src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author huffman
Thu Aug 25 09:17:02 2011 -0700 (2011-08-25)
changeset 44519 ea85d78a994e
parent 44518 219a6fe4cfae
child 44522 2f7e9d890efe
permissions -rw-r--r--
simplify definition of 'interior';
add lemmas interiorI and interiorE;
change lemmas interior_unique and closure_unique to rule_format;
tidy some proofs;
     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" Norm_Arith L2_Norm
    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   by(auto simp add:power2_eq_square)
    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 topology as a value *}
    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 = \<Union>{T. open T \<and> T \<subseteq> S}"
   559 
   560 lemma interiorI [intro?]:
   561   assumes "open T" and "x \<in> T" and "T \<subseteq> S"
   562   shows "x \<in> interior S"
   563   using assms unfolding interior_def by fast
   564 
   565 lemma interiorE [elim?]:
   566   assumes "x \<in> interior S"
   567   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
   568   using assms unfolding interior_def by fast
   569 
   570 lemma open_interior [simp, intro]: "open (interior S)"
   571   by (simp add: interior_def open_Union)
   572 
   573 lemma interior_subset: "interior S \<subseteq> S"
   574   by (auto simp add: interior_def)
   575 
   576 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
   577   by (auto simp add: interior_def)
   578 
   579 lemma interior_open: "open S \<Longrightarrow> interior S = S"
   580   by (intro equalityI interior_subset interior_maximal subset_refl)
   581 
   582 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
   583   by (metis open_interior interior_open)
   584 
   585 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
   586   by (metis interior_maximal interior_subset subset_trans)
   587 
   588 lemma interior_empty [simp]: "interior {} = {}"
   589   using open_empty by (rule interior_open)
   590 
   591 lemma interior_interior [simp]: "interior (interior S) = interior S"
   592   using open_interior by (rule interior_open)
   593 
   594 lemma subset_interior: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
   595   by (auto simp add: interior_def) (* TODO: rename to interior_mono *)
   596 
   597 lemma interior_unique:
   598   assumes "T \<subseteq> S" and "open T"
   599   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
   600   shows "interior S = T"
   601   by (intro equalityI assms interior_subset open_interior interior_maximal)
   602 
   603 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
   604   by (intro equalityI Int_mono Int_greatest subset_interior Int_lower1
   605     Int_lower2 interior_maximal interior_subset open_Int open_interior)
   606 
   607 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
   608   using open_contains_ball_eq [where S="interior S"]
   609   by (simp add: open_subset_interior)
   610 
   611 lemma interior_limit_point [intro]:
   612   fixes x :: "'a::perfect_space"
   613   assumes x: "x \<in> interior S" shows "x islimpt S"
   614   using x islimpt_UNIV [of x]
   615   unfolding interior_def islimpt_def
   616   apply (clarsimp, rename_tac T T')
   617   apply (drule_tac x="T \<inter> T'" in spec)
   618   apply (auto simp add: open_Int)
   619   done
   620 
   621 lemma interior_closed_Un_empty_interior:
   622   assumes cS: "closed S" and iT: "interior T = {}"
   623   shows "interior (S \<union> T) = interior S"
   624 proof
   625   show "interior S \<subseteq> interior (S \<union> T)"
   626     by (rule subset_interior, rule Un_upper1)
   627 next
   628   show "interior (S \<union> T) \<subseteq> interior S"
   629   proof
   630     fix x assume "x \<in> interior (S \<union> T)"
   631     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
   632     show "x \<in> interior S"
   633     proof (rule ccontr)
   634       assume "x \<notin> interior S"
   635       with `x \<in> R` `open R` obtain y where "y \<in> R - S"
   636         unfolding interior_def by fast
   637       from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
   638       from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
   639       from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
   640       show "False" unfolding interior_def by fast
   641     qed
   642   qed
   643 qed
   644 
   645 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
   646 proof (rule interior_unique)
   647   show "interior A \<times> interior B \<subseteq> A \<times> B"
   648     by (intro Sigma_mono interior_subset)
   649   show "open (interior A \<times> interior B)"
   650     by (intro open_Times open_interior)
   651   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"
   652   proof (safe)
   653     fix x y assume "(x, y) \<in> T"
   654     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
   655       using `open T` unfolding open_prod_def by fast
   656     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
   657       using `T \<subseteq> A \<times> B` by auto
   658     thus "x \<in> interior A" and "y \<in> interior B"
   659       by (auto intro: interiorI)
   660   qed
   661 qed
   662 
   663 
   664 subsection {* Closure of a Set *}
   665 
   666 definition "closure S = S \<union> {x | x. x islimpt S}"
   667 
   668 lemma interior_closure: "interior S = - (closure (- S))"
   669   unfolding interior_def closure_def islimpt_def by auto
   670 
   671 lemma closure_interior: "closure S = - interior (- S)"
   672   unfolding interior_closure by simp
   673 
   674 lemma closed_closure[simp, intro]: "closed (closure S)"
   675   unfolding closure_interior by (simp add: closed_Compl)
   676 
   677 lemma closure_subset: "S \<subseteq> closure S"
   678   unfolding closure_def by simp
   679 
   680 lemma closure_hull: "closure S = closed hull S"
   681   unfolding hull_def closure_interior interior_def by auto
   682 
   683 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
   684   unfolding closure_hull using closed_Inter by (rule hull_eq)
   685 
   686 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
   687   unfolding closure_eq .
   688 
   689 lemma closure_closure [simp]: "closure (closure S) = closure S"
   690   unfolding closure_hull by (rule hull_hull)
   691 
   692 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
   693   unfolding closure_hull by (rule hull_mono)
   694 
   695 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
   696   unfolding closure_hull by (rule hull_minimal)
   697 
   698 lemma closure_unique:
   699   assumes "S \<subseteq> T" and "closed T"
   700   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
   701   shows "closure S = T"
   702   using assms unfolding closure_hull by (rule hull_unique)
   703 
   704 lemma closure_empty [simp]: "closure {} = {}"
   705   using closed_empty by (rule closure_closed)
   706 
   707 lemma closure_univ [simp]: "closure UNIV = UNIV"
   708   using closed_UNIV by (rule closure_closed)
   709 
   710 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
   711   unfolding closure_interior by simp
   712 
   713 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
   714   using closure_empty closure_subset[of S]
   715   by blast
   716 
   717 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
   718   using closure_eq[of S] closure_subset[of S]
   719   by simp
   720 
   721 lemma open_inter_closure_eq_empty:
   722   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
   723   using open_subset_interior[of S "- T"]
   724   using interior_subset[of "- T"]
   725   unfolding closure_interior
   726   by auto
   727 
   728 lemma open_inter_closure_subset:
   729   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
   730 proof
   731   fix x
   732   assume as: "open S" "x \<in> S \<inter> closure T"
   733   { assume *:"x islimpt T"
   734     have "x islimpt (S \<inter> T)"
   735     proof (rule islimptI)
   736       fix A
   737       assume "x \<in> A" "open A"
   738       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
   739         by (simp_all add: open_Int)
   740       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
   741         by (rule islimptE)
   742       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
   743         by simp_all
   744       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
   745     qed
   746   }
   747   then show "x \<in> closure (S \<inter> T)" using as
   748     unfolding closure_def
   749     by blast
   750 qed
   751 
   752 lemma closure_complement: "closure (- S) = - interior S"
   753   unfolding closure_interior by simp
   754 
   755 lemma interior_complement: "interior (- S) = - closure S"
   756   unfolding closure_interior by simp
   757 
   758 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
   759 proof (rule closure_unique)
   760   show "A \<times> B \<subseteq> closure A \<times> closure B"
   761     by (intro Sigma_mono closure_subset)
   762   show "closed (closure A \<times> closure B)"
   763     by (intro closed_Times closed_closure)
   764   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
   765     apply (simp add: closed_def open_prod_def, clarify)
   766     apply (rule ccontr)
   767     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
   768     apply (simp add: closure_interior interior_def)
   769     apply (drule_tac x=C in spec)
   770     apply (drule_tac x=D in spec)
   771     apply auto
   772     done
   773 qed
   774 
   775 
   776 subsection {* Frontier (aka boundary) *}
   777 
   778 definition "frontier S = closure S - interior S"
   779 
   780 lemma frontier_closed: "closed(frontier S)"
   781   by (simp add: frontier_def closed_Diff)
   782 
   783 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
   784   by (auto simp add: frontier_def interior_closure)
   785 
   786 lemma frontier_straddle:
   787   fixes a :: "'a::metric_space"
   788   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")
   789 proof
   790   assume "?lhs"
   791   { fix e::real
   792     assume "e > 0"
   793     let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
   794     { assume "a\<in>S"
   795       have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
   796       moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
   797         unfolding frontier_closures closure_def islimpt_def using `e>0`
   798         by (auto, erule_tac x="ball a e" in allE, auto)
   799       ultimately have ?rhse by auto
   800     }
   801     moreover
   802     { assume "a\<notin>S"
   803       hence ?rhse using `?lhs`
   804         unfolding frontier_closures closure_def islimpt_def
   805         using open_ball[of a e] `e > 0`
   806           by simp (metis centre_in_ball mem_ball open_ball) 
   807     }
   808     ultimately have ?rhse by auto
   809   }
   810   thus ?rhs by auto
   811 next
   812   assume ?rhs
   813   moreover
   814   { fix T assume "a\<notin>S" and
   815     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"
   816     from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
   817     then obtain e where "e>0" "ball a e \<subseteq> T" by auto
   818     then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
   819     have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
   820       using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
   821   }
   822   hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
   823   moreover
   824   { fix T assume "a \<in> T"  "open T" "a\<in>S"
   825     then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
   826     obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
   827     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
   828   }
   829   hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
   830   ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
   831 qed
   832 
   833 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
   834   by (metis frontier_def closure_closed Diff_subset)
   835 
   836 lemma frontier_empty[simp]: "frontier {} = {}"
   837   by (simp add: frontier_def)
   838 
   839 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
   840 proof-
   841   { assume "frontier S \<subseteq> S"
   842     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
   843     hence "closed S" using closure_subset_eq by auto
   844   }
   845   thus ?thesis using frontier_subset_closed[of S] ..
