src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author hoelzl
Fri Mar 22 10:41:43 2013 +0100 (2013-03-22)
changeset 51475 ebf9d4fd00ba
parent 51473 1210309fddab
child 51478 270b21f3ae0a
permissions -rw-r--r--
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
     1 (*  title:      HOL/Library/Topology_Euclidian_Space.thy
     2     Author:     Amine Chaieb, University of Cambridge
     3     Author:     Robert Himmelmann, TU Muenchen
     4     Author:     Brian Huffman, Portland State University
     5 *)
     6 
     7 header {* Elementary topology in Euclidean space. *}
     8 
     9 theory Topology_Euclidean_Space
    10 imports
    11   Complex_Main
    12   "~~/src/HOL/Library/Countable_Set"
    13   "~~/src/HOL/Library/Glbs"
    14   "~~/src/HOL/Library/FuncSet"
    15   Linear_Algebra
    16   Norm_Arith
    17 begin
    18 
    19 lemma dist_0_norm:
    20   fixes x :: "'a::real_normed_vector"
    21   shows "dist 0 x = norm x"
    22 unfolding dist_norm by simp
    23 
    24 lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d"
    25   using dist_triangle[of y z x] by (simp add: dist_commute)
    26 
    27 (* LEGACY *)
    28 lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s o r) ----> l"
    29   by (rule LIMSEQ_subseq_LIMSEQ)
    30 
    31 lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
    32 
    33 lemma countable_PiE: 
    34   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"
    35   by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
    36 
    37 subsection {* Topological Basis *}
    38 
    39 context topological_space
    40 begin
    41 
    42 definition "topological_basis B =
    43   ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x)))"
    44 
    45 lemma topological_basis:
    46   "topological_basis B = (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
    47   unfolding topological_basis_def
    48   apply safe
    49      apply fastforce
    50     apply fastforce
    51    apply (erule_tac x="x" in allE)
    52    apply simp
    53    apply (rule_tac x="{x}" in exI)
    54   apply auto
    55   done
    56 
    57 lemma topological_basis_iff:
    58   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
    59   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'))"
    60     (is "_ \<longleftrightarrow> ?rhs")
    61 proof safe
    62   fix O' and x::'a
    63   assume H: "topological_basis B" "open O'" "x \<in> O'"
    64   hence "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
    65   then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
    66   thus "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
    67 next
    68   assume H: ?rhs
    69   show "topological_basis B" using assms unfolding topological_basis_def
    70   proof safe
    71     fix O'::"'a set" assume "open O'"
    72     with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
    73       by (force intro: bchoice simp: Bex_def)
    74     thus "\<exists>B'\<subseteq>B. \<Union>B' = O'"
    75       by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
    76   qed
    77 qed
    78 
    79 lemma topological_basisI:
    80   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
    81   assumes "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
    82   shows "topological_basis B"
    83   using assms by (subst topological_basis_iff) auto
    84 
    85 lemma topological_basisE:
    86   fixes O'
    87   assumes "topological_basis B"
    88   assumes "open O'"
    89   assumes "x \<in> O'"
    90   obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
    91 proof atomize_elim
    92   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" by (simp add: topological_basis_def)
    93   with topological_basis_iff assms
    94   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" using assms by (simp add: Bex_def)
    95 qed
    96 
    97 lemma topological_basis_open:
    98   assumes "topological_basis B"
    99   assumes "X \<in> B"
   100   shows "open X"
   101   using assms
   102   by (simp add: topological_basis_def)
   103 
   104 lemma topological_basis_imp_subbasis:
   105   assumes B: "topological_basis B" shows "open = generate_topology B"
   106 proof (intro ext iffI)
   107   fix S :: "'a set" assume "open S"
   108   with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
   109     unfolding topological_basis_def by blast
   110   then show "generate_topology B S"
   111     by (auto intro: generate_topology.intros dest: topological_basis_open)
   112 next
   113   fix S :: "'a set" assume "generate_topology B S" then show "open S"
   114     by induct (auto dest: topological_basis_open[OF B])
   115 qed
   116 
   117 lemma basis_dense:
   118   fixes B::"'a set set" and f::"'a set \<Rightarrow> 'a"
   119   assumes "topological_basis B"
   120   assumes choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
   121   shows "(\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X))"
   122 proof (intro allI impI)
   123   fix X::"'a set" assume "open X" "X \<noteq> {}"
   124   from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]]
   125   guess B' . note B' = this
   126   thus "\<exists>B'\<in>B. f B' \<in> X" by (auto intro!: choosefrom_basis)
   127 qed
   128 
   129 end
   130 
   131 lemma topological_basis_prod:
   132   assumes A: "topological_basis A" and B: "topological_basis B"
   133   shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
   134   unfolding topological_basis_def
   135 proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
   136   fix S :: "('a \<times> 'b) set" assume "open S"
   137   then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
   138   proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
   139     fix x y assume "(x, y) \<in> S"
   140     from open_prod_elim[OF `open S` this]
   141     obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
   142       by (metis mem_Sigma_iff)
   143     moreover from topological_basisE[OF A a] guess A0 .
   144     moreover from topological_basisE[OF B b] guess B0 .
   145     ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
   146       by (intro UN_I[of "(A0, B0)"]) auto
   147   qed auto
   148 qed (metis A B topological_basis_open open_Times)
   149 
   150 subsection {* Countable Basis *}
   151 
   152 locale countable_basis =
   153   fixes B::"'a::topological_space set set"
   154   assumes is_basis: "topological_basis B"
   155   assumes countable_basis: "countable B"
   156 begin
   157 
   158 lemma open_countable_basis_ex:
   159   assumes "open X"
   160   shows "\<exists>B' \<subseteq> B. X = Union B'"
   161   using assms countable_basis is_basis unfolding topological_basis_def by blast
   162 
   163 lemma open_countable_basisE:
   164   assumes "open X"
   165   obtains B' where "B' \<subseteq> B" "X = Union B'"
   166   using assms open_countable_basis_ex by (atomize_elim) simp
   167 
   168 lemma countable_dense_exists:
   169   shows "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
   170 proof -
   171   let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
   172   have "countable (?f ` B)" using countable_basis by simp
   173   with basis_dense[OF is_basis, of ?f] show ?thesis
   174     by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
   175 qed
   176 
   177 lemma countable_dense_setE:
   178   obtains D :: "'a set"
   179   where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
   180   using countable_dense_exists by blast
   181 
   182 end
   183 
   184 lemma (in first_countable_topology) first_countable_basisE:
   185   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
   186     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
   187   using first_countable_basis[of x]
   188   apply atomize_elim
   189   apply (elim exE)
   190   apply (rule_tac x="range A" in exI)
   191   apply auto
   192   done
   193 
   194 lemma (in first_countable_topology) first_countable_basis_Int_stableE:
   195   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
   196     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
   197     "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"
   198 proof atomize_elim
   199   from first_countable_basisE[of x] guess A' . note A' = this
   200   def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"
   201   thus "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>
   202         (\<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)"
   203   proof (safe intro!: exI[where x=A])
   204     show "countable A" unfolding A_def by (intro countable_image countable_Collect_finite)
   205     fix a assume "a \<in> A"
   206     thus "x \<in> a" "open a" using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
   207   next
   208     let ?int = "\<lambda>N. \<Inter>from_nat_into A' ` N"
   209     fix a b assume "a \<in> A" "b \<in> A"
   210     then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)" by (auto simp: A_def)
   211     thus "a \<inter> b \<in> A" by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
   212   next
   213     fix S assume "open S" "x \<in> S" then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
   214     thus "\<exists>a\<in>A. a \<subseteq> S" using a A'
   215       by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])
   216   qed
   217 qed
   218 
   219 lemma (in topological_space) first_countableI:
   220   assumes "countable A" and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
   221    and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
   222   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))"
   223 proof (safe intro!: exI[of _ "from_nat_into A"])
   224   have "A \<noteq> {}" using 2[of UNIV] by auto
   225   { fix i show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
   226       using range_from_nat_into_subset[OF `A \<noteq> {}`] 1 by auto }
   227   { fix S assume "open S" "x\<in>S" from 2[OF this] show "\<exists>i. from_nat_into A i \<subseteq> S"
   228       using subset_range_from_nat_into[OF `countable A`] by auto }
   229 qed
   230 
   231 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
   232 proof
   233   fix x :: "'a \<times> 'b"
   234   from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
   235   from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
   236   show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) 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))"
   237   proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
   238     fix a b assume x: "a \<in> A" "b \<in> B"
   239     with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
   240       unfolding mem_Times_iff by (auto intro: open_Times)
   241   next
   242     fix S assume "open S" "x \<in> S"
   243     from open_prod_elim[OF this] guess a' b' .
   244     moreover with A(4)[of a'] B(4)[of b']
   245     obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" by auto
   246     ultimately show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
   247       by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
   248   qed (simp add: A B)
   249 qed
   250 
   251 class second_countable_topology = topological_space +
   252   assumes ex_countable_subbasis: "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
   253 begin
   254 
   255 lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
   256 proof -
   257   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" by blast
   258   let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
   259 
   260   show ?thesis
   261   proof (intro exI conjI)
   262     show "countable ?B"
   263       by (intro countable_image countable_Collect_finite_subset B)
   264     { fix S assume "open S"
   265       then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
   266         unfolding B
   267       proof induct
   268         case UNIV show ?case by (intro exI[of _ "{{}}"]) simp
   269       next
   270         case (Int a b)
   271         then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
   272           and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
   273           by blast
   274         show ?case
   275           unfolding x y Int_UN_distrib2
   276           by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
   277       next
   278         case (UN K)
   279         then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
   280         then guess k unfolding bchoice_iff ..
   281         then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
   282           by (intro exI[of _ "UNION K k"]) auto
   283       next
   284         case (Basis S) then show ?case
   285           by (intro exI[of _ "{{S}}"]) auto
   286       qed
   287       then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
   288         unfolding subset_image_iff by blast }
   289     then show "topological_basis ?B"
   290       unfolding topological_space_class.topological_basis_def
   291       by (safe intro!: topological_space_class.open_Inter) 
   292          (simp_all add: B generate_topology.Basis subset_eq)
   293   qed
   294 qed
   295 
   296 end
   297 
   298 sublocale second_countable_topology <
   299   countable_basis "SOME B. countable B \<and> topological_basis B"
   300   using someI_ex[OF ex_countable_basis]
   301   by unfold_locales safe
   302 
   303 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
   304 proof
   305   obtain A :: "'a set set" where "countable A" "topological_basis A"
   306     using ex_countable_basis by auto
   307   moreover
   308   obtain B :: "'b set set" where "countable B" "topological_basis B"
   309     using ex_countable_basis by auto
   310   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
   311     by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
   312       topological_basis_imp_subbasis)
   313 qed
   314 
   315 instance second_countable_topology \<subseteq> first_countable_topology
   316 proof
   317   fix x :: 'a
   318   def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"
   319   then have B: "countable B" "topological_basis B"
   320     using countable_basis is_basis
   321     by (auto simp: countable_basis is_basis)
   322   then show "\<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))"
   323     by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
   324        (fastforce simp: topological_space_class.topological_basis_def)+
   325 qed
   326 
   327 subsection {* Polish spaces *}
   328 
   329 text {* Textbooks define Polish spaces as completely metrizable.
   330   We assume the topology to be complete for a given metric. *}
   331 
   332 class polish_space = complete_space + second_countable_topology
   333 
   334 subsection {* General notion of a topology as a value *}
   335 
   336 definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
   337 typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
   338   morphisms "openin" "topology"
   339   unfolding istopology_def by blast
   340 
   341 lemma istopology_open_in[intro]: "istopology(openin U)"
   342   using openin[of U] by blast
   343 
   344 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
   345   using topology_inverse[unfolded mem_Collect_eq] .
   346 
   347 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
   348   using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
   349 
   350 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
   351 proof-
   352   { assume "T1=T2"
   353     hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }
   354   moreover
   355   { assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
   356     hence "openin T1 = openin T2" by (simp add: fun_eq_iff)
   357     hence "topology (openin T1) = topology (openin T2)" by simp
   358     hence "T1 = T2" unfolding openin_inverse .
   359   }
   360   ultimately show ?thesis by blast
   361 qed
   362 
   363 text{* Infer the "universe" from union of all sets in the topology. *}
   364 
   365 definition "topspace T =  \<Union>{S. openin T S}"
   366 
   367 subsubsection {* Main properties of open sets *}
   368 
   369 lemma openin_clauses:
   370   fixes U :: "'a topology"
   371   shows "openin U {}"
   372   "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
   373   "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
   374   using openin[of U] unfolding istopology_def mem_Collect_eq
   375   by fast+
   376 
   377 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
   378   unfolding topspace_def by blast
   379 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
   380 
   381 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
   382   using openin_clauses by simp
   383 
   384 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
   385   using openin_clauses by simp
   386 
   387 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
   388   using openin_Union[of "{S,T}" U] by auto
   389 
   390 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
   391 
   392 lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
   393   (is "?lhs \<longleftrightarrow> ?rhs")
   394 proof
   395   assume ?lhs
   396   then show ?rhs by auto
   397 next
   398   assume H: ?rhs
   399   let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
   400   have "openin U ?t" by (simp add: openin_Union)
   401   also have "?t = S" using H by auto
   402   finally show "openin U S" .
   403 qed
   404 
   405 
   406 subsubsection {* Closed sets *}
   407 
   408 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
   409 
   410 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
   411 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
   412 lemma closedin_topspace[intro,simp]:
   413   "closedin U (topspace U)" by (simp add: closedin_def)
   414 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
   415   by (auto simp add: Diff_Un closedin_def)
   416 
   417 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
   418 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
   419   shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
   420 
   421 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
   422   using closedin_Inter[of "{S,T}" U] by auto
   423 
   424 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
   425 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
   426   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
   427   apply (metis openin_subset subset_eq)
   428   done
   429 
   430 lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
   431   by (simp add: openin_closedin_eq)
   432 
   433 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
   434 proof-
   435   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
   436     by (auto simp add: topspace_def openin_subset)
   437   then show ?thesis using oS cT by (auto simp add: closedin_def)
   438 qed
   439 
   440 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
   441 proof-
   442   have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
   443     by (auto simp add: topspace_def )
   444   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
   445 qed
   446 
   447 subsubsection {* Subspace topology *}
   448 
   449 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
   450 
   451 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
   452   (is "istopology ?L")
   453 proof-
   454   have "?L {}" by blast
   455   {fix A B assume A: "?L A" and B: "?L B"
   456     from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
   457     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
   458     then have "?L (A \<inter> B)" by blast}
   459   moreover
   460   {fix K assume K: "K \<subseteq> Collect ?L"
   461     have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
   462       apply (rule set_eqI)
   463       apply (simp add: Ball_def image_iff)
   464       by metis
   465     from K[unfolded th0 subset_image_iff]
   466     obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
   467     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
   468     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)
   469     ultimately have "?L (\<Union>K)" by blast}
   470   ultimately show ?thesis
   471     unfolding subset_eq mem_Collect_eq istopology_def by blast
   472 qed
   473 
   474 lemma openin_subtopology:
   475   "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
   476   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
   477   by auto
   478 
   479 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
   480   by (auto simp add: topspace_def openin_subtopology)
   481 
   482 lemma closedin_subtopology:
   483   "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
   484   unfolding closedin_def topspace_subtopology
   485   apply (simp add: openin_subtopology)
   486   apply (rule iffI)
   487   apply clarify
   488   apply (rule_tac x="topspace U - T" in exI)
   489   by auto
   490 
   491 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
   492   unfolding openin_subtopology
   493   apply (rule iffI, clarify)
   494   apply (frule openin_subset[of U])  apply blast
   495   apply (rule exI[where x="topspace U"])
   496   apply auto
   497   done
   498 
   499 lemma subtopology_superset:
   500   assumes UV: "topspace U \<subseteq> V"
   501   shows "subtopology U V = U"
   502 proof-
   503   {fix S
   504     {fix T assume T: "openin U T" "S = T \<inter> V"
   505       from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
   506       have "openin U S" unfolding eq using T by blast}
   507     moreover
   508     {assume S: "openin U S"
   509       hence "\<exists>T. openin U T \<and> S = T \<inter> V"
   510         using openin_subset[OF S] UV by auto}
   511     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
   512   then show ?thesis unfolding topology_eq openin_subtopology by blast
   513 qed
   514 
   515 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
   516   by (simp add: subtopology_superset)
   517 
   518 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
   519   by (simp add: subtopology_superset)
   520 
   521 subsubsection {* The standard Euclidean topology *}
   522 
   523 definition
   524   euclidean :: "'a::topological_space topology" where
   525   "euclidean = topology open"
   526 
   527 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
   528   unfolding euclidean_def
   529   apply (rule cong[where x=S and y=S])
   530   apply (rule topology_inverse[symmetric])
   531   apply (auto simp add: istopology_def)
   532   done
   533 
   534 lemma topspace_euclidean: "topspace euclidean = UNIV"
   535   apply (simp add: topspace_def)
   536   apply (rule set_eqI)
   537   by (auto simp add: open_openin[symmetric])
   538 
   539 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
   540   by (simp add: topspace_euclidean topspace_subtopology)
   541 
   542 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
   543   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
   544 
   545 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
   546   by (simp add: open_openin openin_subopen[symmetric])
   547 
   548 text {* Basic "localization" results are handy for connectedness. *}
   549 
   550 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
   551   by (auto simp add: openin_subtopology open_openin[symmetric])
   552 
   553 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
   554   by (auto simp add: openin_open)
   555 
   556 lemma open_openin_trans[trans]:
   557  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
   558   by (metis Int_absorb1  openin_open_Int)
   559 
   560 lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
   561   by (auto simp add: openin_open)
   562 
   563 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
   564   by (simp add: closedin_subtopology closed_closedin Int_ac)
   565 
   566 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
   567   by (metis closedin_closed)
   568 
   569 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
   570   apply (subgoal_tac "S \<inter> T = T" )
   571   apply auto
   572   apply (frule closedin_closed_Int[of T S])
   573   by simp
   574 
   575 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
   576   by (auto simp add: closedin_closed)
   577 
   578 lemma openin_euclidean_subtopology_iff:
   579   fixes S U :: "'a::metric_space set"
   580   shows "openin (subtopology euclidean U) S
   581   \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
   582 proof
   583   assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
   584 next
   585   def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
   586   have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
   587     unfolding T_def
   588     apply clarsimp
   589     apply (rule_tac x="d - dist x a" in exI)
   590     apply (clarsimp simp add: less_diff_eq)
   591     apply (erule rev_bexI)
   592     apply (rule_tac x=d in exI, clarify)
   593     apply (erule le_less_trans [OF dist_triangle])
   594     done
   595   assume ?rhs hence 2: "S = U \<inter> T"
   596     unfolding T_def
   597     apply auto
   598     apply (drule (1) bspec, erule rev_bexI)
   599     apply auto
   600     done
   601   from 1 2 show ?lhs
   602     unfolding openin_open open_dist by fast
   603 qed
   604 
   605 text {* These "transitivity" results are handy too *}
   606 
   607 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
   608   \<Longrightarrow> openin (subtopology euclidean U) S"
   609   unfolding open_openin openin_open by blast
   610 
   611 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
   612   by (auto simp add: openin_open intro: openin_trans)
   613 
   614 lemma closedin_trans[trans]:
   615  "closedin (subtopology euclidean T) S \<Longrightarrow>
   616            closedin (subtopology euclidean U) T
   617            ==> closedin (subtopology euclidean U) S"
   618   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
   619 
   620 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
   621   by (auto simp add: closedin_closed intro: closedin_trans)
   622 
   623 
   624 subsection {* Open and closed balls *}
   625 
   626 definition
   627   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   628   "ball x e = {y. dist x y < e}"
   629 
   630 definition
   631   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   632   "cball x e = {y. dist x y \<le> e}"
   633 
   634 lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
   635   by (simp add: ball_def)
   636 
   637 lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
   638   by (simp add: cball_def)
   639 
   640 lemma mem_ball_0:
   641   fixes x :: "'a::real_normed_vector"
   642   shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
   643   by (simp add: dist_norm)
   644 
   645 lemma mem_cball_0:
   646   fixes x :: "'a::real_normed_vector"
   647   shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
   648   by (simp add: dist_norm)
   649 
   650 lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"
   651   by simp
   652 
   653 lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
   654   by simp
   655 
   656 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
   657 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
   658 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
   659 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
   660   by (simp add: set_eq_iff) arith
   661 
   662 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
   663   by (simp add: set_eq_iff)
   664 
   665 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
   666   "(a::real) - b < 0 \<longleftrightarrow> a < b"
   667   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
   668 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
   669   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
   670 
   671 lemma open_ball[intro, simp]: "open (ball x e)"
   672   unfolding open_dist ball_def mem_Collect_eq Ball_def
   673   unfolding dist_commute
   674   apply clarify
   675   apply (rule_tac x="e - dist xa x" in exI)
   676   using dist_triangle_alt[where z=x]
   677   apply (clarsimp simp add: diff_less_iff)
   678   apply atomize
   679   apply (erule_tac x="y" in allE)
   680   apply (erule_tac x="xa" in allE)
   681   by arith
   682 
   683 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
   684   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
   685 
   686 lemma openE[elim?]:
   687   assumes "open S" "x\<in>S" 
   688   obtains e where "e>0" "ball x e \<subseteq> S"
   689   using assms unfolding open_contains_ball by auto
   690 
   691 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
   692   by (metis open_contains_ball subset_eq centre_in_ball)
   693 
   694 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
   695   unfolding mem_ball set_eq_iff
   696   apply (simp add: not_less)
   697   by (metis zero_le_dist order_trans dist_self)
   698 
   699 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
   700 
   701 lemma euclidean_dist_l2:
   702   fixes x y :: "'a :: euclidean_space"
   703   shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
   704   unfolding dist_norm norm_eq_sqrt_inner setL2_def
   705   by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
   706 
   707 definition "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
   708 
   709 lemma rational_boxes:
   710   fixes x :: "'a\<Colon>euclidean_space"
   711   assumes "0 < e"
   712   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"
   713 proof -
   714   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
   715   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)
   716   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
   717   proof
   718     fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto
   719   qed
   720   from choice[OF this] guess a .. note a = this
   721   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
   722   proof
   723     fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto
   724   qed
   725   from choice[OF this] guess b .. note b = this
   726   let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
   727   show ?thesis
   728   proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
   729     fix y :: 'a assume *: "y \<in> box ?a ?b"
   730     have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"
   731       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
   732     also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
   733     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
   734       fix i :: "'a" assume i: "i \<in> Basis"
   735       have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)
   736       moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto
   737       moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto
   738       ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'" by auto
   739       then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
   740         unfolding e'_def by (auto simp: dist_real_def)
   741       then have "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
   742         by (rule power_strict_mono) auto
   743       then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
   744         by (simp add: power_divide)
   745     qed auto
   746     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat)
   747     finally show "y \<in> ball x e" by (auto simp: ball_def)
   748   qed (insert a b, auto simp: box_def)
   749 qed
   750 
   751 lemma open_UNION_box:
   752   fixes M :: "'a\<Colon>euclidean_space set"
   753   assumes "open M" 
   754   defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
   755   defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
   756   defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^isub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
   757   shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
   758 proof safe
   759   fix x assume "x \<in> M"
   760   obtain e where e: "e > 0" "ball x e \<subseteq> M"
   761     using openE[OF `open M` `x \<in> M`] by auto
   762   moreover then obtain a b where ab: "x \<in> box a b"
   763     "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"
   764     using rational_boxes[OF e(1)] by metis
   765   ultimately show "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
   766      by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
