src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author huffman
Thu Aug 18 17:32:02 2011 -0700 (2011-08-18)
changeset 44286 8766839efb1b
parent 44282 f0de18b62d63
child 44342 8321948340ea
permissions -rw-r--r--
declare euclidean_component_zero[simp] at the point it is proved
     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 SEQ Linear_Algebra "~~/src/HOL/Library/Glbs"
    11 begin
    12 
    13 (* to be moved elsewhere *)
    14 
    15 lemma euclidean_dist_l2:"dist x (y::'a::euclidean_space) = setL2 (\<lambda>i. dist(x$$i) (y$$i)) {..<DIM('a)}"
    16   unfolding dist_norm norm_eq_sqrt_inner setL2_def apply(subst euclidean_inner)
    17   apply(auto simp add:power2_eq_square) unfolding euclidean_component_diff ..
    18 
    19 lemma dist_nth_le: "dist (x $$ i) (y $$ i) \<le> dist x (y::'a::euclidean_space)"
    20   apply(subst(2) euclidean_dist_l2) apply(cases "i<DIM('a)")
    21   apply(rule member_le_setL2) by auto
    22 
    23 subsection {* General notion of a topologies as values *}
    24 
    25 definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
    26 typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
    27   morphisms "openin" "topology"
    28   unfolding istopology_def by blast
    29 
    30 lemma istopology_open_in[intro]: "istopology(openin U)"
    31   using openin[of U] by blast
    32 
    33 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
    34   using topology_inverse[unfolded mem_Collect_eq] .
    35 
    36 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
    37   using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
    38 
    39 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
    40 proof-
    41   {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
    42   moreover
    43   {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
    44     hence "openin T1 = openin T2" by (simp add: fun_eq_iff)
    45     hence "topology (openin T1) = topology (openin T2)" by simp
    46     hence "T1 = T2" unfolding openin_inverse .}
    47   ultimately show ?thesis by blast
    48 qed
    49 
    50 text{* Infer the "universe" from union of all sets in the topology. *}
    51 
    52 definition "topspace T =  \<Union>{S. openin T S}"
    53 
    54 subsubsection {* Main properties of open sets *}
    55 
    56 lemma openin_clauses:
    57   fixes U :: "'a topology"
    58   shows "openin U {}"
    59   "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
    60   "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
    61   using openin[of U] unfolding istopology_def mem_Collect_eq
    62   by fast+
    63 
    64 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
    65   unfolding topspace_def by blast
    66 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
    67 
    68 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
    69   using openin_clauses by simp
    70 
    71 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
    72   using openin_clauses by simp
    73 
    74 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
    75   using openin_Union[of "{S,T}" U] by auto
    76 
    77 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
    78 
    79 lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)" (is "?lhs \<longleftrightarrow> ?rhs")
    80 proof
    81   assume ?lhs then show ?rhs by auto
    82 next
    83   assume H: ?rhs
    84   let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
    85   have "openin U ?t" by (simp add: openin_Union)
    86   also have "?t = S" using H by auto
    87   finally show "openin U S" .
    88 qed
    89 
    90 subsubsection {* Closed sets *}
    91 
    92 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
    93 
    94 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
    95 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
    96 lemma closedin_topspace[intro,simp]:
    97   "closedin U (topspace U)" by (simp add: closedin_def)
    98 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
    99   by (auto simp add: Diff_Un closedin_def)
   100 
   101 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
   102 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
   103   shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
   104 
   105 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
   106   using closedin_Inter[of "{S,T}" U] by auto
   107 
   108 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
   109 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
   110   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
   111   apply (metis openin_subset subset_eq)
   112   done
   113 
   114 lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
   115   by (simp add: openin_closedin_eq)
   116 
   117 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
   118 proof-
   119   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
   120     by (auto simp add: topspace_def openin_subset)
   121   then show ?thesis using oS cT by (auto simp add: closedin_def)
   122 qed
   123 
   124 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
   125 proof-
   126   have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
   127     by (auto simp add: topspace_def )
   128   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
   129 qed
   130 
   131 subsubsection {* Subspace topology *}
   132 
   133 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
   134 
   135 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
   136   (is "istopology ?L")
   137 proof-
   138   have "?L {}" by blast
   139   {fix A B assume A: "?L A" and B: "?L B"
   140     from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
   141     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
   142     then have "?L (A \<inter> B)" by blast}
   143   moreover
   144   {fix K assume K: "K \<subseteq> Collect ?L"
   145     have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
   146       apply (rule set_eqI)
   147       apply (simp add: Ball_def image_iff)
   148       by metis
   149     from K[unfolded th0 subset_image_iff]
   150     obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
   151     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
   152     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)
   153     ultimately have "?L (\<Union>K)" by blast}
   154   ultimately show ?thesis
   155     unfolding subset_eq mem_Collect_eq istopology_def by blast
   156 qed
   157 
   158 lemma openin_subtopology:
   159   "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
   160   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
   161   by auto
   162 
   163 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
   164   by (auto simp add: topspace_def openin_subtopology)
   165 
   166 lemma closedin_subtopology:
   167   "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
   168   unfolding closedin_def topspace_subtopology
   169   apply (simp add: openin_subtopology)
   170   apply (rule iffI)
   171   apply clarify
   172   apply (rule_tac x="topspace U - T" in exI)
   173   by auto
   174 
   175 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
   176   unfolding openin_subtopology
   177   apply (rule iffI, clarify)
   178   apply (frule openin_subset[of U])  apply blast
   179   apply (rule exI[where x="topspace U"])
   180   by auto
   181 
   182 lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
   183   shows "subtopology U V = U"
   184 proof-
   185   {fix S
   186     {fix T assume T: "openin U T" "S = T \<inter> V"
   187       from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
   188       have "openin U S" unfolding eq using T by blast}
   189     moreover
   190     {assume S: "openin U S"
   191       hence "\<exists>T. openin U T \<and> S = T \<inter> V"
   192         using openin_subset[OF S] UV by auto}
   193     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
   194   then show ?thesis unfolding topology_eq openin_subtopology by blast
   195 qed
   196 
   197 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
   198   by (simp add: subtopology_superset)
   199 
   200 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
   201   by (simp add: subtopology_superset)
   202 
   203 subsubsection {* The standard Euclidean topology *}
   204 
   205 definition
   206   euclidean :: "'a::topological_space topology" where
   207   "euclidean = topology open"
   208 
   209 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
   210   unfolding euclidean_def
   211   apply (rule cong[where x=S and y=S])
   212   apply (rule topology_inverse[symmetric])
   213   apply (auto simp add: istopology_def)
   214   done
   215 
   216 lemma topspace_euclidean: "topspace euclidean = UNIV"
   217   apply (simp add: topspace_def)
   218   apply (rule set_eqI)
   219   by (auto simp add: open_openin[symmetric])
   220 
   221 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
   222   by (simp add: topspace_euclidean topspace_subtopology)
   223 
   224 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
   225   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
   226 
   227 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
   228   by (simp add: open_openin openin_subopen[symmetric])
   229 
   230 text {* Basic "localization" results are handy for connectedness. *}
   231 
   232 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
   233   by (auto simp add: openin_subtopology open_openin[symmetric])
   234 
   235 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
   236   by (auto simp add: openin_open)
   237 
   238 lemma open_openin_trans[trans]:
   239  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
   240   by (metis Int_absorb1  openin_open_Int)
   241 
   242 lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
   243   by (auto simp add: openin_open)
   244 
   245 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
   246   by (simp add: closedin_subtopology closed_closedin Int_ac)
   247 
   248 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
   249   by (metis closedin_closed)
   250 
   251 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
   252   apply (subgoal_tac "S \<inter> T = T" )
   253   apply auto
   254   apply (frule closedin_closed_Int[of T S])
   255   by simp
   256 
   257 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
   258   by (auto simp add: closedin_closed)
   259 
   260 lemma openin_euclidean_subtopology_iff:
   261   fixes S U :: "'a::metric_space set"
   262   shows "openin (subtopology euclidean U) S
   263   \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
   264 proof
   265   assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
   266 next
   267   def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
   268   have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
   269     unfolding T_def
   270     apply clarsimp
   271     apply (rule_tac x="d - dist x a" in exI)
   272     apply (clarsimp simp add: less_diff_eq)
   273     apply (erule rev_bexI)
   274     apply (rule_tac x=d in exI, clarify)
   275     apply (erule le_less_trans [OF dist_triangle])
   276     done
   277   assume ?rhs hence 2: "S = U \<inter> T"
   278     unfolding T_def
   279     apply auto
   280     apply (drule (1) bspec, erule rev_bexI)
   281     apply auto
   282     done
   283   from 1 2 show ?lhs
   284     unfolding openin_open open_dist by fast
   285 qed
   286 
   287 text {* These "transitivity" results are handy too *}
   288 
   289 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
   290   \<Longrightarrow> openin (subtopology euclidean U) S"
   291   unfolding open_openin openin_open by blast
   292 
   293 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
   294   by (auto simp add: openin_open intro: openin_trans)
   295 
   296 lemma closedin_trans[trans]:
   297  "closedin (subtopology euclidean T) S \<Longrightarrow>
   298            closedin (subtopology euclidean U) T
   299            ==> closedin (subtopology euclidean U) S"
   300   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
   301 
   302 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
   303   by (auto simp add: closedin_closed intro: closedin_trans)
   304 
   305 
   306 subsection {* Open and closed balls *}
   307 
   308 definition
   309   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   310   "ball x e = {y. dist x y < e}"
   311 
   312 definition
   313   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   314   "cball x e = {y. dist x y \<le> e}"
   315 
   316 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
   317 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
   318 
   319 lemma mem_ball_0 [simp]:
   320   fixes x :: "'a::real_normed_vector"
   321   shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
   322   by (simp add: dist_norm)
   323 
   324 lemma mem_cball_0 [simp]:
   325   fixes x :: "'a::real_normed_vector"
   326   shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
   327   by (simp add: dist_norm)
   328 
   329 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
   330 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
   331 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
   332 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
   333 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
   334   by (simp add: set_eq_iff) arith
   335 
   336 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
   337   by (simp add: set_eq_iff)
   338 
   339 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
   340   "(a::real) - b < 0 \<longleftrightarrow> a < b"
   341   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
   342 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
   343   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
   344 
   345 lemma open_ball[intro, simp]: "open (ball x e)"
   346   unfolding open_dist ball_def mem_Collect_eq Ball_def
   347   unfolding dist_commute
   348   apply clarify
   349   apply (rule_tac x="e - dist xa x" in exI)
   350   using dist_triangle_alt[where z=x]
   351   apply (clarsimp simp add: diff_less_iff)
   352   apply atomize
   353   apply (erule_tac x="y" in allE)
   354   apply (erule_tac x="xa" in allE)
   355   by arith
   356 
   357 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
   358 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
   359   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
   360 
   361 lemma openE[elim?]:
   362   assumes "open S" "x\<in>S" 
   363   obtains e where "e>0" "ball x e \<subseteq> S"
   364   using assms unfolding open_contains_ball by auto
   365 
   366 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
   367   by (metis open_contains_ball subset_eq centre_in_ball)
   368 
   369 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
   370   unfolding mem_ball set_eq_iff
   371   apply (simp add: not_less)
   372   by (metis zero_le_dist order_trans dist_self)
   373 
   374 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
   375 
   376 
   377 subsection{* Connectedness *}
   378 
   379 definition "connected S \<longleftrightarrow>
   380   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
   381   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
   382 
   383 lemma connected_local:
   384  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
   385                  openin (subtopology euclidean S) e1 \<and>
   386                  openin (subtopology euclidean S) e2 \<and>
   387                  S \<subseteq> e1 \<union> e2 \<and>
   388                  e1 \<inter> e2 = {} \<and>
   389                  ~(e1 = {}) \<and>
   390                  ~(e2 = {}))"
   391 unfolding connected_def openin_open by (safe, blast+)
   392 
   393 lemma exists_diff:
   394   fixes P :: "'a set \<Rightarrow> bool"
   395   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
   396 proof-
   397   {assume "?lhs" hence ?rhs by blast }
   398   moreover
   399   {fix S assume H: "P S"
   400     have "S = - (- S)" by auto
   401     with H have "P (- (- S))" by metis }
   402   ultimately show ?thesis by metis
   403 qed
   404 
   405 lemma connected_clopen: "connected S \<longleftrightarrow>
   406         (\<forall>T. openin (subtopology euclidean S) T \<and>
   407             closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
   408 proof-
   409   have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
   410     unfolding connected_def openin_open closedin_closed
   411     apply (subst exists_diff) by blast
   412   hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
   413     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
   414 
   415   have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
   416     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
   417     unfolding connected_def openin_open closedin_closed by auto
   418   {fix e2
   419     {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
   420         by auto}
   421     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
   422   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
   423   then show ?thesis unfolding th0 th1 by simp
   424 qed
   425 
   426 lemma connected_empty[simp, intro]: "connected {}"
   427   by (simp add: connected_def)
   428 
   429 
   430 subsection{* Limit points *}
   431 
   432 definition (in topological_space)
   433   islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
   434   "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
   435 
   436 lemma islimptI:
   437   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
   438   shows "x islimpt S"
   439   using assms unfolding islimpt_def by auto
   440 
   441 lemma islimptE:
   442   assumes "x islimpt S" and "x \<in> T" and "open T"
   443   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
   444   using assms unfolding islimpt_def by auto
   445 
   446 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
   447 
   448 lemma islimpt_approachable:
   449   fixes x :: "'a::metric_space"
   450   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
   451   unfolding islimpt_def
   452   apply auto
   453   apply(erule_tac x="ball x e" in allE)
   454   apply auto
   455   apply(rule_tac x=y in bexI)
   456   apply (auto simp add: dist_commute)
   457   apply (simp add: open_dist, drule (1) bspec)
   458   apply (clarify, drule spec, drule (1) mp, auto)
   459   done
   460 
   461 lemma islimpt_approachable_le:
   462   fixes x :: "'a::metric_space"
   463   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
   464   unfolding islimpt_approachable
   465   using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
   466   by metis 
   467 
   468 text {* A perfect space has no isolated points. *}
   469 
   470 class perfect_space = topological_space +
   471   assumes islimpt_UNIV [simp, intro]: "x islimpt UNIV"
   472 
   473 lemma perfect_choose_dist:
   474   fixes x :: "'a::{perfect_space, metric_space}"
   475   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
   476 using islimpt_UNIV [of x]
   477 by (simp add: islimpt_approachable)
   478 
   479 instance euclidean_space \<subseteq> perfect_space
   480 proof
   481   fix x :: 'a
   482   { fix e :: real assume "0 < e"
   483     def y \<equiv> "x + scaleR (e/2) (sgn (basis 0))"
   484     from `0 < e` have "y \<noteq> x"
   485       unfolding y_def by (simp add: sgn_zero_iff DIM_positive)
   486     from `0 < e` have "dist y x < e"
   487       unfolding y_def by (simp add: dist_norm norm_sgn)
   488     from `y \<noteq> x` and `dist y x < e`
   489     have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
   490   }
   491   then show "x islimpt UNIV" unfolding islimpt_approachable by blast
   492 qed
   493 
   494 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
   495   unfolding closed_def
   496   apply (subst open_subopen)
   497   apply (simp add: islimpt_def subset_eq)
   498   by (metis ComplE ComplI)
   499 
   500 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
   501   unfolding islimpt_def by auto
   502 
   503 lemma finite_set_avoid:
   504   fixes a :: "'a::metric_space"
   505   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
   506 proof(induct rule: finite_induct[OF fS])
   507   case 1 thus ?case by (auto intro: zero_less_one)
   508 next
   509   case (2 x F)
   510   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
   511   {assume "x = a" hence ?case using d by auto  }
   512   moreover
   513   {assume xa: "x\<noteq>a"
   514     let ?d = "min d (dist a x)"
   515     have dp: "?d > 0" using xa d(1) using dist_nz by auto
   516     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
   517     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
   518   ultimately show ?case by blast
   519 qed
   520 
   521 lemma islimpt_finite:
   522   fixes S :: "'a::metric_space set"
   523   assumes fS: "finite S" shows "\<not> a islimpt S"
   524   unfolding islimpt_approachable
   525   using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
   526 
   527 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
   528   apply (rule iffI)
   529   defer
   530   apply (metis Un_upper1 Un_upper2 islimpt_subset)
   531   unfolding islimpt_def
   532   apply (rule ccontr, clarsimp, rename_tac A B)
   533   apply (drule_tac x="A \<inter> B" in spec)
   534   apply (auto simp add: open_Int)
   535   done
   536 
   537 lemma discrete_imp_closed:
   538   fixes S :: "'a::metric_space set"
   539   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
   540   shows "closed S"
   541 proof-
   542   {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
   543     from e have e2: "e/2 > 0" by arith
   544     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
   545     let ?m = "min (e/2) (dist x y) "
   546     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
   547     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
   548     have th: "dist z y < e" using z y
   549       by (intro dist_triangle_lt [where z=x], simp)
   550     from d[rule_format, OF y(1) z(1) th] y z
   551     have False by (auto simp add: dist_commute)}
   552   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
   553 qed
   554 
   555 
   556 subsection {* Interior of a Set *}
   557 
   558 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
   559 
   560 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
   561   apply (simp add: set_eq_iff interior_def)
   562   apply (subst (2) open_subopen) by (safe, blast+)
   563 
   564 lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
   565 
   566 lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
   567 
   568 lemma open_interior[simp, intro]: "open(interior S)"
   569   apply (simp add: interior_def)
   570   apply (subst open_subopen) by blast
   571 
   572 lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
   573 lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
   574 lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
   575 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
   576 lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T  \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"
   577   by (metis equalityI interior_maximal interior_subset open_interior)
   578 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
   579   apply (simp add: interior_def)
   580   by (metis open_contains_ball centre_in_ball open_ball subset_trans)
   581 
   582 lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
   583   by (metis interior_maximal interior_subset subset_trans)
   584 
   585 lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
   586   apply (rule equalityI, simp)
   587   apply (metis Int_lower1 Int_lower2 subset_interior)
   588   by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
   589 
   590 lemma interior_limit_point [intro]:
   591   fixes x :: "'a::perfect_space"
   592   assumes x: "x \<in> interior S" shows "x islimpt S"
   593   using x islimpt_UNIV [of x]
   594   unfolding interior_def islimpt_def
   595   apply (clarsimp, rename_tac T T')
   596   apply (drule_tac x="T \<inter> T'" in spec)
   597   apply (auto simp add: open_Int)
   598   done
   599 
   600 lemma interior_closed_Un_empty_interior:
   601   assumes cS: "closed S" and iT: "interior T = {}"
   602   shows "interior(S \<union> T) = interior S"
   603 proof
   604   show "interior S \<subseteq> interior (S\<union>T)"
   605     by (rule subset_interior, blast)
   606 next
   607   show "interior (S \<union> T) \<subseteq> interior S"
   608   proof
   609     fix x assume "x \<in> interior (S \<union> T)"
   610     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
   611       unfolding interior_def by fast
   612     show "x \<in> interior S"
   613     proof (rule ccontr)
   614       assume "x \<notin> interior S"
   615       with `x \<in> R` `open R` obtain y where "y \<in> R - S"
   616         unfolding interior_def set_eq_iff by fast
   617       from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
   618       from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
   619       from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
   620       show "False" unfolding interior_def by fast
   621     qed
   622   qed
   623 qed
   624 
   625 
   626 subsection {* Closure of a Set *}
   627 
   628 definition "closure S = S \<union> {x | x. x islimpt S}"
   629 
   630 lemma closure_interior: "closure S = - interior (- S)"
   631 proof-
   632   { fix x
   633     have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
   634     proof
   635       let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"
   636       assume "?lhs"
   637       hence *:"\<not> ?exT x"
   638         unfolding interior_def
   639         by simp
   640       { assume "\<not> ?rhs"
   641         hence False using *
   642           unfolding closure_def islimpt_def
   643           by blast
   644       }
   645       thus "?rhs"
   646         by blast
   647     next
   648       assume "?rhs" thus "?lhs"
   649         unfolding closure_def interior_def islimpt_def
   650         by blast
   651     qed
   652   }
   653   thus ?thesis
   654     by blast
   655 qed
   656 
   657 lemma interior_closure: "interior S = - (closure (- S))"
   658 proof-
   659   { fix x
   660     have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"
   661       unfolding interior_def closure_def islimpt_def
   662       by auto
   663   }
   664   thus ?thesis
   665     by blast
   666 qed
   667 
   668 lemma closed_closure[simp, intro]: "closed (closure S)"
   669 proof-
   670   have "closed (- interior (-S))" by blast
   671   thus ?thesis using closure_interior[of S] by simp
   672 qed
   673 
   674 lemma closure_hull: "closure S = closed hull S"
   675 proof-
   676   have "S \<subseteq> closure S"
   677     unfolding closure_def
   678     by blast
   679   moreover
   680   have "closed (closure S)"
   681     using closed_closure[of S]
   682     by assumption
   683   moreover
   684   { fix t
   685     assume *:"S \<subseteq> t" "closed t"
   686     { fix x
   687       assume "x islimpt S"
   688       hence "x islimpt t" using *(1)
   689         using islimpt_subset[of x, of S, of t]
   690         by blast
   691     }
   692     with * have "closure S \<subseteq> t"
   693       unfolding closure_def
   694       using closed_limpt[of t]
   695       by auto
   696   }
   697   ultimately show ?thesis
   698     using hull_unique[of S, of "closure S", of closed]
   699     by simp
   700 qed
   701 
   702 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
   703   unfolding closure_hull
   704   using hull_eq[of closed, OF  closed_Inter, of S]
   705   by metis
   706 
   707 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
   708   using closure_eq[of S]
   709   by simp
   710 
   711 lemma closure_closure[simp]: "closure (closure S) = closure S"
   712   unfolding closure_hull
   713   using hull_hull[of closed S]
   714   by assumption
   715 
   716 lemma closure_subset: "S \<subseteq> closure S"
   717   unfolding closure_hull
   718   using hull_subset[of S closed]
   719   by assumption
   720 
   721 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
   722   unfolding closure_hull
   723   using hull_mono[of S T closed]
   724   by assumption
   725 
   726 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
   727   using hull_minimal[of S T closed]
   728   unfolding closure_hull
   729   by simp
   730 
   731 lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"
   732   using hull_unique[of S T closed]
   733   unfolding closure_hull
   734   by simp
   735 
   736 lemma closure_empty[simp]: "closure {} = {}"
   737   using closed_empty closure_closed[of "{}"]
   738   by simp
   739 
   740 lemma closure_univ[simp]: "closure UNIV = UNIV"
   741   using closure_closed[of UNIV]
   742   by simp
   743 
   744 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
   745   using closure_empty closure_subset[of S]
   746   by blast
   747 
   748 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
   749   using closure_eq[of S] closure_subset[of S]
   750   by simp
   751 
   752 lemma open_inter_closure_eq_empty:
   753   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
   754   using open_subset_interior[of S "- T"]
   755   using interior_subset[of "- T"]
   756   unfolding closure_interior
   757   by auto
   758 
   759 lemma open_inter_closure_subset:
   760   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
   761 proof
   762   fix x
   763   assume as: "open S" "x \<in> S \<inter> closure T"
   764   { assume *:"x islimpt T"
   765     have "x islimpt (S \<inter> T)"
   766     proof (rule islimptI)
   767       fix A
   768       assume "x \<in> A" "open A"
   769       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
   770         by (simp_all add: open_Int)
   771       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
   772         by (rule islimptE)
   773       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
   774         by simp_all
   775       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
   776     qed
   777   }
   778   then show "x \<in> closure (S \<inter> T)" using as
   779     unfolding closure_def
   780     by blast
   781 qed
   782 
   783 lemma closure_complement: "closure(- S) = - interior(S)"
   784 proof-
   785   have "S = - (- S)"
   786     by auto
   787   thus ?thesis
   788     unfolding closure_interior
   789     by auto
   790 qed
   791 
   792 lemma interior_complement: "interior(- S) = - closure(S)"
   793   unfolding closure_interior
   794   by blast
   795 
   796 
   797 subsection {* Frontier (aka boundary) *}
   798 
   799 definition "frontier S = closure S - interior S"
   800 
   801 lemma frontier_closed: "closed(frontier S)"
   802   by (simp add: frontier_def closed_Diff)
   803 
   804 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
   805   by (auto simp add: frontier_def interior_closure)
   806 
   807 lemma frontier_straddle:
   808   fixes a :: "'a::metric_space"
   809   shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))" (is "?lhs \<longleftrightarrow> ?rhs")
   810 proof
   811   assume "?lhs"
   812   { fix e::real
   813     assume "e > 0"
   814     let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
   815     { assume "a\<in>S"
   816       have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
   817       moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
   818         unfolding frontier_closures closure_def islimpt_def using `e>0`
   819         by (auto, erule_tac x="ball a e" in allE, auto)
   820       ultimately have ?rhse by auto
   821     }
   822     moreover
   823     { assume "a\<notin>S"
   824       hence ?rhse using `?lhs`
   825         unfolding frontier_closures closure_def islimpt_def
   826         using open_ball[of a e] `e > 0`
   827           by simp (metis centre_in_ball mem_ball open_ball) 
   828     }
   829     ultimately have ?rhse by auto
   830   }
   831   thus ?rhs by auto
   832 next
   833   assume ?rhs
   834   moreover
   835   { fix T assume "a\<notin>S" and
   836     as:"\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)" "a \<notin> S" "a \<in> T" "open T"
   837     from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
   838     then obtain e where "e>0" "ball a e \<subseteq> T" by auto
   839     then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
   840     have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
   841       using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
   842   }
   843   hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
   844   moreover
   845   { fix T assume "a \<in> T"  "open T" "a\<in>S"
   846     then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
   847     obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
   848     hence "\<exists>y\<in>- S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
   849   }
   850   hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
   851   ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
   852 qed
   853 
   854 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
   855   by (metis frontier_def closure_closed Diff_subset)
   856 
   857 lemma frontier_empty[simp]: "frontier {} = {}"
   858   by (simp add: frontier_def)
   859 
   860 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
   861 proof-
   862   { assume "frontier S \<subseteq> S"
   863     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
   864     hence "closed S" using closure_subset_eq by auto
   865   }
   866   thus ?thesis using frontier_subset_closed[of S] ..
