (auto simp: dist_real_def)
```
```  1341       then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
```
```  1342         by (rule power_strict_mono) auto
```
```  1343       then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
```
```  1344         by (simp add: power_divide)
```
```  1345     qed auto
```
```  1346     also have "\<dots> = e"
```
```  1347       using \<open>0 < e\<close> by simp
```
```  1348     finally show "y \<in> ball x e"
```
```  1349       by (auto simp: ball_def)
```
```  1350   qed (insert a b, auto simp: box_def)
```
```  1351 qed
```
```  1352
```
```  1353 lemma open_UNION_box:
```
```  1354   fixes M :: "'a::euclidean_space set"
```
```  1355   assumes "open M"
```
```  1356   defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
```
```  1357   defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
```
```  1358   defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
```
```  1359   shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
```
```  1360 proof -
```
```  1361   have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))" if "x \<in> M" for x
```
```  1362   proof -
```
```  1363     obtain e where e: "e > 0" "ball x e \<subseteq> M"
```
```  1364       using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
```
```  1365     moreover obtain a b where ab:
```
```  1366       "x \<in> box a b"
```
```  1367       "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
```
```  1368       "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"
```
```  1369       "box a b \<subseteq> ball x e"
```
```  1370       using rational_boxes[OF e(1)] by metis
```
```  1371     ultimately show ?thesis
```
```  1372        by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
```
```  1373           (auto simp: euclidean_representation I_def a'_def b'_def)
```
```  1374   qed
```
```  1375   then show ?thesis by (auto simp: I_def)
```
```  1376 qed
```
```  1377
```
```  1378 lemma box_eq_empty:
```
```  1379   fixes a :: "'a::euclidean_space"
```
```  1380   shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
```
```  1381     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
```
```  1382 proof -
```
```  1383   {
```
```  1384     fix i x
```
```  1385     assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>box a b"
```
```  1386     then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"
```
```  1387       unfolding mem_box by (auto simp: box_def)
```
```  1388     then have "a\<bullet>i < b\<bullet>i" by auto
```
```  1389     then have False using as by auto
```
```  1390   }
```
```  1391   moreover
```
```  1392   {
```
```  1393     assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
```
```  1394     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1395     {
```
```  1396       fix i :: 'a
```
```  1397       assume i: "i \<in> Basis"
```
```  1398       have "a\<bullet>i < b\<bullet>i"
```
```  1399         using as[THEN bspec[where x=i]] i by auto
```
```  1400       then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
```
```  1401         by (auto simp: inner_add_left)
```
```  1402     }
```
```  1403     then have "box a b \<noteq> {}"
```
```  1404       using mem_box(1)[of "?x" a b] by auto
```
```  1405   }
```
```  1406   ultimately show ?th1 by blast
```
```  1407
```
```  1408   {
```
```  1409     fix i x
```
```  1410     assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>cbox a b"
```
```  1411     then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
```
```  1412       unfolding mem_box by auto
```
```  1413     then have "a\<bullet>i \<le> b\<bullet>i" by auto
```
```  1414     then have False using as by auto
```
```  1415   }
```
```  1416   moreover
```
```  1417   {
```
```  1418     assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
```
```  1419     let ?x = "(1/2) *\<^sub>R (a + b)"
```
```  1420     {
```
```  1421       fix i :: 'a
```
```  1422       assume i:"i \<in> Basis"
```
```  1423       have "a\<bullet>i \<le> b\<bullet>i"
```
```  1424         using as[THEN bspec[where x=i]] i by auto
```
```  1425       then have "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
```
```  1426         by (auto simp: inner_add_left)
```
```  1427     }
```
```  1428     then have "cbox a b \<noteq> {}"
```
```  1429       using mem_box(2)[of "?x" a b] by auto
```
```  1430   }
```
```  1431   ultimately show ?th2 by blast
```
```  1432 qed
```
```  1433
```
```  1434 lemma box_ne_empty:
```
```  1435   fixes a :: "'a::euclidean_space"
```
```  1436   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"
```
```  1437   and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
```
```  1438   unfolding box_eq_empty[of a b] by fastforce+
```
```  1439
```
```  1440 lemma
```
```  1441   fixes a :: "'a::euclidean_space"
```
```  1442   shows cbox_sing: "cbox a a = {a}"
```
```  1443     and box_sing: "box a a = {}"
```
```  1444   unfolding set_eq_iff mem_box eq_iff [symmetric]
```
```  1445   by (auto intro!: euclidean_eqI[where 'a='a])
```
```  1446      (metis all_not_in_conv nonempty_Basis)
```
```  1447
```
```  1448 lemma subset_box_imp:
```
```  1449   fixes a :: "'a::euclidean_space"
```
```  1450   shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
```
```  1451     and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> box a b"
```
```  1452     and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> cbox a b"
```
```  1453      and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> box a b"
```
```  1454   unfolding subset_eq[unfolded Ball_def] unfolding mem_box
```
```  1455   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
```
```  1456
```
```  1457 lemma box_subset_cbox:
```
```  1458   fixes a :: "'a::euclidean_space"
```
```  1459   shows "box a b \<subseteq> cbox a b"
```
```  1460   unfolding subset_eq [unfolded Ball_def] mem_box
```
```  1461   by (fast intro: less_imp_le)
```
```  1462
```
```  1463 lemma subset_box:
```
```  1464   fixes a :: "'a::euclidean_space"
```
```  1465   shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1)
```
```  1466     and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2)
```
```  1467     and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3)
```
```  1468     and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) \<longrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
```
```  1469 proof -
```
```  1470   show ?th1
```
```  1471     unfolding subset_eq and Ball_def and mem_box
```
```  1472     by (auto intro: order_trans)
```
```  1473   show ?th2
```
```  1474     unfolding subset_eq and Ball_def and mem_box
```
```  1475     by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
```
```  1476   {
```
```  1477     assume as: "box c d \<subseteq> cbox a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
```
```  1478     then have "box c d \<noteq> {}"
```
```  1479       unfolding box_eq_empty by auto
```
```  1480     fix i :: 'a
```
```  1481     assume i: "i \<in> Basis"
```
```  1482     (** TODO combine the following two parts as done in the HOL_light version. **)
```
```  1483     {
```
```  1484       let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
```
```  1485       assume as2: "a\<bullet>i > c\<bullet>i"
```
```  1486       {
```
```  1487         fix j :: 'a
```
```  1488         assume j: "j \<in> Basis"
```
```  1489         then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
```
```  1490           apply (cases "j = i")
```
```  1491           using as(2)[THEN bspec[where x=j]] i
```
```  1492           apply (auto simp add: as2)
```
```  1493           done
```
```  1494       }
```
```  1495       then have "?x\<in>box c d"
```
```  1496         using i unfolding mem_box by auto
```
```  1497       moreover
```
```  1498       have "?x \<notin> cbox a b"
```
```  1499         unfolding mem_box
```
```  1500         apply auto
```
```  1501         apply (rule_tac x=i in bexI)
```
```  1502         using as(2)[THEN bspec[where x=i]] and as2 i
```
```  1503         apply auto
```
```  1504         done
```
```  1505       ultimately have False using as by auto
```
```  1506     }
```
```  1507     then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto
```
```  1508     moreover
```
```  1509     {
```
```  1510       let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
```
```  1511       assume as2: "b\<bullet>i < d\<bullet>i"
```
```  1512       {
```
```  1513         fix j :: 'a
```
```  1514         assume "j\<in>Basis"
```
```  1515         then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
```
```  1516           apply (cases "j = i")
```
```  1517           using as(2)[THEN bspec[where x=j]]
```
```  1518           apply (auto simp add: as2)
```
```  1519           done
```
```  1520       }
```
```  1521       then have "?x\<in>box c d"
```
```  1522         unfolding mem_box by auto
```
```  1523       moreover
```
```  1524       have "?x\<notin>cbox a b"
```
```  1525         unfolding mem_box
```
```  1526         apply auto
```
```  1527         apply (rule_tac x=i in bexI)
```
```  1528         using as(2)[THEN bspec[where x=i]] and as2 using i
```
```  1529         apply auto
```
```  1530         done
```
```  1531       ultimately have False using as by auto
```
```  1532     }
```
```  1533     then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto
```
```  1534     ultimately
```
```  1535     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
```
```  1536   } note part1 = this
```
```  1537   show ?th3
```
```  1538     unfolding subset_eq and Ball_def and mem_box
```
```  1539     apply (rule, rule, rule, rule)
```
```  1540     apply (rule part1)
```
```  1541     unfolding subset_eq and Ball_def and mem_box
```
```  1542     prefer 4
```
```  1543     apply auto
```
```  1544     apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
```
```  1545     done
```
```  1546   {
```
```  1547     assume as: "box c d \<subseteq> box a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
```
```  1548     fix i :: 'a
```
```  1549     assume i:"i\<in>Basis"
```
```  1550     from as(1) have "box c d \<subseteq> cbox a b"
```
```  1551       using box_subset_cbox[of a b] by auto
```
```  1552     then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
```
```  1553       using part1 and as(2) using i by auto
```
```  1554   } note * = this
```
```  1555   show ?th4
```
```  1556     unfolding subset_eq and Ball_def and mem_box
```
```  1557     apply (rule, rule, rule, rule)
```
```  1558     apply (rule *)
```
```  1559     unfolding subset_eq and Ball_def and mem_box
```
```  1560     prefer 4
```
```  1561     apply auto
```
```  1562     apply (erule_tac x=xa in allE, simp)+
```
```  1563     done
```
```  1564 qed
```
```  1565
```
```  1566 lemma eq_cbox: "cbox a b = cbox c d \<longleftrightarrow> cbox a b = {} \<and> cbox c d = {} \<or> a = c \<and> b = d"
```
```  1567       (is "?lhs = ?rhs")
```
```  1568 proof
```
```  1569   assume ?lhs
```
```  1570   then have "cbox a b \<subseteq> cbox c d" "cbox c d \<subseteq> cbox a b"
```
```  1571     by auto
```
```  1572   then show ?rhs
```
```  1573     by (force simp add: subset_box box_eq_empty intro: antisym euclidean_eqI)
```
```  1574 next
```
```  1575   assume ?rhs
```
```  1576   then show ?lhs
```
```  1577     by force
```
```  1578 qed
```
```  1579
```
```  1580 lemma eq_cbox_box [simp]: "cbox a b = box c d \<longleftrightarrow> cbox a b = {} \<and> box c d = {}"
```
```  1581   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1582 proof
```
```  1583   assume ?lhs
```
```  1584   then have "cbox a b \<subseteq> box c d" "box c d \<subseteq>cbox a b"
```
```  1585     by auto
```
```  1586   then show ?rhs
```
```  1587     apply (simp add: subset_box)
```
```  1588     using \<open>cbox a b = box c d\<close> box_ne_empty box_sing
```
```  1589     apply (fastforce simp add:)
```
```  1590     done
```
```  1591 next
```
```  1592   assume ?rhs
```
```  1593   then show ?lhs
```
```  1594     by force
```
```  1595 qed
```
```  1596
```
```  1597 lemma eq_box_cbox [simp]: "box a b = cbox c d \<longleftrightarrow> box a b = {} \<and> cbox c d = {}"
```
```  1598   by (metis eq_cbox_box)
```
```  1599
```
```  1600 lemma eq_box: "box a b = box c d \<longleftrightarrow> box a b = {} \<and> box c d = {} \<or> a = c \<and> b = d"
```
```  1601   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1602 proof
```
```  1603   assume ?lhs
```
```  1604   then have "box a b \<subseteq> box c d" "box c d \<subseteq> box a b"
```
```  1605     by auto
```
```  1606   then show ?rhs
```
```  1607     apply (simp add: subset_box)
```
```  1608     using box_ne_empty(2) \<open>box a b = box c d\<close>
```
```  1609     apply auto
```
```  1610      apply (meson euclidean_eqI less_eq_real_def not_less)+
```
```  1611     done
```
```  1612 next
```
```  1613   assume ?rhs
```
```  1614   then show ?lhs
```
```  1615     by force
```
```  1616 qed
```
```  1617
```
```  1618 lemma Int_interval:
```
```  1619   fixes a :: "'a::euclidean_space"
```
```  1620   shows "cbox a b \<inter> cbox c d =
```
```  1621     cbox (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
```
```  1622   unfolding set_eq_iff and Int_iff and mem_box
```
```  1623   by auto
```
```  1624
```
```  1625 lemma disjoint_interval:
```
```  1626   fixes a::"'a::euclidean_space"
```
```  1627   shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1)
```
```  1628     and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2)
```
```  1629     and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3)
```
```  1630     and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
```
```  1631 proof -
```
```  1632   let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
```
```  1633   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
```
```  1634       (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
```
```  1635     by blast
```
```  1636   note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
```
```  1637   show ?th1 unfolding * by (intro **) auto
```
```  1638   show ?th2 unfolding * by (intro **) auto
```
```  1639   show ?th3 unfolding * by (intro **) auto
```
```  1640   show ?th4 unfolding * by (intro **) auto
```
```  1641 qed
```
```  1642
```
```  1643 lemma UN_box_eq_UNIV: "(\<Union>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"
```
```  1644 proof -
```
```  1645   have "\<bar>x \<bullet> b\<bar> < real_of_int (\<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil> + 1)"
```
```  1646     if [simp]: "b \<in> Basis" for x b :: 'a
```
```  1647   proof -
```
```  1648     have "\<bar>x \<bullet> b\<bar> \<le> real_of_int \<lceil>\<bar>x \<bullet> b\<bar>\<rceil>"
```
```  1649       by (rule le_of_int_ceiling)
```
```  1650     also have "\<dots> \<le> real_of_int \<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil>"
```
```  1651       by (auto intro!: ceiling_mono)
```
```  1652     also have "\<dots> < real_of_int (\<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil> + 1)"
```
```  1653       by simp
```
```  1654     finally show ?thesis .
```
```  1655   qed
```
```  1656   then have "\<exists>n::nat. \<forall>b\<in>Basis. \<bar>x \<bullet> b\<bar> < real n" for x :: 'a
```
```  1657     by (metis order.strict_trans reals_Archimedean2)
```
```  1658   moreover have "\<And>x b::'a. \<And>n::nat.  \<bar>x \<bullet> b\<bar> < real n \<longleftrightarrow> - real n < x \<bullet> b \<and> x \<bullet> b < real n"
```
```  1659     by auto
```
```  1660   ultimately show ?thesis
```
```  1661     by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
```
```  1662 qed
```
```  1663
```
```  1664 text \<open>Intervals in general, including infinite and mixtures of open and closed.\<close>
```
```  1665
```
```  1666 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
```
```  1667   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"
```
```  1668
```
```  1669 lemma is_interval_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
```
```  1670   and is_interval_box: "is_interval (box a b)" (is ?th2)
```
```  1671   unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
```
```  1672   by (meson order_trans le_less_trans less_le_trans less_trans)+
```
```  1673
```
```  1674 lemma is_interval_empty [iff]: "is_interval {}"
```
```  1675   unfolding is_interval_def  by simp
```
```  1676
```
```  1677 lemma is_interval_univ [iff]: "is_interval UNIV"
```
```  1678   unfolding is_interval_def  by simp
```
```  1679
```
```  1680 lemma mem_is_intervalI:
```
```  1681   assumes "is_interval s"
```
```  1682     and "a \<in> s" "b \<in> s"
```
```  1683     and "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i"
```
```  1684   shows "x \<in> s"
```
```  1685   by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])
```
```  1686
```
```  1687 lemma interval_subst:
```
```  1688   fixes S::"'a::euclidean_space set"
```
```  1689   assumes "is_interval S"
```
```  1690     and "x \<in> S" "y j \<in> S"
```
```  1691     and "j \<in> Basis"
```
```  1692   shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"
```
```  1693   by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)
```
```  1694
```
```  1695 lemma mem_box_componentwiseI:
```
```  1696   fixes S::"'a::euclidean_space set"
```
```  1697   assumes "is_interval S"
```
```  1698   assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i) ` S)"
```
```  1699   shows "x \<in> S"
```
```  1700 proof -
```
```  1701   from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"
```
```  1702     by auto
```
```  1703   with finite_Basis obtain s and bs::"'a list"
```
```  1704     where s: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i = s i \<bullet> i" "\<And>i. i \<in> Basis \<Longrightarrow> s i \<in> S"
```
```  1705       and bs: "set bs = Basis" "distinct bs"
```
```  1706     by (metis finite_distinct_list)
```
```  1707   from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S"
```
```  1708     by blast
```
```  1709   define y where
```
```  1710     "y = rec_list (s j) (\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"
```
```  1711   have "x = (\<Sum>i\<in>Basis. (if i \<in> set bs then s i \<bullet> i else s j \<bullet> i) *\<^sub>R i)"
```
```  1712     using bs by (auto simp add: s(1)[symmetric] euclidean_representation)
```
```  1713   also have [symmetric]: "y bs = \<dots>"
```
```  1714     using bs(2) bs(1)[THEN equalityD1]
```
```  1715     by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
```
```  1716   also have "y bs \<in> S"
```
```  1717     using bs(1)[THEN equalityD1]
```
```  1718     apply (induct bs)
```
```  1719      apply (auto simp: y_def j)
```
```  1720     apply (rule interval_subst[OF assms(1)])
```
```  1721       apply (auto simp: s)
```
```  1722     done
```
```  1723   finally show ?thesis .
```
```  1724 qed
```
```  1725
```
```  1726 lemma cbox01_nonempty [simp]: "cbox 0 One \<noteq> {}"
```
```  1727   by (simp add: box_ne_empty inner_Basis inner_sum_left sum_nonneg)
```
```  1728
```
```  1729 lemma box01_nonempty [simp]: "box 0 One \<noteq> {}"
```
```  1730   by (simp add: box_ne_empty inner_Basis inner_sum_left) (simp add: sum.remove)
```
```  1731
```
```  1732 lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
```
```  1733   using nonempty_Basis box01_nonempty box_eq_empty(1) box_ne_empty(1) by blast
```
```  1734
```
```  1735
```
```  1736 subsection \<open>Connectedness\<close>
```
```  1737
```
```  1738 lemma connected_local:
```
```  1739  "connected S \<longleftrightarrow>
```
```  1740   \<not> (\<exists>e1 e2.
```
```  1741       openin (subtopology euclidean S) e1 \<and>
```
```  1742       openin (subtopology euclidean S) e2 \<and>
```
```  1743       S \<subseteq> e1 \<union> e2 \<and>
```
```  1744       e1 \<inter> e2 = {} \<and>
```
```  1745       e1 \<noteq> {} \<and>
```
```  1746       e2 \<noteq> {})"
```
```  1747   unfolding connected_def openin_open
```
```  1748   by safe blast+
```
```  1749
```
```  1750 lemma exists_diff:
```
```  1751   fixes P :: "'a set \<Rightarrow> bool"
```
```  1752   shows "(\<exists>S. P (- S)) \<longleftrightarrow> (\<exists>S. P S)"
```
```  1753     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1754 proof -
```
```  1755   have ?rhs if ?lhs
```
```  1756     using that by blast
```
```  1757   moreover have "P (- (- S))" if "P S" for S
```
```  1758   proof -
```
```  1759     have "S = - (- S)" by simp
```
```  1760     with that show ?thesis by metis
```
```  1761   qed
```
```  1762   ultimately show ?thesis by metis
```
```  1763 qed
```
```  1764
```
```  1765 lemma connected_clopen: "connected S \<longleftrightarrow>
```
```  1766   (\<forall>T. openin (subtopology euclidean S) T \<and>
```
```  1767      closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1768 proof -
```
```  1769   have "\<not> connected S \<longleftrightarrow>
```
```  1770     (\<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> {})"
```
```  1771     unfolding connected_def openin_open closedin_closed
```
```  1772     by (metis double_complement)
```
```  1773   then have th0: "connected S \<longleftrightarrow>
```
```  1774     \<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> {})"
```
```  1775     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")
```
```  1776     by (simp add: closed_def) metis
```
```  1777   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'))"
```
```  1778     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
```
```  1779     unfolding connected_def openin_open closedin_closed by auto
```
```  1780   have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" for e2
```
```  1781   proof -
```
```  1782     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
```
```  1783       by auto
```
```  1784     then show ?thesis
```
```  1785       by metis
```
```  1786   qed
```
```  1787   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
```
```  1788     by blast
```
```  1789   then show ?thesis
```
```  1790     by (simp add: th0 th1)
```
```  1791 qed
```
```  1792
```
```  1793
```
```  1794 subsection \<open>Limit points\<close>
```
```  1795
```
```  1796 definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)
```
```  1797   where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
```
```  1798
```
```  1799 lemma islimptI:
```
```  1800   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
```
```  1801   shows "x islimpt S"
```
```  1802   using assms unfolding islimpt_def by auto
```
```  1803
```
```  1804 lemma islimptE:
```
```  1805   assumes "x islimpt S" and "x \<in> T" and "open T"
```
```  1806   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
```
```  1807   using assms unfolding islimpt_def by auto
```
```  1808
```
```  1809 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
```
```  1810   unfolding islimpt_def eventually_at_topological by auto
```
```  1811
```
```  1812 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
```
```  1813   unfolding islimpt_def by fast
```
```  1814
```
```  1815 lemma islimpt_approachable:
```
```  1816   fixes x :: "'a::metric_space"
```
```  1817   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
```
```  1818   unfolding islimpt_iff_eventually eventually_at by fast
```
```  1819
```
```  1820 lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
```
```  1821   for x :: "'a::metric_space"
```
```  1822   unfolding islimpt_approachable
```
```  1823   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
```
```  1824     THEN arg_cong [where f=Not]]
```
```  1825   by (simp add: Bex_def conj_commute conj_left_commute)
```
```  1826
```
```  1827 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
```
```  1828   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
```
```  1829
```
```  1830 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
```
```  1831   unfolding islimpt_def by blast
```
```  1832
```
```  1833 text \<open>A perfect space has no isolated points.\<close>
```
```  1834
```
```  1835 lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV"
```
```  1836   for x :: "'a::perfect_space"
```
```  1837   unfolding islimpt_UNIV_iff by (rule not_open_singleton)
```
```  1838
```
```  1839 lemma perfect_choose_dist: "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
```
```  1840   for x :: "'a::{perfect_space,metric_space}"
```
```  1841   using islimpt_UNIV [of x] by (simp add: islimpt_approachable)
```
```  1842
```
```  1843 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
```
```  1844   unfolding closed_def
```
```  1845   apply (subst open_subopen)
```
```  1846   apply (simp add: islimpt_def subset_eq)
```
```  1847   apply (metis ComplE ComplI)
```
```  1848   done
```
```  1849
```
```  1850 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
```
```  1851   by (auto simp add: islimpt_def)
```
```  1852
```
```  1853 lemma finite_set_avoid:
```
```  1854   fixes a :: "'a::metric_space"
```
```  1855   assumes fS: "finite S"
```
```  1856   shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
```
```  1857 proof (induct rule: finite_induct[OF fS])
```
```  1858   case 1
```
```  1859   then show ?case by (auto intro: zero_less_one)
```
```  1860 next
```
```  1861   case (2 x F)
```
```  1862   from 2 obtain d where d: "d > 0" "\<forall>x\<in>F. x \<noteq> a \<longrightarrow> d \<le> dist a x"
```
```  1863     by blast
```
```  1864   show ?case
```
```  1865   proof (cases "x = a")
```
```  1866     case True
```
```  1867     with d show ?thesis by auto
```
```  1868   next
```
```  1869     case False
```
```  1870     let ?d = "min d (dist a x)"
```
```  1871     from False d(1) have dp: "?d > 0"
```
```  1872       by auto
```
```  1873     from d have d': "\<forall>x\<in>F. x \<noteq> a \<longrightarrow> ?d \<le> dist a x"
```
```  1874       by auto
```
```  1875     with dp False show ?thesis
```
```  1876       by (auto intro!: exI[where x="?d"])
```
```  1877   qed
```
```  1878 qed
```
```  1879
```
```  1880 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
```
```  1881   by (simp add: islimpt_iff_eventually eventually_conj_iff)
```
```  1882
```
```  1883 lemma discrete_imp_closed:
```
```  1884   fixes S :: "'a::metric_space set"
```
```  1885   assumes e: "0 < e"
```
```  1886     and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
```
```  1887   shows "closed S"
```
```  1888 proof -
```
```  1889   have False if C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" for x
```
```  1890   proof -
```
```  1891     from e have e2: "e/2 > 0" by arith
```
```  1892     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
```
```  1893       by blast
```
```  1894     let ?m = "min (e/2) (dist x y) "
```
```  1895     from e2 y(2) have mp: "?m > 0"
```
```  1896       by simp
```
```  1897     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
```
```  1898       by blast
```
```  1899     from z y have "dist z y < e"
```
```  1900       by (intro dist_triangle_lt [where z=x]) simp
```
```  1901     from d[rule_format, OF y(1) z(1) this] y z show ?thesis
```
```  1902       by (auto simp add: dist_commute)
```
```  1903   qed
```
```  1904   then show ?thesis
```
```  1905     by (metis islimpt_approachable closed_limpt [where 'a='a])
```
```  1906 qed
```
```  1907
```
```  1908 lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
```
```  1909   by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
```
```  1910
```
```  1911 lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
```
```  1912   by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
```
```  1913
```
```  1914 lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
```
```  1915   unfolding Nats_def by (rule closed_of_nat_image)
```
```  1916
```
```  1917 lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
```
```  1918   unfolding Ints_def by (rule closed_of_int_image)
```
```  1919
```
```  1920
```
```  1921 subsection \<open>Interior of a Set\<close>
```
```  1922
```
```  1923 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
```
```  1924
```
```  1925 lemma interiorI [intro?]:
```
```  1926   assumes "open T" and "x \<in> T" and "T \<subseteq> S"
```
```  1927   shows "x \<in> interior S"
```
```  1928   using assms unfolding interior_def by fast
```
```  1929
```
```  1930 lemma interiorE [elim?]:
```
```  1931   assumes "x \<in> interior S"
```
```  1932   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
```
```  1933   using assms unfolding interior_def by fast
```
```  1934
```
```  1935 lemma open_interior [simp, intro]: "open (interior S)"
```
```  1936   by (simp add: interior_def open_Union)
```
```  1937
```
```  1938 lemma interior_subset: "interior S \<subseteq> S"
```
```  1939   by (auto simp add: interior_def)
```
```  1940
```
```  1941 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
```
```  1942   by (auto simp add: interior_def)
```
```  1943
```
```  1944 lemma interior_open: "open S \<Longrightarrow> interior S = S"
```
```  1945   by (intro equalityI interior_subset interior_maximal subset_refl)
```
```  1946
```
```  1947 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
```
```  1948   by (metis open_interior interior_open)
```
```  1949
```
```  1950 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
```
```  1951   by (metis interior_maximal interior_subset subset_trans)
```
```  1952
```
```  1953 lemma interior_empty [simp]: "interior {} = {}"
```
```  1954   using open_empty by (rule interior_open)
```
```  1955
```
```  1956 lemma interior_UNIV [simp]: "interior UNIV = UNIV"
```
```  1957   using open_UNIV by (rule interior_open)
```
```  1958
```