   767         (auto simp: euclidean_representation I_def a'_def b'_def)
   768 qed (auto simp: I_def)
   769 
   770 subsection{* Connectedness *}
   771 
   772 definition "connected S \<longleftrightarrow>
   773   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
   774   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
   775 
   776 lemma connected_local:
   777  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
   778                  openin (subtopology euclidean S) e1 \<and>
   779                  openin (subtopology euclidean S) e2 \<and>
   780                  S \<subseteq> e1 \<union> e2 \<and>
   781                  e1 \<inter> e2 = {} \<and>
   782                  ~(e1 = {}) \<and>
   783                  ~(e2 = {}))"
   784 unfolding connected_def openin_open by (safe, blast+)
   785 
   786 lemma exists_diff:
   787   fixes P :: "'a set \<Rightarrow> bool"
   788   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
   789 proof-
   790   {assume "?lhs" hence ?rhs by blast }
   791   moreover
   792   {fix S assume H: "P S"
   793     have "S = - (- S)" by auto
   794     with H have "P (- (- S))" by metis }
   795   ultimately show ?thesis by metis
   796 qed
   797 
   798 lemma connected_clopen: "connected S \<longleftrightarrow>
   799         (\<forall>T. openin (subtopology euclidean S) T \<and>
   800             closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
   801 proof-
   802   have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
   803     unfolding connected_def openin_open closedin_closed
   804     apply (subst exists_diff) by blast
   805   hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
   806     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
   807 
   808   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'))"
   809     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
   810     unfolding connected_def openin_open closedin_closed by auto
   811   {fix e2
   812     {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
   813         by auto}
   814     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
   815   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
   816   then show ?thesis unfolding th0 th1 by simp
   817 qed
   818 
   819 lemma connected_empty[simp, intro]: "connected {}"
   820   by (simp add: connected_def)
   821 
   822 
   823 subsection{* Limit points *}
   824 
   825 definition (in topological_space)
   826   islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
   827   "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
   828 
   829 lemma islimptI:
   830   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
   831   shows "x islimpt S"
   832   using assms unfolding islimpt_def by auto
   833 
   834 lemma islimptE:
   835   assumes "x islimpt S" and "x \<in> T" and "open T"
   836   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
   837   using assms unfolding islimpt_def by auto
   838 
   839 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
   840   unfolding islimpt_def eventually_at_topological by auto
   841 
   842 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"
   843   unfolding islimpt_def by fast
   844 
   845 lemma islimpt_approachable:
   846   fixes x :: "'a::metric_space"
   847   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
   848   unfolding islimpt_iff_eventually eventually_at by fast
   849 
   850 lemma islimpt_approachable_le:
   851   fixes x :: "'a::metric_space"
   852   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
   853   unfolding islimpt_approachable
   854   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
   855     THEN arg_cong [where f=Not]]
   856   by (simp add: Bex_def conj_commute conj_left_commute)
   857 
   858 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
   859   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
   860 
   861 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
   862   unfolding islimpt_def by blast
   863 
   864 text {* A perfect space has no isolated points. *}
   865 
   866 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
   867   unfolding islimpt_UNIV_iff by (rule not_open_singleton)
   868 
   869 lemma perfect_choose_dist:
   870   fixes x :: "'a::{perfect_space, metric_space}"
   871   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
   872 using islimpt_UNIV [of x]
   873 by (simp add: islimpt_approachable)
   874 
   875 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
   876   unfolding closed_def
   877   apply (subst open_subopen)
   878   apply (simp add: islimpt_def subset_eq)
   879   by (metis ComplE ComplI)
   880 
   881 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
   882   unfolding islimpt_def by auto
   883 
   884 lemma finite_set_avoid:
   885   fixes a :: "'a::metric_space"
   886   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
   887 proof(induct rule: finite_induct[OF fS])
   888   case 1 thus ?case by (auto intro: zero_less_one)
   889 next
   890   case (2 x F)
   891   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
   892   {assume "x = a" hence ?case using d by auto  }
   893   moreover
   894   {assume xa: "x\<noteq>a"
   895     let ?d = "min d (dist a x)"
   896     have dp: "?d > 0" using xa d(1) using dist_nz by auto
   897     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
   898     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
   899   ultimately show ?case by blast
   900 qed
   901 
   902 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
   903   by (simp add: islimpt_iff_eventually eventually_conj_iff)
   904 
   905 lemma discrete_imp_closed:
   906   fixes S :: "'a::metric_space set"
   907   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
   908   shows "closed S"
   909 proof-
   910   {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
   911     from e have e2: "e/2 > 0" by arith
   912     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
   913     let ?m = "min (e/2) (dist x y) "
   914     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
   915     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
   916     have th: "dist z y < e" using z y
   917       by (intro dist_triangle_lt [where z=x], simp)
   918     from d[rule_format, OF y(1) z(1) th] y z
   919     have False by (auto simp add: dist_commute)}
   920   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
   921 qed
   922 
   923 
   924 subsection {* Interior of a Set *}
   925 
   926 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
   927 
   928 lemma interiorI [intro?]:
   929   assumes "open T" and "x \<in> T" and "T \<subseteq> S"
   930   shows "x \<in> interior S"
   931   using assms unfolding interior_def by fast
   932 
   933 lemma interiorE [elim?]:
   934   assumes "x \<in> interior S"
   935   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
   936   using assms unfolding interior_def by fast
   937 
   938 lemma open_interior [simp, intro]: "open (interior S)"
   939   by (simp add: interior_def open_Union)
   940 
   941 lemma interior_subset: "interior S \<subseteq> S"
   942   by (auto simp add: interior_def)
   943 
   944 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
   945   by (auto simp add: interior_def)
   946 
   947 lemma interior_open: "open S \<Longrightarrow> interior S = S"
   948   by (intro equalityI interior_subset interior_maximal subset_refl)
   949 
   950 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
   951   by (metis open_interior interior_open)
   952 
   953 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
   954   by (metis interior_maximal interior_subset subset_trans)
   955 
   956 lemma interior_empty [simp]: "interior {} = {}"
   957   using open_empty by (rule interior_open)
   958 
   959 lemma interior_UNIV [simp]: "interior UNIV = UNIV"
   960   using open_UNIV by (rule interior_open)
   961 
   962 lemma interior_interior [simp]: "interior (interior S) = interior S"
   963   using open_interior by (rule interior_open)
   964 
   965 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
   966   by (auto simp add: interior_def)
   967 
   968 lemma interior_unique:
   969   assumes "T \<subseteq> S" and "open T"
   970   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
   971   shows "interior S = T"
   972   by (intro equalityI assms interior_subset open_interior interior_maximal)
   973 
   974 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
   975   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
   976     Int_lower2 interior_maximal interior_subset open_Int open_interior)
   977 
   978 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
   979   using open_contains_ball_eq [where S="interior S"]
   980   by (simp add: open_subset_interior)
   981 
   982 lemma interior_limit_point [intro]:
   983   fixes x :: "'a::perfect_space"
   984   assumes x: "x \<in> interior S" shows "x islimpt S"
   985   using x islimpt_UNIV [of x]
   986   unfolding interior_def islimpt_def
   987   apply (clarsimp, rename_tac T T')
   988   apply (drule_tac x="T \<inter> T'" in spec)
   989   apply (auto simp add: open_Int)
   990   done
   991 
   992 lemma interior_closed_Un_empty_interior:
   993   assumes cS: "closed S" and iT: "interior T = {}"
   994   shows "interior (S \<union> T) = interior S"
   995 proof
   996   show "interior S \<subseteq> interior (S \<union> T)"
   997     by (rule interior_mono, rule Un_upper1)
   998 next
   999   show "interior (S \<union> T) \<subseteq> interior S"
  1000   proof
  1001     fix x assume "x \<in> interior (S \<union> T)"
  1002     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
  1003     show "x \<in> interior S"
  1004     proof (rule ccontr)
  1005       assume "x \<notin> interior S"
  1006       with `x \<in> R` `open R` obtain y where "y \<in> R - S"
  1007         unfolding interior_def by fast
  1008       from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
  1009       from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
  1010       from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
  1011       show "False" unfolding interior_def by fast
  1012     qed
  1013   qed
  1014 qed
  1015 
  1016 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
  1017 proof (rule interior_unique)
  1018   show "interior A \<times> interior B \<subseteq> A \<times> B"
  1019     by (intro Sigma_mono interior_subset)
  1020   show "open (interior A \<times> interior B)"
  1021     by (intro open_Times open_interior)
  1022   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"
  1023   proof (safe)
  1024     fix x y assume "(x, y) \<in> T"
  1025     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
  1026       using `open T` unfolding open_prod_def by fast
  1027     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
  1028       using `T \<subseteq> A \<times> B` by auto
  1029     thus "x \<in> interior A" and "y \<in> interior B"
  1030       by (auto intro: interiorI)
  1031   qed
  1032 qed
  1033 
  1034 
  1035 subsection {* Closure of a Set *}
  1036 
  1037 definition "closure S = S \<union> {x | x. x islimpt S}"
  1038 
  1039 lemma interior_closure: "interior S = - (closure (- S))"
  1040   unfolding interior_def closure_def islimpt_def by auto
  1041 
  1042 lemma closure_interior: "closure S = - interior (- S)"
  1043   unfolding interior_closure by simp
  1044 
  1045 lemma closed_closure[simp, intro]: "closed (closure S)"
  1046   unfolding closure_interior by (simp add: closed_Compl)
  1047 
  1048 lemma closure_subset: "S \<subseteq> closure S"
  1049   unfolding closure_def by simp
  1050 
  1051 lemma closure_hull: "closure S = closed hull S"
  1052   unfolding hull_def closure_interior interior_def by auto
  1053 
  1054 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
  1055   unfolding closure_hull using closed_Inter by (rule hull_eq)
  1056 
  1057 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
  1058   unfolding closure_eq .
  1059 
  1060 lemma closure_closure [simp]: "closure (closure S) = closure S"
  1061   unfolding closure_hull by (rule hull_hull)
  1062 
  1063 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
  1064   unfolding closure_hull by (rule hull_mono)
  1065 
  1066 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
  1067   unfolding closure_hull by (rule hull_minimal)
  1068 
  1069 lemma closure_unique:
  1070   assumes "S \<subseteq> T" and "closed T"
  1071   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
  1072   shows "closure S = T"
  1073   using assms unfolding closure_hull by (rule hull_unique)
  1074 
  1075 lemma closure_empty [simp]: "closure {} = {}"
  1076   using closed_empty by (rule closure_closed)
  1077 
  1078 lemma closure_UNIV [simp]: "closure UNIV = UNIV"
  1079   using closed_UNIV by (rule closure_closed)
  1080 
  1081 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
  1082   unfolding closure_interior by simp
  1083 
  1084 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
  1085   using closure_empty closure_subset[of S]
  1086   by blast
  1087 
  1088 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
  1089   using closure_eq[of S] closure_subset[of S]
  1090   by simp
  1091 
  1092 lemma open_inter_closure_eq_empty:
  1093   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
  1094   using open_subset_interior[of S "- T"]
  1095   using interior_subset[of "- T"]
  1096   unfolding closure_interior
  1097   by auto
  1098 
  1099 lemma open_inter_closure_subset:
  1100   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
  1101 proof
  1102   fix x
  1103   assume as: "open S" "x \<in> S \<inter> closure T"
  1104   { assume *:"x islimpt T"
  1105     have "x islimpt (S \<inter> T)"
  1106     proof (rule islimptI)
  1107       fix A
  1108       assume "x \<in> A" "open A"
  1109       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
  1110         by (simp_all add: open_Int)
  1111       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
  1112         by (rule islimptE)
  1113       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
  1114         by simp_all
  1115       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
  1116     qed
  1117   }
  1118   then show "x \<in> closure (S \<inter> T)" using as
  1119     unfolding closure_def
  1120     by blast
  1121 qed
  1122 
  1123 lemma closure_complement: "closure (- S) = - interior S"
  1124   unfolding closure_interior by simp
  1125 
  1126 lemma interior_complement: "interior (- S) = - closure S"
  1127   unfolding closure_interior by simp
  1128 
  1129 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
  1130 proof (rule closure_unique)
  1131   show "A \<times> B \<subseteq> closure A \<times> closure B"
  1132     by (intro Sigma_mono closure_subset)
  1133   show "closed (closure A \<times> closure B)"
  1134     by (intro closed_Times closed_closure)
  1135   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
  1136     apply (simp add: closed_def open_prod_def, clarify)
  1137     apply (rule ccontr)
  1138     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
  1139     apply (simp add: closure_interior interior_def)
  1140     apply (drule_tac x=C in spec)
  1141     apply (drule_tac x=D in spec)
  1142     apply auto
  1143     done
  1144 qed
  1145 
  1146 
  1147 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"
  1148   unfolding closure_def using islimpt_punctured by blast
  1149 
  1150 
  1151 subsection {* Frontier (aka boundary) *}
  1152 
  1153 definition "frontier S = closure S - interior S"
  1154 
  1155 lemma frontier_closed: "closed(frontier S)"
  1156   by (simp add: frontier_def closed_Diff)
  1157 
  1158 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
  1159   by (auto simp add: frontier_def interior_closure)
  1160 
  1161 lemma frontier_straddle:
  1162   fixes a :: "'a::metric_space"
  1163   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))"
  1164   unfolding frontier_def closure_interior
  1165   by (auto simp add: mem_interior subset_eq ball_def)
  1166 
  1167 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
  1168   by (metis frontier_def closure_closed Diff_subset)
  1169 
  1170 lemma frontier_empty[simp]: "frontier {} = {}"
  1171   by (simp add: frontier_def)
  1172 
  1173 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
  1174 proof-
  1175   { assume "frontier S \<subseteq> S"
  1176     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
  1177     hence "closed S" using closure_subset_eq by auto
  1178   }
  1179   thus ?thesis using frontier_subset_closed[of S] ..
  1180 qed
  1181 
  1182 lemma frontier_complement: "frontier(- S) = frontier S"
  1183   by (auto simp add: frontier_def closure_complement interior_complement)
  1184 
  1185 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
  1186   using frontier_complement frontier_subset_eq[of "- S"]
  1187   unfolding open_closed by auto
  1188 
  1189 subsection {* Filters and the ``eventually true'' quantifier *}
  1190 
  1191 definition
  1192   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
  1193     (infixr "indirection" 70) where
  1194   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
  1195 
  1196 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
  1197 
  1198 lemma trivial_limit_within:
  1199   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
  1200 proof
  1201   assume "trivial_limit (at a within S)"
  1202   thus "\<not> a islimpt S"
  1203     unfolding trivial_limit_def
  1204     unfolding eventually_within eventually_at_topological
  1205     unfolding islimpt_def
  1206     apply (clarsimp simp add: set_eq_iff)
  1207     apply (rename_tac T, rule_tac x=T in exI)
  1208     apply (clarsimp, drule_tac x=y in bspec, simp_all)
  1209     done
  1210 next
  1211   assume "\<not> a islimpt S"
  1212   thus "trivial_limit (at a within S)"
  1213     unfolding trivial_limit_def
  1214     unfolding eventually_within eventually_at_topological
  1215     unfolding islimpt_def
  1216     apply clarsimp
  1217     apply (rule_tac x=T in exI)
  1218     apply auto
  1219     done
  1220 qed
  1221 
  1222 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
  1223   using trivial_limit_within [of a UNIV] by simp
  1224 
  1225 lemma trivial_limit_at:
  1226   fixes a :: "'a::perfect_space"
  1227   shows "\<not> trivial_limit (at a)"
  1228   by (rule at_neq_bot)
  1229 
  1230 lemma trivial_limit_at_infinity:
  1231   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
  1232   unfolding trivial_limit_def eventually_at_infinity
  1233   apply clarsimp
  1234   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
  1235    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
  1236   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
  1237   apply (drule_tac x=UNIV in spec, simp)
  1238   done
  1239 
  1240 lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"
  1241   using islimpt_in_closure by (metis trivial_limit_within)
  1242 
  1243 text {* Some property holds "sufficiently close" to the limit point. *}
  1244 
  1245 lemma eventually_at: (* FIXME: this replaces Metric_Spaces.eventually_at *)
  1246   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
  1247 unfolding eventually_at dist_nz by auto
  1248 
  1249 lemma eventually_within: (* FIXME: this replaces Topological_Spaces.eventually_within *)
  1250   "eventually P (at a within S) \<longleftrightarrow>
  1251         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
  1252   by (rule eventually_within_less)
  1253 
  1254 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
  1255   unfolding trivial_limit_def
  1256   by (auto elim: eventually_rev_mp)
  1257 
  1258 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
  1259   by simp
  1260 
  1261 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
  1262   by (simp add: filter_eq_iff)
  1263 
  1264 text{* Combining theorems for "eventually" *}
  1265 
  1266 lemma eventually_rev_mono:
  1267   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
  1268 using eventually_mono [of P Q] by fast
  1269 
  1270 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
  1271   by (simp add: eventually_False)
  1272 
  1273 
  1274 subsection {* Limits *}
  1275 
  1276 text{* Notation Lim to avoid collition with lim defined in analysis *}
  1277 
  1278 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
  1279   where "Lim A f = (THE l. (f ---> l) A)"
  1280 
  1281 text{* Uniqueness of the limit, when nontrivial. *}
  1282 
  1283 lemma tendsto_Lim:
  1284   fixes f :: "'a \<Rightarrow> 'b::t2_space"
  1285   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
  1286   unfolding Lim_def using tendsto_unique[of net f] by auto
  1287 
  1288 lemma Lim:
  1289  "(f ---> l) net \<longleftrightarrow>
  1290         trivial_limit net \<or>
  1291         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
  1292   unfolding tendsto_iff trivial_limit_eq by auto
  1293 
  1294 text{* Show that they yield usual definitions in the various cases. *}
  1295 
  1296 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
  1297            (\<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)"
  1298   by (auto simp add: tendsto_iff eventually_within_le)
  1299 
  1300 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
  1301         (\<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)"
  1302   by (auto simp add: tendsto_iff eventually_within)
  1303 
  1304 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
  1305         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
  1306   by (auto simp add: tendsto_iff eventually_at)
  1307 
  1308 lemma Lim_at_infinity:
  1309   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
  1310   by (auto simp add: tendsto_iff eventually_at_infinity)
  1311 
  1312 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
  1313   by (rule topological_tendstoI, auto elim: eventually_rev_mono)
  1314 
  1315 text{* The expected monotonicity property. *}
  1316 
  1317 lemma Lim_within_empty: "(f ---> l) (net within {})"
  1318   unfolding tendsto_def Limits.eventually_within by simp
  1319 
  1320 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
  1321   unfolding tendsto_def Topological_Spaces.eventually_within
  1322   by (auto elim!: eventually_elim1)
  1323 
  1324 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
  1325   shows "(f ---> l) (net within (S \<union> T))"
  1326   using assms unfolding tendsto_def Limits.eventually_within
  1327   apply clarify
  1328   apply (drule spec, drule (1) mp, drule (1) mp)
  1329   apply (drule spec, drule (1) mp, drule (1) mp)
  1330   apply (auto elim: eventually_elim2)
  1331   done
  1332 
  1333 lemma Lim_Un_univ:
  1334  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
  1335         ==> (f ---> l) net"
  1336   by (metis Lim_Un within_UNIV)
  1337 
  1338 text{* Interrelations between restricted and unrestricted limits. *}
  1339 
  1340 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
  1341   (* FIXME: rename *)
  1342   unfolding tendsto_def Limits.eventually_within
  1343   apply (clarify, drule spec, drule (1) mp, drule (1) mp)
  1344   by (auto elim!: eventually_elim1)
  1345 
  1346 lemma eventually_within_interior:
  1347   assumes "x \<in> interior S"
  1348   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
  1349 proof-
  1350   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
  1351   { assume "?lhs"
  1352     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
  1353       unfolding Limits.eventually_within eventually_at_topological
  1354       by auto
  1355     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
  1356       by auto
  1357     then have "?rhs"
  1358       unfolding eventually_at_topological by auto
  1359   } moreover
  1360   { assume "?rhs" hence "?lhs"
  1361       unfolding Limits.eventually_within
  1362       by (auto elim: eventually_elim1)
  1363   } ultimately
  1364   show "?thesis" ..
  1365 qed
  1366 
  1367 lemma at_within_interior:
  1368   "x \<in> interior S \<Longrightarrow> at x within S = at x"
  1369   by (simp add: filter_eq_iff eventually_within_interior)
  1370 
  1371 lemma at_within_open:
  1372   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
  1373   by (simp only: at_within_interior interior_open)
  1374 
  1375 lemma Lim_within_open:
  1376   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
  1377   assumes"a \<in> S" "open S"
  1378   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
  1379   using assms by (simp only: at_within_open)
  1380 
  1381 lemma Lim_within_LIMSEQ:
  1382   fixes a :: "'a::metric_space"
  1383   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
  1384   shows "(X ---> L) (at a within T)"
  1385   using assms unfolding tendsto_def [where l=L]
  1386   by (simp add: sequentially_imp_eventually_within)
  1387 
  1388 lemma Lim_right_bound:
  1389   fixes f :: "real \<Rightarrow> real"
  1390   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"
  1391   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
  1392   shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
  1393 proof cases
  1394   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
  1395 next
  1396   assume [simp]: "{x<..} \<inter> I \<noteq> {}"
  1397   show ?thesis
  1398   proof (rule Lim_within_LIMSEQ, safe)
  1399     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
  1400     
  1401     show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
  1402     proof (rule LIMSEQ_I, rule ccontr)
  1403       fix r :: real assume "0 < r"
  1404       with cInf_close[of "f ` ({x<..} \<inter> I)" r]
  1405       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
  1406       from `x < y` have "0 < y - x" by auto
  1407       from S(2)[THEN LIMSEQ_D, OF this]
  1408       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
  1409       
  1410       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
  1411       moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1412         using S bnd by (intro cInf_lower[where z=K]) auto
  1413       ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1414         by (auto simp: not_less field_simps)
  1415       with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
  1416       show False by auto
  1417     qed
  1418   qed
  1419 qed
  1420 
  1421 text{* Another limit point characterization. *}
  1422 
  1423 lemma islimpt_sequential:
  1424   fixes x :: "'a::first_countable_topology"
  1425   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"
  1426     (is "?lhs = ?rhs")
  1427 proof
  1428   assume ?lhs
  1429   from countable_basis_at_decseq[of x] guess A . note A = this
  1430   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
  1431   { fix n
  1432     from `?lhs` have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
  1433       unfolding islimpt_def using A(1,2)[of n] by auto
  1434     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
  1435       unfolding f_def by (rule someI_ex)
  1436     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }
  1437   then have "\<forall>n. f n \<in> S - {x}" by auto
  1438   moreover have "(\<lambda>n. f n) ----> x"
  1439   proof (rule topological_tendstoI)
  1440     fix S assume "open S" "x \<in> S"
  1441     from A(3)[OF this] `\<And>n. f n \<in> A n`
  1442     show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)
  1443   qed
  1444   ultimately show ?rhs by fast
  1445 next
  1446   assume ?rhs
  1447   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto
  1448   show ?lhs
  1449     unfolding islimpt_def
  1450   proof safe
  1451     fix T assume "open T" "x \<in> T"
  1452     from lim[THEN topological_tendstoD, OF this] f
  1453     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
  1454       unfolding eventually_sequentially by auto
  1455   qed
  1456 qed
  1457 
  1458 lemma Lim_inv: (* TODO: delete *)
  1459   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
  1460   assumes "(f ---> l) A" and "l \<noteq> 0"
  1461   shows "((inverse o f) ---> inverse l) A"
  1462   unfolding o_def using assms by (rule tendsto_inverse)
  1463 
  1464 lemma Lim_null:
  1465   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1466   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
  1467   by (simp add: Lim dist_norm)
  1468 
  1469 lemma Lim_null_comparison:
  1470   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1471   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
  1472   shows "(f ---> 0) net"
  1473 proof (rule metric_tendsto_imp_tendsto)
  1474   show "(g ---> 0) net" by fact
  1475   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
  1476     using assms(1) by (rule eventually_elim1, simp add: dist_norm)
  1477 qed
  1478 
  1479 lemma Lim_transform_bound:
  1480   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1481   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
  1482   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
  1483   shows "(f ---> 0) net"
  1484   using assms(1) tendsto_norm_zero [OF assms(2)]
  1485   by (rule Lim_null_comparison)
  1486 
  1487 text{* Deducing things about the limit from the elements. *}
  1488 
  1489 lemma Lim_in_closed_set:
  1490   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
  1491   shows "l \<in> S"
  1492 proof (rule ccontr)
  1493   assume "l \<notin> S"
  1494   with `closed S` have "open (- S)" "l \<in> - S"
  1495     by (simp_all add: open_Compl)
  1496   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
  1497     by (rule topological_tendstoD)
  1498   with assms(2) have "eventually (\<lambda>x. False) net"
  1499     by (rule eventually_elim2) simp
  1500   with assms(3) show "False"
  1501     by (simp add: eventually_False)
  1502 qed
  1503 
  1504 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
  1505 
  1506 lemma Lim_dist_ubound:
  1507   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
  1508   shows "dist a l <= e"
  1509 proof-
  1510   have "dist a l \<in> {..e}"
  1511   proof (rule Lim_in_closed_set)
  1512     show "closed {..e}" by simp
  1513     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
  1514     show "\<not> trivial_limit net" by fact
  1515     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
  1516   qed
  1517   thus ?thesis by simp
  1518 qed
  1519 
  1520 lemma Lim_norm_ubound:
  1521   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1522   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
  1523   shows "norm(l) <= e"
  1524 proof-
  1525   have "norm l \<in> {..e}"
  1526   proof (rule Lim_in_closed_set)
  1527     show "closed {..e}" by simp
  1528     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
  1529     show "\<not> trivial_limit net" by fact
  1530     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1531   qed
  1532   thus ?thesis by simp
  1533 qed
  1534 
  1535 lemma Lim_norm_lbound:
  1536   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1537   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
  1538   shows "e \<le> norm l"
  1539 proof-
  1540   have "norm l \<in> {e..}"
  1541   proof (rule Lim_in_closed_set)
  1542     show "closed {e..}" by simp
  1543     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
  1544     show "\<not> trivial_limit net" by fact
  1545     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1546   qed
  1547   thus ?thesis by simp
  1548 qed
  1549 
  1550 text{* Limit under bilinear function *}
  1551 
  1552 lemma Lim_bilinear:
  1553   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
  1554   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
  1555 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
  1556 by (rule bounded_bilinear.tendsto)
  1557 
  1558 text{* These are special for limits out of the same vector space. *}
  1559 
  1560 lemma Lim_within_id: "(id ---> a) (at a within s)"
  1561   unfolding id_def by (rule tendsto_ident_at_within)
  1562 
  1563 lemma Lim_at_id: "(id ---> a) (at a)"
  1564   unfolding id_def by (rule tendsto_ident_at)
  1565 
  1566 lemma Lim_at_zero:
  1567   fixes a :: "'a::real_normed_vector"
  1568   fixes l :: "'b::topological_space"
  1569   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
  1570   using LIM_offset_zero LIM_offset_zero_cancel ..