   867 qed
   868 
   869 lemma frontier_complement: "frontier(- S) = frontier S"
   870   by (auto simp add: frontier_def closure_complement interior_complement)
   871 
   872 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
   873   using frontier_complement frontier_subset_eq[of "- S"]
   874   unfolding open_closed by auto
   875 
   876 
   877 subsection {* Filters and the ``eventually true'' quantifier *}
   878 
   879 definition
   880   at_infinity :: "'a::real_normed_vector filter" where
   881   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
   882 
   883 definition
   884   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
   885     (infixr "indirection" 70) where
   886   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
   887 
   888 text{* Prove That They are all filters. *}
   889 
   890 lemma eventually_at_infinity:
   891   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
   892 unfolding at_infinity_def
   893 proof (rule eventually_Abs_filter, rule is_filter.intro)
   894   fix P Q :: "'a \<Rightarrow> bool"
   895   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
   896   then obtain r s where
   897     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
   898   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
   899   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
   900 qed auto
   901 
   902 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
   903 
   904 lemma trivial_limit_within:
   905   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
   906 proof
   907   assume "trivial_limit (at a within S)"
   908   thus "\<not> a islimpt S"
   909     unfolding trivial_limit_def
   910     unfolding eventually_within eventually_at_topological
   911     unfolding islimpt_def
   912     apply (clarsimp simp add: set_eq_iff)
   913     apply (rename_tac T, rule_tac x=T in exI)
   914     apply (clarsimp, drule_tac x=y in bspec, simp_all)
   915     done
   916 next
   917   assume "\<not> a islimpt S"
   918   thus "trivial_limit (at a within S)"
   919     unfolding trivial_limit_def
   920     unfolding eventually_within eventually_at_topological
   921     unfolding islimpt_def
   922     apply clarsimp
   923     apply (rule_tac x=T in exI)
   924     apply auto
   925     done
   926 qed
   927 
   928 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
   929   using trivial_limit_within [of a UNIV]
   930   by (simp add: within_UNIV)
   931 
   932 lemma trivial_limit_at:
   933   fixes a :: "'a::perfect_space"
   934   shows "\<not> trivial_limit (at a)"
   935   by (simp add: trivial_limit_at_iff)
   936 
   937 lemma trivial_limit_at_infinity:
   938   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
   939   unfolding trivial_limit_def eventually_at_infinity
   940   apply clarsimp
   941   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
   942    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
   943   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
   944   apply (drule_tac x=UNIV in spec, simp)
   945   done
   946 
   947 text {* Some property holds "sufficiently close" to the limit point. *}
   948 
   949 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
   950   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
   951 unfolding eventually_at dist_nz by auto
   952 
   953 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
   954         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
   955 unfolding eventually_within eventually_at dist_nz by auto
   956 
   957 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
   958         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
   959 unfolding eventually_within
   960 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl) 
   961 
   962 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
   963   unfolding trivial_limit_def
   964   by (auto elim: eventually_rev_mp)
   965 
   966 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
   967   unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
   968 
   969 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
   970   unfolding trivial_limit_def ..
   971 
   972 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
   973   apply (safe elim!: trivial_limit_eventually)
   974   apply (simp add: eventually_False [symmetric])
   975   done
   976 
   977 text{* Combining theorems for "eventually" *}
   978 
   979 lemma eventually_rev_mono:
   980   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
   981 using eventually_mono [of P Q] by fast
   982 
   983 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
   984   by (simp add: eventually_False)
   985 
   986 
   987 subsection {* Limits *}
   988 
   989 text{* Notation Lim to avoid collition with lim defined in analysis *}
   990 
   991 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
   992   where "Lim A f = (THE l. (f ---> l) A)"
   993 
   994 lemma Lim:
   995  "(f ---> l) net \<longleftrightarrow>
   996         trivial_limit net \<or>
   997         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
   998   unfolding tendsto_iff trivial_limit_eq by auto
   999 
  1000 text{* Show that they yield usual definitions in the various cases. *}
  1001 
  1002 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
  1003            (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
  1004   by (auto simp add: tendsto_iff eventually_within_le)
  1005 
  1006 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
  1007         (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
  1008   by (auto simp add: tendsto_iff eventually_within)
  1009 
  1010 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
  1011         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
  1012   by (auto simp add: tendsto_iff eventually_at)
  1013 
  1014 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
  1015   unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
  1016 
  1017 lemma Lim_at_infinity:
  1018   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
  1019   by (auto simp add: tendsto_iff eventually_at_infinity)
  1020 
  1021 lemma Lim_sequentially:
  1022  "(S ---> l) sequentially \<longleftrightarrow>
  1023           (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
  1024   by (rule LIMSEQ_def) (* FIXME: redundant *)
  1025 
  1026 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
  1027   by (rule topological_tendstoI, auto elim: eventually_rev_mono)
  1028 
  1029 text{* The expected monotonicity property. *}
  1030 
  1031 lemma Lim_within_empty: "(f ---> l) (net within {})"
  1032   unfolding tendsto_def Limits.eventually_within by simp
  1033 
  1034 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
  1035   unfolding tendsto_def Limits.eventually_within
  1036   by (auto elim!: eventually_elim1)
  1037 
  1038 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
  1039   shows "(f ---> l) (net within (S \<union> T))"
  1040   using assms unfolding tendsto_def Limits.eventually_within
  1041   apply clarify
  1042   apply (drule spec, drule (1) mp, drule (1) mp)
  1043   apply (drule spec, drule (1) mp, drule (1) mp)
  1044   apply (auto elim: eventually_elim2)
  1045   done
  1046 
  1047 lemma Lim_Un_univ:
  1048  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
  1049         ==> (f ---> l) net"
  1050   by (metis Lim_Un within_UNIV)
  1051 
  1052 text{* Interrelations between restricted and unrestricted limits. *}
  1053 
  1054 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
  1055   (* FIXME: rename *)
  1056   unfolding tendsto_def Limits.eventually_within
  1057   apply (clarify, drule spec, drule (1) mp, drule (1) mp)
  1058   by (auto elim!: eventually_elim1)
  1059 
  1060 lemma eventually_within_interior:
  1061   assumes "x \<in> interior S"
  1062   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
  1063 proof-
  1064   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
  1065     unfolding interior_def by fast
  1066   { assume "?lhs"
  1067     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
  1068       unfolding Limits.eventually_within Limits.eventually_at_topological
  1069       by auto
  1070     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
  1071       by auto
  1072     then have "?rhs"
  1073       unfolding Limits.eventually_at_topological by auto
  1074   } moreover
  1075   { assume "?rhs" hence "?lhs"
  1076       unfolding Limits.eventually_within
  1077       by (auto elim: eventually_elim1)
  1078   } ultimately
  1079   show "?thesis" ..
  1080 qed
  1081 
  1082 lemma at_within_interior:
  1083   "x \<in> interior S \<Longrightarrow> at x within S = at x"
  1084   by (simp add: filter_eq_iff eventually_within_interior)
  1085 
  1086 lemma at_within_open:
  1087   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
  1088   by (simp only: at_within_interior interior_open)
  1089 
  1090 lemma Lim_within_open:
  1091   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
  1092   assumes"a \<in> S" "open S"
  1093   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
  1094   using assms by (simp only: at_within_open)
  1095 
  1096 lemma Lim_within_LIMSEQ:
  1097   fixes a :: real and L :: "'a::metric_space"
  1098   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
  1099   shows "(X ---> L) (at a within T)"
  1100 proof (rule ccontr)
  1101   assume "\<not> (X ---> L) (at a within T)"
  1102   hence "\<exists>r>0. \<forall>s>0. \<exists>x\<in>T. x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> r \<le> dist (X x) L"
  1103     unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
  1104   then obtain r where r: "r > 0" "\<And>s. s > 0 \<Longrightarrow> \<exists>x\<in>T-{a}. \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r" by blast
  1105 
  1106   let ?F = "\<lambda>n::nat. SOME x. x \<in> T \<and> x \<noteq> a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
  1107   have "\<And>n. \<exists>x. x \<in> T \<and> x \<noteq> a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
  1108     using r by (simp add: Bex_def)
  1109   hence F: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"
  1110     by (rule someI_ex)
  1111   hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
  1112     and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
  1113     and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
  1114     by fast+
  1115 
  1116   have "?F ----> a"
  1117   proof (rule LIMSEQ_I, unfold real_norm_def)
  1118       fix e::real
  1119       assume "0 < e"
  1120         (* choose no such that inverse (real (Suc n)) < e *)
  1121       then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
  1122       then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
  1123       show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
  1124       proof (intro exI allI impI)
  1125         fix n
  1126         assume mlen: "m \<le> n"
  1127         have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
  1128           by (rule F2)
  1129         also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
  1130           using mlen by auto
  1131         also from nodef have
  1132           "inverse (real (Suc m)) < e" .
  1133         finally show "\<bar>?F n - a\<bar> < e" .
  1134       qed
  1135   qed
  1136   moreover note `\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
  1137   ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
  1138   
  1139   moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
  1140   proof -
  1141     {
  1142       fix no::nat
  1143       obtain n where "n = no + 1" by simp
  1144       then have nolen: "no \<le> n" by simp
  1145         (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
  1146       have "dist (X (?F n)) L \<ge> r"
  1147         by (rule F3)
  1148       with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
  1149     }
  1150     then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
  1151     with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
  1152     thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
  1153   qed
  1154   ultimately show False by simp
  1155 qed
  1156 
  1157 lemma Lim_right_bound:
  1158   fixes f :: "real \<Rightarrow> real"
  1159   assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
  1160   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
  1161   shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
  1162 proof cases
  1163   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
  1164 next
  1165   assume [simp]: "{x<..} \<inter> I \<noteq> {}"
  1166   show ?thesis
  1167   proof (rule Lim_within_LIMSEQ, safe)
  1168     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
  1169     
  1170     show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
  1171     proof (rule LIMSEQ_I, rule ccontr)
  1172       fix r :: real assume "0 < r"
  1173       with Inf_close[of "f ` ({x<..} \<inter> I)" r]
  1174       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
  1175       from `x < y` have "0 < y - x" by auto
  1176       from S(2)[THEN LIMSEQ_D, OF this]
  1177       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
  1178       
  1179       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
  1180       moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1181         using S bnd by (intro Inf_lower[where z=K]) auto
  1182       ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1183         by (auto simp: not_less field_simps)
  1184       with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
  1185       show False by auto
  1186     qed
  1187   qed
  1188 qed
  1189 
  1190 text{* Another limit point characterization. *}
  1191 
  1192 lemma islimpt_sequential:
  1193   fixes x :: "'a::metric_space"
  1194   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
  1195     (is "?lhs = ?rhs")
  1196 proof
  1197   assume ?lhs
  1198   then obtain f where f:"\<forall>y. y>0 \<longrightarrow> f y \<in> S \<and> f y \<noteq> x \<and> dist (f y) x < y"
  1199     unfolding islimpt_approachable using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto
  1200   { fix n::nat
  1201     have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
  1202   }
  1203   moreover
  1204   { fix e::real assume "e>0"
  1205     hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
  1206     then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  1207     hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
  1208     moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
  1209     ultimately have "\<exists>N::nat. \<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < e" apply(rule_tac x=N in exI) apply auto apply(erule_tac x=n in allE)+ by auto
  1210   }
  1211   hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
  1212     unfolding Lim_sequentially using f by auto
  1213   ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
  1214 next
  1215   assume ?rhs
  1216   then obtain f::"nat\<Rightarrow>'a"  where f:"(\<forall>n. f n \<in> S - {x})" "(\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f n) x < e)" unfolding Lim_sequentially by auto
  1217   { fix e::real assume "e>0"
  1218     then obtain N where "dist (f N) x < e" using f(2) by auto
  1219     moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
  1220     ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
  1221   }
  1222   thus ?lhs unfolding islimpt_approachable by auto
  1223 qed
  1224 
  1225 lemma Lim_inv: (* TODO: delete *)
  1226   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
  1227   assumes "(f ---> l) A" and "l \<noteq> 0"
  1228   shows "((inverse o f) ---> inverse l) A"
  1229   unfolding o_def using assms by (rule tendsto_inverse)
  1230 
  1231 lemma Lim_null:
  1232   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1233   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
  1234   by (simp add: Lim dist_norm)
  1235 
  1236 lemma Lim_null_comparison:
  1237   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1238   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
  1239   shows "(f ---> 0) net"
  1240 proof (rule metric_tendsto_imp_tendsto)
  1241   show "(g ---> 0) net" by fact
  1242   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
  1243     using assms(1) by (rule eventually_elim1, simp add: dist_norm)
  1244 qed
  1245 
  1246 lemma Lim_transform_bound:
  1247   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1248   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
  1249   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
  1250   shows "(f ---> 0) net"
  1251   using assms(1) tendsto_norm_zero [OF assms(2)]
  1252   by (rule Lim_null_comparison)
  1253 
  1254 text{* Deducing things about the limit from the elements. *}
  1255 
  1256 lemma Lim_in_closed_set:
  1257   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
  1258   shows "l \<in> S"
  1259 proof (rule ccontr)
  1260   assume "l \<notin> S"
  1261   with `closed S` have "open (- S)" "l \<in> - S"
  1262     by (simp_all add: open_Compl)
  1263   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
  1264     by (rule topological_tendstoD)
  1265   with assms(2) have "eventually (\<lambda>x. False) net"
  1266     by (rule eventually_elim2) simp
  1267   with assms(3) show "False"
  1268     by (simp add: eventually_False)
  1269 qed
  1270 
  1271 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
  1272 
  1273 lemma Lim_dist_ubound:
  1274   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
  1275   shows "dist a l <= e"
  1276 proof-
  1277   have "dist a l \<in> {..e}"
  1278   proof (rule Lim_in_closed_set)
  1279     show "closed {..e}" by simp
  1280     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
  1281     show "\<not> trivial_limit net" by fact
  1282     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
  1283   qed
  1284   thus ?thesis by simp
  1285 qed
  1286 
  1287 lemma Lim_norm_ubound:
  1288   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1289   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
  1290   shows "norm(l) <= e"
  1291 proof-
  1292   have "norm l \<in> {..e}"
  1293   proof (rule Lim_in_closed_set)
  1294     show "closed {..e}" by simp
  1295     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
  1296     show "\<not> trivial_limit net" by fact
  1297     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1298   qed
  1299   thus ?thesis by simp
  1300 qed
  1301 
  1302 lemma Lim_norm_lbound:
  1303   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1304   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
  1305   shows "e \<le> norm l"
  1306 proof-
  1307   have "norm l \<in> {e..}"
  1308   proof (rule Lim_in_closed_set)
  1309     show "closed {e..}" by simp
  1310     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
  1311     show "\<not> trivial_limit net" by fact
  1312     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1313   qed
  1314   thus ?thesis by simp
  1315 qed
  1316 
  1317 text{* Uniqueness of the limit, when nontrivial. *}
  1318 
  1319 lemma tendsto_Lim:
  1320   fixes f :: "'a \<Rightarrow> 'b::t2_space"
  1321   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
  1322   unfolding Lim_def using tendsto_unique[of net f] by auto
  1323 
  1324 text{* Limit under bilinear function *}
  1325 
  1326 lemma Lim_bilinear:
  1327   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
  1328   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
  1329 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
  1330 by (rule bounded_bilinear.tendsto)
  1331 
  1332 text{* These are special for limits out of the same vector space. *}
  1333 
  1334 lemma Lim_within_id: "(id ---> a) (at a within s)"
  1335   unfolding tendsto_def Limits.eventually_within eventually_at_topological
  1336   by auto
  1337 
  1338 lemma Lim_at_id: "(id ---> a) (at a)"
  1339 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
  1340 
  1341 lemma Lim_at_zero:
  1342   fixes a :: "'a::real_normed_vector"
  1343   fixes l :: "'b::topological_space"
  1344   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
  1345   using LIM_offset_zero LIM_offset_zero_cancel ..