```  1959 lemma interior_interior [simp]: "interior (interior S) = interior S"
```
```  1960   using open_interior by (rule interior_open)
```
```  1961
```
```  1962 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
```
```  1963   by (auto simp add: interior_def)
```
```  1964
```
```  1965 lemma interior_unique:
```
```  1966   assumes "T \<subseteq> S" and "open T"
```
```  1967   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
```
```  1968   shows "interior S = T"
```
```  1969   by (intro equalityI assms interior_subset open_interior interior_maximal)
```
```  1970
```
```  1971 lemma interior_singleton [simp]: "interior {a} = {}"
```
```  1972   for a :: "'a::perfect_space"
```
```  1973   apply (rule interior_unique)
```
```  1974     apply simp_all
```
```  1975   using not_open_singleton subset_singletonD
```
```  1976   apply fastforce
```
```  1977   done
```
```  1978
```
```  1979 lemma interior_Int [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
```
```  1980   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
```
```  1981     Int_lower2 interior_maximal interior_subset open_Int open_interior)
```
```  1982
```
```  1983 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
```
```  1984   using open_contains_ball_eq [where S="interior S"]
```
```  1985   by (simp add: open_subset_interior)
```
```  1986
```
```  1987 lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
```
```  1988   using interior_subset[of s] by (subst eventually_nhds) blast
```
```  1989
```
```  1990 lemma interior_limit_point [intro]:
```
```  1991   fixes x :: "'a::perfect_space"
```
```  1992   assumes x: "x \<in> interior S"
```
```  1993   shows "x islimpt S"
```
```  1994   using x islimpt_UNIV [of x]
```
```  1995   unfolding interior_def islimpt_def
```
```  1996   apply (clarsimp, rename_tac T T')
```
```  1997   apply (drule_tac x="T \<inter> T'" in spec)
```
```  1998   apply (auto simp add: open_Int)
```
```  1999   done
```
```  2000
```
```  2001 lemma interior_closed_Un_empty_interior:
```
```  2002   assumes cS: "closed S"
```
```  2003     and iT: "interior T = {}"
```
```  2004   shows "interior (S \<union> T) = interior S"
```
```  2005 proof
```
```  2006   show "interior S \<subseteq> interior (S \<union> T)"
```
```  2007     by (rule interior_mono) (rule Un_upper1)
```
```  2008   show "interior (S \<union> T) \<subseteq> interior S"
```
```  2009   proof
```
```  2010     fix x
```
```  2011     assume "x \<in> interior (S \<union> T)"
```
```  2012     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
```
```  2013     show "x \<in> interior S"
```
```  2014     proof (rule ccontr)
```
```  2015       assume "x \<notin> interior S"
```
```  2016       with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
```
```  2017         unfolding interior_def by fast
```
```  2018       from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
```
```  2019         by (rule open_Diff)
```
```  2020       from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
```
```  2021         by fast
```
```  2022       from \<open>y \<in> R - S\<close> \<open>open (R - S)\<close> \<open>R - S \<subseteq> T\<close> \<open>interior T = {}\<close> show False
```
```  2023         unfolding interior_def by fast
```
```  2024     qed
```
```  2025   qed
```
```  2026 qed
```
```  2027
```
```  2028 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
```
```  2029 proof (rule interior_unique)
```
```  2030   show "interior A \<times> interior B \<subseteq> A \<times> B"
```
```  2031     by (intro Sigma_mono interior_subset)
```
```  2032   show "open (interior A \<times> interior B)"
```
```  2033     by (intro open_Times open_interior)
```
```  2034   fix T
```
```  2035   assume "T \<subseteq> A \<times> B" and "open T"
```
```  2036   then show "T \<subseteq> interior A \<times> interior B"
```
```  2037   proof safe
```
```  2038     fix x y
```
```  2039     assume "(x, y) \<in> T"
```
```  2040     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
```
```  2041       using \<open>open T\<close> unfolding open_prod_def by fast
```
```  2042     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
```
```  2043       using \<open>T \<subseteq> A \<times> B\<close> by auto
```
```  2044     then show "x \<in> interior A" and "y \<in> interior B"
```
```  2045       by (auto intro: interiorI)
```
```  2046   qed
```
```  2047 qed
```
```  2048
```
```  2049 lemma interior_Ici:
```
```  2050   fixes x :: "'a :: {dense_linorder,linorder_topology}"
```
```  2051   assumes "b < x"
```
```  2052   shows "interior {x ..} = {x <..}"
```
```  2053 proof (rule interior_unique)
```
```  2054   fix T
```
```  2055   assume "T \<subseteq> {x ..}" "open T"
```
```  2056   moreover have "x \<notin> T"
```
```  2057   proof
```
```  2058     assume "x \<in> T"
```
```  2059     obtain y where "y < x" "{y <.. x} \<subseteq> T"
```
```  2060       using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
```
```  2061     with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
```
```  2062       by (auto simp: subset_eq Ball_def)
```
```  2063     with \<open>T \<subseteq> {x ..}\<close> show False by auto
```
```  2064   qed
```
```  2065   ultimately show "T \<subseteq> {x <..}"
```
```  2066     by (auto simp: subset_eq less_le)
```
```  2067 qed auto
```
```  2068
```
```  2069 lemma interior_Iic:
```
```  2070   fixes x :: "'a ::{dense_linorder,linorder_topology}"
```
```  2071   assumes "x < b"
```
```  2072   shows "interior {.. x} = {..< x}"
```
```  2073 proof (rule interior_unique)
```
```  2074   fix T
```
```  2075   assume "T \<subseteq> {.. x}" "open T"
```
```  2076   moreover have "x \<notin> T"
```
```  2077   proof
```
```  2078     assume "x \<in> T"
```
```  2079     obtain y where "x < y" "{x ..< y} \<subseteq> T"
```
```  2080       using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
```
```  2081     with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
```
```  2082       by (auto simp: subset_eq Ball_def less_le)
```
```  2083     with \<open>T \<subseteq> {.. x}\<close> show False by auto
```
```  2084   qed
```
```  2085   ultimately show "T \<subseteq> {..< x}"
```
```  2086     by (auto simp: subset_eq less_le)
```
```  2087 qed auto
```
```  2088
```
```  2089
```
```  2090 subsection \<open>Closure of a Set\<close>
```
```  2091
```
```  2092 definition "closure S = S \<union> {x | x. x islimpt S}"
```
```  2093
```
```  2094 lemma interior_closure: "interior S = - (closure (- S))"
```
```  2095   by (auto simp add: interior_def closure_def islimpt_def)
```
```  2096
```
```  2097 lemma closure_interior: "closure S = - interior (- S)"
```
```  2098   by (simp add: interior_closure)
```
```  2099
```
```  2100 lemma closed_closure[simp, intro]: "closed (closure S)"
```
```  2101   by (simp add: closure_interior closed_Compl)
```
```  2102
```
```  2103 lemma closure_subset: "S \<subseteq> closure S"
```
```  2104   by (simp add: closure_def)
```
```  2105
```
```  2106 lemma closure_hull: "closure S = closed hull S"
```
```  2107   by (auto simp add: hull_def closure_interior interior_def)
```
```  2108
```
```  2109 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
```
```  2110   unfolding closure_hull using closed_Inter by (rule hull_eq)
```
```  2111
```
```  2112 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
```
```  2113   by (simp only: closure_eq)
```
```  2114
```
```  2115 lemma closure_closure [simp]: "closure (closure S) = closure S"
```
```  2116   unfolding closure_hull by (rule hull_hull)
```
```  2117
```
```  2118 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
```
```  2119   unfolding closure_hull by (rule hull_mono)
```
```  2120
```
```  2121 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
```
```  2122   unfolding closure_hull by (rule hull_minimal)
```
```  2123
```
```  2124 lemma closure_unique:
```
```  2125   assumes "S \<subseteq> T"
```
```  2126     and "closed T"
```
```  2127     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
```
```  2128   shows "closure S = T"
```
```  2129   using assms unfolding closure_hull by (rule hull_unique)
```
```  2130
```
```  2131 lemma closure_empty [simp]: "closure {} = {}"
```
```  2132   using closed_empty by (rule closure_closed)
```
```  2133
```
```  2134 lemma closure_UNIV [simp]: "closure UNIV = UNIV"
```
```  2135   using closed_UNIV by (rule closure_closed)
```
```  2136
```
```  2137 lemma closure_Un [simp]: "closure (S \<union> T) = closure S \<union> closure T"
```
```  2138   by (simp add: closure_interior)
```
```  2139
```
```  2140 lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
```
```  2141   using closure_empty closure_subset[of S] by blast
```
```  2142
```
```  2143 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
```
```  2144   using closure_eq[of S] closure_subset[of S] by simp
```
```  2145
```
```  2146 lemma open_Int_closure_eq_empty: "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
```
```  2147   using open_subset_interior[of S "- T"]
```
```  2148   using interior_subset[of "- T"]
```
```  2149   by (auto simp: closure_interior)
```
```  2150
```
```  2151 lemma open_Int_closure_subset: "open S \<Longrightarrow> S \<inter> closure T \<subseteq> closure (S \<inter> T)"
```
```  2152 proof
```
```  2153   fix x
```
```  2154   assume *: "open S" "x \<in> S \<inter> closure T"
```
```  2155   have "x islimpt (S \<inter> T)" if **: "x islimpt T"
```
```  2156   proof (rule islimptI)
```
```  2157     fix A
```
```  2158     assume "x \<in> A" "open A"
```
```  2159     with * have "x \<in> A \<inter> S" "open (A \<inter> S)"
```
```  2160       by (simp_all add: open_Int)
```
```  2161     with ** obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
```
```  2162       by (rule islimptE)
```
```  2163     then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
```
```  2164       by simp_all
```
```  2165     then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
```
```  2166   qed
```
```  2167   with * show "x \<in> closure (S \<inter> T)"
```
```  2168     unfolding closure_def by blast
```
```  2169 qed
```
```  2170
```
```  2171 lemma closure_complement: "closure (- S) = - interior S"
```
```  2172   by (simp add: closure_interior)
```
```  2173
```
```  2174 lemma interior_complement: "interior (- S) = - closure S"
```
```  2175   by (simp add: closure_interior)
```
```  2176
```
```  2177 lemma interior_diff: "interior(S - T) = interior S - closure T"
```
```  2178   by (simp add: Diff_eq interior_complement)
```
```  2179
```
```  2180 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
```
```  2181 proof (rule closure_unique)
```
```  2182   show "A \<times> B \<subseteq> closure A \<times> closure B"
```
```  2183     by (intro Sigma_mono closure_subset)
```
```  2184   show "closed (closure A \<times> closure B)"
```
```  2185     by (intro closed_Times closed_closure)
```
```  2186   fix T
```
```  2187   assume "A \<times> B \<subseteq> T" and "closed T"
```
```  2188   then show "closure A \<times> closure B \<subseteq> T"
```
```  2189     apply (simp add: closed_def open_prod_def)
```
```  2190     apply clarify
```
```  2191     apply (rule ccontr)
```
```  2192     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
```
```  2193     apply (simp add: closure_interior interior_def)
```
```  2194     apply (drule_tac x=C in spec)
```
```  2195     apply (drule_tac x=D in spec)
```
```  2196     apply auto
```
```  2197     done
```
```  2198 qed
```
```  2199
```
```  2200 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"
```
```  2201   unfolding closure_def using islimpt_punctured by blast
```
```  2202
```
```  2203 lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
```
```  2204   by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
```
```  2205
```
```  2206 lemma limpt_of_limpts: "x islimpt {y. y islimpt S} \<Longrightarrow> x islimpt S"
```
```  2207   for x :: "'a::metric_space"
```
```  2208   apply (clarsimp simp add: islimpt_approachable)
```
```  2209   apply (drule_tac x="e/2" in spec)
```
```  2210   apply (auto simp: simp del: less_divide_eq_numeral1)
```
```  2211   apply (drule_tac x="dist x' x" in spec)
```
```  2212   apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
```
```  2213   apply (erule rev_bexI)
```
```  2214   apply (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
```
```  2215   done
```
```  2216
```
```  2217 lemma closed_limpts:  "closed {x::'a::metric_space. x islimpt S}"
```
```  2218   using closed_limpt limpt_of_limpts by blast
```
```  2219
```
```  2220 lemma limpt_of_closure: "x islimpt closure S \<longleftrightarrow> x islimpt S"
```
```  2221   for x :: "'a::metric_space"
```
```  2222   by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
```
```  2223
```
```  2224 lemma closedin_limpt:
```
```  2225   "closedin (subtopology euclidean T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
```
```  2226   apply (simp add: closedin_closed, safe)
```
```  2227    apply (simp add: closed_limpt islimpt_subset)
```
```  2228   apply (rule_tac x="closure S" in exI)
```
```  2229   apply simp
```
```  2230   apply (force simp: closure_def)
```
```  2231   done
```
```  2232
```
```  2233 lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
```
```  2234   by (meson closedin_limpt closed_subset closedin_closed_trans)
```
```  2235
```
```  2236 lemma closedin_subset_trans:
```
```  2237   "closedin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
```
```  2238     closedin (subtopology euclidean T) S"
```
```  2239   by (meson closedin_limpt subset_iff)
```
```  2240
```
```  2241 lemma openin_subset_trans:
```
```  2242   "openin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
```
```  2243     openin (subtopology euclidean T) S"
```
```  2244   by (auto simp: openin_open)
```
```  2245
```
```  2246 lemma openin_Times:
```
```  2247   "openin (subtopology euclidean S) S' \<Longrightarrow> openin (subtopology euclidean T) T' \<Longrightarrow>
```
```  2248     openin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
```
```  2249   unfolding openin_open using open_Times by blast
```
```  2250
```
```  2251 lemma Times_in_interior_subtopology:
```
```  2252   fixes U :: "('a::metric_space \<times> 'b::metric_space) set"
```
```  2253   assumes "(x, y) \<in> U" "openin (subtopology euclidean (S \<times> T)) U"
```
```  2254   obtains V W where "openin (subtopology euclidean S) V" "x \<in> V"
```
```  2255                     "openin (subtopology euclidean T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
```
```  2256 proof -
```
```  2257   from assms obtain e where "e > 0" and "U \<subseteq> S \<times> T"
```
```  2258     and e: "\<And>x' y'. \<lbrakk>x'\<in>S; y'\<in>T; dist (x', y') (x, y) < e\<rbrakk> \<Longrightarrow> (x', y') \<in> U"
```
```  2259     by (force simp: openin_euclidean_subtopology_iff)
```
```  2260   with assms have "x \<in> S" "y \<in> T"
```
```  2261     by auto
```
```  2262   show ?thesis
```
```  2263   proof
```
```  2264     show "openin (subtopology euclidean S) (ball x (e/2) \<inter> S)"
```
```  2265       by (simp add: Int_commute openin_open_Int)
```
```  2266     show "x \<in> ball x (e / 2) \<inter> S"
```
```  2267       by (simp add: \<open>0 < e\<close> \<open>x \<in> S\<close>)
```
```  2268     show "openin (subtopology euclidean T) (ball y (e/2) \<inter> T)"
```
```  2269       by (simp add: Int_commute openin_open_Int)
```
```  2270     show "y \<in> ball y (e / 2) \<inter> T"
```
```  2271       by (simp add: \<open>0 < e\<close> \<open>y \<in> T\<close>)
```
```  2272     show "(ball x (e / 2) \<inter> S) \<times> (ball y (e / 2) \<inter> T) \<subseteq> U"
```
```  2273       by clarify (simp add: e dist_Pair_Pair \<open>0 < e\<close> dist_commute sqrt_sum_squares_half_less)
```
```  2274   qed
```
```  2275 qed
```
```  2276
```
```  2277 lemma openin_Times_eq:
```
```  2278   fixes S :: "'a::metric_space set" and T :: "'b::metric_space set"
```
```  2279   shows
```
```  2280     "openin (subtopology euclidean (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
```
```  2281       S' = {} \<or> T' = {} \<or> openin (subtopology euclidean S) S' \<and> openin (subtopology euclidean T) T'"
```
```  2282     (is "?lhs = ?rhs")
```
```  2283 proof (cases "S' = {} \<or> T' = {}")
```
```  2284   case True
```
```  2285   then show ?thesis by auto
```
```  2286 next
```
```  2287   case False
```
```  2288   then obtain x y where "x \<in> S'" "y \<in> T'"
```
```  2289     by blast
```
```  2290   show ?thesis
```
```  2291   proof
```
```  2292     assume ?lhs
```
```  2293     have "openin (subtopology euclidean S) S'"
```
```  2294       apply (subst openin_subopen, clarify)
```
```  2295       apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
```
```  2296       using \<open>y \<in> T'\<close>
```
```  2297        apply auto
```
```  2298       done
```
```  2299     moreover have "openin (subtopology euclidean T) T'"
```
```  2300       apply (subst openin_subopen, clarify)
```
```  2301       apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
```
```  2302       using \<open>x \<in> S'\<close>
```
```  2303        apply auto
```
```  2304       done
```
```  2305     ultimately show ?rhs
```
```  2306       by simp
```
```  2307   next
```
```  2308     assume ?rhs
```
```  2309     with False show ?lhs
```
```  2310       by (simp add: openin_Times)
```
```  2311   qed
```
```  2312 qed
```
```  2313
```
```  2314 lemma closedin_Times:
```
```  2315   "closedin (subtopology euclidean S) S' \<Longrightarrow> closedin (subtopology euclidean T) T' \<Longrightarrow>
```
```  2316     closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
```
```  2317   unfolding closedin_closed using closed_Times by blast
```
```  2318
```
```  2319 lemma bdd_below_closure:
```
```  2320   fixes A :: "real set"
```
```  2321   assumes "bdd_below A"
```
```  2322   shows "bdd_below (closure A)"
```
```  2323 proof -
```
```  2324   from assms obtain m where "\<And>x. x \<in> A \<Longrightarrow> m \<le> x"
```
```  2325     by (auto simp: bdd_below_def)
```
```  2326   then have "A \<subseteq> {m..}" by auto
```
```  2327   then have "closure A \<subseteq> {m..}"
```
```  2328     using closed_real_atLeast by (rule closure_minimal)
```
```  2329   then show ?thesis
```
```  2330     by (auto simp: bdd_below_def)
```
```  2331 qed
```
```  2332
```
```  2333
```
```  2334 subsection \<open>Connected components, considered as a connectedness relation or a set\<close>
```
```  2335
```
```  2336 definition "connected_component s x y \<equiv> \<exists>t. connected t \<and> t \<subseteq> s \<and> x \<in> t \<and> y \<in> t"
```
```  2337
```
```  2338 abbreviation "connected_component_set s x \<equiv> Collect (connected_component s x)"
```
```  2339
```
```  2340 lemma connected_componentI:
```
```  2341   "connected t \<Longrightarrow> t \<subseteq> s \<Longrightarrow> x \<in> t \<Longrightarrow> y \<in> t \<Longrightarrow> connected_component s x y"
```
```  2342   by (auto simp: connected_component_def)
```
```  2343
```
```  2344 lemma connected_component_in: "connected_component s x y \<Longrightarrow> x \<in> s \<and> y \<in> s"
```
```  2345   by (auto simp: connected_component_def)
```
```  2346
```
```  2347 lemma connected_component_refl: "x \<in> s \<Longrightarrow> connected_component s x x"
```
```  2348   by (auto simp: connected_component_def) (use connected_sing in blast)
```
```  2349
```
```  2350 lemma connected_component_refl_eq [simp]: "connected_component s x x \<longleftrightarrow> x \<in> s"
```
```  2351   by (auto simp: connected_component_refl) (auto simp: connected_component_def)
```
```  2352
```
```  2353 lemma connected_component_sym: "connected_component s x y \<Longrightarrow> connected_component s y x"
```
```  2354   by (auto simp: connected_component_def)
```
```  2355
```
```  2356 lemma connected_component_trans:
```
```  2357   "connected_component s x y \<Longrightarrow> connected_component s y z \<Longrightarrow> connected_component s x z"
```
```  2358   unfolding connected_component_def
```
```  2359   by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)
```
```  2360
```
```  2361 lemma connected_component_of_subset:
```
```  2362   "connected_component s x y \<Longrightarrow> s \<subseteq> t \<Longrightarrow> connected_component t x y"
```
```  2363   by (auto simp: connected_component_def)
```
```  2364
```
```  2365 lemma connected_component_Union: "connected_component_set s x = \<Union>{t. connected t \<and> x \<in> t \<and> t \<subseteq> s}"
```
```  2366   by (auto simp: connected_component_def)
```
```  2367
```
```  2368 lemma connected_connected_component [iff]: "connected (connected_component_set s x)"
```
```  2369   by (auto simp: connected_component_Union intro: connected_Union)
```
```  2370
```
```  2371 lemma connected_iff_eq_connected_component_set:
```
```  2372   "connected s \<longleftrightarrow> (\<forall>x \<in> s. connected_component_set s x = s)"
```
```  2373 proof (cases "s = {}")
```
```  2374   case True
```
```  2375   then show ?thesis by simp
```
```  2376 next
```
```  2377   case False
```
```  2378   then obtain x where "x \<in> s" by auto
```
```  2379   show ?thesis
```
```  2380   proof
```
```  2381     assume "connected s"
```
```  2382     then show "\<forall>x \<in> s. connected_component_set s x = s"
```
```  2383       by (force simp: connected_component_def)
```
```  2384   next
```
```  2385     assume "\<forall>x \<in> s. connected_component_set s x = s"
```
```  2386     then show "connected s"
```
```  2387       by (metis \<open>x \<in> s\<close> connected_connected_component)
```
```  2388   qed
```
```  2389 qed
```
```  2390
```
```  2391 lemma connected_component_subset: "connected_component_set s x \<subseteq> s"
```
```  2392   using connected_component_in by blast
```
```  2393
```
```  2394 lemma connected_component_eq_self: "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> connected_component_set s x = s"
```
```  2395   by (simp add: connected_iff_eq_connected_component_set)
```
```  2396
```
```  2397 lemma connected_iff_connected_component:
```
```  2398   "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component s x y)"
```
```  2399   using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)
```
```  2400
```
```  2401 lemma connected_component_maximal:
```
```  2402   "x \<in> t \<Longrightarrow> connected t \<Longrightarrow> t \<subseteq> s \<Longrightarrow> t \<subseteq> (connected_component_set s x)"
```
```  2403   using connected_component_eq_self connected_component_of_subset by blast
```
```  2404
```
```  2405 lemma connected_component_mono:
```
```  2406   "s \<subseteq> t \<Longrightarrow> connected_component_set s x \<subseteq> connected_component_set t x"
```
```  2407   by (simp add: Collect_mono connected_component_of_subset)
```
```  2408
```
```  2409 lemma connected_component_eq_empty [simp]: "connected_component_set s x = {} \<longleftrightarrow> x \<notin> s"
```
```  2410   using connected_component_refl by (fastforce simp: connected_component_in)
```
```  2411
```
```  2412 lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
```
```  2413   using connected_component_eq_empty by blast
```
```  2414
```
```  2415 lemma connected_component_eq:
```
```  2416   "y \<in> connected_component_set s x \<Longrightarrow> (connected_component_set s y = connected_component_set s x)"
```
```  2417   by (metis (no_types, lifting)
```
```  2418       Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)
```
```  2419
```
```  2420 lemma closed_connected_component:
```
```  2421   assumes s: "closed s"
```
```  2422   shows "closed (connected_component_set s x)"
```
```  2423 proof (cases "x \<in> s")
```
```  2424   case False
```
```  2425   then show ?thesis
```
```  2426     by (metis connected_component_eq_empty closed_empty)
```
```  2427 next
```
```  2428   case True
```
```  2429   show ?thesis
```
```  2430     unfolding closure_eq [symmetric]
```
```  2431   proof
```
```  2432     show "closure (connected_component_set s x) \<subseteq> connected_component_set s x"
```
```  2433       apply (rule connected_component_maximal)
```
```  2434         apply (simp add: closure_def True)
```
```  2435        apply (simp add: connected_imp_connected_closure)
```
```  2436       apply (simp add: s closure_minimal connected_component_subset)
```
```  2437       done
```
```  2438   next
```
```  2439     show "connected_component_set s x \<subseteq> closure (connected_component_set s x)"
```
```  2440       by (simp add: closure_subset)
```
```  2441   qed
```
```  2442 qed
```
```  2443
```
```  2444 lemma connected_component_disjoint:
```
```  2445   "connected_component_set s a \<inter> connected_component_set s b = {} \<longleftrightarrow>
```
```  2446     a \<notin> connected_component_set s b"
```
```  2447   apply (auto simp: connected_component_eq)
```
```  2448   using connected_component_eq connected_component_sym
```
```  2449   apply blast
```
```  2450   done
```
```  2451
```
```  2452 lemma connected_component_nonoverlap:
```
```  2453   "connected_component_set s a \<inter> connected_component_set s b = {} \<longleftrightarrow>
```
```  2454     a \<notin> s \<or> b \<notin> s \<or> connected_component_set s a \<noteq> connected_component_set s b"
```
```  2455   apply (auto simp: connected_component_in)
```
```  2456   using connected_component_refl_eq
```
```  2457     apply blast
```
```  2458    apply (metis connected_component_eq mem_Collect_eq)
```
```  2459   apply (metis connected_component_eq mem_Collect_eq)
```
```  2460   done
```
```  2461
```
```  2462 lemma connected_component_overlap:
```
```  2463   "connected_component_set s a \<inter> connected_component_set s b \<noteq> {} \<longleftrightarrow>
```
```  2464     a \<in> s \<and> b \<in> s \<and> connected_component_set s a = connected_component_set s b"
```
```  2465   by (auto simp: connected_component_nonoverlap)
```
```  2466
```
```  2467 lemma connected_component_sym_eq: "connected_component s x y \<longleftrightarrow> connected_component s y x"
```
```  2468   using connected_component_sym by blast
```
```  2469
```
```  2470 lemma connected_component_eq_eq:
```
```  2471   "connected_component_set s x = connected_component_set s y \<longleftrightarrow>
```
```  2472     x \<notin> s \<and> y \<notin> s \<or> x \<in> s \<and> y \<in> s \<and> connected_component s x y"
```
```  2473   apply (cases "y \<in> s")
```
```  2474    apply (simp add:)
```
```  2475    apply (metis connected_component_eq connected_component_eq_empty connected_component_refl_eq mem_Collect_eq)
```
```  2476   apply (cases "x \<in> s")
```
```  2477    apply (simp add:)
```
```  2478    apply (metis connected_component_eq_empty)
```
```  2479   using connected_component_eq_empty
```
```  2480   apply blast
```
```  2481   done
```
```  2482
```
```  2483 lemma connected_iff_connected_component_eq:
```
```  2484   "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component_set s x = connected_component_set s y)"
```
```  2485   by (simp add: connected_component_eq_eq connected_iff_connected_component)
```
```  2486
```
```  2487 lemma connected_component_idemp:
```
```  2488   "connected_component_set (connected_component_set s x) x = connected_component_set s x"
```
```  2489   apply (rule subset_antisym)
```
```  2490    apply (simp add: connected_component_subset)
```
```  2491   apply (metis connected_component_eq_empty connected_component_maximal
```
```  2492       connected_component_refl_eq connected_connected_component mem_Collect_eq set_eq_subset)
```
```  2493   done
```
```  2494
```
```  2495 lemma connected_component_unique:
```
```  2496   "\<lbrakk>x \<in> c; c \<subseteq> s; connected c;
```
```  2497     \<And>c'. x \<in> c' \<and> c' \<subseteq> s \<and> connected c'
```
```  2498               \<Longrightarrow> c' \<subseteq> c\<rbrakk>
```
```  2499         \<Longrightarrow> connected_component_set s x = c"
```
```  2500 apply (rule subset_antisym)
```
```  2501 apply (meson connected_component_maximal connected_component_subset connected_connected_component contra_subsetD)
```
```  2502 by (simp add: connected_component_maximal)
```
```  2503
```
```  2504 lemma joinable_connected_component_eq:
```
```  2505   "\<lbrakk>connected t; t \<subseteq> s;
```
```  2506     connected_component_set s x \<inter> t \<noteq> {};
```
```  2507     connected_component_set s y \<inter> t \<noteq> {}\<rbrakk>
```
```  2508     \<Longrightarrow> connected_component_set s x = connected_component_set s y"
```
```  2509 apply (simp add: ex_in_conv [symmetric])
```
```  2510 apply (rule connected_component_eq)
```
```  2511 by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in connected_component_maximal subsetD mem_Collect_eq)
```
```  2512
```
```  2513
```
```  2514 lemma Union_connected_component: "\<Union>(connected_component_set s ` s) = s"
```
```  2515   apply (rule subset_antisym)
```
```  2516   apply (simp add: SUP_least connected_component_subset)
```
```  2517   using connected_component_refl_eq
```
```  2518   by force
```
```  2519
```
```  2520
```
```  2521 lemma complement_connected_component_unions:
```
```  2522     "s - connected_component_set s x =
```
```  2523      \<Union>(connected_component_set s ` s - {connected_component_set s x})"
```
```  2524   apply (subst Union_connected_component [symmetric], auto)
```
```  2525   apply (metis connected_component_eq_eq connected_component_in)
```
```  2526   by (metis connected_component_eq mem_Collect_eq)
```
```  2527
```
```  2528 lemma connected_component_intermediate_subset:
```
```  2529         "\<lbrakk>connected_component_set u a \<subseteq> t; t \<subseteq> u\<rbrakk>
```
```  2530         \<Longrightarrow> connected_component_set t a = connected_component_set u a"
```
```  2531   apply (case_tac "a \<in> u")
```
```  2532   apply (simp add: connected_component_maximal connected_component_mono subset_antisym)
```
```  2533   using connected_component_eq_empty by blast
```
```  2534
```
```  2535 proposition connected_Times:
```
```  2536   assumes S: "connected S" and T: "connected T"
```
```  2537   shows "connected (S \<times> T)"
```
```  2538 proof (clarsimp simp add: connected_iff_connected_component)
```
```  2539   fix x y x' y'
```
```  2540   assume xy: "x \<in> S" "y \<in> T" "x' \<in> S" "y' \<in> T"
```
```  2541   with xy obtain U V where U: "connected U" "U \<subseteq> S" "x \<in> U" "x' \<in> U"
```
```  2542                        and V: "connected V" "V \<subseteq> T" "y \<in> V" "y' \<in> V"
```
```  2543     using S T \<open>x \<in> S\<close> \<open>x' \<in> S\<close> by blast+
```
```  2544   show "connected_component (S \<times> T) (x, y) (x', y')"
```
```  2545     unfolding connected_component_def
```
```  2546   proof (intro exI conjI)
```
```  2547     show "connected ((\<lambda>x. (x, y)) ` U \<union> Pair x' ` V)"
```
```  2548     proof (rule connected_Un)
```
```  2549       have "continuous_on U (\<lambda>x. (x, y))"
```
```  2550         by (intro continuous_intros)
```
```  2551       then show "connected ((\<lambda>x. (x, y)) ` U)"
```
```  2552         by (rule connected_continuous_image) (rule \<open>connected U\<close>)
```
```  2553       have "continuous_on V (Pair x')"
```
```  2554         by (intro continuous_intros)
```
```  2555       then show "connected (Pair x' ` V)"
```
```  2556         by (rule connected_continuous_image) (rule \<open>connected V\<close>)
```
```  2557     qed (use U V in auto)
```
```  2558   qed (use U V in auto)
```
```  2559 qed
```
```  2560
```
```  2561 corollary connected_Times_eq [simp]:
```
```  2562    "connected (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> connected S \<and> connected T"  (is "?lhs = ?rhs")
```
```  2563 proof
```
```  2564   assume L: ?lhs
```
```  2565   show ?rhs
```
```  2566   proof (cases "S = {} \<or> T = {}")
```
```  2567     case True
```
```  2568     then show ?thesis by auto
```
```  2569   next
```
```  2570     case False
```
```  2571     have "connected (fst ` (S \<times> T))" "connected (snd ` (S \<times> T))"
```
```  2572       using continuous_on_fst continuous_on_snd continuous_on_id
```
```  2573       by (blast intro: connected_continuous_image [OF _ L])+
```
```  2574     with False show ?thesis
```
```  2575       by auto
```
```  2576   qed
```
```  2577 next
```
```  2578   assume ?rhs
```
```  2579   then show ?lhs
```
```  2580     by (auto simp: connected_Times)
```
```  2581 qed
```
```  2582
```
```  2583
```
```  2584 subsection \<open>The set of connected components of a set\<close>
```
```  2585
```
```  2586 definition components:: "'a::topological_space set \<Rightarrow> 'a set set"
```
```  2587   where "components s \<equiv> connected_component_set s ` s"
```
```  2588
```
```  2589 lemma components_iff: "s \<in> components u \<longleftrightarrow> (\<exists>x. x \<in> u \<and> s = connected_component_set u x)"
```
```  2590   by (auto simp: components_def)
```
```  2591
```
```  2592 lemma componentsI: "x \<in> u \<Longrightarrow> connected_component_set u x \<in> components u"
```
```  2593   by (auto simp: components_def)
```
```  2594
```
```  2595 lemma componentsE:
```
```  2596   assumes "s \<in> components u"
```
```  2597   obtains x where "x \<in> u" "s = connected_component_set u x"
```
```  2598   using assms by (auto simp: components_def)
```
```  2599
```
```  2600 lemma Union_components [simp]: "\<Union>(components u) = u"
```
```  2601   apply (rule subset_antisym)
```
```  2602   using Union_connected_component components_def apply fastforce
```
```  2603   apply (metis Union_connected_component components_def set_eq_subset)
```
```  2604   done
```
```  2605
```
```  2606 lemma pairwise_disjoint_components: "pairwise (\<lambda>X Y. X \<inter> Y = {}) (components u)"
```
```  2607   apply (simp add: pairwise_def)
```
```  2608   apply (auto simp: components_iff)
```
```  2609   apply (metis connected_component_eq_eq connected_component_in)+
```
```  2610   done
```
```  2611
```
```  2612 lemma in_components_nonempty: "c \<in> components s \<Longrightarrow> c \<noteq> {}"
```
```  2613     by (metis components_iff connected_component_eq_empty)
```
```  2614
```
```  2615 lemma in_components_subset: "c \<in> components s \<Longrightarrow> c \<subseteq> s"
```
```  2616   using Union_components by blast
```
```  2617
```
```  2618 lemma in_components_connected: "c \<in> components s \<Longrightarrow> connected c"
```
```  2619   by (metis components_iff connected_connected_component)
```
```  2620
```
```  2621 lemma in_components_maximal:
```
```  2622   "c \<in> components s \<longleftrightarrow>
```
```  2623     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)"
```
```  2624   apply (rule iffI)
```
```  2625    apply (simp add: in_components_nonempty in_components_connected)
```
```  2626    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)
```
```  2627   apply (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)
```
```  2628   done
```
```  2629
```
```  2630 lemma joinable_components_eq:
```
```  2631   "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"
```
```  2632   by (metis (full_types) components_iff joinable_connected_component_eq)
```
```  2633
```
```  2634 lemma closed_components: "\<lbrakk>closed s; c \<in> components s\<rbrakk> \<Longrightarrow> closed c"
```
```  2635   by (metis closed_connected_component components_iff)
```
```  2636
```
```  2637 lemma components_nonoverlap:
```
```  2638     "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c \<inter> c' = {}) \<longleftrightarrow> (c \<noteq> c')"
```
```  2639   apply (auto simp: in_components_nonempty components_iff)
```
```  2640     using connected_component_refl apply blast
```
```  2641    apply (metis connected_component_eq_eq connected_component_in)
```
```  2642   by (metis connected_component_eq mem_Collect_eq)
```
```  2643
```
```  2644 lemma components_eq: "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c = c' \<longleftrightarrow> c \<inter> c' \<noteq> {})"
```
```  2645   by (metis components_nonoverlap)
```
```  2646
```
```  2647 lemma components_eq_empty [simp]: "components s = {} \<longleftrightarrow> s = {}"
```
```  2648   by (simp add: components_def)
```
```  2649
```
```  2650 lemma components_empty [simp]: "components {} = {}"
```
```  2651   by simp
```
```  2652
```
```  2653 lemma connected_eq_connected_components_eq: "connected s \<longleftrightarrow> (\<forall>c \<in> components s. \<forall>c' \<in> components s. c = c')"
```
```  2654   by (metis (no_types, hide_lams) components_iff connected_component_eq_eq connected_iff_connected_component)
```
```  2655
```
```  2656 lemma components_eq_sing_iff: "components s = {s} \<longleftrightarrow> connected s \<and> s \<noteq> {}"
```
```  2657   apply (rule iffI)
```
```  2658   using in_components_connected apply fastforce
```
```  2659   apply safe
```
```  2660   using Union_components apply fastforce
```
```  2661    apply (metis components_iff connected_component_eq_self)
```
```  2662   using in_components_maximal
```
```  2663   apply auto
```
```  2664   done
```
```  2665
```
```  2666 lemma components_eq_sing_exists: "(\<exists>a. components s = {a}) \<longleftrightarrow> connected s \<and> s \<noteq> {}"
```
```  2667   apply (rule iffI)
```
```  2668   using connected_eq_connected_components_eq apply fastforce
```
```  2669   apply (metis components_eq_sing_iff)
```
```  2670   done
```
```  2671
```
```  2672 lemma connected_eq_components_subset_sing: "connected s \<longleftrightarrow> components s \<subseteq> {s}"
```
```  2673   by (metis Union_components components_empty components_eq_sing_iff connected_empty insert_subset order_refl subset_singletonD)
```
```  2674
```
```  2675 lemma connected_eq_components_subset_sing_exists: "connected s \<longleftrightarrow> (\<exists>a. components s \<subseteq> {a})"
```
```  2676   by (metis components_eq_sing_exists connected_eq_components_subset_sing empty_iff subset_iff subset_singletonD)
```
```  2677
```
```  2678 lemma in_components_self: "s \<in> components s \<longleftrightarrow> connected s \<and> s \<noteq> {}"
```
```  2679   by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)
```
```  2680
```
```  2681 lemma components_maximal: "\<lbrakk>c \<in> components s; connected t; t \<subseteq> s; c \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> t \<subseteq> c"
```
```  2682   apply (simp add: components_def ex_in_conv [symmetric], clarify)
```
```  2683   by (meson connected_component_def connected_component_trans)
```
```  2684
```
```  2685 lemma exists_component_superset: "\<lbrakk>t \<subseteq> s; s \<noteq> {}; connected t\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> components s \<and> t \<subseteq> c"
```
```  2686   apply (cases "t = {}")
```
```  2687    apply force
```
```  2688   apply (metis components_def ex_in_conv connected_component_maximal contra_subsetD image_eqI)
```
```  2689   done
```
```  2690
```
```  2691 lemma components_intermediate_subset: "\<lbrakk>s \<in> components u; s \<subseteq> t; t \<subseteq> u\<rbrakk> \<Longrightarrow> s \<in> components t"
```
```  2692   apply (auto simp: components_iff)
```
```  2693   apply (metis connected_component_eq_empty connected_component_intermediate_subset)
```
```  2694   done
```
```  2695
```
```  2696 lemma in_components_unions_complement: "c \<in> components s \<Longrightarrow> s - c = \<Union>(components s - {c})"
```
```  2697   by (metis complement_connected_component_unions components_def components_iff)
```
```  2698
```
```  2699 lemma connected_intermediate_closure:
```
```  2700   assumes cs: "connected s" and st: "s \<subseteq> t" and ts: "t \<subseteq> closure s"
```
```  2701   shows "connected t"
```
```  2702 proof (rule connectedI)
```
```  2703   fix A B
```
```  2704   assume A: "open A" and B: "open B" and Alap: "A \<inter> t \<noteq> {}" and Blap: "B \<inter> t \<noteq> {}"
```
```  2705     and disj: "A \<inter> B \<inter> t = {}" and cover: "t \<subseteq> A \<union> B"
```
```  2706   have disjs: "A \<inter> B \<inter> s = {}"
```
```  2707     using disj st by auto
```
```  2708   have "A \<inter> closure s \<noteq> {}"
```
```  2709     using Alap Int_absorb1 ts by blast
```
```  2710   then have Alaps: "A \<inter> s \<noteq> {}"
```
```  2711     by (simp add: A open_Int_closure_eq_empty)
```
```  2712   have "B \<inter> closure s \<noteq> {}"
```
```  2713     using Blap Int_absorb1 ts by blast
```
```  2714   then have Blaps: "B \<inter> s \<noteq> {}"
```
```  2715     by (simp add: B open_Int_closure_eq_empty)
```
```  2716   then show False
```
```  2717     using cs [unfolded connected_def] A B disjs Alaps Blaps cover st
```
```  2718     by blast
```
```  2719 qed
```
```  2720
```
```  2721 lemma closedin_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"
```
```  2722 proof (cases "connected_component_set s x = {}")
```
```  2723   case True
```
```  2724   then show ?thesis
```
```  2725     by (metis closedin_empty)
```
```  2726 next
```
```  2727   case False
```
```  2728   then obtain y where y: "connected_component s x y"
```
```  2729     by blast
```
```  2730   have *: "connected_component_set s x \<subseteq> s \<inter> closure (connected_component_set s x)"
```
```  2731     by (auto simp: closure_def connected_component_in)
```
```  2732   have "connected_component s x y \<Longrightarrow> s \<inter> closure (connected_component_set s x) \<subseteq> connected_component_set s x"
```
```  2733     apply (rule connected_component_maximal)
```
```  2734       apply simp
```
```  2735     using closure_subset connected_component_in apply fastforce
```
```  2736     using * connected_intermediate_closure apply blast+
```
```  2737     done
```
```  2738   with y * show ?thesis
```
```  2739     by (auto simp add: Topology_Euclidean_Space.closedin_closed)
```
```  2740 qed
```
```  2741
```
```  2742
```
```  2743 subsection \<open>Frontier (also known as boundary)\<close>
```
```  2744
```
```  2745 definition "frontier S = closure S - interior S"
```
```  2746
```
```  2747 lemma frontier_closed [iff]: "closed (frontier S)"
```
```  2748   by (simp add: frontier_def closed_Diff)
```
```  2749
```
```  2750 lemma frontier_closures: "frontier S = closure S \<inter> closure (- S)"
```
```  2751   by (auto simp add: frontier_def interior_closure)
```
```  2752
```
```  2753 lemma frontier_straddle:
```
```  2754   fixes a :: "'a::metric_space"
```
```  2755   shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
```
```  2756   unfolding frontier_def closure_interior
```
```  2757   by (auto simp add: mem_interior subset_eq ball_def)
```
```  2758
```
```  2759 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
```
```  2760   by (metis frontier_def closure_closed Diff_subset)
```
```  2761
```
```  2762 lemma frontier_empty [simp]: "frontier {} = {}"
```
```  2763   by (simp add: frontier_def)
```
```  2764
```
```  2765 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
```
```  2766 proof -
```
```  2767   {
```
```  2768     assume "frontier S \<subseteq> S"
```
```  2769     then have "closure S \<subseteq> S"
```
```  2770       using interior_subset unfolding frontier_def by auto
```
```  2771     then have "closed S"
```
```  2772       using closure_subset_eq by auto
```
```  2773   }
```
```  2774   then show ?thesis using frontier_subset_closed[of S] ..
```
```  2775 qed
```
```  2776
```
```  2777 lemma frontier_complement [simp]: "frontier (- S) = frontier S"
```
```  2778   by (auto simp add: frontier_def closure_complement interior_complement)
```
```  2779
```
```  2780 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
```
```  2781   using frontier_complement frontier_subset_eq[of "- S"]
```
```  2782   unfolding open_closed by auto
```
```  2783
```
```  2784 lemma frontier_UNIV [simp]: "frontier UNIV = {}"
```
```  2785   using frontier_complement frontier_empty by fastforce
```
```  2786
```
```  2787 lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
```
```  2788   by (simp add: Int_commute frontier_def interior_closure)
```
```  2789
```
```  2790 lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
```
```  2791   by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
```
```  2792
```
```  2793 lemma connected_Int_frontier:
```
```  2794      "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
```
```  2795   apply (simp add: frontier_interiors connected_openin, safe)
```
```  2796   apply (drule_tac x="s \<inter> interior t" in spec, safe)
```
```  2797    apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
```
```  2798    apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
```
```  2799   done
```
```  2800
```
```  2801 lemma closure_Un_frontier: "closure S = S \<union> frontier S"
```
```  2802 proof -
```
```  2803   have "S \<union> interior S = S"
```
```  2804     using interior_subset by auto
```
```  2805   then show ?thesis
```
```  2806     using closure_subset by (auto simp: frontier_def)
```
```  2807 qed
```
```  2808
```
```  2809
```
```  2810 subsection \<open>Filters and the ``eventually true'' quantifier\<close>
```
```  2811
```
```  2812 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"  (infixr "indirection" 70)
```
```  2813   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
```
```  2814
```
```  2815 text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
```
```  2816
```
```  2817 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
```
```  2818 proof
```
```  2819   assume "trivial_limit (at a within S)"
```
```  2820   then show "\<not> a islimpt S"
```
```  2821     unfolding trivial_limit_def
```
```  2822     unfolding eventually_at_topological
```
```  2823     unfolding islimpt_def
```
```  2824     apply (clarsimp simp add: set_eq_iff)
```
```  2825     apply (rename_tac T, rule_tac x=T in exI)
```
```  2826     apply (clarsimp, drule_tac x=y in bspec, simp_all)
```
```  2827     done
```
```  2828 next
```
```  2829   assume "\<not> a islimpt S"
```
```  2830   then show "trivial_limit (at a within S)"
```
```  2831     unfolding trivial_limit_def eventually_at_topological islimpt_def
```
```  2832     by metis
```
```  2833 qed
```
```  2834
```
```  2835 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
```
```  2836   using trivial_limit_within [of a UNIV] by simp
```
```  2837
```
```  2838 lemma trivial_limit_at: "\<not> trivial_limit (at a)"
```
```  2839   for a :: "'a::perfect_space"
```
```  2840   by (rule at_neq_bot)
```
```  2841
```
```  2842 lemma trivial_limit_at_infinity:
```
```  2843   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
```
```  2844   unfolding trivial_limit_def eventually_at_infinity
```
```  2845   apply clarsimp
```
```  2846   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
```
```  2847    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
```
```  2848   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
```
```  2849   apply (drule_tac x=UNIV in spec, simp)
```
```  2850   done
```
```  2851
```
```  2852 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
```
```  2853   using islimpt_in_closure by (metis trivial_limit_within)
```
```  2854
```
```  2855 lemma at_within_eq_bot_iff: "at c within A = bot \<longleftrightarrow> c \<notin> closure (A - {c})"
```
```  2856   using not_trivial_limit_within[of c A] by blast
```
```  2857
```
```  2858 text \<open>Some property holds "sufficiently close" to the limit point.\<close>
```
```  2859
```
```  2860 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
```
```  2861   by simp
```
```  2862
```
```  2863 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
```
```  2864   by (simp add: filter_eq_iff)
```
```  2865
```
```  2866
```
```  2867 subsection \<open>Limits\<close>
```
```  2868
```
```  2869 lemma Lim: "(f \<longlongrightarrow> l) net \<longleftrightarrow> trivial_limit net \<or> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
```
```  2870   by (auto simp: tendsto_iff trivial_limit_eq)
```
```  2871
```
```  2872 text \<open>Show that they yield usual definitions in the various cases.\<close>
```
```  2873
```
```  2874 lemma Lim_within_le: "(f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow>
```
```  2875     (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
```
```  2876   by (auto simp add: tendsto_iff eventually_at_le)
```
```  2877
```
```  2878 lemma Lim_within: "(f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow>
```
```  2879     (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a  < d \<longrightarrow> dist (f x) l < e)"
```
```  2880   by (auto simp add: tendsto_iff eventually_at)
```
```  2881
```
```  2882 corollary Lim_withinI [intro?]:
```
```  2883   assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e"
```
```  2884   shows "(f \<longlongrightarrow> l) (at a within S)"
```
```  2885   apply (simp add: Lim_within, clarify)
```
```  2886   apply (rule ex_forward [OF assms [OF half_gt_zero]])
```
```  2887   apply auto
```
```  2888   done
```
```  2889
```
```  2890 lemma Lim_at: "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow>
```
```  2891     (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"
```
```  2892   by (auto simp add: tendsto_iff eventually_at)
```
```  2893
```
```  2894 lemma Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
```
```  2895   by (auto simp add: tendsto_iff eventually_at_infinity)
```
```  2896
```
```  2897 corollary Lim_at_infinityI [intro?]:
```
```  2898   assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
```
```  2899   shows "(f \<longlongrightarrow> l) at_infinity"
```
```  2900   apply (simp add: Lim_at_infinity, clarify)
```
```  2901   apply (rule ex_forward [OF assms [OF half_gt_zero]])
```
```  2902    apply auto
```
```  2903   done
```
```  2904
```
```  2905 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f \<longlongrightarrow> l) net"
```
```  2906   by (rule topological_tendstoI) (auto elim: eventually_mono)
```
```  2907
```
```  2908 lemma Lim_transform_within_set:
```
```  2909   fixes a :: "'a::metric_space" and l :: "'b::metric_space"
```
```  2910   shows "\<lbrakk>(f \<longlongrightarrow> l) (at a within S); eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)\<rbrakk>
```
```  2911          \<Longrightarrow> (f \<longlongrightarrow> l) (at a within T)"
```
```  2912 apply (clarsimp simp: eventually_at Lim_within)
```
```  2913 apply (drule_tac x=e in spec, clarify)
```
```  2914 apply (rename_tac k)
```
```  2915 apply (rule_tac x="min d k" in exI, simp)
```
```  2916 done
```
```  2917
```
```  2918 lemma Lim_transform_within_set_eq:
```
```  2919   fixes a l :: "'a::real_normed_vector"
```
```  2920   shows "eventually (\<lambda>x. x \<in> s \<longleftrightarrow> x \<in> t) (at a)
```
```  2921          \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within s) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within t))"
```
```  2922   by (force intro: Lim_transform_within_set elim: eventually_mono)
```
```  2923
```
```  2924 lemma Lim_transform_within_openin:
```
```  2925   fixes a :: "'a::metric_space"
```
```  2926   assumes f: "(f \<longlongrightarrow> l) (at a within T)"
```
```  2927     and "openin (subtopology euclidean T) S" "a \<in> S"
```
```  2928     and eq: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
```
```  2929   shows "(g \<longlongrightarrow> l) (at a within T)"
```
```  2930 proof -
```
```  2931   obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball a \<epsilon> \<inter> T \<subseteq> S"
```
```  2932     using assms by (force simp: openin_contains_ball)
```
```  2933   then have "a \<in> ball a \<epsilon>"
```
```  2934     by simp
```
```  2935   show ?thesis
```
```  2936     by (rule Lim_transform_within [OF f \<open>0 < \<epsilon>\<close> eq]) (use \<epsilon> in \<open>auto simp: dist_commute subset_iff\<close>)
```
```  2937 qed
```
```  2938
```
```  2939 lemma continuous_transform_within_openin:
```
```  2940   fixes a :: "'a::metric_space"
```
```  2941   assumes "continuous (at a within T) f"
```
```  2942     and "openin (subtopology euclidean T) S" "a \<in> S"
```
```  2943     and eq: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
```
```  2944   shows "continuous (at a within T) g"
```
```  2945   using assms by (simp add: Lim_transform_within_openin continuous_within)
```
```  2946
```
```  2947 text \<open>The expected monotonicity property.\<close>
```
```  2948
```
```  2949 lemma Lim_Un:
```
```  2950   assumes "(f \<longlongrightarrow> l) (at x within S)" "(f \<longlongrightarrow> l) (at x within T)"
```
```  2951   shows "(f \<longlongrightarrow> l) (at x within (S \<union> T))"
```
```  2952   using assms unfolding at_within_union by (rule filterlim_sup)
```
```  2953
```
```  2954 lemma Lim_Un_univ:
```
```  2955   "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T) \<Longrightarrow>
```
```  2956     S \<union> T = UNIV \<Longrightarrow> (f \<longlongrightarrow> l) (at x)"
```
```  2957   by (metis Lim_Un)
```
```  2958
```
```  2959 text \<open>Interrelations between restricted and unrestricted limits.\<close>
```
```  2960
```
```  2961 lemma Lim_at_imp_Lim_at_within: "(f \<longlongrightarrow> l) (at x) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S)"
```
```  2962   by (metis order_refl filterlim_mono subset_UNIV at_le)
```
```  2963
```
```  2964 lemma eventually_within_interior:
```
```  2965   assumes "x \<in> interior S"
```
```  2966   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"
```
```  2967   (is "?lhs = ?rhs")
```
```  2968 proof
```
```  2969   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
```
```  2970   {
```
```  2971     assume ?lhs
```
```  2972     then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
```
```  2973       by (auto simp: eventually_at_topological)
```
```  2974     with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
```
```  2975       by auto
```
```  2976     then show ?rhs
```
```  2977       by (auto simp: eventually_at_topological)
```
```  2978   next
```
```  2979     assume ?rhs
```
```  2980     then show ?lhs
```
```  2981       by (auto elim: eventually_mono simp: eventually_at_filter)
```
```  2982   }
```
```  2983 qed
```
```  2984
```
```  2985 lemma at_within_interior: "x \<in> interior S \<Longrightarrow> at x within S = at x"
```
```  2986   unfolding filter_eq_iff by (intro allI eventually_within_interior)
```
```  2987
```
```  2988 lemma Lim_within_LIMSEQ:
```
```  2989   fixes a :: "'a::first_countable_topology"
```
```  2990   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
```
```  2991   shows "(X \<longlongrightarrow> L) (at a within T)"
```
```  2992   using assms unfolding tendsto_def [where l=L]
```
```  2993   by (simp add: sequentially_imp_eventually_within)
```
```  2994
```
```  2995 lemma Lim_right_bound:
```
```  2996   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
```
```  2997     'b::{linorder_topology, conditionally_complete_linorder}"
```
```  2998   assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
```
```  2999     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
```
```  3000   shows "(f \<longlongrightarrow> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
```
```  3001 proof (cases "{x<..} \<inter> I = {}")
```
```  3002   case True
```
```  3003   then show ?thesis by simp
```
```  3004 next
```
```  3005   case False
```
```  3006   show ?thesis
```
```  3007   proof (rule order_tendstoI)
```
```  3008     fix a
```
```  3009     assume a: "a < Inf (f ` ({x<..} \<inter> I))"
```
```  3010     {
```
```  3011       fix y
```
```  3012       assume "y \<in> {x<..} \<inter> I"
```
```  3013       with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
```
```  3014         by (auto intro!: cInf_lower bdd_belowI2)
```
```  3015       with a have "a < f y"
```
```  3016         by (blast intro: less_le_trans)
```
```  3017     }
```
```  3018     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
```
```  3019       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
```
```  3020   next
```
```  3021     fix a
```
```  3022     assume "Inf (f ` ({x<..} \<inter> I)) < a"
```
```  3023     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
```
```  3024       by auto
```
```  3025     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
```
```  3026       unfolding eventually_at_right[OF \<open>x < y\<close>] by (metis less_imp_le le_less_trans mono)
```
```  3027     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
```
```  3028       unfolding eventually_at_filter by eventually_elim simp
```
```  3029   qed
```
```  3030 qed
```
```  3031
```
```  3032 text \<open>Another limit point characterization.\<close>
```
```  3033
```
```  3034 lemma limpt_sequential_inj:
```
```  3035   fixes x :: "'a::metric_space"
```
```  3036   shows "x islimpt S \<longleftrightarrow>
```
```  3037          (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> inj f \<and> (f \<longlongrightarrow> x) sequentially)"
```
```  3038          (is "?lhs = ?rhs")
```
```  3039 proof
```
```  3040   assume ?lhs
```
```  3041   then have "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
```
```  3042     by (force simp: islimpt_approachable)
```
```  3043   then obtain y where y: "\<And>e. e>0 \<Longrightarrow> y e \<in> S \<and> y e \<noteq> x \<and> dist (y e) x < e"
```
```  3044     by metis
```
```  3045   define f where "f \<equiv> rec_nat (y 1) (\<lambda>n fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))"
```
```  3046   have [simp]: "f 0 = y 1"
```
```  3047                "f(Suc n) = y (min (inverse(2 ^ (Suc n))) (dist (f n) x))" for n
```
```  3048     by (simp_all add: f_def)
```
```  3049   have f: "f n \<in> S \<and> (f n \<noteq> x) \<and> dist (f n) x < inverse(2 ^ n)" for n
```
```  3050   proof (induction n)
```
```  3051     case 0 show ?case
```
```  3052       by (simp add: y)
```
```  3053   next
```
```  3054     case (Suc n) then show ?case
```
```  3055       apply (auto simp: y)
```
```  3056       by (metis half_gt_zero_iff inverse_positive_iff_positive less_divide_eq_numeral1(1) min_less_iff_conj y zero_less_dist_iff zero_less_numeral zero_less_power)
```
```  3057   qed
```
```  3058   show ?rhs
```
```  3059   proof (rule_tac x=f in exI, intro conjI allI)
```
```  3060     show "\<And>n. f n \<in> S - {x}"
```
```  3061       using f by blast
```
```  3062     have "dist (f n) x < dist (f m) x" if "m < n" for m n
```
```  3063     using that
```
```  3064     proof (induction n)
```
```  3065       case 0 then show ?case by simp
```
```  3066     next
```
```  3067       case (Suc n)
```
```  3068       then consider "m < n" | "m = n" using less_Suc_eq by blast
```
```  3069       then show ?case
```
```  3070       proof cases
```
```  3071         assume "m < n"
```
```  3072         have "dist (f(Suc n)) x = dist (y (min (inverse(2 ^ (Suc n))) (dist (f n) x))) x"
```
```  3073           by simp
```
```  3074         also have "... < dist (f n) x"
```
```  3075           by (metis dist_pos_lt f min.strict_order_iff min_less_iff_conj y)
```
```  3076         also have "... < dist (f m) x"
```
```  3077           using Suc.IH \<open>m < n\<close> by blast
```
```  3078         finally show ?thesis .