  1571 
  1572 text{* It's also sometimes useful to extract the limit point from the filter. *}
  1573 
  1574 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
  1575   "netlimit F \<equiv> Lim F (\<lambda>x. x)"
  1576 
  1577 lemma netlimit_within:
  1578   "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
  1579   by (rule tendsto_Lim) (auto intro: tendsto_intros)
  1580 
  1581 lemma netlimit_at:
  1582   fixes a :: "'a::{perfect_space,t2_space}"
  1583   shows "netlimit (at a) = a"
  1584   using netlimit_within [of a UNIV] by simp
  1585 
  1586 lemma lim_within_interior:
  1587   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
  1588   by (simp add: at_within_interior)
  1589 
  1590 lemma netlimit_within_interior:
  1591   fixes x :: "'a::{t2_space,perfect_space}"
  1592   assumes "x \<in> interior S"
  1593   shows "netlimit (at x within S) = x"
  1594 using assms by (simp add: at_within_interior netlimit_at)
  1595 
  1596 text{* Transformation of limit. *}
  1597 
  1598 lemma Lim_transform:
  1599   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
  1600   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
  1601   shows "(g ---> l) net"
  1602   using tendsto_diff [OF assms(2) assms(1)] by simp
  1603 
  1604 lemma Lim_transform_eventually:
  1605   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
  1606   apply (rule topological_tendstoI)
  1607   apply (drule (2) topological_tendstoD)
  1608   apply (erule (1) eventually_elim2, simp)
  1609   done
  1610 
  1611 lemma Lim_transform_within:
  1612   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1613   and "(f ---> l) (at x within S)"
  1614   shows "(g ---> l) (at x within S)"
  1615 proof (rule Lim_transform_eventually)
  1616   show "eventually (\<lambda>x. f x = g x) (at x within S)"
  1617     unfolding eventually_within
  1618     using assms(1,2) by auto
  1619   show "(f ---> l) (at x within S)" by fact
  1620 qed
  1621 
  1622 lemma Lim_transform_at:
  1623   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1624   and "(f ---> l) (at x)"
  1625   shows "(g ---> l) (at x)"
  1626 proof (rule Lim_transform_eventually)
  1627   show "eventually (\<lambda>x. f x = g x) (at x)"
  1628     unfolding eventually_at
  1629     using assms(1,2) by auto
  1630   show "(f ---> l) (at x)" by fact
  1631 qed
  1632 
  1633 text{* Common case assuming being away from some crucial point like 0. *}
  1634 
  1635 lemma Lim_transform_away_within:
  1636   fixes a b :: "'a::t1_space"
  1637   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1638   and "(f ---> l) (at a within S)"
  1639   shows "(g ---> l) (at a within S)"
  1640 proof (rule Lim_transform_eventually)
  1641   show "(f ---> l) (at a within S)" by fact
  1642   show "eventually (\<lambda>x. f x = g x) (at a within S)"
  1643     unfolding Limits.eventually_within eventually_at_topological
  1644     by (rule exI [where x="- {b}"], simp add: open_Compl assms)
  1645 qed
  1646 
  1647 lemma Lim_transform_away_at:
  1648   fixes a b :: "'a::t1_space"
  1649   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1650   and fl: "(f ---> l) (at a)"
  1651   shows "(g ---> l) (at a)"
  1652   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
  1653   by simp
  1654 
  1655 text{* Alternatively, within an open set. *}
  1656 
  1657 lemma Lim_transform_within_open:
  1658   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
  1659   and "(f ---> l) (at a)"
  1660   shows "(g ---> l) (at a)"
  1661 proof (rule Lim_transform_eventually)
  1662   show "eventually (\<lambda>x. f x = g x) (at a)"
  1663     unfolding eventually_at_topological
  1664     using assms(1,2,3) by auto
  1665   show "(f ---> l) (at a)" by fact
  1666 qed
  1667 
  1668 text{* A congruence rule allowing us to transform limits assuming not at point. *}
  1669 
  1670 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
  1671 
  1672 lemma Lim_cong_within(*[cong add]*):
  1673   assumes "a = b" "x = y" "S = T"
  1674   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
  1675   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
  1676   unfolding tendsto_def Limits.eventually_within eventually_at_topological
  1677   using assms by simp
  1678 
  1679 lemma Lim_cong_at(*[cong add]*):
  1680   assumes "a = b" "x = y"
  1681   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
  1682   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
  1683   unfolding tendsto_def eventually_at_topological
  1684   using assms by simp
  1685 
  1686 text{* Useful lemmas on closure and set of possible sequential limits.*}
  1687 
  1688 lemma closure_sequential:
  1689   fixes l :: "'a::first_countable_topology"
  1690   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
  1691 proof
  1692   assume "?lhs" moreover
  1693   { assume "l \<in> S"
  1694     hence "?rhs" using tendsto_const[of l sequentially] by auto
  1695   } moreover
  1696   { assume "l islimpt S"
  1697     hence "?rhs" unfolding islimpt_sequential by auto
  1698   } ultimately
  1699   show "?rhs" unfolding closure_def by auto
  1700 next
  1701   assume "?rhs"
  1702   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
  1703 qed
  1704 
  1705 lemma closed_sequential_limits:
  1706   fixes S :: "'a::first_countable_topology set"
  1707   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
  1708   unfolding closed_limpt
  1709   using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
  1710   by metis
  1711 
  1712 lemma closure_approachable:
  1713   fixes S :: "'a::metric_space set"
  1714   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
  1715   apply (auto simp add: closure_def islimpt_approachable)
  1716   by (metis dist_self)
  1717 
  1718 lemma closed_approachable:
  1719   fixes S :: "'a::metric_space set"
  1720   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
  1721   by (metis closure_closed closure_approachable)
  1722 
  1723 lemma closure_contains_Inf:
  1724   fixes S :: "real set"
  1725   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
  1726   shows "Inf S \<in> closure S"
  1727   unfolding closure_approachable
  1728 proof safe
  1729   have *: "\<forall>x\<in>S. Inf S \<le> x"
  1730     using cInf_lower_EX[of _ S] assms by metis
  1731 
  1732   fix e :: real assume "0 < e"
  1733   then obtain x where x: "x \<in> S" "x < Inf S + e"
  1734     using cInf_close `S \<noteq> {}` by auto
  1735   moreover then have "x > Inf S - e" using * by auto
  1736   ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
  1737   then show "\<exists>x\<in>S. dist x (Inf S) < e"
  1738     using x by (auto simp: dist_norm)
  1739 qed
  1740 
  1741 lemma closed_contains_Inf:
  1742   fixes S :: "real set"
  1743   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
  1744     and "closed S"
  1745   shows "Inf S \<in> S"
  1746   by (metis closure_contains_Inf closure_closed assms)
  1747 
  1748 
  1749 lemma not_trivial_limit_within_ball:
  1750   "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
  1751   (is "?lhs = ?rhs")
  1752 proof -
  1753   { assume "?lhs"
  1754     { fix e :: real
  1755       assume "e>0"
  1756       then obtain y where "y:(S-{x}) & dist y x < e"
  1757         using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
  1758         by auto
  1759       then have "y : (S Int ball x e - {x})"
  1760         unfolding ball_def by (simp add: dist_commute)
  1761       then have "S Int ball x e - {x} ~= {}" by blast
  1762     } then have "?rhs" by auto
  1763   }
  1764   moreover
  1765   { assume "?rhs"
  1766     { fix e :: real
  1767       assume "e>0"
  1768       then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
  1769       then have "y:(S-{x}) & dist y x < e"
  1770         unfolding ball_def by (simp add: dist_commute)
  1771       then have "EX y:(S-{x}). dist y x < e" by auto
  1772     }
  1773     then have "?lhs"
  1774       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
  1775   }
  1776   ultimately show ?thesis by auto
  1777 qed
  1778 
  1779 subsection {* Infimum Distance *}
  1780 
  1781 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"
  1782 
  1783 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"
  1784   by (simp add: infdist_def)
  1785 
  1786 lemma infdist_nonneg:
  1787   shows "0 \<le> infdist x A"
  1788   using assms by (auto simp add: infdist_def intro: cInf_greatest)
  1789 
  1790 lemma infdist_le:
  1791   assumes "a \<in> A"
  1792   assumes "d = dist x a"
  1793   shows "infdist x A \<le> d"
  1794   using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)
  1795 
  1796 lemma infdist_zero[simp]:
  1797   assumes "a \<in> A" shows "infdist a A = 0"
  1798 proof -
  1799   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto
  1800   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto
  1801 qed
  1802 
  1803 lemma infdist_triangle:
  1804   shows "infdist x A \<le> infdist y A + dist x y"
  1805 proof cases
  1806   assume "A = {}" thus ?thesis by (simp add: infdist_def)
  1807 next
  1808   assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto
  1809   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
  1810   proof (rule cInf_greatest)
  1811     from `A \<noteq> {}` show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp
  1812     fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
  1813     then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto
  1814     show "infdist x A \<le> d"
  1815       unfolding infdist_notempty[OF `A \<noteq> {}`]
  1816     proof (rule cInf_lower2)
  1817       show "dist x a \<in> {dist x a |a. a \<in> A}" using `a \<in> A` by auto
  1818       show "dist x a \<le> d" unfolding d by (rule dist_triangle)
  1819       fix d assume "d \<in> {dist x a |a. a \<in> A}"
  1820       then obtain a where "a \<in> A" "d = dist x a" by auto
  1821       thus "infdist x A \<le> d" by (rule infdist_le)
  1822     qed
  1823   qed
  1824   also have "\<dots> = dist x y + infdist y A"
  1825   proof (rule cInf_eq, safe)
  1826     fix a assume "a \<in> A"
  1827     thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)
  1828   next
  1829     fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
  1830     hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF `A \<noteq> {}`] using `a \<in> A`
  1831       by (intro cInf_greatest) (auto simp: field_simps)
  1832     thus "i \<le> dist x y + infdist y A" by simp
  1833   qed
  1834   finally show ?thesis by simp
  1835 qed
  1836 
  1837 lemma in_closure_iff_infdist_zero:
  1838   assumes "A \<noteq> {}"
  1839   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
  1840 proof
  1841   assume "x \<in> closure A"
  1842   show "infdist x A = 0"
  1843   proof (rule ccontr)
  1844     assume "infdist x A \<noteq> 0"
  1845     with infdist_nonneg[of x A] have "infdist x A > 0" by auto
  1846     hence "ball x (infdist x A) \<inter> closure A = {}" apply auto
  1847       by (metis `0 < infdist x A` `x \<in> closure A` closure_approachable dist_commute
  1848         eucl_less_not_refl euclidean_trans(2) infdist_le)
  1849     hence "x \<notin> closure A" by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal)
  1850     thus False using `x \<in> closure A` by simp
  1851   qed
  1852 next
  1853   assume x: "infdist x A = 0"
  1854   then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)
  1855   show "x \<in> closure A" unfolding closure_approachable
  1856   proof (safe, rule ccontr)
  1857     fix e::real assume "0 < e"
  1858     assume "\<not> (\<exists>y\<in>A. dist y x < e)"
  1859     hence "infdist x A \<ge> e" using `a \<in> A`
  1860       unfolding infdist_def
  1861       by (force simp: dist_commute intro: cInf_greatest)
  1862     with x `0 < e` show False by auto
  1863   qed
  1864 qed
  1865 
  1866 lemma in_closed_iff_infdist_zero:
  1867   assumes "closed A" "A \<noteq> {}"
  1868   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"
  1869 proof -
  1870   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
  1871     by (rule in_closure_iff_infdist_zero) fact
  1872   with assms show ?thesis by simp
  1873 qed
  1874 
  1875 lemma tendsto_infdist [tendsto_intros]:
  1876   assumes f: "(f ---> l) F"
  1877   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"
  1878 proof (rule tendstoI)
  1879   fix e ::real assume "0 < e"
  1880   from tendstoD[OF f this]
  1881   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
  1882   proof (eventually_elim)
  1883     fix x
  1884     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
  1885     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"
  1886       by (simp add: dist_commute dist_real_def)
  1887     also assume "dist (f x) l < e"
  1888     finally show "dist (infdist (f x) A) (infdist l A) < e" .
  1889   qed
  1890 qed
  1891 
  1892 text{* Some other lemmas about sequences. *}
  1893 
  1894 lemma sequentially_offset:
  1895   assumes "eventually (\<lambda>i. P i) sequentially"
  1896   shows "eventually (\<lambda>i. P (i + k)) sequentially"
  1897   using assms unfolding eventually_sequentially by (metis trans_le_add1)
  1898 
  1899 lemma seq_offset:
  1900   assumes "(f ---> l) sequentially"
  1901   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
  1902   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)
  1903 
  1904 lemma seq_offset_neg:
  1905   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
  1906   apply (rule topological_tendstoI)
  1907   apply (drule (2) topological_tendstoD)
  1908   apply (simp only: eventually_sequentially)
  1909   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
  1910   apply metis
  1911   by arith
  1912 
  1913 lemma seq_offset_rev:
  1914   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
  1915   by (rule LIMSEQ_offset) (* FIXME: redundant *)
  1916 
  1917 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
  1918   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)
  1919 
  1920 subsection {* More properties of closed balls *}
  1921 
  1922 lemma closed_cball: "closed (cball x e)"
  1923 unfolding cball_def closed_def
  1924 unfolding Collect_neg_eq [symmetric] not_le
  1925 apply (clarsimp simp add: open_dist, rename_tac y)
  1926 apply (rule_tac x="dist x y - e" in exI, clarsimp)
  1927 apply (rename_tac x')
  1928 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
  1929 apply simp
  1930 done
  1931 
  1932 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
  1933 proof-
  1934   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
  1935     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
  1936   } moreover
  1937   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
  1938     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
  1939   } ultimately
  1940   show ?thesis unfolding open_contains_ball by auto
  1941 qed
  1942 
  1943 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
  1944   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
  1945 
  1946 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
  1947   apply (simp add: interior_def, safe)
  1948   apply (force simp add: open_contains_cball)
  1949   apply (rule_tac x="ball x e" in exI)
  1950   apply (simp add: subset_trans [OF ball_subset_cball])
  1951   done
  1952 
  1953 lemma islimpt_ball:
  1954   fixes x y :: "'a::{real_normed_vector,perfect_space}"
  1955   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
  1956 proof
  1957   assume "?lhs"
  1958   { assume "e \<le> 0"
  1959     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
  1960     have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
  1961   }
  1962   hence "e > 0" by (metis not_less)
  1963   moreover
  1964   have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto
  1965   ultimately show "?rhs" by auto
  1966 next
  1967   assume "?rhs" hence "e>0"  by auto
  1968   { fix d::real assume "d>0"
  1969     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1970     proof(cases "d \<le> dist x y")
  1971       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1972       proof(cases "x=y")
  1973         case True hence False using `d \<le> dist x y` `d>0` by auto
  1974         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
  1975       next
  1976         case False
  1977 
  1978         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
  1979               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1980           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
  1981         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
  1982           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
  1983           unfolding scaleR_minus_left scaleR_one
  1984           by (auto simp add: norm_minus_commute)
  1985         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
  1986           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
  1987           unfolding distrib_right using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
  1988         also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
  1989         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
  1990 
  1991         moreover
  1992 
  1993         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
  1994           using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
  1995         moreover
  1996         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
  1997           using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
  1998           unfolding dist_norm by auto
  1999         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
  2000       qed
  2001     next
  2002       case False hence "d > dist x y" by auto
  2003       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  2004       proof(cases "x=y")
  2005         case True
  2006         obtain z where **: "z \<noteq> y" "dist z y < min e d"
  2007           using perfect_choose_dist[of "min e d" y]
  2008           using `d > 0` `e>0` by auto
  2009         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  2010           unfolding `x = y`
  2011           using `z \<noteq> y` **
  2012           by (rule_tac x=z in bexI, auto simp add: dist_commute)
  2013       next
  2014         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  2015           using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
  2016       qed
  2017     qed  }
  2018   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
  2019 qed
  2020 
  2021 lemma closure_ball_lemma:
  2022   fixes x y :: "'a::real_normed_vector"
  2023   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
  2024 proof (rule islimptI)
  2025   fix T assume "y \<in> T" "open T"
  2026   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
  2027     unfolding open_dist by fast
  2028   (* choose point between x and y, within distance r of y. *)
  2029   def k \<equiv> "min 1 (r / (2 * dist x y))"
  2030   def z \<equiv> "y + scaleR k (x - y)"
  2031   have z_def2: "z = x + scaleR (1 - k) (y - x)"
  2032     unfolding z_def by (simp add: algebra_simps)
  2033   have "dist z y < r"
  2034     unfolding z_def k_def using `0 < r`
  2035     by (simp add: dist_norm min_def)
  2036   hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
  2037   have "dist x z < dist x y"
  2038     unfolding z_def2 dist_norm
  2039     apply (simp add: norm_minus_commute)
  2040     apply (simp only: dist_norm [symmetric])
  2041     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
  2042     apply (rule mult_strict_right_mono)
  2043     apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
  2044     apply (simp add: zero_less_dist_iff `x \<noteq> y`)
  2045     done
  2046   hence "z \<in> ball x (dist x y)" by simp
  2047   have "z \<noteq> y"
  2048     unfolding z_def k_def using `x \<noteq> y` `0 < r`
  2049     by (simp add: min_def)
  2050   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
  2051     using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
  2052     by fast
  2053 qed
  2054 
  2055 lemma closure_ball:
  2056   fixes x :: "'a::real_normed_vector"
  2057   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
  2058 apply (rule equalityI)
  2059 apply (rule closure_minimal)
  2060 apply (rule ball_subset_cball)
  2061 apply (rule closed_cball)
  2062 apply (rule subsetI, rename_tac y)
  2063 apply (simp add: le_less [where 'a=real])
  2064 apply (erule disjE)
  2065 apply (rule subsetD [OF closure_subset], simp)
  2066 apply (simp add: closure_def)
  2067 apply clarify
  2068 apply (rule closure_ball_lemma)
  2069 apply (simp add: zero_less_dist_iff)
  2070 done
  2071 
  2072 (* In a trivial vector space, this fails for e = 0. *)
  2073 lemma interior_cball:
  2074   fixes x :: "'a::{real_normed_vector, perfect_space}"
  2075   shows "interior (cball x e) = ball x e"
  2076 proof(cases "e\<ge>0")
  2077   case False note cs = this
  2078   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
  2079   { fix y assume "y \<in> cball x e"
  2080     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
  2081   hence "cball x e = {}" by auto
  2082   hence "interior (cball x e) = {}" using interior_empty by auto
  2083   ultimately show ?thesis by blast
  2084 next
  2085   case True note cs = this
  2086   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
  2087   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
  2088     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
  2089 
  2090     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
  2091       using perfect_choose_dist [of d] by auto
  2092     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
  2093     hence xa_cball:"xa \<in> cball x e" using as(1) by auto
  2094 
  2095     hence "y \<in> ball x e" proof(cases "x = y")
  2096       case True
  2097       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
  2098       thus "y \<in> ball x e" using `x = y ` by simp
  2099     next
  2100       case False
  2101       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
  2102         using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
  2103       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
  2104       have "y - x \<noteq> 0" using `x \<noteq> y` by auto
  2105       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
  2106         using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
  2107 
  2108       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
  2109         by (auto simp add: dist_norm algebra_simps)
  2110       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  2111         by (auto simp add: algebra_simps)
  2112       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
  2113         using ** by auto
  2114       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)
  2115       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
  2116       thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
  2117     qed  }
  2118   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
  2119   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
  2120 qed
  2121 
  2122 lemma frontier_ball:
  2123   fixes a :: "'a::real_normed_vector"
  2124   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
  2125   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
  2126   apply (simp add: set_eq_iff)
  2127   by arith
  2128 
  2129 lemma frontier_cball:
  2130   fixes a :: "'a::{real_normed_vector, perfect_space}"
  2131   shows "frontier(cball a e) = {x. dist a x = e}"
  2132   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
  2133   apply (simp add: set_eq_iff)
  2134   by arith
  2135 
  2136 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
  2137   apply (simp add: set_eq_iff not_le)
  2138   by (metis zero_le_dist dist_self order_less_le_trans)
  2139 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
  2140 
  2141 lemma cball_eq_sing:
  2142   fixes x :: "'a::{metric_space,perfect_space}"
  2143   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
  2144 proof (rule linorder_cases)
  2145   assume e: "0 < e"
  2146   obtain a where "a \<noteq> x" "dist a x < e"
  2147     using perfect_choose_dist [OF e] by auto
  2148   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
  2149   with e show ?thesis by (auto simp add: set_eq_iff)
  2150 qed auto
  2151 
  2152 lemma cball_sing:
  2153   fixes x :: "'a::metric_space"
  2154   shows "e = 0 ==> cball x e = {x}"
  2155   by (auto simp add: set_eq_iff)
  2156 
  2157 
  2158 subsection {* Boundedness *}
  2159 
  2160   (* FIXME: This has to be unified with BSEQ!! *)
  2161 definition (in metric_space)
  2162   bounded :: "'a set \<Rightarrow> bool" where
  2163   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
  2164 
  2165 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"
  2166   unfolding bounded_def subset_eq by auto
  2167 
  2168 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
  2169 unfolding bounded_def
  2170 apply safe
  2171 apply (rule_tac x="dist a x + e" in exI, clarify)
  2172 apply (drule (1) bspec)
  2173 apply (erule order_trans [OF dist_triangle add_left_mono])
  2174 apply auto
  2175 done
  2176 
  2177 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
  2178 unfolding bounded_any_center [where a=0]
  2179 by (simp add: dist_norm)
  2180 
  2181 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
  2182   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
  2183   using assms by auto
  2184 
  2185 lemma bounded_empty [simp]: "bounded {}"
  2186   by (simp add: bounded_def)
  2187 
  2188 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
  2189   by (metis bounded_def subset_eq)
  2190 
  2191 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
  2192   by (metis bounded_subset interior_subset)
  2193 
  2194 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
  2195 proof-
  2196   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
  2197   { fix y assume "y \<in> closure S"
  2198     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
  2199       unfolding closure_sequential by auto
  2200     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
  2201     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
  2202       by (rule eventually_mono, simp add: f(1))
  2203     have "dist x y \<le> a"
  2204       apply (rule Lim_dist_ubound [of sequentially f])
  2205       apply (rule trivial_limit_sequentially)
  2206       apply (rule f(2))
  2207       apply fact
  2208       done
  2209   }
  2210   thus ?thesis unfolding bounded_def by auto
  2211 qed
  2212 
  2213 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
  2214   apply (simp add: bounded_def)
  2215   apply (rule_tac x=x in exI)
  2216   apply (rule_tac x=e in exI)
  2217   apply auto
  2218   done
  2219 
  2220 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
  2221   by (metis ball_subset_cball bounded_cball bounded_subset)
  2222 
  2223 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
  2224   apply (auto simp add: bounded_def)
  2225   apply (rename_tac x y r s)
  2226   apply (rule_tac x=x in exI)
  2227   apply (rule_tac x="max r (dist x y + s)" in exI)
  2228   apply (rule ballI, rename_tac z, safe)
  2229   apply (drule (1) bspec, simp)
  2230   apply (drule (1) bspec)
  2231   apply (rule min_max.le_supI2)
  2232   apply (erule order_trans [OF dist_triangle add_left_mono])
  2233   done
  2234 
  2235 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
  2236   by (induct rule: finite_induct[of F], auto)
  2237 
  2238 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
  2239   by (induct set: finite, auto)
  2240 
  2241 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
  2242 proof -
  2243   have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp
  2244   hence "bounded {x}" unfolding bounded_def by fast
  2245   thus ?thesis by (metis insert_is_Un bounded_Un)
  2246 qed
  2247 
  2248 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
  2249   by (induct set: finite, simp_all)
  2250 
  2251 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
  2252   apply (simp add: bounded_iff)
  2253   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
  2254   by metis arith
  2255 
  2256 lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"
  2257   unfolding Bseq_def bounded_pos by auto
  2258 
  2259 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
  2260   by (metis Int_lower1 Int_lower2 bounded_subset)
  2261 
  2262 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
  2263 apply (metis Diff_subset bounded_subset)
  2264 done
  2265 
  2266 lemma not_bounded_UNIV[simp, intro]:
  2267   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
  2268 proof(auto simp add: bounded_pos not_le)
  2269   obtain x :: 'a where "x \<noteq> 0"
  2270     using perfect_choose_dist [OF zero_less_one] by fast
  2271   fix b::real  assume b: "b >0"
  2272   have b1: "b +1 \<ge> 0" using b by simp
  2273   with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
  2274     by (simp add: norm_sgn)
  2275   then show "\<exists>x::'a. b < norm x" ..