  1346 
  1347 text{* It's also sometimes useful to extract the limit point from the filter. *}
  1348 
  1349 definition
  1350   netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
  1351   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
  1352 
  1353 lemma netlimit_within:
  1354   assumes "\<not> trivial_limit (at a within S)"
  1355   shows "netlimit (at a within S) = a"
  1356 unfolding netlimit_def
  1357 apply (rule some_equality)
  1358 apply (rule Lim_at_within)
  1359 apply (rule LIM_ident)
  1360 apply (erule tendsto_unique [OF assms])
  1361 apply (rule Lim_at_within)
  1362 apply (rule LIM_ident)
  1363 done
  1364 
  1365 lemma netlimit_at:
  1366   fixes a :: "'a::{perfect_space,t2_space}"
  1367   shows "netlimit (at a) = a"
  1368   apply (subst within_UNIV[symmetric])
  1369   using netlimit_within[of a UNIV]
  1370   by (simp add: trivial_limit_at within_UNIV)
  1371 
  1372 lemma lim_within_interior:
  1373   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
  1374   by (simp add: at_within_interior)
  1375 
  1376 lemma netlimit_within_interior:
  1377   fixes x :: "'a::{t2_space,perfect_space}"
  1378   assumes "x \<in> interior S"
  1379   shows "netlimit (at x within S) = x"
  1380 using assms by (simp add: at_within_interior netlimit_at)
  1381 
  1382 text{* Transformation of limit. *}
  1383 
  1384 lemma Lim_transform:
  1385   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
  1386   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
  1387   shows "(g ---> l) net"
  1388   using tendsto_diff [OF assms(2) assms(1)] by simp
  1389 
  1390 lemma Lim_transform_eventually:
  1391   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
  1392   apply (rule topological_tendstoI)
  1393   apply (drule (2) topological_tendstoD)
  1394   apply (erule (1) eventually_elim2, simp)
  1395   done
  1396 
  1397 lemma Lim_transform_within:
  1398   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1399   and "(f ---> l) (at x within S)"
  1400   shows "(g ---> l) (at x within S)"
  1401 proof (rule Lim_transform_eventually)
  1402   show "eventually (\<lambda>x. f x = g x) (at x within S)"
  1403     unfolding eventually_within
  1404     using assms(1,2) by auto
  1405   show "(f ---> l) (at x within S)" by fact
  1406 qed
  1407 
  1408 lemma Lim_transform_at:
  1409   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1410   and "(f ---> l) (at x)"
  1411   shows "(g ---> l) (at x)"
  1412 proof (rule Lim_transform_eventually)
  1413   show "eventually (\<lambda>x. f x = g x) (at x)"
  1414     unfolding eventually_at
  1415     using assms(1,2) by auto
  1416   show "(f ---> l) (at x)" by fact
  1417 qed
  1418 
  1419 text{* Common case assuming being away from some crucial point like 0. *}
  1420 
  1421 lemma Lim_transform_away_within:
  1422   fixes a b :: "'a::t1_space"
  1423   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1424   and "(f ---> l) (at a within S)"
  1425   shows "(g ---> l) (at a within S)"
  1426 proof (rule Lim_transform_eventually)
  1427   show "(f ---> l) (at a within S)" by fact
  1428   show "eventually (\<lambda>x. f x = g x) (at a within S)"
  1429     unfolding Limits.eventually_within eventually_at_topological
  1430     by (rule exI [where x="- {b}"], simp add: open_Compl assms)
  1431 qed
  1432 
  1433 lemma Lim_transform_away_at:
  1434   fixes a b :: "'a::t1_space"
  1435   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1436   and fl: "(f ---> l) (at a)"
  1437   shows "(g ---> l) (at a)"
  1438   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
  1439   by (auto simp add: within_UNIV)
  1440 
  1441 text{* Alternatively, within an open set. *}
  1442 
  1443 lemma Lim_transform_within_open:
  1444   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
  1445   and "(f ---> l) (at a)"
  1446   shows "(g ---> l) (at a)"
  1447 proof (rule Lim_transform_eventually)
  1448   show "eventually (\<lambda>x. f x = g x) (at a)"
  1449     unfolding eventually_at_topological
  1450     using assms(1,2,3) by auto
  1451   show "(f ---> l) (at a)" by fact
  1452 qed
  1453 
  1454 text{* A congruence rule allowing us to transform limits assuming not at point. *}
  1455 
  1456 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
  1457 
  1458 lemma Lim_cong_within(*[cong add]*):
  1459   assumes "a = b" "x = y" "S = T"
  1460   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
  1461   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
  1462   unfolding tendsto_def Limits.eventually_within eventually_at_topological
  1463   using assms by simp
  1464 
  1465 lemma Lim_cong_at(*[cong add]*):
  1466   assumes "a = b" "x = y"
  1467   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
  1468   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
  1469   unfolding tendsto_def eventually_at_topological
  1470   using assms by simp
  1471 
  1472 text{* Useful lemmas on closure and set of possible sequential limits.*}
  1473 
  1474 lemma closure_sequential:
  1475   fixes l :: "'a::metric_space"
  1476   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
  1477 proof
  1478   assume "?lhs" moreover
  1479   { assume "l \<in> S"
  1480     hence "?rhs" using tendsto_const[of l sequentially] by auto
  1481   } moreover
  1482   { assume "l islimpt S"
  1483     hence "?rhs" unfolding islimpt_sequential by auto
  1484   } ultimately
  1485   show "?rhs" unfolding closure_def by auto
  1486 next
  1487   assume "?rhs"
  1488   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
  1489 qed
  1490 
  1491 lemma closed_sequential_limits:
  1492   fixes S :: "'a::metric_space set"
  1493   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
  1494   unfolding closed_limpt
  1495   using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
  1496   by metis
  1497 
  1498 lemma closure_approachable:
  1499   fixes S :: "'a::metric_space set"
  1500   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
  1501   apply (auto simp add: closure_def islimpt_approachable)
  1502   by (metis dist_self)
  1503 
  1504 lemma closed_approachable:
  1505   fixes S :: "'a::metric_space set"
  1506   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
  1507   by (metis closure_closed closure_approachable)
  1508 
  1509 text{* Some other lemmas about sequences. *}
  1510 
  1511 lemma sequentially_offset:
  1512   assumes "eventually (\<lambda>i. P i) sequentially"
  1513   shows "eventually (\<lambda>i. P (i + k)) sequentially"
  1514   using assms unfolding eventually_sequentially by (metis trans_le_add1)
  1515 
  1516 lemma seq_offset:
  1517   assumes "(f ---> l) sequentially"
  1518   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
  1519   using assms unfolding tendsto_def
  1520   by clarify (rule sequentially_offset, simp)
  1521 
  1522 lemma seq_offset_neg:
  1523   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
  1524   apply (rule topological_tendstoI)
  1525   apply (drule (2) topological_tendstoD)
  1526   apply (simp only: eventually_sequentially)
  1527   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
  1528   apply metis
  1529   by arith
  1530 
  1531 lemma seq_offset_rev:
  1532   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
  1533   apply (rule topological_tendstoI)
  1534   apply (drule (2) topological_tendstoD)
  1535   apply (simp only: eventually_sequentially)
  1536   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
  1537   by metis arith
  1538 
  1539 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
  1540 proof-
  1541   { fix e::real assume "e>0"
  1542     hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
  1543       using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
  1544       by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
  1545   }
  1546   thus ?thesis unfolding Lim_sequentially dist_norm by simp
  1547 qed
  1548 
  1549 subsection {* More properties of closed balls *}
  1550 
  1551 lemma closed_cball: "closed (cball x e)"
  1552 unfolding cball_def closed_def
  1553 unfolding Collect_neg_eq [symmetric] not_le
  1554 apply (clarsimp simp add: open_dist, rename_tac y)
  1555 apply (rule_tac x="dist x y - e" in exI, clarsimp)
  1556 apply (rename_tac x')
  1557 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
  1558 apply simp
  1559 done
  1560 
  1561 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
  1562 proof-
  1563   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
  1564     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
  1565   } moreover
  1566   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
  1567     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
  1568   } ultimately
  1569   show ?thesis unfolding open_contains_ball by auto
  1570 qed
  1571 
  1572 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
  1573   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
  1574 
  1575 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
  1576   apply (simp add: interior_def, safe)
  1577   apply (force simp add: open_contains_cball)
  1578   apply (rule_tac x="ball x e" in exI)
  1579   apply (simp add: subset_trans [OF ball_subset_cball])
  1580   done
  1581 
  1582 lemma islimpt_ball:
  1583   fixes x y :: "'a::{real_normed_vector,perfect_space}"
  1584   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
  1585 proof
  1586   assume "?lhs"
  1587   { assume "e \<le> 0"
  1588     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
  1589     have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
  1590   }
  1591   hence "e > 0" by (metis not_less)
  1592   moreover
  1593   have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto
  1594   ultimately show "?rhs" by auto
  1595 next
  1596   assume "?rhs" hence "e>0"  by auto
  1597   { fix d::real assume "d>0"
  1598     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1599     proof(cases "d \<le> dist x y")
  1600       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1601       proof(cases "x=y")
  1602         case True hence False using `d \<le> dist x y` `d>0` by auto
  1603         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
  1604       next
  1605         case False
  1606 
  1607         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
  1608               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1609           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
  1610         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
  1611           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
  1612           unfolding scaleR_minus_left scaleR_one
  1613           by (auto simp add: norm_minus_commute)
  1614         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
  1615           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
  1616           unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
  1617         also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
  1618         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
  1619 
  1620         moreover
  1621 
  1622         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
  1623           using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
  1624         moreover
  1625         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
  1626           using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
  1627           unfolding dist_norm by auto
  1628         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
  1629       qed
  1630     next
  1631       case False hence "d > dist x y" by auto
  1632       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1633       proof(cases "x=y")
  1634         case True
  1635         obtain z where **: "z \<noteq> y" "dist z y < min e d"
  1636           using perfect_choose_dist[of "min e d" y]
  1637           using `d > 0` `e>0` by auto
  1638         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1639           unfolding `x = y`
  1640           using `z \<noteq> y` **
  1641           by (rule_tac x=z in bexI, auto simp add: dist_commute)
  1642       next
  1643         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1644           using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
  1645       qed
  1646     qed  }
  1647   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
  1648 qed
  1649 
  1650 lemma closure_ball_lemma:
  1651   fixes x y :: "'a::real_normed_vector"
  1652   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
  1653 proof (rule islimptI)
  1654   fix T assume "y \<in> T" "open T"
  1655   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
  1656     unfolding open_dist by fast
  1657   (* choose point between x and y, within distance r of y. *)
  1658   def k \<equiv> "min 1 (r / (2 * dist x y))"
  1659   def z \<equiv> "y + scaleR k (x - y)"
  1660   have z_def2: "z = x + scaleR (1 - k) (y - x)"
  1661     unfolding z_def by (simp add: algebra_simps)
  1662   have "dist z y < r"
  1663     unfolding z_def k_def using `0 < r`
  1664     by (simp add: dist_norm min_def)
  1665   hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
  1666   have "dist x z < dist x y"
  1667     unfolding z_def2 dist_norm
  1668     apply (simp add: norm_minus_commute)
  1669     apply (simp only: dist_norm [symmetric])
  1670     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
  1671     apply (rule mult_strict_right_mono)
  1672     apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
  1673     apply (simp add: zero_less_dist_iff `x \<noteq> y`)
  1674     done
  1675   hence "z \<in> ball x (dist x y)" by simp
  1676   have "z \<noteq> y"
  1677     unfolding z_def k_def using `x \<noteq> y` `0 < r`
  1678     by (simp add: min_def)
  1679   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
  1680     using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
  1681     by fast
  1682 qed
  1683 
  1684 lemma closure_ball:
  1685   fixes x :: "'a::real_normed_vector"
  1686   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
  1687 apply (rule equalityI)
  1688 apply (rule closure_minimal)
  1689 apply (rule ball_subset_cball)
  1690 apply (rule closed_cball)
  1691 apply (rule subsetI, rename_tac y)
  1692 apply (simp add: le_less [where 'a=real])
  1693 apply (erule disjE)
  1694 apply (rule subsetD [OF closure_subset], simp)
  1695 apply (simp add: closure_def)
  1696 apply clarify
  1697 apply (rule closure_ball_lemma)
  1698 apply (simp add: zero_less_dist_iff)
  1699 done
  1700 
  1701 (* In a trivial vector space, this fails for e = 0. *)
  1702 lemma interior_cball:
  1703   fixes x :: "'a::{real_normed_vector, perfect_space}"
  1704   shows "interior (cball x e) = ball x e"
  1705 proof(cases "e\<ge>0")
  1706   case False note cs = this
  1707   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
  1708   { fix y assume "y \<in> cball x e"
  1709     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
  1710   hence "cball x e = {}" by auto
  1711   hence "interior (cball x e) = {}" using interior_empty by auto
  1712   ultimately show ?thesis by blast
  1713 next
  1714   case True note cs = this
  1715   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
  1716   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
  1717     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
  1718 
  1719     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
  1720       using perfect_choose_dist [of d] by auto
  1721     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
  1722     hence xa_cball:"xa \<in> cball x e" using as(1) by auto
  1723 
  1724     hence "y \<in> ball x e" proof(cases "x = y")
  1725       case True
  1726       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
  1727       thus "y \<in> ball x e" using `x = y ` by simp
  1728     next
  1729       case False
  1730       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
  1731         using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
  1732       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
  1733       have "y - x \<noteq> 0" using `x \<noteq> y` by auto
  1734       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
  1735         using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
  1736 
  1737       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
  1738         by (auto simp add: dist_norm algebra_simps)
  1739       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1740         by (auto simp add: algebra_simps)
  1741       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
  1742         using ** by auto
  1743       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
  1744       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
  1745       thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
  1746     qed  }
  1747   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
  1748   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
  1749 qed
  1750 
  1751 lemma frontier_ball:
  1752   fixes a :: "'a::real_normed_vector"
  1753   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
  1754   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
  1755   apply (simp add: set_eq_iff)
  1756   by arith
  1757 
  1758 lemma frontier_cball:
  1759   fixes a :: "'a::{real_normed_vector, perfect_space}"
  1760   shows "frontier(cball a e) = {x. dist a x = e}"
  1761   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
  1762   apply (simp add: set_eq_iff)
  1763   by arith
  1764 
  1765 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
  1766   apply (simp add: set_eq_iff not_le)
  1767   by (metis zero_le_dist dist_self order_less_le_trans)
  1768 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
  1769 
  1770 lemma cball_eq_sing:
  1771   fixes x :: "'a::{metric_space,perfect_space}"
  1772   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
  1773 proof (rule linorder_cases)
  1774   assume e: "0 < e"
  1775   obtain a where "a \<noteq> x" "dist a x < e"
  1776     using perfect_choose_dist [OF e] by auto
  1777   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
  1778   with e show ?thesis by (auto simp add: set_eq_iff)
  1779 qed auto
  1780 
  1781 lemma cball_sing:
  1782   fixes x :: "'a::metric_space"
  1783   shows "e = 0 ==> cball x e = {x}"
  1784   by (auto simp add: set_eq_iff)
  1785 
  1786 
  1787 subsection {* Boundedness *}
  1788 
  1789   (* FIXME: This has to be unified with BSEQ!! *)
  1790 definition (in metric_space)
  1791   bounded :: "'a set \<Rightarrow> bool" where
  1792   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
  1793 
  1794 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
  1795 unfolding bounded_def
  1796 apply safe
  1797 apply (rule_tac x="dist a x + e" in exI, clarify)
  1798 apply (drule (1) bspec)
  1799 apply (erule order_trans [OF dist_triangle add_left_mono])
  1800 apply auto
  1801 done
  1802 
  1803 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
  1804 unfolding bounded_any_center [where a=0]
  1805 by (simp add: dist_norm)
  1806 
  1807 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
  1808 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
  1809   by (metis bounded_def subset_eq)
  1810 
  1811 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
  1812   by (metis bounded_subset interior_subset)
  1813 
  1814 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
  1815 proof-
  1816   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
  1817   { fix y assume "y \<in> closure S"
  1818     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
  1819       unfolding closure_sequential by auto
  1820     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
  1821     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
  1822       by (rule eventually_mono, simp add: f(1))
  1823     have "dist x y \<le> a"
  1824       apply (rule Lim_dist_ubound [of sequentially f])
  1825       apply (rule trivial_limit_sequentially)
  1826       apply (rule f(2))
  1827       apply fact
  1828       done
  1829   }
  1830   thus ?thesis unfolding bounded_def by auto
  1831 qed
  1832 
  1833 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
  1834   apply (simp add: bounded_def)
  1835   apply (rule_tac x=x in exI)
  1836   apply (rule_tac x=e in exI)
  1837   apply auto
  1838   done
  1839 
  1840 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
  1841   by (metis ball_subset_cball bounded_cball bounded_subset)
  1842 
  1843 lemma finite_imp_bounded[intro]:
  1844   fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
  1845 proof-
  1846   { fix a and F :: "'a set" assume as:"bounded F"
  1847     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
  1848     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
  1849     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
  1850   }
  1851   thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto
  1852 qed
  1853 
  1854 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
  1855   apply (auto simp add: bounded_def)
  1856   apply (rename_tac x y r s)
  1857   apply (rule_tac x=x in exI)
  1858   apply (rule_tac x="max r (dist x y + s)" in exI)
  1859   apply (rule ballI, rename_tac z, safe)
  1860   apply (drule (1) bspec, simp)
  1861   apply (drule (1) bspec)
  1862   apply (rule min_max.le_supI2)
  1863   apply (erule order_trans [OF dist_triangle add_left_mono])
  1864   done
  1865 
  1866 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
  1867   by (induct rule: finite_induct[of F], auto)
  1868 
  1869 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
  1870   apply (simp add: bounded_iff)
  1871   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
  1872   by metis arith
  1873 
  1874 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
  1875   by (metis Int_lower1 Int_lower2 bounded_subset)
  1876 
  1877 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
  1878 apply (metis Diff_subset bounded_subset)
  1879 done
  1880 
  1881 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
  1882   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
  1883 
  1884 lemma not_bounded_UNIV[simp, intro]:
  1885   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
  1886 proof(auto simp add: bounded_pos not_le)
  1887   obtain x :: 'a where "x \<noteq> 0"
  1888     using perfect_choose_dist [OF zero_less_one] by fast
  1889   fix b::real  assume b: "b >0"
  1890   have b1: "b +1 \<ge> 0" using b by simp
  1891   with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
  1892     by (simp add: norm_sgn)
  1893   then show "\<exists>x::'a. b < norm x" ..
  1894 qed
  1895 
  1896 lemma bounded_linear_image:
  1897   assumes "bounded S" "bounded_linear f"
  1898   shows "bounded(f ` S)"
  1899 proof-
  1900   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  1901   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)
  1902   { fix x assume "x\<in>S"
  1903     hence "norm x \<le> b" using b by auto
  1904     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
  1905       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
  1906   }
  1907   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
  1908     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
  1909 qed
  1910 
  1911 lemma bounded_scaling:
  1912   fixes S :: "'a::real_normed_vector set"
  1913   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
  1914   apply (rule bounded_linear_image, assumption)
  1915   apply (rule bounded_linear_scaleR_right)
  1916   done
  1917 
  1918 lemma bounded_translation:
  1919   fixes S :: "'a::real_normed_vector set"
  1920   assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
  1921 proof-
  1922   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  1923   { fix x assume "x\<in>S"
  1924     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
  1925   }
  1926   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
  1927     by (auto intro!: add exI[of _ "b + norm a"])
  1928 qed
  1929 
  1930 
  1931 text{* Some theorems on sups and infs using the notion "bounded". *}
  1932 
  1933 lemma bounded_real:
  1934   fixes S :: "real set"
  1935   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
  1936   by (simp add: bounded_iff)
  1937 
  1938 lemma bounded_has_Sup:
  1939   fixes S :: "real set"
  1940   assumes "bounded S" "S \<noteq> {}"
  1941   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
  1942 proof
  1943   fix x assume "x\<in>S"
  1944   thus "x \<le> Sup S"
  1945     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
  1946 next
  1947   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
  1948     by (metis SupInf.Sup_least)
  1949 qed
  1950 
  1951 lemma Sup_insert:
  1952   fixes S :: "real set"
  1953   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" 
  1954 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) 
  1955 
  1956 lemma Sup_insert_finite:
  1957   fixes S :: "real set"
  1958   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
  1959   apply (rule Sup_insert)
  1960   apply (rule finite_imp_bounded)
  1961   by simp
  1962 
  1963 lemma bounded_has_Inf:
  1964   fixes S :: "real set"
  1965   assumes "bounded S"  "S \<noteq> {}"
  1966   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
  1967 proof
  1968   fix x assume "x\<in>S"
  1969   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
  1970   thus "x \<ge> Inf S" using `x\<in>S`
  1971     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
  1972 next
  1973   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
  1974     by (metis SupInf.Inf_greatest)
  1975 qed
  1976 
  1977 lemma Inf_insert:
  1978   fixes S :: "real set"
  1979   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" 
  1980 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal) 
  1981 lemma Inf_insert_finite:
  1982   fixes S :: "real set"
  1983   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
  1984   by (rule Inf_insert, rule finite_imp_bounded, simp)
  1985 
  1986 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
  1987 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
  1988   apply (frule isGlb_isLb)
  1989   apply (frule_tac x = y in isGlb_isLb)
  1990   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
  1991   done
  1992 
  1993 
  1994 subsection {* Equivalent versions of compactness *}
  1995 
  1996 subsubsection{* Sequential compactness *}
  1997 
  1998 definition
  1999   compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
  2000   "compact S \<longleftrightarrow>
  2001    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
  2002        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
  2003 
  2004 lemma compactI:
  2005   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
  2006   shows "compact S"
  2007   unfolding compact_def using assms by fast
  2008 
  2009 lemma compactE:
  2010   assumes "compact S" "\<forall>n. f n \<in> S"
  2011   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
  2012   using assms unfolding compact_def by fast
  2013 
  2014 text {*
  2015   A metric space (or topological vector space) is said to have the
  2016   Heine-Borel property if every closed and bounded subset is compact.
  2017 *}
  2018 
  2019 class heine_borel = metric_space +
  2020   assumes bounded_imp_convergent_subsequence:
  2021     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
  2022       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2023 
  2024 lemma bounded_closed_imp_compact:
  2025   fixes s::"'a::heine_borel set"
  2026   assumes "bounded s" and "closed s" shows "compact s"
  2027 proof (unfold compact_def, clarify)
  2028   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  2029   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  2030     using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
  2031   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
  2032   have "l \<in> s" using `closed s` fr l
  2033     unfolding closed_sequential_limits by blast
  2034   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2035     using `l \<in> s` r l by blast
  2036 qed
  2037 
  2038 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
  2039 proof(induct n)
  2040   show "0 \<le> r 0" by auto
  2041 next
  2042   fix n assume "n \<le> r n"
  2043   moreover have "r n < r (Suc n)"
  2044     using assms [unfolded subseq_def] by auto
  2045   ultimately show "Suc n \<le> r (Suc n)" by auto
  2046 qed
  2047 
  2048 lemma eventually_subseq:
  2049   assumes r: "subseq r"
  2050   shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
  2051 unfolding eventually_sequentially
  2052 by (metis subseq_bigger [OF r] le_trans)
  2053 
  2054 lemma lim_subseq:
  2055   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
  2056 unfolding tendsto_def eventually_sequentially o_def
  2057 by (metis subseq_bigger le_trans)
  2058 
  2059 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
  2060   unfolding Ex1_def
  2061   apply (rule_tac x="nat_rec e f" in exI)
  2062   apply (rule conjI)+
  2063 apply (rule def_nat_rec_0, simp)
  2064 apply (rule allI, rule def_nat_rec_Suc, simp)
  2065 apply (rule allI, rule impI, rule ext)
  2066 apply (erule conjE)
  2067 apply (induct_tac x)
  2068 apply simp
  2069 apply (erule_tac x="n" in allE)
  2070 apply (simp)
  2071 done
  2072 
  2073 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
  2074   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
  2075   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"
  2076 proof-
  2077   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
  2078   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
  2079   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
  2080     { fix n::nat
  2081       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
  2082       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
  2083       with n have "s N \<le> t - e" using `e>0` by auto
  2084       hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto  }
  2085     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
  2086     hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto  }
  2087   thus ?thesis by blast
  2088 qed
  2089 
  2090 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
  2091   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
  2092   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
  2093   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
  2094   unfolding monoseq_def incseq_def
  2095   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
  2096   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
  2097 
  2098 (* TODO: merge this lemma with the ones above *)
  2099 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
  2100   assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"
  2101   shows "\<exists>l. (s ---> l) sequentially"
  2102 proof-
  2103   obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto
  2104   { fix m::nat
  2105     have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"
  2106       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
  2107       apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }
  2108   hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
  2109   then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar> (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a]
  2110     unfolding monoseq_def by auto
  2111   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
  2112     unfolding dist_norm  by auto
  2113 qed
  2114 
  2115 lemma compact_real_lemma:
  2116   assumes "\<forall>n::nat. abs(s n) \<le> b"
  2117   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
  2118 proof-
  2119   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
  2120     using seq_monosub[of s] by auto
  2121   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
  2122     unfolding tendsto_iff dist_norm eventually_sequentially by auto
  2123 qed
  2124 
  2125 instance real :: heine_borel
  2126 proof
  2127   fix s :: "real set" and f :: "nat \<Rightarrow> real"
  2128   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2129   then obtain b where b: "\<forall>n. abs (f n) \<le> b"
  2130     unfolding bounded_iff by auto
  2131   obtain l :: real and r :: "nat \<Rightarrow> nat" where
  2132     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  2133     using compact_real_lemma [OF b] by auto
  2134   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2135     by auto
  2136 qed
  2137 
  2138 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
  2139   apply (erule bounded_linear_image)
  2140   apply (rule bounded_linear_euclidean_component)
  2141   done
  2142 
  2143 lemma compact_lemma:
  2144   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
  2145   assumes "bounded s" and "\<forall>n. f n \<in> s"
  2146   shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
  2147         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
  2148 proof
  2149   fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
  2150   have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
  2151   hence "\<exists>l::'a. \<exists>r. subseq r \<and>
  2152       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
  2153   proof(induct d) case empty thus ?case unfolding subseq_def by auto
  2154   next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
  2155     have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
  2156     obtain l1::"'a" and r1 where r1:"subseq r1" and
  2157       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
  2158       using insert(3) using insert(4) by auto
  2159     have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
  2160     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
  2161       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
  2162     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
  2163       using r1 and r2 unfolding r_def o_def subseq_def by auto
  2164     moreover
  2165     def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
  2166     { fix e::real assume "e>0"
  2167       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
  2168       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
  2169       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
  2170         by (rule eventually_subseq)
  2171       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
  2172         using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
  2173         using insert.prems by auto
  2174     }
  2175     ultimately show ?case by auto
  2176   qed
  2177   thus "\<exists>l::'a. \<exists>r. subseq r \<and>
  2178       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
  2179     apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
  2180     apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe 
  2181     apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
  2182     apply(erule_tac x=i in ballE) 
  2183   proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
  2184     assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
  2185     hence *:"i\<ge>DIM('a)" by auto
  2186     thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
  2187   qed
  2188 qed
  2189 
  2190 instance euclidean_space \<subseteq> heine_borel
  2191 proof
  2192   fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
  2193   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2194   then obtain l::'a and r where r: "subseq r"
  2195     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
  2196     using compact_lemma [OF s f] by blast
  2197   let ?d = "{..<DIM('a)}"
  2198   { fix e::real assume "e>0"
  2199     hence "0 < e / (real_of_nat (card ?d))"
  2200       using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
  2201     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
  2202       by simp
  2203     moreover
  2204     { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
  2205       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
  2206         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
  2207       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
  2208         apply(rule setsum_strict_mono) using n by auto
  2209       finally have "dist (f (r n)) l < e" unfolding setsum_constant
  2210         using DIM_positive[where 'a='a] by auto
  2211     }
  2212     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
  2213       by (rule eventually_elim1)
  2214   }
  2215   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
  2216   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
  2217 qed
  2218 
  2219 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
  2220 unfolding bounded_def
  2221 apply clarify
  2222 apply (rule_tac x="a" in exI)
  2223 apply (rule_tac x="e" in exI)
  2224 apply clarsimp
  2225 apply (drule (1) bspec)
  2226 apply (simp add: dist_Pair_Pair)
  2227 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
  2228 done
  2229 
  2230 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
  2231 unfolding bounded_def
  2232 apply clarify
  2233 apply (rule_tac x="b" in exI)
  2234 apply (rule_tac x="e" in exI)
  2235 apply clarsimp
  2236 apply (drule (1) bspec)
  2237 apply (simp add: dist_Pair_Pair)
  2238 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
  2239 done
  2240 
  2241 instance prod :: (heine_borel, heine_borel) heine_borel
  2242 proof
  2243   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
  2244   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2245   from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
  2246   from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
  2247   obtain l1 r1 where r1: "subseq r1"
  2248     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
  2249     using bounded_imp_convergent_subsequence [OF s1 f1]
  2250     unfolding o_def by fast
  2251   from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
  2252   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
  2253   obtain l2 r2 where r2: "subseq r2"
  2254     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
  2255     using bounded_imp_convergent_subsequence [OF s2 f2]
  2256     unfolding o_def by fast
  2257   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
  2258     using lim_subseq [OF r2 l1] unfolding o_def .
  2259   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
  2260     using tendsto_Pair [OF l1' l2] unfolding o_def by simp
  2261   have r: "subseq (r1 \<circ> r2)"
  2262     using r1 r2 unfolding subseq_def by simp
  2263   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2264     using l r by fast
  2265 qed
  2266 
  2267 subsubsection{* Completeness *}
  2268 
  2269 lemma cauchy_def:
  2270   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
  2271 unfolding Cauchy_def by blast
  2272 
  2273 definition
  2274   complete :: "'a::metric_space set \<Rightarrow> bool" where
  2275   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
  2276                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"
  2277 
  2278 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
  2279 proof-
  2280   { assume ?rhs
  2281     { fix e::real
  2282       assume "e>0"
  2283       with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
  2284         by (erule_tac x="e/2" in allE) auto
  2285       { fix n m
  2286         assume nm:"N \<le> m \<and> N \<le> n"
  2287         hence "dist (s m) (s n) < e" using N
  2288           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
  2289           by blast
  2290       }
  2291       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
  2292         by blast
  2293     }
  2294     hence ?lhs
  2295       unfolding cauchy_def
  2296       by blast
  2297   }
  2298   thus ?thesis
  2299     unfolding cauchy_def
  2300     using dist_triangle_half_l
  2301     by blast
  2302 qed
  2303 
  2304 lemma convergent_imp_cauchy:
  2305  "(s ---> l) sequentially ==> Cauchy s"
  2306 proof(simp only: cauchy_def, rule, rule)
  2307   fix e::real assume "e>0" "(s ---> l) sequentially"
  2308   then obtain N::nat where N:"\<forall>n\<ge>N. dist (s n) l < e/2" unfolding Lim_sequentially by(erule_tac x="e/2" in allE) auto
  2309   thus "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"  using dist_triangle_half_l[of _ l e _] by (rule_tac x=N in exI) auto
  2310 qed
  2311 
  2312 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
  2313 proof-
  2314   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto
  2315   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
  2316   moreover
  2317   have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
  2318   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
  2319     unfolding bounded_any_center [where a="s N"] by auto
  2320   ultimately show "?thesis"
  2321     unfolding bounded_any_center [where a="s N"]
  2322     apply(rule_tac x="max a 1" in exI) apply auto
  2323     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
  2324 qed
  2325 
  2326 lemma compact_imp_complete: assumes "compact s" shows "complete s"
  2327 proof-
  2328   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
  2329     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast
  2330 
  2331     note lr' = subseq_bigger [OF lr(2)]
  2332 
  2333     { fix e::real assume "e>0"
  2334       from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using `e>0` apply (erule_tac x="e/2" in allE) by auto
  2335       from lr(3)[unfolded Lim_sequentially, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using `e>0` by auto
  2336       { fix n::nat assume n:"n \<ge> max N M"
  2337         have "dist ((f \<circ> r) n) l < e/2" using n M by auto
  2338         moreover have "r n \<ge> N" using lr'[of n] n by auto
  2339         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
  2340         ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }
  2341       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }
  2342     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto  }
  2343   thus ?thesis unfolding complete_def by auto
  2344 qed
  2345 
  2346 instance heine_borel < complete_space
  2347 proof
  2348   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  2349   hence "bounded (range f)"
  2350     by (rule cauchy_imp_bounded)
  2351   hence "compact (closure (range f))"
  2352     using bounded_closed_imp_compact [of "closure (range f)"] by auto
  2353   hence "complete (closure (range f))"
  2354     by (rule compact_imp_complete)
  2355   moreover have "\<forall>n. f n \<in> closure (range f)"
  2356     using closure_subset [of "range f"] by auto
  2357   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
  2358     using `Cauchy f` unfolding complete_def by auto
  2359   then show "convergent f"
  2360     unfolding convergent_def by auto
  2361 qed
  2362 
  2363 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
  2364 proof(simp add: complete_def, rule, rule)
  2365   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  2366   hence "convergent f" by (rule Cauchy_convergent)
  2367   thus "\<exists>l. f ----> l" unfolding convergent_def .  