   846 qed
   847 
   848 lemma frontier_complement: "frontier(- S) = frontier S"
   849   by (auto simp add: frontier_def closure_complement interior_complement)
   850 
   851 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
   852   using frontier_complement frontier_subset_eq[of "- S"]
   853   unfolding open_closed by auto
   854 
   855 
   856 subsection {* Filters and the ``eventually true'' quantifier *}
   857 
   858 definition
   859   at_infinity :: "'a::real_normed_vector filter" where
   860   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
   861 
   862 definition
   863   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
   864     (infixr "indirection" 70) where
   865   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
   866 
   867 text{* Prove That They are all filters. *}
   868 
   869 lemma eventually_at_infinity:
   870   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
   871 unfolding at_infinity_def
   872 proof (rule eventually_Abs_filter, rule is_filter.intro)
   873   fix P Q :: "'a \<Rightarrow> bool"
   874   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
   875   then obtain r s where
   876     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
   877   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
   878   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
   879 qed auto
   880 
   881 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
   882 
   883 lemma trivial_limit_within:
   884   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
   885 proof
   886   assume "trivial_limit (at a within S)"
   887   thus "\<not> a islimpt S"
   888     unfolding trivial_limit_def
   889     unfolding eventually_within eventually_at_topological
   890     unfolding islimpt_def
   891     apply (clarsimp simp add: set_eq_iff)
   892     apply (rename_tac T, rule_tac x=T in exI)
   893     apply (clarsimp, drule_tac x=y in bspec, simp_all)
   894     done
   895 next
   896   assume "\<not> a islimpt S"
   897   thus "trivial_limit (at a within S)"
   898     unfolding trivial_limit_def
   899     unfolding eventually_within eventually_at_topological
   900     unfolding islimpt_def
   901     apply clarsimp
   902     apply (rule_tac x=T in exI)
   903     apply auto
   904     done
   905 qed
   906 
   907 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
   908   using trivial_limit_within [of a UNIV]
   909   by (simp add: within_UNIV)
   910 
   911 lemma trivial_limit_at:
   912   fixes a :: "'a::perfect_space"
   913   shows "\<not> trivial_limit (at a)"
   914   by (simp add: trivial_limit_at_iff)
   915 
   916 lemma trivial_limit_at_infinity:
   917   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
   918   unfolding trivial_limit_def eventually_at_infinity
   919   apply clarsimp
   920   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
   921    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
   922   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
   923   apply (drule_tac x=UNIV in spec, simp)
   924   done
   925 
   926 text {* Some property holds "sufficiently close" to the limit point. *}
   927 
   928 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
   929   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
   930 unfolding eventually_at dist_nz by auto
   931 
   932 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
   933         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
   934 unfolding eventually_within eventually_at dist_nz by auto
   935 
   936 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
   937         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
   938 unfolding eventually_within
   939 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl) 
   940 
   941 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
   942   unfolding trivial_limit_def
   943   by (auto elim: eventually_rev_mp)
   944 
   945 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
   946   unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
   947 
   948 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
   949   by (simp add: filter_eq_iff)
   950 
   951 text{* Combining theorems for "eventually" *}
   952 
   953 lemma eventually_rev_mono:
   954   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
   955 using eventually_mono [of P Q] by fast
   956 
   957 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
   958   by (simp add: eventually_False)
   959 
   960 
   961 subsection {* Limits *}
   962 
   963 text{* Notation Lim to avoid collition with lim defined in analysis *}
   964 
   965 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
   966   where "Lim A f = (THE l. (f ---> l) A)"
   967 
   968 lemma Lim:
   969  "(f ---> l) net \<longleftrightarrow>
   970         trivial_limit net \<or>
   971         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
   972   unfolding tendsto_iff trivial_limit_eq by auto
   973 
   974 text{* Show that they yield usual definitions in the various cases. *}
   975 
   976 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
   977            (\<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)"
   978   by (auto simp add: tendsto_iff eventually_within_le)
   979 
   980 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
   981         (\<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)"
   982   by (auto simp add: tendsto_iff eventually_within)
   983 
   984 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
   985         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
   986   by (auto simp add: tendsto_iff eventually_at)
   987 
   988 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
   989   unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
   990 
   991 lemma Lim_at_infinity:
   992   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
   993   by (auto simp add: tendsto_iff eventually_at_infinity)
   994 
   995 lemma Lim_sequentially:
   996  "(S ---> l) sequentially \<longleftrightarrow>
   997           (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
   998   by (rule LIMSEQ_def) (* FIXME: redundant *)
   999 
  1000 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
  1001   by (rule topological_tendstoI, auto elim: eventually_rev_mono)
  1002 
  1003 text{* The expected monotonicity property. *}
  1004 
  1005 lemma Lim_within_empty: "(f ---> l) (net within {})"
  1006   unfolding tendsto_def Limits.eventually_within by simp
  1007 
  1008 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
  1009   unfolding tendsto_def Limits.eventually_within
  1010   by (auto elim!: eventually_elim1)
  1011 
  1012 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
  1013   shows "(f ---> l) (net within (S \<union> T))"
  1014   using assms unfolding tendsto_def Limits.eventually_within
  1015   apply clarify
  1016   apply (drule spec, drule (1) mp, drule (1) mp)
  1017   apply (drule spec, drule (1) mp, drule (1) mp)
  1018   apply (auto elim: eventually_elim2)
  1019   done
  1020 
  1021 lemma Lim_Un_univ:
  1022  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
  1023         ==> (f ---> l) net"
  1024   by (metis Lim_Un within_UNIV)
  1025 
  1026 text{* Interrelations between restricted and unrestricted limits. *}
  1027 
  1028 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
  1029   (* FIXME: rename *)
  1030   unfolding tendsto_def Limits.eventually_within
  1031   apply (clarify, drule spec, drule (1) mp, drule (1) mp)
  1032   by (auto elim!: eventually_elim1)
  1033 
  1034 lemma eventually_within_interior:
  1035   assumes "x \<in> interior S"
  1036   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
  1037 proof-
  1038   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
  1039   { assume "?lhs"
  1040     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
  1041       unfolding Limits.eventually_within Limits.eventually_at_topological
  1042       by auto
  1043     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
  1044       by auto
  1045     then have "?rhs"
  1046       unfolding Limits.eventually_at_topological by auto
  1047   } moreover
  1048   { assume "?rhs" hence "?lhs"
  1049       unfolding Limits.eventually_within
  1050       by (auto elim: eventually_elim1)
  1051   } ultimately
  1052   show "?thesis" ..
  1053 qed
  1054 
  1055 lemma at_within_interior:
  1056   "x \<in> interior S \<Longrightarrow> at x within S = at x"
  1057   by (simp add: filter_eq_iff eventually_within_interior)
  1058 
  1059 lemma at_within_open:
  1060   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
  1061   by (simp only: at_within_interior interior_open)
  1062 
  1063 lemma Lim_within_open:
  1064   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
  1065   assumes"a \<in> S" "open S"
  1066   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
  1067   using assms by (simp only: at_within_open)
  1068 
  1069 lemma Lim_within_LIMSEQ:
  1070   fixes a :: real and L :: "'a::metric_space"
  1071   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
  1072   shows "(X ---> L) (at a within T)"
  1073 proof (rule ccontr)
  1074   assume "\<not> (X ---> L) (at a within T)"
  1075   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"
  1076     unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
  1077   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
  1078 
  1079   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"
  1080   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"
  1081     using r by (simp add: Bex_def)
  1082   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"
  1083     by (rule someI_ex)
  1084   hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
  1085     and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
  1086     and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
  1087     by fast+
  1088 
  1089   have "?F ----> a"
  1090   proof (rule LIMSEQ_I, unfold real_norm_def)
  1091       fix e::real
  1092       assume "0 < e"
  1093         (* choose no such that inverse (real (Suc n)) < e *)
  1094       then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
  1095       then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
  1096       show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
  1097       proof (intro exI allI impI)
  1098         fix n
  1099         assume mlen: "m \<le> n"
  1100         have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
  1101           by (rule F2)
  1102         also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
  1103           using mlen by auto
  1104         also from nodef have
  1105           "inverse (real (Suc m)) < e" .
  1106         finally show "\<bar>?F n - a\<bar> < e" .
  1107       qed
  1108   qed
  1109   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`
  1110   ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
  1111   
  1112   moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
  1113   proof -
  1114     {
  1115       fix no::nat
  1116       obtain n where "n = no + 1" by simp
  1117       then have nolen: "no \<le> n" by simp
  1118         (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
  1119       have "dist (X (?F n)) L \<ge> r"
  1120         by (rule F3)
  1121       with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
  1122     }
  1123     then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
  1124     with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
  1125     thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
  1126   qed
  1127   ultimately show False by simp
  1128 qed
  1129 
  1130 lemma Lim_right_bound:
  1131   fixes f :: "real \<Rightarrow> real"
  1132   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"
  1133   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
  1134   shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
  1135 proof cases
  1136   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
  1137 next
  1138   assume [simp]: "{x<..} \<inter> I \<noteq> {}"
  1139   show ?thesis
  1140   proof (rule Lim_within_LIMSEQ, safe)
  1141     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
  1142     
  1143     show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
  1144     proof (rule LIMSEQ_I, rule ccontr)
  1145       fix r :: real assume "0 < r"
  1146       with Inf_close[of "f ` ({x<..} \<inter> I)" r]
  1147       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
  1148       from `x < y` have "0 < y - x" by auto
  1149       from S(2)[THEN LIMSEQ_D, OF this]
  1150       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
  1151       
  1152       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
  1153       moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1154         using S bnd by (intro Inf_lower[where z=K]) auto
  1155       ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1156         by (auto simp: not_less field_simps)
  1157       with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
  1158       show False by auto
  1159     qed
  1160   qed
  1161 qed
  1162 
  1163 text{* Another limit point characterization. *}
  1164 
  1165 lemma islimpt_sequential:
  1166   fixes x :: "'a::metric_space"
  1167   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
  1168     (is "?lhs = ?rhs")
  1169 proof
  1170   assume ?lhs
  1171   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"
  1172     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
  1173   { fix n::nat
  1174     have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
  1175   }
  1176   moreover
  1177   { fix e::real assume "e>0"
  1178     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
  1179     then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  1180     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)
  1181     moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
  1182     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
  1183   }
  1184   hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
  1185     unfolding Lim_sequentially using f by auto
  1186   ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
  1187 next
  1188   assume ?rhs
  1189   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
  1190   { fix e::real assume "e>0"
  1191     then obtain N where "dist (f N) x < e" using f(2) by auto
  1192     moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
  1193     ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
  1194   }
  1195   thus ?lhs unfolding islimpt_approachable by auto
  1196 qed
  1197 
  1198 lemma Lim_inv: (* TODO: delete *)
  1199   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
  1200   assumes "(f ---> l) A" and "l \<noteq> 0"
  1201   shows "((inverse o f) ---> inverse l) A"
  1202   unfolding o_def using assms by (rule tendsto_inverse)
  1203 
  1204 lemma Lim_null:
  1205   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1206   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
  1207   by (simp add: Lim dist_norm)
  1208 
  1209 lemma Lim_null_comparison:
  1210   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1211   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
  1212   shows "(f ---> 0) net"
  1213 proof (rule metric_tendsto_imp_tendsto)
  1214   show "(g ---> 0) net" by fact
  1215   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
  1216     using assms(1) by (rule eventually_elim1, simp add: dist_norm)
  1217 qed
  1218 
  1219 lemma Lim_transform_bound:
  1220   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1221   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
  1222   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
  1223   shows "(f ---> 0) net"
  1224   using assms(1) tendsto_norm_zero [OF assms(2)]
  1225   by (rule Lim_null_comparison)
  1226 
  1227 text{* Deducing things about the limit from the elements. *}
  1228 
  1229 lemma Lim_in_closed_set:
  1230   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
  1231   shows "l \<in> S"
  1232 proof (rule ccontr)
  1233   assume "l \<notin> S"
  1234   with `closed S` have "open (- S)" "l \<in> - S"
  1235     by (simp_all add: open_Compl)
  1236   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
  1237     by (rule topological_tendstoD)
  1238   with assms(2) have "eventually (\<lambda>x. False) net"
  1239     by (rule eventually_elim2) simp
  1240   with assms(3) show "False"
  1241     by (simp add: eventually_False)
  1242 qed
  1243 
  1244 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
  1245 
  1246 lemma Lim_dist_ubound:
  1247   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
  1248   shows "dist a l <= e"
  1249 proof-
  1250   have "dist a l \<in> {..e}"
  1251   proof (rule Lim_in_closed_set)
  1252     show "closed {..e}" by simp
  1253     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
  1254     show "\<not> trivial_limit net" by fact
  1255     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
  1256   qed
  1257   thus ?thesis by simp
  1258 qed
  1259 
  1260 lemma Lim_norm_ubound:
  1261   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1262   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
  1263   shows "norm(l) <= e"
  1264 proof-
  1265   have "norm l \<in> {..e}"
  1266   proof (rule Lim_in_closed_set)
  1267     show "closed {..e}" by simp
  1268     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
  1269     show "\<not> trivial_limit net" by fact
  1270     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1271   qed
  1272   thus ?thesis by simp
  1273 qed
  1274 
  1275 lemma Lim_norm_lbound:
  1276   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1277   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
  1278   shows "e \<le> norm l"
  1279 proof-
  1280   have "norm l \<in> {e..}"
  1281   proof (rule Lim_in_closed_set)
  1282     show "closed {e..}" by simp
  1283     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
  1284     show "\<not> trivial_limit net" by fact
  1285     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1286   qed
  1287   thus ?thesis by simp
  1288 qed
  1289 
  1290 text{* Uniqueness of the limit, when nontrivial. *}
  1291 
  1292 lemma tendsto_Lim:
  1293   fixes f :: "'a \<Rightarrow> 'b::t2_space"
  1294   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
  1295   unfolding Lim_def using tendsto_unique[of net f] by auto
  1296 
  1297 text{* Limit under bilinear function *}
  1298 
  1299 lemma Lim_bilinear:
  1300   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
  1301   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
  1302 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
  1303 by (rule bounded_bilinear.tendsto)
  1304 
  1305 text{* These are special for limits out of the same vector space. *}
  1306 
  1307 lemma Lim_within_id: "(id ---> a) (at a within s)"
  1308   unfolding tendsto_def Limits.eventually_within eventually_at_topological
  1309   by auto
  1310 
  1311 lemma Lim_at_id: "(id ---> a) (at a)"
  1312 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
  1313 
  1314 lemma Lim_at_zero:
  1315   fixes a :: "'a::real_normed_vector"
  1316   fixes l :: "'b::topological_space"
  1317   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
  1318   using LIM_offset_zero LIM_offset_zero_cancel ..