```
```  3079       next
```
```  3080         assume "m = n" then show ?case
```
```  3081           by simp (metis dist_pos_lt f half_gt_zero_iff inverse_positive_iff_positive min_less_iff_conj y zero_less_numeral zero_less_power)
```
```  3082       qed
```
```  3083     qed
```
```  3084     then show "inj f"
```
```  3085       by (metis less_irrefl linorder_injI)
```
```  3086     show "f \<longlonglongrightarrow> x"
```
```  3087       apply (rule tendstoI)
```
```  3088       apply (rule_tac c="nat (ceiling(1/e))" in eventually_sequentiallyI)
```
```  3089       apply (rule less_trans [OF f [THEN conjunct2, THEN conjunct2]])
```
```  3090       apply (simp add: field_simps)
```
```  3091       by (meson le_less_trans mult_less_cancel_left not_le of_nat_less_two_power)
```
```  3092   qed
```
```  3093 next
```
```  3094   assume ?rhs
```
```  3095   then show ?lhs
```
```  3096     by (fastforce simp add: islimpt_approachable lim_sequentially)
```
```  3097 qed
```
```  3098
```
```  3099 (*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
```
```  3100 lemma islimpt_sequential:
```
```  3101   fixes x :: "'a::first_countable_topology"
```
```  3102   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f \<longlongrightarrow> x) sequentially)"
```
```  3103     (is "?lhs = ?rhs")
```
```  3104 proof
```
```  3105   assume ?lhs
```
```  3106   from countable_basis_at_decseq[of x] obtain A where A:
```
```  3107       "\<And>i. open (A i)"
```
```  3108       "\<And>i. x \<in> A i"
```
```  3109       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  3110     by blast
```
```  3111   define f where "f n = (SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y)" for n
```
```  3112   {
```
```  3113     fix n
```
```  3114     from \<open>?lhs\<close> have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
```
```  3115       unfolding islimpt_def using A(1,2)[of n] by auto
```
```  3116     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
```
```  3117       unfolding f_def by (rule someI_ex)
```
```  3118     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
```
```  3119   }
```
```  3120   then have "\<forall>n. f n \<in> S - {x}" by auto
```
```  3121   moreover have "(\<lambda>n. f n) \<longlonglongrightarrow> x"
```
```  3122   proof (rule topological_tendstoI)
```
```  3123     fix S
```
```  3124     assume "open S" "x \<in> S"
```
```  3125     from A(3)[OF this] \<open>\<And>n. f n \<in> A n\<close>
```
```  3126     show "eventually (\<lambda>x. f x \<in> S) sequentially"
```
```  3127       by (auto elim!: eventually_mono)
```
```  3128   qed
```
```  3129   ultimately show ?rhs by fast
```
```  3130 next
```
```  3131   assume ?rhs
```
```  3132   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f \<longlonglongrightarrow> x"
```
```  3133     by auto
```
```  3134   show ?lhs
```
```  3135     unfolding islimpt_def
```
```  3136   proof safe
```
```  3137     fix T
```
```  3138     assume "open T" "x \<in> T"
```
```  3139     from lim[THEN topological_tendstoD, OF this] f
```
```  3140     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
```
```  3141       unfolding eventually_sequentially by auto
```
```  3142   qed
```
```  3143 qed
```
```  3144
```
```  3145 lemma Lim_null:
```
```  3146   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  3147   shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
```
```  3148   by (simp add: Lim dist_norm)
```
```  3149
```
```  3150 lemma Lim_null_comparison:
```
```  3151   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  3152   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
```
```  3153   shows "(f \<longlongrightarrow> 0) net"
```
```  3154   using assms(2)
```
```  3155 proof (rule metric_tendsto_imp_tendsto)
```
```  3156   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
```
```  3157     using assms(1) by (rule eventually_mono) (simp add: dist_norm)
```
```  3158 qed
```
```  3159
```
```  3160 lemma Lim_transform_bound:
```
```  3161   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  3162     and g :: "'a \<Rightarrow> 'c::real_normed_vector"
```
```  3163   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
```
```  3164     and "(g \<longlongrightarrow> 0) net"
```
```  3165   shows "(f \<longlongrightarrow> 0) net"
```
```  3166   using assms(1) tendsto_norm_zero [OF assms(2)]
```
```  3167   by (rule Lim_null_comparison)
```
```  3168
```
```  3169 lemma lim_null_mult_right_bounded:
```
```  3170   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
```
```  3171   assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
```
```  3172     shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
```
```  3173 proof -
```
```  3174   have *: "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
```
```  3175     by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
```
```  3176   have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
```
```  3177     apply (rule Lim_null_comparison [OF _ *])
```
```  3178     apply (simp add: eventually_mono [OF g] mult_left_mono)
```
```  3179     done
```
```  3180   then show ?thesis
```
```  3181     by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
```
```  3182 qed
```
```  3183
```
```  3184 lemma lim_null_mult_left_bounded:
```
```  3185   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
```
```  3186   assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
```
```  3187     shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
```
```  3188 proof -
```
```  3189   have *: "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
```
```  3190     by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
```
```  3191   have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
```
```  3192     apply (rule Lim_null_comparison [OF _ *])
```
```  3193     apply (simp add: eventually_mono [OF g] mult_right_mono)
```
```  3194     done
```
```  3195   then show ?thesis
```
```  3196     by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
```
```  3197 qed
```
```  3198
```
```  3199 lemma lim_null_scaleR_bounded:
```
```  3200   assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
```
```  3201     shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
```
```  3202 proof
```
```  3203   fix \<epsilon>::real
```
```  3204   assume "0 < \<epsilon>"
```
```  3205   then have B: "0 < \<epsilon> / (abs B + 1)" by simp
```
```  3206   have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
```
```  3207   proof -
```
```  3208     have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
```
```  3209       by (simp add: mult_left_mono g)
```
```  3210     also have "... \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
```
```  3211       by (simp add: mult_left_mono)
```
```  3212     also have "... < \<epsilon>"
```
```  3213       by (rule f)
```
```  3214     finally show ?thesis .
```
```  3215   qed
```
```  3216   show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
```
```  3217     apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
```
```  3218     apply (auto simp: \<open>0 < \<epsilon>\<close> divide_simps * split: if_split_asm)
```
```  3219     done
```
```  3220 qed
```
```  3221
```
```  3222 text\<open>Deducing things about the limit from the elements.\<close>
```
```  3223
```
```  3224 lemma Lim_in_closed_set:
```
```  3225   assumes "closed S"
```
```  3226     and "eventually (\<lambda>x. f(x) \<in> S) net"
```
```  3227     and "\<not> trivial_limit net" "(f \<longlongrightarrow> l) net"
```
```  3228   shows "l \<in> S"
```
```  3229 proof (rule ccontr)
```
```  3230   assume "l \<notin> S"
```
```  3231   with \<open>closed S\<close> have "open (- S)" "l \<in> - S"
```
```  3232     by (simp_all add: open_Compl)
```
```  3233   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
```
```  3234     by (rule topological_tendstoD)
```
```  3235   with assms(2) have "eventually (\<lambda>x. False) net"
```
```  3236     by (rule eventually_elim2) simp
```
```  3237   with assms(3) show "False"
```
```  3238     by (simp add: eventually_False)
```
```  3239 qed
```
```  3240
```
```  3241 text\<open>Need to prove closed(cball(x,e)) before deducing this as a corollary.\<close>
```
```  3242
```
```  3243 lemma Lim_dist_ubound:
```
```  3244   assumes "\<not>(trivial_limit net)"
```
```  3245     and "(f \<longlongrightarrow> l) net"
```
```  3246     and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
```
```  3247   shows "dist a l \<le> e"
```
```  3248   using assms by (fast intro: tendsto_le tendsto_intros)
```
```  3249
```
```  3250 lemma Lim_norm_ubound:
```
```  3251   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  3252   assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
```
```  3253   shows "norm(l) \<le> e"
```
```  3254   using assms by (fast intro: tendsto_le tendsto_intros)
```
```  3255
```
```  3256 lemma Lim_norm_lbound:
```
```  3257   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  3258   assumes "\<not> trivial_limit net"
```
```  3259     and "(f \<longlongrightarrow> l) net"
```
```  3260     and "eventually (\<lambda>x. e \<le> norm (f x)) net"
```
```  3261   shows "e \<le> norm l"
```
```  3262   using assms by (fast intro: tendsto_le tendsto_intros)
```
```  3263
```
```  3264 text\<open>Limit under bilinear function\<close>
```
```  3265
```
```  3266 lemma Lim_bilinear:
```
```  3267   assumes "(f \<longlongrightarrow> l) net"
```
```  3268     and "(g \<longlongrightarrow> m) net"
```
```  3269     and "bounded_bilinear h"
```
```  3270   shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
```
```  3271   using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
```
```  3272   by (rule bounded_bilinear.tendsto)
```
```  3273
```
```  3274 text\<open>These are special for limits out of the same vector space.\<close>
```
```  3275
```
```  3276 lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
```
```  3277   unfolding id_def by (rule tendsto_ident_at)
```
```  3278
```
```  3279 lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
```
```  3280   unfolding id_def by (rule tendsto_ident_at)
```
```  3281
```
```  3282 lemma Lim_at_zero:
```
```  3283   fixes a :: "'a::real_normed_vector"
```
```  3284     and l :: "'b::topological_space"
```
```  3285   shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
```
```  3286   using LIM_offset_zero LIM_offset_zero_cancel ..
```
```  3287
```
```  3288 text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
```
```  3289
```
```  3290 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
```
```  3291   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
```
```  3292
```
```  3293 lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
```
```  3294   by (rule tendsto_Lim) (auto intro: tendsto_intros)
```
```  3295
```
```  3296 lemma netlimit_at:
```
```  3297   fixes a :: "'a::{perfect_space,t2_space}"
```
```  3298   shows "netlimit (at a) = a"
```
```  3299   using netlimit_within [of a UNIV] by simp
```
```  3300
```
```  3301 lemma lim_within_interior:
```
```  3302   "x \<in> interior S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at x)"
```
```  3303   by (metis at_within_interior)
```
```  3304
```
```  3305 lemma netlimit_within_interior:
```
```  3306   fixes x :: "'a::{t2_space,perfect_space}"
```
```  3307   assumes "x \<in> interior S"
```
```  3308   shows "netlimit (at x within S) = x"
```
```  3309   using assms by (metis at_within_interior netlimit_at)
```
```  3310
```
```  3311 lemma netlimit_at_vector:
```
```  3312   fixes a :: "'a::real_normed_vector"
```
```  3313   shows "netlimit (at a) = a"
```
```  3314 proof (cases "\<exists>x. x \<noteq> a")
```
```  3315   case True then obtain x where x: "x \<noteq> a" ..
```
```  3316   have "\<not> trivial_limit (at a)"
```
```  3317     unfolding trivial_limit_def eventually_at dist_norm
```
```  3318     apply clarsimp
```
```  3319     apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
```
```  3320     apply (simp add: norm_sgn sgn_zero_iff x)
```
```  3321     done
```
```  3322   then show ?thesis
```
```  3323     by (rule netlimit_within [of a UNIV])
```
```  3324 qed simp
```
```  3325
```
```  3326
```
```  3327 text\<open>Useful lemmas on closure and set of possible sequential limits.\<close>
```
```  3328
```
```  3329 lemma closure_sequential:
```
```  3330   fixes l :: "'a::first_countable_topology"
```
```  3331   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially)"
```
```  3332   (is "?lhs = ?rhs")
```
```  3333 proof
```
```  3334   assume "?lhs"
```
```  3335   moreover
```
```  3336   {
```
```  3337     assume "l \<in> S"
```
```  3338     then have "?rhs" using tendsto_const[of l sequentially] by auto
```
```  3339   }
```
```  3340   moreover
```
```  3341   {
```
```  3342     assume "l islimpt S"
```
```  3343     then have "?rhs" unfolding islimpt_sequential by auto
```
```  3344   }
```
```  3345   ultimately show "?rhs"
```
```  3346     unfolding closure_def by auto
```
```  3347 next
```
```  3348   assume "?rhs"
```
```  3349   then show "?lhs" unfolding closure_def islimpt_sequential by auto
```
```  3350 qed
```
```  3351
```
```  3352 lemma closed_sequential_limits:
```
```  3353   fixes S :: "'a::first_countable_topology set"
```
```  3354   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
```
```  3355 by (metis closure_sequential closure_subset_eq subset_iff)
```
```  3356
```
```  3357 lemma closure_approachable:
```
```  3358   fixes S :: "'a::metric_space set"
```
```  3359   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
```
```  3360   apply (auto simp add: closure_def islimpt_approachable)
```
```  3361   apply (metis dist_self)
```
```  3362   done
```
```  3363
```
```  3364 lemma closed_approachable:
```
```  3365   fixes S :: "'a::metric_space set"
```
```  3366   shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
```
```  3367   by (metis closure_closed closure_approachable)
```
```  3368
```
```  3369 lemma closure_contains_Inf:
```
```  3370   fixes S :: "real set"
```
```  3371   assumes "S \<noteq> {}" "bdd_below S"
```
```  3372   shows "Inf S \<in> closure S"
```
```  3373 proof -
```
```  3374   have *: "\<forall>x\<in>S. Inf S \<le> x"
```
```  3375     using cInf_lower[of _ S] assms by metis
```
```  3376   {
```
```  3377     fix e :: real
```
```  3378     assume "e > 0"
```
```  3379     then have "Inf S < Inf S + e" by simp
```
```  3380     with assms obtain x where "x \<in> S" "x < Inf S + e"
```
```  3381       by (subst (asm) cInf_less_iff) auto
```
```  3382     with * have "\<exists>x\<in>S. dist x (Inf S) < e"
```
```  3383       by (intro bexI[of _ x]) (auto simp add: dist_real_def)
```
```  3384   }
```
```  3385   then show ?thesis unfolding closure_approachable by auto
```
```  3386 qed
```
```  3387
```
```  3388 lemma closed_contains_Inf:
```
```  3389   fixes S :: "real set"
```
```  3390   shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"
```
```  3391   by (metis closure_contains_Inf closure_closed)
```
```  3392
```
```  3393 lemma closed_subset_contains_Inf:
```
```  3394   fixes A C :: "real set"
```
```  3395   shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<in> C"
```
```  3396   by (metis closure_contains_Inf closure_minimal subset_eq)
```
```  3397
```
```  3398 lemma atLeastAtMost_subset_contains_Inf:
```
```  3399   fixes A :: "real set" and a b :: real
```
```  3400   shows "A \<noteq> {} \<Longrightarrow> a \<le> b \<Longrightarrow> A \<subseteq> {a..b} \<Longrightarrow> Inf A \<in> {a..b}"
```
```  3401   by (rule closed_subset_contains_Inf)
```
```  3402      (auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a])
```
```  3403
```
```  3404 lemma not_trivial_limit_within_ball:
```
```  3405   "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
```
```  3406   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3407 proof
```
```  3408   show ?rhs if ?lhs
```
```  3409   proof -
```
```  3410     {
```
```  3411       fix e :: real
```
```  3412       assume "e > 0"
```
```  3413       then obtain y where "y \<in> S - {x}" and "dist y x < e"
```
```  3414         using \<open>?lhs\<close> not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
```
```  3415         by auto
```
```  3416       then have "y \<in> S \<inter> ball x e - {x}"
```
```  3417         unfolding ball_def by (simp add: dist_commute)
```
```  3418       then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
```
```  3419     }
```
```  3420     then show ?thesis by auto
```
```  3421   qed
```
```  3422   show ?lhs if ?rhs
```
```  3423   proof -
```
```  3424     {
```
```  3425       fix e :: real
```
```  3426       assume "e > 0"
```
```  3427       then obtain y where "y \<in> S \<inter> ball x e - {x}"
```
```  3428         using \<open>?rhs\<close> by blast
```
```  3429       then have "y \<in> S - {x}" and "dist y x < e"
```
```  3430         unfolding ball_def by (simp_all add: dist_commute)
```
```  3431       then have "\<exists>y \<in> S - {x}. dist y x < e"
```
```  3432         by auto
```
```  3433     }
```
```  3434     then show ?thesis
```
```  3435       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
```
```  3436       by auto
```
```  3437   qed
```
```  3438 qed
```
```  3439
```
```  3440
```
```  3441 subsection \<open>Infimum Distance\<close>
```
```  3442
```
```  3443 definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"
```
```  3444
```
```  3445 lemma bdd_below_infdist[intro, simp]: "bdd_below (dist x`A)"
```
```  3446   by (auto intro!: zero_le_dist)
```
```  3447
```
```  3448 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"
```
```  3449   by (simp add: infdist_def)
```
```  3450
```
```  3451 lemma infdist_nonneg: "0 \<le> infdist x A"
```
```  3452   by (auto simp add: infdist_def intro: cINF_greatest)
```
```  3453
```
```  3454 lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"
```
```  3455   by (auto intro: cINF_lower simp add: infdist_def)
```
```  3456
```
```  3457 lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"
```
```  3458   by (auto intro!: cINF_lower2 simp add: infdist_def)
```
```  3459
```
```  3460 lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"
```
```  3461   by (auto intro!: antisym infdist_nonneg infdist_le2)
```
```  3462
```
```  3463 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
```
```  3464 proof (cases "A = {}")
```
```  3465   case True
```
```  3466   then show ?thesis by (simp add: infdist_def)
```
```  3467 next
```
```  3468   case False
```
```  3469   then obtain a where "a \<in> A" by auto
```
```  3470   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
```
```  3471   proof (rule cInf_greatest)
```
```  3472     from \<open>A \<noteq> {}\<close> show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"
```
```  3473       by simp
```
```  3474     fix d
```
```  3475     assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
```
```  3476     then obtain a where d: "d = dist x y + dist y a" "a \<in> A"
```
```  3477       by auto
```
```  3478     show "infdist x A \<le> d"
```
```  3479       unfolding infdist_notempty[OF \<open>A \<noteq> {}\<close>]
```
```  3480     proof (rule cINF_lower2)
```
```  3481       show "a \<in> A" by fact
```
```  3482       show "dist x a \<le> d"
```
```  3483         unfolding d by (rule dist_triangle)
```
```  3484     qed simp
```
```  3485   qed
```
```  3486   also have "\<dots> = dist x y + infdist y A"
```
```  3487   proof (rule cInf_eq, safe)
```
```  3488     fix a
```
```  3489     assume "a \<in> A"
```
```  3490     then show "dist x y + infdist y A \<le> dist x y + dist y a"
```
```  3491       by (auto intro: infdist_le)
```
```  3492   next
```
```  3493     fix i
```
```  3494     assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
```
```  3495     then have "i - dist x y \<le> infdist y A"
```
```  3496       unfolding infdist_notempty[OF \<open>A \<noteq> {}\<close>] using \<open>a \<in> A\<close>
```
```  3497       by (intro cINF_greatest) (auto simp: field_simps)
```
```  3498     then show "i \<le> dist x y + infdist y A"
```
```  3499       by simp
```
```  3500   qed
```
```  3501   finally show ?thesis by simp
```
```  3502 qed
```
```  3503
```
```  3504 lemma in_closure_iff_infdist_zero:
```
```  3505   assumes "A \<noteq> {}"
```
```  3506   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
```
```  3507 proof
```
```  3508   assume "x \<in> closure A"
```
```  3509   show "infdist x A = 0"
```
```  3510   proof (rule ccontr)
```
```  3511     assume "infdist x A \<noteq> 0"
```
```  3512     with infdist_nonneg[of x A] have "infdist x A > 0"
```
```  3513       by auto
```
```  3514     then have "ball x (infdist x A) \<inter> closure A = {}"
```
```  3515       apply auto
```
```  3516       apply (metis \<open>x \<in> closure A\<close> closure_approachable dist_commute infdist_le not_less)
```
```  3517       done
```
```  3518     then have "x \<notin> closure A"
```
```  3519       by (metis \<open>0 < infdist x A\<close> centre_in_ball disjoint_iff_not_equal)
```
```  3520     then show False using \<open>x \<in> closure A\<close> by simp
```
```  3521   qed
```
```  3522 next
```
```  3523   assume x: "infdist x A = 0"
```
```  3524   then obtain a where "a \<in> A"
```
```  3525     by atomize_elim (metis all_not_in_conv assms)
```
```  3526   show "x \<in> closure A"
```
```  3527     unfolding closure_approachable
```
```  3528     apply safe
```
```  3529   proof (rule ccontr)
```
```  3530     fix e :: real
```
```  3531     assume "e > 0"
```
```  3532     assume "\<not> (\<exists>y\<in>A. dist y x < e)"
```
```  3533     then have "infdist x A \<ge> e" using \<open>a \<in> A\<close>
```
```  3534       unfolding infdist_def
```
```  3535       by (force simp: dist_commute intro: cINF_greatest)
```
```  3536     with x \<open>e > 0\<close> show False by auto
```
```  3537   qed
```
```  3538 qed
```
```  3539
```
```  3540 lemma in_closed_iff_infdist_zero:
```
```  3541   assumes "closed A" "A \<noteq> {}"
```
```  3542   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"
```
```  3543 proof -
```
```  3544   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
```
```  3545     by (rule in_closure_iff_infdist_zero) fact
```
```  3546   with assms show ?thesis by simp
```
```  3547 qed
```
```  3548
```
```  3549 lemma tendsto_infdist [tendsto_intros]:
```
```  3550   assumes f: "(f \<longlongrightarrow> l) F"
```
```  3551   shows "((\<lambda>x. infdist (f x) A) \<longlongrightarrow> infdist l A) F"
```
```  3552 proof (rule tendstoI)
```
```  3553   fix e ::real
```
```  3554   assume "e > 0"
```
```  3555   from tendstoD[OF f this]
```
```  3556   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
```
```  3557   proof (eventually_elim)
```
```  3558     fix x
```
```  3559     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
```
```  3560     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"
```
```  3561       by (simp add: dist_commute dist_real_def)
```
```  3562     also assume "dist (f x) l < e"
```
```  3563     finally show "dist (infdist (f x) A) (infdist l A) < e" .