  2368 qed
  2369 
  2370 lemma complete_imp_closed: assumes "complete s" shows "closed s"
  2371 proof -
  2372   { fix x assume "x islimpt s"
  2373     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
  2374       unfolding islimpt_sequential by auto
  2375     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
  2376       using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
  2377     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
  2378   }
  2379   thus "closed s" unfolding closed_limpt by auto
  2380 qed
  2381 
  2382 lemma complete_eq_closed:
  2383   fixes s :: "'a::complete_space set"
  2384   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
  2385 proof
  2386   assume ?lhs thus ?rhs by (rule complete_imp_closed)
  2387 next
  2388   assume ?rhs
  2389   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
  2390     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
  2391     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `?rhs`[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }
  2392   thus ?lhs unfolding complete_def by auto
  2393 qed
  2394 
  2395 lemma convergent_eq_cauchy:
  2396   fixes s :: "nat \<Rightarrow> 'a::complete_space"
  2397   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
  2398 proof
  2399   assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
  2400   thus ?rhs using convergent_imp_cauchy by auto
  2401 next
  2402   assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
  2403 qed
  2404 
  2405 lemma convergent_imp_bounded:
  2406   fixes s :: "nat \<Rightarrow> 'a::metric_space"
  2407   shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
  2408   using convergent_imp_cauchy[of s]
  2409   using cauchy_imp_bounded[of s]
  2410   unfolding image_def
  2411   by auto
  2412 
  2413 subsubsection{* Total boundedness *}
  2414 
  2415 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
  2416   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
  2417 declare helper_1.simps[simp del]
  2418 
  2419 lemma compact_imp_totally_bounded:
  2420   assumes "compact s"
  2421   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
  2422 proof(rule, rule, rule ccontr)
  2423   fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k)"
  2424   def x \<equiv> "helper_1 s e"
  2425   { fix n
  2426     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
  2427     proof(induct_tac rule:nat_less_induct)
  2428       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
  2429       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
  2430       have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x ` {0 ..< n}" in allE) using as by auto
  2431       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
  2432       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
  2433         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
  2434       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
  2435     qed }
  2436   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
  2437   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto
  2438   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
  2439   then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using `e>0` by auto
  2440   show False
  2441     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
  2442     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
  2443     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
  2444 qed
  2445 
  2446 subsubsection{* Heine-Borel theorem *}
  2447 
  2448 text {* Following Burkill \& Burkill vol. 2. *}
  2449 
  2450 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
  2451   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"
  2452   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
  2453 proof(rule ccontr)
  2454   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
  2455   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
  2456   { fix n::nat
  2457     have "1 / real (n + 1) > 0" by auto
  2458     hence "\<exists>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> (ball x (inverse (real (n+1))) \<subseteq> xa))" using cont unfolding Bex_def by auto }
  2459   hence "\<forall>n::nat. \<exists>x. x \<in> s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)" by auto
  2460   then obtain f where f:"\<forall>n::nat. f n \<in> s \<and> (\<forall>xa\<in>t. \<not> ball (f n) (inverse (real (n + 1))) \<subseteq> xa)"
  2461     using choice[of "\<lambda>n::nat. \<lambda>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)"] by auto
  2462 
  2463   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
  2464     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
  2465 
  2466   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
  2467   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
  2468     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
  2469 
  2470   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
  2471     using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
  2472 
  2473   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
  2474   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
  2475     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
  2476     using subseq_bigger[OF r, of "N1 + N2"] by auto
  2477 
  2478   def x \<equiv> "(f (r (N1 + N2)))"
  2479   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
  2480     using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
  2481   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
  2482   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
  2483 
  2484   have "dist x l < e/2" using N1 unfolding x_def o_def by auto
  2485   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
  2486 
  2487   thus False using e and `y\<notin>b` by auto
  2488 qed
  2489 
  2490 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
  2491                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
  2492 proof clarify
  2493   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
  2494   then obtain e::real where "e>0" and "\<forall>x\<in>s. \<exists>b\<in>f. ball x e \<subseteq> b" using heine_borel_lemma[of s f] by auto
  2495   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
  2496   hence "\<exists>bb. \<forall>x\<in>s. bb x \<in>f \<and> ball x e \<subseteq> bb x" using bchoice[of s "\<lambda>x b. b\<in>f \<and> ball x e \<subseteq> b"] by auto
  2497   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
  2498 
  2499   from `compact s` have  "\<exists> k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" using compact_imp_totally_bounded[of s] `e>0` by auto
  2500   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
  2501 
  2502   have "finite (bb ` k)" using k(1) by auto
  2503   moreover
  2504   { fix x assume "x\<in>s"
  2505     hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3)  unfolding subset_eq by auto
  2506     hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
  2507     hence "x \<in> \<Union>(bb ` k)" using  Union_iff[of x "bb ` k"] by auto
  2508   }
  2509   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f'" using bb k(2) by (rule_tac x="bb ` k" in exI) auto
  2510 qed
  2511 
  2512 subsubsection {* Bolzano-Weierstrass property *}
  2513 
  2514 lemma heine_borel_imp_bolzano_weierstrass:
  2515   assumes "\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f) --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f'))"
  2516           "infinite t"  "t \<subseteq> s"
  2517   shows "\<exists>x \<in> s. x islimpt t"
  2518 proof(rule ccontr)
  2519   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
  2520   then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)" unfolding islimpt_def
  2521     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto
  2522   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
  2523     using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
  2524   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
  2525   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
  2526     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
  2527     hence "x = y" using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto  }
  2528   hence "inj_on f t" unfolding inj_on_def by simp
  2529   hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
  2530   moreover
  2531   { fix x assume "x\<in>t" "f x \<notin> g"
  2532     from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
  2533     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
  2534     hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
  2535     hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto  }
  2536   hence "f ` t \<subseteq> g" by auto
  2537   ultimately show False using g(2) using finite_subset by auto
  2538 qed
  2539 
  2540 subsubsection {* Complete the chain of compactness variants *}
  2541 
  2542 lemma islimpt_range_imp_convergent_subsequence:
  2543   fixes f :: "nat \<Rightarrow> 'a::metric_space"
  2544   assumes "l islimpt (range f)"
  2545   shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2546 proof (intro exI conjI)
  2547   have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
  2548     using assms unfolding islimpt_def
  2549     by (drule_tac x="ball l e" in spec)
  2550        (auto simp add: zero_less_dist_iff dist_commute)
  2551 
  2552   def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
  2553   have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
  2554     unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
  2555   have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
  2556     unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
  2557   have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
  2558     unfolding t_def by (simp add: Least_le)
  2559   have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
  2560     unfolding t_def by (drule not_less_Least) simp
  2561   have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
  2562     apply (rule t_le)
  2563     apply (erule f_t_neq)
  2564     apply (erule (1) less_le_trans [OF f_t_closer])
  2565     done
  2566   have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
  2567     by (drule f_t_closer) auto
  2568   have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
  2569     apply (subst less_le)
  2570     apply (rule conjI)
  2571     apply (rule t_antimono)
  2572     apply (erule f_t_neq)
  2573     apply (erule f_t_closer [THEN less_imp_le])
  2574     apply (rule t_dist_f_neq [symmetric])
  2575     apply (erule f_t_neq)
  2576     done
  2577   have dist_f_t_less':
  2578     "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
  2579     apply (simp add: le_less)
  2580     apply (erule disjE)
  2581     apply (rule less_trans)
  2582     apply (erule f_t_closer)
  2583     apply (rule le_less_trans)
  2584     apply (erule less_tD)
  2585     apply (erule f_t_neq)
  2586     apply (erule f_t_closer)
  2587     apply (erule subst)
  2588     apply (erule f_t_closer)
  2589     done
  2590 
  2591   def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
  2592   have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
  2593     unfolding r_def by simp_all
  2594   have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
  2595     by (induct_tac n) (simp_all add: r_simps f_t_neq)
  2596 
  2597   show "subseq r"
  2598     unfolding subseq_Suc_iff
  2599     apply (rule allI)
  2600     apply (case_tac n)
  2601     apply (simp_all add: r_simps)
  2602     apply (rule t_less, rule zero_less_one)
  2603     apply (rule t_less, rule f_r_neq)
  2604     done
  2605   show "((f \<circ> r) ---> l) sequentially"
  2606     unfolding Lim_sequentially o_def
  2607     apply (clarify, rule_tac x="t e" in exI, clarify)
  2608     apply (drule le_trans, rule seq_suble [OF `subseq r`])
  2609     apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
  2610     done
  2611 qed
  2612 
  2613 lemma finite_range_imp_infinite_repeats:
  2614   fixes f :: "nat \<Rightarrow> 'a"
  2615   assumes "finite (range f)"
  2616   shows "\<exists>k. infinite {n. f n = k}"
  2617 proof -
  2618   { fix A :: "'a set" assume "finite A"
  2619     hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
  2620     proof (induct)
  2621       case empty thus ?case by simp
  2622     next
  2623       case (insert x A)
  2624      show ?case
  2625       proof (cases "finite {n. f n = x}")
  2626         case True
  2627         with `infinite {n. f n \<in> insert x A}`
  2628         have "infinite {n. f n \<in> A}" by simp
  2629         thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
  2630       next
  2631         case False thus "\<exists>k. infinite {n. f n = k}" ..
  2632       qed
  2633     qed
  2634   } note H = this
  2635   from assms show "\<exists>k. infinite {n. f n = k}"
  2636     by (rule H) simp
  2637 qed
  2638 
  2639 lemma bolzano_weierstrass_imp_compact:
  2640   fixes s :: "'a::metric_space set"
  2641   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2642   shows "compact s"
  2643 proof -
  2644   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  2645     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2646     proof (cases "finite (range f)")
  2647       case True
  2648       hence "\<exists>l. infinite {n. f n = l}"
  2649         by (rule finite_range_imp_infinite_repeats)
  2650       then obtain l where "infinite {n. f n = l}" ..
  2651       hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
  2652         by (rule infinite_enumerate)
  2653       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
  2654       hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2655         unfolding o_def by (simp add: fr tendsto_const)
  2656       hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2657         by - (rule exI)
  2658       from f have "\<forall>n. f (r n) \<in> s" by simp
  2659       hence "l \<in> s" by (simp add: fr)
  2660       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2661         by (rule rev_bexI) fact
  2662     next
  2663       case False
  2664       with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
  2665       then obtain l where "l \<in> s" "l islimpt (range f)" ..
  2666       have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2667         using `l islimpt (range f)`
  2668         by (rule islimpt_range_imp_convergent_subsequence)
  2669       with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
  2670     qed
  2671   }
  2672   thus ?thesis unfolding compact_def by auto
  2673 qed
  2674 
  2675 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
  2676   "helper_2 beyond 0 = beyond 0" |
  2677   "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
  2678 
  2679 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
  2680   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2681   shows "bounded s"
  2682 proof(rule ccontr)
  2683   assume "\<not> bounded s"
  2684   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
  2685     unfolding bounded_any_center [where a=undefined]
  2686     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
  2687   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
  2688     unfolding linorder_not_le by auto
  2689   def x \<equiv> "helper_2 beyond"
  2690 
  2691   { fix m n ::nat assume "m<n"
  2692     hence "dist undefined (x m) + 1 < dist undefined (x n)"
  2693     proof(induct n)
  2694       case 0 thus ?case by auto
  2695     next
  2696       case (Suc n)
  2697       have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
  2698         unfolding x_def and helper_2.simps
  2699         using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
  2700       thus ?case proof(cases "m < n")
  2701         case True thus ?thesis using Suc and * by auto
  2702       next
  2703         case False hence "m = n" using Suc(2) by auto
  2704         thus ?thesis using * by auto
  2705       qed
  2706     qed  } note * = this
  2707   { fix m n ::nat assume "m\<noteq>n"
  2708     have "1 < dist (x m) (x n)"
  2709     proof(cases "m<n")
  2710       case True
  2711       hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
  2712       thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
  2713     next
  2714       case False hence "n<m" using `m\<noteq>n` by auto
  2715       hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
  2716       thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
  2717     qed  } note ** = this
  2718   { fix a b assume "x a = x b" "a \<noteq> b"
  2719     hence False using **[of a b] by auto  }
  2720   hence "inj x" unfolding inj_on_def by auto
  2721   moreover
  2722   { fix n::nat
  2723     have "x n \<in> s"
  2724     proof(cases "n = 0")
  2725       case True thus ?thesis unfolding x_def using beyond by auto
  2726     next
  2727       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
  2728       thus ?thesis unfolding x_def using beyond by auto
  2729     qed  }
  2730   ultimately have "infinite (range x) \<and> range x \<subseteq> s" unfolding x_def using range_inj_infinite[of "helper_2 beyond"] using beyond(1) by auto
  2731 
  2732   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
  2733   then obtain y where "x y \<noteq> l" and y:"dist (x y) l < 1/2" unfolding islimpt_approachable apply(erule_tac x="1/2" in allE) by auto
  2734   then obtain z where "x z \<noteq> l" and z:"dist (x z) l < dist (x y) l" using l[unfolded islimpt_approachable, THEN spec[where x="dist (x y) l"]]
  2735     unfolding dist_nz by auto
  2736   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
  2737 qed
  2738 
  2739 lemma sequence_infinite_lemma:
  2740   fixes f :: "nat \<Rightarrow> 'a::t1_space"
  2741   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
  2742   shows "infinite (range f)"
  2743 proof
  2744   assume "finite (range f)"
  2745   hence "closed (range f)" by (rule finite_imp_closed)
  2746   hence "open (- range f)" by (rule open_Compl)
  2747   from assms(1) have "l \<in> - range f" by auto
  2748   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
  2749     using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
  2750   thus False unfolding eventually_sequentially by auto
  2751 qed
  2752 
  2753 lemma closure_insert:
  2754   fixes x :: "'a::t1_space"
  2755   shows "closure (insert x s) = insert x (closure s)"
  2756 apply (rule closure_unique)
  2757 apply (rule conjI [OF insert_mono [OF closure_subset]])
  2758 apply (rule conjI [OF closed_insert [OF closed_closure]])
  2759 apply (simp add: closure_minimal)
  2760 done
  2761 
  2762 lemma islimpt_insert:
  2763   fixes x :: "'a::t1_space"
  2764   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
  2765 proof
  2766   assume *: "x islimpt (insert a s)"
  2767   show "x islimpt s"
  2768   proof (rule islimptI)
  2769     fix t assume t: "x \<in> t" "open t"
  2770     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
  2771     proof (cases "x = a")
  2772       case True
  2773       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
  2774         using * t by (rule islimptE)
  2775       with `x = a` show ?thesis by auto
  2776     next
  2777       case False
  2778       with t have t': "x \<in> t - {a}" "open (t - {a})"
  2779         by (simp_all add: open_Diff)
  2780       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
  2781         using * t' by (rule islimptE)
  2782       thus ?thesis by auto
  2783     qed
  2784   qed
  2785 next
  2786   assume "x islimpt s" thus "x islimpt (insert a s)"
  2787     by (rule islimpt_subset) auto
  2788 qed
  2789 
  2790 lemma islimpt_union_finite:
  2791   fixes x :: "'a::t1_space"
  2792   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
  2793 by (induct set: finite, simp_all add: islimpt_insert)
  2794  
  2795 lemma sequence_unique_limpt:
  2796   fixes f :: "nat \<Rightarrow> 'a::t2_space"
  2797   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
  2798   shows "l' = l"
  2799 proof (rule ccontr)
  2800   assume "l' \<noteq> l"
  2801   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
  2802     using hausdorff [OF `l' \<noteq> l`] by auto
  2803   have "eventually (\<lambda>n. f n \<in> t) sequentially"
  2804     using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
  2805   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
  2806     unfolding eventually_sequentially by auto
  2807 
  2808   have "UNIV = {..<N} \<union> {N..}" by auto
  2809   hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
  2810   hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
  2811   hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
  2812   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
  2813     using `l' \<in> s` `open s` by (rule islimptE)
  2814   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
  2815   with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
  2816   with `s \<inter> t = {}` show False by simp
  2817 qed
  2818 
  2819 lemma bolzano_weierstrass_imp_closed:
  2820   fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
  2821   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2822   shows "closed s"
  2823 proof-
  2824   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
  2825     hence "l \<in> s"
  2826     proof(cases "\<forall>n. x n \<noteq> l")
  2827       case False thus "l\<in>s" using as(1) by auto
  2828     next
  2829       case True note cas = this
  2830       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
  2831       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
  2832       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
  2833     qed  }
  2834   thus ?thesis unfolding closed_sequential_limits by fast
  2835 qed
  2836 
  2837 text {* Hence express everything as an equivalence. *}
  2838 
  2839 lemma compact_eq_heine_borel:
  2840   fixes s :: "'a::metric_space set"
  2841   shows "compact s \<longleftrightarrow>
  2842            (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
  2843                --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
  2844 proof
  2845   assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
  2846 next
  2847   assume ?rhs
  2848   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
  2849     by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
  2850   thus ?lhs by (rule bolzano_weierstrass_imp_compact)
  2851 qed
  2852 
  2853 lemma compact_eq_bolzano_weierstrass:
  2854   fixes s :: "'a::metric_space set"
  2855   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
  2856 proof
  2857   assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2858 next
  2859   assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
  2860 qed
  2861 
  2862 lemma compact_eq_bounded_closed:
  2863   fixes s :: "'a::heine_borel set"
  2864   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")
  2865 proof
  2866   assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
  2867 next
  2868   assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
  2869 qed
  2870 
  2871 lemma compact_imp_bounded:
  2872   fixes s :: "'a::metric_space set"
  2873   shows "compact s ==> bounded s"
  2874 proof -
  2875   assume "compact s"
  2876   hence "\<forall>f. (\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"
  2877     by (rule compact_imp_heine_borel)
  2878   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
  2879     using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2880   thus "bounded s"
  2881     by (rule bolzano_weierstrass_imp_bounded)
  2882 qed
  2883 
  2884 lemma compact_imp_closed:
  2885   fixes s :: "'a::metric_space set"
  2886   shows "compact s ==> closed s"
  2887 proof -
  2888   assume "compact s"
  2889   hence "\<forall>f. (\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"
  2890     by (rule compact_imp_heine_borel)
  2891   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
  2892     using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2893   thus "closed s"
  2894     by (rule bolzano_weierstrass_imp_closed)
  2895 qed
  2896 
  2897 text{* In particular, some common special cases. *}
  2898 
  2899 lemma compact_empty[simp]:
  2900  "compact {}"
  2901   unfolding compact_def
  2902   by simp
  2903 
  2904 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
  2905   unfolding subseq_def by simp (* TODO: move somewhere else *)
  2906 
  2907 lemma compact_union [intro]:
  2908   assumes "compact s" and "compact t"
  2909   shows "compact (s \<union> t)"
  2910 proof (rule compactI)
  2911   fix f :: "nat \<Rightarrow> 'a"
  2912   assume "\<forall>n. f n \<in> s \<union> t"
  2913   hence "infinite {n. f n \<in> s \<union> t}" by simp
  2914   hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
  2915   thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2916   proof
  2917     assume "infinite {n. f n \<in> s}"
  2918     from infinite_enumerate [OF this]
  2919     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
  2920     obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
  2921       using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
  2922     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
  2923       using `subseq q` by (simp_all add: subseq_o o_assoc)
  2924     thus ?thesis by auto
  2925   next
  2926     assume "infinite {n. f n \<in> t}"
  2927     from infinite_enumerate [OF this]
  2928     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
  2929     obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
  2930       using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
  2931     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
  2932       using `subseq q` by (simp_all add: subseq_o o_assoc)
  2933     thus ?thesis by auto
  2934   qed
  2935 qed
  2936 
  2937 lemma compact_inter_closed [intro]:
  2938   assumes "compact s" and "closed t"
  2939   shows "compact (s \<inter> t)"
  2940 proof (rule compactI)
  2941   fix f :: "nat \<Rightarrow> 'a"
  2942   assume "\<forall>n. f n \<in> s \<inter> t"
  2943   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
  2944   obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
  2945     using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
  2946   moreover
  2947   from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
  2948     unfolding closed_sequential_limits o_def by fast
  2949   ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2950     by auto
  2951 qed
  2952 
  2953 lemma closed_inter_compact [intro]:
  2954   assumes "closed s" and "compact t"
  2955   shows "compact (s \<inter> t)"
  2956   using compact_inter_closed [of t s] assms
  2957   by (simp add: Int_commute)
  2958 
  2959 lemma compact_inter [intro]:
  2960   assumes "compact s" and "compact t"
  2961   shows "compact (s \<inter> t)"
  2962   using assms by (intro compact_inter_closed compact_imp_closed)
  2963 
  2964 lemma compact_sing [simp]: "compact {a}"
  2965   unfolding compact_def o_def subseq_def
  2966   by (auto simp add: tendsto_const)
  2967 
  2968 lemma compact_insert [simp]:
  2969   assumes "compact s" shows "compact (insert x s)"
  2970 proof -
  2971   have "compact ({x} \<union> s)"
  2972     using compact_sing assms by (rule compact_union)
  2973   thus ?thesis by simp
  2974 qed
  2975 
  2976 lemma finite_imp_compact:
  2977   shows "finite s \<Longrightarrow> compact s"
  2978   by (induct set: finite) simp_all
  2979 
  2980 lemma compact_cball[simp]:
  2981   fixes x :: "'a::heine_borel"
  2982   shows "compact(cball x e)"
  2983   using compact_eq_bounded_closed bounded_cball closed_cball
  2984   by blast
  2985 
  2986 lemma compact_frontier_bounded[intro]:
  2987   fixes s :: "'a::heine_borel set"
  2988   shows "bounded s ==> compact(frontier s)"
  2989   unfolding frontier_def
  2990   using compact_eq_bounded_closed
  2991   by blast
  2992 
  2993 lemma compact_frontier[intro]:
  2994   fixes s :: "'a::heine_borel set"
  2995   shows "compact s ==> compact (frontier s)"
  2996   using compact_eq_bounded_closed compact_frontier_bounded
  2997   by blast
  2998 
  2999 lemma frontier_subset_compact:
  3000   fixes s :: "'a::heine_borel set"
  3001   shows "compact s ==> frontier s \<subseteq> s"
  3002   using frontier_subset_closed compact_eq_bounded_closed
  3003   by blast
  3004 
  3005 lemma open_delete:
  3006   fixes s :: "'a::t1_space set"
  3007   shows "open s \<Longrightarrow> open (s - {x})"
  3008   by (simp add: open_Diff)
  3009 
  3010 text{* Finite intersection property. I could make it an equivalence in fact. *}
  3011 
  3012 lemma compact_imp_fip:
  3013   assumes "compact s"  "\<forall>t \<in> f. closed t"
  3014         "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
  3015   shows "s \<inter> (\<Inter> f) \<noteq> {}"
  3016 proof
  3017   assume as:"s \<inter> (\<Inter> f) = {}"
  3018   hence "s \<subseteq> \<Union> uminus ` f" by auto
  3019   moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
  3020   ultimately obtain f' where f':"f' \<subseteq> uminus ` f"  "finite f'"  "s \<subseteq> \<Union>f'" using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="(\<lambda>t. - t) ` f"]] by auto
  3021   hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
  3022   hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
  3023   thus False using f'(3) unfolding subset_eq and Union_iff by blast
  3024 qed
  3025 
  3026 
  3027 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}
  3028 
  3029 lemma bounded_closed_nest:
  3030   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
  3031   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
  3032   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
  3033 proof-
  3034   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto
  3035   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
  3036 
  3037   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
  3038     unfolding compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast
  3039 
  3040   { fix n::nat
  3041     { fix e::real assume "e>0"
  3042       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
  3043       hence "dist ((x \<circ> r) (max N n)) l < e" by auto
  3044       moreover
  3045       have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
  3046       hence "(x \<circ> r) (max N n) \<in> s n"
  3047         using x apply(erule_tac x=n in allE)
  3048         using x apply(erule_tac x="r (max N n)" in allE)
  3049         using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
  3050       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
  3051     }
  3052     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
  3053   }
  3054   thus ?thesis by auto
  3055 qed
  3056 
  3057 text {* Decreasing case does not even need compactness, just completeness. *}
  3058 
  3059 lemma decreasing_closed_nest:
  3060   assumes "\<forall>n. closed(s n)"
  3061           "\<forall>n. (s n \<noteq> {})"
  3062           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3063           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
  3064   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
  3065 proof-
  3066   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
  3067   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
  3068   then obtain t where t: "\<forall>n. t n \<in> s n" by auto
  3069   { fix e::real assume "e>0"
  3070     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
  3071     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
  3072       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+
  3073       hence "dist (t m) (t n) < e" using N by auto
  3074     }
  3075     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
  3076   }
  3077   hence  "Cauchy t" unfolding cauchy_def by auto
  3078   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
  3079   { fix n::nat
  3080     { fix e::real assume "e>0"
  3081       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
  3082       have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto
  3083       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
  3084     }
  3085     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
  3086   }
  3087   then show ?thesis by auto
  3088 qed
  3089 
  3090 text {* Strengthen it to the intersection actually being a singleton. *}
  3091 
  3092 lemma decreasing_closed_nest_sing:
  3093   fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
  3094   assumes "\<forall>n. closed(s n)"
  3095           "\<forall>n. s n \<noteq> {}"
  3096           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3097           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
  3098   shows "\<exists>a. \<Inter>(range s) = {a}"
  3099 proof-
  3100   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
  3101   { fix b assume b:"b \<in> \<Inter>(range s)"
  3102     { fix e::real assume "e>0"
  3103       hence "dist a b < e" using assms(4 )using b using a by blast
  3104     }
  3105     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
  3106   }
  3107   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
  3108   thus ?thesis ..