  2276 qed
  2277 
  2278 lemma bounded_linear_image:
  2279   assumes "bounded S" "bounded_linear f"
  2280   shows "bounded(f ` S)"
  2281 proof-
  2282   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  2283   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)
  2284   { fix x assume "x\<in>S"
  2285     hence "norm x \<le> b" using b by auto
  2286     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
  2287       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
  2288   }
  2289   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
  2290     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
  2291 qed
  2292 
  2293 lemma bounded_scaling:
  2294   fixes S :: "'a::real_normed_vector set"
  2295   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
  2296   apply (rule bounded_linear_image, assumption)
  2297   apply (rule bounded_linear_scaleR_right)
  2298   done
  2299 
  2300 lemma bounded_translation:
  2301   fixes S :: "'a::real_normed_vector set"
  2302   assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
  2303 proof-
  2304   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  2305   { fix x assume "x\<in>S"
  2306     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
  2307   }
  2308   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
  2309     by (auto intro!: exI[of _ "b + norm a"])
  2310 qed
  2311 
  2312 
  2313 text{* Some theorems on sups and infs using the notion "bounded". *}
  2314 
  2315 lemma bounded_real:
  2316   fixes S :: "real set"
  2317   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
  2318   by (simp add: bounded_iff)
  2319 
  2320 lemma bounded_has_Sup:
  2321   fixes S :: "real set"
  2322   assumes "bounded S" "S \<noteq> {}"
  2323   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
  2324 proof
  2325   fix x assume "x\<in>S"
  2326   thus "x \<le> Sup S"
  2327     by (metis cSup_upper abs_le_D1 assms(1) bounded_real)
  2328 next
  2329   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
  2330     by (metis cSup_least)
  2331 qed
  2332 
  2333 lemma Sup_insert:
  2334   fixes S :: "real set"
  2335   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" 
  2336   apply (subst cSup_insert_If)
  2337   apply (rule bounded_has_Sup(1)[of S, rule_format])
  2338   apply (auto simp: sup_max)
  2339   done
  2340 
  2341 lemma Sup_insert_finite:
  2342   fixes S :: "real set"
  2343   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
  2344   apply (rule Sup_insert)
  2345   apply (rule finite_imp_bounded)
  2346   by simp
  2347 
  2348 lemma bounded_has_Inf:
  2349   fixes S :: "real set"
  2350   assumes "bounded S"  "S \<noteq> {}"
  2351   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
  2352 proof
  2353   fix x assume "x\<in>S"
  2354   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
  2355   thus "x \<ge> Inf S" using `x\<in>S`
  2356     by (metis cInf_lower_EX abs_le_D2 minus_le_iff)
  2357 next
  2358   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
  2359     by (metis cInf_greatest)
  2360 qed
  2361 
  2362 lemma Inf_insert:
  2363   fixes S :: "real set"
  2364   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" 
  2365   apply (subst cInf_insert_if)
  2366   apply (rule bounded_has_Inf(1)[of S, rule_format])
  2367   apply (auto simp: inf_min)
  2368   done
  2369 
  2370 lemma Inf_insert_finite:
  2371   fixes S :: "real set"
  2372   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
  2373   by (rule Inf_insert, rule finite_imp_bounded, simp)
  2374 
  2375 subsection {* Compactness *}
  2376 
  2377 subsubsection{* Open-cover compactness *}
  2378 
  2379 definition compact :: "'a::topological_space set \<Rightarrow> bool" where
  2380   compact_eq_heine_borel: -- "This name is used for backwards compatibility"
  2381     "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
  2382 
  2383 lemma compactI:
  2384   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union> C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union> C'"
  2385   shows "compact s"
  2386   unfolding compact_eq_heine_borel using assms by metis
  2387 
  2388 lemma compactE:
  2389   assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
  2390   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
  2391   using assms unfolding compact_eq_heine_borel by metis
  2392 
  2393 lemma compactE_image:
  2394   assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
  2395   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
  2396   using assms unfolding ball_simps[symmetric] SUP_def
  2397   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
  2398 
  2399 subsubsection {* Bolzano-Weierstrass property *}
  2400 
  2401 lemma heine_borel_imp_bolzano_weierstrass:
  2402   assumes "compact s" "infinite t"  "t \<subseteq> s"
  2403   shows "\<exists>x \<in> s. x islimpt t"
  2404 proof(rule ccontr)
  2405   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
  2406   then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)" unfolding islimpt_def
  2407     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto
  2408   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
  2409     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
  2410   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
  2411   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
  2412     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
  2413     hence "x = y" using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto  }
  2414   hence "inj_on f t" unfolding inj_on_def by simp
  2415   hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
  2416   moreover
  2417   { fix x assume "x\<in>t" "f x \<notin> g"
  2418     from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
  2419     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
  2420     hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
  2421     hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto  }
  2422   hence "f ` t \<subseteq> g" by auto
  2423   ultimately show False using g(2) using finite_subset by auto
  2424 qed
  2425 
  2426 lemma acc_point_range_imp_convergent_subsequence:
  2427   fixes l :: "'a :: first_countable_topology"
  2428   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
  2429   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  2430 proof -
  2431   from countable_basis_at_decseq[of l] guess A . note A = this
  2432 
  2433   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"
  2434   { fix n i
  2435     have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
  2436       using l A by auto
  2437     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
  2438       unfolding ex_in_conv by (intro notI) simp
  2439     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
  2440       by auto
  2441     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
  2442       by (auto simp: not_le)
  2443     then have "i < s n i" "f (s n i) \<in> A (Suc n)"
  2444       unfolding s_def by (auto intro: someI2_ex) }
  2445   note s = this
  2446   def r \<equiv> "nat_rec (s 0 0) s"
  2447   have "subseq r"
  2448     by (auto simp: r_def s subseq_Suc_iff)
  2449   moreover
  2450   have "(\<lambda>n. f (r n)) ----> l"
  2451   proof (rule topological_tendstoI)
  2452     fix S assume "open S" "l \<in> S"
  2453     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
  2454     moreover
  2455     { fix i assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"
  2456         by (cases i) (simp_all add: r_def s) }
  2457     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)
  2458     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
  2459       by eventually_elim auto
  2460   qed
  2461   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  2462     by (auto simp: convergent_def comp_def)
  2463 qed
  2464 
  2465 lemma sequence_infinite_lemma:
  2466   fixes f :: "nat \<Rightarrow> 'a::t1_space"
  2467   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
  2468   shows "infinite (range f)"
  2469 proof
  2470   assume "finite (range f)"
  2471   hence "closed (range f)" by (rule finite_imp_closed)
  2472   hence "open (- range f)" by (rule open_Compl)
  2473   from assms(1) have "l \<in> - range f" by auto
  2474   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
  2475     using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
  2476   thus False unfolding eventually_sequentially by auto
  2477 qed
  2478 
  2479 lemma closure_insert:
  2480   fixes x :: "'a::t1_space"
  2481   shows "closure (insert x s) = insert x (closure s)"
  2482 apply (rule closure_unique)
  2483 apply (rule insert_mono [OF closure_subset])
  2484 apply (rule closed_insert [OF closed_closure])
  2485 apply (simp add: closure_minimal)
  2486 done
  2487 
  2488 lemma islimpt_insert:
  2489   fixes x :: "'a::t1_space"
  2490   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
  2491 proof
  2492   assume *: "x islimpt (insert a s)"
  2493   show "x islimpt s"
  2494   proof (rule islimptI)
  2495     fix t assume t: "x \<in> t" "open t"
  2496     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
  2497     proof (cases "x = a")
  2498       case True
  2499       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
  2500         using * t by (rule islimptE)
  2501       with `x = a` show ?thesis by auto
  2502     next
  2503       case False
  2504       with t have t': "x \<in> t - {a}" "open (t - {a})"
  2505         by (simp_all add: open_Diff)
  2506       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
  2507         using * t' by (rule islimptE)
  2508       thus ?thesis by auto
  2509     qed
  2510   qed
  2511 next
  2512   assume "x islimpt s" thus "x islimpt (insert a s)"
  2513     by (rule islimpt_subset) auto
  2514 qed
  2515 
  2516 lemma islimpt_finite:
  2517   fixes x :: "'a::t1_space"
  2518   shows "finite s \<Longrightarrow> \<not> x islimpt s"
  2519 by (induct set: finite, simp_all add: islimpt_insert)
  2520 
  2521 lemma islimpt_union_finite:
  2522   fixes x :: "'a::t1_space"
  2523   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
  2524 by (simp add: islimpt_Un islimpt_finite)
  2525 
  2526 lemma islimpt_eq_acc_point:
  2527   fixes l :: "'a :: t1_space"
  2528   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
  2529 proof (safe intro!: islimptI)
  2530   fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
  2531   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
  2532     by (auto intro: finite_imp_closed)
  2533   then show False
  2534     by (rule islimptE) auto
  2535 next
  2536   fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
  2537   then have "infinite (T \<inter> S - {l})" by auto
  2538   then have "\<exists>x. x \<in> (T \<inter> S - {l})"
  2539     unfolding ex_in_conv by (intro notI) simp
  2540   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
  2541     by auto
  2542 qed
  2543 
  2544 lemma islimpt_range_imp_convergent_subsequence:
  2545   fixes l :: "'a :: {t1_space, first_countable_topology}"
  2546   assumes l: "l islimpt (range f)"
  2547   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  2548   using l unfolding islimpt_eq_acc_point
  2549   by (rule acc_point_range_imp_convergent_subsequence)
  2550 
  2551 lemma sequence_unique_limpt:
  2552   fixes f :: "nat \<Rightarrow> 'a::t2_space"
  2553   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
  2554   shows "l' = l"
  2555 proof (rule ccontr)
  2556   assume "l' \<noteq> l"
  2557   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
  2558     using hausdorff [OF `l' \<noteq> l`] by auto
  2559   have "eventually (\<lambda>n. f n \<in> t) sequentially"
  2560     using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
  2561   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
  2562     unfolding eventually_sequentially by auto
  2563 
  2564   have "UNIV = {..<N} \<union> {N..}" by auto
  2565   hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
  2566   hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
  2567   hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
  2568   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
  2569     using `l' \<in> s` `open s` by (rule islimptE)
  2570   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
  2571   with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
  2572   with `s \<inter> t = {}` show False by simp
  2573 qed
  2574 
  2575 lemma bolzano_weierstrass_imp_closed:
  2576   fixes s :: "'a::{first_countable_topology, t2_space} set"
  2577   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2578   shows "closed s"
  2579 proof-
  2580   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
  2581     hence "l \<in> s"
  2582     proof(cases "\<forall>n. x n \<noteq> l")
  2583       case False thus "l\<in>s" using as(1) by auto
  2584     next
  2585       case True note cas = this
  2586       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
  2587       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
  2588       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
  2589     qed  }
  2590   thus ?thesis unfolding closed_sequential_limits by fast
  2591 qed
  2592 
  2593 lemma compact_imp_closed:
  2594   fixes s :: "'a::t2_space set"
  2595   assumes "compact s" shows "closed s"
  2596 unfolding closed_def
  2597 proof (rule openI)
  2598   fix y assume "y \<in> - s"
  2599   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
  2600   note `compact s`
  2601   moreover have "\<forall>u\<in>?C. open u" by simp
  2602   moreover have "s \<subseteq> \<Union>?C"
  2603   proof
  2604     fix x assume "x \<in> s"
  2605     with `y \<in> - s` have "x \<noteq> y" by clarsimp
  2606     hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
  2607       by (rule hausdorff)
  2608     with `x \<in> s` show "x \<in> \<Union>?C"
  2609       unfolding eventually_nhds by auto
  2610   qed
  2611   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
  2612     by (rule compactE)
  2613   from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
  2614   with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
  2615     by (simp add: eventually_Ball_finite)
  2616   with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
  2617     by (auto elim!: eventually_mono [rotated])
  2618   thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
  2619     by (simp add: eventually_nhds subset_eq)
  2620 qed
  2621 
  2622 lemma compact_imp_bounded:
  2623   assumes "compact U" shows "bounded U"
  2624 proof -
  2625   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto
  2626   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
  2627     by (elim compactE_image)
  2628   from `finite D` have "bounded (\<Union>x\<in>D. ball x 1)"
  2629     by (simp add: bounded_UN)
  2630   thus "bounded U" using `U \<subseteq> (\<Union>x\<in>D. ball x 1)` 
  2631     by (rule bounded_subset)
  2632 qed
  2633 
  2634 text{* In particular, some common special cases. *}
  2635 
  2636 lemma compact_empty[simp]:
  2637  "compact {}"
  2638   unfolding compact_eq_heine_borel
  2639   by auto
  2640 
  2641 lemma compact_union [intro]:
  2642   assumes "compact s" "compact t" shows " compact (s \<union> t)"
  2643 proof (rule compactI)
  2644   fix f assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
  2645   from * `compact s` obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
  2646     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis
  2647   moreover from * `compact t` obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
  2648     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis
  2649   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
  2650     by (auto intro!: exI[of _ "s' \<union> t'"])
  2651 qed
  2652 
  2653 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
  2654   by (induct set: finite) auto
  2655 
  2656 lemma compact_UN [intro]:
  2657   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
  2658   unfolding SUP_def by (rule compact_Union) auto
  2659 
  2660 lemma compact_inter_closed [intro]:
  2661   assumes "compact s" and "closed t"
  2662   shows "compact (s \<inter> t)"
  2663 proof (rule compactI)
  2664   fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
  2665   from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto
  2666   moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
  2667   ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
  2668     using `compact s` unfolding compact_eq_heine_borel by auto
  2669   then guess D ..
  2670   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
  2671     by (intro exI[of _ "D - {-t}"]) auto
  2672 qed
  2673 
  2674 lemma closed_inter_compact [intro]:
  2675   assumes "closed s" and "compact t"
  2676   shows "compact (s \<inter> t)"
  2677   using compact_inter_closed [of t s] assms
  2678   by (simp add: Int_commute)
  2679 
  2680 lemma compact_inter [intro]:
  2681   fixes s t :: "'a :: t2_space set"
  2682   assumes "compact s" and "compact t"
  2683   shows "compact (s \<inter> t)"
  2684   using assms by (intro compact_inter_closed compact_imp_closed)
  2685 
  2686 lemma compact_sing [simp]: "compact {a}"
  2687   unfolding compact_eq_heine_borel by auto
  2688 
  2689 lemma compact_insert [simp]:
  2690   assumes "compact s" shows "compact (insert x s)"
  2691 proof -
  2692   have "compact ({x} \<union> s)"
  2693     using compact_sing assms by (rule compact_union)
  2694   thus ?thesis by simp
  2695 qed
  2696 
  2697 lemma finite_imp_compact:
  2698   shows "finite s \<Longrightarrow> compact s"
  2699   by (induct set: finite) simp_all
  2700 
  2701 lemma open_delete:
  2702   fixes s :: "'a::t1_space set"
  2703   shows "open s \<Longrightarrow> open (s - {x})"
  2704   by (simp add: open_Diff)
  2705 
  2706 text{* Finite intersection property *}
  2707 
  2708 lemma inj_setminus: "inj_on uminus (A::'a set set)"
  2709   by (auto simp: inj_on_def)
  2710 
  2711 lemma compact_fip:
  2712   "compact U \<longleftrightarrow>
  2713     (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"
  2714   (is "_ \<longleftrightarrow> ?R")
  2715 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
  2716   fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
  2717     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
  2718   from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>uminus`A"
  2719     by auto
  2720   with `compact U` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
  2721     unfolding compact_eq_heine_borel by (metis subset_image_iff)
  2722   with fi[THEN spec, of B] show False
  2723     by (auto dest: finite_imageD intro: inj_setminus)
  2724 next
  2725   fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
  2726   from cover have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
  2727     by auto
  2728   with `?R` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>uminus`B = {}"
  2729     by (metis subset_image_iff)
  2730   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
  2731     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
  2732 qed
  2733 
  2734 lemma compact_imp_fip:
  2735   "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>
  2736     s \<inter> (\<Inter> f) \<noteq> {}"
  2737   unfolding compact_fip by auto
  2738 
  2739 text{*Compactness expressed with filters*}
  2740 
  2741 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"
  2742 
  2743 lemma eventually_filter_from_subbase:
  2744   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"
  2745     (is "_ \<longleftrightarrow> ?R P")
  2746   unfolding filter_from_subbase_def
  2747 proof (rule eventually_Abs_filter is_filter.intro)+
  2748   show "?R (\<lambda>x. True)"
  2749     by (rule exI[of _ "{}"]) (simp add: le_fun_def)
  2750 next
  2751   fix P Q assume "?R P" then guess X ..
  2752   moreover assume "?R Q" then guess Y ..
  2753   ultimately show "?R (\<lambda>x. P x \<and> Q x)"
  2754     by (intro exI[of _ "X \<union> Y"]) auto
  2755 next
  2756   fix P Q
  2757   assume "?R P" then guess X ..
  2758   moreover assume "\<forall>x. P x \<longrightarrow> Q x"
  2759   ultimately show "?R Q"
  2760     by (intro exI[of _ X]) auto
  2761 qed
  2762 
  2763 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"
  2764   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])
  2765 
  2766 lemma filter_from_subbase_not_bot:
  2767   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"
  2768   unfolding trivial_limit_def eventually_filter_from_subbase by auto
  2769 
  2770 lemma closure_iff_nhds_not_empty:
  2771   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
  2772 proof safe
  2773   assume x: "x \<in> closure X"
  2774   fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
  2775   then have "x \<notin> closure (-S)" 
  2776     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
  2777   with x have "x \<in> closure X - closure (-S)"
  2778     by auto
  2779   also have "\<dots> \<subseteq> closure (X \<inter> S)"
  2780     using `open S` open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
  2781   finally have "X \<inter> S \<noteq> {}" by auto
  2782   then show False using `X \<inter> A = {}` `S \<subseteq> A` by auto
  2783 next
  2784   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
  2785   from this[THEN spec, of "- X", THEN spec, of "- closure X"]
  2786   show "x \<in> closure X"
  2787     by (simp add: closure_subset open_Compl)
  2788 qed
  2789 
  2790 lemma compact_filter:
  2791   "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))"
  2792 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
  2793   fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
  2794   from F have "U \<noteq> {}"
  2795     by (auto simp: eventually_False)
  2796 
  2797   def Z \<equiv> "closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
  2798   then have "\<forall>z\<in>Z. closed z"
  2799     by auto
  2800   moreover 
  2801   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
  2802     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])
  2803   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
  2804   proof (intro allI impI)
  2805     fix B assume "finite B" "B \<subseteq> Z"
  2806     with `finite B` ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"
  2807       by (auto intro!: eventually_Ball_finite)
  2808     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
  2809       by eventually_elim auto
  2810     with F show "U \<inter> \<Inter>B \<noteq> {}"
  2811       by (intro notI) (simp add: eventually_False)
  2812   qed
  2813   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
  2814     using `compact U` unfolding compact_fip by blast
  2815   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto
  2816 
  2817   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
  2818     unfolding eventually_inf eventually_nhds
  2819   proof safe
  2820     fix P Q R S
  2821     assume "eventually R F" "open S" "x \<in> S"
  2822     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
  2823     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
  2824     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
  2825     ultimately show False by (auto simp: set_eq_iff)
  2826   qed
  2827   with `x \<in> U` show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
  2828     by (metis eventually_bot)
  2829 next
  2830   fix A assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
  2831 
  2832   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"
  2833   then have inj_P': "\<And>A. inj_on P' A"
  2834     by (auto intro!: inj_onI simp: fun_eq_iff)
  2835   def F \<equiv> "filter_from_subbase (P' ` insert U A)"
  2836   have "F \<noteq> bot"
  2837     unfolding F_def
  2838   proof (safe intro!: filter_from_subbase_not_bot)
  2839     fix X assume "X \<subseteq> P' ` insert U A" "finite X" "Inf X = bot"
  2840     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P' ` B) = bot"
  2841       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)
  2842     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto
  2843     with B show False by (auto simp: P'_def fun_eq_iff)
  2844   qed
  2845   moreover have "eventually (\<lambda>x. x \<in> U) F"
  2846     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)
  2847   moreover 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)"
  2848   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
  2849     by auto
  2850 
  2851   { fix V assume "V \<in> A"
  2852     then have V: "eventually (\<lambda>x. x \<in> V) F"
  2853       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)
  2854     have "x \<in> closure V"
  2855       unfolding closure_iff_nhds_not_empty
  2856     proof (intro impI allI)
  2857       fix S A assume "open S" "x \<in> S" "S \<subseteq> A"
  2858       then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)
  2859       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
  2860         by (auto simp: eventually_inf)
  2861       with x show "V \<inter> A \<noteq> {}"
  2862         by (auto simp del: Int_iff simp add: trivial_limit_def)
  2863     qed
  2864     then have "x \<in> V"
  2865       using `V \<in> A` A(1) by simp }
  2866   with `x\<in>U` have "x \<in> U \<inter> \<Inter>A" by auto
  2867   with `U \<inter> \<Inter>A = {}` show False by auto
  2868 qed
  2869 
  2870 definition "countably_compact U \<longleftrightarrow>
  2871     (\<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))"
  2872 
  2873 lemma countably_compactE:
  2874   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
  2875   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
  2876   using assms unfolding countably_compact_def by metis
  2877 
  2878 lemma countably_compactI:
  2879   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')"
  2880   shows "countably_compact s"
  2881   using assms unfolding countably_compact_def by metis
  2882 
  2883 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
  2884   by (auto simp: compact_eq_heine_borel countably_compact_def)
  2885 
  2886 lemma countably_compact_imp_compact:
  2887   assumes "countably_compact U"
  2888   assumes ccover: "countable B" "\<forall>b\<in>B. open b"
  2889   assumes 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"
  2890   shows "compact U"
  2891   using `countably_compact U` unfolding compact_eq_heine_borel countably_compact_def
  2892 proof safe
  2893   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
  2894   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)"
  2895 
  2896   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
  2897   ultimately have "countable C" "\<forall>a\<in>C. open a"
  2898     unfolding C_def using ccover by auto
  2899   moreover
  2900   have "\<Union>A \<inter> U \<subseteq> \<Union>C"
  2901   proof safe
  2902     fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"
  2903     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast
  2904     with `a \<in> A` show "x \<in> \<Union>C" unfolding C_def
  2905       by auto
  2906   qed
  2907   then have "U \<subseteq> \<Union>C" using `U \<subseteq> \<Union>A` by auto
  2908   ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
  2909     using * by metis
  2910   moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
  2911     by (auto simp: C_def)
  2912   then guess f unfolding bchoice_iff Bex_def ..
  2913   ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
  2914     unfolding C_def by (intro exI[of _ "f`T"]) fastforce
  2915 qed
  2916 
  2917 lemma countably_compact_imp_compact_second_countable:
  2918   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
  2919 proof (rule countably_compact_imp_compact)
  2920   fix T and x :: 'a assume "open T" "x \<in> T"
  2921   from topological_basisE[OF is_basis this] guess b .