```
```  3564   qed
```
```  3565 qed
```
```  3566
```
```  3567 text\<open>Some other lemmas about sequences.\<close>
```
```  3568
```
```  3569 lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)
```
```  3570   assumes "eventually (\<lambda>i. P i) sequentially"
```
```  3571   shows "eventually (\<lambda>i. P (i + k)) sequentially"
```
```  3572   using assms by (rule eventually_sequentially_seg [THEN iffD2])
```
```  3573
```
```  3574 lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)
```
```  3575   "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) \<longlongrightarrow> l) sequentially"
```
```  3576   apply (erule filterlim_compose)
```
```  3577   apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)
```
```  3578   apply arith
```
```  3579   done
```
```  3580
```
```  3581 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) \<longlongrightarrow> 0) sequentially"
```
```  3582   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)
```
```  3583
```
```  3584 subsection \<open>More properties of closed balls\<close>
```
```  3585
```
```  3586 lemma closed_cball [iff]: "closed (cball x e)"
```
```  3587 proof -
```
```  3588   have "closed (dist x -` {..e})"
```
```  3589     by (intro closed_vimage closed_atMost continuous_intros)
```
```  3590   also have "dist x -` {..e} = cball x e"
```
```  3591     by auto
```
```  3592   finally show ?thesis .
```
```  3593 qed
```
```  3594
```
```  3595 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
```
```  3596 proof -
```
```  3597   {
```
```  3598     fix x and e::real
```
```  3599     assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
```
```  3600     then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
```
```  3601   }
```
```  3602   moreover
```
```  3603   {
```
```  3604     fix x and e::real
```
```  3605     assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
```
```  3606     then have "\<exists>d>0. ball x d \<subseteq> S"
```
```  3607       unfolding subset_eq
```
```  3608       apply(rule_tac x="e/2" in exI)
```
```  3609       apply auto
```
```  3610       done
```
```  3611   }
```
```  3612   ultimately show ?thesis
```
```  3613     unfolding open_contains_ball by auto
```
```  3614 qed
```
```  3615
```
```  3616 lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
```
```  3617   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
```
```  3618
```
```  3619 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
```
```  3620   apply (simp add: interior_def, safe)
```
```  3621   apply (force simp add: open_contains_cball)
```
```  3622   apply (rule_tac x="ball x e" in exI)
```
```  3623   apply (simp add: subset_trans [OF ball_subset_cball])
```
```  3624   done
```
```  3625
```
```  3626 lemma islimpt_ball:
```
```  3627   fixes x y :: "'a::{real_normed_vector,perfect_space}"
```
```  3628   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"
```
```  3629   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3630 proof
```
```  3631   show ?rhs if ?lhs
```
```  3632   proof
```
```  3633     {
```
```  3634       assume "e \<le> 0"
```
```  3635       then have *: "ball x e = {}"
```
```  3636         using ball_eq_empty[of x e] by auto
```
```  3637       have False using \<open>?lhs\<close>
```
```  3638         unfolding * using islimpt_EMPTY[of y] by auto
```
```  3639     }
```
```  3640     then show "e > 0" by (metis not_less)
```
```  3641     show "y \<in> cball x e"
```
```  3642       using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
```
```  3643         ball_subset_cball[of x e] \<open>?lhs\<close>
```
```  3644       unfolding closed_limpt by auto
```
```  3645   qed
```
```  3646   show ?lhs if ?rhs
```
```  3647   proof -
```
```  3648     from that have "e > 0" by auto
```
```  3649     {
```
```  3650       fix d :: real
```
```  3651       assume "d > 0"
```
```  3652       have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  3653       proof (cases "d \<le> dist x y")
```
```  3654         case True
```
```  3655         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  3656         proof (cases "x = y")
```
```  3657           case True
```
```  3658           then have False
```
```  3659             using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto
```
```  3660           then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  3661             by auto
```
```  3662         next
```
```  3663           case False
```
```  3664           have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
```
```  3665             norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
```
```  3666             unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
```
```  3667             by auto
```
```  3668           also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
```
```  3669             using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
```
```  3670             unfolding scaleR_minus_left scaleR_one
```
```  3671             by (auto simp add: norm_minus_commute)
```
```  3672           also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
```
```  3673             unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
```
```  3674             unfolding distrib_right using \<open>x\<noteq>y\<close>  by auto
```
```  3675           also have "\<dots> \<le> e - d/2" using \<open>d \<le> dist x y\<close> and \<open>d>0\<close> and \<open>?rhs\<close>
```
```  3676             by (auto simp add: dist_norm)
```
```  3677           finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using \<open>d>0\<close>
```
```  3678             by auto
```
```  3679           moreover
```
```  3680           have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
```
```  3681             using \<open>x\<noteq>y\<close>[unfolded dist_nz] \<open>d>0\<close> unfolding scaleR_eq_0_iff
```
```  3682             by (auto simp add: dist_commute)
```
```  3683           moreover
```
```  3684           have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
```
```  3685             unfolding dist_norm
```
```  3686             apply simp
```
```  3687             unfolding norm_minus_cancel
```
```  3688             using \<open>d > 0\<close> \<open>x\<noteq>y\<close>[unfolded dist_nz] dist_commute[of x y]
```
```  3689             unfolding dist_norm
```
```  3690             apply auto
```
```  3691             done
```
```  3692           ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  3693             apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)
```
```  3694             apply auto
```
```  3695             done
```
```  3696         qed
```
```  3697       next
```
```  3698         case False
```
```  3699         then have "d > dist x y" by auto
```
```  3700         show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  3701         proof (cases "x = y")
```
```  3702           case True
```
```  3703           obtain z where **: "z \<noteq> y" "dist z y < min e d"
```
```  3704             using perfect_choose_dist[of "min e d" y]
```
```  3705             using \<open>d > 0\<close> \<open>e>0\<close> by auto
```
```  3706           show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  3707             unfolding \<open>x = y\<close>
```
```  3708             using \<open>z \<noteq> y\<close> **
```
```  3709             apply (rule_tac x=z in bexI)
```
```  3710             apply (auto simp add: dist_commute)
```
```  3711             done
```
```  3712         next
```
```  3713           case False
```
```  3714           then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  3715             using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close>
```
```  3716             apply (rule_tac x=x in bexI)
```
```  3717             apply auto
```
```  3718             done
```
```  3719         qed
```
```  3720       qed
```
```  3721     }
```
```  3722     then show ?thesis
```
```  3723       unfolding mem_cball islimpt_approachable mem_ball by auto
```
```  3724   qed
```
```  3725 qed
```
```  3726
```
```  3727 lemma closure_ball_lemma:
```
```  3728   fixes x y :: "'a::real_normed_vector"
```
```  3729   assumes "x \<noteq> y"
```
```  3730   shows "y islimpt ball x (dist x y)"
```
```  3731 proof (rule islimptI)
```
```  3732   fix T
```
```  3733   assume "y \<in> T" "open T"
```
```  3734   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
```
```  3735     unfolding open_dist by fast
```
```  3736   (* choose point between x and y, within distance r of y. *)
```
```  3737   define k where "k = min 1 (r / (2 * dist x y))"
```
```  3738   define z where "z = y + scaleR k (x - y)"
```
```  3739   have z_def2: "z = x + scaleR (1 - k) (y - x)"
```
```  3740     unfolding z_def by (simp add: algebra_simps)
```
```  3741   have "dist z y < r"
```
```  3742     unfolding z_def k_def using \<open>0 < r\<close>
```
```  3743     by (simp add: dist_norm min_def)
```
```  3744   then have "z \<in> T"
```
```  3745     using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp
```
```  3746   have "dist x z < dist x y"
```
```  3747     unfolding z_def2 dist_norm
```
```  3748     apply (simp add: norm_minus_commute)
```
```  3749     apply (simp only: dist_norm [symmetric])
```
```  3750     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
```
```  3751     apply (rule mult_strict_right_mono)
```
```  3752     apply (simp add: k_def \<open>0 < r\<close> \<open>x \<noteq> y\<close>)
```
```  3753     apply (simp add: \<open>x \<noteq> y\<close>)
```
```  3754     done
```
```  3755   then have "z \<in> ball x (dist x y)"
```
```  3756     by simp
```
```  3757   have "z \<noteq> y"
```
```  3758     unfolding z_def k_def using \<open>x \<noteq> y\<close> \<open>0 < r\<close>
```
```  3759     by (simp add: min_def)
```
```  3760   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
```
```  3761     using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>
```
```  3762     by fast
```
```  3763 qed
```
```  3764
```
```  3765 lemma closure_ball [simp]:
```
```  3766   fixes x :: "'a::real_normed_vector"
```
```  3767   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
```
```  3768   apply (rule equalityI)
```
```  3769   apply (rule closure_minimal)
```
```  3770   apply (rule ball_subset_cball)
```
```  3771   apply (rule closed_cball)
```
```  3772   apply (rule subsetI, rename_tac y)
```
```  3773   apply (simp add: le_less [where 'a=real])
```
```  3774   apply (erule disjE)
```
```  3775   apply (rule subsetD [OF closure_subset], simp)
```
```  3776   apply (simp add: closure_def)
```
```  3777   apply clarify
```
```  3778   apply (rule closure_ball_lemma)
```
```  3779   apply (simp add: zero_less_dist_iff)
```
```  3780   done
```
```  3781
```
```  3782 (* In a trivial vector space, this fails for e = 0. *)
```
```  3783 lemma interior_cball [simp]:
```
```  3784   fixes x :: "'a::{real_normed_vector, perfect_space}"
```
```  3785   shows "interior (cball x e) = ball x e"
```
```  3786 proof (cases "e \<ge> 0")
```
```  3787   case False note cs = this
```
```  3788   from cs have null: "ball x e = {}"
```
```  3789     using ball_empty[of e x] by auto
```
```  3790   moreover
```
```  3791   {
```
```  3792     fix y
```
```  3793     assume "y \<in> cball x e"
```
```  3794     then have False
```
```  3795       by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball subset_antisym subset_ball)
```
```  3796   }
```
```  3797   then have "cball x e = {}" by auto
```
```  3798   then have "interior (cball x e) = {}"
```
```  3799     using interior_empty by auto
```
```  3800   ultimately show ?thesis by blast
```
```  3801 next
```
```  3802   case True note cs = this
```
```  3803   have "ball x e \<subseteq> cball x e"
```
```  3804     using ball_subset_cball by auto
```
```  3805   moreover
```
```  3806   {
```
```  3807     fix S y
```
```  3808     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
```
```  3809     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"
```
```  3810       unfolding open_dist by blast
```
```  3811     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
```
```  3812       using perfect_choose_dist [of d] by auto
```
```  3813     have "xa \<in> S"
```
```  3814       using d[THEN spec[where x = xa]]
```
```  3815       using xa by (auto simp add: dist_commute)
```
```  3816     then have xa_cball: "xa \<in> cball x e"
```
```  3817       using as(1) by auto
```
```  3818     then have "y \<in> ball x e"
```
```  3819     proof (cases "x = y")
```
```  3820       case True
```
```  3821       then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
```
```  3822       then show "y \<in> ball x e"
```
```  3823         using \<open>x = y \<close> by simp
```
```  3824     next
```
```  3825       case False
```
```  3826       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"
```
```  3827         unfolding dist_norm
```
```  3828         using \<open>d>0\<close> norm_ge_zero[of "y - x"] \<open>x \<noteq> y\<close> by auto
```
```  3829       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"
```
```  3830         using d as(1)[unfolded subset_eq] by blast
```
```  3831       have "y - x \<noteq> 0" using \<open>x \<noteq> y\<close> by auto
```
```  3832       hence **:"d / (2 * norm (y - x)) > 0"
```
```  3833         unfolding zero_less_norm_iff[symmetric] using \<open>d>0\<close> by auto
```
```  3834       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =
```
```  3835         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
```
```  3836         by (auto simp add: dist_norm algebra_simps)
```
```  3837       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
```
```  3838         by (auto simp add: algebra_simps)
```
```  3839       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
```
```  3840         using ** by auto
```
```  3841       also have "\<dots> = (dist y x) + d/2"
```
```  3842         using ** by (auto simp add: distrib_right dist_norm)
```
```  3843       finally have "e \<ge> dist x y +d/2"
```
```  3844         using *[unfolded mem_cball] by (auto simp add: dist_commute)
```
```  3845       then show "y \<in> ball x e"
```
```  3846         unfolding mem_ball using \<open>d>0\<close> by auto
```
```  3847     qed
```
```  3848   }
```
```  3849   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"
```
```  3850     by auto
```
```  3851   ultimately show ?thesis
```
```  3852     using interior_unique[of "ball x e" "cball x e"]
```
```  3853     using open_ball[of x e]
```
```  3854     by auto
```
```  3855 qed
```
```  3856
```
```  3857 lemma interior_ball [simp]: "interior (ball x e) = ball x e"
```
```  3858   by (simp add: interior_open)
```
```  3859
```
```  3860 lemma frontier_ball [simp]:
```
```  3861   fixes a :: "'a::real_normed_vector"
```
```  3862   shows "0 < e \<Longrightarrow> frontier (ball a e) = sphere a e"
```
```  3863   by (force simp: frontier_def)
```
```  3864
```
```  3865 lemma frontier_cball [simp]:
```
```  3866   fixes a :: "'a::{real_normed_vector, perfect_space}"
```
```  3867   shows "frontier (cball a e) = sphere a e"
```
```  3868   by (force simp: frontier_def)
```
```  3869
```
```  3870 lemma cball_eq_empty [simp]: "cball x e = {} \<longleftrightarrow> e < 0"
```
```  3871   apply (simp add: set_eq_iff not_le)
```
```  3872   apply (metis zero_le_dist dist_self order_less_le_trans)
```
```  3873   done
```
```  3874
```
```  3875 lemma cball_empty [simp]: "e < 0 \<Longrightarrow> cball x e = {}"
```
```  3876   by (simp add: cball_eq_empty)
```
```  3877
```
```  3878 lemma cball_eq_sing:
```
```  3879   fixes x :: "'a::{metric_space,perfect_space}"
```
```  3880   shows "cball x e = {x} \<longleftrightarrow> e = 0"
```
```  3881 proof (rule linorder_cases)
```
```  3882   assume e: "0 < e"
```
```  3883   obtain a where "a \<noteq> x" "dist a x < e"
```
```  3884     using perfect_choose_dist [OF e] by auto
```
```  3885   then have "a \<noteq> x" "dist x a \<le> e"
```
```  3886     by (auto simp add: dist_commute)
```
```  3887   with e show ?thesis by (auto simp add: set_eq_iff)
```
```  3888 qed auto
```
```  3889
```
```  3890 lemma cball_sing:
```
```  3891   fixes x :: "'a::metric_space"
```
```  3892   shows "e = 0 \<Longrightarrow> cball x e = {x}"
```
```  3893   by (auto simp add: set_eq_iff)
```
```  3894
```
```  3895 lemma ball_divide_subset: "d \<ge> 1 \<Longrightarrow> ball x (e/d) \<subseteq> ball x e"
```
```  3896   apply (cases "e \<le> 0")
```
```  3897   apply (simp add: ball_empty divide_simps)
```
```  3898   apply (rule subset_ball)
```
```  3899   apply (simp add: divide_simps)
```
```  3900   done
```
```  3901
```
```  3902 lemma ball_divide_subset_numeral: "ball x (e / numeral w) \<subseteq> ball x e"
```
```  3903   using ball_divide_subset one_le_numeral by blast
```
```  3904
```
```  3905 lemma cball_divide_subset: "d \<ge> 1 \<Longrightarrow> cball x (e/d) \<subseteq> cball x e"
```
```  3906   apply (cases "e < 0")
```
```  3907   apply (simp add: divide_simps)
```
```  3908   apply (rule subset_cball)
```
```  3909   apply (metis div_by_1 frac_le not_le order_refl zero_less_one)
```
```  3910   done
```
```  3911
```
```  3912 lemma cball_divide_subset_numeral: "cball x (e / numeral w) \<subseteq> cball x e"
```
```  3913   using cball_divide_subset one_le_numeral by blast
```
```  3914
```
```  3915
```
```  3916 subsection \<open>Boundedness\<close>
```
```  3917
```
```  3918   (* FIXME: This has to be unified with BSEQ!! *)
```
```  3919 definition (in metric_space) bounded :: "'a set \<Rightarrow> bool"
```
```  3920   where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
```
```  3921
```
```  3922 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
```
```  3923   unfolding bounded_def subset_eq  by auto (meson order_trans zero_le_dist)
```
```  3924
```
```  3925 lemma bounded_subset_ballD:
```
```  3926   assumes "bounded S" shows "\<exists>r. 0 < r \<and> S \<subseteq> ball x r"
```
```  3927 proof -
```
```  3928   obtain e::real and y where "S \<subseteq> cball y e"  "0 \<le> e"
```
```  3929     using assms by (auto simp: bounded_subset_cball)
```
```  3930   then show ?thesis
```
```  3931     apply (rule_tac x="dist x y + e + 1" in exI)
```
```  3932     apply (simp add: add.commute add_pos_nonneg)
```
```  3933     apply (erule subset_trans)
```
```  3934     apply (clarsimp simp add: cball_def)
```
```  3935     by (metis add_le_cancel_right add_strict_increasing dist_commute dist_triangle_le zero_less_one)
```
```  3936 qed
```
```  3937
```
```  3938 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
```
```  3939   unfolding bounded_def
```
```  3940   by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
```
```  3941
```
```  3942 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
```
```  3943   unfolding bounded_any_center [where a=0]
```
```  3944   by (simp add: dist_norm)
```
```  3945
```
```  3946 lemma bdd_above_norm: "bdd_above (norm ` X) \<longleftrightarrow> bounded X"
```
```  3947   by (simp add: bounded_iff bdd_above_def)
```
```  3948
```
```  3949 lemma bounded_realI:
```
```  3950   assumes "\<forall>x\<in>s. \<bar>x::real\<bar> \<le> B"
```
```  3951   shows "bounded s"
```
```  3952   unfolding bounded_def dist_real_def
```
```  3953   by (metis abs_minus_commute assms diff_0_right)
```
```  3954
```
```  3955 lemma bounded_empty [simp]: "bounded {}"
```
```  3956   by (simp add: bounded_def)
```
```  3957
```
```  3958 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"
```
```  3959   by (metis bounded_def subset_eq)
```
```  3960
```
```  3961 lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"
```
```  3962   by (metis bounded_subset interior_subset)
```
```  3963
```
```  3964 lemma bounded_closure[intro]:
```
```  3965   assumes "bounded S"
```
```  3966   shows "bounded (closure S)"
```
```  3967 proof -
```
```  3968   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"
```
```  3969     unfolding bounded_def by auto
```
```  3970   {
```
```  3971     fix y
```
```  3972     assume "y \<in> closure S"
```
```  3973     then obtain f where f: "\<forall>n. f n \<in> S"  "(f \<longlongrightarrow> y) sequentially"
```
```  3974       unfolding closure_sequential by auto
```
```  3975     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
```
```  3976     then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
```
```  3977       by (simp add: f(1))
```
```  3978     have "dist x y \<le> a"
```
```  3979       apply (rule Lim_dist_ubound [of sequentially f])
```
```  3980       apply (rule trivial_limit_sequentially)
```
```  3981       apply (rule f(2))
```
```  3982       apply fact
```
```  3983       done
```
```  3984   }
```
```  3985   then show ?thesis
```
```  3986     unfolding bounded_def by auto
```
```  3987 qed
```
```  3988
```
```  3989 lemma bounded_closure_image: "bounded (f ` closure S) \<Longrightarrow> bounded (f ` S)"
```
```  3990   by (simp add: bounded_subset closure_subset image_mono)
```
```  3991
```
```  3992 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
```
```  3993   apply (simp add: bounded_def)
```
```  3994   apply (rule_tac x=x in exI)
```
```  3995   apply (rule_tac x=e in exI)
```
```  3996   apply auto
```
```  3997   done
```
```  3998
```
```  3999 lemma bounded_ball[simp,intro]: "bounded (ball x e)"
```
```  4000   by (metis ball_subset_cball bounded_cball bounded_subset)
```
```  4001
```
```  4002 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
```
```  4003   by (auto simp add: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj)
```
```  4004
```
```  4005 lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"
```
```  4006   by (induct rule: finite_induct[of F]) auto
```
```  4007
```
```  4008 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
```
```  4009   by (induct set: finite) auto
```
```  4010
```
```  4011 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
```
```  4012 proof -
```
```  4013   have "\<forall>y\<in>{x}. dist x y \<le> 0"
```
```  4014     by simp
```
```  4015   then have "bounded {x}"
```
```  4016     unfolding bounded_def by fast
```
```  4017   then show ?thesis
```
```  4018     by (metis insert_is_Un bounded_Un)
```
```  4019 qed
```
```  4020
```
```  4021 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
```
```  4022   by (induct set: finite) simp_all
```
```  4023
```
```  4024 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
```
```  4025   apply (simp add: bounded_iff)
```
```  4026   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + \<bar>y\<bar> \<and> (x \<le> y \<longrightarrow> x \<le> 1 + \<bar>y\<bar>)")
```
```  4027   apply metis
```
```  4028   apply arith
```
```  4029   done
```
```  4030
```
```  4031 lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
```
```  4032   apply (simp add: bounded_pos)
```
```  4033   apply (safe; rule_tac x="b+1" in exI; force)
```
```  4034   done
```
```  4035
```
```  4036 lemma Bseq_eq_bounded:
```
```  4037   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  4038   shows "Bseq f \<longleftrightarrow> bounded (range f)"
```
```  4039   unfolding Bseq_def bounded_pos by auto
```
```  4040
```
```  4041 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
```
```  4042   by (metis Int_lower1 Int_lower2 bounded_subset)
```
```  4043
```
```  4044 lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"
```
```  4045   by (metis Diff_subset bounded_subset)
```
```  4046
```
```  4047 lemma not_bounded_UNIV[simp]:
```
```  4048   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
```
```  4049 proof (auto simp add: bounded_pos not_le)
```
```  4050   obtain x :: 'a where "x \<noteq> 0"
```
```  4051     using perfect_choose_dist [OF zero_less_one] by fast
```
```  4052   fix b :: real
```
```  4053   assume b: "b >0"
```
```  4054   have b1: "b +1 \<ge> 0"
```
```  4055     using b by simp
```
```  4056   with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
```
```  4057     by (simp add: norm_sgn)
```
```  4058   then show "\<exists>x::'a. b < norm x" ..