  3109 qed
  3110 
  3111 text{* Cauchy-type criteria for uniform convergence. *}
  3112 
  3113 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
  3114  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
  3115   (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")
  3116 proof(rule)
  3117   assume ?lhs
  3118   then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e" by auto
  3119   { fix e::real assume "e>0"
  3120     then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto
  3121     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
  3122       hence "dist (s m x) (s n x) < e"
  3123         using N[THEN spec[where x=m], THEN spec[where x=x]]
  3124         using N[THEN spec[where x=n], THEN spec[where x=x]]
  3125         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }
  3126     hence "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e"  by auto  }
  3127   thus ?rhs by auto
  3128 next
  3129   assume ?rhs
  3130   hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto
  3131   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
  3132     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
  3133   { fix e::real assume "e>0"
  3134     then obtain N where N:"\<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e/2"
  3135       using `?rhs`[THEN spec[where x="e/2"]] by auto
  3136     { fix x assume "P x"
  3137       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
  3138         using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
  3139       fix n::nat assume "n\<ge>N"
  3140       hence "dist(s n x)(l x) < e"  using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
  3141         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute)  }
  3142     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
  3143   thus ?lhs by auto
  3144 qed
  3145 
  3146 lemma uniformly_cauchy_imp_uniformly_convergent:
  3147   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
  3148   assumes "\<forall>e>0.\<exists>N. \<forall>m (n::nat) x. N \<le> m \<and> N \<le> n \<and> P x --> dist(s m x)(s n x) < e"
  3149           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
  3150   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
  3151 proof-
  3152   obtain l' where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"
  3153     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
  3154   moreover
  3155   { fix x assume "P x"
  3156     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
  3157       using l and assms(2) unfolding Lim_sequentially by blast  }
  3158   ultimately show ?thesis by auto
  3159 qed
  3160 
  3161 
  3162 subsection {* Continuity *}
  3163 
  3164 text {* Define continuity over a net to take in restrictions of the set. *}
  3165 
  3166 definition
  3167   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3168   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
  3169 
  3170 lemma continuous_trivial_limit:
  3171  "trivial_limit net ==> continuous net f"
  3172   unfolding continuous_def tendsto_def trivial_limit_eq by auto
  3173 
  3174 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
  3175   unfolding continuous_def
  3176   unfolding tendsto_def
  3177   using netlimit_within[of x s]
  3178   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
  3179 
  3180 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
  3181   using continuous_within [of x UNIV f] by (simp add: within_UNIV)
  3182 
  3183 lemma continuous_at_within:
  3184   assumes "continuous (at x) f"  shows "continuous (at x within s) f"
  3185   using assms unfolding continuous_at continuous_within
  3186   by (rule Lim_at_within)
  3187 
  3188 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
  3189 
  3190 lemma continuous_within_eps_delta:
  3191   "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s.  dist x' x < d --> dist (f x') (f x) < e)"
  3192   unfolding continuous_within and Lim_within
  3193   apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
  3194 
  3195 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.
  3196                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
  3197   using continuous_within_eps_delta[of x UNIV f]
  3198   unfolding within_UNIV by blast
  3199 
  3200 text{* Versions in terms of open balls. *}
  3201 
  3202 lemma continuous_within_ball:
  3203  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  3204                             f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3205 proof
  3206   assume ?lhs
  3207   { fix e::real assume "e>0"
  3208     then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
  3209       using `?lhs`[unfolded continuous_within Lim_within] by auto
  3210     { fix y assume "y\<in>f ` (ball x d \<inter> s)"
  3211       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
  3212         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
  3213     }
  3214     hence "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e" using `d>0` unfolding subset_eq ball_def by (auto simp add: dist_commute)  }
  3215   thus ?rhs by auto
  3216 next
  3217   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
  3218     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
  3219 qed
  3220 
  3221 lemma continuous_at_ball:
  3222   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3223 proof
  3224   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3225     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)
  3226     unfolding dist_nz[THEN sym] by auto
  3227 next
  3228   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3229     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)
  3230 qed
  3231 
  3232 text{* Define setwise continuity in terms of limits within the set. *}
  3233 
  3234 definition
  3235   continuous_on ::
  3236     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3237 where
  3238   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
  3239 
  3240 lemma continuous_on_topological:
  3241   "continuous_on s f \<longleftrightarrow>
  3242     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  3243       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  3244 unfolding continuous_on_def tendsto_def
  3245 unfolding Limits.eventually_within eventually_at_topological
  3246 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  3247 
  3248 lemma continuous_on_iff:
  3249   "continuous_on s f \<longleftrightarrow>
  3250     (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
  3251 unfolding continuous_on_def Lim_within
  3252 apply (intro ball_cong [OF refl] all_cong ex_cong)
  3253 apply (rename_tac y, case_tac "y = x", simp)
  3254 apply (simp add: dist_nz)
  3255 done
  3256 
  3257 definition
  3258   uniformly_continuous_on ::
  3259     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
  3260 where
  3261   "uniformly_continuous_on s f \<longleftrightarrow>
  3262     (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
  3263 
  3264 text{* Some simple consequential lemmas. *}
  3265 
  3266 lemma uniformly_continuous_imp_continuous:
  3267  " uniformly_continuous_on s f ==> continuous_on s f"
  3268   unfolding uniformly_continuous_on_def continuous_on_iff by blast
  3269 
  3270 lemma continuous_at_imp_continuous_within:
  3271  "continuous (at x) f ==> continuous (at x within s) f"
  3272   unfolding continuous_within continuous_at using Lim_at_within by auto
  3273 
  3274 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
  3275 unfolding tendsto_def by (simp add: trivial_limit_eq)
  3276 
  3277 lemma continuous_at_imp_continuous_on:
  3278   assumes "\<forall>x\<in>s. continuous (at x) f"
  3279   shows "continuous_on s f"
  3280 unfolding continuous_on_def
  3281 proof
  3282   fix x assume "x \<in> s"
  3283   with assms have *: "(f ---> f (netlimit (at x))) (at x)"
  3284     unfolding continuous_def by simp
  3285   have "(f ---> f x) (at x)"
  3286   proof (cases "trivial_limit (at x)")
  3287     case True thus ?thesis
  3288       by (rule Lim_trivial_limit)
  3289   next
  3290     case False
  3291     hence 1: "netlimit (at x) = x"
  3292       using netlimit_within [of x UNIV]
  3293       by (simp add: within_UNIV)
  3294     with * show ?thesis by simp
  3295   qed
  3296   thus "(f ---> f x) (at x within s)"
  3297     by (rule Lim_at_within)
  3298 qed
  3299 
  3300 lemma continuous_on_eq_continuous_within:
  3301   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
  3302 unfolding continuous_on_def continuous_def
  3303 apply (rule ball_cong [OF refl])
  3304 apply (case_tac "trivial_limit (at x within s)")
  3305 apply (simp add: Lim_trivial_limit)
  3306 apply (simp add: netlimit_within)
  3307 done
  3308 
  3309 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
  3310 
  3311 lemma continuous_on_eq_continuous_at:
  3312   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
  3313   by (auto simp add: continuous_on continuous_at Lim_within_open)
  3314 
  3315 lemma continuous_within_subset:
  3316  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
  3317              ==> continuous (at x within t) f"
  3318   unfolding continuous_within by(metis Lim_within_subset)
  3319 
  3320 lemma continuous_on_subset:
  3321   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
  3322   unfolding continuous_on by (metis subset_eq Lim_within_subset)
  3323 
  3324 lemma continuous_on_interior:
  3325   shows "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
  3326 unfolding interior_def
  3327 apply simp
  3328 by (meson continuous_on_eq_continuous_at continuous_on_subset)
  3329 
  3330 lemma continuous_on_eq:
  3331   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
  3332   unfolding continuous_on_def tendsto_def Limits.eventually_within
  3333   by simp
  3334 
  3335 text {* Characterization of various kinds of continuity in terms of sequences. *}
  3336 
  3337 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
  3338 lemma continuous_within_sequentially:
  3339   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3340   shows "continuous (at a within s) f \<longleftrightarrow>
  3341                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
  3342                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
  3343 proof
  3344   assume ?lhs
  3345   { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (x n) a < e"
  3346     fix e::real assume "e>0"
  3347     from `?lhs` obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) (f a) < e" unfolding continuous_within Lim_within using `e>0` by auto
  3348     from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
  3349     hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
  3350       apply(rule_tac  x=N in exI) using N d  apply auto using x(1)
  3351       apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
  3352       apply(erule_tac x="x n" in ballE)  apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
  3353   }
  3354   thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
  3355 next
  3356   assume ?rhs
  3357   { fix e::real assume "e>0"
  3358     assume "\<not> (\<exists>d>0. \<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) (f a) < e)"
  3359     hence "\<forall>d. \<exists>x. d>0 \<longrightarrow> x\<in>s \<and> (0 < dist x a \<and> dist x a < d \<and> \<not> dist (f x) (f a) < e)" by blast
  3360     then obtain x where x:"\<forall>d>0. x d \<in> s \<and> (0 < dist (x d) a \<and> dist (x d) a < d \<and> \<not> dist (f (x d)) (f a) < e)"
  3361       using choice[of "\<lambda>d x.0<d \<longrightarrow> x\<in>s \<and> (0 < dist x a \<and> dist x a < d \<and> \<not> dist (f x) (f a) < e)"] by auto
  3362     { fix d::real assume "d>0"
  3363       hence "\<exists>N::nat. inverse (real (N + 1)) < d" using real_arch_inv[of d] by (auto, rule_tac x="n - 1" in exI)auto
  3364       then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
  3365       { fix n::nat assume n:"n\<ge>N"
  3366         hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
  3367         moreover have "inverse (real (n + 1)) < d" using N n by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
  3368         ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
  3369       }
  3370       hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
  3371     }
  3372     hence "(\<forall>n::nat. x (inverse (real (n + 1))) \<in> s) \<and> (\<forall>e>0. \<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < e)" using x by auto
  3373     hence "\<forall>e>0. \<exists>N::nat. \<forall>n\<ge>N. dist (f (x (inverse (real (n + 1))))) (f a) < e"  using `?rhs`[THEN spec[where x="\<lambda>n::nat. x (inverse (real (n+1)))"], unfolded Lim_sequentially] by auto
  3374     hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
  3375   }
  3376   thus ?lhs  unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
  3377 qed
  3378 
  3379 lemma continuous_at_sequentially:
  3380   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3381   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
  3382                   --> ((f o x) ---> f a) sequentially)"
  3383   using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
  3384 
  3385 lemma continuous_on_sequentially:
  3386   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3387   shows "continuous_on s f \<longleftrightarrow>
  3388     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
  3389                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
  3390 proof
  3391   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
  3392 next
  3393   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
  3394 qed
  3395 
  3396 lemma uniformly_continuous_on_sequentially':
  3397   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
  3398                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
  3399                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
  3400 proof
  3401   assume ?lhs
  3402   { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
  3403     { fix e::real assume "e>0"
  3404       then obtain d where "d>0" and d:"\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
  3405         using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
  3406       obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded Lim_sequentially dist_norm] and `d>0` by auto
  3407       { fix n assume "n\<ge>N"
  3408         hence "dist (f (x n)) (f (y n)) < e"
  3409           using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y
  3410           unfolding dist_commute by simp  }
  3411       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }
  3412     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto  }
  3413   thus ?rhs by auto
  3414 next
  3415   assume ?rhs
  3416   { assume "\<not> ?lhs"
  3417     then obtain e where "e>0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto
  3418     then obtain fa where fa:"\<forall>x.  0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
  3419       using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def
  3420       by (auto simp add: dist_commute)
  3421     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
  3422     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
  3423     have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
  3424       unfolding x_def and y_def using fa by auto
  3425     { fix e::real assume "e>0"
  3426       then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e]   by auto
  3427       { fix n::nat assume "n\<ge>N"
  3428         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
  3429         also have "\<dots> < e" using N by auto
  3430         finally have "inverse (real n + 1) < e" by auto
  3431         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }
  3432       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }
  3433     hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding Lim_sequentially dist_real_def by auto
  3434     hence False using fxy and `e>0` by auto  }
  3435   thus ?lhs unfolding uniformly_continuous_on_def by blast
  3436 qed
  3437 
  3438 lemma uniformly_continuous_on_sequentially:
  3439   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  3440   shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
  3441                     ((\<lambda>n. x n - y n) ---> 0) sequentially
  3442                     \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
  3443 (* BH: maybe the previous lemma should replace this one? *)
  3444 unfolding uniformly_continuous_on_sequentially'
  3445 unfolding dist_norm tendsto_norm_zero_iff ..
  3446 
  3447 text{* The usual transformation theorems. *}
  3448 
  3449 lemma continuous_transform_within:
  3450   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3451   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
  3452           "continuous (at x within s) f"
  3453   shows "continuous (at x within s) g"
  3454 unfolding continuous_within
  3455 proof (rule Lim_transform_within)
  3456   show "0 < d" by fact
  3457   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  3458     using assms(3) by auto
  3459   have "f x = g x"
  3460     using assms(1,2,3) by auto
  3461   thus "(f ---> g x) (at x within s)"
  3462     using assms(4) unfolding continuous_within by simp
  3463 qed
  3464 
  3465 lemma continuous_transform_at:
  3466   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3467   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
  3468           "continuous (at x) f"
  3469   shows "continuous (at x) g"
  3470   using continuous_transform_within [of d x UNIV f g] assms
  3471   by (simp add: within_UNIV)
  3472 
  3473 text{* Combination results for pointwise continuity. *}
  3474 
  3475 lemma continuous_const: "continuous net (\<lambda>x. c)"
  3476   by (auto simp add: continuous_def tendsto_const)
  3477 
  3478 lemma continuous_cmul:
  3479   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3480   shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
  3481   by (auto simp add: continuous_def intro: tendsto_intros)
  3482 
  3483 lemma continuous_neg:
  3484   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3485   shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
  3486   by (auto simp add: continuous_def tendsto_minus)
  3487 
  3488 lemma continuous_add:
  3489   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3490   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
  3491   by (auto simp add: continuous_def tendsto_add)
  3492 
  3493 lemma continuous_sub:
  3494   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3495   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
  3496   by (auto simp add: continuous_def tendsto_diff)
  3497 
  3498 
  3499 text{* Same thing for setwise continuity. *}
  3500 
  3501 lemma continuous_on_const:
  3502  "continuous_on s (\<lambda>x. c)"
  3503   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3504 
  3505 lemma continuous_on_cmul:
  3506   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3507   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
  3508   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3509 
  3510 lemma continuous_on_neg:
  3511   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3512   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
  3513   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3514 
  3515 lemma continuous_on_add:
  3516   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3517   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3518            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
  3519   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3520 
  3521 lemma continuous_on_sub:
  3522   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3523   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3524            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
  3525   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3526 
  3527 text{* Same thing for uniform continuity, using sequential formulations. *}
  3528 
  3529 lemma uniformly_continuous_on_const:
  3530  "uniformly_continuous_on s (\<lambda>x. c)"
  3531   unfolding uniformly_continuous_on_def by simp
  3532 
  3533 lemma uniformly_continuous_on_cmul:
  3534   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3535   assumes "uniformly_continuous_on s f"
  3536   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
  3537 proof-
  3538   { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
  3539     hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
  3540       using tendsto_scaleR [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
  3541       unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
  3542   }
  3543   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
  3544     unfolding dist_norm tendsto_norm_zero_iff by auto
  3545 qed
  3546 
  3547 lemma dist_minus:
  3548   fixes x y :: "'a::real_normed_vector"
  3549   shows "dist (- x) (- y) = dist x y"
  3550   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
  3551 
  3552 lemma uniformly_continuous_on_neg:
  3553   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3554   shows "uniformly_continuous_on s f
  3555          ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
  3556   unfolding uniformly_continuous_on_def dist_minus .