  1319 
  1320 text{* It's also sometimes useful to extract the limit point from the filter. *}
  1321 
  1322 definition
  1323   netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
  1324   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
  1325 
  1326 lemma netlimit_within:
  1327   assumes "\<not> trivial_limit (at a within S)"
  1328   shows "netlimit (at a within S) = a"
  1329 unfolding netlimit_def
  1330 apply (rule some_equality)
  1331 apply (rule Lim_at_within)
  1332 apply (rule LIM_ident)
  1333 apply (erule tendsto_unique [OF assms])
  1334 apply (rule Lim_at_within)
  1335 apply (rule LIM_ident)
  1336 done
  1337 
  1338 lemma netlimit_at:
  1339   fixes a :: "'a::{perfect_space,t2_space}"
  1340   shows "netlimit (at a) = a"
  1341   apply (subst within_UNIV[symmetric])
  1342   using netlimit_within[of a UNIV]
  1343   by (simp add: trivial_limit_at within_UNIV)
  1344 
  1345 lemma lim_within_interior:
  1346   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
  1347   by (simp add: at_within_interior)
  1348 
  1349 lemma netlimit_within_interior:
  1350   fixes x :: "'a::{t2_space,perfect_space}"
  1351   assumes "x \<in> interior S"
  1352   shows "netlimit (at x within S) = x"
  1353 using assms by (simp add: at_within_interior netlimit_at)
  1354 
  1355 text{* Transformation of limit. *}
  1356 
  1357 lemma Lim_transform:
  1358   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
  1359   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
  1360   shows "(g ---> l) net"
  1361   using tendsto_diff [OF assms(2) assms(1)] by simp
  1362 
  1363 lemma Lim_transform_eventually:
  1364   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
  1365   apply (rule topological_tendstoI)
  1366   apply (drule (2) topological_tendstoD)
  1367   apply (erule (1) eventually_elim2, simp)
  1368   done
  1369 
  1370 lemma Lim_transform_within:
  1371   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1372   and "(f ---> l) (at x within S)"
  1373   shows "(g ---> l) (at x within S)"
  1374 proof (rule Lim_transform_eventually)
  1375   show "eventually (\<lambda>x. f x = g x) (at x within S)"
  1376     unfolding eventually_within
  1377     using assms(1,2) by auto
  1378   show "(f ---> l) (at x within S)" by fact
  1379 qed
  1380 
  1381 lemma Lim_transform_at:
  1382   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1383   and "(f ---> l) (at x)"
  1384   shows "(g ---> l) (at x)"
  1385 proof (rule Lim_transform_eventually)
  1386   show "eventually (\<lambda>x. f x = g x) (at x)"
  1387     unfolding eventually_at
  1388     using assms(1,2) by auto
  1389   show "(f ---> l) (at x)" by fact
  1390 qed
  1391 
  1392 text{* Common case assuming being away from some crucial point like 0. *}
  1393 
  1394 lemma Lim_transform_away_within:
  1395   fixes a b :: "'a::t1_space"
  1396   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1397   and "(f ---> l) (at a within S)"
  1398   shows "(g ---> l) (at a within S)"
  1399 proof (rule Lim_transform_eventually)
  1400   show "(f ---> l) (at a within S)" by fact
  1401   show "eventually (\<lambda>x. f x = g x) (at a within S)"
  1402     unfolding Limits.eventually_within eventually_at_topological
  1403     by (rule exI [where x="- {b}"], simp add: open_Compl assms)
  1404 qed
  1405 
  1406 lemma Lim_transform_away_at:
  1407   fixes a b :: "'a::t1_space"
  1408   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1409   and fl: "(f ---> l) (at a)"
  1410   shows "(g ---> l) (at a)"
  1411   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
  1412   by (auto simp add: within_UNIV)
  1413 
  1414 text{* Alternatively, within an open set. *}
  1415 
  1416 lemma Lim_transform_within_open:
  1417   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
  1418   and "(f ---> l) (at a)"
  1419   shows "(g ---> l) (at a)"
  1420 proof (rule Lim_transform_eventually)
  1421   show "eventually (\<lambda>x. f x = g x) (at a)"
  1422     unfolding eventually_at_topological
  1423     using assms(1,2,3) by auto
  1424   show "(f ---> l) (at a)" by fact
  1425 qed
  1426 
  1427 text{* A congruence rule allowing us to transform limits assuming not at point. *}
  1428 
  1429 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
  1430 
  1431 lemma Lim_cong_within(*[cong add]*):
  1432   assumes "a = b" "x = y" "S = T"
  1433   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
  1434   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
  1435   unfolding tendsto_def Limits.eventually_within eventually_at_topological
  1436   using assms by simp
  1437 
  1438 lemma Lim_cong_at(*[cong add]*):
  1439   assumes "a = b" "x = y"
  1440   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
  1441   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
  1442   unfolding tendsto_def eventually_at_topological
  1443   using assms by simp
  1444 
  1445 text{* Useful lemmas on closure and set of possible sequential limits.*}
  1446 
  1447 lemma closure_sequential:
  1448   fixes l :: "'a::metric_space"
  1449   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
  1450 proof
  1451   assume "?lhs" moreover
  1452   { assume "l \<in> S"
  1453     hence "?rhs" using tendsto_const[of l sequentially] by auto
  1454   } moreover
  1455   { assume "l islimpt S"
  1456     hence "?rhs" unfolding islimpt_sequential by auto
  1457   } ultimately
  1458   show "?rhs" unfolding closure_def by auto
  1459 next
  1460   assume "?rhs"
  1461   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
  1462 qed
  1463 
  1464 lemma closed_sequential_limits:
  1465   fixes S :: "'a::metric_space set"
  1466   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
  1467   unfolding closed_limpt
  1468   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]
  1469   by metis
  1470 
  1471 lemma closure_approachable:
  1472   fixes S :: "'a::metric_space set"
  1473   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
  1474   apply (auto simp add: closure_def islimpt_approachable)
  1475   by (metis dist_self)
  1476 
  1477 lemma closed_approachable:
  1478   fixes S :: "'a::metric_space set"
  1479   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
  1480   by (metis closure_closed closure_approachable)
  1481 
  1482 text{* Some other lemmas about sequences. *}
  1483 
  1484 lemma sequentially_offset:
  1485   assumes "eventually (\<lambda>i. P i) sequentially"
  1486   shows "eventually (\<lambda>i. P (i + k)) sequentially"
  1487   using assms unfolding eventually_sequentially by (metis trans_le_add1)
  1488 
  1489 lemma seq_offset:
  1490   assumes "(f ---> l) sequentially"
  1491   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
  1492   using assms unfolding tendsto_def
  1493   by clarify (rule sequentially_offset, simp)
  1494 
  1495 lemma seq_offset_neg:
  1496   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
  1497   apply (rule topological_tendstoI)
  1498   apply (drule (2) topological_tendstoD)
  1499   apply (simp only: eventually_sequentially)
  1500   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
  1501   apply metis
  1502   by arith
  1503 
  1504 lemma seq_offset_rev:
  1505   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
  1506   apply (rule topological_tendstoI)
  1507   apply (drule (2) topological_tendstoD)
  1508   apply (simp only: eventually_sequentially)
  1509   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
  1510   by metis arith
  1511 
  1512 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
  1513 proof-
  1514   { fix e::real assume "e>0"
  1515     hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
  1516       using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
  1517       by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
  1518   }
  1519   thus ?thesis unfolding Lim_sequentially dist_norm by simp
  1520 qed
  1521 
  1522 subsection {* More properties of closed balls *}
  1523 
  1524 lemma closed_cball: "closed (cball x e)"
  1525 unfolding cball_def closed_def
  1526 unfolding Collect_neg_eq [symmetric] not_le
  1527 apply (clarsimp simp add: open_dist, rename_tac y)
  1528 apply (rule_tac x="dist x y - e" in exI, clarsimp)
  1529 apply (rename_tac x')
  1530 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
  1531 apply simp
  1532 done
  1533 
  1534 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
  1535 proof-
  1536   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
  1537     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
  1538   } moreover
  1539   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
  1540     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
  1541   } ultimately
  1542   show ?thesis unfolding open_contains_ball by auto
  1543 qed
  1544 
  1545 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
  1546   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
  1547 
  1548 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
  1549   apply (simp add: interior_def, safe)
  1550   apply (force simp add: open_contains_cball)
  1551   apply (rule_tac x="ball x e" in exI)
  1552   apply (simp add: subset_trans [OF ball_subset_cball])
  1553   done
  1554 
  1555 lemma islimpt_ball:
  1556   fixes x y :: "'a::{real_normed_vector,perfect_space}"
  1557   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
  1558 proof
  1559   assume "?lhs"
  1560   { assume "e \<le> 0"
  1561     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
  1562     have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
  1563   }
  1564   hence "e > 0" by (metis not_less)
  1565   moreover
  1566   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
  1567   ultimately show "?rhs" by auto
  1568 next
  1569   assume "?rhs" hence "e>0"  by auto
  1570   { fix d::real assume "d>0"
  1571     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1572     proof(cases "d \<le> dist x y")
  1573       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1574       proof(cases "x=y")
  1575         case True hence False using `d \<le> dist x y` `d>0` by auto
  1576         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
  1577       next
  1578         case False
  1579 
  1580         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
  1581               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1582           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
  1583         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
  1584           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
  1585           unfolding scaleR_minus_left scaleR_one
  1586           by (auto simp add: norm_minus_commute)
  1587         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
  1588           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
  1589           unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
  1590         also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
  1591         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
  1592 
  1593         moreover
  1594 
  1595         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
  1596           using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
  1597         moreover
  1598         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
  1599           using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
  1600           unfolding dist_norm by auto
  1601         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
  1602       qed
  1603     next
  1604       case False hence "d > dist x y" by auto
  1605       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1606       proof(cases "x=y")
  1607         case True
  1608         obtain z where **: "z \<noteq> y" "dist z y < min e d"
  1609           using perfect_choose_dist[of "min e d" y]
  1610           using `d > 0` `e>0` by auto
  1611         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1612           unfolding `x = y`
  1613           using `z \<noteq> y` **
  1614           by (rule_tac x=z in bexI, auto simp add: dist_commute)
  1615       next
  1616         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1617           using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
  1618       qed
  1619     qed  }
  1620   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
  1621 qed
  1622 
  1623 lemma closure_ball_lemma:
  1624   fixes x y :: "'a::real_normed_vector"
  1625   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
  1626 proof (rule islimptI)
  1627   fix T assume "y \<in> T" "open T"
  1628   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
  1629     unfolding open_dist by fast
  1630   (* choose point between x and y, within distance r of y. *)
  1631   def k \<equiv> "min 1 (r / (2 * dist x y))"
  1632   def z \<equiv> "y + scaleR k (x - y)"
  1633   have z_def2: "z = x + scaleR (1 - k) (y - x)"
  1634     unfolding z_def by (simp add: algebra_simps)
  1635   have "dist z y < r"
  1636     unfolding z_def k_def using `0 < r`
  1637     by (simp add: dist_norm min_def)
  1638   hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
  1639   have "dist x z < dist x y"
  1640     unfolding z_def2 dist_norm
  1641     apply (simp add: norm_minus_commute)
  1642     apply (simp only: dist_norm [symmetric])
  1643     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
  1644     apply (rule mult_strict_right_mono)
  1645     apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