  2922   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto
  2923 qed (insert countable_basis topological_basis_open[OF is_basis], auto)
  2924 
  2925 lemma countably_compact_eq_compact:
  2926   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
  2927   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
  2928   
  2929 subsubsection{* Sequential compactness *}
  2930 
  2931 definition
  2932   seq_compact :: "'a::topological_space set \<Rightarrow> bool" where
  2933   "seq_compact S \<longleftrightarrow>
  2934    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
  2935        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
  2936 
  2937 lemma seq_compact_imp_countably_compact:
  2938   fixes U :: "'a :: first_countable_topology set"
  2939   assumes "seq_compact U"
  2940   shows "countably_compact U"
  2941 proof (safe intro!: countably_compactI)
  2942   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
  2943   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"
  2944     using `seq_compact U` by (fastforce simp: seq_compact_def subset_eq)
  2945   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
  2946   proof cases
  2947     assume "finite A" with A show ?thesis by auto
  2948   next
  2949     assume "infinite A"
  2950     then have "A \<noteq> {}" by auto
  2951     show ?thesis
  2952     proof (rule ccontr)
  2953       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
  2954       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto
  2955       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis
  2956       def X \<equiv> "\<lambda>n. X' (from_nat_into A ` {.. n})"
  2957       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
  2958         using `A \<noteq> {}` unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)
  2959       then have "range X \<subseteq> U" by auto
  2960       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto
  2961       from `x\<in>U` `U \<subseteq> \<Union>A` from_nat_into_surj[OF `countable A`]
  2962       obtain n where "x \<in> from_nat_into A n" by auto
  2963       with r(2) A(1) from_nat_into[OF `A \<noteq> {}`, of n]
  2964       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
  2965         unfolding tendsto_def by (auto simp: comp_def)
  2966       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
  2967         by (auto simp: eventually_sequentially)
  2968       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
  2969         by auto
  2970       moreover from `subseq r`[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
  2971         by (auto intro!: exI[of _ "max n N"])
  2972       ultimately show False
  2973         by auto
  2974     qed
  2975   qed
  2976 qed
  2977 
  2978 lemma compact_imp_seq_compact:
  2979   fixes U :: "'a :: first_countable_topology set"
  2980   assumes "compact U" shows "seq_compact U"
  2981   unfolding seq_compact_def
  2982 proof safe
  2983   fix X :: "nat \<Rightarrow> 'a" assume "\<forall>n. X n \<in> U"
  2984   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
  2985     by (auto simp: eventually_filtermap)
  2986   moreover have "filtermap X sequentially \<noteq> bot"
  2987     by (simp add: trivial_limit_def eventually_filtermap)
  2988   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
  2989     using `compact U` by (auto simp: compact_filter)
  2990 
  2991   from countable_basis_at_decseq[of x] guess A . note A = this
  2992   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"
  2993   { fix n i
  2994     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
  2995     proof (rule ccontr)
  2996       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
  2997       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto
  2998       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
  2999         by (auto simp: eventually_filtermap eventually_sequentially)
  3000       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
  3001         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
  3002       ultimately have "eventually (\<lambda>x. False) ?F"
  3003         by (auto simp add: eventually_inf)
  3004       with x show False
  3005         by (simp add: eventually_False)
  3006     qed
  3007     then have "i < s n i" "X (s n i) \<in> A (Suc n)"
  3008       unfolding s_def by (auto intro: someI2_ex) }
  3009   note s = this
  3010   def r \<equiv> "nat_rec (s 0 0) s"
  3011   have "subseq r"
  3012     by (auto simp: r_def s subseq_Suc_iff)
  3013   moreover
  3014   have "(\<lambda>n. X (r n)) ----> x"
  3015   proof (rule topological_tendstoI)
  3016     fix S assume "open S" "x \<in> S"
  3017     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
  3018     moreover
  3019     { fix i assume "Suc 0 \<le> i" then have "X (r i) \<in> A i"
  3020         by (cases i) (simp_all add: r_def s) }
  3021     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)
  3022     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
  3023       by eventually_elim auto
  3024   qed
  3025   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"
  3026     using `x \<in> U` by (auto simp: convergent_def comp_def)
  3027 qed
  3028 
  3029 lemma seq_compactI:
  3030   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
  3031   shows "seq_compact S"
  3032   unfolding seq_compact_def using assms by fast
  3033 
  3034 lemma seq_compactE:
  3035   assumes "seq_compact S" "\<forall>n. f n \<in> S"
  3036   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
  3037   using assms unfolding seq_compact_def by fast
  3038 
  3039 lemma countably_compact_imp_acc_point:
  3040   assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"
  3041   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
  3042 proof (rule ccontr)
  3043   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"  
  3044   note `countably_compact s`
  3045   moreover have "\<forall>t\<in>C. open t" 
  3046     by (auto simp: C_def)
  3047   moreover
  3048   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
  3049   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
  3050   have "s \<subseteq> \<Union>C"
  3051     using `t \<subseteq> s`
  3052     unfolding C_def Union_image_eq
  3053     apply (safe dest!: s)
  3054     apply (rule_tac a="U \<inter> t" in UN_I)
  3055     apply (auto intro!: interiorI simp add: finite_subset)
  3056     done
  3057   moreover
  3058   from `countable t` have "countable C"
  3059     unfolding C_def by (auto intro: countable_Collect_finite_subset)
  3060   ultimately guess D by (rule countably_compactE)
  3061   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and
  3062     s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
  3063     by (metis (lifting) Union_image_eq finite_subset_image C_def)
  3064   from s `t \<subseteq> s` have "t \<subseteq> \<Union>E"
  3065     using interior_subset by blast
  3066   moreover have "finite (\<Union>E)"
  3067     using E by auto
  3068   ultimately show False using `infinite t` by (auto simp: finite_subset)
  3069 qed
  3070 
  3071 lemma countable_acc_point_imp_seq_compact:
  3072   fixes s :: "'a::first_countable_topology set"
  3073   assumes "\<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))"
  3074   shows "seq_compact s"
  3075 proof -
  3076   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  3077     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3078     proof (cases "finite (range f)")
  3079       case True
  3080       obtain l where "infinite {n. f n = f l}"
  3081         using pigeonhole_infinite[OF _ True] by auto
  3082       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"
  3083         using infinite_enumerate by blast
  3084       hence "subseq r \<and> (f \<circ> r) ----> f l"
  3085         by (simp add: fr tendsto_const o_def)
  3086       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  3087         by auto
  3088     next
  3089       case False
  3090       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto
  3091       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
  3092       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3093         using acc_point_range_imp_convergent_subsequence[of l f] by auto
  3094       with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
  3095     qed
  3096   }
  3097   thus ?thesis unfolding seq_compact_def by auto
  3098 qed
  3099 
  3100 lemma seq_compact_eq_countably_compact:
  3101   fixes U :: "'a :: first_countable_topology set"
  3102   shows "seq_compact U \<longleftrightarrow> countably_compact U"
  3103   using
  3104     countable_acc_point_imp_seq_compact
  3105     countably_compact_imp_acc_point
  3106     seq_compact_imp_countably_compact
  3107   by metis
  3108 
  3109 lemma seq_compact_eq_acc_point:
  3110   fixes s :: "'a :: first_countable_topology set"
  3111   shows "seq_compact s \<longleftrightarrow> (\<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)))"
  3112   using
  3113     countable_acc_point_imp_seq_compact[of s]
  3114     countably_compact_imp_acc_point[of s]
  3115     seq_compact_imp_countably_compact[of s]
  3116   by metis
  3117 
  3118 lemma seq_compact_eq_compact:
  3119   fixes U :: "'a :: second_countable_topology set"
  3120   shows "seq_compact U \<longleftrightarrow> compact U"
  3121   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
  3122 
  3123 lemma bolzano_weierstrass_imp_seq_compact:
  3124   fixes s :: "'a::{t1_space, first_countable_topology} set"
  3125   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
  3126   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
  3127 
  3128 subsubsection{* Total boundedness *}
  3129 
  3130 lemma cauchy_def:
  3131   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
  3132 unfolding Cauchy_def by metis
  3133 
  3134 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
  3135   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
  3136 declare helper_1.simps[simp del]
  3137 
  3138 lemma seq_compact_imp_totally_bounded:
  3139   assumes "seq_compact s"
  3140   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
  3141 proof(rule, rule, rule ccontr)
  3142   fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k)"
  3143   def x \<equiv> "helper_1 s e"
  3144   { fix n
  3145     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
  3146     proof(induct_tac rule:nat_less_induct)
  3147       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
  3148       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
  3149       have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x ` {0 ..< n}" in allE) using as by auto
  3150       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
  3151       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
  3152         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
  3153       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
  3154     qed }
  3155   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
  3156   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto
  3157   from this(3) have "Cauchy (x \<circ> r)" using LIMSEQ_imp_Cauchy by auto
  3158   then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using `e>0` by auto
  3159   show False
  3160     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
  3161     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
  3162     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
  3163 qed
  3164 
  3165 subsubsection{* Heine-Borel theorem *}
  3166 
  3167 lemma seq_compact_imp_heine_borel:
  3168   fixes s :: "'a :: metric_space set"
  3169   assumes "seq_compact s" shows "compact s"
  3170 proof -
  3171   from seq_compact_imp_totally_bounded[OF `seq_compact s`]
  3172   guess f unfolding choice_iff' .. note f = this
  3173   def K \<equiv> "(\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
  3174   have "countably_compact s"
  3175     using `seq_compact s` by (rule seq_compact_imp_countably_compact)
  3176   then show "compact s"
  3177   proof (rule countably_compact_imp_compact)
  3178     show "countable K"
  3179       unfolding K_def using f
  3180       by (auto intro: countable_finite countable_subset countable_rat
  3181                intro!: countable_image countable_SIGMA countable_UN)
  3182     show "\<forall>b\<in>K. open b" by (auto simp: K_def)
  3183   next
  3184     fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"
  3185     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto
  3186     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto
  3187     from Rats_dense_in_real[OF `0 < e / 2`] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto
  3188     from f[rule_format, of r] `0 < r` `x \<in> s` obtain k where "k \<in> f r" "x \<in> ball k r"
  3189       unfolding Union_image_eq by auto
  3190     from `r \<in> \<rat>` `0 < r` `k \<in> f r` have "ball k r \<in> K" by (auto simp: K_def)
  3191     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
  3192     proof (rule bexI[rotated], safe)
  3193       fix y assume "y \<in> ball k r"
  3194       with `r < e / 2` `x \<in> ball k r` have "dist x y < e"
  3195         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)
  3196       with `ball x e \<subseteq> T` show "y \<in> T" by auto
  3197     qed (rule `x \<in> ball k r`)
  3198   qed
  3199 qed
  3200 
  3201 lemma compact_eq_seq_compact_metric:
  3202   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
  3203   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
  3204 
  3205 lemma compact_def:
  3206   "compact (S :: 'a::metric_space set) \<longleftrightarrow>
  3207    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"
  3208   unfolding compact_eq_seq_compact_metric seq_compact_def by auto
  3209 
  3210 subsubsection {* Complete the chain of compactness variants *}
  3211 
  3212 lemma compact_eq_bolzano_weierstrass:
  3213   fixes s :: "'a::metric_space set"
  3214   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
  3215 proof
  3216   assume ?lhs thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto
  3217 next
  3218   assume ?rhs thus ?lhs
  3219     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
  3220 qed
  3221 
  3222 lemma bolzano_weierstrass_imp_bounded:
  3223   "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
  3224   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .
  3225 
  3226 text {*
  3227   A metric space (or topological vector space) is said to have the
  3228   Heine-Borel property if every closed and bounded subset is compact.
  3229 *}
  3230 
  3231 class heine_borel = metric_space +
  3232   assumes bounded_imp_convergent_subsequence:
  3233     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3234 
  3235 lemma bounded_closed_imp_seq_compact:
  3236   fixes s::"'a::heine_borel set"
  3237   assumes "bounded s" and "closed s" shows "seq_compact s"
  3238 proof (unfold seq_compact_def, clarify)
  3239   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  3240   with `bounded s` have "bounded (range f)" by (auto intro: bounded_subset)
  3241   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  3242     using bounded_imp_convergent_subsequence [OF `bounded (range f)`] by auto
  3243   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
  3244   have "l \<in> s" using `closed s` fr l
  3245     unfolding closed_sequential_limits by blast
  3246   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3247     using `l \<in> s` r l by blast
  3248 qed
  3249 
  3250 lemma compact_eq_bounded_closed:
  3251   fixes s :: "'a::heine_borel set"
  3252   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")
  3253 proof
  3254   assume ?lhs thus ?rhs
  3255     using compact_imp_closed compact_imp_bounded by blast
  3256 next
  3257   assume ?rhs thus ?lhs
  3258     using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto
  3259 qed
  3260 
  3261 (* TODO: is this lemma necessary? *)
  3262 lemma bounded_increasing_convergent:
  3263   fixes s :: "nat \<Rightarrow> real"
  3264   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"
  3265   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]
  3266   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)
  3267 
  3268 instance real :: heine_borel
  3269 proof
  3270   fix f :: "nat \<Rightarrow> real"
  3271   assume f: "bounded (range f)"
  3272   obtain r where r: "subseq r" "monoseq (f \<circ> r)"
  3273     unfolding comp_def by (metis seq_monosub)
  3274   moreover
  3275   then have "Bseq (f \<circ> r)"
  3276     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)
  3277   ultimately show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"
  3278     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
  3279 qed
  3280 
  3281 lemma compact_lemma:
  3282   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
  3283   assumes "bounded (range f)"
  3284   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>
  3285         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
  3286 proof safe
  3287   fix d :: "'a set" assume d: "d \<subseteq> Basis" 
  3288   with finite_Basis have "finite d" by (blast intro: finite_subset)
  3289   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>
  3290       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
  3291   proof(induct d) case empty thus ?case unfolding subseq_def by auto
  3292   next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto
  3293     have s': "bounded ((\<lambda>x. x \<bullet> k) ` range f)" using `bounded (range f)`
  3294       by (auto intro!: bounded_linear_image bounded_linear_inner_left)
  3295     obtain l1::"'a" and r1 where r1:"subseq r1" and
  3296       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
  3297       using insert(3) using insert(4) by auto
  3298     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k) ` range f" by simp
  3299     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"
  3300       by (metis (lifting) bounded_subset f' image_subsetI s')
  3301     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"
  3302       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"] by (auto simp: o_def)
  3303     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
  3304       using r1 and r2 unfolding r_def o_def subseq_def by auto
  3305     moreover
  3306     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"
  3307     { fix e::real assume "e>0"
  3308       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast
  3309       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)
  3310       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
  3311         by (rule eventually_subseq)
  3312       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
  3313         using N1' N2 
  3314         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)
  3315     }
  3316     ultimately show ?case by auto
  3317   qed
  3318 qed
  3319 
  3320 instance euclidean_space \<subseteq> heine_borel
  3321 proof
  3322   fix f :: "nat \<Rightarrow> 'a"
  3323   assume f: "bounded (range f)"
  3324   then obtain l::'a and r where r: "subseq r"
  3325     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
  3326     using compact_lemma [OF f] by blast
  3327   { fix e::real assume "e>0"
  3328     hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive)
  3329     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"
  3330       by simp
  3331     moreover
  3332     { fix n assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"
  3333       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"
  3334         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
  3335       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
  3336         apply(rule setsum_strict_mono) using n by auto
  3337       finally have "dist (f (r n)) l < e" 
  3338         by auto
  3339     }
  3340     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
  3341       by (rule eventually_elim1)
  3342   }
  3343   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
  3344   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
  3345 qed
  3346 
  3347 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
  3348 unfolding bounded_def
  3349 apply clarify
  3350 apply (rule_tac x="a" in exI)
  3351 apply (rule_tac x="e" in exI)
  3352 apply clarsimp
  3353 apply (drule (1) bspec)
  3354 apply (simp add: dist_Pair_Pair)
  3355 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
  3356 done
  3357 
  3358 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
  3359 unfolding bounded_def
  3360 apply clarify
  3361 apply (rule_tac x="b" in exI)
  3362 apply (rule_tac x="e" in exI)
  3363 apply clarsimp
  3364 apply (drule (1) bspec)
  3365 apply (simp add: dist_Pair_Pair)
  3366 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
  3367 done
  3368 
  3369 instance prod :: (heine_borel, heine_borel) heine_borel
  3370 proof
  3371   fix f :: "nat \<Rightarrow> 'a \<times> 'b"
  3372   assume f: "bounded (range f)"
  3373   from f have s1: "bounded (range (fst \<circ> f))" unfolding image_comp by (rule bounded_fst)
  3374   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"
  3375     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
  3376   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
  3377     by (auto simp add: image_comp intro: bounded_snd bounded_subset)
  3378   obtain l2 r2 where r2: "subseq r2"
  3379     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
  3380     using bounded_imp_convergent_subsequence [OF s2]
  3381     unfolding o_def by fast
  3382   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
  3383     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
  3384   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
  3385     using tendsto_Pair [OF l1' l2] unfolding o_def by simp
  3386   have r: "subseq (r1 \<circ> r2)"
  3387     using r1 r2 unfolding subseq_def by simp
  3388   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3389     using l r by fast
  3390 qed
  3391 
  3392 subsubsection{* Completeness *}
  3393 
  3394 definition complete :: "'a::metric_space set \<Rightarrow> bool" where
  3395   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"
  3396 
  3397 lemma compact_imp_complete: assumes "compact s" shows "complete s"
  3398 proof-
  3399   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
  3400     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"
  3401       using assms unfolding compact_def by blast
  3402 
  3403     note lr' = seq_suble [OF lr(2)]
  3404 
  3405     { fix e::real assume "e>0"
  3406       from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using `e>0` apply (erule_tac x="e/2" in allE) by auto
  3407       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using `e>0` by auto
  3408       { fix n::nat assume n:"n \<ge> max N M"
  3409         have "dist ((f \<circ> r) n) l < e/2" using n M by auto
  3410         moreover have "r n \<ge> N" using lr'[of n] n by auto
  3411         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
  3412         ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }
  3413       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }
  3414     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding LIMSEQ_def by auto  }
  3415   thus ?thesis unfolding complete_def by auto
  3416 qed
  3417 
  3418 lemma nat_approx_posE:
  3419   fixes e::real
  3420   assumes "0 < e"
  3421   obtains n::nat where "1 / (Suc n) < e"
  3422 proof atomize_elim
  3423   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"
  3424     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: `0 < e`)
  3425   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"
  3426     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: `0 < e`)
  3427   also have "\<dots> = e" by simp
  3428   finally show  "\<exists>n. 1 / real (Suc n) < e" ..
  3429 qed
  3430 
  3431 lemma compact_eq_totally_bounded:
  3432   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k)))"
  3433     (is "_ \<longleftrightarrow> ?rhs")
  3434 proof
  3435   assume assms: "?rhs"
  3436   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)"
  3437     by (auto simp: choice_iff')
  3438 
  3439   show "compact s"
  3440   proof cases
  3441     assume "s = {}" thus "compact s" by (simp add: compact_def)
  3442   next
  3443     assume "s \<noteq> {}"
  3444     show ?thesis
  3445       unfolding compact_def
  3446     proof safe
  3447       fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  3448       
  3449       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"
  3450       then have [simp]: "\<And>n. 0 < e n" by auto
  3451       def B \<equiv> "\<lambda>n U. SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
  3452       { fix n U assume "infinite {n. f n \<in> U}"
  3453         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
  3454           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
  3455         then guess a ..
  3456         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
  3457           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
  3458         from someI_ex[OF this]
  3459         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
  3460           unfolding B_def by auto }
  3461       note B = this
  3462 
  3463       def F \<equiv> "nat_rec (B 0 UNIV) B"
  3464       { fix n have "infinite {i. f i \<in> F n}"
  3465           by (induct n) (auto simp: F_def B) }
  3466       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
  3467         using B by (simp add: F_def)
  3468       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
  3469         using decseq_SucI[of F] by (auto simp: decseq_def)
  3470 
  3471       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
  3472       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
  3473         fix k i
  3474         have "infinite ({n. f n \<in> F k} - {.. i})"
  3475           using `infinite {n. f n \<in> F k}` by auto
  3476         from infinite_imp_nonempty[OF this]
  3477         show "\<exists>x>i. f x \<in> F k"
  3478           by (simp add: set_eq_iff not_le conj_commute)
  3479       qed
  3480 
  3481       def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
  3482       have "subseq t"
  3483         unfolding subseq_Suc_iff by (simp add: t_def sel)
  3484       moreover have "\<forall>i. (f \<circ> t) i \<in> s"
  3485         using f by auto
  3486       moreover
  3487       { fix n have "(f \<circ> t) n \<in> F n"
  3488           by (cases n) (simp_all add: t_def sel) }
  3489       note t = this
  3490 
  3491       have "Cauchy (f \<circ> t)"
  3492       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
  3493         fix r :: real and N n m assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
  3494         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
  3495           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)
  3496         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
  3497           by (auto simp: subset_eq)
  3498         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] `2 * e N < r`
  3499         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
  3500           by (simp add: dist_commute)
  3501       qed
  3502 
  3503       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  3504         using assms unfolding complete_def by blast
  3505     qed
  3506   qed
  3507 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
  3508 
  3509 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
  3510 proof-
  3511   { assume ?rhs
  3512     { fix e::real
  3513       assume "e>0"
  3514       with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
  3515         by (erule_tac x="e/2" in allE) auto
  3516       { fix n m
  3517         assume nm:"N \<le> m \<and> N \<le> n"
  3518         hence "dist (s m) (s n) < e" using N
  3519           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
  3520           by blast
  3521       }
  3522       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
  3523         by blast
  3524     }
  3525     hence ?lhs
  3526       unfolding cauchy_def
  3527       by blast
  3528   }
  3529   thus ?thesis
  3530     unfolding cauchy_def
  3531     using dist_triangle_half_l
  3532     by blast
  3533 qed
  3534 
  3535 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
  3536 proof-
  3537   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto
  3538   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
  3539   moreover
  3540   have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
  3541   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
  3542     unfolding bounded_any_center [where a="s N"] by auto
  3543   ultimately show "?thesis"
  3544     unfolding bounded_any_center [where a="s N"]
  3545     apply(rule_tac x="max a 1" in exI) apply auto
  3546     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
  3547 qed
  3548 
  3549 instance heine_borel < complete_space
  3550 proof
  3551   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  3552   hence "bounded (range f)"
  3553     by (rule cauchy_imp_bounded)
  3554   hence "compact (closure (range f))"
  3555     unfolding compact_eq_bounded_closed by auto
  3556   hence "complete (closure (range f))"
  3557     by (rule compact_imp_complete)
  3558   moreover have "\<forall>n. f n \<in> closure (range f)"
  3559     using closure_subset [of "range f"] by auto
  3560   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
  3561     using `Cauchy f` unfolding complete_def by auto
  3562   then show "convergent f"
  3563     unfolding convergent_def by auto
  3564 qed
  3565 
  3566 instance euclidean_space \<subseteq> banach ..
  3567 
  3568 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
  3569 proof(simp add: complete_def, rule, rule)
  3570   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  3571   hence "convergent f" by (rule Cauchy_convergent)
  3572   thus "\<exists>l. f ----> l" unfolding convergent_def .  
  3573 qed
  3574 
  3575 lemma complete_imp_closed: assumes "complete s" shows "closed s"
  3576 proof -
  3577   { fix x assume "x islimpt s"
  3578     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
  3579       unfolding islimpt_sequential by auto
  3580     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
  3581       using `complete s`[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto
  3582     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
  3583   }
  3584   thus "closed s" unfolding closed_limpt by auto
  3585 qed
  3586 
  3587 lemma complete_eq_closed:
  3588   fixes s :: "'a::complete_space set"
  3589   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
  3590 proof
  3591   assume ?lhs thus ?rhs by (rule complete_imp_closed)
  3592 next
  3593   assume ?rhs
  3594   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
  3595     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
  3596     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `?rhs`[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }
  3597   thus ?lhs unfolding complete_def by auto
  3598 qed
  3599 
  3600 lemma convergent_eq_cauchy:
  3601   fixes s :: "nat \<Rightarrow> 'a::complete_space"
  3602   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"
  3603   unfolding Cauchy_convergent_iff convergent_def ..