  3557 
  3558 lemma uniformly_continuous_on_add:
  3559   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3560   assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
  3561   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
  3562 proof-
  3563   {  fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
  3564                     "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
  3565     hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
  3566       using tendsto_add[of "\<lambda> n. f (x n) - f (y n)" 0  sequentially "\<lambda> n. g (x n) - g (y n)" 0] by auto
  3567     hence "((\<lambda>n. f (x n) + g (x n) - (f (y n) + g (y n))) ---> 0) sequentially" unfolding Lim_sequentially and add_diff_add [symmetric] by auto  }
  3568   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
  3569     unfolding dist_norm tendsto_norm_zero_iff by auto
  3570 qed
  3571 
  3572 lemma uniformly_continuous_on_sub:
  3573   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3574   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
  3575            ==> uniformly_continuous_on s  (\<lambda>x. f x - g x)"
  3576   unfolding ab_diff_minus
  3577   using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
  3578   using uniformly_continuous_on_neg[of s g] by auto
  3579 
  3580 text{* Identity function is continuous in every sense. *}
  3581 
  3582 lemma continuous_within_id:
  3583  "continuous (at a within s) (\<lambda>x. x)"
  3584   unfolding continuous_within by (rule Lim_at_within [OF LIM_ident])
  3585 
  3586 lemma continuous_at_id:
  3587  "continuous (at a) (\<lambda>x. x)"
  3588   unfolding continuous_at by (rule LIM_ident)
  3589 
  3590 lemma continuous_on_id:
  3591  "continuous_on s (\<lambda>x. x)"
  3592   unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)
  3593 
  3594 lemma uniformly_continuous_on_id:
  3595  "uniformly_continuous_on s (\<lambda>x. x)"
  3596   unfolding uniformly_continuous_on_def by auto
  3597 
  3598 text{* Continuity of all kinds is preserved under composition. *}
  3599 
  3600 lemma continuous_within_topological:
  3601   "continuous (at x within s) f \<longleftrightarrow>
  3602     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  3603       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  3604 unfolding continuous_within
  3605 unfolding tendsto_def Limits.eventually_within eventually_at_topological
  3606 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  3607 
  3608 lemma continuous_within_compose:
  3609   assumes "continuous (at x within s) f"
  3610   assumes "continuous (at (f x) within f ` s) g"
  3611   shows "continuous (at x within s) (g o f)"
  3612 using assms unfolding continuous_within_topological by simp metis
  3613 
  3614 lemma continuous_at_compose:
  3615   assumes "continuous (at x) f"  "continuous (at (f x)) g"
  3616   shows "continuous (at x) (g o f)"
  3617 proof-
  3618   have " continuous (at (f x) within range f) g" using assms(2) using continuous_within_subset[of "f x" UNIV g "range f", unfolded within_UNIV] by auto
  3619   thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
  3620 qed
  3621 
  3622 lemma continuous_on_compose:
  3623   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
  3624   unfolding continuous_on_topological by simp metis
  3625 
  3626 lemma uniformly_continuous_on_compose:
  3627   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f ` s) g"
  3628   shows "uniformly_continuous_on s (g o f)"
  3629 proof-
  3630   { fix e::real assume "e>0"
  3631     then obtain d where "d>0" and d:"\<forall>x\<in>f ` s. \<forall>x'\<in>f ` s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto
  3632     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using `d>0` using assms(1) unfolding uniformly_continuous_on_def by auto
  3633     hence "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e" using `d>0` using d by auto  }
  3634   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
  3635 qed
  3636 
  3637 text{* Continuity in terms of open preimages. *}
  3638 
  3639 lemma continuous_at_open:
  3640   shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
  3641 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
  3642 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
  3643 
  3644 lemma continuous_on_open:
  3645   shows "continuous_on s f \<longleftrightarrow>
  3646         (\<forall>t. openin (subtopology euclidean (f ` s)) t
  3647             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  3648 proof (safe)
  3649   fix t :: "'b set"
  3650   assume 1: "continuous_on s f"
  3651   assume 2: "openin (subtopology euclidean (f ` s)) t"
  3652   from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
  3653     unfolding openin_open by auto
  3654   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
  3655   have "open U" unfolding U_def by (simp add: open_Union)
  3656   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
  3657   proof (intro ballI iffI)
  3658     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
  3659       unfolding U_def t by auto
  3660   next
  3661     fix x assume "x \<in> s" and "f x \<in> t"
  3662     hence "x \<in> s" and "f x \<in> B"
  3663       unfolding t by auto
  3664     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
  3665       unfolding t continuous_on_topological by metis
  3666     then show "x \<in> U"
  3667       unfolding U_def by auto
  3668   qed
  3669   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
  3670   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3671     unfolding openin_open by fast
  3672 next
  3673   assume "?rhs" show "continuous_on s f"
  3674   unfolding continuous_on_topological
  3675   proof (clarify)
  3676     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
  3677     have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
  3678       unfolding openin_open using `open B` by auto
  3679     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
  3680       using `?rhs` by fast
  3681     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
  3682       unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
  3683   qed
  3684 qed
  3685 
  3686 text {* Similarly in terms of closed sets. *}
  3687 
  3688 lemma continuous_on_closed:
  3689   shows "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f ` s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  3690 proof
  3691   assume ?lhs
  3692   { fix t
  3693     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  3694     have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
  3695     assume as:"closedin (subtopology euclidean (f ` s)) t"
  3696     hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
  3697     hence "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?lhs`[unfolded continuous_on_open, THEN spec[where x="(f ` s) - t"]]
  3698       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }
  3699   thus ?rhs by auto
  3700 next
  3701   assume ?rhs
  3702   { fix t
  3703     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  3704     assume as:"openin (subtopology euclidean (f ` s)) t"
  3705     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
  3706       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
  3707   thus ?lhs unfolding continuous_on_open by auto
  3708 qed
  3709 
  3710 text {* Half-global and completely global cases. *}
  3711 
  3712 lemma continuous_open_in_preimage:
  3713   assumes "continuous_on s f"  "open t"
  3714   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3715 proof-
  3716   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
  3717   have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  3718     using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
  3719   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  3720 qed
  3721 
  3722 lemma continuous_closed_in_preimage:
  3723   assumes "continuous_on s f"  "closed t"
  3724   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3725 proof-
  3726   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
  3727   have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  3728     using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
  3729   thus ?thesis
  3730     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  3731 qed
  3732 
  3733 lemma continuous_open_preimage:
  3734   assumes "continuous_on s f" "open s" "open t"
  3735   shows "open {x \<in> s. f x \<in> t}"
  3736 proof-
  3737   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  3738     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
  3739   thus ?thesis using open_Int[of s T, OF assms(2)] by auto
  3740 qed
  3741 
  3742 lemma continuous_closed_preimage:
  3743   assumes "continuous_on s f" "closed s" "closed t"
  3744   shows "closed {x \<in> s. f x \<in> t}"
  3745 proof-
  3746   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  3747     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
  3748   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
  3749 qed
  3750 
  3751 lemma continuous_open_preimage_univ:
  3752   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
  3753   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
  3754 
  3755 lemma continuous_closed_preimage_univ:
  3756   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
  3757   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
  3758 
  3759 lemma continuous_open_vimage:
  3760   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
  3761   unfolding vimage_def by (rule continuous_open_preimage_univ)
  3762 
  3763 lemma continuous_closed_vimage:
  3764   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
  3765   unfolding vimage_def by (rule continuous_closed_preimage_univ)
  3766 
  3767 lemma interior_image_subset:
  3768   assumes "\<forall>x. continuous (at x) f" "inj f"
  3769   shows "interior (f ` s) \<subseteq> f ` (interior s)"
  3770   apply rule unfolding interior_def mem_Collect_eq image_iff apply safe
  3771 proof- fix x T assume as:"open T" "x \<in> T" "T \<subseteq> f ` s" 
  3772   hence "x \<in> f ` s" by auto then guess y unfolding image_iff .. note y=this
  3773   thus "\<exists>xa\<in>{x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> s}. x = f xa" apply(rule_tac x=y in bexI) using assms as
  3774     apply safe apply(rule_tac x="{x. f x \<in> T}" in exI) apply(safe,rule continuous_open_preimage_univ)
  3775   proof- fix x assume "f x \<in> T" hence "f x \<in> f ` s" using as by auto
  3776     thus "x \<in> s" unfolding inj_image_mem_iff[OF assms(2)] . qed auto qed
  3777 
  3778 text {* Equality of continuous functions on closure and related results. *}
  3779 
  3780 lemma continuous_closed_in_preimage_constant:
  3781   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3782   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
  3783   using continuous_closed_in_preimage[of s f "{a}"] by auto
  3784 
  3785 lemma continuous_closed_preimage_constant:
  3786   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3787   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
  3788   using continuous_closed_preimage[of s f "{a}"] by auto
  3789 
  3790 lemma continuous_constant_on_closure:
  3791   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3792   assumes "continuous_on (closure s) f"
  3793           "\<forall>x \<in> s. f x = a"
  3794   shows "\<forall>x \<in> (closure s). f x = a"
  3795     using continuous_closed_preimage_constant[of "closure s" f a]
  3796     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
  3797 
  3798 lemma image_closure_subset:
  3799   assumes "continuous_on (closure s) f"  "closed t"  "(f ` s) \<subseteq> t"
  3800   shows "f ` (closure s) \<subseteq> t"
  3801 proof-
  3802   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
  3803   moreover have "closed {x \<in> closure s. f x \<in> t}"
  3804     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
  3805   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
  3806     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
  3807   thus ?thesis by auto
  3808 qed
  3809 
  3810 lemma continuous_on_closure_norm_le:
  3811   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3812   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"
  3813   shows "norm(f x) \<le> b"
  3814 proof-
  3815   have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
  3816   show ?thesis
  3817     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
  3818     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
  3819 qed
  3820 
  3821 text {* Making a continuous function avoid some value in a neighbourhood. *}
  3822 
  3823 lemma continuous_within_avoid:
  3824   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3825   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"
  3826   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
  3827 proof-
  3828   obtain d where "d>0" and d:"\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < dist (f x) a"
  3829     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
  3830   { fix y assume " y\<in>s"  "dist x y < d"
  3831     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
  3832       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
  3833   thus ?thesis using `d>0` by auto
  3834 qed
  3835 
  3836 lemma continuous_at_avoid:
  3837   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3838   assumes "continuous (at x) f"  "f x \<noteq> a"
  3839   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  3840 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
  3841 
  3842 lemma continuous_on_avoid:
  3843   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
  3844   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"
  3845   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
  3846 using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)]  continuous_within_avoid[of x s f a]  assms(2,3) by auto
  3847 
  3848 lemma continuous_on_open_avoid:
  3849   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
  3850   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"
  3851   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  3852 using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]  continuous_at_avoid[of x f a]  assms(3,4) by auto
  3853 
  3854 text {* Proving a function is constant by proving open-ness of level set. *}
  3855 
  3856 lemma continuous_levelset_open_in_cases:
  3857   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3858   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  3859         openin (subtopology euclidean s) {x \<in> s. f x = a}
  3860         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
  3861 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
  3862 
  3863 lemma continuous_levelset_open_in:
  3864   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3865   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  3866         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
  3867         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"
  3868 using continuous_levelset_open_in_cases[of s f ]
  3869 by meson
  3870 
  3871 lemma continuous_levelset_open:
  3872   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3873   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
  3874   shows "\<forall>x \<in> s. f x = a"
  3875 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
  3876 
  3877 text {* Some arithmetical combinations (more to prove). *}
  3878 
  3879 lemma open_scaling[intro]:
  3880   fixes s :: "'a::real_normed_vector set"
  3881   assumes "c \<noteq> 0"  "open s"
  3882   shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
  3883 proof-
  3884   { fix x assume "x \<in> s"
  3885     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto
  3886     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
  3887     moreover
  3888     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
  3889       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
  3890         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
  3891           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
  3892       hence "y \<in> op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]  e[THEN spec[where x="(1 / c) *\<^sub>R y"]]  assms(1) unfolding dist_norm scaleR_scaleR by auto  }
  3893     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto  }
  3894   thus ?thesis unfolding open_dist by auto
  3895 qed
  3896 
  3897 lemma minus_image_eq_vimage:
  3898   fixes A :: "'a::ab_group_add set"
  3899   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
  3900   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
  3901 
  3902 lemma open_negations:
  3903   fixes s :: "'a::real_normed_vector set"
  3904   shows "open s ==> open ((\<lambda> x. -x) ` s)"
  3905   unfolding scaleR_minus1_left [symmetric]
  3906   by (rule open_scaling, auto)
  3907 
  3908 lemma open_translation:
  3909   fixes s :: "'a::real_normed_vector set"
  3910   assumes "open s"  shows "open((\<lambda>x. a + x) ` s)"
  3911 proof-
  3912   { fix x have "continuous (at x) (\<lambda>x. x - a)" using continuous_sub[of "at x" "\<lambda>x. x" "\<lambda>x. a"] continuous_at_id[of x] continuous_const[of "at x" a] by auto  }
  3913   moreover have "{x. x - a \<in> s}  = op + a ` s" apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
  3914   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
  3915 qed
  3916 
  3917 lemma open_affinity:
  3918   fixes s :: "'a::real_normed_vector set"
  3919   assumes "open s"  "c \<noteq> 0"
  3920   shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  3921 proof-
  3922   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
  3923   have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
  3924   thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
  3925 qed
  3926 
  3927 lemma interior_translation:
  3928   fixes s :: "'a::real_normed_vector set"
  3929   shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
  3930 proof (rule set_eqI, rule)
  3931   fix x assume "x \<in> interior (op + a ` s)"
  3932   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
  3933   hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto
  3934   thus "x \<in> op + a ` interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using `e > 0` by auto
  3935 next
  3936   fix x assume "x \<in> op + a ` interior s"
  3937   then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto
  3938   { fix z have *:"a + y - z = y + a - z" by auto
  3939     assume "z\<in>ball x e"
  3940     hence "z - a \<in> s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y ab_group_add_class.diff_diff_eq2 * by auto
  3941     hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }
  3942   hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
  3943   thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
  3944 qed
  3945 
  3946 text {* We can now extend limit compositions to consider the scalar multiplier. *}
  3947 
  3948 lemma continuous_vmul:
  3949   fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
  3950   shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
  3951   unfolding continuous_def by (intro tendsto_intros)
  3952 
  3953 lemma continuous_mul:
  3954   fixes c :: "'a::metric_space \<Rightarrow> real"
  3955   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3956   shows "continuous net c \<Longrightarrow> continuous net f
  3957              ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
  3958   unfolding continuous_def by (intro tendsto_intros)
  3959 
  3960 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul
  3961   continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
  3962 
  3963 lemma continuous_on_vmul:
  3964   fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
  3965   shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
  3966   unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
  3967 
  3968 lemma continuous_on_mul:
  3969   fixes c :: "'a::metric_space \<Rightarrow> real"
  3970   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3971   shows "continuous_on s c \<Longrightarrow> continuous_on s f
  3972              ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
  3973   unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
  3974 
  3975 lemma continuous_on_mul_real:
  3976   fixes f :: "'a::metric_space \<Rightarrow> real"
  3977   fixes g :: "'a::metric_space \<Rightarrow> real"
  3978   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3979              ==> continuous_on s (\<lambda>x. f x * g x)"
  3980   using continuous_on_mul[of s f g] unfolding real_scaleR_def .
  3981 
  3982 lemmas continuous_on_intros = continuous_on_add continuous_on_const
  3983   continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg
  3984   continuous_on_sub continuous_on_mul continuous_on_vmul continuous_on_mul_real
  3985   uniformly_continuous_on_add uniformly_continuous_on_const
  3986   uniformly_continuous_on_id uniformly_continuous_on_compose
  3987   uniformly_continuous_on_cmul uniformly_continuous_on_neg
  3988   uniformly_continuous_on_sub
  3989 
  3990 text {* And so we have continuity of inverse. *}
  3991 
  3992 lemma continuous_inv:
  3993   fixes f :: "'a::metric_space \<Rightarrow> real"
  3994   shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
  3995            ==> continuous net (inverse o f)"
  3996   unfolding continuous_def using Lim_inv by auto
  3997 
  3998 lemma continuous_at_within_inv:
  3999   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
  4000   assumes "continuous (at a within s) f" "f a \<noteq> 0"
  4001   shows "continuous (at a within s) (inverse o f)"
  4002   using assms unfolding continuous_within o_def
  4003   by (intro tendsto_intros)
  4004 
  4005 lemma continuous_at_inv:
  4006   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
  4007   shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
  4008          ==> continuous (at a) (inverse o f) "
  4009   using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
  4010 
  4011 text {* Topological properties of linear functions. *}
  4012 
  4013 lemma linear_lim_0:
  4014   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
  4015 proof-
  4016   interpret f: bounded_linear f by fact
  4017   have "(f ---> f 0) (at 0)"
  4018     using tendsto_ident_at by (rule f.tendsto)
  4019   thus ?thesis unfolding f.zero .
  4020 qed
  4021 
  4022 lemma linear_continuous_at:
  4023   assumes "bounded_linear f"  shows "continuous (at a) f"
  4024   unfolding continuous_at using assms
  4025   apply (rule bounded_linear.tendsto)
  4026   apply (rule tendsto_ident_at)
  4027   done
  4028 
  4029 lemma linear_continuous_within:
  4030   shows "bounded_linear f ==> continuous (at x within s) f"
  4031   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
  4032 
  4033 lemma linear_continuous_on:
  4034   shows "bounded_linear f ==> continuous_on s f"
  4035   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
  4036 
  4037 text {* Also bilinear functions, in composition form. *}
  4038 
  4039 lemma bilinear_continuous_at_compose:
  4040   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
  4041         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
  4042   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
  4043 
  4044 lemma bilinear_continuous_within_compose:
  4045   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
  4046         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
  4047   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
  4048 
  4049 lemma bilinear_continuous_on_compose:
  4050   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
  4051              ==> continuous_on s (\<lambda>x. h (f x) (g x))"
  4052   unfolding continuous_on_def
  4053   by (fast elim: bounded_bilinear.tendsto)
  4054 
  4055 text {* Preservation of compactness and connectedness under continuous function. *}
  4056 
  4057 lemma compact_continuous_image:
  4058   assumes "continuous_on s f"  "compact s"
  4059   shows "compact(f ` s)"
  4060 proof-
  4061   { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
  4062     then obtain y where y:"\<forall>n. y n \<in> s \<and> x n = f (y n)" unfolding image_iff Bex_def using choice[of "\<lambda>n xa. xa \<in> s \<and> x n = f xa"] by auto
  4063     then obtain l r where "l\<in>s" and r:"subseq r" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto
  4064     { fix e::real assume "e>0"
  4065       then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=l], OF `l\<in>s`] by auto
  4066       then obtain N::nat where N:"\<forall>n\<ge>N. dist ((y \<circ> r) n) l < d" using lr[unfolded Lim_sequentially, THEN spec[where x=d]] by auto
  4067       { fix n::nat assume "n\<ge>N" hence "dist ((x \<circ> r) n) (f l) < e" using N[THEN spec[where x=n]] d[THEN bspec[where x="y (r n)"]] y[THEN spec[where x="r n"]] by auto  }
  4068       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }
  4069     hence "\<exists>l\<in>f ` s. \<exists>r. subseq r \<and> ((x \<circ> r) ---> l) sequentially" unfolding Lim_sequentially using r lr `l\<in>s` by auto  }
  4070   thus ?thesis unfolding compact_def by auto
  4071 qed
  4072 
  4073 lemma connected_continuous_image:
  4074   assumes "continuous_on s f"  "connected s"
  4075   shows "connected(f ` s)"
  4076 proof-
  4077   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f ` s"  "openin (subtopology euclidean (f ` s)) T"  "closedin (subtopology euclidean (f ` s)) T"
  4078     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
  4079       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
  4080       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
  4081       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
  4082     hence False using as(1,2)
  4083       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
  4084   thus ?thesis unfolding connected_clopen by auto
  4085 qed
  4086 
  4087 text {* Continuity implies uniform continuity on a compact domain. *}
  4088 
  4089 lemma compact_uniformly_continuous:
  4090   assumes "continuous_on s f"  "compact s"
  4091   shows "uniformly_continuous_on s f"
  4092 proof-
  4093     { fix x assume x:"x\<in>s"
  4094       hence "\<forall>xa. \<exists>y. 0 < xa \<longrightarrow> (y > 0 \<and> (\<forall>x'\<in>s. dist x' x < y \<longrightarrow> dist (f x') (f x) < xa))" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=x]] by auto
  4095       hence "\<exists>fa. \<forall>xa>0. \<forall>x'\<in>s. fa xa > 0 \<and> (dist x' x < fa xa \<longrightarrow> dist (f x') (f x) < xa)" using choice[of "\<lambda>e d. e>0 \<longrightarrow> d>0 \<and>(\<forall>x'\<in>s. (dist x' x < d \<longrightarrow> dist (f x') (f x) < e))"] by auto  }
  4096     then have "\<forall>x\<in>s. \<exists>y. \<forall>xa. 0 < xa \<longrightarrow> (\<forall>x'\<in>s. y xa > 0 \<and> (dist x' x < y xa \<longrightarrow> dist (f x') (f x) < xa))" by auto
  4097     then obtain d where d:"\<forall>e>0. \<forall>x\<in>s. \<forall>x'\<in>s. d x e > 0 \<and> (dist x' x < d x e \<longrightarrow> dist (f x') (f x) < e)"
  4098       using bchoice[of s "\<lambda>x fa. \<forall>xa>0. \<forall>x'\<in>s. fa xa > 0 \<and> (dist x' x < fa xa \<longrightarrow> dist (f x') (f x) < xa)"] by blast
  4099 
  4100   { fix e::real assume "e>0"
  4101 
  4102     { fix x assume "x\<in>s" hence "x \<in> ball x (d x (e / 2))" unfolding centre_in_ball using d[THEN spec[where x="e/2"]] using `e>0` by auto  }
  4103     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
  4104     moreover
  4105     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }
  4106     ultimately obtain ea where "ea>0" and ea:"\<forall>x\<in>s. \<exists>b\<in>{ball x (d x (e / 2)) |x. x \<in> s}. ball x ea \<subseteq> b" using heine_borel_lemma[OF assms(2), of "{ball x (d x (e / 2)) | x. x\<in>s }"] by auto
  4107 
  4108     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
  4109       obtain z where "z\<in>s" and z:"ball x ea \<subseteq> ball z (d z (e / 2))" using ea[THEN bspec[where x=x]] and `x\<in>s` by auto
  4110       hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
  4111       hence "dist (f z) (f x) < e / 2" using d[THEN spec[where x="e/2"]] and `e>0` and `x\<in>s` and `z\<in>s`
  4112         by (auto  simp add: dist_commute)
  4113       moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
  4114         by (auto simp add: dist_commute)
  4115       hence "dist (f z) (f y) < e / 2" using d[THEN spec[where x="e/2"]] and `e>0` and `y\<in>s` and `z\<in>s`
  4116         by (auto  simp add: dist_commute)
  4117       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
  4118         by (auto simp add: dist_commute)  }
  4119     then have "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using `ea>0` by auto  }
  4120   thus ?thesis unfolding uniformly_continuous_on_def by auto
  4121 qed
  4122 
  4123 text{* Continuity of inverse function on compact domain. *}
  4124 
  4125 lemma continuous_on_inverse:
  4126   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  4127     (* TODO: can this be generalized more? *)
  4128   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"
  4129   shows "continuous_on (f ` s) g"
  4130 proof-
  4131   have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
  4132   { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
  4133     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
  4134     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
  4135       unfolding T(2) and Int_left_absorb by auto
  4136     moreover have "compact (s \<inter> T)"
  4137       using assms(2) unfolding compact_eq_bounded_closed
  4138       using bounded_subset[of s "s \<inter> T"] and T(1) by auto
  4139     ultimately have "closed (f ` t)" using T(1) unfolding T(2)
  4140       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
  4141     moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
  4142     ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
  4143       unfolding closedin_closed by auto  }
  4144   thus ?thesis unfolding continuous_on_closed by auto
  4145 qed
  4146 
  4147 text {* A uniformly convergent limit of continuous functions is continuous. *}
  4148 
  4149 lemma continuous_uniform_limit:
  4150   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
  4151   assumes "\<not> trivial_limit F"
  4152   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
  4153   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
  4154   shows "continuous_on s g"
  4155 proof-
  4156   { fix x and e::real assume "x\<in>s" "e>0"
  4157     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
  4158       using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
  4159     from eventually_happens [OF eventually_conj [OF this assms(2)]]
  4160     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"
  4161       using assms(1) by blast
  4162     have "e / 3 > 0" using `e>0` by auto
  4163     then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
  4164       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
  4165     { fix y assume "y \<in> s" and "dist y x < d"
  4166       hence "dist (f n y) (f n x) < e / 3"
  4167         by (rule d [rule_format])
  4168       hence "dist (f n y) (g x) < 2 * e / 3"
  4169         using dist_triangle [of "f n y" "g x" "f n x"]
  4170         using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
  4171         by auto
  4172       hence "dist (g y) (g x) < e"
  4173         using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
  4174         using dist_triangle3 [of "g y" "g x" "f n y"]
  4175         by auto }
  4176     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
  4177       using `d>0` by auto }
  4178   thus ?thesis unfolding continuous_on_iff by auto
  4179 qed
  4180 
  4181 
  4182 subsection {* Topological stuff lifted from and dropped to R *}
  4183 
  4184 lemma open_real:
  4185   fixes s :: "real set" shows
  4186  "open s \<longleftrightarrow>
  4187         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
  4188   unfolding open_dist dist_norm by simp
  4189 
  4190 lemma islimpt_approachable_real:
  4191   fixes s :: "real set"
  4192   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
  4193   unfolding islimpt_approachable dist_norm by simp
  4194 
  4195 lemma closed_real:
  4196   fixes s :: "real set"
  4197   shows "closed s \<longleftrightarrow>
  4198         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
  4199             --> x \<in> s)"
  4200   unfolding closed_limpt islimpt_approachable dist_norm by simp
  4201 
  4202 lemma continuous_at_real_range:
  4203   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4204   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  4205         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
  4206   unfolding continuous_at unfolding Lim_at
  4207   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
  4208   apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
  4209   apply(erule_tac x=e in allE) by auto
  4210 
  4211 lemma continuous_on_real_range:
  4212   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4213   shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
  4214   unfolding continuous_on_iff dist_norm by simp
  4215 
  4216 lemma continuous_at_norm: "continuous (at x) norm"
  4217   unfolding continuous_at by (intro tendsto_intros)
  4218 
  4219 lemma continuous_on_norm: "continuous_on s norm"
  4220 unfolding continuous_on by (intro ballI tendsto_intros)
  4221 
  4222 lemma continuous_at_infnorm: "continuous (at x) infnorm"
  4223   unfolding continuous_at Lim_at o_def unfolding dist_norm
  4224   apply auto apply (rule_tac x=e in exI) apply auto
  4225   using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
  4226 
  4227 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
  4228 
  4229 lemma compact_attains_sup:
  4230   fixes s :: "real set"
  4231   assumes "compact s"  "s \<noteq> {}"
  4232   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
  4233 proof-
  4234   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4235   { fix e::real assume as: "\<forall>x\<in>s. x \<le> Sup s" "Sup s \<notin> s"  "0 < e" "\<forall>x'\<in>s. x' = Sup s \<or> \<not> Sup s - x' < e"
  4236     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
  4237     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
  4238     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto  }
  4239   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
  4240     apply(rule_tac x="Sup s" in bexI) by auto
  4241 qed
  4242 
  4243 lemma Inf:
  4244   fixes S :: "real set"
  4245   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
  4246 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def) 
  4247 
  4248 lemma compact_attains_inf:
  4249   fixes s :: "real set"
  4250   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
  4251 proof-
  4252   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4253   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"
  4254       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
  4255     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
  4256     moreover
  4257     { fix x assume "x \<in> s"
  4258       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
  4259       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
  4260     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
  4261     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto  }
  4262   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
  4263     apply(rule_tac x="Inf s" in bexI) by auto
  4264 qed
  4265 
  4266 lemma continuous_attains_sup:
  4267   fixes f :: "'a::metric_space \<Rightarrow> real"
  4268   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4269         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"
  4270   using compact_attains_sup[of "f ` s"]
  4271   using compact_continuous_image[of s f] by auto
  4272 
  4273 lemma continuous_attains_inf:
  4274   fixes f :: "'a::metric_space \<Rightarrow> real"
  4275   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4276         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
  4277   using compact_attains_inf[of "f ` s"]
  4278   using compact_continuous_image[of s f] by auto
  4279 
  4280 lemma distance_attains_sup:
  4281   assumes "compact s" "s \<noteq> {}"
  4282   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
  4283 proof (rule continuous_attains_sup [OF assms])
  4284   { fix x assume "x\<in>s"
  4285     have "(dist a ---> dist a x) (at x within s)"
  4286       by (intro tendsto_dist tendsto_const Lim_at_within LIM_ident)
  4287   }
  4288   thus "continuous_on s (dist a)"
  4289     unfolding continuous_on ..