  1646     apply (simp add: zero_less_dist_iff `x \<noteq> y`)
  1647     done
  1648   hence "z \<in> ball x (dist x y)" by simp
  1649   have "z \<noteq> y"
  1650     unfolding z_def k_def using `x \<noteq> y` `0 < r`
  1651     by (simp add: min_def)
  1652   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
  1653     using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
  1654     by fast
  1655 qed
  1656 
  1657 lemma closure_ball:
  1658   fixes x :: "'a::real_normed_vector"
  1659   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
  1660 apply (rule equalityI)
  1661 apply (rule closure_minimal)
  1662 apply (rule ball_subset_cball)
  1663 apply (rule closed_cball)
  1664 apply (rule subsetI, rename_tac y)
  1665 apply (simp add: le_less [where 'a=real])
  1666 apply (erule disjE)
  1667 apply (rule subsetD [OF closure_subset], simp)
  1668 apply (simp add: closure_def)
  1669 apply clarify
  1670 apply (rule closure_ball_lemma)
  1671 apply (simp add: zero_less_dist_iff)
  1672 done
  1673 
  1674 (* In a trivial vector space, this fails for e = 0. *)
  1675 lemma interior_cball:
  1676   fixes x :: "'a::{real_normed_vector, perfect_space}"
  1677   shows "interior (cball x e) = ball x e"
  1678 proof(cases "e\<ge>0")
  1679   case False note cs = this
  1680   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
  1681   { fix y assume "y \<in> cball x e"
  1682     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
  1683   hence "cball x e = {}" by auto
  1684   hence "interior (cball x e) = {}" using interior_empty by auto
  1685   ultimately show ?thesis by blast
  1686 next
  1687   case True note cs = this
  1688   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
  1689   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
  1690     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
  1691 
  1692     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
  1693       using perfect_choose_dist [of d] by auto
  1694     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
  1695     hence xa_cball:"xa \<in> cball x e" using as(1) by auto
  1696 
  1697     hence "y \<in> ball x e" proof(cases "x = y")
  1698       case True
  1699       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
  1700       thus "y \<in> ball x e" using `x = y ` by simp
  1701     next
  1702       case False
  1703       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
  1704         using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
  1705       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
  1706       have "y - x \<noteq> 0" using `x \<noteq> y` by auto
  1707       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
  1708         using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
  1709 
  1710       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)"
  1711         by (auto simp add: dist_norm algebra_simps)
  1712       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1713         by (auto simp add: algebra_simps)
  1714       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
  1715         using ** by auto
  1716       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
  1717       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
  1718       thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
  1719     qed  }
  1720   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
  1721   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
  1722 qed
  1723 
  1724 lemma frontier_ball:
  1725   fixes a :: "'a::real_normed_vector"
  1726   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
  1727   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
  1728   apply (simp add: set_eq_iff)
  1729   by arith
  1730 
  1731 lemma frontier_cball:
  1732   fixes a :: "'a::{real_normed_vector, perfect_space}"
  1733   shows "frontier(cball a e) = {x. dist a x = e}"
  1734   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
  1735   apply (simp add: set_eq_iff)
  1736   by arith
  1737 
  1738 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
  1739   apply (simp add: set_eq_iff not_le)
  1740   by (metis zero_le_dist dist_self order_less_le_trans)
  1741 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
  1742 
  1743 lemma cball_eq_sing:
  1744   fixes x :: "'a::{metric_space,perfect_space}"
  1745   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
  1746 proof (rule linorder_cases)
  1747   assume e: "0 < e"
  1748   obtain a where "a \<noteq> x" "dist a x < e"
  1749     using perfect_choose_dist [OF e] by auto
  1750   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
  1751   with e show ?thesis by (auto simp add: set_eq_iff)
  1752 qed auto
  1753 
  1754 lemma cball_sing:
  1755   fixes x :: "'a::metric_space"
  1756   shows "e = 0 ==> cball x e = {x}"
  1757   by (auto simp add: set_eq_iff)
  1758 
  1759 
  1760 subsection {* Boundedness *}
  1761 
  1762   (* FIXME: This has to be unified with BSEQ!! *)
  1763 definition (in metric_space)
  1764   bounded :: "'a set \<Rightarrow> bool" where
  1765   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
  1766 
  1767 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
  1768 unfolding bounded_def
  1769 apply safe
  1770 apply (rule_tac x="dist a x + e" in exI, clarify)
  1771 apply (drule (1) bspec)
  1772 apply (erule order_trans [OF dist_triangle add_left_mono])
  1773 apply auto
  1774 done
  1775 
  1776 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
  1777 unfolding bounded_any_center [where a=0]
  1778 by (simp add: dist_norm)
  1779 
  1780 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
  1781 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
  1782   by (metis bounded_def subset_eq)
  1783 
  1784 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
  1785   by (metis bounded_subset interior_subset)
  1786 
  1787 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
  1788 proof-
  1789   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
  1790   { fix y assume "y \<in> closure S"
  1791     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
  1792       unfolding closure_sequential by auto
  1793     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
  1794     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
  1795       by (rule eventually_mono, simp add: f(1))
  1796     have "dist x y \<le> a"
  1797       apply (rule Lim_dist_ubound [of sequentially f])
  1798       apply (rule trivial_limit_sequentially)
  1799       apply (rule f(2))
  1800       apply fact
  1801       done
  1802   }
  1803   thus ?thesis unfolding bounded_def by auto
  1804 qed
  1805 
  1806 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
  1807   apply (simp add: bounded_def)
  1808   apply (rule_tac x=x in exI)
  1809   apply (rule_tac x=e in exI)
  1810   apply auto
  1811   done
  1812 
  1813 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
  1814   by (metis ball_subset_cball bounded_cball bounded_subset)
  1815 
  1816 lemma finite_imp_bounded[intro]:
  1817   fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
  1818 proof-
  1819   { fix a and F :: "'a set" assume as:"bounded F"
  1820     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
  1821     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
  1822     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
  1823   }
  1824   thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto
  1825 qed
  1826 
  1827 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
  1828   apply (auto simp add: bounded_def)
  1829   apply (rename_tac x y r s)
  1830   apply (rule_tac x=x in exI)
  1831   apply (rule_tac x="max r (dist x y + s)" in exI)
  1832   apply (rule ballI, rename_tac z, safe)
  1833   apply (drule (1) bspec, simp)
  1834   apply (drule (1) bspec)
  1835   apply (rule min_max.le_supI2)
  1836   apply (erule order_trans [OF dist_triangle add_left_mono])
  1837   done
  1838 
  1839 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
  1840   by (induct rule: finite_induct[of F], auto)
  1841 
  1842 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
  1843   apply (simp add: bounded_iff)
  1844   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
  1845   by metis arith
  1846 
  1847 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
  1848   by (metis Int_lower1 Int_lower2 bounded_subset)
  1849 
  1850 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
  1851 apply (metis Diff_subset bounded_subset)
  1852 done
  1853 
  1854 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
  1855   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
  1856 
  1857 lemma not_bounded_UNIV[simp, intro]:
  1858   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
  1859 proof(auto simp add: bounded_pos not_le)
  1860   obtain x :: 'a where "x \<noteq> 0"
  1861     using perfect_choose_dist [OF zero_less_one] by fast
  1862   fix b::real  assume b: "b >0"
  1863   have b1: "b +1 \<ge> 0" using b by simp
  1864   with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
  1865     by (simp add: norm_sgn)
  1866   then show "\<exists>x::'a. b < norm x" ..
  1867 qed
  1868 
  1869 lemma bounded_linear_image:
  1870   assumes "bounded S" "bounded_linear f"
  1871   shows "bounded(f ` S)"
  1872 proof-
  1873   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  1874   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)
  1875   { fix x assume "x\<in>S"
  1876     hence "norm x \<le> b" using b by auto
  1877     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
  1878       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
  1879   }
  1880   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
  1881     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
  1882 qed
  1883 
  1884 lemma bounded_scaling:
  1885   fixes S :: "'a::real_normed_vector set"
  1886   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
  1887   apply (rule bounded_linear_image, assumption)
  1888   apply (rule bounded_linear_scaleR_right)
  1889   done
  1890 
  1891 lemma bounded_translation:
  1892   fixes S :: "'a::real_normed_vector set"
  1893   assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
  1894 proof-
  1895   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  1896   { fix x assume "x\<in>S"
  1897     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
  1898   }
  1899   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
  1900     by (auto intro!: add exI[of _ "b + norm a"])
  1901 qed
  1902 
  1903 
  1904 text{* Some theorems on sups and infs using the notion "bounded". *}
  1905 
  1906 lemma bounded_real:
  1907   fixes S :: "real set"
  1908   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
  1909   by (simp add: bounded_iff)
  1910 
  1911 lemma bounded_has_Sup:
  1912   fixes S :: "real set"
  1913   assumes "bounded S" "S \<noteq> {}"
  1914   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
  1915 proof
  1916   fix x assume "x\<in>S"
  1917   thus "x \<le> Sup S"
  1918     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
  1919 next
  1920   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
  1921     by (metis SupInf.Sup_least)
  1922 qed
  1923 
  1924 lemma Sup_insert:
  1925   fixes S :: "real set"
  1926   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" 
  1927 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) 
  1928 
  1929 lemma Sup_insert_finite:
  1930   fixes S :: "real set"
  1931   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
  1932   apply (rule Sup_insert)
  1933   apply (rule finite_imp_bounded)
  1934   by simp
  1935 
  1936 lemma bounded_has_Inf:
  1937   fixes S :: "real set"
  1938   assumes "bounded S"  "S \<noteq> {}"
  1939   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
  1940 proof
  1941   fix x assume "x\<in>S"
  1942   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
  1943   thus "x \<ge> Inf S" using `x\<in>S`
  1944     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
  1945 next
  1946   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
  1947     by (metis SupInf.Inf_greatest)
  1948 qed
  1949 
  1950 lemma Inf_insert:
  1951   fixes S :: "real set"
  1952   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" 
  1953 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal) 
  1954 lemma Inf_insert_finite:
  1955   fixes S :: "real set"
  1956   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
  1957   by (rule Inf_insert, rule finite_imp_bounded, simp)
  1958 
  1959 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
  1960 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
  1961   apply (frule isGlb_isLb)
  1962   apply (frule_tac x = y in isGlb_isLb)
  1963   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
  1964   done
  1965 
  1966 
  1967 subsection {* Equivalent versions of compactness *}
  1968 
  1969 subsubsection{* Sequential compactness *}
  1970 
  1971 definition
  1972   compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
  1973   "compact S \<longleftrightarrow>
  1974    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
  1975        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
  1976 
  1977 lemma compactI:
  1978   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
  1979   shows "compact S"
  1980   unfolding compact_def using assms by fast
  1981 
  1982 lemma compactE:
  1983   assumes "compact S" "\<forall>n. f n \<in> S"