  3604 
  3605 lemma convergent_imp_bounded:
  3606   fixes s :: "nat \<Rightarrow> 'a::metric_space"
  3607   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"
  3608   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
  3609 
  3610 lemma compact_cball[simp]:
  3611   fixes x :: "'a::heine_borel"
  3612   shows "compact(cball x e)"
  3613   using compact_eq_bounded_closed bounded_cball closed_cball
  3614   by blast
  3615 
  3616 lemma compact_frontier_bounded[intro]:
  3617   fixes s :: "'a::heine_borel set"
  3618   shows "bounded s ==> compact(frontier s)"
  3619   unfolding frontier_def
  3620   using compact_eq_bounded_closed
  3621   by blast
  3622 
  3623 lemma compact_frontier[intro]:
  3624   fixes s :: "'a::heine_borel set"
  3625   shows "compact s ==> compact (frontier s)"
  3626   using compact_eq_bounded_closed compact_frontier_bounded
  3627   by blast
  3628 
  3629 lemma frontier_subset_compact:
  3630   fixes s :: "'a::heine_borel set"
  3631   shows "compact s ==> frontier s \<subseteq> s"
  3632   using frontier_subset_closed compact_eq_bounded_closed
  3633   by blast
  3634 
  3635 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}
  3636 
  3637 lemma bounded_closed_nest:
  3638   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
  3639   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
  3640   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
  3641 proof-
  3642   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto
  3643   from assms(4,1) have *:"seq_compact (s 0)" using bounded_closed_imp_seq_compact[of "s 0"] by auto
  3644 
  3645   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
  3646     unfolding seq_compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast
  3647 
  3648   { fix n::nat
  3649     { fix e::real assume "e>0"
  3650       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto
  3651       hence "dist ((x \<circ> r) (max N n)) l < e" by auto
  3652       moreover
  3653       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto
  3654       hence "(x \<circ> r) (max N n) \<in> s n"
  3655         using x apply(erule_tac x=n in allE)
  3656         using x apply(erule_tac x="r (max N n)" in allE)
  3657         using assms(3) apply(erule_tac x=n in allE) apply(erule_tac x="r (max N n)" in allE) by auto
  3658       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
  3659     }
  3660     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
  3661   }
  3662   thus ?thesis by auto
  3663 qed
  3664 
  3665 text {* Decreasing case does not even need compactness, just completeness. *}
  3666 
  3667 lemma decreasing_closed_nest:
  3668   assumes "\<forall>n. closed(s n)"
  3669           "\<forall>n. (s n \<noteq> {})"
  3670           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3671           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
  3672   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"
  3673 proof-
  3674   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
  3675   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
  3676   then obtain t where t: "\<forall>n. t n \<in> s n" by auto
  3677   { fix e::real assume "e>0"
  3678     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
  3679     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
  3680       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+
  3681       hence "dist (t m) (t n) < e" using N by auto
  3682     }
  3683     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
  3684   }
  3685   hence  "Cauchy t" unfolding cauchy_def by auto
  3686   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
  3687   { fix n::nat
  3688     { fix e::real assume "e>0"
  3689       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto
  3690       have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto
  3691       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
  3692     }
  3693     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
  3694   }
  3695   then show ?thesis by auto
  3696 qed
  3697 
  3698 text {* Strengthen it to the intersection actually being a singleton. *}
  3699 
  3700 lemma decreasing_closed_nest_sing:
  3701   fixes s :: "nat \<Rightarrow> 'a::complete_space set"
  3702   assumes "\<forall>n. closed(s n)"
  3703           "\<forall>n. s n \<noteq> {}"
  3704           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3705           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
  3706   shows "\<exists>a. \<Inter>(range s) = {a}"
  3707 proof-
  3708   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
  3709   { fix b assume b:"b \<in> \<Inter>(range s)"
  3710     { fix e::real assume "e>0"
  3711       hence "dist a b < e" using assms(4 )using b using a by blast
  3712     }
  3713     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
  3714   }
  3715   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
  3716   thus ?thesis ..
  3717 qed
  3718 
  3719 text{* Cauchy-type criteria for uniform convergence. *}
  3720 
  3721 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space" shows
  3722  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
  3723   (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")
  3724 proof(rule)
  3725   assume ?lhs
  3726   then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e" by auto
  3727   { fix e::real assume "e>0"
  3728     then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto
  3729     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
  3730       hence "dist (s m x) (s n x) < e"
  3731         using N[THEN spec[where x=m], THEN spec[where x=x]]
  3732         using N[THEN spec[where x=n], THEN spec[where x=x]]
  3733         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }
  3734     hence "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e"  by auto  }
  3735   thus ?rhs by auto
  3736 next
  3737   assume ?rhs
  3738   hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto
  3739   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
  3740     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
  3741   { fix e::real assume "e>0"
  3742     then obtain N where N:"\<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e/2"
  3743       using `?rhs`[THEN spec[where x="e/2"]] by auto
  3744     { fix x assume "P x"
  3745       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
  3746         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using `e>0` by(auto elim!: allE[where x="e/2"])
  3747       fix n::nat assume "n\<ge>N"
  3748       hence "dist(s n x)(l x) < e"  using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
  3749         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute)  }
  3750     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
  3751   thus ?lhs by auto
  3752 qed
  3753 
  3754 lemma uniformly_cauchy_imp_uniformly_convergent:
  3755   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"
  3756   assumes "\<forall>e>0.\<exists>N. \<forall>m (n::nat) x. N \<le> m \<and> N \<le> n \<and> P x --> dist(s m x)(s n x) < e"
  3757           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
  3758   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
  3759 proof-
  3760   obtain l' where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"
  3761     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
  3762   moreover
  3763   { fix x assume "P x"
  3764     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
  3765       using l and assms(2) unfolding LIMSEQ_def by blast  }
  3766   ultimately show ?thesis by auto
  3767 qed
  3768 
  3769 
  3770 subsection {* Continuity *}
  3771 
  3772 text {* Define continuity over a net to take in restrictions of the set. *}
  3773 
  3774 definition
  3775   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3776   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
  3777 
  3778 lemma continuous_trivial_limit:
  3779  "trivial_limit net ==> continuous net f"
  3780   unfolding continuous_def tendsto_def trivial_limit_eq by auto
  3781 
  3782 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
  3783   unfolding continuous_def
  3784   unfolding tendsto_def
  3785   using netlimit_within[of x s]
  3786   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
  3787 
  3788 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
  3789   using continuous_within [of x UNIV f] by simp
  3790 
  3791 lemma continuous_isCont: "isCont f x = continuous (at x) f"
  3792   unfolding isCont_def LIM_def
  3793   unfolding continuous_at Lim_at unfolding dist_nz by auto
  3794 
  3795 lemma continuous_at_within:
  3796   assumes "continuous (at x) f"  shows "continuous (at x within s) f"
  3797   using assms unfolding continuous_at continuous_within
  3798   by (rule Lim_at_within)
  3799 
  3800 
  3801 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
  3802 
  3803 lemma continuous_within_eps_delta:
  3804   "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s.  dist x' x < d --> dist (f x') (f x) < e)"
  3805   unfolding continuous_within and Lim_within
  3806   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto
  3807 
  3808 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.
  3809                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
  3810   using continuous_within_eps_delta [of x UNIV f] by simp
  3811 
  3812 text{* Versions in terms of open balls. *}
  3813 
  3814 lemma continuous_within_ball:
  3815  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  3816                             f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3817 proof
  3818   assume ?lhs
  3819   { fix e::real assume "e>0"
  3820     then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
  3821       using `?lhs`[unfolded continuous_within Lim_within] by auto
  3822     { fix y assume "y\<in>f ` (ball x d \<inter> s)"
  3823       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
  3824         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
  3825     }
  3826     hence "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e" using `d>0` unfolding subset_eq ball_def by (auto simp add: dist_commute)  }
  3827   thus ?rhs by auto
  3828 next
  3829   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
  3830     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
  3831 qed
  3832 
  3833 lemma continuous_at_ball:
  3834   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3835 proof
  3836   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3837     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)
  3838     unfolding dist_nz[THEN sym] by auto
  3839 next
  3840   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3841     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)
  3842 qed
  3843 
  3844 text{* Define setwise continuity in terms of limits within the set. *}
  3845 
  3846 definition
  3847   continuous_on ::
  3848     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3849 where
  3850   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
  3851 
  3852 lemma continuous_on_topological:
  3853   "continuous_on s f \<longleftrightarrow>
  3854     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  3855       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  3856 unfolding continuous_on_def tendsto_def
  3857 unfolding Limits.eventually_within eventually_at_topological
  3858 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  3859 
  3860 lemma continuous_on_iff:
  3861   "continuous_on s f \<longleftrightarrow>
  3862     (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
  3863 unfolding continuous_on_def Lim_within
  3864 apply (intro ball_cong [OF refl] all_cong ex_cong)
  3865 apply (rename_tac y, case_tac "y = x", simp)
  3866 apply (simp add: dist_nz)
  3867 done
  3868 
  3869 definition
  3870   uniformly_continuous_on ::
  3871     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
  3872 where
  3873   "uniformly_continuous_on s f \<longleftrightarrow>
  3874     (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
  3875 
  3876 text{* Some simple consequential lemmas. *}
  3877 
  3878 lemma uniformly_continuous_imp_continuous:
  3879  " uniformly_continuous_on s f ==> continuous_on s f"
  3880   unfolding uniformly_continuous_on_def continuous_on_iff by blast
  3881 
  3882 lemma continuous_at_imp_continuous_within:
  3883  "continuous (at x) f ==> continuous (at x within s) f"
  3884   unfolding continuous_within continuous_at using Lim_at_within by auto
  3885 
  3886 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
  3887 unfolding tendsto_def by (simp add: trivial_limit_eq)
  3888 
  3889 lemma continuous_at_imp_continuous_on:
  3890   assumes "\<forall>x\<in>s. continuous (at x) f"
  3891   shows "continuous_on s f"
  3892 unfolding continuous_on_def
  3893 proof
  3894   fix x assume "x \<in> s"
  3895   with assms have *: "(f ---> f (netlimit (at x))) (at x)"
  3896     unfolding continuous_def by simp
  3897   have "(f ---> f x) (at x)"
  3898   proof (cases "trivial_limit (at x)")
  3899     case True thus ?thesis
  3900       by (rule Lim_trivial_limit)
  3901   next
  3902     case False
  3903     hence 1: "netlimit (at x) = x"
  3904       using netlimit_within [of x UNIV] by simp
  3905     with * show ?thesis by simp
  3906   qed
  3907   thus "(f ---> f x) (at x within s)"
  3908     by (rule Lim_at_within)
  3909 qed
  3910 
  3911 lemma continuous_on_eq_continuous_within:
  3912   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
  3913 unfolding continuous_on_def continuous_def
  3914 apply (rule ball_cong [OF refl])
  3915 apply (case_tac "trivial_limit (at x within s)")
  3916 apply (simp add: Lim_trivial_limit)
  3917 apply (simp add: netlimit_within)
  3918 done
  3919 
  3920 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
  3921 
  3922 lemma continuous_on_eq_continuous_at:
  3923   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
  3924   by (auto simp add: continuous_on continuous_at Lim_within_open)
  3925 
  3926 lemma continuous_within_subset:
  3927  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
  3928              ==> continuous (at x within t) f"
  3929   unfolding continuous_within by(metis Lim_within_subset)
  3930 
  3931 lemma continuous_on_subset:
  3932   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
  3933   unfolding continuous_on by (metis subset_eq Lim_within_subset)
  3934 
  3935 lemma continuous_on_interior:
  3936   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
  3937   by (erule interiorE, drule (1) continuous_on_subset,
  3938     simp add: continuous_on_eq_continuous_at)
  3939 
  3940 lemma continuous_on_eq:
  3941   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
  3942   unfolding continuous_on_def tendsto_def Limits.eventually_within
  3943   by simp
  3944 
  3945 text {* Characterization of various kinds of continuity in terms of sequences. *}
  3946 
  3947 lemma continuous_within_sequentially:
  3948   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3949   shows "continuous (at a within s) f \<longleftrightarrow>
  3950                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
  3951                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
  3952 proof
  3953   assume ?lhs
  3954   { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"
  3955     fix T::"'b set" assume "open T" and "f a \<in> T"
  3956     with `?lhs` obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> f x \<in> T"
  3957       unfolding continuous_within tendsto_def eventually_within by auto
  3958     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"
  3959       using x(2) `d>0` by simp
  3960     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"
  3961     proof eventually_elim
  3962       case (elim n) thus ?case
  3963         using d x(1) `f a \<in> T` unfolding dist_nz[THEN sym] by auto
  3964     qed
  3965   }
  3966   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp
  3967 next
  3968   assume ?rhs thus ?lhs
  3969     unfolding continuous_within tendsto_def [where l="f a"]
  3970     by (simp add: sequentially_imp_eventually_within)
  3971 qed
  3972 
  3973 lemma continuous_at_sequentially:
  3974   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3975   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
  3976                   --> ((f o x) ---> f a) sequentially)"
  3977   using continuous_within_sequentially[of a UNIV f] by simp
  3978 
  3979 lemma continuous_on_sequentially:
  3980   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3981   shows "continuous_on s f \<longleftrightarrow>
  3982     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
  3983                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
  3984 proof
  3985   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
  3986 next
  3987   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
  3988 qed
  3989 
  3990 lemma uniformly_continuous_on_sequentially:
  3991   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
  3992                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
  3993                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
  3994 proof
  3995   assume ?lhs
  3996   { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
  3997     { fix e::real assume "e>0"
  3998       then obtain d where "d>0" and d:"\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
  3999         using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
  4000       obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded LIMSEQ_def dist_norm] and `d>0` by auto
  4001       { fix n assume "n\<ge>N"
  4002         hence "dist (f (x n)) (f (y n)) < e"
  4003           using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y
  4004           unfolding dist_commute by simp  }
  4005       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }
  4006     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }
  4007   thus ?rhs by auto
  4008 next
  4009   assume ?rhs
  4010   { assume "\<not> ?lhs"
  4011     then obtain e where "e>0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto
  4012     then obtain fa where fa:"\<forall>x.  0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
  4013       using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def
  4014       by (auto simp add: dist_commute)
  4015     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
  4016     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
  4017     have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
  4018       unfolding x_def and y_def using fa by auto
  4019     { fix e::real assume "e>0"
  4020       then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e]   by auto
  4021       { fix n::nat assume "n\<ge>N"
  4022         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
  4023         also have "\<dots> < e" using N by auto
  4024         finally have "inverse (real n + 1) < e" by auto
  4025         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }
  4026       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }
  4027     hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding LIMSEQ_def dist_real_def by auto
  4028     hence False using fxy and `e>0` by auto  }
  4029   thus ?lhs unfolding uniformly_continuous_on_def by blast
  4030 qed
  4031 
  4032 text{* The usual transformation theorems. *}
  4033 
  4034 lemma continuous_transform_within:
  4035   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  4036   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
  4037           "continuous (at x within s) f"
  4038   shows "continuous (at x within s) g"
  4039 unfolding continuous_within
  4040 proof (rule Lim_transform_within)
  4041   show "0 < d" by fact
  4042   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  4043     using assms(3) by auto
  4044   have "f x = g x"
  4045     using assms(1,2,3) by auto
  4046   thus "(f ---> g x) (at x within s)"
  4047     using assms(4) unfolding continuous_within by simp
  4048 qed
  4049 
  4050 lemma continuous_transform_at:
  4051   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  4052   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
  4053           "continuous (at x) f"
  4054   shows "continuous (at x) g"
  4055   using continuous_transform_within [of d x UNIV f g] assms by simp
  4056 
  4057 subsubsection {* Structural rules for pointwise continuity *}
  4058 
  4059 ML {*
  4060 
  4061 structure Continuous_Intros = Named_Thms
  4062 (
  4063   val name = @{binding continuous_intros}
  4064   val description = "Structural introduction rules for pointwise continuity"
  4065 )
  4066 
  4067 *}
  4068 
  4069 setup Continuous_Intros.setup
  4070 
  4071 lemma continuous_within_id[continuous_intros]: "continuous (at a within s) (\<lambda>x. x)"
  4072   unfolding continuous_within by (rule tendsto_ident_at_within)
  4073 
  4074 lemma continuous_at_id[continuous_intros]: "continuous (at a) (\<lambda>x. x)"
  4075   unfolding continuous_at by (rule tendsto_ident_at)
  4076 
  4077 lemma continuous_const[continuous_intros]: "continuous F (\<lambda>x. c)"
  4078   unfolding continuous_def by (rule tendsto_const)
  4079 
  4080 lemma continuous_fst[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. fst (f x))"
  4081   unfolding continuous_def by (rule tendsto_fst)
  4082 
  4083 lemma continuous_snd[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))"
  4084   unfolding continuous_def by (rule tendsto_snd)
  4085 
  4086 lemma continuous_Pair[continuous_intros]: "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
  4087   unfolding continuous_def by (rule tendsto_Pair)
  4088 
  4089 lemma continuous_dist[continuous_intros]:
  4090   assumes "continuous F f" and "continuous F g"
  4091   shows "continuous F (\<lambda>x. dist (f x) (g x))"
  4092   using assms unfolding continuous_def by (rule tendsto_dist)
  4093 
  4094 lemma continuous_infdist[continuous_intros]:
  4095   assumes "continuous F f"
  4096   shows "continuous F (\<lambda>x. infdist (f x) A)"
  4097   using assms unfolding continuous_def by (rule tendsto_infdist)
  4098 
  4099 lemma continuous_norm[continuous_intros]:
  4100   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
  4101   unfolding continuous_def by (rule tendsto_norm)
  4102 
  4103 lemma continuous_infnorm[continuous_intros]:
  4104   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
  4105   unfolding continuous_def by (rule tendsto_infnorm)
  4106 
  4107 lemma continuous_add[continuous_intros]:
  4108   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4109   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
  4110   unfolding continuous_def by (rule tendsto_add)
  4111 
  4112 lemma continuous_minus[continuous_intros]:
  4113   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4114   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
  4115   unfolding continuous_def by (rule tendsto_minus)
  4116 
  4117 lemma continuous_diff[continuous_intros]:
  4118   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4119   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
  4120   unfolding continuous_def by (rule tendsto_diff)
  4121 
  4122 lemma continuous_scaleR[continuous_intros]:
  4123   fixes g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4124   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x *\<^sub>R g x)"
  4125   unfolding continuous_def by (rule tendsto_scaleR)
  4126 
  4127 lemma continuous_mult[continuous_intros]:
  4128   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
  4129   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"
  4130   unfolding continuous_def by (rule tendsto_mult)
  4131 
  4132 lemma continuous_inner[continuous_intros]:
  4133   assumes "continuous F f" and "continuous F g"
  4134   shows "continuous F (\<lambda>x. inner (f x) (g x))"
  4135   using assms unfolding continuous_def by (rule tendsto_inner)
  4136 
  4137 lemma continuous_inverse[continuous_intros]:
  4138   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  4139   assumes "continuous F f" and "f (netlimit F) \<noteq> 0"
  4140   shows "continuous F (\<lambda>x. inverse (f x))"
  4141   using assms unfolding continuous_def by (rule tendsto_inverse)
  4142 
  4143 lemma continuous_at_within_inverse[continuous_intros]:
  4144   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  4145   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
  4146   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
  4147   using assms unfolding continuous_within by (rule tendsto_inverse)
  4148 
  4149 lemma continuous_at_inverse[continuous_intros]:
  4150   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  4151   assumes "continuous (at a) f" and "f a \<noteq> 0"
  4152   shows "continuous (at a) (\<lambda>x. inverse (f x))"
  4153   using assms unfolding continuous_at by (rule tendsto_inverse)
  4154 
  4155 subsubsection {* Structural rules for setwise continuity *}
  4156 
  4157 ML {*
  4158 
  4159 structure Continuous_On_Intros = Named_Thms
  4160 (
  4161   val name = @{binding continuous_on_intros}
  4162   val description = "Structural introduction rules for setwise continuity"
  4163 )
  4164 
  4165 *}
  4166 
  4167 setup Continuous_On_Intros.setup
  4168 
  4169 lemma continuous_on_id[continuous_on_intros]: "continuous_on s (\<lambda>x. x)"
  4170   unfolding continuous_on_def by (fast intro: tendsto_ident_at_within)
  4171 
  4172 lemma continuous_on_const[continuous_on_intros]: "continuous_on s (\<lambda>x. c)"
  4173   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4174 
  4175 lemma continuous_on_norm[continuous_on_intros]:
  4176   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
  4177   unfolding continuous_on_def by (fast intro: tendsto_norm)
  4178 
  4179 lemma continuous_on_infnorm[continuous_on_intros]:
  4180   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
  4181   unfolding continuous_on by (fast intro: tendsto_infnorm)
  4182 
  4183 lemma continuous_on_minus[continuous_on_intros]:
  4184   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4185   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
  4186   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4187 
  4188 lemma continuous_on_add[continuous_on_intros]:
  4189   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4190   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  4191            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
  4192   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4193 
  4194 lemma continuous_on_diff[continuous_on_intros]:
  4195   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4196   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  4197            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
  4198   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4199 
  4200 lemma (in bounded_linear) continuous_on[continuous_on_intros]:
  4201   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
  4202   unfolding continuous_on_def by (fast intro: tendsto)
  4203 
  4204 lemma (in bounded_bilinear) continuous_on[continuous_on_intros]:
  4205   "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
  4206   unfolding continuous_on_def by (fast intro: tendsto)
  4207 
  4208 lemma continuous_on_scaleR[continuous_on_intros]:
  4209   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4210   assumes "continuous_on s f" and "continuous_on s g"
  4211   shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"
  4212   using bounded_bilinear_scaleR assms
  4213   by (rule bounded_bilinear.continuous_on)
  4214 
  4215 lemma continuous_on_mult[continuous_on_intros]:
  4216   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
  4217   assumes "continuous_on s f" and "continuous_on s g"
  4218   shows "continuous_on s (\<lambda>x. f x * g x)"
  4219   using bounded_bilinear_mult assms
  4220   by (rule bounded_bilinear.continuous_on)
  4221 
  4222 lemma continuous_on_inner[continuous_on_intros]:
  4223   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"
  4224   assumes "continuous_on s f" and "continuous_on s g"
  4225   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"
  4226   using bounded_bilinear_inner assms
  4227   by (rule bounded_bilinear.continuous_on)
  4228 
  4229 lemma continuous_on_inverse[continuous_on_intros]:
  4230   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
  4231   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
  4232   shows "continuous_on s (\<lambda>x. inverse (f x))"
  4233   using assms unfolding continuous_on by (fast intro: tendsto_inverse)
  4234 
  4235 subsubsection {* Structural rules for uniform continuity *}
  4236 
  4237 lemma uniformly_continuous_on_id[continuous_on_intros]:
  4238   shows "uniformly_continuous_on s (\<lambda>x. x)"
  4239   unfolding uniformly_continuous_on_def by auto
  4240 
  4241 lemma uniformly_continuous_on_const[continuous_on_intros]:
  4242   shows "uniformly_continuous_on s (\<lambda>x. c)"
  4243   unfolding uniformly_continuous_on_def by simp
  4244 
  4245 lemma uniformly_continuous_on_dist[continuous_on_intros]:
  4246   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  4247   assumes "uniformly_continuous_on s f"
  4248   assumes "uniformly_continuous_on s g"
  4249   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
  4250 proof -
  4251   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
  4252       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
  4253       using dist_triangle3 [of c d a] dist_triangle [of a d b]
  4254       by arith
  4255   } note le = this
  4256   { fix x y
  4257     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"
  4258     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"
  4259     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"
  4260       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
  4261         simp add: le)
  4262   }
  4263   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially
  4264     unfolding dist_real_def by simp
  4265 qed
  4266 
  4267 lemma uniformly_continuous_on_norm[continuous_on_intros]:
  4268   assumes "uniformly_continuous_on s f"
  4269   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
  4270   unfolding norm_conv_dist using assms
  4271   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
  4272 
  4273 lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:
  4274   assumes "uniformly_continuous_on s g"
  4275   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
  4276   using assms unfolding uniformly_continuous_on_sequentially
  4277   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
  4278   by (auto intro: tendsto_zero)
  4279 
  4280 lemma uniformly_continuous_on_cmul[continuous_on_intros]:
  4281   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4282   assumes "uniformly_continuous_on s f"
  4283   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
  4284   using bounded_linear_scaleR_right assms
  4285   by (rule bounded_linear.uniformly_continuous_on)
  4286 
  4287 lemma dist_minus:
  4288   fixes x y :: "'a::real_normed_vector"
  4289   shows "dist (- x) (- y) = dist x y"
  4290   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
  4291 
  4292 lemma uniformly_continuous_on_minus[continuous_on_intros]:
  4293   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4294   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
  4295   unfolding uniformly_continuous_on_def dist_minus .