```
```  4059 qed
```
```  4060
```
```  4061 corollary cobounded_imp_unbounded:
```
```  4062     fixes S :: "'a::{real_normed_vector, perfect_space} set"
```
```  4063     shows "bounded (- S) \<Longrightarrow> ~ (bounded S)"
```
```  4064   using bounded_Un [of S "-S"]  by (simp add: sup_compl_top)
```
```  4065
```
```  4066 lemma bounded_linear_image:
```
```  4067   assumes "bounded S"
```
```  4068     and "bounded_linear f"
```
```  4069   shows "bounded (f ` S)"
```
```  4070 proof -
```
```  4071   from assms(1) obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
```
```  4072     unfolding bounded_pos by auto
```
```  4073   from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"
```
```  4074     using bounded_linear.pos_bounded by (auto simp add: ac_simps)
```
```  4075   {
```
```  4076     fix x
```
```  4077     assume "x \<in> S"
```
```  4078     then have "norm x \<le> b"
```
```  4079       using b by auto
```
```  4080     then have "norm (f x) \<le> B * b"
```
```  4081       using B(2)
```
```  4082       apply (erule_tac x=x in allE)
```
```  4083       apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
```
```  4084       done
```
```  4085   }
```
```  4086   then show ?thesis
```
```  4087     unfolding bounded_pos
```
```  4088     apply (rule_tac x="b*B" in exI)
```
```  4089     using b B by (auto simp add: mult.commute)
```
```  4090 qed
```
```  4091
```
```  4092 lemma bounded_scaling:
```
```  4093   fixes S :: "'a::real_normed_vector set"
```
```  4094   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
```
```  4095   apply (rule bounded_linear_image)
```
```  4096   apply assumption
```
```  4097   apply (rule bounded_linear_scaleR_right)
```
```  4098   done
```
```  4099
```
```  4100 lemma bounded_translation:
```
```  4101   fixes S :: "'a::real_normed_vector set"
```
```  4102   assumes "bounded S"
```
```  4103   shows "bounded ((\<lambda>x. a + x) ` S)"
```
```  4104 proof -
```
```  4105   from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
```
```  4106     unfolding bounded_pos by auto
```
```  4107   {
```
```  4108     fix x
```
```  4109     assume "x \<in> S"
```
```  4110     then have "norm (a + x) \<le> b + norm a"
```
```  4111       using norm_triangle_ineq[of a x] b by auto
```
```  4112   }
```
```  4113   then show ?thesis
```
```  4114     unfolding bounded_pos
```
```  4115     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
```
```  4116     by (auto intro!: exI[of _ "b + norm a"])
```
```  4117 qed
```
```  4118
```
```  4119 lemma bounded_translation_minus:
```
```  4120   fixes S :: "'a::real_normed_vector set"
```
```  4121   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
```
```  4122 using bounded_translation [of S "-a"] by simp
```
```  4123
```
```  4124 lemma bounded_uminus [simp]:
```
```  4125   fixes X :: "'a::real_normed_vector set"
```
```  4126   shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
```
```  4127 by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp add: add.commute norm_minus_commute)
```
```  4128
```
```  4129
```
```  4130 subsection\<open>Some theorems on sups and infs using the notion "bounded".\<close>
```
```  4131
```
```  4132 lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"
```
```  4133   by (simp add: bounded_iff)
```
```  4134
```
```  4135 lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"
```
```  4136   by (auto simp: bounded_def bdd_above_def dist_real_def)
```
```  4137      (metis abs_le_D1 abs_minus_commute diff_le_eq)
```
```  4138
```
```  4139 lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"
```
```  4140   by (auto simp: bounded_def bdd_below_def dist_real_def)
```
```  4141      (metis abs_le_D1 add.commute diff_le_eq)
```
```  4142
```
```  4143 lemma bounded_inner_imp_bdd_above:
```
```  4144   assumes "bounded s"
```
```  4145     shows "bdd_above ((\<lambda>x. x \<bullet> a) ` s)"
```
```  4146 by (simp add: assms bounded_imp_bdd_above bounded_linear_image bounded_linear_inner_left)
```
```  4147
```
```  4148 lemma bounded_inner_imp_bdd_below:
```
```  4149   assumes "bounded s"
```
```  4150     shows "bdd_below ((\<lambda>x. x \<bullet> a) ` s)"
```
```  4151 by (simp add: assms bounded_imp_bdd_below bounded_linear_image bounded_linear_inner_left)
```
```  4152
```
```  4153 lemma bounded_has_Sup:
```
```  4154   fixes S :: "real set"
```
```  4155   assumes "bounded S"
```
```  4156     and "S \<noteq> {}"
```
```  4157   shows "\<forall>x\<in>S. x \<le> Sup S"
```
```  4158     and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
```
```  4159 proof
```
```  4160   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
```
```  4161     using assms by (metis cSup_least)
```
```  4162 qed (metis cSup_upper assms(1) bounded_imp_bdd_above)
```
```  4163
```
```  4164 lemma Sup_insert:
```
```  4165   fixes S :: "real set"
```
```  4166   shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
```
```  4167   by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)
```
```  4168
```
```  4169 lemma Sup_insert_finite:
```
```  4170   fixes S :: "'a::conditionally_complete_linorder set"
```
```  4171   shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
```
```  4172 by (simp add: cSup_insert sup_max)
```
```  4173
```
```  4174 lemma bounded_has_Inf:
```
```  4175   fixes S :: "real set"
```
```  4176   assumes "bounded S"
```
```  4177     and "S \<noteq> {}"
```
```  4178   shows "\<forall>x\<in>S. x \<ge> Inf S"
```
```  4179     and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
```
```  4180 proof
```
```  4181   show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
```
```  4182     using assms by (metis cInf_greatest)
```
```  4183 qed (metis cInf_lower assms(1) bounded_imp_bdd_below)
```
```  4184
```
```  4185 lemma Inf_insert:
```
```  4186   fixes S :: "real set"
```
```  4187   shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
```
```  4188   by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)
```
```  4189
```
```  4190 lemma Inf_insert_finite:
```
```  4191   fixes S :: "'a::conditionally_complete_linorder set"
```
```  4192   shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
```
```  4193 by (simp add: cInf_eq_Min)
```
```  4194
```
```  4195 lemma finite_imp_less_Inf:
```
```  4196   fixes a :: "'a::conditionally_complete_linorder"
```
```  4197   shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a < x\<rbrakk> \<Longrightarrow> a < Inf X"
```
```  4198   by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)
```
```  4199
```
```  4200 lemma finite_less_Inf_iff:
```
```  4201   fixes a :: "'a :: conditionally_complete_linorder"
```
```  4202   shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a < Inf X \<longleftrightarrow> (\<forall>x \<in> X. a < x)"
```
```  4203   by (auto simp: cInf_eq_Min)
```
```  4204
```
```  4205 lemma finite_imp_Sup_less:
```
```  4206   fixes a :: "'a::conditionally_complete_linorder"
```
```  4207   shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a > x\<rbrakk> \<Longrightarrow> a > Sup X"
```
```  4208   by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)
```
```  4209
```
```  4210 lemma finite_Sup_less_iff:
```
```  4211   fixes a :: "'a :: conditionally_complete_linorder"
```
```  4212   shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a > Sup X \<longleftrightarrow> (\<forall>x \<in> X. a > x)"
```
```  4213   by (auto simp: cSup_eq_Max)
```
```  4214
```
```  4215 subsection \<open>Compactness\<close>
```
```  4216
```
```  4217 subsubsection \<open>Bolzano-Weierstrass property\<close>
```
```  4218
```
```  4219 lemma heine_borel_imp_bolzano_weierstrass:
```
```  4220   assumes "compact s"
```
```  4221     and "infinite t"
```
```  4222     and "t \<subseteq> s"
```
```  4223   shows "\<exists>x \<in> s. x islimpt t"
```
```  4224 proof (rule ccontr)
```
```  4225   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
```
```  4226   then obtain f where f: "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
```
```  4227     unfolding islimpt_def
```
```  4228     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
```
```  4229     by auto
```
```  4230   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
```
```  4231     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
```
```  4232     using f by auto
```
```  4233   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"
```
```  4234     by auto
```
```  4235   {
```
```  4236     fix x y
```
```  4237     assume "x \<in> t" "y \<in> t" "f x = f y"
```
```  4238     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"
```
```  4239       using f[THEN bspec[where x=x]] and \<open>t \<subseteq> s\<close> by auto
```
```  4240     then have "x = y"
```
```  4241       using \<open>f x = f y\<close> and f[THEN bspec[where x=y]] and \<open>y \<in> t\<close> and \<open>t \<subseteq> s\<close>
```
```  4242       by auto
```
```  4243   }
```
```  4244   then have "inj_on f t"
```
```  4245     unfolding inj_on_def by simp
```
```  4246   then have "infinite (f ` t)"
```
```  4247     using assms(2) using finite_imageD by auto
```
```  4248   moreover
```
```  4249   {
```
```  4250     fix x
```
```  4251     assume "x \<in> t" "f x \<notin> g"
```
```  4252     from g(3) assms(3) \<open>x \<in> t\<close> obtain h where "h \<in> g" and "x \<in> h"
```
```  4253       by auto
```
```  4254     then obtain y where "y \<in> s" "h = f y"
```
```  4255       using g'[THEN bspec[where x=h]] by auto
```
```  4256     then have "y = x"
```
```  4257       using f[THEN bspec[where x=y]] and \<open>x\<in>t\<close> and \<open>x\<in>h\<close>[unfolded \<open>h = f y\<close>]
```
```  4258       by auto
```
```  4259     then have False
```
```  4260       using \<open>f x \<notin> g\<close> \<open>h \<in> g\<close> unfolding \<open>h = f y\<close>
```
```  4261       by auto
```
```  4262   }
```
```  4263   then have "f ` t \<subseteq> g" by auto
```
```  4264   ultimately show False
```
```  4265     using g(2) using finite_subset by auto
```
```  4266 qed
```
```  4267
```
```  4268 lemma acc_point_range_imp_convergent_subsequence:
```
```  4269   fixes l :: "'a :: first_countable_topology"
```
```  4270   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
```
```  4271   shows "\<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
```
```  4272 proof -
```
```  4273   from countable_basis_at_decseq[of l]
```
```  4274   obtain A where A:
```
```  4275       "\<And>i. open (A i)"
```
```  4276       "\<And>i. l \<in> A i"
```
```  4277       "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  4278     by blast
```
```  4279   define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
```
```  4280   {
```
```  4281     fix n i
```
```  4282     have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
```
```  4283       using l A by auto
```
```  4284     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
```
```  4285       unfolding ex_in_conv by (intro notI) simp
```
```  4286     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
```
```  4287       by auto
```
```  4288     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
```
```  4289       by (auto simp: not_le)
```
```  4290     then have "i < s n i" "f (s n i) \<in> A (Suc n)"
```
```  4291       unfolding s_def by (auto intro: someI2_ex)
```
```  4292   }
```
```  4293   note s = this
```
```  4294   define r where "r = rec_nat (s 0 0) s"
```
```  4295   have "subseq r"
```
```  4296     by (auto simp: r_def s subseq_Suc_iff)
```
```  4297   moreover
```
```  4298   have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
```
```  4299   proof (rule topological_tendstoI)
```
```  4300     fix S
```
```  4301     assume "open S" "l \<in> S"
```
```  4302     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  4303       by auto
```
```  4304     moreover
```
```  4305     {
```
```  4306       fix i
```
```  4307       assume "Suc 0 \<le> i"
```
```  4308       then have "f (r i) \<in> A i"
```
```  4309         by (cases i) (simp_all add: r_def s)
```
```  4310     }
```
```  4311     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
```
```  4312       by (auto simp: eventually_sequentially)
```
```  4313     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
```
```  4314       by eventually_elim auto
```
```  4315   qed
```
```  4316   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
```
```  4317     by (auto simp: convergent_def comp_def)
```
```  4318 qed
```
```  4319
```
```  4320 lemma sequence_infinite_lemma:
```
```  4321   fixes f :: "nat \<Rightarrow> 'a::t1_space"
```
```  4322   assumes "\<forall>n. f n \<noteq> l"
```
```  4323     and "(f \<longlongrightarrow> l) sequentially"
```
```  4324   shows "infinite (range f)"
```
```  4325 proof
```
```  4326   assume "finite (range f)"
```
```  4327   then have "closed (range f)"
```
```  4328     by (rule finite_imp_closed)
```
```  4329   then have "open (- range f)"
```
```  4330     by (rule open_Compl)
```
```  4331   from assms(1) have "l \<in> - range f"
```
```  4332     by auto
```
```  4333   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
```
```  4334     using \<open>open (- range f)\<close> \<open>l \<in> - range f\<close>
```
```  4335     by (rule topological_tendstoD)
```
```  4336   then show False
```
```  4337     unfolding eventually_sequentially
```
```  4338     by auto
```
```  4339 qed
```
```  4340
```
```  4341 lemma closure_insert:
```
```  4342   fixes x :: "'a::t1_space"
```
```  4343   shows "closure (insert x s) = insert x (closure s)"
```
```  4344   apply (rule closure_unique)
```
```  4345   apply (rule insert_mono [OF closure_subset])
```
```  4346   apply (rule closed_insert [OF closed_closure])
```
```  4347   apply (simp add: closure_minimal)
```
```  4348   done
```
```  4349
```
```  4350 lemma islimpt_insert:
```
```  4351   fixes x :: "'a::t1_space"
```
```  4352   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
```
```  4353 proof
```
```  4354   assume *: "x islimpt (insert a s)"
```
```  4355   show "x islimpt s"
```
```  4356   proof (rule islimptI)
```
```  4357     fix t
```
```  4358     assume t: "x \<in> t" "open t"
```
```  4359     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
```
```  4360     proof (cases "x = a")
```
```  4361       case True
```
```  4362       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
```
```  4363         using * t by (rule islimptE)
```
```  4364       with \<open>x = a\<close> show ?thesis by auto
```
```  4365     next
```
```  4366       case False
```
```  4367       with t have t': "x \<in> t - {a}" "open (t - {a})"
```
```  4368         by (simp_all add: open_Diff)
```
```  4369       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
```
```  4370         using * t' by (rule islimptE)
```
```  4371       then show ?thesis by auto
```
```  4372     qed
```
```  4373   qed
```
```  4374 next
```
```  4375   assume "x islimpt s"
```
```  4376   then show "x islimpt (insert a s)"
```
```  4377     by (rule islimpt_subset) auto
```
```  4378 qed
```
```  4379
```
```  4380 lemma islimpt_finite:
```
```  4381   fixes x :: "'a::t1_space"
```
```  4382   shows "finite s \<Longrightarrow> \<not> x islimpt s"
```
```  4383   by (induct set: finite) (simp_all add: islimpt_insert)
```
```  4384
```
```  4385 lemma islimpt_Un_finite:
```
```  4386   fixes x :: "'a::t1_space"
```
```  4387   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
```
```  4388   by (simp add: islimpt_Un islimpt_finite)
```
```  4389
```
```  4390 lemma islimpt_eq_acc_point:
```
```  4391   fixes l :: "'a :: t1_space"
```
```  4392   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
```
```  4393 proof (safe intro!: islimptI)
```
```  4394   fix U
```
```  4395   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
```
```  4396   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
```
```  4397     by (auto intro: finite_imp_closed)
```
```  4398   then show False
```
```  4399     by (rule islimptE) auto
```
```  4400 next
```
```  4401   fix T
```
```  4402   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
```
```  4403   then have "infinite (T \<inter> S - {l})"
```
```  4404     by auto
```
```  4405   then have "\<exists>x. x \<in> (T \<inter> S - {l})"
```
```  4406     unfolding ex_in_conv by (intro notI) simp
```
```  4407   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
```
```  4408     by auto
```
```  4409 qed
```
```  4410
```
```  4411 corollary infinite_openin:
```
```  4412   fixes S :: "'a :: t1_space set"
```
```  4413   shows "\<lbrakk>openin (subtopology euclidean U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
```
```  4414   by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
```
```  4415
```
```  4416 lemma islimpt_range_imp_convergent_subsequence:
```
```  4417   fixes l :: "'a :: {t1_space, first_countable_topology}"
```
```  4418   assumes l: "l islimpt (range f)"
```
```  4419   shows "\<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
```
```  4420   using l unfolding islimpt_eq_acc_point
```
```  4421   by (rule acc_point_range_imp_convergent_subsequence)
```
```  4422
```
```  4423 lemma islimpt_eq_infinite_ball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> ball x e))"
```
```  4424   apply (simp add: islimpt_eq_acc_point, safe)
```
```  4425    apply (metis Int_commute open_ball centre_in_ball)
```
```  4426   by (metis open_contains_ball Int_mono finite_subset inf_commute subset_refl)
```
```  4427
```
```  4428 lemma islimpt_eq_infinite_cball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> cball x e))"
```
```  4429   apply (simp add: islimpt_eq_infinite_ball, safe)
```
```  4430    apply (meson Int_mono ball_subset_cball finite_subset order_refl)
```
```  4431   by (metis open_ball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq)
```
```  4432
```
```  4433 lemma sequence_unique_limpt:
```
```  4434   fixes f :: "nat \<Rightarrow> 'a::t2_space"
```
```  4435   assumes "(f \<longlongrightarrow> l) sequentially"
```
```  4436     and "l' islimpt (range f)"
```
```  4437   shows "l' = l"
```
```  4438 proof (rule ccontr)
```
```  4439   assume "l' \<noteq> l"
```
```  4440   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
```
```  4441     using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
```
```  4442   have "eventually (\<lambda>n. f n \<in> t) sequentially"
```
```  4443     using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
```
```  4444   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
```
```  4445     unfolding eventually_sequentially by auto
```
```  4446
```
```  4447   have "UNIV = {..<N} \<union> {N..}"
```
```  4448     by auto
```
```  4449   then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
```
```  4450     using assms(2) by simp
```
```  4451   then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
```
```  4452     by (simp add: image_Un)
```
```  4453   then have "l' islimpt (f ` {N..})"
```
```  4454     by (simp add: islimpt_Un_finite)
```
```  4455   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
```
```  4456     using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
```
```  4457   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
```
```  4458     by auto
```
```  4459   with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
```
```  4460     by simp
```
```  4461   with \<open>s \<inter> t = {}\<close> show False
```
```  4462     by simp
```
```  4463 qed
```
```  4464
```
```  4465 lemma bolzano_weierstrass_imp_closed:
```
```  4466   fixes s :: "'a::{first_countable_topology,t2_space} set"
```
```  4467   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
```
```  4468   shows "closed s"
```
```  4469 proof -
```
```  4470   {
```
```  4471     fix x l
```
```  4472     assume as: "\<forall>n::nat. x n \<in> s" "(x \<longlongrightarrow> l) sequentially"
```
```  4473     then have "l \<in> s"
```
```  4474     proof (cases "\<forall>n. x n \<noteq> l")
```
```  4475       case False
```
```  4476       then show "l\<in>s" using as(1) by auto
```
```  4477     next
```
```  4478       case True note cas = this
```
```  4479       with as(2) have "infinite (range x)"
```
```  4480         using sequence_infinite_lemma[of x l] by auto
```
```  4481       then obtain l' where "l'\<in>s" "l' islimpt (range x)"
```
```  4482         using assms[THEN spec[where x="range x"]] as(1) by auto
```
```  4483       then show "l\<in>s" using sequence_unique_limpt[of x l l']
```
```  4484         using as cas by auto
```
```  4485     qed
```
```  4486   }
```
```  4487   then show ?thesis
```
```  4488     unfolding closed_sequential_limits by fast
```
```  4489 qed
```
```  4490
```
```  4491 lemma compact_imp_bounded:
```
```  4492   assumes "compact U"
```
```  4493   shows "bounded U"
```
```  4494 proof -
```
```  4495   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"
```
```  4496     using assms by auto
```
```  4497   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
```
```  4498     by (rule compactE_image)
```
```  4499   from \<open>finite D\<close> have "bounded (\<Union>x\<in>D. ball x 1)"
```
```  4500     by (simp add: bounded_UN)
```
```  4501   then show "bounded U" using \<open>U \<subseteq> (\<Union>x\<in>D. ball x 1)\<close>
```
```  4502     by (rule bounded_subset)
```
```  4503 qed
```
```  4504
```
```  4505 text\<open>In particular, some common special cases.\<close>
```
```  4506
```
```  4507 lemma compact_Un [intro]:
```
```  4508   assumes "compact s"
```
```  4509     and "compact t"
```
```  4510   shows " compact (s \<union> t)"
```
```  4511 proof (rule compactI)
```
```  4512   fix f
```
```  4513   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
```
```  4514   from * \<open>compact s\<close> obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
```
```  4515     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
```
```  4516   moreover
```
```  4517   from * \<open>compact t\<close> obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
```
```  4518     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
```
```  4519   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
```
```  4520     by (auto intro!: exI[of _ "s' \<union> t'"])
```
```  4521 qed
```
```  4522
```
```  4523 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
```
```  4524   by (induct set: finite) auto
```
```  4525
```
```  4526 lemma compact_UN [intro]:
```
```  4527   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
```
```  4528   by (rule compact_Union) auto
```
```  4529
```
```  4530 lemma closed_Int_compact [intro]:
```
```  4531   assumes "closed s"
```
```  4532     and "compact t"
```
```  4533   shows "compact (s \<inter> t)"
```
```  4534   using compact_Int_closed [of t s] assms
```
```  4535   by (simp add: Int_commute)
```
```  4536
```
```  4537 lemma compact_Int [intro]:
```
```  4538   fixes s t :: "'a :: t2_space set"
```
```  4539   assumes "compact s"
```
```  4540     and "compact t"
```
```  4541   shows "compact (s \<inter> t)"
```
```  4542   using assms by (intro compact_Int_closed compact_imp_closed)
```
```  4543
```
```  4544 lemma compact_sing [simp]: "compact {a}"
```
```  4545   unfolding compact_eq_heine_borel by auto
```
```  4546
```
```  4547 lemma compact_insert [simp]:
```
```  4548   assumes "compact s"
```
```  4549   shows "compact (insert x s)"
```
```  4550 proof -
```
```  4551   have "compact ({x} \<union> s)"
```
```  4552     using compact_sing assms by (rule compact_Un)
```
```  4553   then show ?thesis by simp
```
```  4554 qed
```
```  4555
```
```  4556 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"
```
```  4557   by (induct set: finite) simp_all
```
```  4558
```
```  4559 lemma open_delete:
```
```  4560   fixes s :: "'a::t1_space set"
```
```  4561   shows "open s \<Longrightarrow> open (s - {x})"
```
```  4562   by (simp add: open_Diff)
```
```  4563
```
```  4564 lemma openin_delete:
```
```  4565   fixes a :: "'a :: t1_space"
```
```  4566   shows "openin (subtopology euclidean u) s
```
```  4567          \<Longrightarrow> openin (subtopology euclidean u) (s - {a})"
```
```  4568 by (metis Int_Diff open_delete openin_open)
```
```  4569
```
```  4570
```
```  4571 text\<open>Compactness expressed with filters\<close>
```
```  4572
```
```  4573 lemma closure_iff_nhds_not_empty:
```
```  4574   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
```
```  4575 proof safe
```
```  4576   assume x: "x \<in> closure X"
```
```  4577   fix S A
```
```  4578   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
```
```  4579   then have "x \<notin> closure (-S)"
```
```  4580     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
```
```  4581   with x have "x \<in> closure X - closure (-S)"
```
```  4582     by auto
```
```  4583   also have "\<dots> \<subseteq> closure (X \<inter> S)"
```
```  4584     using \<open>open S\<close> open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
```
```  4585   finally have "X \<inter> S \<noteq> {}" by auto
```
```  4586   then show False using \<open>X \<inter> A = {}\<close> \<open>S \<subseteq> A\<close> by auto
```
```  4587 next
```
```  4588   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
```
```  4589   from this[THEN spec, of "- X", THEN spec, of "- closure X"]
```
```  4590   show "x \<in> closure X"
```
```  4591     by (simp add: closure_subset open_Compl)
```
```  4592 qed
```
```  4593
```
```  4594 lemma compact_filter:
```
```  4595   "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
```
```  4596 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
```
```  4597   fix F
```
```  4598   assume "compact U"
```
```  4599   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
```
```  4600   then have "U \<noteq> {}"
```
```  4601     by (auto simp: eventually_False)
```
```  4602
```
```  4603   define Z where "Z = closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
```
```  4604   then have "\<forall>z\<in>Z. closed z"
```
```  4605     by auto
```
```  4606   moreover
```
```  4607   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
```
```  4608     unfolding Z_def by (auto elim: eventually_mono intro: set_mp[OF closure_subset])
```
```  4609   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
```
```  4610   proof (intro allI impI)
```
```  4611     fix B assume "finite B" "B \<subseteq> Z"
```
```  4612     with \<open>finite B\<close> ev_Z F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
```
```  4613       by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
```
```  4614     with F show "U \<inter> \<Inter>B \<noteq> {}"
```
```  4615       by (intro notI) (simp add: eventually_False)
```
```  4616   qed
```
```  4617   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
```
```  4618     using \<open>compact U\<close> unfolding compact_fip by blast
```
```  4619   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"
```
```  4620     by auto
```
```  4621
```
```  4622   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
```
```  4623     unfolding eventually_inf eventually_nhds
```
```  4624   proof safe
```
```  4625     fix P Q R S
```
```  4626     assume "eventually R F" "open S" "x \<in> S"
```
```  4627     with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
```
```  4628     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
```
```  4629     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
```
```  4630     ultimately show False by (auto simp: set_eq_iff)
```
```  4631   qed
```
```  4632   with \<open>x \<in> U\<close> show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
```
```  4633     by (metis eventually_bot)
```
```  4634 next
```
```  4635   fix A
```
```  4636   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
```
```  4637   define F where "F = (INF a:insert U A. principal a)"
```
```  4638   have "F \<noteq> bot"
```
```  4639     unfolding F_def
```
```  4640   proof (rule INF_filter_not_bot)
```
```  4641     fix X
```
```  4642     assume X: "X \<subseteq> insert U A" "finite X"
```
```  4643     with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"
```
```  4644       by auto
```
```  4645     with X show "(INF a:X. principal a) \<noteq> bot"
```
```  4646       by (auto simp add: INF_principal_finite principal_eq_bot_iff)
```
```  4647   qed
```
```  4648   moreover
```
```  4649   have "F \<le> principal U"
```
```  4650     unfolding F_def by auto
```
```  4651   then have "eventually (\<lambda>x. x \<in> U) F"
```
```  4652     by (auto simp: le_filter_def eventually_principal)
```
```  4653   moreover
```
```  4654   assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
```
```  4655   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
```
```  4656     by auto
```
```  4657
```
```  4658   { fix V assume "V \<in> A"
```
```  4659     then have "F \<le> principal V"
```
```  4660       unfolding F_def by (intro INF_lower2[of V]) auto
```
```  4661     then have V: "eventually (\<lambda>x. x \<in> V) F"
```
```  4662       by (auto simp: le_filter_def eventually_principal)
```
```  4663     have "x \<in> closure V"
```
```  4664       unfolding closure_iff_nhds_not_empty
```
```  4665     proof (intro impI allI)
```
```  4666       fix S A
```
```  4667       assume "open S" "x \<in> S" "S \<subseteq> A"
```
```  4668       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"
```
```  4669         by (auto simp: eventually_nhds)
```
```  4670       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
```
```  4671         by (auto simp: eventually_inf)
```
```  4672       with x show "V \<inter> A \<noteq> {}"
```
```  4673         by (auto simp del: Int_iff simp add: trivial_limit_def)
```
```  4674     qed
```
```  4675     then have "x \<in> V"
```
```  4676       using \<open>V \<in> A\<close> A(1) by simp
```
```  4677   }
```
```  4678   with \<open>x\<in>U\<close> have "x \<in> U \<inter> \<Inter>A" by auto
```
```  4679   with \<open>U \<inter> \<Inter>A = {}\<close> show False by auto
```
```  4680 qed
```
```  4681
```
```  4682 definition "countably_compact U \<longleftrightarrow>
```
```  4683     (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"
```
```  4684
```
```  4685 lemma countably_compactE:
```
```  4686   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
```
```  4687   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
```
```  4688   using assms unfolding countably_compact_def by metis
```
```  4689
```
```  4690 lemma countably_compactI:
```
```  4691   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
```
```  4692   shows "countably_compact s"
```
```  4693   using assms unfolding countably_compact_def by metis
```
```  4694
```
```  4695 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
```
```  4696   by (auto simp: compact_eq_heine_borel countably_compact_def)
```
```  4697
```
```  4698 lemma countably_compact_imp_compact:
```
```  4699   assumes "countably_compact U"
```
```  4700     and ccover: "countable B" "\<forall>b\<in>B. open b"
```
```  4701     and basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
```
```  4702   shows "compact U"
```
```  4703   using \<open>countably_compact U\<close>
```
```  4704   unfolding compact_eq_heine_borel countably_compact_def
```
```  4705 proof safe
```
```  4706   fix A
```
```  4707   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
```
```  4708   assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
```
```  4709   moreover define C where "C = {b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
```
```  4710   ultimately have "countable C" "\<forall>a\<in>C. open a"
```
```  4711     unfolding C_def using ccover by auto
```
```  4712   moreover
```
```  4713   have "\<Union>A \<inter> U \<subseteq> \<Union>C"
```
```  4714   proof safe
```
```  4715     fix x a
```
```  4716     assume "x \<in> U" "x \<in> a" "a \<in> A"
```
```  4717     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"
```
```  4718       by blast
```
```  4719     with \<open>a \<in> A\<close> show "x \<in> \<Union>C"
```
```  4720       unfolding C_def by auto
```
```  4721   qed
```
```  4722   then have "U \<subseteq> \<Union>C" using \<open>U \<subseteq> \<Union>A\<close> by auto
```
```  4723   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
```
```  4724     using * by metis
```
```  4725   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
```
```  4726     by (auto simp: C_def)
```
```  4727   then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"
```
```  4728     unfolding bchoice_iff Bex_def ..
```
```  4729   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
```
```  4730     unfolding C_def by (intro exI[of _ "f`T"]) fastforce
```
```  4731 qed
```
```  4732
```
```  4733 lemma countably_compact_imp_compact_second_countable:
```
```  4734   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
```
```  4735 proof (rule countably_compact_imp_compact)
```
```  4736   fix T and x :: 'a
```
```  4737   assume "open T" "x \<in> T"
```
```  4738   from topological_basisE[OF is_basis this] obtain b where
```
```  4739     "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .
```
```  4740   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"
```
```  4741     by blast
```
```  4742 qed (insert countable_basis topological_basis_open[OF is_basis], auto)
```
```  4743
```
```  4744 lemma countably_compact_eq_compact:
```
```  4745   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
```
```  4746   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
```
```  4747
```
```  4748 subsubsection\<open>Sequential compactness\<close>
```
```  4749
```
```  4750 definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"
```
```  4751   where "seq_compact S \<longleftrightarrow>
```
```  4752     (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially))"
```
```  4753
```
```  4754 lemma seq_compactI:
```
```  4755   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  4756   shows "seq_compact S"
```
```  4757   unfolding seq_compact_def using assms by fast
```
```  4758
```
```  4759 lemma seq_compactE:
```
```  4760   assumes "seq_compact S" "\<forall>n. f n \<in> S"
```
```  4761   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  4762   using assms unfolding seq_compact_def by fast
```
```  4763
```
```  4764 lemma closed_sequentially: (* TODO: move upwards *)
```
```  4765   assumes "closed s" and "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> l"
```
```  4766   shows "l \<in> s"
```
```  4767 proof (rule ccontr)
```
```  4768   assume "l \<notin> s"
```
```  4769   with \<open>closed s\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
```
```  4770     by (fast intro: topological_tendstoD)
```
```  4771   with \<open>\<forall>n. f n \<in> s\<close> show "False"
```
```  4772     by simp
```
```  4773 qed
```
```  4774
```
```  4775 lemma seq_compact_Int_closed:
```
```  4776   assumes "seq_compact s" and "closed t"
```
```  4777   shows "seq_compact (s \<inter> t)"
```
```  4778 proof (rule seq_compactI)
```
```  4779   fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"
```
```  4780   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
```
```  4781     by simp_all
```
```  4782   from \<open>seq_compact s\<close> and \<open>\<forall>n. f n \<in> s\<close>
```
```  4783   obtain l r where "l \<in> s" and r: "subseq r" and l: "(f \<circ> r) \<longlonglongrightarrow> l"
```
```  4784     by (rule seq_compactE)
```
```  4785   from \<open>\<forall>n. f n \<in> t\<close> have "\<forall>n. (f \<circ> r) n \<in> t"
```
```  4786     by simp
```
```  4787   from \<open>closed t\<close> and this and l have "l \<in> t"
```
```  4788     by (rule closed_sequentially)
```
```  4789   with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
```
```  4790     by fast
```
```  4791 qed
```
```  4792
```
```  4793 lemma seq_compact_closed_subset:
```
```  4794   assumes "closed s" and "s \<subseteq> t" and "seq_compact t"
```
```  4795   shows "seq_compact s"
```
```  4796   using assms seq_compact_Int_closed [of t s] by (simp add: Int_absorb1)
```
```  4797
```
```  4798 lemma seq_compact_imp_countably_compact:
```
```  4799   fixes U :: "'a :: first_countable_topology set"
```
```  4800   assumes "seq_compact U"
```
```  4801   shows "countably_compact U"
```
```  4802 proof (safe intro!: countably_compactI)
```
```  4803   fix A
```
```  4804   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
```
```  4805   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
```
```  4806     using \<open>seq_compact U\<close> by (fastforce simp: seq_compact_def subset_eq)
```
```  4807   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
```
```  4808   proof cases
```
```  4809     assume "finite A"
```
```  4810     with A show ?thesis by auto
```
```  4811   next
```
```  4812     assume "infinite A"
```
```  4813     then have "A \<noteq> {}" by auto
```
```  4814     show ?thesis
```
```  4815     proof (rule ccontr)
```
```  4816       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
```
```  4817       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"
```
```  4818         by auto
```
```  4819       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"
```
```  4820         by metis
```
```  4821       define X where "X n = X' (from_nat_into A ` {.. n})" for n
```
```  4822       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
```
```  4823         using \<open>A \<noteq> {}\<close> unfolding X_def by (intro T) (auto intro: from_nat_into)
```
```  4824       then have "range X \<subseteq> U"
```
```  4825         by auto
```
```  4826       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) \<longlonglongrightarrow> x"
```
```  4827         by auto
```
```  4828       from \<open>x\<in>U\<close> \<open>U \<subseteq> \<Union>A\<close> from_nat_into_surj[OF \<open>countable A\<close>]
```
```  4829       obtain n where "x \<in> from_nat_into A n" by auto
```
```  4830       with r(2) A(1) from_nat_into[OF \<open>A \<noteq> {}\<close>, of n]
```
```  4831       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
```
```  4832         unfolding tendsto_def by (auto simp: comp_def)
```
```  4833       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
```
```  4834         by (auto simp: eventually_sequentially)
```
```  4835       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
```
```  4836         by auto
```
```  4837       moreover from \<open>subseq r\<close>[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
```
```  4838         by (auto intro!: exI[of _ "max n N"])
```
```  4839       ultimately show False
```
```  4840         by auto
```
```  4841     qed
```
```  4842   qed
```
```  4843 qed
```
```  4844
```
```  4845 lemma compact_imp_seq_compact:
```
```  4846   fixes U :: "'a :: first_countable_topology set"
```
```  4847   assumes "compact U"
```
```  4848   shows "seq_compact U"
```
```  4849   unfolding seq_compact_def
```
```  4850 proof safe
```
```  4851   fix X :: "nat \<Rightarrow> 'a"
```
```  4852   assume "\<forall>n. X n \<in> U"
```
```  4853   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
```
```  4854     by (auto simp: eventually_filtermap)
```
```  4855   moreover
```
```  4856   have "filtermap X sequentially \<noteq> bot"
```
```  4857     by (simp add: trivial_limit_def eventually_filtermap)
```
```  4858   ultimately
```
```  4859   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
```
```  4860     using \<open>compact U\<close> by (auto simp: compact_filter)
```
```  4861
```
```  4862   from countable_basis_at_decseq[of x]
```
```  4863   obtain A where A:
```
```  4864       "\<And>i. open (A i)"
```
```  4865       "\<And>i. x \<in> A i"
```
```  4866       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  4867     by blast
```
```  4868   define s where "s n i = (SOME j. i < j \<and> X j \<in> A (Suc n))" for n i
```
```  4869   {
```
```  4870     fix n i
```
```  4871     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
```
```  4872     proof (rule ccontr)
```
```  4873       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
```
```  4874       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"
```
```  4875         by auto
```
```  4876       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
```
```  4877         by (auto simp: eventually_filtermap eventually_sequentially)
```
```  4878       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
```
```  4879         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
```
```  4880       ultimately have "eventually (\<lambda>x. False) ?F"
```
```  4881         by (auto simp add: eventually_inf)
```
```  4882       with x show False
```
```  4883         by (simp add: eventually_False)
```
```  4884     qed
```
```  4885     then have "i < s n i" "X (s n i) \<in> A (Suc n)"
```
```  4886       unfolding s_def by (auto intro: someI2_ex)
```
```  4887   }
```
```  4888   note s = this
```
```  4889   define r where "r = rec_nat (s 0 0) s"
```
```  4890   have "subseq r"
```
```  4891     by (auto simp: r_def s subseq_Suc_iff)
```
```  4892   moreover
```
```  4893   have "(\<lambda>n. X (r n)) \<longlonglongrightarrow> x"
```
```  4894   proof (rule topological_tendstoI)
```
```  4895     fix S
```
```  4896     assume "open S" "x \<in> S"
```
```  4897     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  4898       by auto
```
```  4899     moreover
```
```  4900     {
```
```  4901       fix i
```
```  4902       assume "Suc 0 \<le> i"
```
```  4903       then have "X (r i) \<in> A i"
```
```  4904         by (cases i) (simp_all add: r_def s)
```
```  4905     }
```
```  4906     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"
```
```  4907       by (auto simp: eventually_sequentially)
```
```  4908     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
```
```  4909       by eventually_elim auto
```
```  4910   qed
```
```  4911   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
```
```  4912     using \<open>x \<in> U\<close> by (auto simp: convergent_def comp_def)
```
```  4913 qed
```
```  4914
```
```  4915 lemma countably_compact_imp_acc_point:
```
```  4916   assumes "countably_compact s"
```
```  4917     and "countable t"
```
```  4918     and "infinite t"
```
```  4919     and "t \<subseteq> s"
```
```  4920   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
```
```  4921 proof (rule ccontr)
```
```  4922   define C where "C = (\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
```
```  4923   note \<open>countably_compact s\<close>
```
```  4924   moreover have "\<forall>t\<in>C. open t"
```
```  4925     by (auto simp: C_def)
```
```  4926   moreover
```
```  4927   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
```
```  4928   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
```
```  4929   have "s \<subseteq> \<Union>C"
```
```  4930     using \<open>t \<subseteq> s\<close>
```
```  4931     unfolding C_def
```
```  4932     apply (safe dest!: s)
```
```  4933     apply (rule_tac a="U \<inter> t" in UN_I)
```
```  4934     apply (auto intro!: interiorI simp add: finite_subset)
```
```  4935     done
```
```  4936   moreover
```
```  4937   from \<open>countable t\<close> have "countable C"
```
```  4938     unfolding C_def by (auto intro: countable_Collect_finite_subset)
```
```  4939   ultimately
```
```  4940   obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"
```
```  4941     by (rule countably_compactE)
```
```  4942   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"
```
```  4943     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
```
```  4944     by (metis (lifting) finite_subset_image C_def)
```
```  4945   from s \<open>t \<subseteq> s\<close> have "t \<subseteq> \<Union>E"
```
```  4946     using interior_subset by blast
```
```  4947   moreover have "finite (\<Union>E)"
```
```  4948     using E by auto
```
```  4949   ultimately show False using \<open>infinite t\<close>
```
```  4950     by (auto simp: finite_subset)
```
```  4951 qed
```
```  4952
```
```  4953 lemma countable_acc_point_imp_seq_compact:
```
```  4954   fixes s :: "'a::first_countable_topology set"
```
```  4955   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>
```
```  4956     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
```
```  4957   shows "seq_compact s"
```
```  4958 proof -
```
```  4959   {
```
```  4960     fix f :: "nat \<Rightarrow> 'a"
```
```  4961     assume f: "\<forall>n. f n \<in> s"
```
```  4962     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  4963     proof (cases "finite (range f)")
```
```  4964       case True
```
```  4965       obtain l where "infinite {n. f n = f l}"
```
```  4966         using pigeonhole_infinite[OF _ True] by auto
```
```  4967       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"
```
```  4968         using infinite_enumerate by blast
```
```  4969       then have "subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> f l"
```
```  4970         by (simp add: fr o_def)
```
```  4971       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
```
```  4972         by auto
```
```  4973     next
```
```  4974       case False
```
```  4975       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
```
```  4976         by auto
```
```  4977       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
```
```  4978       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  4979         using acc_point_range_imp_convergent_subsequence[of l f] by auto
```
```  4980       with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" ..
```
```  4981     qed
```
```  4982   }
```
```  4983   then show ?thesis
```
```  4984     unfolding seq_compact_def by auto
```
```  4985 qed
```
```  4986
```
```  4987 lemma seq_compact_eq_countably_compact:
```
```  4988   fixes U :: "'a :: first_countable_topology set"
```
```  4989   shows "seq_compact U \<longleftrightarrow> countably_compact U"
```
```  4990   using
```
```  4991     countable_acc_point_imp_seq_compact
```
```  4992     countably_compact_imp_acc_point
```
```  4993     seq_compact_imp_countably_compact
```
```  4994   by metis
```
```  4995
```
```  4996 lemma seq_compact_eq_acc_point:
```
```  4997   fixes s :: "'a :: first_countable_topology set"
```
```  4998   shows "seq_compact s \<longleftrightarrow>
```
```  4999     (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
```
```  5000   using
```
```  5001     countable_acc_point_imp_seq_compact[of s]
```
```  5002     countably_compact_imp_acc_point[of s]
```
```  5003     seq_compact_imp_countably_compact[of s]
```
```  5004   by metis
```
```  5005
```
```  5006 lemma seq_compact_eq_compact:
```
```  5007   fixes U :: "'a :: second_countable_topology set"
```
```  5008   shows "seq_compact U \<longleftrightarrow> compact U"
```
```  5009   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
```
```  5010
```
```  5011 lemma bolzano_weierstrass_imp_seq_compact:
```
```  5012   fixes s :: "'a::{t1_space, first_countable_topology} set"
```
```  5013   shows "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
```
```  5014   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
```
```  5015
```
```  5016
```
```  5017 subsubsection\<open>Totally bounded\<close>
```
```  5018
```
```  5019 lemma cauchy_def: "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N \<longrightarrow> dist (s m) (s n) < e)"
```
```  5020   unfolding Cauchy_def by metis
```
```  5021
```
```  5022 lemma seq_compact_imp_totally_bounded:
```
```  5023   assumes "seq_compact s"
```
```  5024   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)"
```
```  5025 proof -
```
```  5026   { fix e::real assume "e > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> s \<Longrightarrow> \<not> s \<subseteq> (\<Union>x\<in>k. ball x e)"
```
```  5027     let ?Q = "\<lambda>x n r. r \<in> s \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < e))"
```
```  5028     have "\<exists>x. \<forall>n::nat. ?Q x n (x n)"
```
```  5029     proof (rule dependent_wellorder_choice)
```
```  5030       fix n x assume "\<And>y. y < n \<Longrightarrow> ?Q x y (x y)"
```
```  5031       then have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)"
```
```  5032         using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq)
```
```  5033       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)"
```
```  5034         unfolding subset_eq by auto
```
```  5035       show "\<exists>r. ?Q x n r"
```
```  5036         using z by auto
```
```  5037     qed simp
```
```  5038     then obtain x where "\<forall>n::nat. x n \<in> s" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"
```
```  5039       by blast
```
```  5040     then obtain l r where "l \<in> s" and r:"subseq r" and "((x \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  5041       using assms by (metis seq_compact_def)
```
```  5042     from this(3) have "Cauchy (x \<circ> r)"
```
```  5043       using LIMSEQ_imp_Cauchy by auto
```
```  5044     then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"
```
```  5045       unfolding cauchy_def using \<open>e > 0\<close> by blast
```
```  5046     then have False
```
```  5047       using x[of "r N" "r (N+1)"] r by (auto simp: subseq_def) }
```
```  5048   then show ?thesis
```
```  5049     by metis
```
```  5050 qed
```
```  5051
```
```  5052 subsubsection\<open>Heine-Borel theorem\<close>
```
```  5053
```
```  5054 lemma seq_compact_imp_heine_borel:
```
```  5055   fixes s :: "'a :: metric_space set"
```
```  5056   assumes "seq_compact s"
```
```  5057   shows "compact s"
```
```  5058 proof -
```
```  5059   from seq_compact_imp_totally_bounded[OF \<open>seq_compact s\<close>]
```
```  5060   obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>f e. ball x e)"
```
```  5061     unfolding choice_iff' ..
```
```  5062   define K where "K = (\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
```
```  5063   have "countably_compact s"
```
```  5064     using \<open>seq_compact s\<close> by (rule seq_compact_imp_countably_compact)
```
```  5065   then show "compact s"
```
```  5066   proof (rule countably_compact_imp_compact)
```
```  5067     show "countable K"
```
```  5068       unfolding K_def using f
```
```  5069       by (auto intro: countable_finite countable_subset countable_rat
```
```  5070                intro!: countable_image countable_SIGMA countable_UN)
```
```  5071     show "\<forall>b\<in>K. open b" by (auto simp: K_def)
```
```  5072   next
```
```  5073     fix T x
```
```  5074     assume T: "open T" "x \<in> T" and x: "x \<in> s"
```
```  5075     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"
```
```  5076       by auto
```
```  5077     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"
```
```  5078       by auto
```
```  5079     from Rats_dense_in_real[OF \<open>0 < e / 2\<close>] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"
```
```  5080       by auto
```
```  5081     from f[rule_format, of r] \<open>0 < r\<close> \<open>x \<in> s\<close> obtain k where "k \<in> f r" "x \<in> ball k r"
```
```  5082       by auto
```
```  5083     from \<open>r \<in> \<rat>\<close> \<open>0 < r\<close> \<open>k \<in> f r\<close> have "ball k r \<in> K"
```
```  5084       by (auto simp: K_def)
```
```  5085     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
```
```  5086     proof (rule bexI[rotated], safe)
```
```  5087       fix y
```
```  5088       assume "y \<in> ball k r"
```
```  5089       with \<open>r < e / 2\<close> \<open>x \<in> ball k r\<close> have "dist x y < e"
```
```  5090         by (intro dist_triangle_half_r [of k _ e]) (auto simp: dist_commute)
```
```  5091       with \<open>ball x e \<subseteq> T\<close> show "y \<in> T"
```
```  5092         by auto
```
```  5093     next
```
```  5094       show "x \<in> ball k r" by fact
```
```  5095     qed
```
```  5096   qed
```
```  5097 qed
```
```  5098
```
```  5099 lemma compact_eq_seq_compact_metric:
```
```  5100   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
```
```  5101   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
```
```  5102
```
```  5103 lemma compact_def:
```
```  5104   "compact (S :: 'a::metric_space set) \<longleftrightarrow>
```
```  5105    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
```
```  5106   unfolding compact_eq_seq_compact_metric seq_compact_def by auto
```
```  5107
```
```  5108 subsubsection \<open>Complete the chain of compactness variants\<close>
```
```  5109
```
```  5110 lemma compact_eq_bolzano_weierstrass:
```
```  5111   fixes s :: "'a::metric_space set"
```
```  5112   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"
```
```  5113   (is "?lhs = ?rhs")
```
```  5114 proof
```
```  5115   assume ?lhs
```
```  5116   then show ?rhs
```
```  5117     using heine_borel_imp_bolzano_weierstrass[of s] by auto
```
```  5118 next
```
```  5119   assume ?rhs
```
```  5120   then show ?lhs
```
```  5121     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
```
```  5122 qed
```
```  5123
```
```  5124 lemma bolzano_weierstrass_imp_bounded:
```
```  5125   "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
```
```  5126   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .
```
```  5127
```
```  5128
```
```  5129 subsection \<open>Metric spaces with the Heine-Borel property\<close>
```
```  5130
```
```  5131 text \<open>
```
```  5132   A metric space (or topological vector space) is said to have the
```
```  5133   Heine-Borel property if every closed and bounded subset is compact.
```
```  5134 \<close>
```
```  5135
```
```  5136 class heine_borel = metric_space +
```
```  5137   assumes bounded_imp_convergent_subsequence:
```
```  5138     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  5139
```
```  5140 lemma bounded_closed_imp_seq_compact:
```
```  5141   fixes s::"'a::heine_borel set"
```
```  5142   assumes "bounded s"
```
```  5143     and "closed s"
```
```  5144   shows "seq_compact s"
```
```  5145 proof (unfold seq_compact_def, clarify)
```
```  5146   fix f :: "nat \<Rightarrow> 'a"
```
```  5147   assume f: "\<forall>n. f n \<in> s"
```
```  5148   with \<open>bounded s\<close> have "bounded (range f)"
```
```  5149     by (auto intro: bounded_subset)
```
```  5150   obtain l r where r: "subseq r" and l: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  5151     using bounded_imp_convergent_subsequence [OF \<open>bounded (range f)\<close>] by auto
```
```  5152   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"
```
```  5153     by simp
```
```  5154   have "l \<in> s" using \<open>closed s\<close> fr l
```
```  5155     by (rule closed_sequentially)
```
```  5156   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  5157     using \<open>l \<in> s\<close> r l by blast
```
```  5158 qed
```
```  5159
```
```  5160 lemma compact_eq_bounded_closed:
```
```  5161   fixes s :: "'a::heine_borel set"
```
```  5162   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"
```
```  5163   (is "?lhs = ?rhs")
```
```  5164 proof
```
```  5165   assume ?lhs
```
```  5166   then show ?rhs
```
```  5167     using compact_imp_closed compact_imp_bounded
```
```  5168     by blast
```
```  5169 next
```
```  5170   assume ?rhs
```
```  5171   then show ?lhs
```
```  5172     using bounded_closed_imp_seq_compact[of s]
```
```  5173     unfolding compact_eq_seq_compact_metric
```
```  5174     by auto
```
```  5175 qed
```
```  5176
```
```  5177 lemma compact_closure [simp]:
```
```  5178   fixes S :: "'a::heine_borel set"
```
```  5179   shows "compact(closure S) \<longleftrightarrow> bounded S"
```
```  5180 by (meson bounded_closure bounded_subset closed_closure closure_subset compact_eq_bounded_closed)
```
```  5181
```
```  5182 lemma compact_components:
```
```  5183   fixes s :: "'a::heine_borel set"
```
```  5184   shows "\<lbrakk>compact s; c \<in> components s\<rbrakk> \<Longrightarrow> compact c"
```
```  5185 by (meson bounded_subset closed_components in_components_subset compact_eq_bounded_closed)
```
```  5186
```
```  5187 lemma not_compact_UNIV[simp]:
```
```  5188   fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
```
```  5189   shows "~ compact (UNIV::'a set)"
```
```  5190     by (simp add: compact_eq_bounded_closed)
```
```  5191
```
```  5192 (* TODO: is this lemma necessary? *)
```
```  5193 lemma bounded_increasing_convergent:
```
```  5194   fixes s :: "nat \<Rightarrow> real"
```
```  5195   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s \<longlonglongrightarrow> l"
```
```  5196   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]
```
```  5197   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)
```
```  5198
```
```  5199 instance real :: heine_borel
```
```  5200 proof
```
```  5201   fix f :: "nat \<Rightarrow> real"
```
```  5202   assume f: "bounded (range f)"
```
```  5203   obtain r where r: "subseq r" "monoseq (f \<circ> r)"
```
```  5204     unfolding comp_def by (metis seq_monosub)
```
```  5205   then have "Bseq (f \<circ> r)"
```
```  5206     unfolding Bseq_eq_bounded using f by (force intro: bounded_subset)
```
```  5207   with r show "\<exists>l r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
```
```  5208     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
```
```  5209 qed
```
```  5210
```
```  5211 lemma compact_lemma_general:
```
```  5212   fixes f :: "nat \<Rightarrow> 'a"
```
```  5213   fixes proj::"'a \<Rightarrow> 'b \<Rightarrow> 'c::heine_borel" (infixl "proj" 60)
```
```  5214   fixes unproj:: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"
```
```  5215   assumes finite_basis: "finite basis"
```
```  5216   assumes bounded_proj: "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. x proj k) ` range f)"
```
```  5217   assumes proj_unproj: "\<And>e k. k \<in> basis \<Longrightarrow> (unproj e) proj k = e k"
```
```  5218   assumes unproj_proj: "\<And>x. unproj (\<lambda>k. x proj k) = x"
```
```  5219   shows "\<forall>d\<subseteq>basis. \<exists>l::'a. \<exists> r.
```
```  5220     subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
```
```  5221 proof safe
```
```  5222   fix d :: "'b set"
```
```  5223   assume d: "d \<subseteq> basis"
```
```  5224   with finite_basis have "finite d"
```
```  5225     by (blast intro: finite_subset)
```
```  5226   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>
```
```  5227     (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
```
```  5228   proof (induct d)
```
```  5229     case empty
```
```  5230     then show ?case
```
```  5231       unfolding subseq_def by auto
```
```  5232   next
```
```  5233     case (insert k d)
```
```  5234     have k[intro]: "k \<in> basis"
```
```  5235       using insert by auto
```
```  5236     have s': "bounded ((\<lambda>x. x proj k) ` range f)"
```
```  5237       using k
```
```  5238       by (rule bounded_proj)
```
```  5239     obtain l1::"'a" and r1 where r1: "subseq r1"
```
```  5240       and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
```
```  5241       using insert(3) using insert(4) by auto
```
```  5242     have f': "\<forall>n. f (r1 n) proj k \<in> (\<lambda>x. x proj k) ` range f"
```
```  5243       by simp
```
```  5244     have "bounded (range (\<lambda>i. f (r1 i) proj k))"
```
```  5245       by (metis (lifting) bounded_subset f' image_subsetI s')
```
```  5246     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) proj k) \<longlongrightarrow> l2) sequentially"
```
```  5247       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) proj k"]
```
```  5248       by (auto simp: o_def)
```
```  5249     define r where "r = r1 \<circ> r2"
```
```  5250     have r:"subseq r"
```
```  5251       using r1 and r2 unfolding r_def o_def subseq_def by auto
```
```  5252     moreover
```
```  5253     define l where "l = unproj (\<lambda>i. if i = k then l2 else l1 proj i)"
```
```  5254     {
```
```  5255       fix e::real
```
```  5256       assume "e > 0"
```
```  5257       from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
```
```  5258         by blast
```
```  5259       from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) proj k) l2 < e) sequentially"
```
```  5260         by (rule tendstoD)
```
```  5261       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e) sequentially"
```
```  5262         by (rule eventually_subseq)
```
```  5263       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) proj i) (l proj i) < e) sequentially"
```
```  5264         using N1' N2
```
```  5265         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def proj_unproj)
```
```  5266     }
```
```  5267     ultimately show ?case by auto
```
```  5268   qed
```
```  5269 qed
```
```  5270
```
```  5271 lemma compact_lemma:
```
```  5272   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
```
```  5273   assumes "bounded (range f)"
```
```  5274   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.