  4290 qed
  4291 
  4292 text {* For \emph{minimal} distance, we only need closure, not compactness. *}
  4293 
  4294 lemma distance_attains_inf:
  4295   fixes a :: "'a::heine_borel"
  4296   assumes "closed s"  "s \<noteq> {}"
  4297   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
  4298 proof-
  4299   from assms(2) obtain b where "b\<in>s" by auto
  4300   let ?B = "cball a (dist b a) \<inter> s"
  4301   have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
  4302   hence "?B \<noteq> {}" by auto
  4303   moreover
  4304   { fix x assume "x\<in>?B"
  4305     fix e::real assume "e>0"
  4306     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
  4307       from as have "\<bar>dist a x' - dist a x\<bar> < e"
  4308         unfolding abs_less_iff minus_diff_eq
  4309         using dist_triangle2 [of a x' x]
  4310         using dist_triangle [of a x x']
  4311         by arith
  4312     }
  4313     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
  4314       using `e>0` by auto
  4315   }
  4316   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
  4317     unfolding continuous_on Lim_within dist_norm real_norm_def
  4318     by fast
  4319   moreover have "compact ?B"
  4320     using compact_cball[of a "dist b a"]
  4321     unfolding compact_eq_bounded_closed
  4322     using bounded_Int and closed_Int and assms(1) by auto
  4323   ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y"
  4324     using continuous_attains_inf[of ?B "dist a"] by fastsimp
  4325   thus ?thesis by fastsimp
  4326 qed
  4327 
  4328 
  4329 subsection {* Pasted sets *}
  4330 
  4331 lemma bounded_Times:
  4332   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
  4333 proof-
  4334   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
  4335     using assms [unfolded bounded_def] by auto
  4336   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
  4337     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
  4338   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
  4339 qed
  4340 
  4341 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
  4342 by (induct x) simp
  4343 
  4344 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
  4345 unfolding compact_def
  4346 apply clarify
  4347 apply (drule_tac x="fst \<circ> f" in spec)
  4348 apply (drule mp, simp add: mem_Times_iff)
  4349 apply (clarify, rename_tac l1 r1)
  4350 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
  4351 apply (drule mp, simp add: mem_Times_iff)
  4352 apply (clarify, rename_tac l2 r2)
  4353 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
  4354 apply (rule_tac x="r1 \<circ> r2" in exI)
  4355 apply (rule conjI, simp add: subseq_def)
  4356 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
  4357 apply (drule (1) tendsto_Pair) back
  4358 apply (simp add: o_def)
  4359 done
  4360 
  4361 text{* Hence some useful properties follow quite easily. *}
  4362 
  4363 lemma compact_scaling:
  4364   fixes s :: "'a::real_normed_vector set"
  4365   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
  4366 proof-
  4367   let ?f = "\<lambda>x. scaleR c x"
  4368   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)
  4369   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
  4370     using linear_continuous_at[OF *] assms by auto
  4371 qed
  4372 
  4373 lemma compact_negations:
  4374   fixes s :: "'a::real_normed_vector set"
  4375   assumes "compact s"  shows "compact ((\<lambda>x. -x) ` s)"
  4376   using compact_scaling [OF assms, of "- 1"] by auto
  4377 
  4378 lemma compact_sums:
  4379   fixes s t :: "'a::real_normed_vector set"
  4380   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
  4381 proof-
  4382   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
  4383     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
  4384   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
  4385     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  4386   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
  4387 qed
  4388 
  4389 lemma compact_differences:
  4390   fixes s t :: "'a::real_normed_vector set"
  4391   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
  4392 proof-
  4393   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
  4394     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4395   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
  4396 qed
  4397 
  4398 lemma compact_translation:
  4399   fixes s :: "'a::real_normed_vector set"
  4400   assumes "compact s"  shows "compact ((\<lambda>x. a + x) ` s)"
  4401 proof-
  4402   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
  4403   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
  4404 qed
  4405 
  4406 lemma compact_affinity:
  4407   fixes s :: "'a::real_normed_vector set"
  4408   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  4409 proof-
  4410   have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  4411   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
  4412 qed
  4413 
  4414 text {* Hence we get the following. *}
  4415 
  4416 lemma compact_sup_maxdistance:
  4417   fixes s :: "'a::real_normed_vector set"
  4418   assumes "compact s"  "s \<noteq> {}"
  4419   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
  4420 proof-
  4421   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
  4422   then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}"  "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"
  4423     using compact_differences[OF assms(1) assms(1)]
  4424     using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by auto
  4425   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
  4426   thus ?thesis using x(2)[unfolded `x = a - b`] by blast
  4427 qed
  4428 
  4429 text {* We can state this in terms of diameter of a set. *}
  4430 
  4431 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
  4432   (* TODO: generalize to class metric_space *)
  4433 
  4434 lemma diameter_bounded:
  4435   assumes "bounded s"
  4436   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
  4437         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
  4438 proof-
  4439   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
  4440   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
  4441   { fix x y assume "x \<in> s" "y \<in> s"
  4442     hence "norm (x - y) \<le> 2 * a" using norm_triangle_ineq[of x "-y", unfolded norm_minus_cancel] a[THEN bspec[where x=x]] a[THEN bspec[where x=y]] by (auto simp add: field_simps)  }
  4443   note * = this
  4444   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto
  4445     have "norm(x - y) \<le> diameter s" unfolding diameter_def using `s\<noteq>{}` *[OF `x\<in>s` `y\<in>s`] `x\<in>s` `y\<in>s`
  4446       by simp (blast intro!: Sup_upper *) }
  4447   moreover
  4448   { fix d::real assume "d>0" "d < diameter s"
  4449     hence "s\<noteq>{}" unfolding diameter_def by auto
  4450     have "\<exists>d' \<in> ?D. d' > d"
  4451     proof(rule ccontr)
  4452       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
  4453       hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE) 
  4454       thus False using `d < diameter s` `s\<noteq>{}` 
  4455         apply (auto simp add: diameter_def) 
  4456         apply (drule Sup_real_iff [THEN [2] rev_iffD2])
  4457         apply (auto, force) 
  4458         done
  4459     qed
  4460     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }
  4461   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
  4462         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
  4463 qed
  4464 
  4465 lemma diameter_bounded_bound:
  4466  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
  4467   using diameter_bounded by blast
  4468 
  4469 lemma diameter_compact_attained:
  4470   fixes s :: "'a::real_normed_vector set"
  4471   assumes "compact s"  "s \<noteq> {}"
  4472   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
  4473 proof-
  4474   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
  4475   then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto
  4476   hence "diameter s \<le> norm (x - y)"
  4477     unfolding diameter_def by clarsimp (rule Sup_least, fast+)
  4478   thus ?thesis
  4479     by (metis b diameter_bounded_bound order_antisym xys)
  4480 qed
  4481 
  4482 text {* Related results with closure as the conclusion. *}
  4483 
  4484 lemma closed_scaling:
  4485   fixes s :: "'a::real_normed_vector set"
  4486   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
  4487 proof(cases "s={}")
  4488   case True thus ?thesis by auto
  4489 next
  4490   case False
  4491   show ?thesis
  4492   proof(cases "c=0")
  4493     have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
  4494     case True thus ?thesis apply auto unfolding * by auto
  4495   next
  4496     case False
  4497     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s"  "(x ---> l) sequentially"
  4498       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
  4499           using as(1)[THEN spec[where x=n]]
  4500           using `c\<noteq>0` by auto
  4501       }
  4502       moreover
  4503       { fix e::real assume "e>0"
  4504         hence "0 < e *\<bar>c\<bar>"  using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
  4505         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
  4506           using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
  4507         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
  4508           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
  4509           using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto  }
  4510       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
  4511       ultimately have "l \<in> scaleR c ` s"
  4512         using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]
  4513         unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }
  4514     thus ?thesis unfolding closed_sequential_limits by fast
  4515   qed
  4516 qed
  4517 
  4518 lemma closed_negations:
  4519   fixes s :: "'a::real_normed_vector set"
  4520   assumes "closed s"  shows "closed ((\<lambda>x. -x) ` s)"
  4521   using closed_scaling[OF assms, of "- 1"] by simp
  4522 
  4523 lemma compact_closed_sums:
  4524   fixes s :: "'a::real_normed_vector set"
  4525   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  4526 proof-
  4527   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
  4528   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"
  4529     from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> s"  "\<forall>n. snd (f n) \<in> t"
  4530       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
  4531     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
  4532       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
  4533     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
  4534       using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
  4535     hence "l - l' \<in> t"
  4536       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
  4537       using f(3) by auto
  4538     hence "l \<in> ?S" using `l' \<in> s` apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto
  4539   }
  4540   thus ?thesis unfolding closed_sequential_limits by fast
  4541 qed
  4542 
  4543 lemma closed_compact_sums:
  4544   fixes s t :: "'a::real_normed_vector set"
  4545   assumes "closed s"  "compact t"
  4546   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  4547 proof-
  4548   have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto
  4549     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
  4550   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
  4551 qed
  4552 
  4553 lemma compact_closed_differences:
  4554   fixes s t :: "'a::real_normed_vector set"
  4555   assumes "compact s"  "closed t"
  4556   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  4557 proof-
  4558   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"
  4559     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4560   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
  4561 qed
  4562 
  4563 lemma closed_compact_differences:
  4564   fixes s t :: "'a::real_normed_vector set"
  4565   assumes "closed s" "compact t"
  4566   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  4567 proof-
  4568   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
  4569     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4570  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
  4571 qed
  4572 
  4573 lemma closed_translation:
  4574   fixes a :: "'a::real_normed_vector"
  4575   assumes "closed s"  shows "closed ((\<lambda>x. a + x) ` s)"
  4576 proof-
  4577   have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
  4578   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
  4579 qed
  4580 
  4581 lemma translation_Compl:
  4582   fixes a :: "'a::ab_group_add"
  4583   shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
  4584   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
  4585 
  4586 lemma translation_UNIV:
  4587   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
  4588   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
  4589 
  4590 lemma translation_diff:
  4591   fixes a :: "'a::ab_group_add"
  4592   shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
  4593   by auto
  4594 
  4595 lemma closure_translation:
  4596   fixes a :: "'a::real_normed_vector"
  4597   shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
  4598 proof-
  4599   have *:"op + a ` (- s) = - op + a ` s"
  4600     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
  4601   show ?thesis unfolding closure_interior translation_Compl
  4602     using interior_translation[of a "- s"] unfolding * by auto
  4603 qed
  4604 
  4605 lemma frontier_translation:
  4606   fixes a :: "'a::real_normed_vector"
  4607   shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
  4608   unfolding frontier_def translation_diff interior_translation closure_translation by auto
  4609 
  4610 
  4611 subsection {* Separation between points and sets *}
  4612 
  4613 lemma separate_point_closed:
  4614   fixes s :: "'a::heine_borel set"
  4615   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
  4616 proof(cases "s = {}")
  4617   case True
  4618   thus ?thesis by(auto intro!: exI[where x=1])
  4619 next
  4620   case False
  4621   assume "closed s" "a \<notin> s"
  4622   then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using `s \<noteq> {}` distance_attains_inf [of s a] by blast
  4623   with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
  4624 qed
  4625 
  4626 lemma separate_compact_closed:
  4627   fixes s t :: "'a::{heine_borel, real_normed_vector} set"
  4628     (* TODO: does this generalize to heine_borel? *)
  4629   assumes "compact s" and "closed t" and "s \<inter> t = {}"
  4630   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  4631 proof-
  4632   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
  4633   then obtain d where "d>0" and d:"\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. d \<le> dist 0 x"
  4634     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
  4635   { fix x y assume "x\<in>s" "y\<in>t"
  4636     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
  4637     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
  4638       by (auto  simp add: dist_commute)
  4639     hence "d \<le> dist x y" unfolding dist_norm by auto  }
  4640   thus ?thesis using `d>0` by auto
  4641 qed
  4642 
  4643 lemma separate_closed_compact:
  4644   fixes s t :: "'a::{heine_borel, real_normed_vector} set"
  4645   assumes "closed s" and "compact t" and "s \<inter> t = {}"
  4646   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  4647 proof-
  4648   have *:"t \<inter> s = {}" using assms(3) by auto
  4649   show ?thesis using separate_compact_closed[OF assms(2,1) *]
  4650     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
  4651     by (auto simp add: dist_commute)
  4652 qed
  4653 
  4654 
  4655 subsection {* Intervals *}
  4656   
  4657 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
  4658   "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
  4659   "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
  4660   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  4661 
  4662 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4663   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
  4664   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
  4665   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  4666 
  4667 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
  4668  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
  4669  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
  4670 proof-
  4671   { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
  4672     hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
  4673     hence "a$$i < b$$i" by auto
  4674     hence False using as by auto  }
  4675   moreover
  4676   { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
  4677     let ?x = "(1/2) *\<^sub>R (a + b)"
  4678     { fix i assume i:"i<DIM('a)" 
  4679       have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
  4680       hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
  4681         unfolding euclidean_simps by auto }
  4682     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
  4683   ultimately show ?th1 by blast
  4684 
  4685   { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
  4686     hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
  4687     hence "a$$i \<le> b$$i" by auto
  4688     hence False using as by auto  }
  4689   moreover
  4690   { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
  4691     let ?x = "(1/2) *\<^sub>R (a + b)"
  4692     { fix i assume i:"i<DIM('a)"
  4693       have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
  4694       hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
  4695         unfolding euclidean_simps by auto }
  4696     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
  4697   ultimately show ?th2 by blast
  4698 qed
  4699 
  4700 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
  4701   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
  4702   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
  4703   unfolding interval_eq_empty[of a b] by fastsimp+
  4704 
  4705 lemma interval_sing: fixes a :: "'a::ordered_euclidean_space" shows
  4706  "{a .. a} = {a}" "{a<..<a} = {}"
  4707   apply(auto simp add: set_eq_iff euclidean_eq[where 'a='a] eucl_less[where 'a='a] eucl_le[where 'a='a])
  4708   apply (simp add: order_eq_iff) apply(rule_tac x=0 in exI) by (auto simp add: not_less less_imp_le)
  4709 
  4710 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
  4711  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
  4712  "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
  4713  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
  4714  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
  4715   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval 
  4716   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
  4717 
  4718 lemma interval_open_subset_closed:  fixes a :: "'a::ordered_euclidean_space" shows
  4719  "{a<..<b} \<subseteq> {a .. b}"
  4720 proof(simp add: subset_eq, rule)
  4721   fix x
  4722   assume x:"x \<in>{a<..<b}"
  4723   { fix i assume "i<DIM('a)"
  4724     hence "a $$ i \<le> x $$ i"
  4725       using x order_less_imp_le[of "a$$i" "x$$i"] 
  4726       by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
  4727   }
  4728   moreover
  4729   { fix i assume "i<DIM('a)"
  4730     hence "x $$ i \<le> b $$ i"
  4731       using x order_less_imp_le[of "x$$i" "b$$i"]
  4732       by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
  4733   }
  4734   ultimately
  4735   show "a \<le> x \<and> x \<le> b"
  4736     by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
  4737 qed
  4738 
  4739 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4740  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and
  4741  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and
  4742  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and
  4743  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)
  4744 proof-
  4745   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
  4746   show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
  4747   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
  4748     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
  4749     fix i assume i:"i<DIM('a)"
  4750     (** TODO combine the following two parts as done in the HOL_light version. **)
  4751     { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
  4752       assume as2: "a$$i > c$$i"
  4753       { fix j assume j:"j<DIM('a)"
  4754         hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
  4755           apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
  4756           by (auto simp add: as2)  }
  4757       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
  4758       moreover
  4759       have "?x\<notin>{a .. b}"
  4760         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
  4761         using as(2)[THEN spec[where x=i]] and as2 i
  4762         by auto
  4763       ultimately have False using as by auto  }
  4764     hence "a$$i \<le> c$$i" by(rule ccontr)auto
  4765     moreover
  4766     { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
  4767       assume as2: "b$$i < d$$i"
  4768       { fix j assume "j<DIM('a)"
  4769         hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j" 
  4770           apply(cases "j=i") using as(2)[THEN spec[where x=j]]
  4771           by (auto simp add: as2)  }
  4772       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
  4773       moreover
  4774       have "?x\<notin>{a .. b}"
  4775         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
  4776         using as(2)[THEN spec[where x=i]] and as2 using i
  4777         by auto
  4778       ultimately have False using as by auto  }
  4779     hence "b$$i \<ge> d$$i" by(rule ccontr)auto
  4780     ultimately
  4781     have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
  4782   } note part1 = this
  4783   show ?th3 unfolding subset_eq and Ball_def and mem_interval 
  4784     apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
  4785     prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+ 
  4786   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
  4787     fix i assume i:"i<DIM('a)"
  4788     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
  4789     hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto  } note * = this
  4790   show ?th4 unfolding subset_eq and Ball_def and mem_interval 
  4791     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
  4792     apply auto by(erule_tac x=i in allE, simp)+ 
  4793 qed
  4794 
  4795 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
  4796   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and
  4797   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and
  4798   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and
  4799   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)
  4800 proof-
  4801   let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
  4802   note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
  4803   show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4804     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4805   show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4806     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4807   show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4808     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4809   show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4810     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4811 qed
  4812 
  4813 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4814  "{a .. b} \<inter> {c .. d} =  {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
  4815   unfolding set_eq_iff and Int_iff and mem_interval
  4816   by auto
  4817 
  4818 (* Moved interval_open_subset_closed a bit upwards *)
  4819 
  4820 lemma open_interval[intro]:
  4821   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
  4822 proof-
  4823   have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i})"
  4824     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
  4825       linear_continuous_at bounded_linear_euclidean_component
  4826       open_real_greaterThanLessThan)
  4827   also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i}) = {a<..<b}"
  4828     by (auto simp add: eucl_less [where 'a='a])
  4829   finally show "open {a<..<b}" .
  4830 qed
  4831 
  4832 lemma closed_interval[intro]:
  4833   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
  4834 proof-
  4835   have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i})"
  4836     by (intro closed_INT ballI continuous_closed_vimage allI
  4837       linear_continuous_at bounded_linear_euclidean_component
  4838       closed_real_atLeastAtMost)
  4839   also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i}) = {a .. b}"
  4840     by (auto simp add: eucl_le [where 'a='a])
  4841   finally show "closed {a .. b}" .
  4842 qed
  4843 
  4844 lemma interior_closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows
  4845  "interior {a .. b} = {a<..<b}" (is "?L = ?R")
  4846 proof(rule subset_antisym)
  4847   show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
  4848 next
  4849   { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
  4850     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
  4851     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
  4852     { fix i assume i:"i<DIM('a)"
  4853       have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
  4854            "dist (x + (e / 2) *\<^sub>R basis i) x < e"
  4855         unfolding dist_norm apply auto
  4856         unfolding norm_minus_cancel using norm_basis and `e>0` by auto
  4857       hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
  4858                      "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
  4859         using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
  4860         and   e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
  4861         unfolding mem_interval by (auto elim!: allE[where x=i])
  4862       hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
  4863         unfolding basis_component using `e>0` i by auto  }
  4864     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
  4865   thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
  4866 qed
  4867 
  4868 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
  4869 proof-
  4870   let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
  4871   { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
  4872     { fix i assume "i<DIM('a)"
  4873       hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto  }
  4874     hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
  4875     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
  4876   thus ?thesis unfolding interval and bounded_iff by auto
  4877 qed
  4878 
  4879 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4880  "bounded {a .. b} \<and> bounded {a<..<b}"
  4881   using bounded_closed_interval[of a b]
  4882   using interval_open_subset_closed[of a b]
  4883   using bounded_subset[of "{a..b}" "{a<..<b}"]
  4884   by simp
  4885 
  4886 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
  4887  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
  4888   using bounded_interval[of a b] by auto
  4889 
  4890 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
  4891   using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
  4892   by auto
  4893 
  4894 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
  4895   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
  4896 proof-
  4897   { fix i assume "i<DIM('a)"
  4898     hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
  4899       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
  4900       unfolding euclidean_simps by auto  }
  4901   thus ?thesis unfolding mem_interval by auto
  4902 qed
  4903 
  4904 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
  4905   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
  4906   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
  4907 proof-
  4908   { fix i assume i:"i<DIM('a)"
  4909     have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
  4910     also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
  4911       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  4912       using x unfolding mem_interval using i apply simp
  4913       using y unfolding mem_interval using i apply simp
  4914       done
  4915     finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
  4916     moreover {
  4917     have "b $$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp
  4918     also have "\<dots> > e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
  4919       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  4920       using x unfolding mem_interval using i apply simp
  4921       using y unfolding mem_interval using i apply simp
  4922       done
  4923     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
  4924     } ultimately have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" by auto }
  4925   thus ?thesis unfolding mem_interval by auto
  4926 qed
  4927 
  4928 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
  4929   assumes "{a<..<b} \<noteq> {}"
  4930   shows "closure {a<..<b} = {a .. b}"
  4931 proof-
  4932   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
  4933   let ?c = "(1 / 2) *\<^sub>R (a + b)"
  4934   { fix x assume as:"x \<in> {a .. b}"
  4935     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
  4936     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
  4937       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
  4938       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
  4939         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
  4940         by (auto simp add: algebra_simps)
  4941       hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
  4942       hence False using fn unfolding f_def using xc by auto  }
  4943     moreover
  4944     { assume "\<not> (f ---> x) sequentially"
  4945       { fix e::real assume "e>0"
  4946         hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
  4947         then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  4948         hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
  4949         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }
  4950       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
  4951         unfolding Lim_sequentially by(auto simp add: dist_norm)
  4952       hence "(f ---> x) sequentially" unfolding f_def
  4953         using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
  4954         using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto  }
  4955     ultimately have "x \<in> closure {a<..<b}"
  4956       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }
  4957   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
  4958 qed
  4959 
  4960 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
  4961   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
  4962 proof-
  4963   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
  4964   def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
  4965   { fix x assume "x\<in>s"
  4966     fix i assume i:"i<DIM('a)"
  4967     hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
  4968       and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto  }
  4969   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
  4970 qed
  4971 
  4972 lemma bounded_subset_open_interval:
  4973   fixes s :: "('a::ordered_euclidean_space) set"
  4974   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
  4975   by (auto dest!: bounded_subset_open_interval_symmetric)
  4976 
  4977 lemma bounded_subset_closed_interval_symmetric:
  4978   fixes s :: "('a::ordered_euclidean_space) set"
  4979   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
  4980 proof-
  4981   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
  4982   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
  4983 qed
  4984 
  4985 lemma bounded_subset_closed_interval:
  4986   fixes s :: "('a::ordered_euclidean_space) set"
  4987   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
  4988   using bounded_subset_closed_interval_symmetric[of s] by auto
  4989 
  4990 lemma frontier_closed_interval:
  4991   fixes a b :: "'a::ordered_euclidean_space"
  4992   shows "frontier {a .. b} = {a .. b} - {a<..<b}"
  4993   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
  4994 
  4995 lemma frontier_open_interval:
  4996   fixes a b :: "'a::ordered_euclidean_space"
  4997   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
  4998 proof(cases "{a<..<b} = {}")
  4999   case True thus ?thesis using frontier_empty by auto
  5000 next
  5001   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
  5002 qed
  5003 
  5004 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
  5005   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
  5006   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
  5007 
  5008 
  5009 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)
  5010 
  5011 lemma closed_interval_left: fixes b::"'a::euclidean_space"
  5012   shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
  5013 proof-
  5014   { fix i assume i:"i<DIM('a)"
  5015     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$ i \<le> b $$ i}. x' \<noteq> x \<and> dist x' x < e"
  5016     { assume "x$$i > b$$i"
  5017       then obtain y where "y $$ i \<le> b $$ i"  "y \<noteq> x"  "dist y x < x$$i - b$$i"
  5018         using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
  5019       hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i 
  5020         by auto   }
  5021     hence "x$$i \<le> b$$i" by(rule ccontr)auto  }
  5022   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  5023 qed
  5024 
  5025 lemma closed_interval_right: fixes a::"'a::euclidean_space"
  5026   shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
  5027 proof-
  5028   { fix i assume i:"i<DIM('a)"
  5029     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$ i \<le> x $$ i}. x' \<noteq> x \<and> dist x' x < e"
  5030     { assume "a$$i > x$$i"
  5031       then obtain y where "a $$ i \<le> y $$ i"  "y \<noteq> x"  "dist y x < a$$i - x$$i"
  5032         using x[THEN spec[where x="a$$i - x$$i"]] i by auto
  5033       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto   }
  5034     hence "a$$i \<le> x$$i" by(rule ccontr)auto  }
  5035   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  5036 qed
  5037 
  5038 text {* Intervals in general, including infinite and mixtures of open and closed. *}
  5039 
  5040 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
  5041   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$i \<le> x$$i \<and> x$$i \<le> b$$i) \<or> (b$$i \<le> x$$i \<and> x$$i \<le> a$$i))) \<longrightarrow> x \<in> s)"
  5042 
  5043 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
  5044   "is_interval {a<..<b}" (is ?th2) proof -
  5045   have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
  5046   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
  5047     by(meson order_trans le_less_trans less_le_trans *)+ qed
  5048 
  5049 lemma is_interval_empty:
  5050  "is_interval {}"
  5051   unfolding is_interval_def
  5052   by simp
  5053 
  5054 lemma is_interval_univ:
  5055  "is_interval UNIV"
  5056   unfolding is_interval_def
  5057   by simp
  5058 
  5059 
  5060 subsection {* Closure of halfspaces and hyperplanes *}
  5061 
  5062 lemma isCont_open_vimage:
  5063   assumes "\<And>x. isCont f x" and "open s" shows "open (f -` s)"
  5064 proof -
  5065   from assms(1) have "continuous_on UNIV f"
  5066     unfolding isCont_def continuous_on_def within_UNIV by simp
  5067   hence "open {x \<in> UNIV. f x \<in> s}"
  5068     using open_UNIV `open s` by (rule continuous_open_preimage)
  5069   thus "open (f -` s)"
  5070     by (simp add: vimage_def)
  5071 qed
  5072 
  5073 lemma isCont_closed_vimage:
  5074   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f -` s)"
  5075   using assms unfolding closed_def vimage_Compl [symmetric]
  5076   by (rule isCont_open_vimage)
  5077 
  5078 lemma open_Collect_less:
  5079   fixes f g :: "'a::topological_space \<Rightarrow> real"
  5080   assumes f: "\<And>x. isCont f x"
  5081   assumes g: "\<And>x. isCont g x"
  5082   shows "open {x. f x < g x}"
  5083 proof -
  5084   have "open ((\<lambda>x. g x - f x) -` {0<..})"
  5085     using isCont_diff [OF g f] open_real_greaterThan
  5086     by (rule isCont_open_vimage)
  5087   also have "((\<lambda>x. g x - f x) -` {0<..}) = {x. f x < g x}"
  5088     by auto
  5089   finally show ?thesis .