  4296 
  4297 lemma uniformly_continuous_on_add[continuous_on_intros]:
  4298   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4299   assumes "uniformly_continuous_on s f"
  4300   assumes "uniformly_continuous_on s g"
  4301   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
  4302   using assms unfolding uniformly_continuous_on_sequentially
  4303   unfolding dist_norm tendsto_norm_zero_iff add_diff_add
  4304   by (auto intro: tendsto_add_zero)
  4305 
  4306 lemma uniformly_continuous_on_diff[continuous_on_intros]:
  4307   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4308   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"
  4309   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
  4310   unfolding ab_diff_minus using assms
  4311   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)
  4312 
  4313 text{* Continuity of all kinds is preserved under composition. *}
  4314 
  4315 lemma continuous_within_topological:
  4316   "continuous (at x within s) f \<longleftrightarrow>
  4317     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  4318       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  4319 unfolding continuous_within
  4320 unfolding tendsto_def Limits.eventually_within eventually_at_topological
  4321 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  4322 
  4323 lemma continuous_within_compose[continuous_intros]:
  4324   assumes "continuous (at x within s) f"
  4325   assumes "continuous (at (f x) within f ` s) g"
  4326   shows "continuous (at x within s) (g o f)"
  4327 using assms unfolding continuous_within_topological by simp metis
  4328 
  4329 lemma continuous_at_compose[continuous_intros]:
  4330   assumes "continuous (at x) f" and "continuous (at (f x)) g"
  4331   shows "continuous (at x) (g o f)"
  4332 proof-
  4333   have "continuous (at (f x) within range f) g" using assms(2)
  4334     using continuous_within_subset[of "f x" UNIV g "range f"] by simp
  4335   thus ?thesis using assms(1)
  4336     using continuous_within_compose[of x UNIV f g] by simp
  4337 qed
  4338 
  4339 lemma continuous_on_compose[continuous_on_intros]:
  4340   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
  4341   unfolding continuous_on_topological by simp metis
  4342 
  4343 lemma uniformly_continuous_on_compose[continuous_on_intros]:
  4344   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f ` s) g"
  4345   shows "uniformly_continuous_on s (g o f)"
  4346 proof-
  4347   { fix e::real assume "e>0"
  4348     then obtain d where "d>0" and d:"\<forall>x\<in>f ` s. \<forall>x'\<in>f ` s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto
  4349     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using `d>0` using assms(1) unfolding uniformly_continuous_on_def by auto
  4350     hence "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e" using `d>0` using d by auto  }
  4351   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
  4352 qed
  4353 
  4354 text{* Continuity in terms of open preimages. *}
  4355 
  4356 lemma continuous_at_open:
  4357   shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
  4358 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
  4359 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
  4360 
  4361 lemma continuous_imp_tendsto:
  4362   assumes "continuous (at x0) f" and "x ----> x0"
  4363   shows "(f \<circ> x) ----> (f x0)"
  4364 proof (rule topological_tendstoI)
  4365   fix S
  4366   assume "open S" "f x0 \<in> S"
  4367   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"
  4368      using assms continuous_at_open by metis
  4369   then have "eventually (\<lambda>n. x n \<in> T) sequentially"
  4370     using assms T_def by (auto simp: tendsto_def)
  4371   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"
  4372     using T_def by (auto elim!: eventually_elim1)
  4373 qed
  4374 
  4375 lemma continuous_on_open:
  4376   shows "continuous_on s f \<longleftrightarrow>
  4377         (\<forall>t. openin (subtopology euclidean (f ` s)) t
  4378             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  4379 proof (safe)
  4380   fix t :: "'b set"
  4381   assume 1: "continuous_on s f"
  4382   assume 2: "openin (subtopology euclidean (f ` s)) t"
  4383   from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
  4384     unfolding openin_open by auto
  4385   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
  4386   have "open U" unfolding U_def by (simp add: open_Union)
  4387   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
  4388   proof (intro ballI iffI)
  4389     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
  4390       unfolding U_def t by auto
  4391   next
  4392     fix x assume "x \<in> s" and "f x \<in> t"
  4393     hence "x \<in> s" and "f x \<in> B"
  4394       unfolding t by auto
  4395     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
  4396       unfolding t continuous_on_topological by metis
  4397     then show "x \<in> U"
  4398       unfolding U_def by auto
  4399   qed
  4400   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
  4401   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  4402     unfolding openin_open by fast
  4403 next
  4404   assume "?rhs" show "continuous_on s f"
  4405   unfolding continuous_on_topological
  4406   proof (clarify)
  4407     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
  4408     have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
  4409       unfolding openin_open using `open B` by auto
  4410     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
  4411       using `?rhs` by fast
  4412     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
  4413       unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
  4414   qed
  4415 qed
  4416 
  4417 text {* Similarly in terms of closed sets. *}
  4418 
  4419 lemma continuous_on_closed:
  4420   shows "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f ` s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  4421 proof
  4422   assume ?lhs
  4423   { fix t
  4424     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  4425     have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
  4426     assume as:"closedin (subtopology euclidean (f ` s)) t"
  4427     hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
  4428     hence "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?lhs`[unfolded continuous_on_open, THEN spec[where x="(f ` s) - t"]]
  4429       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }
  4430   thus ?rhs by auto
  4431 next
  4432   assume ?rhs
  4433   { fix t
  4434     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  4435     assume as:"openin (subtopology euclidean (f ` s)) t"
  4436     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
  4437       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
  4438   thus ?lhs unfolding continuous_on_open by auto
  4439 qed
  4440 
  4441 text {* Half-global and completely global cases. *}
  4442 
  4443 lemma continuous_open_in_preimage:
  4444   assumes "continuous_on s f"  "open t"
  4445   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  4446 proof-
  4447   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
  4448   have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  4449     using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
  4450   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  4451 qed
  4452 
  4453 lemma continuous_closed_in_preimage:
  4454   assumes "continuous_on s f"  "closed t"
  4455   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  4456 proof-
  4457   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
  4458   have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  4459     using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
  4460   thus ?thesis
  4461     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  4462 qed
  4463 
  4464 lemma continuous_open_preimage:
  4465   assumes "continuous_on s f" "open s" "open t"
  4466   shows "open {x \<in> s. f x \<in> t}"
  4467 proof-
  4468   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  4469     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
  4470   thus ?thesis using open_Int[of s T, OF assms(2)] by auto
  4471 qed
  4472 
  4473 lemma continuous_closed_preimage:
  4474   assumes "continuous_on s f" "closed s" "closed t"
  4475   shows "closed {x \<in> s. f x \<in> t}"
  4476 proof-
  4477   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  4478     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
  4479   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
  4480 qed
  4481 
  4482 lemma continuous_open_preimage_univ:
  4483   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
  4484   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
  4485 
  4486 lemma continuous_closed_preimage_univ:
  4487   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
  4488   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
  4489 
  4490 lemma continuous_open_vimage:
  4491   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
  4492   unfolding vimage_def by (rule continuous_open_preimage_univ)
  4493 
  4494 lemma continuous_closed_vimage:
  4495   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
  4496   unfolding vimage_def by (rule continuous_closed_preimage_univ)
  4497 
  4498 lemma interior_image_subset:
  4499   assumes "\<forall>x. continuous (at x) f" "inj f"
  4500   shows "interior (f ` s) \<subseteq> f ` (interior s)"
  4501 proof
  4502   fix x assume "x \<in> interior (f ` s)"
  4503   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` s" ..
  4504   hence "x \<in> f ` s" by auto
  4505   then obtain y where y: "y \<in> s" "x = f y" by auto
  4506   have "open (vimage f T)"
  4507     using assms(1) `open T` by (rule continuous_open_vimage)
  4508   moreover have "y \<in> vimage f T"
  4509     using `x = f y` `x \<in> T` by simp
  4510   moreover have "vimage f T \<subseteq> s"
  4511     using `T \<subseteq> image f s` `inj f` unfolding inj_on_def subset_eq by auto
  4512   ultimately have "y \<in> interior s" ..
  4513   with `x = f y` show "x \<in> f ` interior s" ..
  4514 qed
  4515 
  4516 text {* Equality of continuous functions on closure and related results. *}
  4517 
  4518 lemma continuous_closed_in_preimage_constant:
  4519   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4520   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
  4521   using continuous_closed_in_preimage[of s f "{a}"] by auto
  4522 
  4523 lemma continuous_closed_preimage_constant:
  4524   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4525   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
  4526   using continuous_closed_preimage[of s f "{a}"] by auto
  4527 
  4528 lemma continuous_constant_on_closure:
  4529   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4530   assumes "continuous_on (closure s) f"
  4531           "\<forall>x \<in> s. f x = a"
  4532   shows "\<forall>x \<in> (closure s). f x = a"
  4533     using continuous_closed_preimage_constant[of "closure s" f a]
  4534     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
  4535 
  4536 lemma image_closure_subset:
  4537   assumes "continuous_on (closure s) f"  "closed t"  "(f ` s) \<subseteq> t"
  4538   shows "f ` (closure s) \<subseteq> t"
  4539 proof-
  4540   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
  4541   moreover have "closed {x \<in> closure s. f x \<in> t}"
  4542     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
  4543   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
  4544     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
  4545   thus ?thesis by auto
  4546 qed
  4547 
  4548 lemma continuous_on_closure_norm_le:
  4549   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4550   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"
  4551   shows "norm(f x) \<le> b"
  4552 proof-
  4553   have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
  4554   show ?thesis
  4555     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
  4556     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
  4557 qed
  4558 
  4559 text {* Making a continuous function avoid some value in a neighbourhood. *}
  4560 
  4561 lemma continuous_within_avoid:
  4562   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4563   assumes "continuous (at x within s) f" and "f x \<noteq> a"
  4564   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
  4565 proof-
  4566   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"
  4567     using t1_space [OF `f x \<noteq> a`] by fast
  4568   have "(f ---> f x) (at x within s)"
  4569     using assms(1) by (simp add: continuous_within)
  4570   hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"
  4571     using `open U` and `f x \<in> U`
  4572     unfolding tendsto_def by fast
  4573   hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"
  4574     using `a \<notin> U` by (fast elim: eventually_mono [rotated])
  4575   thus ?thesis
  4576     unfolding Limits.eventually_within Metric_Spaces.eventually_at
  4577     by (rule ex_forward, cut_tac `f x \<noteq> a`, auto simp: dist_commute)
  4578 qed
  4579 
  4580 lemma continuous_at_avoid:
  4581   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4582   assumes "continuous (at x) f" and "f x \<noteq> a"
  4583   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  4584   using assms continuous_within_avoid[of x UNIV f a] by simp
  4585 
  4586 lemma continuous_on_avoid:
  4587   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4588   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"
  4589   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
  4590 using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)]  continuous_within_avoid[of x s f a]  assms(3) by auto
  4591 
  4592 lemma continuous_on_open_avoid:
  4593   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4594   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"
  4595   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  4596 using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]  continuous_at_avoid[of x f a]  assms(4) by auto
  4597 
  4598 text {* Proving a function is constant by proving open-ness of level set. *}
  4599 
  4600 lemma continuous_levelset_open_in_cases:
  4601   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4602   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  4603         openin (subtopology euclidean s) {x \<in> s. f x = a}
  4604         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
  4605 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
  4606 
  4607 lemma continuous_levelset_open_in:
  4608   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4609   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  4610         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
  4611         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"
  4612 using continuous_levelset_open_in_cases[of s f ]
  4613 by meson
  4614 
  4615 lemma continuous_levelset_open:
  4616   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4617   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
  4618   shows "\<forall>x \<in> s. f x = a"
  4619 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
  4620 
  4621 text {* Some arithmetical combinations (more to prove). *}
  4622 
  4623 lemma open_scaling[intro]:
  4624   fixes s :: "'a::real_normed_vector set"
  4625   assumes "c \<noteq> 0"  "open s"
  4626   shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
  4627 proof-
  4628   { fix x assume "x \<in> s"
  4629     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto
  4630     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
  4631     moreover
  4632     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
  4633       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
  4634         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
  4635           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
  4636       hence "y \<in> op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]  e[THEN spec[where x="(1 / c) *\<^sub>R y"]]  assms(1) unfolding dist_norm scaleR_scaleR by auto  }
  4637     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto  }
  4638   thus ?thesis unfolding open_dist by auto
  4639 qed
  4640 
  4641 lemma minus_image_eq_vimage:
  4642   fixes A :: "'a::ab_group_add set"
  4643   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
  4644   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
  4645 
  4646 lemma open_negations:
  4647   fixes s :: "'a::real_normed_vector set"
  4648   shows "open s ==> open ((\<lambda> x. -x) ` s)"
  4649   unfolding scaleR_minus1_left [symmetric]
  4650   by (rule open_scaling, auto)
  4651 
  4652 lemma open_translation:
  4653   fixes s :: "'a::real_normed_vector set"
  4654   assumes "open s"  shows "open((\<lambda>x. a + x) ` s)"
  4655 proof-
  4656   { fix x have "continuous (at x) (\<lambda>x. x - a)"
  4657       by (intro continuous_diff continuous_at_id continuous_const) }
  4658   moreover have "{x. x - a \<in> s} = op + a ` s" by force
  4659   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
  4660 qed
  4661 
  4662 lemma open_affinity:
  4663   fixes s :: "'a::real_normed_vector set"
  4664   assumes "open s"  "c \<noteq> 0"
  4665   shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  4666 proof-
  4667   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
  4668   have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
  4669   thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
  4670 qed
  4671 
  4672 lemma interior_translation:
  4673   fixes s :: "'a::real_normed_vector set"
  4674   shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
  4675 proof (rule set_eqI, rule)
  4676   fix x assume "x \<in> interior (op + a ` s)"
  4677   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
  4678   hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto
  4679   thus "x \<in> op + a ` interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using `e > 0` by auto
  4680 next
  4681   fix x assume "x \<in> op + a ` interior s"
  4682   then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto
  4683   { fix z have *:"a + y - z = y + a - z" by auto
  4684     assume "z\<in>ball x e"
  4685     hence "z - a \<in> s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 * by auto
  4686     hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }
  4687   hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
  4688   thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
  4689 qed
  4690 
  4691 text {* Topological properties of linear functions. *}
  4692 
  4693 lemma linear_lim_0:
  4694   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
  4695 proof-
  4696   interpret f: bounded_linear f by fact
  4697   have "(f ---> f 0) (at 0)"
  4698     using tendsto_ident_at by (rule f.tendsto)
  4699   thus ?thesis unfolding f.zero .
  4700 qed
  4701 
  4702 lemma linear_continuous_at:
  4703   assumes "bounded_linear f"  shows "continuous (at a) f"
  4704   unfolding continuous_at using assms
  4705   apply (rule bounded_linear.tendsto)
  4706   apply (rule tendsto_ident_at)
  4707   done
  4708 
  4709 lemma linear_continuous_within:
  4710   shows "bounded_linear f ==> continuous (at x within s) f"
  4711   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
  4712 
  4713 lemma linear_continuous_on:
  4714   shows "bounded_linear f ==> continuous_on s f"
  4715   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
  4716 
  4717 text {* Also bilinear functions, in composition form. *}
  4718 
  4719 lemma bilinear_continuous_at_compose:
  4720   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
  4721         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
  4722   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
  4723 
  4724 lemma bilinear_continuous_within_compose:
  4725   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
  4726         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
  4727   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
  4728 
  4729 lemma bilinear_continuous_on_compose:
  4730   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
  4731              ==> continuous_on s (\<lambda>x. h (f x) (g x))"
  4732   unfolding continuous_on_def
  4733   by (fast elim: bounded_bilinear.tendsto)
  4734 
  4735 text {* Preservation of compactness and connectedness under continuous function. *}
  4736 
  4737 lemma compact_eq_openin_cover:
  4738   "compact S \<longleftrightarrow>
  4739     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
  4740       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
  4741 proof safe
  4742   fix C
  4743   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"
  4744   hence "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
  4745     unfolding openin_open by force+
  4746   with `compact S` obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
  4747     by (rule compactE)
  4748   hence "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"
  4749     by auto
  4750   thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
  4751 next
  4752   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
  4753         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"
  4754   show "compact S"
  4755   proof (rule compactI)
  4756     fix C
  4757     let ?C = "image (\<lambda>T. S \<inter> T) C"
  4758     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"
  4759     hence "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"
  4760       unfolding openin_open by auto
  4761     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"
  4762       by metis
  4763     let ?D = "inv_into C (\<lambda>T. S \<inter> T) ` D"
  4764     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"
  4765     proof (intro conjI)
  4766       from `D \<subseteq> ?C` show "?D \<subseteq> C"
  4767         by (fast intro: inv_into_into)
  4768       from `finite D` show "finite ?D"
  4769         by (rule finite_imageI)
  4770       from `S \<subseteq> \<Union>D` show "S \<subseteq> \<Union>?D"
  4771         apply (rule subset_trans)
  4772         apply clarsimp
  4773         apply (frule subsetD [OF `D \<subseteq> ?C`, THEN f_inv_into_f])
  4774         apply (erule rev_bexI, fast)
  4775         done
  4776     qed
  4777     thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
  4778   qed
  4779 qed
  4780 
  4781 lemma compact_continuous_image:
  4782   assumes "continuous_on s f" and "compact s"
  4783   shows "compact (f ` s)"
  4784 using assms (* FIXME: long unstructured proof *)
  4785 unfolding continuous_on_open
  4786 unfolding compact_eq_openin_cover
  4787 apply clarify
  4788 apply (drule_tac x="image (\<lambda>t. {x \<in> s. f x \<in> t}) C" in spec)
  4789 apply (drule mp)
  4790 apply (rule conjI)
  4791 apply simp
  4792 apply clarsimp
  4793 apply (drule subsetD)
  4794 apply (erule imageI)
  4795 apply fast
  4796 apply (erule thin_rl)
  4797 apply clarify
  4798 apply (rule_tac x="image (inv_into C (\<lambda>t. {x \<in> s. f x \<in> t})) D" in exI)
  4799 apply (intro conjI)
  4800 apply clarify
  4801 apply (rule inv_into_into)
  4802 apply (erule (1) subsetD)
  4803 apply (erule finite_imageI)
  4804 apply (clarsimp, rename_tac x)
  4805 apply (drule (1) subsetD, clarify)
  4806 apply (drule (1) subsetD, clarify)
  4807 apply (rule rev_bexI)
  4808 apply assumption
  4809 apply (subgoal_tac "{x \<in> s. f x \<in> t} \<in> (\<lambda>t. {x \<in> s. f x \<in> t}) ` C")
  4810 apply (drule f_inv_into_f)
  4811 apply fast
  4812 apply (erule imageI)
  4813 done
  4814 
  4815 lemma connected_continuous_image:
  4816   assumes "continuous_on s f"  "connected s"
  4817   shows "connected(f ` s)"
  4818 proof-
  4819   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f ` s"  "openin (subtopology euclidean (f ` s)) T"  "closedin (subtopology euclidean (f ` s)) T"
  4820     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
  4821       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
  4822       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
  4823       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
  4824     hence False using as(1,2)
  4825       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
  4826   thus ?thesis unfolding connected_clopen by auto
  4827 qed
  4828 
  4829 text {* Continuity implies uniform continuity on a compact domain. *}
  4830   
  4831 lemma compact_uniformly_continuous:
  4832   assumes f: "continuous_on s f" and s: "compact s"
  4833   shows "uniformly_continuous_on s f"
  4834   unfolding uniformly_continuous_on_def
  4835 proof (cases, safe)
  4836   fix e :: real assume "0 < e" "s \<noteq> {}"
  4837   def [simp]: R \<equiv> "{(y, d). y \<in> s \<and> 0 < d \<and> ball y d \<inter> s \<subseteq> {x \<in> s. f x \<in> ball (f y) (e/2) } }"
  4838   let ?b = "(\<lambda>(y, d). ball y (d/2))"
  4839   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"
  4840   proof safe
  4841     fix y assume "y \<in> s"
  4842     from continuous_open_in_preimage[OF f open_ball]
  4843     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"
  4844       unfolding openin_subtopology open_openin by metis
  4845     then obtain d where "ball y d \<subseteq> T" "0 < d"
  4846       using `0 < e` `y \<in> s` by (auto elim!: openE)
  4847     with T `y \<in> s` show "y \<in> (\<Union>r\<in>R. ?b r)"
  4848       by (intro UN_I[of "(y, d)"]) auto
  4849   qed auto
  4850   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"
  4851     by (rule compactE_image)
  4852   with `s \<noteq> {}` have [simp]: "\<And>x. x < Min (snd ` D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"
  4853     by (subst Min_gr_iff) auto
  4854   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
  4855   proof (rule, safe)
  4856     fix x x' assume in_s: "x' \<in> s" "x \<in> s"
  4857     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"
  4858       by blast
  4859     moreover assume "dist x x' < Min (snd`D) / 2"
  4860     ultimately have "dist y x' < d"
  4861       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)
  4862     with D x in_s show  "dist (f x) (f x') < e"
  4863       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)
  4864   qed (insert D, auto)
  4865 qed auto
  4866 
  4867 text{* Continuity of inverse function on compact domain. *}
  4868 
  4869 lemma continuous_on_inv:
  4870   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
  4871   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"
  4872   shows "continuous_on (f ` s) g"
  4873 unfolding continuous_on_topological
  4874 proof (clarsimp simp add: assms(3))
  4875   fix x :: 'a and B :: "'a set"
  4876   assume "x \<in> s" and "open B" and "x \<in> B"
  4877   have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
  4878     using assms(3) by (auto, metis)
  4879   have "continuous_on (s - B) f"
  4880     using `continuous_on s f` Diff_subset
  4881     by (rule continuous_on_subset)
  4882   moreover have "compact (s - B)"
  4883     using `open B` and `compact s`
  4884     unfolding Diff_eq by (intro compact_inter_closed closed_Compl)
  4885   ultimately have "compact (f ` (s - B))"
  4886     by (rule compact_continuous_image)
  4887   hence "closed (f ` (s - B))"
  4888     by (rule compact_imp_closed)
  4889   hence "open (- f ` (s - B))"
  4890     by (rule open_Compl)
  4891   moreover have "f x \<in> - f ` (s - B)"
  4892     using `x \<in> s` and `x \<in> B` by (simp add: 1)
  4893   moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
  4894     by (simp add: 1)
  4895   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"
  4896     by fast
  4897 qed
  4898 
  4899 text {* A uniformly convergent limit of continuous functions is continuous. *}
  4900 
  4901 lemma continuous_uniform_limit:
  4902   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
  4903   assumes "\<not> trivial_limit F"
  4904   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
  4905   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
  4906   shows "continuous_on s g"
  4907 proof-
  4908   { fix x and e::real assume "x\<in>s" "e>0"
  4909     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
  4910       using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
  4911     from eventually_happens [OF eventually_conj [OF this assms(2)]]
  4912     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"
  4913       using assms(1) by blast
  4914     have "e / 3 > 0" using `e>0` by auto
  4915     then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
  4916       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
  4917     { fix y assume "y \<in> s" and "dist y x < d"
  4918       hence "dist (f n y) (f n x) < e / 3"
  4919         by (rule d [rule_format])
  4920       hence "dist (f n y) (g x) < 2 * e / 3"
  4921         using dist_triangle [of "f n y" "g x" "f n x"]
  4922         using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
  4923         by auto
  4924       hence "dist (g y) (g x) < e"
  4925         using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
  4926         using dist_triangle3 [of "g y" "g x" "f n y"]
  4927         by auto }
  4928     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
  4929       using `d>0` by auto }
  4930   thus ?thesis unfolding continuous_on_iff by auto
  4931 qed
  4932 
  4933 
  4934 subsection {* Topological stuff lifted from and dropped to R *}
  4935 
  4936 lemma open_real:
  4937   fixes s :: "real set" shows
  4938  "open s \<longleftrightarrow>
  4939         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
  4940   unfolding open_dist dist_norm by simp
  4941 
  4942 lemma islimpt_approachable_real:
  4943   fixes s :: "real set"
  4944   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
  4945   unfolding islimpt_approachable dist_norm by simp
  4946 
  4947 lemma closed_real:
  4948   fixes s :: "real set"
  4949   shows "closed s \<longleftrightarrow>
  4950         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
  4951             --> x \<in> s)"
  4952   unfolding closed_limpt islimpt_approachable dist_norm by simp
  4953 
  4954 lemma continuous_at_real_range:
  4955   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4956   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  4957         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
  4958   unfolding continuous_at unfolding Lim_at
  4959   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
  4960   apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
  4961   apply(erule_tac x=e in allE) by auto
  4962 
  4963 lemma continuous_on_real_range:
  4964   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4965   shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
  4966   unfolding continuous_on_iff dist_norm by simp
  4967 
  4968 lemma compact_attains_sup:
  4969   fixes S :: "'a::linorder_topology set"
  4970   assumes "compact S" "S \<noteq> {}"
  4971   shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
  4972 proof (rule classical)
  4973   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
  4974   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
  4975     by (metis not_le)
  4976   then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
  4977     by auto
  4978   with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
  4979     by (erule compactE_image)
  4980   with `S \<noteq> {}` have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
  4981     by (auto intro!: Max_in)
  4982   with C have "S \<subseteq> {..< Max (t`C)}"
  4983     by (auto intro: less_le_trans simp: subset_eq)
  4984   with t Max `C \<subseteq> S` show ?thesis
  4985     by fastforce
  4986 qed
  4987 
  4988 lemma compact_attains_inf:
  4989   fixes S :: "'a::linorder_topology set"
  4990   assumes "compact S" "S \<noteq> {}"
  4991   shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
  4992 proof (rule classical)
  4993   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
  4994   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
  4995     by (metis not_le)
  4996   then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
  4997     by auto
  4998   with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
  4999     by (erule compactE_image)
  5000   with `S \<noteq> {}` have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
  5001     by (auto intro!: Min_in)
  5002   with C have "S \<subseteq> {Min (t`C) <..}"
  5003     by (auto intro: le_less_trans simp: subset_eq)
  5004   with t Min `C \<subseteq> S` show ?thesis
  5005     by fastforce
  5006 qed
  5007 
  5008 lemma continuous_attains_sup:
  5009   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
  5010   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s.  f y \<le> f x)"
  5011   using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
  5012 
  5013 lemma continuous_attains_inf:
  5014   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
  5015   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y)"
  5016   using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
  5017 
  5018 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
  5019 
  5020 lemma distance_attains_sup:
  5021   assumes "compact s" "s \<noteq> {}"
  5022   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"
  5023 proof (rule continuous_attains_sup [OF assms])
  5024   { fix x assume "x\<in>s"
  5025     have "(dist a ---> dist a x) (at x within s)"
  5026       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)
  5027   }
  5028   thus "continuous_on s (dist a)"
  5029     unfolding continuous_on ..