  1984   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
  1985   using assms unfolding compact_def by fast
  1986 
  1987 text {*
  1988   A metric space (or topological vector space) is said to have the
  1989   Heine-Borel property if every closed and bounded subset is compact.
  1990 *}
  1991 
  1992 class heine_borel = metric_space +
  1993   assumes bounded_imp_convergent_subsequence:
  1994     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
  1995       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  1996 
  1997 lemma bounded_closed_imp_compact:
  1998   fixes s::"'a::heine_borel set"
  1999   assumes "bounded s" and "closed s" shows "compact s"
  2000 proof (unfold compact_def, clarify)
  2001   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  2002   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  2003     using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
  2004   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
  2005   have "l \<in> s" using `closed s` fr l
  2006     unfolding closed_sequential_limits by blast
  2007   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2008     using `l \<in> s` r l by blast
  2009 qed
  2010 
  2011 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
  2012 proof(induct n)
  2013   show "0 \<le> r 0" by auto
  2014 next
  2015   fix n assume "n \<le> r n"
  2016   moreover have "r n < r (Suc n)"
  2017     using assms [unfolded subseq_def] by auto
  2018   ultimately show "Suc n \<le> r (Suc n)" by auto
  2019 qed
  2020 
  2021 lemma eventually_subseq:
  2022   assumes r: "subseq r"
  2023   shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
  2024 unfolding eventually_sequentially
  2025 by (metis subseq_bigger [OF r] le_trans)
  2026 
  2027 lemma lim_subseq:
  2028   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
  2029 unfolding tendsto_def eventually_sequentially o_def
  2030 by (metis subseq_bigger le_trans)
  2031 
  2032 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
  2033   unfolding Ex1_def
  2034   apply (rule_tac x="nat_rec e f" in exI)
  2035   apply (rule conjI)+
  2036 apply (rule def_nat_rec_0, simp)
  2037 apply (rule allI, rule def_nat_rec_Suc, simp)
  2038 apply (rule allI, rule impI, rule ext)
  2039 apply (erule conjE)
  2040 apply (induct_tac x)
  2041 apply simp
  2042 apply (erule_tac x="n" in allE)
  2043 apply (simp)
  2044 done
  2045 
  2046 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
  2047   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
  2048   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"
  2049 proof-
  2050   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
  2051   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
  2052   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
  2053     { fix n::nat
  2054       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
  2055       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
  2056       with n have "s N \<le> t - e" using `e>0` by auto
  2057       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  }
  2058     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
  2059     hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto  }
  2060   thus ?thesis by blast
  2061 qed
  2062 
  2063 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
  2064   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
  2065   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
  2066   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
  2067   unfolding monoseq_def incseq_def
  2068   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
  2069   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
  2070 
  2071 (* TODO: merge this lemma with the ones above *)
  2072 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
  2073   assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"
  2074   shows "\<exists>l. (s ---> l) sequentially"
  2075 proof-
  2076   obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto
  2077   { fix m::nat
  2078     have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"
  2079       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
  2080       apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }
  2081   hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
  2082   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]
  2083     unfolding monoseq_def by auto
  2084   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
  2085     unfolding dist_norm  by auto
  2086 qed
  2087 
  2088 lemma compact_real_lemma:
  2089   assumes "\<forall>n::nat. abs(s n) \<le> b"
  2090   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
  2091 proof-
  2092   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
  2093     using seq_monosub[of s] by auto
  2094   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
  2095     unfolding tendsto_iff dist_norm eventually_sequentially by auto
  2096 qed
  2097 
  2098 instance real :: heine_borel
  2099 proof
  2100   fix s :: "real set" and f :: "nat \<Rightarrow> real"
  2101   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2102   then obtain b where b: "\<forall>n. abs (f n) \<le> b"
  2103     unfolding bounded_iff by auto
  2104   obtain l :: real and r :: "nat \<Rightarrow> nat" where
  2105     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  2106     using compact_real_lemma [OF b] by auto
  2107   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2108     by auto
  2109 qed
  2110 
  2111 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
  2112   apply (erule bounded_linear_image)
  2113   apply (rule bounded_linear_euclidean_component)
  2114   done
  2115 
  2116 lemma compact_lemma:
  2117   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
  2118   assumes "bounded s" and "\<forall>n. f n \<in> s"
  2119   shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
  2120         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
  2121 proof
  2122   fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
  2123   have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
  2124   hence "\<exists>l::'a. \<exists>r. subseq r \<and>
  2125       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
  2126   proof(induct d) case empty thus ?case unfolding subseq_def by auto
  2127   next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
  2128     have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
  2129     obtain l1::"'a" and r1 where r1:"subseq r1" and
  2130       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
  2131       using insert(3) using insert(4) by auto
  2132     have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
  2133     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
  2134       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
  2135     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
  2136       using r1 and r2 unfolding r_def o_def subseq_def by auto
  2137     moreover
  2138     def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
  2139     { fix e::real assume "e>0"
  2140       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
  2141       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
  2142       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
  2143         by (rule eventually_subseq)
  2144       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
  2145         using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
  2146         using insert.prems by auto
  2147     }
  2148     ultimately show ?case by auto
  2149   qed
  2150   thus "\<exists>l::'a. \<exists>r. subseq r \<and>
  2151       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
  2152     apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
  2153     apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe 
  2154     apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
  2155     apply(erule_tac x=i in ballE) 
  2156   proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
  2157     assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
  2158     hence *:"i\<ge>DIM('a)" by auto
  2159     thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
  2160   qed
  2161 qed
  2162 
  2163 instance euclidean_space \<subseteq> heine_borel
  2164 proof
  2165   fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
  2166   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2167   then obtain l::'a and r where r: "subseq r"
  2168     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
  2169     using compact_lemma [OF s f] by blast
  2170   let ?d = "{..<DIM('a)}"
  2171   { fix e::real assume "e>0"
  2172     hence "0 < e / (real_of_nat (card ?d))"
  2173       using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
  2174     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
  2175       by simp
  2176     moreover
  2177     { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
  2178       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
  2179         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
  2180       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
  2181         apply(rule setsum_strict_mono) using n by auto
  2182       finally have "dist (f (r n)) l < e" unfolding setsum_constant
  2183         using DIM_positive[where 'a='a] by auto
  2184     }
  2185     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
  2186       by (rule eventually_elim1)
  2187   }
  2188   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
  2189   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
  2190 qed
  2191 
  2192 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
  2193 unfolding bounded_def
  2194 apply clarify
  2195 apply (rule_tac x="a" in exI)
  2196 apply (rule_tac x="e" in exI)
  2197 apply clarsimp
  2198 apply (drule (1) bspec)
  2199 apply (simp add: dist_Pair_Pair)
  2200 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
  2201 done
  2202 
  2203 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
  2204 unfolding bounded_def
  2205 apply clarify
  2206 apply (rule_tac x="b" in exI)
  2207 apply (rule_tac x="e" in exI)
  2208 apply clarsimp
  2209 apply (drule (1) bspec)
  2210 apply (simp add: dist_Pair_Pair)
  2211 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
  2212 done
  2213 
  2214 instance prod :: (heine_borel, heine_borel) heine_borel
  2215 proof
  2216   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
  2217   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2218   from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
  2219   from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
  2220   obtain l1 r1 where r1: "subseq r1"
  2221     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
  2222     using bounded_imp_convergent_subsequence [OF s1 f1]
  2223     unfolding o_def by fast
  2224   from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
  2225   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
  2226   obtain l2 r2 where r2: "subseq r2"
  2227     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
  2228     using bounded_imp_convergent_subsequence [OF s2 f2]
  2229     unfolding o_def by fast
  2230   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
  2231     using lim_subseq [OF r2 l1] unfolding o_def .
  2232   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
  2233     using tendsto_Pair [OF l1' l2] unfolding o_def by simp
  2234   have r: "subseq (r1 \<circ> r2)"
  2235     using r1 r2 unfolding subseq_def by simp
  2236   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2237     using l r by fast
  2238 qed
  2239 
  2240 subsubsection{* Completeness *}
  2241 
  2242 lemma cauchy_def:
  2243   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
  2244 unfolding Cauchy_def by blast
  2245 
  2246 definition
  2247   complete :: "'a::metric_space set \<Rightarrow> bool" where
  2248   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
  2249                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"
  2250 
  2251 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
  2252 proof-
  2253   { assume ?rhs
  2254     { fix e::real
  2255       assume "e>0"
  2256       with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
  2257         by (erule_tac x="e/2" in allE) auto
  2258       { fix n m
  2259         assume nm:"N \<le> m \<and> N \<le> n"
  2260         hence "dist (s m) (s n) < e" using N
  2261           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
  2262           by blast
  2263       }
  2264       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
  2265         by blast
  2266     }
  2267     hence ?lhs
  2268       unfolding cauchy_def
  2269       by blast
  2270   }
  2271   thus ?thesis
  2272     unfolding cauchy_def
  2273     using dist_triangle_half_l
  2274     by blast
  2275 qed
  2276 
  2277 lemma convergent_imp_cauchy:
  2278  "(s ---> l) sequentially ==> Cauchy s"
  2279 proof(simp only: cauchy_def, rule, rule)
  2280   fix e::real assume "e>0" "(s ---> l) sequentially"
  2281   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
  2282   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
  2283 qed
  2284 
  2285 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
  2286 proof-
  2287   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
  2288   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
  2289   moreover
  2290   have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
  2291   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
  2292     unfolding bounded_any_center [where a="s N"] by auto
  2293   ultimately show "?thesis"
  2294     unfolding bounded_any_center [where a="s N"]
  2295     apply(rule_tac x="max a 1" in exI) apply auto
  2296     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
  2297 qed
  2298 
  2299 lemma compact_imp_complete: assumes "compact s" shows "complete s"
  2300 proof-
  2301   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
  2302     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
  2303 
  2304     note lr' = subseq_bigger [OF lr(2)]
  2305 
  2306     { fix e::real assume "e>0"
  2307       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
  2308       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
  2309       { fix n::nat assume n:"n \<ge> max N M"
  2310         have "dist ((f \<circ> r) n) l < e/2" using n M by auto
  2311         moreover have "r n \<ge> N" using lr'[of n] n by auto
  2312         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
  2313         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)  }
  2314       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }
  2315     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto  }
  2316   thus ?thesis unfolding complete_def by auto
  2317 qed
  2318 
  2319 instance heine_borel < complete_space
  2320 proof
  2321   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  2322   hence "bounded (range f)"
  2323     by (rule cauchy_imp_bounded)
  2324   hence "compact (closure (range f))"
  2325     using bounded_closed_imp_compact [of "closure (range f)"] by auto
  2326   hence "complete (closure (range f))"
  2327     by (rule compact_imp_complete)
  2328   moreover have "\<forall>n. f n \<in> closure (range f)"
  2329     using closure_subset [of "range f"] by auto
  2330   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
  2331     using `Cauchy f` unfolding complete_def by auto
  2332   then show "convergent f"
  2333     unfolding convergent_def by auto
  2334 qed
  2335 
  2336 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
  2337 proof(simp add: complete_def, rule, rule)
  2338   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  2339   hence "convergent f" by (rule Cauchy_convergent)
  2340   thus "\<exists>l. f ----> l" unfolding convergent_def .  