  5090 qed
  5091 
  5092 lemma closed_Collect_le:
  5093   fixes f g :: "'a::topological_space \<Rightarrow> real"
  5094   assumes f: "\<And>x. isCont f x"
  5095   assumes g: "\<And>x. isCont g x"
  5096   shows "closed {x. f x \<le> g x}"
  5097 proof -
  5098   have "closed ((\<lambda>x. g x - f x) -` {0..})"
  5099     using isCont_diff [OF g f] closed_real_atLeast
  5100     by (rule isCont_closed_vimage)
  5101   also have "((\<lambda>x. g x - f x) -` {0..}) = {x. f x \<le> g x}"
  5102     by auto
  5103   finally show ?thesis .
  5104 qed
  5105 
  5106 lemma closed_Collect_eq:
  5107   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
  5108   assumes f: "\<And>x. isCont f x"
  5109   assumes g: "\<And>x. isCont g x"
  5110   shows "closed {x. f x = g x}"
  5111 proof -
  5112   have "open {(x::'b, y::'b). x \<noteq> y}"
  5113     unfolding open_prod_def by (auto dest!: hausdorff)
  5114   hence "closed {(x::'b, y::'b). x = y}"
  5115     unfolding closed_def split_def Collect_neg_eq .
  5116   with isCont_Pair [OF f g]
  5117   have "closed ((\<lambda>x. (f x, g x)) -` {(x, y). x = y})"
  5118     by (rule isCont_closed_vimage)
  5119   also have "\<dots> = {x. f x = g x}" by auto
  5120   finally show ?thesis .
  5121 qed
  5122 
  5123 lemma Lim_inner:
  5124   assumes "(f ---> l) net"  shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
  5125   by (intro tendsto_intros assms)
  5126 
  5127 lemma continuous_at_inner: "continuous (at x) (inner a)"
  5128   unfolding continuous_at by (intro tendsto_intros)
  5129 
  5130 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
  5131   unfolding euclidean_component_def by (rule continuous_at_inner)
  5132 
  5133 lemma continuous_on_inner:
  5134   fixes s :: "'a::real_inner set"
  5135   shows "continuous_on s (inner a)"
  5136   unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  5137 
  5138 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
  5139   by (simp add: closed_Collect_le)
  5140 
  5141 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
  5142   by (simp add: closed_Collect_le)
  5143 
  5144 lemma closed_hyperplane: "closed {x. inner a x = b}"
  5145   by (simp add: closed_Collect_eq)
  5146 
  5147 lemma closed_halfspace_component_le:
  5148   shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
  5149   by (simp add: closed_Collect_le)
  5150 
  5151 lemma closed_halfspace_component_ge:
  5152   shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
  5153   by (simp add: closed_Collect_le)
  5154 
  5155 text {* Openness of halfspaces. *}
  5156 
  5157 lemma open_halfspace_lt: "open {x. inner a x < b}"
  5158   by (simp add: open_Collect_less)
  5159 
  5160 lemma open_halfspace_gt: "open {x. inner a x > b}"
  5161   by (simp add: open_Collect_less)
  5162 
  5163 lemma open_halfspace_component_lt:
  5164   shows "open {x::'a::euclidean_space. x$$i < a}"
  5165   by (simp add: open_Collect_less)
  5166 
  5167 lemma open_halfspace_component_gt:
  5168   shows "open {x::'a::euclidean_space. x$$i > a}"
  5169   by (simp add: open_Collect_less)
  5170 
  5171 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
  5172 
  5173 lemma eucl_lessThan_eq_halfspaces:
  5174   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5175   shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
  5176  by (auto simp: eucl_less[where 'a='a])
  5177 
  5178 lemma eucl_greaterThan_eq_halfspaces:
  5179   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5180   shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
  5181  by (auto simp: eucl_less[where 'a='a])
  5182 
  5183 lemma eucl_atMost_eq_halfspaces:
  5184   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5185   shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
  5186  by (auto simp: eucl_le[where 'a='a])
  5187 
  5188 lemma eucl_atLeast_eq_halfspaces:
  5189   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5190   shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
  5191  by (auto simp: eucl_le[where 'a='a])
  5192 
  5193 lemma open_eucl_lessThan[simp, intro]:
  5194   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5195   shows "open {..< a}"
  5196   by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
  5197 
  5198 lemma open_eucl_greaterThan[simp, intro]:
  5199   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5200   shows "open {a <..}"
  5201   by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
  5202 
  5203 lemma closed_eucl_atMost[simp, intro]:
  5204   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5205   shows "closed {.. a}"
  5206   unfolding eucl_atMost_eq_halfspaces
  5207   by (simp add: closed_INT closed_Collect_le)
  5208 
  5209 lemma closed_eucl_atLeast[simp, intro]:
  5210   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5211   shows "closed {a ..}"
  5212   unfolding eucl_atLeast_eq_halfspaces
  5213   by (simp add: closed_INT closed_Collect_le)
  5214 
  5215 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
  5216   by (auto intro!: continuous_open_vimage)
  5217 
  5218 text {* This gives a simple derivation of limit component bounds. *}
  5219 
  5220 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
  5221   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$$i \<le> b) net"
  5222   shows "l$$i \<le> b"
  5223 proof-
  5224   { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
  5225       unfolding euclidean_component_def by auto  } note * = this
  5226   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
  5227     using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
  5228 qed
  5229 
  5230 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
  5231   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$$i) net"
  5232   shows "b \<le> l$$i"
  5233 proof-
  5234   { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
  5235       unfolding euclidean_component_def by auto  } note * = this
  5236   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
  5237     using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
  5238 qed
  5239 
  5240 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
  5241   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
  5242   shows "l$$i = b"
  5243   using ev[unfolded order_eq_iff eventually_conj_iff] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
  5244 text{* Limits relative to a union.                                               *}
  5245 
  5246 lemma eventually_within_Un:
  5247   "eventually P (net within (s \<union> t)) \<longleftrightarrow>
  5248     eventually P (net within s) \<and> eventually P (net within t)"
  5249   unfolding Limits.eventually_within
  5250   by (auto elim!: eventually_rev_mp)
  5251 
  5252 lemma Lim_within_union:
  5253  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
  5254   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
  5255   unfolding tendsto_def
  5256   by (auto simp add: eventually_within_Un)
  5257 
  5258 lemma Lim_topological:
  5259  "(f ---> l) net \<longleftrightarrow>
  5260         trivial_limit net \<or>
  5261         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
  5262   unfolding tendsto_def trivial_limit_eq by auto
  5263 
  5264 lemma continuous_on_union:
  5265   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
  5266   shows "continuous_on (s \<union> t) f"
  5267   using assms unfolding continuous_on Lim_within_union
  5268   unfolding Lim_topological trivial_limit_within closed_limpt by auto
  5269 
  5270 lemma continuous_on_cases:
  5271   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
  5272           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
  5273   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
  5274 proof-
  5275   let ?h = "(\<lambda>x. if P x then f x else g x)"
  5276   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
  5277   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
  5278   moreover
  5279   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
  5280   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
  5281   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
  5282 qed
  5283 
  5284 
  5285 text{* Some more convenient intermediate-value theorem formulations.             *}
  5286 
  5287 lemma connected_ivt_hyperplane:
  5288   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
  5289   shows "\<exists>z \<in> s. inner a z = b"
  5290 proof(rule ccontr)
  5291   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
  5292   let ?A = "{x. inner a x < b}"
  5293   let ?B = "{x. inner a x > b}"
  5294   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
  5295   moreover have "?A \<inter> ?B = {}" by auto
  5296   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
  5297   ultimately show False using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5) by auto
  5298 qed
  5299 
  5300 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
  5301  "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$$k \<le> a \<Longrightarrow> a \<le> y$$k \<Longrightarrow> (\<exists>z\<in>s.  z$$k = a)"
  5302   using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
  5303   unfolding euclidean_component_def by auto
  5304 
  5305 
  5306 subsection {* Homeomorphisms *}
  5307 
  5308 definition "homeomorphism s t f g \<equiv>
  5309      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
  5310      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
  5311 
  5312 definition
  5313   homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
  5314     (infixr "homeomorphic" 60) where
  5315   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
  5316 
  5317 lemma homeomorphic_refl: "s homeomorphic s"
  5318   unfolding homeomorphic_def
  5319   unfolding homeomorphism_def
  5320   using continuous_on_id
  5321   apply(rule_tac x = "(\<lambda>x. x)" in exI)
  5322   apply(rule_tac x = "(\<lambda>x. x)" in exI)
  5323   by blast
  5324 
  5325 lemma homeomorphic_sym:
  5326  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
  5327 unfolding homeomorphic_def
  5328 unfolding homeomorphism_def
  5329 by blast 
  5330 
  5331 lemma homeomorphic_trans:
  5332   assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
  5333 proof-
  5334   obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x"  "f1 ` s = t" "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1 ` t = s" "continuous_on t g1"
  5335     using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
  5336   obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x"  "f2 ` t = u" "continuous_on t f2" "\<forall>y\<in>u. f2 (g2 y) = y" "g2 ` u = t" "continuous_on u g2"
  5337     using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
  5338 
  5339   { fix x assume "x\<in>s" hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x" using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2) by auto }
  5340   moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
  5341   moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
  5342   moreover { fix y assume "y\<in>u" hence "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y" using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5) by auto }
  5343   moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
  5344   moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6)  unfolding fg2(5) by auto
  5345   ultimately show ?thesis unfolding homeomorphic_def homeomorphism_def apply(rule_tac x="f2 \<circ> f1" in exI) apply(rule_tac x="g1 \<circ> g2" in exI) by auto
  5346 qed
  5347 
  5348 lemma homeomorphic_minimal:
  5349  "s homeomorphic t \<longleftrightarrow>
  5350     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
  5351            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
  5352            continuous_on s f \<and> continuous_on t g)"
  5353 unfolding homeomorphic_def homeomorphism_def
  5354 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
  5355 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
  5356 unfolding image_iff
  5357 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
  5358 apply auto apply(rule_tac x="g x" in bexI) apply auto
  5359 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
  5360 apply auto apply(rule_tac x="f x" in bexI) by auto
  5361 
  5362 text {* Relatively weak hypotheses if a set is compact. *}
  5363 
  5364 lemma homeomorphism_compact:
  5365   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  5366     (* class constraint due to continuous_on_inverse *)
  5367   assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
  5368   shows "\<exists>g. homeomorphism s t f g"
  5369 proof-
  5370   def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
  5371   have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
  5372   { fix y assume "y\<in>t"
  5373     then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
  5374     hence "g (f x) = x" using g by auto
  5375     hence "f (g y) = y" unfolding x(1)[THEN sym] by auto  }
  5376   hence g':"\<forall>x\<in>t. f (g x) = x" by auto
  5377   moreover
  5378   { fix x
  5379     have "x\<in>s \<Longrightarrow> x \<in> g ` t" using g[THEN bspec[where x=x]] unfolding image_iff using assms(3) by(auto intro!: bexI[where x="f x"])
  5380     moreover
  5381     { assume "x\<in>g ` t"
  5382       then obtain y where y:"y\<in>t" "g y = x" by auto
  5383       then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
  5384       hence "x \<in> s" unfolding g_def using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"] unfolding y(2)[THEN sym] and g_def by auto }
  5385     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..  }
  5386   hence "g ` t = s" by auto
  5387   ultimately
  5388   show ?thesis unfolding homeomorphism_def homeomorphic_def
  5389     apply(rule_tac x=g in exI) using g and assms(3) and continuous_on_inverse[OF assms(2,1), of g, unfolded assms(3)] and assms(2) by auto
  5390 qed
  5391 
  5392 lemma homeomorphic_compact:
  5393   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  5394     (* class constraint due to continuous_on_inverse *)
  5395   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
  5396           \<Longrightarrow> s homeomorphic t"
  5397   unfolding homeomorphic_def by (metis homeomorphism_compact)
  5398 
  5399 text{* Preservation of topological properties.                                   *}
  5400 
  5401 lemma homeomorphic_compactness:
  5402  "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
  5403 unfolding homeomorphic_def homeomorphism_def
  5404 by (metis compact_continuous_image)
  5405 
  5406 text{* Results on translation, scaling etc.                                      *}
  5407 
  5408 lemma homeomorphic_scaling:
  5409   fixes s :: "'a::real_normed_vector set"
  5410   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
  5411   unfolding homeomorphic_minimal
  5412   apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
  5413   apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
  5414   using assms apply auto
  5415   using continuous_on_cmul[OF continuous_on_id] by auto
  5416 
  5417 lemma homeomorphic_translation:
  5418   fixes s :: "'a::real_normed_vector set"
  5419   shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
  5420   unfolding homeomorphic_minimal
  5421   apply(rule_tac x="\<lambda>x. a + x" in exI)
  5422   apply(rule_tac x="\<lambda>x. -a + x" in exI)
  5423   using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
  5424 
  5425 lemma homeomorphic_affinity:
  5426   fixes s :: "'a::real_normed_vector set"
  5427   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  5428 proof-
  5429   have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  5430   show ?thesis
  5431     using homeomorphic_trans
  5432     using homeomorphic_scaling[OF assms, of s]
  5433     using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
  5434 qed
  5435 
  5436 lemma homeomorphic_balls:
  5437   fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
  5438   assumes "0 < d"  "0 < e"
  5439   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
  5440         "(cball a d) homeomorphic (cball b e)" (is ?cth)
  5441 proof-
  5442   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
  5443   show ?th unfolding homeomorphic_minimal
  5444     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
  5445     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
  5446     using assms apply (auto simp add: dist_commute)
  5447     unfolding dist_norm
  5448     apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
  5449     unfolding continuous_on
  5450     by (intro ballI tendsto_intros, simp)+
  5451 next
  5452   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
  5453   show ?cth unfolding homeomorphic_minimal
  5454     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
  5455     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
  5456     using assms apply (auto simp add: dist_commute)
  5457     unfolding dist_norm
  5458     apply (auto simp add: pos_divide_le_eq)
  5459     unfolding continuous_on
  5460     by (intro ballI tendsto_intros, simp)+
  5461 qed
  5462 
  5463 text{* "Isometry" (up to constant bounds) of injective linear map etc.           *}
  5464 
  5465 lemma cauchy_isometric:
  5466   fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
  5467   assumes e:"0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and xs:"\<forall>n::nat. x n \<in> s" and cf:"Cauchy(f o x)"
  5468   shows "Cauchy x"
  5469 proof-
  5470   interpret f: bounded_linear f by fact
  5471   { fix d::real assume "d>0"
  5472     then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
  5473       using cf[unfolded cauchy o_def dist_norm, THEN spec[where x="e*d"]] and e and mult_pos_pos[of e d] by auto
  5474     { fix n assume "n\<ge>N"
  5475       hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
  5476       moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
  5477         using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
  5478         using normf[THEN bspec[where x="x n - x N"]] by auto
  5479       ultimately have "norm (x n - x N) < d" using `e>0`
  5480         using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto   }
  5481     hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
  5482   thus ?thesis unfolding cauchy and dist_norm by auto
  5483 qed
  5484 
  5485 lemma complete_isometric_image:
  5486   fixes f :: "'a::euclidean_space => 'b::euclidean_space"
  5487   assumes "0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and cs:"complete s"
  5488   shows "complete(f ` s)"
  5489 proof-
  5490   { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
  5491     then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)" 
  5492       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
  5493     hence x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)" by auto
  5494     hence "f \<circ> x = g" unfolding fun_eq_iff by auto
  5495     then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
  5496       using cs[unfolded complete_def, THEN spec[where x="x"]]
  5497       using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
  5498     hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
  5499       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
  5500       unfolding `f \<circ> x = g` by auto  }
  5501   thus ?thesis unfolding complete_def by auto
  5502 qed
  5503 
  5504 lemma dist_0_norm:
  5505   fixes x :: "'a::real_normed_vector"
  5506   shows "dist 0 x = norm x"
  5507 unfolding dist_norm by simp
  5508 
  5509 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  5510   assumes s:"closed s"  "subspace s"  and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
  5511   shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
  5512 proof(cases "s \<subseteq> {0::'a}")
  5513   case True
  5514   { fix x assume "x \<in> s"
  5515     hence "x = 0" using True by auto
  5516     hence "norm x \<le> norm (f x)" by auto  }
  5517   thus ?thesis by(auto intro!: exI[where x=1])
  5518 next
  5519   interpret f: bounded_linear f by fact
  5520   case False
  5521   then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
  5522   from False have "s \<noteq> {}" by auto
  5523   let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
  5524   let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
  5525   let ?S'' = "{x::'a. norm x = norm a}"
  5526 
  5527   have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
  5528   hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
  5529   moreover have "?S' = s \<inter> ?S''" by auto
  5530   ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
  5531   moreover have *:"f ` ?S' = ?S" by auto
  5532   ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
  5533   hence "closed ?S" using compact_imp_closed by auto
  5534   moreover have "?S \<noteq> {}" using a by auto
  5535   ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y" using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
  5536   then obtain b where "b\<in>s" and ba:"norm b = norm a" and b:"\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)" unfolding *[THEN sym] unfolding image_iff by auto
  5537 
  5538   let ?e = "norm (f b) / norm b"
  5539   have "norm b > 0" using ba and a and norm_ge_zero by auto
  5540   moreover have "norm (f b) > 0" using f(2)[THEN bspec[where x=b], OF `b\<in>s`] using `norm b >0` unfolding zero_less_norm_iff by auto
  5541   ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
  5542   moreover
  5543   { fix x assume "x\<in>s"
  5544     hence "norm (f b) / norm b * norm x \<le> norm (f x)"
  5545     proof(cases "x=0")
  5546       case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
  5547     next
  5548       case False
  5549       hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
  5550       have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
  5551       hence "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}" using `x\<in>s` and `x\<noteq>0` by auto
  5552       thus "norm (f b) / norm b * norm x \<le> norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
  5553         unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
  5554         by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
  5555     qed }
  5556   ultimately
  5557   show ?thesis by auto
  5558 qed
  5559 
  5560 lemma closed_injective_image_subspace:
  5561   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  5562   assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
  5563   shows "closed(f ` s)"
  5564 proof-
  5565   obtain e where "e>0" and e:"\<forall>x\<in>s. e * norm x \<le> norm (f x)" using injective_imp_isometric[OF assms(4,1,2,3)] by auto
  5566   show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
  5567     unfolding complete_eq_closed[THEN sym] by auto
  5568 qed
  5569 
  5570 
  5571 subsection {* Some properties of a canonical subspace *}
  5572 
  5573 lemma subspace_substandard:
  5574   "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
  5575   unfolding subspace_def by(auto simp add: euclidean_simps) (* FIXME: duplicate rewrite rule *)
  5576 
  5577 lemma closed_substandard:
  5578  "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
  5579 proof-
  5580   let ?D = "{i. P i} \<inter> {..<DIM('a)}"
  5581   let ?Bs = "{{x::'a. inner (basis i) x = 0}| i. i \<in> ?D}"
  5582   { fix x
  5583     { assume "x\<in>?A"
  5584       hence x:"\<forall>i\<in>?D. x $$ i = 0" by auto
  5585       hence "x\<in> \<Inter> ?Bs" by(auto simp add: x euclidean_component_def) }
  5586     moreover
  5587     { assume x:"x\<in>\<Inter>?Bs"
  5588       { fix i assume i:"i \<in> ?D"
  5589         then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::'a. inner (basis i) x = 0}" by auto
  5590         hence "x $$ i = 0" unfolding B using x unfolding euclidean_component_def by auto  }
  5591       hence "x\<in>?A" by auto }
  5592     ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
  5593   hence "?A = \<Inter> ?Bs" by auto
  5594   thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
  5595 qed
  5596 
  5597 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  5598   shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
  5599 proof-
  5600   let ?D = "{..<DIM('a)}"
  5601   let ?B = "(basis::nat => 'a) ` d"
  5602   let ?bas = "basis::nat \<Rightarrow> 'a"
  5603   have "?B \<subseteq> ?A" by auto
  5604   moreover
  5605   { fix x::"'a" assume "x\<in>?A"
  5606     hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
  5607     hence "x\<in> span ?B"
  5608     proof(induct d arbitrary: x)
  5609       case empty hence "x=0" apply(subst euclidean_eq) by auto
  5610       thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
  5611     next
  5612       case (insert k F)
  5613       hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
  5614       have **:"F \<subseteq> insert k F" by auto
  5615       def y \<equiv> "x - x$$k *\<^sub>R basis k"
  5616       have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
  5617       { fix i assume i':"i \<notin> F"
  5618         hence "y $$ i = 0" unfolding y_def 
  5619           using *[THEN spec[where x=i]] by(auto simp add: euclidean_simps) }
  5620       hence "y \<in> span (basis ` F)" using insert(3) by auto
  5621       hence "y \<in> span (basis ` (insert k F))"
  5622         using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
  5623         using image_mono[OF **, of basis] using assms by auto
  5624       moreover
  5625       have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
  5626       hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
  5627         using span_mul by auto
  5628       ultimately
  5629       have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
  5630         using span_add by auto
  5631       thus ?case using y by auto
  5632     qed
  5633   }
  5634   hence "?A \<subseteq> span ?B" by auto
  5635   moreover
  5636   { fix x assume "x \<in> ?B"
  5637     hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto  }
  5638   hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
  5639   moreover
  5640   have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
  5641   hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
  5642   have "card ?B = card d" unfolding card_image[OF *] by auto
  5643   ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
  5644 qed
  5645 
  5646 text{* Hence closure and completeness of all subspaces.                          *}
  5647 
  5648 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
  5649 apply (induct n)
  5650 apply (rule_tac x="{}" in exI, simp)
  5651 apply clarsimp
  5652 apply (subgoal_tac "\<exists>x. x \<notin> A")
  5653 apply (erule exE)
  5654 apply (rule_tac x="insert x A" in exI, simp)
  5655 apply (subgoal_tac "A \<noteq> UNIV", auto)
  5656 done
  5657 
  5658 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
  5659   assumes "subspace s" shows "closed s"
  5660 proof-
  5661   have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
  5662   def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
  5663   let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
  5664   have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
  5665       inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
  5666     apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
  5667     using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
  5668   then guess f apply-by(erule exE conjE)+ note f = this
  5669   interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
  5670   have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
  5671     by(erule_tac x=0 in ballE) auto
  5672   moreover have "closed ?t" using closed_substandard .
  5673   moreover have "subspace ?t" using subspace_substandard .
  5674   ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
  5675     unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
  5676 qed
  5677 
  5678 lemma complete_subspace:
  5679   fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
  5680   using complete_eq_closed closed_subspace
  5681   by auto
  5682 
  5683 lemma dim_closure:
  5684   fixes s :: "('a::euclidean_space) set"
  5685   shows "dim(closure s) = dim s" (is "?dc = ?d")
  5686 proof-
  5687   have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
  5688     using closed_subspace[OF subspace_span, of s]
  5689     using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
  5690   thus ?thesis using dim_subset[OF closure_subset, of s] by auto
  5691 qed
  5692 
  5693 
  5694 subsection {* Affine transformations of intervals *}
  5695 
  5696 lemma real_affinity_le:
  5697  "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
  5698   by (simp add: field_simps inverse_eq_divide)
  5699 
  5700 lemma real_le_affinity:
  5701  "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
  5702   by (simp add: field_simps inverse_eq_divide)
  5703 
  5704 lemma real_affinity_lt:
  5705  "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
  5706   by (simp add: field_simps inverse_eq_divide)
  5707 
  5708 lemma real_lt_affinity:
  5709  "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
  5710   by (simp add: field_simps inverse_eq_divide)
  5711 
  5712 lemma real_affinity_eq:
  5713  "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
  5714   by (simp add: field_simps inverse_eq_divide)
  5715 
  5716 lemma real_eq_affinity:
  5717  "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c  \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
  5718   by (simp add: field_simps inverse_eq_divide)
  5719 
  5720 lemma image_affinity_interval: fixes m::real
  5721   fixes a b c :: "'a::ordered_euclidean_space"
  5722   shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
  5723             (if {a .. b} = {} then {}
  5724             else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
  5725             else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
  5726 proof(cases "m=0")  
  5727   { fix x assume "x \<le> c" "c \<le> x"
  5728     hence "x=c" unfolding eucl_le[where 'a='a] apply-
  5729       apply(subst euclidean_eq) by (auto intro: order_antisym) }
  5730   moreover case True
  5731   moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: eucl_le[where 'a='a])
  5732   ultimately show ?thesis by auto
  5733 next
  5734   case False
  5735   { fix y assume "a \<le> y" "y \<le> b" "m > 0"