  5030 qed
  5031 
  5032 text {* For \emph{minimal} distance, we only need closure, not compactness. *}
  5033 
  5034 lemma distance_attains_inf:
  5035   fixes a :: "'a::heine_borel"
  5036   assumes "closed s"  "s \<noteq> {}"
  5037   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"
  5038 proof-
  5039   from assms(2) obtain b where "b \<in> s" by auto
  5040   let ?B = "s \<inter> cball a (dist b a)"
  5041   have "?B \<noteq> {}" using `b \<in> s` by (auto simp add: dist_commute)
  5042   moreover have "continuous_on ?B (dist a)"
  5043     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)
  5044   moreover have "compact ?B"
  5045     by (intro closed_inter_compact `closed s` compact_cball)
  5046   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"
  5047     by (metis continuous_attains_inf)
  5048   thus ?thesis by fastforce
  5049 qed
  5050 
  5051 
  5052 subsection {* Pasted sets *}
  5053 
  5054 lemma bounded_Times:
  5055   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
  5056 proof-
  5057   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
  5058     using assms [unfolded bounded_def] by auto
  5059   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
  5060     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
  5061   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
  5062 qed
  5063 
  5064 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
  5065 by (induct x) simp
  5066 
  5067 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
  5068 unfolding seq_compact_def
  5069 apply clarify
  5070 apply (drule_tac x="fst \<circ> f" in spec)
  5071 apply (drule mp, simp add: mem_Times_iff)
  5072 apply (clarify, rename_tac l1 r1)
  5073 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
  5074 apply (drule mp, simp add: mem_Times_iff)
  5075 apply (clarify, rename_tac l2 r2)
  5076 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
  5077 apply (rule_tac x="r1 \<circ> r2" in exI)
  5078 apply (rule conjI, simp add: subseq_def)
  5079 apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
  5080 apply (drule (1) tendsto_Pair) back
  5081 apply (simp add: o_def)
  5082 done
  5083 
  5084 lemma compact_Times: 
  5085   assumes "compact s" "compact t"
  5086   shows "compact (s \<times> t)"
  5087 proof (rule compactI)
  5088   fix C assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
  5089   have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
  5090   proof
  5091     fix x assume "x \<in> s"
  5092     have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
  5093     proof 
  5094       fix y assume "y \<in> t"
  5095       with `x \<in> s` C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
  5096       then show "?P y" by (auto elim!: open_prod_elim)
  5097     qed
  5098     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"
  5099       and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"
  5100       by metis
  5101     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto
  5102     from compactE_image[OF `compact t` this] obtain D where "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
  5103       by auto
  5104     moreover with c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"
  5105       by (fastforce simp: subset_eq)
  5106     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
  5107       using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>a`D"] conjI) (auto intro!: open_INT)
  5108   qed
  5109   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
  5110     and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>d x"
  5111     unfolding subset_eq UN_iff by metis
  5112   moreover from compactE_image[OF `compact s` a] obtain e where e: "e \<subseteq> s" "finite e"
  5113     and s: "s \<subseteq> (\<Union>x\<in>e. a x)" by auto
  5114   moreover
  5115   { from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)" by auto
  5116     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)" using d `e \<subseteq> s` by (intro UN_mono) auto
  5117     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" . }
  5118   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
  5119     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)
  5120 qed
  5121 
  5122 text{* Hence some useful properties follow quite easily. *}
  5123 
  5124 lemma compact_scaling:
  5125   fixes s :: "'a::real_normed_vector set"
  5126   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
  5127 proof-
  5128   let ?f = "\<lambda>x. scaleR c x"
  5129   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)
  5130   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
  5131     using linear_continuous_at[OF *] assms by auto
  5132 qed
  5133 
  5134 lemma compact_negations:
  5135   fixes s :: "'a::real_normed_vector set"
  5136   assumes "compact s"  shows "compact ((\<lambda>x. -x) ` s)"
  5137   using compact_scaling [OF assms, of "- 1"] by auto
  5138 
  5139 lemma compact_sums:
  5140   fixes s t :: "'a::real_normed_vector set"
  5141   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
  5142 proof-
  5143   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
  5144     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
  5145   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
  5146     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  5147   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
  5148 qed
  5149 
  5150 lemma compact_differences:
  5151   fixes s t :: "'a::real_normed_vector set"
  5152   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
  5153 proof-
  5154   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
  5155     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  5156   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
  5157 qed
  5158 
  5159 lemma compact_translation:
  5160   fixes s :: "'a::real_normed_vector set"
  5161   assumes "compact s"  shows "compact ((\<lambda>x. a + x) ` s)"
  5162 proof-
  5163   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
  5164   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
  5165 qed
  5166 
  5167 lemma compact_affinity:
  5168   fixes s :: "'a::real_normed_vector set"
  5169   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  5170 proof-
  5171   have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  5172   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
  5173 qed
  5174 
  5175 text {* Hence we get the following. *}
  5176 
  5177 lemma compact_sup_maxdistance:
  5178   fixes s :: "'a::metric_space set"
  5179   assumes "compact s"  "s \<noteq> {}"
  5180   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
  5181 proof-
  5182   have "compact (s \<times> s)" using `compact s` by (intro compact_Times)
  5183   moreover have "s \<times> s \<noteq> {}" using `s \<noteq> {}` by auto
  5184   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"
  5185     by (intro continuous_at_imp_continuous_on ballI continuous_dist
  5186       continuous_isCont[THEN iffD1] isCont_fst isCont_snd isCont_ident)
  5187   ultimately show ?thesis
  5188     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto
  5189 qed
  5190 
  5191 text {* We can state this in terms of diameter of a set. *}
  5192 
  5193 definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"
  5194 
  5195 lemma diameter_bounded_bound:
  5196   fixes s :: "'a :: metric_space set"
  5197   assumes s: "bounded s" "x \<in> s" "y \<in> s"
  5198   shows "dist x y \<le> diameter s"
  5199 proof -
  5200   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
  5201   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
  5202     unfolding bounded_def by auto
  5203   have "dist x y \<le> Sup ?D"
  5204   proof (rule cSup_upper, safe)
  5205     fix a b assume "a \<in> s" "b \<in> s"
  5206     with z[of a] z[of b] dist_triangle[of a b z]
  5207     show "dist a b \<le> 2 * d"
  5208       by (simp add: dist_commute)
  5209   qed (insert s, auto)
  5210   with `x \<in> s` show ?thesis
  5211     by (auto simp add: diameter_def)
  5212 qed
  5213 
  5214 lemma diameter_lower_bounded:
  5215   fixes s :: "'a :: metric_space set"
  5216   assumes s: "bounded s" and d: "0 < d" "d < diameter s"
  5217   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"
  5218 proof (rule ccontr)
  5219   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
  5220   assume contr: "\<not> ?thesis"
  5221   moreover
  5222   from d have "s \<noteq> {}"
  5223     by (auto simp: diameter_def)
  5224   then have "?D \<noteq> {}" by auto
  5225   ultimately have "Sup ?D \<le> d"
  5226     by (intro cSup_least) (auto simp: not_less)
  5227   with `d < diameter s` `s \<noteq> {}` show False
  5228     by (auto simp: diameter_def)
  5229 qed
  5230 
  5231 lemma diameter_bounded:
  5232   assumes "bounded s"
  5233   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"
  5234         "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"
  5235   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
  5236   by auto
  5237 
  5238 lemma diameter_compact_attained:
  5239   assumes "compact s"  "s \<noteq> {}"
  5240   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"
  5241 proof -
  5242   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
  5243   then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
  5244     using compact_sup_maxdistance[OF assms] by auto
  5245   hence "diameter s \<le> dist x y"
  5246     unfolding diameter_def by clarsimp (rule cSup_least, fast+)
  5247   thus ?thesis
  5248     by (metis b diameter_bounded_bound order_antisym xys)
  5249 qed
  5250 
  5251 text {* Related results with closure as the conclusion. *}
  5252 
  5253 lemma closed_scaling:
  5254   fixes s :: "'a::real_normed_vector set"
  5255   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
  5256 proof(cases "s={}")
  5257   case True thus ?thesis by auto
  5258 next
  5259   case False
  5260   show ?thesis
  5261   proof(cases "c=0")
  5262     have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
  5263     case True thus ?thesis apply auto unfolding * by auto
  5264   next
  5265     case False
  5266     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s"  "(x ---> l) sequentially"
  5267       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
  5268           using as(1)[THEN spec[where x=n]]
  5269           using `c\<noteq>0` by auto
  5270       }
  5271       moreover
  5272       { fix e::real assume "e>0"
  5273         hence "0 < e *\<bar>c\<bar>"  using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
  5274         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
  5275           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto
  5276         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
  5277           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
  5278           using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto  }
  5279       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto
  5280       ultimately have "l \<in> scaleR c ` s"
  5281         using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]
  5282         unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }
  5283     thus ?thesis unfolding closed_sequential_limits by fast
  5284   qed
  5285 qed
  5286 
  5287 lemma closed_negations:
  5288   fixes s :: "'a::real_normed_vector set"
  5289   assumes "closed s"  shows "closed ((\<lambda>x. -x) ` s)"
  5290   using closed_scaling[OF assms, of "- 1"] by simp
  5291 
  5292 lemma compact_closed_sums:
  5293   fixes s :: "'a::real_normed_vector set"
  5294   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  5295 proof-
  5296   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
  5297   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"
  5298     from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> s"  "\<forall>n. snd (f n) \<in> t"
  5299       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
  5300     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
  5301       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
  5302     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
  5303       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto
  5304     hence "l - l' \<in> t"
  5305       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
  5306       using f(3) by auto
  5307     hence "l \<in> ?S" using `l' \<in> s` apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto
  5308   }
  5309   thus ?thesis unfolding closed_sequential_limits by fast
  5310 qed
  5311 
  5312 lemma closed_compact_sums:
  5313   fixes s t :: "'a::real_normed_vector set"
  5314   assumes "closed s"  "compact t"
  5315   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  5316 proof-
  5317   have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto
  5318     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
  5319   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
  5320 qed
  5321 
  5322 lemma compact_closed_differences:
  5323   fixes s t :: "'a::real_normed_vector set"
  5324   assumes "compact s"  "closed t"
  5325   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  5326 proof-
  5327   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"
  5328     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  5329   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
  5330 qed
  5331 
  5332 lemma closed_compact_differences:
  5333   fixes s t :: "'a::real_normed_vector set"
  5334   assumes "closed s" "compact t"
  5335   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  5336 proof-
  5337   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
  5338     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  5339  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
  5340 qed
  5341 
  5342 lemma closed_translation:
  5343   fixes a :: "'a::real_normed_vector"
  5344   assumes "closed s"  shows "closed ((\<lambda>x. a + x) ` s)"
  5345 proof-
  5346   have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
  5347   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
  5348 qed
  5349 
  5350 lemma translation_Compl:
  5351   fixes a :: "'a::ab_group_add"
  5352   shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
  5353   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
  5354 
  5355 lemma translation_UNIV:
  5356   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
  5357   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
  5358 
  5359 lemma translation_diff:
  5360   fixes a :: "'a::ab_group_add"
  5361   shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
  5362   by auto
  5363 
  5364 lemma closure_translation:
  5365   fixes a :: "'a::real_normed_vector"
  5366   shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
  5367 proof-
  5368   have *:"op + a ` (- s) = - op + a ` s"
  5369     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
  5370   show ?thesis unfolding closure_interior translation_Compl
  5371     using interior_translation[of a "- s"] unfolding * by auto
  5372 qed
  5373 
  5374 lemma frontier_translation:
  5375   fixes a :: "'a::real_normed_vector"
  5376   shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
  5377   unfolding frontier_def translation_diff interior_translation closure_translation by auto
  5378 
  5379 
  5380 subsection {* Separation between points and sets *}
  5381 
  5382 lemma separate_point_closed:
  5383   fixes s :: "'a::heine_borel set"
  5384   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
  5385 proof(cases "s = {}")
  5386   case True
  5387   thus ?thesis by(auto intro!: exI[where x=1])
  5388 next
  5389   case False
  5390   assume "closed s" "a \<notin> s"
  5391   then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using `s \<noteq> {}` distance_attains_inf [of s a] by blast
  5392   with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
  5393 qed
  5394 
  5395 lemma separate_compact_closed:
  5396   fixes s t :: "'a::heine_borel set"
  5397   assumes "compact s" and t: "closed t" "s \<inter> t = {}"
  5398   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  5399 proof cases
  5400   assume "s \<noteq> {} \<and> t \<noteq> {}"
  5401   then have "s \<noteq> {}" "t \<noteq> {}" by auto
  5402   let ?inf = "\<lambda>x. infdist x t"
  5403   have "continuous_on s ?inf"
  5404     by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id)
  5405   then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"
  5406     using continuous_attains_inf[OF `compact s` `s \<noteq> {}`] by auto
  5407   then have "0 < ?inf x"
  5408     using t `t \<noteq> {}` in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
  5409   moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"
  5410     using x by (auto intro: order_trans infdist_le)
  5411   ultimately show ?thesis
  5412     by auto
  5413 qed (auto intro!: exI[of _ 1])
  5414 
  5415 lemma separate_closed_compact:
  5416   fixes s t :: "'a::heine_borel set"
  5417   assumes "closed s" and "compact t" and "s \<inter> t = {}"
  5418   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  5419 proof-
  5420   have *:"t \<inter> s = {}" using assms(3) by auto
  5421   show ?thesis using separate_compact_closed[OF assms(2,1) *]
  5422     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
  5423     by (auto simp add: dist_commute)
  5424 qed
  5425 
  5426 
  5427 subsection {* Intervals *}
  5428   
  5429 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
  5430   "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and
  5431   "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"
  5432   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  5433 
  5434 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5435   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"
  5436   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"
  5437   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  5438 
  5439 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
  5440  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and
  5441  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
  5442 proof-
  5443   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"
  5444     hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto
  5445     hence "a\<bullet>i < b\<bullet>i" by auto
  5446     hence False using as by auto  }
  5447   moreover
  5448   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
  5449     let ?x = "(1/2) *\<^sub>R (a + b)"
  5450     { fix i :: 'a assume i:"i\<in>Basis" 
  5451       have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
  5452       hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
  5453         by (auto simp: inner_add_left) }
  5454     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
  5455   ultimately show ?th1 by blast
  5456 
  5457   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"
  5458     hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto
  5459     hence "a\<bullet>i \<le> b\<bullet>i" by auto
  5460     hence False using as by auto  }
  5461   moreover
  5462   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
  5463     let ?x = "(1/2) *\<^sub>R (a + b)"
  5464     { fix i :: 'a assume i:"i\<in>Basis"
  5465       have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
  5466       hence "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
  5467         by (auto simp: inner_add_left) }
  5468     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
  5469   ultimately show ?th2 by blast
  5470 qed
  5471 
  5472 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
  5473   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and
  5474   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
  5475   unfolding interval_eq_empty[of a b] by fastforce+
  5476 
  5477 lemma interval_sing:
  5478   fixes a :: "'a::ordered_euclidean_space"
  5479   shows "{a .. a} = {a}" and "{a<..<a} = {}"
  5480   unfolding set_eq_iff mem_interval eq_iff [symmetric]
  5481   by (auto intro: euclidean_eqI simp: ex_in_conv)
  5482 
  5483 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
  5484  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
  5485  "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
  5486  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
  5487  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
  5488   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
  5489   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
  5490 
  5491 lemma interval_open_subset_closed:
  5492   fixes a :: "'a::ordered_euclidean_space"
  5493   shows "{a<..<b} \<subseteq> {a .. b}"
  5494   unfolding subset_eq [unfolded Ball_def] mem_interval
  5495   by (fast intro: less_imp_le)
  5496 
  5497 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5498  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1) and
  5499  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2) and
  5500  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3) and
  5501  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
  5502 proof-
  5503   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
  5504   show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
  5505   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
  5506     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
  5507     fix i :: 'a assume i:"i\<in>Basis"
  5508     (** TODO combine the following two parts as done in the HOL_light version. **)
  5509     { 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"
  5510       assume as2: "a\<bullet>i > c\<bullet>i"
  5511       { fix j :: 'a assume j:"j\<in>Basis"
  5512         hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
  5513           apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i
  5514           by (auto simp add: as2)  }
  5515       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
  5516       moreover
  5517       have "?x\<notin>{a .. b}"
  5518         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
  5519         using as(2)[THEN bspec[where x=i]] and as2 i
  5520         by auto
  5521       ultimately have False using as by auto  }
  5522     hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto
  5523     moreover
  5524     { 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"
  5525       assume as2: "b\<bullet>i < d\<bullet>i"
  5526       { fix j :: 'a assume "j\<in>Basis"
  5527         hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j" 
  5528           apply(cases "j=i") using as(2)[THEN bspec[where x=j]]
  5529           by (auto simp add: as2) }
  5530       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
  5531       moreover
  5532       have "?x\<notin>{a .. b}"
  5533         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
  5534         using as(2)[THEN bspec[where x=i]] and as2 using i
  5535         by auto
  5536       ultimately have False using as by auto  }
  5537     hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto
  5538     ultimately
  5539     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
  5540   } note part1 = this
  5541   show ?th3
  5542     unfolding subset_eq and Ball_def and mem_interval 
  5543     apply(rule,rule,rule,rule) 
  5544     apply(rule part1)
  5545     unfolding subset_eq and Ball_def and mem_interval
  5546     prefer 4
  5547     apply auto 
  5548     by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+ 
  5549   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
  5550     fix i :: 'a assume i:"i\<in>Basis"
  5551     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
  5552     hence "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" using part1 and as(2) using i by auto  } note * = this
  5553   show ?th4 unfolding subset_eq and Ball_def and mem_interval 
  5554     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
  5555     apply auto by(erule_tac x=xa in allE, simp)+ 
  5556 qed
  5557 
  5558 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5559  "{a .. b} \<inter> {c .. d} =  {(\<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)}"
  5560   unfolding set_eq_iff and Int_iff and mem_interval by auto
  5561 
  5562 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
  5563   "{a .. b} \<inter> {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) and
  5564   "{a .. b} \<inter> {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) and
  5565   "{a<..<b} \<inter> {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) and
  5566   "{a<..<b} \<inter> {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)
  5567 proof-
  5568   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"
  5569   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
  5570       (\<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)" 
  5571     by blast
  5572   note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)
  5573   show ?th1 unfolding * by (intro **) auto
  5574   show ?th2 unfolding * by (intro **) auto
  5575   show ?th3 unfolding * by (intro **) auto
  5576   show ?th4 unfolding * by (intro **) auto
  5577 qed
  5578 
  5579 (* Moved interval_open_subset_closed a bit upwards *)
  5580 
  5581 lemma open_interval[intro]:
  5582   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
  5583 proof-
  5584   have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i<..<b\<bullet>i})"
  5585     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
  5586       linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)
  5587   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"
  5588     by (auto simp add: eucl_less [where 'a='a])
  5589   finally show "open {a<..<b}" .
  5590 qed
  5591 
  5592 lemma closed_interval[intro]:
  5593   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
  5594 proof-
  5595   have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i})"
  5596     by (intro closed_INT ballI continuous_closed_vimage allI
  5597       linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
  5598   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i}) = {a .. b}"
  5599     by (auto simp add: eucl_le [where 'a='a])
  5600   finally show "closed {a .. b}" .
  5601 qed
  5602 
  5603 lemma interior_closed_interval [intro]:
  5604   fixes a b :: "'a::ordered_euclidean_space"
  5605   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")
  5606 proof(rule subset_antisym)
  5607   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval
  5608     by (rule interior_maximal)
  5609 next
  5610   { fix x assume "x \<in> interior {a..b}"
  5611     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..
  5612     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
  5613     { fix i :: 'a assume i:"i\<in>Basis"
  5614       have "dist (x - (e / 2) *\<^sub>R i) x < e"
  5615            "dist (x + (e / 2) *\<^sub>R i) x < e"
  5616         unfolding dist_norm apply auto
  5617         unfolding norm_minus_cancel using norm_Basis[OF i] `e>0` by auto
  5618       hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"
  5619                      "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"
  5620         using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]
  5621         and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
  5622         unfolding mem_interval using i by blast+
  5623       hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
  5624         using `e>0` i by (auto simp: inner_diff_left inner_Basis inner_add_left) }
  5625     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
  5626   thus "?L \<subseteq> ?R" ..
  5627 qed
  5628 
  5629 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
  5630 proof-
  5631   let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
  5632   { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
  5633     { fix i :: 'a assume "i\<in>Basis"
  5634       hence "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>" using x[THEN bspec[where x=i]] by auto  }
  5635     hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
  5636     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
  5637   thus ?thesis unfolding interval and bounded_iff by auto
  5638 qed
  5639 
  5640 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5641  "bounded {a .. b} \<and> bounded {a<..<b}"
  5642   using bounded_closed_interval[of a b]
  5643   using interval_open_subset_closed[of a b]
  5644   using bounded_subset[of "{a..b}" "{a<..<b}"]
  5645   by simp
  5646 
  5647 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
  5648  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
  5649   using bounded_interval[of a b] by auto
  5650 
  5651 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
  5652   using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]
  5653   by (auto simp: compact_eq_seq_compact_metric)
  5654 
  5655 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
  5656   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
  5657 proof-
  5658   { fix i :: 'a assume "i\<in>Basis"
  5659     hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"
  5660       using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }
  5661   thus ?thesis unfolding mem_interval by auto
  5662 qed
  5663 
  5664 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
  5665   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
  5666   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
  5667 proof-
  5668   { fix i :: 'a assume i:"i\<in>Basis"
  5669     have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp
  5670     also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
  5671       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  5672       using x unfolding mem_interval using i apply simp
  5673       using y unfolding mem_interval using i apply simp
  5674       done
  5675     finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto
  5676     moreover {
  5677     have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp
  5678     also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
  5679       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  5680       using x unfolding mem_interval using i apply simp
  5681       using y unfolding mem_interval using i apply simp
  5682       done
  5683     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto
  5684     } ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" by auto }
  5685   thus ?thesis unfolding mem_interval by auto
  5686 qed
  5687 
  5688 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
  5689   assumes "{a<..<b} \<noteq> {}"
  5690   shows "closure {a<..<b} = {a .. b}"
  5691 proof-
  5692   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
  5693   let ?c = "(1 / 2) *\<^sub>R (a + b)"
  5694   { fix x assume as:"x \<in> {a .. b}"
  5695     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
  5696     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
  5697       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
  5698       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
  5699         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
  5700         by (auto simp add: algebra_simps)
  5701       hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
  5702       hence False using fn unfolding f_def using xc by auto  }
  5703     moreover
  5704     { assume "\<not> (f ---> x) sequentially"
  5705       { fix e::real assume "e>0"
  5706         hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
  5707         then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  5708         hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
  5709         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }
  5710       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
  5711         unfolding LIMSEQ_def by(auto simp add: dist_norm)
  5712       hence "(f ---> x) sequentially" unfolding f_def
  5713         using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
  5714         using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto  }
  5715     ultimately have "x \<in> closure {a<..<b}"
  5716       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }
  5717   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
  5718 qed
  5719 
  5720 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
  5721   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
  5722 proof-
  5723   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
  5724   def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"
  5725   { fix x assume "x\<in>s"
  5726     fix i :: 'a assume i:"i\<in>Basis"
  5727     hence "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i" using b[THEN bspec[where x=x], OF `x\<in>s`]
  5728       and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }
  5729   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
  5730 qed
  5731 
  5732 lemma bounded_subset_open_interval:
  5733   fixes s :: "('a::ordered_euclidean_space) set"