  2341 qed
  2342 
  2343 lemma complete_imp_closed: assumes "complete s" shows "closed s"
  2344 proof -
  2345   { fix x assume "x islimpt s"
  2346     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
  2347       unfolding islimpt_sequential by auto
  2348     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
  2349       using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
  2350     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
  2351   }
  2352   thus "closed s" unfolding closed_limpt by auto
  2353 qed
  2354 
  2355 lemma complete_eq_closed:
  2356   fixes s :: "'a::complete_space set"
  2357   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
  2358 proof
  2359   assume ?lhs thus ?rhs by (rule complete_imp_closed)
  2360 next
  2361   assume ?rhs
  2362   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
  2363     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
  2364     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  }
  2365   thus ?lhs unfolding complete_def by auto
  2366 qed
  2367 
  2368 lemma convergent_eq_cauchy:
  2369   fixes s :: "nat \<Rightarrow> 'a::complete_space"
  2370   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
  2371 proof
  2372   assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
  2373   thus ?rhs using convergent_imp_cauchy by auto
  2374 next
  2375   assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
  2376 qed
  2377 
  2378 lemma convergent_imp_bounded:
  2379   fixes s :: "nat \<Rightarrow> 'a::metric_space"
  2380   shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
  2381   using convergent_imp_cauchy[of s]
  2382   using cauchy_imp_bounded[of s]
  2383   unfolding image_def
  2384   by auto
  2385 
  2386 subsubsection{* Total boundedness *}
  2387 
  2388 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
  2389   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
  2390 declare helper_1.simps[simp del]
  2391 
  2392 lemma compact_imp_totally_bounded:
  2393   assumes "compact s"
  2394   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
  2395 proof(rule, rule, rule ccontr)
  2396   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)"
  2397   def x \<equiv> "helper_1 s e"
  2398   { fix n
  2399     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
  2400     proof(induct_tac rule:nat_less_induct)
  2401       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
  2402       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
  2403       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
  2404       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
  2405       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
  2406         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
  2407       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
  2408     qed }
  2409   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
  2410   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
  2411   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
  2412   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
  2413   show False
  2414     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
  2415     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
  2416     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
  2417 qed
  2418 
  2419 subsubsection{* Heine-Borel theorem *}
  2420 
  2421 text {* Following Burkill \& Burkill vol. 2. *}
  2422 
  2423 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
  2424   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"
  2425   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
  2426 proof(rule ccontr)
  2427   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
  2428   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
  2429   { fix n::nat
  2430     have "1 / real (n + 1) > 0" by auto
  2431     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 }
  2432   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
  2433   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)"
  2434     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
  2435 
  2436   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
  2437     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
  2438 
  2439   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
  2440   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
  2441     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
  2442 
  2443   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
  2444     using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
  2445 
  2446   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
  2447   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
  2448     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
  2449     using subseq_bigger[OF r, of "N1 + N2"] by auto
  2450 
  2451   def x \<equiv> "(f (r (N1 + N2)))"
  2452   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
  2453     using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
  2454   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
  2455   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
  2456 
  2457   have "dist x l < e/2" using N1 unfolding x_def o_def by auto
  2458   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
  2459 
  2460   thus False using e and `y\<notin>b` by auto
  2461 qed
  2462 
  2463 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
  2464                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
  2465 proof clarify
  2466   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
  2467   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
  2468   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
  2469   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
  2470   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
  2471 
  2472   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
  2473   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
  2474 
  2475   have "finite (bb ` k)" using k(1) by auto
  2476   moreover
  2477   { fix x assume "x\<in>s"
  2478     hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3)  unfolding subset_eq by auto
  2479     hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
  2480     hence "x \<in> \<Union>(bb ` k)" using  Union_iff[of x "bb ` k"] by auto
  2481   }
  2482   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
  2483 qed
  2484 
  2485 subsubsection {* Bolzano-Weierstrass property *}
  2486 
  2487 lemma heine_borel_imp_bolzano_weierstrass:
  2488   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'))"
  2489           "infinite t"  "t \<subseteq> s"
  2490   shows "\<exists>x \<in> s. x islimpt t"
  2491 proof(rule ccontr)
  2492   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
  2493   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
  2494     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
  2495   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
  2496     using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
  2497   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
  2498   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
  2499     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
  2500     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  }
  2501   hence "inj_on f t" unfolding inj_on_def by simp
  2502   hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
  2503   moreover
  2504   { fix x assume "x\<in>t" "f x \<notin> g"
  2505     from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
  2506     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
  2507     hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
  2508     hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto  }
  2509   hence "f ` t \<subseteq> g" by auto
  2510   ultimately show False using g(2) using finite_subset by auto
  2511 qed
  2512 
  2513 subsubsection {* Complete the chain of compactness variants *}
  2514 
  2515 lemma islimpt_range_imp_convergent_subsequence:
  2516   fixes f :: "nat \<Rightarrow> 'a::metric_space"
  2517   assumes "l islimpt (range f)"
  2518   shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2519 proof (intro exI conjI)
  2520   have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
  2521     using assms unfolding islimpt_def
  2522     by (drule_tac x="ball l e" in spec)
  2523        (auto simp add: zero_less_dist_iff dist_commute)
  2524 
  2525   def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
  2526   have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
  2527     unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
  2528   have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
  2529     unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
  2530   have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
  2531     unfolding t_def by (simp add: Least_le)
  2532   have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
  2533     unfolding t_def by (drule not_less_Least) simp
  2534   have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
  2535     apply (rule t_le)
  2536     apply (erule f_t_neq)
  2537     apply (erule (1) less_le_trans [OF f_t_closer])
  2538     done
  2539   have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
  2540     by (drule f_t_closer) auto
  2541   have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
  2542     apply (subst less_le)
  2543     apply (rule conjI)
  2544     apply (rule t_antimono)
  2545     apply (erule f_t_neq)
  2546     apply (erule f_t_closer [THEN less_imp_le])
  2547     apply (rule t_dist_f_neq [symmetric])
  2548     apply (erule f_t_neq)
  2549     done
  2550   have dist_f_t_less':
  2551     "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
  2552     apply (simp add: le_less)
  2553     apply (erule disjE)
  2554     apply (rule less_trans)
  2555     apply (erule f_t_closer)
  2556     apply (rule le_less_trans)
  2557     apply (erule less_tD)
  2558     apply (erule f_t_neq)
  2559     apply (erule f_t_closer)
  2560     apply (erule subst)
  2561     apply (erule f_t_closer)
  2562     done
  2563 
  2564   def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
  2565   have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
  2566     unfolding r_def by simp_all
  2567   have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
  2568     by (induct_tac n) (simp_all add: r_simps f_t_neq)
  2569 
  2570   show "subseq r"
  2571     unfolding subseq_Suc_iff
  2572     apply (rule allI)
  2573     apply (case_tac n)
  2574     apply (simp_all add: r_simps)
  2575     apply (rule t_less, rule zero_less_one)
  2576     apply (rule t_less, rule f_r_neq)
  2577     done
  2578   show "((f \<circ> r) ---> l) sequentially"
  2579     unfolding Lim_sequentially o_def
  2580     apply (clarify, rule_tac x="t e" in exI, clarify)
  2581     apply (drule le_trans, rule seq_suble [OF `subseq r`])
  2582     apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
  2583     done
  2584 qed
  2585 
  2586 lemma finite_range_imp_infinite_repeats:
  2587   fixes f :: "nat \<Rightarrow> 'a"
  2588   assumes "finite (range f)"
  2589   shows "\<exists>k. infinite {n. f n = k}"
  2590 proof -
  2591   { fix A :: "'a set" assume "finite A"
  2592     hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
  2593     proof (induct)
  2594       case empty thus ?case by simp
  2595     next
  2596       case (insert x A)
  2597      show ?case
  2598       proof (cases "finite {n. f n = x}")
  2599         case True
  2600         with `infinite {n. f n \<in> insert x A}`
  2601         have "infinite {n. f n \<in> A}" by simp
  2602         thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
  2603       next
  2604         case False thus "\<exists>k. infinite {n. f n = k}" ..
  2605       qed
  2606     qed
  2607   } note H = this
  2608   from assms show "\<exists>k. infinite {n. f n = k}"
  2609     by (rule H) simp
  2610 qed
  2611 
  2612 lemma bolzano_weierstrass_imp_compact:
  2613   fixes s :: "'a::metric_space set"
  2614   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2615   shows "compact s"
  2616 proof -
  2617   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  2618     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2619     proof (cases "finite (range f)")
  2620       case True
  2621       hence "\<exists>l. infinite {n. f n = l}"
  2622         by (rule finite_range_imp_infinite_repeats)
  2623       then obtain l where "infinite {n. f n = l}" ..
  2624       hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
  2625         by (rule infinite_enumerate)
  2626       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
  2627       hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2628         unfolding o_def by (simp add: fr tendsto_const)
  2629       hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2630         by - (rule exI)
  2631       from f have "\<forall>n. f (r n) \<in> s" by simp
  2632       hence "l \<in> s" by (simp add: fr)
  2633       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2634         by (rule rev_bexI) fact
  2635     next
  2636       case False
  2637       with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
  2638       then obtain l where "l \<in> s" "l islimpt (range f)" ..