```
```  5275     subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
```
```  5276   by (rule compact_lemma_general[where unproj="\<lambda>e. \<Sum>i\<in>Basis. e i *\<^sub>R i"])
```
```  5277      (auto intro!: assms bounded_linear_inner_left bounded_linear_image
```
```  5278        simp: euclidean_representation)
```
```  5279
```
```  5280 instance euclidean_space \<subseteq> heine_borel
```
```  5281 proof
```
```  5282   fix f :: "nat \<Rightarrow> 'a"
```
```  5283   assume f: "bounded (range f)"
```
```  5284   then obtain l::'a and r where r: "subseq r"
```
```  5285     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
```
```  5286     using compact_lemma [OF f] by blast
```
```  5287   {
```
```  5288     fix e::real
```
```  5289     assume "e > 0"
```
```  5290     hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive)
```
```  5291     with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))) sequentially"
```
```  5292       by simp
```
```  5293     moreover
```
```  5294     {
```
```  5295       fix n
```
```  5296       assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"
```
```  5297       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"
```
```  5298         apply (subst euclidean_dist_l2)
```
```  5299         using zero_le_dist
```
```  5300         apply (rule setL2_le_sum)
```
```  5301         done
```
```  5302       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
```
```  5303         apply (rule sum_strict_mono)
```
```  5304         using n
```
```  5305         apply auto
```
```  5306         done
```
```  5307       finally have "dist (f (r n)) l < e"
```
```  5308         by auto
```
```  5309     }
```
```  5310     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```  5311       by (rule eventually_mono)
```
```  5312   }
```
```  5313   then have *: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  5314     unfolding o_def tendsto_iff by simp
```
```  5315   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  5316     by auto
```
```  5317 qed
```
```  5318
```
```  5319 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
```
```  5320   unfolding bounded_def
```
```  5321   by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)
```
```  5322
```
```  5323 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
```
```  5324   unfolding bounded_def
```
```  5325   by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)
```
```  5326
```
```  5327 instance prod :: (heine_borel, heine_borel) heine_borel
```
```  5328 proof
```
```  5329   fix f :: "nat \<Rightarrow> 'a \<times> 'b"
```
```  5330   assume f: "bounded (range f)"
```
```  5331   then have "bounded (fst ` range f)"
```
```  5332     by (rule bounded_fst)
```
```  5333   then have s1: "bounded (range (fst \<circ> f))"
```
```  5334     by (simp add: image_comp)
```
```  5335   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) \<longlonglongrightarrow> l1"
```
```  5336     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
```
```  5337   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
```
```  5338     by (auto simp add: image_comp intro: bounded_snd bounded_subset)
```
```  5339   obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) \<longlongrightarrow> l2) sequentially"
```
```  5340     using bounded_imp_convergent_subsequence [OF s2]
```
```  5341     unfolding o_def by fast
```
```  5342   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) \<longlongrightarrow> l1) sequentially"
```
```  5343     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
```
```  5344   have l: "((f \<circ> (r1 \<circ> r2)) \<longlongrightarrow> (l1, l2)) sequentially"
```
```  5345     using tendsto_Pair [OF l1' l2] unfolding o_def by simp
```
```  5346   have r: "subseq (r1 \<circ> r2)"
```
```  5347     using r1 r2 unfolding subseq_def by simp
```
```  5348   show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```  5349     using l r by fast
```
```  5350 qed
```
```  5351
```
```  5352 subsubsection \<open>Intersecting chains of compact sets\<close>
```
```  5353
```
```  5354 proposition bounded_closed_chain:
```
```  5355   fixes \<F> :: "'a::heine_borel set set"
```
```  5356   assumes "B \<in> \<F>" "bounded B" and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" and "{} \<notin> \<F>"
```
```  5357       and chain: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
```
```  5358     shows "\<Inter>\<F> \<noteq> {}"
```
```  5359 proof -
```
```  5360   have "B \<inter> \<Inter>\<F> \<noteq> {}"
```
```  5361   proof (rule compact_imp_fip)
```
```  5362     show "compact B" "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
```
```  5363       by (simp_all add: assms compact_eq_bounded_closed)
```
```  5364     show "\<lbrakk>finite \<G>; \<G> \<subseteq> \<F>\<rbrakk> \<Longrightarrow> B \<inter> \<Inter>\<G> \<noteq> {}" for \<G>
```
```  5365     proof (induction \<G> rule: finite_induct)
```
```  5366       case empty
```
```  5367       with assms show ?case by force
```
```  5368     next
```
```  5369       case (insert U \<G>)
```
```  5370       then have "U \<in> \<F>" and ne: "B \<inter> \<Inter>\<G> \<noteq> {}" by auto
```
```  5371       then consider "B \<subseteq> U" | "U \<subseteq> B"
```
```  5372           using \<open>B \<in> \<F>\<close> chain by blast
```
```  5373         then show ?case
```
```  5374         proof cases
```
```  5375           case 1
```
```  5376           then show ?thesis
```
```  5377             using Int_left_commute ne by auto
```
```  5378         next
```
```  5379           case 2
```
```  5380           have "U \<noteq> {}"
```
```  5381             using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast
```
```  5382           moreover
```
```  5383           have False if "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. x \<notin> Y"
```
```  5384           proof -
```
```  5385             have "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. Y \<subseteq> U"
```
```  5386               by (metis chain contra_subsetD insert.prems insert_subset that)
```
```  5387             then obtain Y where "Y \<in> \<G>" "Y \<subseteq> U"
```
```  5388               by (metis all_not_in_conv \<open>U \<noteq> {}\<close>)
```
```  5389             moreover obtain x where "x \<in> \<Inter>\<G>"
```
```  5390               by (metis Int_emptyI ne)
```
```  5391             ultimately show ?thesis
```
```  5392               by (metis Inf_lower subset_eq that)
```
```  5393           qed
```
```  5394           with 2 show ?thesis
```
```  5395             by blast
```
```  5396         qed
```
```  5397       qed
```
```  5398   qed
```
```  5399   then show ?thesis by blast
```
```  5400 qed
```
```  5401
```
```  5402 corollary compact_chain:
```
```  5403   fixes \<F> :: "'a::heine_borel set set"
```
```  5404   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" "{} \<notin> \<F>"
```
```  5405           "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
```
```  5406     shows "\<Inter> \<F> \<noteq> {}"
```
```  5407 proof (cases "\<F> = {}")
```
```  5408   case True
```
```  5409   then show ?thesis by auto
```
```  5410 next
```
```  5411   case False
```
```  5412   show ?thesis
```
```  5413     by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
```
```  5414 qed
```
```  5415
```
```  5416 lemma compact_nest:
```
```  5417   fixes F :: "'a::linorder \<Rightarrow> 'b::heine_borel set"
```
```  5418   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"
```
```  5419   shows "\<Inter>range F \<noteq> {}"
```
```  5420 proof -
```
```  5421   have *: "\<And>S T. S \<in> range F \<and> T \<in> range F \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
```
```  5422     by (metis mono image_iff le_cases)
```
```  5423   show ?thesis
```
```  5424     apply (rule compact_chain [OF _ _ *])
```
```  5425     using F apply (blast intro: dest: *)+
```
```  5426     done
```
```  5427 qed
```
```  5428
```
```  5429
```
```  5430 subsubsection \<open>Completeness\<close>
```
```  5431
```
```  5432 lemma (in metric_space) completeI:
```
```  5433   assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f \<longlonglongrightarrow> l"
```
```  5434   shows "complete s"
```
```  5435   using assms unfolding complete_def by fast
```
```  5436
```
```  5437 lemma (in metric_space) completeE:
```
```  5438   assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"
```
```  5439   obtains l where "l \<in> s" and "f \<longlonglongrightarrow> l"
```
```  5440   using assms unfolding complete_def by fast
```
```  5441
```
```  5442 (* TODO: generalize to uniform spaces *)
```
```  5443 lemma compact_imp_complete:
```
```  5444   fixes s :: "'a::metric_space set"
```
```  5445   assumes "compact s"
```
```  5446   shows "complete s"
```
```  5447 proof -
```
```  5448   {
```
```  5449     fix f
```
```  5450     assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
```
```  5451     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) \<longlonglongrightarrow> l"
```
```  5452       using assms unfolding compact_def by blast
```
```  5453
```
```  5454     note lr' = seq_suble [OF lr(2)]
```
```  5455     {
```
```  5456       fix e :: real
```
```  5457       assume "e > 0"
```
```  5458       from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2"
```
```  5459         unfolding cauchy_def
```
```  5460         using \<open>e > 0\<close>
```
```  5461         apply (erule_tac x="e/2" in allE)
```
```  5462         apply auto
```
```  5463         done
```
```  5464       from lr(3)[unfolded lim_sequentially, THEN spec[where x="e/2"]]
```
```  5465       obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"
```
```  5466         using \<open>e > 0\<close> by auto
```
```  5467       {
```
```  5468         fix n :: nat
```
```  5469         assume n: "n \<ge> max N M"
```
```  5470         have "dist ((f \<circ> r) n) l < e/2"
```
```  5471           using n M by auto
```
```  5472         moreover have "r n \<ge> N"
```
```  5473           using lr'[of n] n by auto
```
```  5474         then have "dist (f n) ((f \<circ> r) n) < e / 2"
```
```  5475           using N and n by auto
```
```  5476         ultimately have "dist (f n) l < e"
```
```  5477           using dist_triangle_half_r[of "f (r n)" "f n" e l]
```
```  5478           by (auto simp add: dist_commute)
```
```  5479       }
```
```  5480       then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
```
```  5481     }
```
```  5482     then have "\<exists>l\<in>s. (f \<longlongrightarrow> l) sequentially" using \<open>l\<in>s\<close>
```
```  5483       unfolding lim_sequentially by auto
```
```  5484   }
```
```  5485   then show ?thesis unfolding complete_def by auto
```
```  5486 qed
```
```  5487
```
```  5488 lemma nat_approx_posE:
```
```  5489   fixes e::real
```
```  5490   assumes "0 < e"
```
```  5491   obtains n :: nat where "1 / (Suc n) < e"
```
```  5492 proof atomize_elim
```
```  5493   have "1 / real (Suc (nat \<lceil>1/e\<rceil>)) < 1 / \<lceil>1/e\<rceil>"
```
```  5494     by (rule divide_strict_left_mono) (auto simp: \<open>0 < e\<close>)
```
```  5495   also have "1 / \<lceil>1/e\<rceil> \<le> 1 / (1/e)"
```
```  5496     by (rule divide_left_mono) (auto simp: \<open>0 < e\<close> ceiling_correct)
```
```  5497   also have "\<dots> = e" by simp
```
```  5498   finally show  "\<exists>n. 1 / real (Suc n) < e" ..
```
```  5499 qed
```
```  5500
```
```  5501 lemma compact_eq_totally_bounded:
```
```  5502   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e))"
```
```  5503     (is "_ \<longleftrightarrow> ?rhs")
```
```  5504 proof
```
```  5505   assume assms: "?rhs"
```
```  5506   then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"
```
```  5507     by (auto simp: choice_iff')
```
```  5508
```
```  5509   show "compact s"
```
```  5510   proof cases
```
```  5511     assume "s = {}"
```
```  5512     then show "compact s" by (simp add: compact_def)
```
```  5513   next
```
```  5514     assume "s \<noteq> {}"
```
```  5515     show ?thesis
```
```  5516       unfolding compact_def
```
```  5517     proof safe
```
```  5518       fix f :: "nat \<Rightarrow> 'a"
```
```  5519       assume f: "\<forall>n. f n \<in> s"
```
```  5520
```
```  5521       define e where "e n = 1 / (2 * Suc n)" for n
```
```  5522       then have [simp]: "\<And>n. 0 < e n" by auto
```
```  5523       define B where "B n U = (SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U))" for n U
```
```  5524       {
```
```  5525         fix n U
```
```  5526         assume "infinite {n. f n \<in> U}"
```
```  5527         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
```
```  5528           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
```
```  5529         then obtain a where
```
```  5530           "a \<in> k (e n)"
```
```  5531           "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..
```
```  5532         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
```
```  5533           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
```
```  5534         from someI_ex[OF this]
```
```  5535         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
```
```  5536           unfolding B_def by auto
```
```  5537       }
```
```  5538       note B = this
```
```  5539
```
```  5540       define F where "F = rec_nat (B 0 UNIV) B"
```
```  5541       {
```
```  5542         fix n
```
```  5543         have "infinite {i. f i \<in> F n}"
```
```  5544           by (induct n) (auto simp: F_def B)
```
```  5545       }
```
```  5546       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
```
```  5547         using B by (simp add: F_def)
```
```  5548       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
```
```  5549         using decseq_SucI[of F] by (auto simp: decseq_def)
```
```  5550
```
```  5551       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
```
```  5552       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
```
```  5553         fix k i
```
```  5554         have "infinite ({n. f n \<in> F k} - {.. i})"
```
```  5555           using \<open>infinite {n. f n \<in> F k}\<close> by auto
```
```  5556         from infinite_imp_nonempty[OF this]
```
```  5557         show "\<exists>x>i. f x \<in> F k"
```
```  5558           by (simp add: set_eq_iff not_le conj_commute)
```
```  5559       qed
```
```  5560
```
```  5561       define t where "t = rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
```
```  5562       have "subseq t"
```
```  5563         unfolding subseq_Suc_iff by (simp add: t_def sel)
```
```  5564       moreover have "\<forall>i. (f \<circ> t) i \<in> s"
```
```  5565         using f by auto
```
```  5566       moreover
```
```  5567       {
```
```  5568         fix n
```
```  5569         have "(f \<circ> t) n \<in> F n"
```
```  5570           by (cases n) (simp_all add: t_def sel)
```
```  5571       }
```
```  5572       note t = this
```
```  5573
```
```  5574       have "Cauchy (f \<circ> t)"
```
```  5575       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
```
```  5576         fix r :: real and N n m
```
```  5577         assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
```
```  5578         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
```
```  5579           using F_dec t by (auto simp: e_def field_simps of_nat_Suc)
```
```  5580         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
```
```  5581           by (auto simp: subset_eq)
```
```  5582         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] \<open>2 * e N < r\<close>
```
```  5583         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
```
```  5584           by (simp add: dist_commute)
```
```  5585       qed
```
```  5586
```
```  5587       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
```
```  5588         using assms unfolding complete_def by blast
```
```  5589     qed
```
```  5590   qed
```
```  5591 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
```
```  5592
```
```  5593 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
```
```  5594 proof -
```
```  5595   {
```
```  5596     assume ?rhs
```
```  5597     {
```
```  5598       fix e::real
```
```  5599       assume "e>0"
```
```  5600       with \<open>?rhs\<close> obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
```
```  5601         by (erule_tac x="e/2" in allE) auto
```
```  5602       {
```
```  5603         fix n m
```
```  5604         assume nm:"N \<le> m \<and> N \<le> n"
```
```  5605         then have "dist (s m) (s n) < e" using N
```
```  5606           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
```
```  5607           by blast
```
```  5608       }
```
```  5609       then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
```
```  5610         by blast
```
```  5611     }
```
```  5612     then have ?lhs
```
```  5613       unfolding cauchy_def
```
```  5614       by blast
```
```  5615   }
```
```  5616   then show ?thesis
```
```  5617     unfolding cauchy_def
```
```  5618     using dist_triangle_half_l
```
```  5619     by blast
```
```  5620 qed
```
```  5621
```
```  5622 lemma cauchy_imp_bounded:
```
```  5623   assumes "Cauchy s"
```
```  5624   shows "bounded (range s)"
```
```  5625 proof -
```
```  5626   from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"
```
```  5627     unfolding cauchy_def
```
```  5628     apply (erule_tac x= 1 in allE)
```
```  5629     apply auto
```
```  5630     done
```
```  5631   then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
```
```  5632   moreover
```
```  5633   have "bounded (s ` {0..N})"
```
```  5634     using finite_imp_bounded[of "s ` {1..N}"] by auto
```
```  5635   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
```
```  5636     unfolding bounded_any_center [where a="s N"] by auto
```
```  5637   ultimately show "?thesis"
```
```  5638     unfolding bounded_any_center [where a="s N"]
```
```  5639     apply (rule_tac x="max a 1" in exI)
```
```  5640     apply auto
```
```  5641     apply (erule_tac x=y in allE)
```
```  5642     apply (erule_tac x=y in ballE)
```
```  5643     apply auto
```
```  5644     done
```
```  5645 qed
```
```  5646
```
```  5647 instance heine_borel < complete_space
```
```  5648 proof
```
```  5649   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
```
```  5650   then have "bounded (range f)"
```
```  5651     by (rule cauchy_imp_bounded)
```
```  5652   then have "compact (closure (range f))"
```
```  5653     unfolding compact_eq_bounded_closed by auto
```
```  5654   then have "complete (closure (range f))"
```
```  5655     by (rule compact_imp_complete)
```
```  5656   moreover have "\<forall>n. f n \<in> closure (range f)"
```
```  5657     using closure_subset [of "range f"] by auto
```
```  5658   ultimately have "\<exists>l\<in>closure (range f). (f \<longlongrightarrow> l) sequentially"
```
```  5659     using \<open>Cauchy f\<close> unfolding complete_def by auto
```
```  5660   then show "convergent f"
```
```  5661     unfolding convergent_def by auto
```
```  5662 qed
```
```  5663
```
```  5664 instance euclidean_space \<subseteq> banach ..
```
```  5665
```
```  5666 lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"
```
```  5667 proof (rule completeI)
```
```  5668   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
```
```  5669   then have "convergent f" by (rule Cauchy_convergent)
```
```  5670   then show "\<exists>l\<in>UNIV. f \<longlonglongrightarrow> l" unfolding convergent_def by simp
```
```  5671 qed
```
```  5672
```
```  5673 lemma complete_imp_closed:
```
```  5674   fixes S :: "'a::metric_space set"
```
```  5675   assumes "complete S"
```
```  5676   shows "closed S"
```
```  5677 proof (unfold closed_sequential_limits, clarify)
```
```  5678   fix f x assume "\<forall>n. f n \<in> S" and "f \<longlonglongrightarrow> x"
```
```  5679   from \<open>f \<longlonglongrightarrow> x\<close> have "Cauchy f"
```
```  5680     by (rule LIMSEQ_imp_Cauchy)
```
```  5681   with \<open>complete S\<close> and \<open>\<forall>n. f n \<in> S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
```
```  5682     by (rule completeE)
```
```  5683   from \<open>f \<longlonglongrightarrow> x\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "x = l"
```
```  5684     by (rule LIMSEQ_unique)
```
```  5685   with \<open>l \<in> S\<close> show "x \<in> S"
```
```  5686     by simp
```
```  5687 qed
```
```  5688
```
```  5689 lemma complete_Int_closed:
```
```  5690   fixes S :: "'a::metric_space set"
```
```  5691   assumes "complete S" and "closed t"
```
```  5692   shows "complete (S \<inter> t)"
```
```  5693 proof (rule completeI)
```
```  5694   fix f assume "\<forall>n. f n \<in> S \<inter> t" and "Cauchy f"
```
```  5695   then have "\<forall>n. f n \<in> S" and "\<forall>n. f n \<in> t"
```
```  5696     by simp_all
```
```  5697   from \<open>complete S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
```
```  5698     using \<open>\<forall>n. f n \<in> S\<close> and \<open>Cauchy f\<close> by (rule completeE)
```
```  5699   from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "l \<in> t"
```
```  5700     by (rule closed_sequentially)
```
```  5701   with \<open>l \<in> S\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>S \<inter> t. f \<longlonglongrightarrow> l"
```
```  5702     by fast
```
```  5703 qed
```
```  5704
```
```  5705 lemma complete_closed_subset:
```
```  5706   fixes S :: "'a::metric_space set"
```
```  5707   assumes "closed S" and "S \<subseteq> t" and "complete t"
```
```  5708   shows "complete S"
```
```  5709   using assms complete_Int_closed [of t S] by (simp add: Int_absorb1)
```
```  5710
```
```  5711 lemma complete_eq_closed:
```
```  5712   fixes S :: "('a::complete_space) set"
```
```  5713   shows "complete S \<longleftrightarrow> closed S"
```
```  5714 proof
```
```  5715   assume "closed S" then show "complete S"
```
```  5716     using subset_UNIV complete_UNIV by (rule complete_closed_subset)
```
```  5717 next
```
```  5718   assume "complete S" then show "closed S"
```
```  5719     by (rule complete_imp_closed)
```
```  5720 qed
```
```  5721
```
```  5722 lemma convergent_eq_Cauchy:
```
```  5723   fixes S :: "nat \<Rightarrow> 'a::complete_space"
```
```  5724   shows "(\<exists>l. (S \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy S"
```
```  5725   unfolding Cauchy_convergent_iff convergent_def ..
```
```  5726
```
```  5727 lemma convergent_imp_bounded:
```
```  5728   fixes S :: "nat \<Rightarrow> 'a::metric_space"
```
```  5729   shows "(S \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range S)"
```
```  5730   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
```
```  5731
```
```  5732 lemma compact_cball[simp]:
```
```  5733   fixes x :: "'a::heine_borel"
```
```  5734   shows "compact (cball x e)"
```
```  5735   using compact_eq_bounded_closed bounded_cball closed_cball
```
```  5736   by blast
```
```  5737
```
```  5738 lemma compact_frontier_bounded[intro]:
```
```  5739   fixes S :: "'a::heine_borel set"
```
```  5740   shows "bounded S \<Longrightarrow> compact (frontier S)"
```
```  5741   unfolding frontier_def
```
```  5742   using compact_eq_bounded_closed
```
```  5743   by blast
```
```  5744
```
```  5745 lemma compact_frontier[intro]:
```
```  5746   fixes S :: "'a::heine_borel set"
```
```  5747   shows "compact S \<Longrightarrow> compact (frontier S)"
```
```  5748   using compact_eq_bounded_closed compact_frontier_bounded
```
```  5749   by blast
```
```  5750
```
```  5751 corollary compact_sphere [simp]:
```
```  5752   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
```
```  5753   shows "compact (sphere a r)"
```
```  5754 using compact_frontier [of "cball a r"] by simp
```
```  5755
```
```  5756 corollary bounded_sphere [simp]:
```
```  5757   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
```
```  5758   shows "bounded (sphere a r)"
```
```  5759 by (simp add: compact_imp_bounded)
```
```  5760
```
```  5761 corollary closed_sphere  [simp]:
```
```  5762   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
```
```  5763   shows "closed (sphere a r)"
```
```  5764 by (simp add: compact_imp_closed)
```
```  5765
```
```  5766 lemma frontier_subset_compact:
```
```  5767   fixes S :: "'a::heine_borel set"
```
```  5768   shows "compact S \<Longrightarrow> frontier S \<subseteq> S"
```
```  5769   using frontier_subset_closed compact_eq_bounded_closed
```
```  5770   by blast
```
```  5771
```
```  5772 subsection\<open>Relations among convergence and absolute convergence for power series.\<close>
```
```  5773
```
```  5774 lemma summable_imp_bounded:
```
```  5775   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  5776   shows "summable f \<Longrightarrow> bounded (range f)"
```
```  5777 by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
```
```  5778
```
```  5779 lemma summable_imp_sums_bounded:
```
```  5780    "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"
```
```  5781 by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
```
```  5782
```
```  5783 lemma power_series_conv_imp_absconv_weak:
```
```  5784   fixes a:: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" and w :: 'a
```
```  5785   assumes sum: "summable (\<lambda>n. a n * z ^ n)" and no: "norm w < norm z"
```
```  5786     shows "summable (\<lambda>n. of_real(norm(a n)) * w ^ n)"
```
```  5787 proof -
```
```  5788   obtain M where M: "\<And>x. norm (a x * z ^ x) \<le> M"
```
```  5789     using summable_imp_bounded [OF sum] by (force simp add: bounded_iff)
```
```  5790   then have *: "summable (\<lambda>n. norm (a n) * norm w ^ n)"
```
```  5791     by (rule_tac M=M in Abel_lemma) (auto simp: norm_mult norm_power intro: no)
```
```  5792   show ?thesis
```
```  5793     apply (rule series_comparison_complex [of "(\<lambda>n. of_real(norm(a n) * norm w ^ n))"])
```
```  5794     apply (simp only: summable_complex_of_real *)
```
```  5795     apply (auto simp: norm_mult norm_power)
```
```  5796     done
```
```  5797 qed
```
```  5798
```
```  5799 subsection \<open>Bounded closed nest property (proof does not use Heine-Borel)\<close>
```
```  5800
```
```  5801 lemma bounded_closed_nest:
```
```  5802   fixes s :: "nat \<Rightarrow> ('a::heine_borel) set"
```
```  5803   assumes "\<forall>n. closed (s n)"
```
```  5804     and "\<forall>n. s n \<noteq> {}"
```
```  5805     and "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"
```
```  5806     and "bounded (s 0)"
```
```  5807   shows "\<exists>a. \<forall>n. a \<in> s n"
```
```  5808 proof -
```
```  5809   from assms(2) obtain x where x: "\<forall>n. x n \<in> s n"
```
```  5810     using choice[of "\<lambda>n x. x \<in> s n"] by auto
```
```  5811   from assms(4,1) have "seq_compact (s 0)"
```
```  5812     by (simp add: bounded_closed_imp_seq_compact)
```
```  5813   then obtain l r where lr: "l \<in> s 0" "subseq r" "(x \<circ> r) \<longlonglongrightarrow> l"
```
```  5814     using x and assms(3) unfolding seq_compact_def by blast
```
```  5815   have "\<forall>n. l \<in> s n"
```
```  5816   proof
```
```  5817     fix n :: nat
```
```  5818     have "closed (s n)"
```
```  5819       using assms(1) by simp
```
```  5820     moreover have "\<forall>i. (x \<circ> r) i \<in> s i"
```
```  5821       using x and assms(3) and lr(2) [THEN seq_suble] by auto
```
```  5822     then have "\<forall>i. (x \<circ> r) (i + n) \<in> s n"
```
```  5823       using assms(3) by (fast intro!: le_add2)
```
```  5824     moreover have "(\<lambda>i. (x \<circ> r) (i + n)) \<longlonglongrightarrow> l"
```
```  5825       using lr(3) by (rule LIMSEQ_ignore_initial_segment)
```
```  5826     ultimately show "l \<in> s n"
```
```  5827       by (rule closed_sequentially)
```
```  5828   qed
```
```  5829   then show ?thesis ..
```
```  5830 qed
```
```  5831
```
```  5832 text \<open>Decreasing case does not even need compactness, just completeness.\<close>
```
```  5833
```
```  5834 lemma decreasing_closed_nest:
```
```  5835   fixes s :: "nat \<Rightarrow> ('a::complete_space) set"
```
```  5836   assumes
```
```  5837     "\<forall>n. closed (s n)"
```
```  5838     "\<forall>n. s n \<noteq> {}"
```
```  5839     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"
```
```  5840     "\<forall>e>0. \<exists>n. \<forall>x\<in>s n. \<forall>y\<in>s n. dist x y < e"
```
```  5841   shows "\<exists>a. \<forall>n. a \<in> s n"
```
```  5842 proof -
```
```  5843   have "\<forall>n. \<exists>x. x \<in> s n"
```
```  5844     using assms(2) by auto
```
```  5845   then have "\<exists>t. \<forall>n. t n \<in> s n"
```
```  5846     using choice[of "\<lambda>n x. x \<in> s n"] by auto
```
```  5847   then obtain t where t: "\<forall>n. t n \<in> s n" by auto
```
```  5848   {
```
```  5849     fix e :: real
```
```  5850     assume "e > 0"
```
```  5851     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"
```
```  5852       using assms(4) by auto
```
```  5853     {
```
```  5854       fix m n :: nat
```
```  5855       assume "N \<le> m \<and> N \<le> n"
```
```  5856       then have "t m \<in> s N" "t n \<in> s N"
```
```  5857         using assms(3) t unfolding  subset_eq t by blast+
```
```  5858       then have "dist (t m) (t n) < e"
```
```  5859         using N by auto
```
```  5860     }
```
```  5861     then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"
```
```  5862       by auto
```
```  5863   }
```
```  5864   then have "Cauchy t"
```
```  5865     unfolding cauchy_def by auto
```
```  5866   then obtain l where l:"(t \<longlongrightarrow> l) sequentially"
```
```  5867     using complete_UNIV unfolding complete_def by auto
```
```  5868   {
```
```  5869     fix n :: nat
```
```  5870     {
```
```  5871       fix e :: real
```
```  5872       assume "e > 0"
```
```  5873       then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"
```
```  5874         using l[unfolded lim_sequentially] by auto
```
```  5875       have "t (max n N) \<in> s n"
```
```  5876         using assms(3)
```
```  5877         unfolding subset_eq
```
```  5878         apply (erule_tac x=n in allE)
```
```  5879         apply (erule_tac x="max n N" in allE)
```
```  5880         using t
```
```  5881         apply auto
```
```  5882         done
```
```  5883       then have "\<exists>y\<in>s n. dist y l < e"
```
```  5884         apply (rule_tac x="t (max n N)" in bexI)
```
```  5885         using N
```
```  5886         apply auto
```
```  5887         done
```
```  5888     }
```
```  5889     then have "l \<in> s n"
```
```  5890       using closed_approachable[of "s n" l] assms(1) by auto
```
```  5891   }
```
```  5892   then show ?thesis by auto
```
```  5893 qed
```
```  5894
```
```  5895 text \<open>Strengthen it to the intersection actually being a singleton.\<close>
```
```  5896
```
```  5897 lemma decreasing_closed_nest_sing:
```
```  5898   fixes s :: "nat \<Rightarrow> 'a::complete_space set"
```
```  5899   assumes
```
```  5900     "\<forall>n. closed(s n)"
```
```  5901     "\<forall>n. s n \<noteq> {}"
```
```  5902     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"
```
```  5903     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
```
```  5904   shows "\<exists>a. \<Inter>(range s) = {a}"
```
```  5905 proof -
```
```  5906   obtain a where a: "\<forall>n. a \<in> s n"
```
```  5907     using decreasing_closed_nest[of s] using assms by auto
```
```  5908   {
```
```  5909     fix b
```
```  5910     assume b: "b \<in> \<Inter>(range s)"
```
```  5911     {
```
```  5912       fix e :: real
```
```  5913       assume "e > 0"
```
```  5914       then have "dist a b < e"
```