  2639       have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2640         using `l islimpt (range f)`
  2641         by (rule islimpt_range_imp_convergent_subsequence)
  2642       with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
  2643     qed
  2644   }
  2645   thus ?thesis unfolding compact_def by auto
  2646 qed
  2647 
  2648 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
  2649   "helper_2 beyond 0 = beyond 0" |
  2650   "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
  2651 
  2652 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
  2653   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2654   shows "bounded s"
  2655 proof(rule ccontr)
  2656   assume "\<not> bounded s"
  2657   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
  2658     unfolding bounded_any_center [where a=undefined]
  2659     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
  2660   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
  2661     unfolding linorder_not_le by auto
  2662   def x \<equiv> "helper_2 beyond"
  2663 
  2664   { fix m n ::nat assume "m<n"
  2665     hence "dist undefined (x m) + 1 < dist undefined (x n)"
  2666     proof(induct n)
  2667       case 0 thus ?case by auto
  2668     next
  2669       case (Suc n)
  2670       have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
  2671         unfolding x_def and helper_2.simps
  2672         using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
  2673       thus ?case proof(cases "m < n")
  2674         case True thus ?thesis using Suc and * by auto
  2675       next
  2676         case False hence "m = n" using Suc(2) by auto
  2677         thus ?thesis using * by auto
  2678       qed
  2679     qed  } note * = this
  2680   { fix m n ::nat assume "m\<noteq>n"
  2681     have "1 < dist (x m) (x n)"
  2682     proof(cases "m<n")
  2683       case True
  2684       hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
  2685       thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
  2686     next
  2687       case False hence "n<m" using `m\<noteq>n` by auto
  2688       hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
  2689       thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
  2690     qed  } note ** = this
  2691   { fix a b assume "x a = x b" "a \<noteq> b"
  2692     hence False using **[of a b] by auto  }
  2693   hence "inj x" unfolding inj_on_def by auto
  2694   moreover
  2695   { fix n::nat
  2696     have "x n \<in> s"
  2697     proof(cases "n = 0")
  2698       case True thus ?thesis unfolding x_def using beyond by auto
  2699     next
  2700       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
  2701       thus ?thesis unfolding x_def using beyond by auto
  2702     qed  }
  2703   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
  2704 
  2705   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
  2706   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
  2707   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"]]
  2708     unfolding dist_nz by auto
  2709   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
  2710 qed
  2711 
  2712 lemma sequence_infinite_lemma:
  2713   fixes f :: "nat \<Rightarrow> 'a::t1_space"
  2714   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
  2715   shows "infinite (range f)"
  2716 proof
  2717   assume "finite (range f)"
  2718   hence "closed (range f)" by (rule finite_imp_closed)
  2719   hence "open (- range f)" by (rule open_Compl)
  2720   from assms(1) have "l \<in> - range f" by auto
  2721   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
  2722     using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
  2723   thus False unfolding eventually_sequentially by auto
  2724 qed
  2725 
  2726 lemma closure_insert:
  2727   fixes x :: "'a::t1_space"
  2728   shows "closure (insert x s) = insert x (closure s)"
  2729 apply (rule closure_unique)
  2730 apply (rule insert_mono [OF closure_subset])
  2731 apply (rule closed_insert [OF closed_closure])
  2732 apply (simp add: closure_minimal)
  2733 done
  2734 
  2735 lemma islimpt_insert:
  2736   fixes x :: "'a::t1_space"
  2737   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
  2738 proof
  2739   assume *: "x islimpt (insert a s)"
  2740   show "x islimpt s"
  2741   proof (rule islimptI)
  2742     fix t assume t: "x \<in> t" "open t"
  2743     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
  2744     proof (cases "x = a")
  2745       case True
  2746       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
  2747         using * t by (rule islimptE)
  2748       with `x = a` show ?thesis by auto
  2749     next
  2750       case False
  2751       with t have t': "x \<in> t - {a}" "open (t - {a})"
  2752         by (simp_all add: open_Diff)
  2753       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
  2754         using * t' by (rule islimptE)
  2755       thus ?thesis by auto
  2756     qed
  2757   qed
  2758 next
  2759   assume "x islimpt s" thus "x islimpt (insert a s)"
  2760     by (rule islimpt_subset) auto
  2761 qed
  2762 
  2763 lemma islimpt_union_finite:
  2764   fixes x :: "'a::t1_space"
  2765   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
  2766 by (induct set: finite, simp_all add: islimpt_insert)
  2767  
  2768 lemma sequence_unique_limpt:
  2769   fixes f :: "nat \<Rightarrow> 'a::t2_space"
  2770   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
  2771   shows "l' = l"
  2772 proof (rule ccontr)
  2773   assume "l' \<noteq> l"
  2774   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
  2775     using hausdorff [OF `l' \<noteq> l`] by auto
  2776   have "eventually (\<lambda>n. f n \<in> t) sequentially"
  2777     using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
  2778   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
  2779     unfolding eventually_sequentially by auto
  2780 
  2781   have "UNIV = {..<N} \<union> {N..}" by auto
  2782   hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
  2783   hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
  2784   hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
  2785   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
  2786     using `l' \<in> s` `open s` by (rule islimptE)
  2787   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
  2788   with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
  2789   with `s \<inter> t = {}` show False by simp
  2790 qed
  2791 
  2792 lemma bolzano_weierstrass_imp_closed:
  2793   fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
  2794   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2795   shows "closed s"
  2796 proof-
  2797   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
  2798     hence "l \<in> s"
  2799     proof(cases "\<forall>n. x n \<noteq> l")
  2800       case False thus "l\<in>s" using as(1) by auto
  2801     next
  2802       case True note cas = this
  2803       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
  2804       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
  2805       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
  2806     qed  }
  2807   thus ?thesis unfolding closed_sequential_limits by fast
  2808 qed
  2809 
  2810 text {* Hence express everything as an equivalence. *}
  2811 
  2812 lemma compact_eq_heine_borel:
  2813   fixes s :: "'a::metric_space set"
  2814   shows "compact s \<longleftrightarrow>
  2815            (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
  2816                --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
  2817 proof
  2818   assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
  2819 next
  2820   assume ?rhs
  2821   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
  2822     by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
  2823   thus ?lhs by (rule bolzano_weierstrass_imp_compact)
  2824 qed
  2825 
  2826 lemma compact_eq_bolzano_weierstrass:
  2827   fixes s :: "'a::metric_space set"
  2828   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
  2829 proof
  2830   assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2831 next
  2832   assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
  2833 qed
  2834 
  2835 lemma compact_eq_bounded_closed:
  2836   fixes s :: "'a::heine_borel set"
  2837   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")
  2838 proof
  2839   assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
  2840 next
  2841   assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
  2842 qed
  2843 
  2844 lemma compact_imp_bounded:
  2845   fixes s :: "'a::metric_space set"
  2846   shows "compact s ==> bounded s"
  2847 proof -
  2848   assume "compact s"
  2849   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')"
  2850     by (rule compact_imp_heine_borel)
  2851   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
  2852     using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2853   thus "bounded s"
  2854     by (rule bolzano_weierstrass_imp_bounded)
  2855 qed
  2856 
  2857 lemma compact_imp_closed:
  2858   fixes s :: "'a::metric_space set"
  2859   shows "compact s ==> closed s"
  2860 proof -
  2861   assume "compact s"
  2862   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')"
  2863     by (rule compact_imp_heine_borel)
  2864   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
  2865     using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2866   thus "closed s"
  2867     by (rule bolzano_weierstrass_imp_closed)
  2868 qed
  2869 
  2870 text{* In particular, some common special cases. *}
  2871 
  2872 lemma compact_empty[simp]:
  2873  "compact {}"
  2874   unfolding compact_def
  2875   by simp
  2876 
  2877 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
  2878   unfolding subseq_def by simp (* TODO: move somewhere else *)
  2879 
  2880 lemma compact_union [intro]:
  2881   assumes "compact s" and "compact t"
  2882   shows "compact (s \<union> t)"
  2883 proof (rule compactI)
  2884   fix f :: "nat \<Rightarrow> 'a"
  2885   assume "\<forall>n. f n \<in> s \<union> t"
  2886   hence "infinite {n. f n \<in> s \<union> t}" by simp
  2887   hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
  2888   thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2889   proof
  2890     assume "infinite {n. f n \<in> s}"
  2891     from infinite_enumerate [OF this]
  2892     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
  2893     obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
  2894       using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
  2895     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
  2896       using `subseq q` by (simp_all add: subseq_o o_assoc)
  2897     thus ?thesis by auto
  2898   next
  2899     assume "infinite {n. f n \<in> t}"
  2900     from infinite_enumerate [OF this]
  2901     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
  2902     obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
  2903       using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
  2904     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
  2905       using `subseq q` by (simp_all add: subseq_o o_assoc)
  2906     thus ?thesis by auto
  2907   qed
  2908 qed
  2909 
  2910 lemma compact_inter_closed [intro]:
  2911   assumes "compact s" and "closed t"
  2912   shows "compact (s \<inter> t)"
  2913 proof (rule compactI)
  2914   fix f :: "nat \<Rightarrow> 'a"
  2915   assume "\<forall>n. f n \<in> s \<inter> t"
  2916   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
  2917   obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
  2918     using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
  2919   moreover
  2920   from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
  2921     unfolding closed_sequential_limits o_def by fast
  2922   ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2923     by auto
  2924 qed
  2925 
  2926 lemma closed_inter_compact [intro]:
  2927   assumes "closed s" and "compact t"
  2928   shows "compact (s \<inter> t)"
  2929   using compact_inter_closed [of t s] assms
  2930   by (simp add: Int_commute)
  2931 
  2932 lemma compact_inter [intro]:
  2933   assumes "compact s" and "compact t"
  2934   shows "compact (s \<inter> t)"
  2935   using assms by (intro compact_inter_closed compact_imp_closed)
  2936 
  2937 lemma compact_sing [simp]: "compact {a}"
  2938   unfolding compact_def o_def subseq_def
  2939   by (auto simp add: tendsto_const)
  2940 
  2941 lemma compact_insert [simp]:
  2942   assumes "compact s" shows "compact (insert x s)"
  2943 proof -
  2944   have "compact ({x} \<union> s)"
  2945     using compact_sing assms by (rule compact_union)
  2946   thus ?thesis by simp
  2947 qed
  2948 
  2949 lemma finite_imp_compact:
  2950   shows "finite s \<Longrightarrow> compact s"
  2951   by (induct set: finite) simp_all
  2952 
  2953 lemma compact_cball[simp]:
  2954   fixes x :: "'a::heine_borel"
  2955   shows "compact(cball x e)"
  2956   using compact_eq_bounded_closed bounded_cball closed_cball
  2957   by blast
  2958 
  2959 lemma compact_frontier_bounded[intro]:
  2960   fixes s :: "'a::heine_borel set"
  2961   shows "bounded s ==> compact(frontier s)"
  2962   unfolding frontier_def
  2963   using compact_eq_bounded_closed
  2964   by blast
  2965 
  2966 lemma compact_frontier[intro]:
  2967   fixes s :: "'a::heine_borel set"
  2968   shows "compact s ==> compact (frontier s)"
  2969   using compact_eq_bounded_closed compact_frontier_bounded
  2970   by blast
  2971 
  2972 lemma frontier_subset_compact:
  2973   fixes s :: "'a::heine_borel set"
  2974   shows "compact s ==> frontier s \<subseteq> s"
  2975   using frontier_subset_closed compact_eq_bounded_closed
  2976   by blast
  2977 
  2978 lemma open_delete:
  2979   fixes s :: "'a::t1_space set"
  2980   shows "open s \<Longrightarrow> open (s - {x})"
  2981   by (simp add: open_Diff)
  2982 
  2983 text{* Finite intersection property. I could make it an equivalence in fact. *}
  2984 
  2985 lemma compact_imp_fip:
  2986   assumes "compact s"  "\<forall>t \<in> f. closed t"
  2987         "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
  2988   shows "s \<inter> (\<Inter> f) \<noteq> {}"
  2989 proof
  2990   assume as:"s \<inter> (\<Inter> f) = {}"
  2991   hence "s \<subseteq> \<Union> uminus ` f" by auto
  2992   moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
  2993   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
  2994   hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
  2995   hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
  2996   thus False using f'(3) unfolding subset_eq and Union_iff by blast
  2997 qed
  2998 
  2999 
  3000 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}
  3001 
  3002 lemma bounded_closed_nest:
  3003   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
  3004   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
  3005   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
  3006 proof-
  3007   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
  3008   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
  3009 
  3010   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
  3011     unfolding compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast
  3012 
  3013   { fix n::nat
  3014     { fix e::real assume "e>0"
  3015       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
  3016       hence "dist ((x \<circ> r) (max N n)) l < e" by auto
  3017       moreover
  3018       have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
  3019       hence "(x \<circ> r) (max N n) \<in> s n"
  3020         using x apply(erule_tac x=n in allE)
  3021         using x apply(erule_tac x="r (max N n)" in allE)
  3022         using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
  3023       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
  3024     }
  3025     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
  3026   }
  3027   thus ?thesis by auto
  3028 qed
  3029 
  3030 text {* Decreasing case does not even need compactness, just completeness. *}
  3031 
  3032 lemma decreasing_closed_nest:
  3033   assumes "\<forall>n. closed(s n)"
  3034           "\<forall>n. (s n \<noteq> {})"
  3035           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3036           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
  3037   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
  3038 proof-
  3039   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
  3040   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
  3041   then obtain t where t: "\<forall>n. t n \<in> s n" by auto
  3042   { fix e::real assume "e>0"
  3043     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
  3044     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
  3045       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+
  3046       hence "dist (t m) (t n) < e" using N by auto
  3047     }
  3048     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
  3049   }
  3050   hence  "Cauchy t" unfolding cauchy_def by auto
  3051   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
  3052   { fix n::nat
  3053     { fix e::real assume "e>0"
  3054       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
  3055       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
  3056       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
  3057     }
  3058     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
  3059   }
  3060   then show ?thesis by auto
  3061 qed
  3062 
  3063 text {* Strengthen it to the intersection actually being a singleton. *}
  3064 
  3065 lemma decreasing_closed_nest_sing:
  3066   fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
  3067   assumes "\<forall>n. closed(s n)"
  3068           "\<forall>n. s n \<noteq> {}"
  3069           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3070           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
  3071   shows "\<exists>a. \<Inter>(range s) = {a}"
  3072 proof-
  3073   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
  3074   { fix b assume b:"b \<in> \<Inter>(range s)"
  3075     { fix e::real assume "e>0"
  3076       hence "dist a b < e" using assms(4 )using b using a by blast
  3077     }
  3078     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
  3079   }
  3080   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
  3081   thus ?thesis ..
  3082 qed
  3083 
  3084 text{* Cauchy-type criteria for uniform convergence. *}
  3085 
  3086 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
  3087  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
  3088   (\<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")
  3089 proof(rule)
  3090   assume ?lhs
  3091   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
  3092   { fix e::real assume "e>0"
  3093     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
  3094     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
  3095       hence "dist (s m x) (s n x) < e"
  3096         using N[THEN spec[where x=m], THEN spec[where x=x]]
  3097         using N[THEN spec[where x=n], THEN spec[where x=x]]
  3098         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }
  3099     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  }
  3100   thus ?rhs by auto
  3101 next
  3102   assume ?rhs
  3103   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
  3104   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
  3105     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
  3106   { fix e::real assume "e>0"
  3107     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"
  3108       using `?rhs`[THEN spec[where x="e/2"]] by auto
  3109     { fix x assume "P x"
  3110       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
  3111         using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
  3112       fix n::nat assume "n\<ge>N"
  3113       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]]
  3114         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)  }
  3115     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
  3116   thus ?lhs by auto
  3117 qed
  3118 
  3119 lemma uniformly_cauchy_imp_uniformly_convergent:
  3120   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
  3121   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"
  3122           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
  3123   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
  3124 proof-
  3125   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"
  3126     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
  3127   moreover
  3128   { fix x assume "P x"
  3129     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
  3130       using l and assms(2) unfolding Lim_sequentially by blast  }
  3131   ultimately show ?thesis by auto
  3132 qed
  3133 
  3134 
  3135 subsection {* Continuity *}
  3136 
  3137 text {* Define continuity over a net to take in restrictions of the set. *}
  3138 
  3139 definition
  3140   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3141   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
  3142 
  3143 lemma continuous_trivial_limit:
  3144  "trivial_limit net ==> continuous net f"
  3145   unfolding continuous_def tendsto_def trivial_limit_eq by auto
  3146 
  3147 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
  3148   unfolding continuous_def
  3149   unfolding tendsto_def
  3150   using netlimit_within[of x s]
  3151   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
  3152 
  3153 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
  3154   using continuous_within [of x UNIV f] by (simp add: within_UNIV)
  3155 
  3156 lemma continuous_at_within:
  3157   assumes "continuous (at x) f"  shows "continuous (at x within s) f"
  3158   using assms unfolding continuous_at continuous_within
  3159   by (rule Lim_at_within)
  3160 
  3161 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
  3162 
  3163 lemma continuous_within_eps_delta:
  3164   "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)"
  3165   unfolding continuous_within and Lim_within
  3166   apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
  3167 
  3168 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.
  3169                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
  3170   using continuous_within_eps_delta[of x UNIV f]
  3171   unfolding within_UNIV by blast
  3172 
  3173 text{* Versions in terms of open balls. *}
  3174 
  3175 lemma continuous_within_ball:
  3176  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  3177                             f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3178 proof
  3179   assume ?lhs
  3180   { fix e::real assume "e>0"
  3181     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"
  3182       using `?lhs`[unfolded continuous_within Lim_within] by auto
  3183     { fix y assume "y\<in>f ` (ball x d \<inter> s)"
  3184       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
  3185         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
  3186     }
  3187     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)  }
  3188   thus ?rhs by auto
  3189 next
  3190   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
  3191     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
  3192 qed
  3193 
  3194 lemma continuous_at_ball:
  3195   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3196 proof
  3197   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3198     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)
  3199     unfolding dist_nz[THEN sym] by auto
  3200 next
  3201   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3202     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)
  3203 qed
  3204 
  3205 text{* Define setwise continuity in terms of limits within the set. *}
  3206 
  3207 definition
  3208   continuous_on ::
  3209     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3210 where
  3211   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
  3212 
  3213 lemma continuous_on_topological:
  3214   "continuous_on s f \<longleftrightarrow>
  3215     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  3216       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  3217 unfolding continuous_on_def tendsto_def
  3218 unfolding Limits.eventually_within eventually_at_topological
  3219 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  3220 
  3221 lemma continuous_on_iff:
  3222   "continuous_on s f \<longleftrightarrow>
  3223     (\<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)"
  3224 unfolding continuous_on_def Lim_within
  3225 apply (intro ball_cong [OF refl] all_cong ex_cong)
  3226 apply (rename_tac y, case_tac "y = x", simp)
  3227 apply (simp add: dist_nz)
  3228 done
  3229 
  3230 definition
  3231   uniformly_continuous_on ::
  3232     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
  3233 where
  3234   "uniformly_continuous_on s f \<longleftrightarrow>
  3235     (\<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)"
  3236 
  3237 text{* Some simple consequential lemmas. *}
  3238 
  3239 lemma uniformly_continuous_imp_continuous:
  3240  " uniformly_continuous_on s f ==> continuous_on s f"
  3241   unfolding uniformly_continuous_on_def continuous_on_iff by blast
  3242 
  3243 lemma continuous_at_imp_continuous_within:
  3244  "continuous (at x) f ==> continuous (at x within s) f"
  3245   unfolding continuous_within continuous_at using Lim_at_within by auto
  3246 
  3247 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
  3248 unfolding tendsto_def by (simp add: trivial_limit_eq)
  3249 
  3250 lemma continuous_at_imp_continuous_on:
  3251   assumes "\<forall>x\<in>s. continuous (at x) f"
  3252   shows "continuous_on s f"
  3253 unfolding continuous_on_def
  3254 proof
  3255   fix x assume "x \<in> s"
  3256   with assms have *: "(f ---> f (netlimit (at x))) (at x)"
  3257     unfolding continuous_def by simp
  3258   have "(f ---> f x) (at x)"
  3259   proof (cases "trivial_limit (at x)")
  3260     case True thus ?thesis
  3261       by (rule Lim_trivial_limit)
  3262   next
  3263     case False
  3264     hence 1: "netlimit (at x) = x"
  3265       using netlimit_within [of x UNIV]
  3266       by (simp add: within_UNIV)
  3267     with * show ?thesis by simp
  3268   qed
  3269   thus "(f ---> f x) (at x within s)"
  3270     by (rule Lim_at_within)
  3271 qed
  3272 
  3273 lemma continuous_on_eq_continuous_within:
  3274   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
  3275 unfolding continuous_on_def continuous_def
  3276 apply (rule ball_cong [OF refl])
  3277 apply (case_tac "trivial_limit (at x within s)")
  3278 apply (simp add: Lim_trivial_limit)
  3279 apply (simp add: netlimit_within)
  3280 done
  3281 
  3282 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
  3283 
  3284 lemma continuous_on_eq_continuous_at:
  3285   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
  3286   by (auto simp add: continuous_on continuous_at Lim_within_open)
  3287 
  3288 lemma continuous_within_subset:
  3289  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
  3290              ==> continuous (at x within t) f"
  3291   unfolding continuous_within by(metis Lim_within_subset)
  3292 
  3293 lemma continuous_on_subset:
  3294   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
  3295   unfolding continuous_on by (metis subset_eq Lim_within_subset)
  3296 
  3297 lemma continuous_on_interior:
  3298   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
  3299   by (erule interiorE, drule (1) continuous_on_subset,
  3300     simp add: continuous_on_eq_continuous_at)
  3301 
  3302 lemma continuous_on_eq:
  3303   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
  3304   unfolding continuous_on_def tendsto_def Limits.eventually_within
  3305   by simp
  3306 
  3307 text {* Characterization of various kinds of continuity in terms of sequences. *}
  3308 
  3309 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
  3310 lemma continuous_within_sequentially:
  3311   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3312   shows "continuous (at a within s) f \<longleftrightarrow>
  3313                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
  3314                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
  3315 proof
  3316   assume ?lhs
  3317   { 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"
  3318     fix e::real assume "e>0"
  3319     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
  3320     from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
  3321     hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
  3322       apply(rule_tac  x=N in exI) using N d  apply auto using x(1)
  3323       apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
  3324       apply(erule_tac x="x n" in ballE)  apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
  3325   }
  3326   thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
  3327 next
  3328   assume ?rhs
  3329   { fix e::real assume "e>0"
  3330     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)"
  3331     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
  3332     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)"
  3333       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
  3334     { fix d::real assume "d>0"
  3335       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
  3336       then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
  3337       { fix n::nat assume n:"n\<ge>N"
  3338         hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
  3339         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)
  3340         ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
  3341       }
  3342       hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
  3343     }
  3344     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
  3345     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
  3346     hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
  3347   }
  3348   thus ?lhs  unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
  3349 qed
  3350 
  3351 lemma continuous_at_sequentially:
  3352   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3353   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
  3354                   --> ((f o x) ---> f a) sequentially)"
  3355   using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
  3356 
  3357 lemma continuous_on_sequentially:
  3358   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3359   shows "continuous_on s f \<longleftrightarrow>
  3360     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
  3361                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
  3362 proof
  3363   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
  3364 next
  3365   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
  3366 qed
  3367 
  3368 lemma uniformly_continuous_on_sequentially':
  3369   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
  3370                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
  3371                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
  3372 proof
  3373   assume ?lhs
  3374   { 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"
  3375     { fix e::real assume "e>0"
  3376       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"
  3377         using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
  3378       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
  3379       { fix n assume "n\<ge>N"
  3380         hence "dist (f (x n)) (f (y n)) < e"
  3381           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
  3382           unfolding dist_commute by simp  }
  3383       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }
  3384     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto  }
  3385   thus ?rhs by auto
  3386 next
  3387   assume ?rhs
  3388   { assume "\<not> ?lhs"
  3389     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
  3390     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"
  3391       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
  3392       by (auto simp add: dist_commute)
  3393     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
  3394     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
  3395     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"
  3396       unfolding x_def and y_def using fa by auto
  3397     { fix e::real assume "e>0"
  3398       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
  3399       { fix n::nat assume "n\<ge>N"
  3400         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
  3401         also have "\<dots> < e" using N by auto
  3402         finally have "inverse (real n + 1) < e" by auto
  3403         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }
  3404       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }
  3405     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
  3406     hence False using fxy and `e>0` by auto  }
  3407   thus ?lhs unfolding uniformly_continuous_on_def by blast
  3408 qed
  3409 
  3410 lemma uniformly_continuous_on_sequentially:
  3411   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  3412   shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
  3413                     ((\<lambda>n. x n - y n) ---> 0) sequentially
  3414                     \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
  3415 (* BH: maybe the previous lemma should replace this one? *)
  3416 unfolding uniformly_continuous_on_sequentially'
  3417 unfolding dist_norm tendsto_norm_zero_iff ..
  3418 
  3419 text{* The usual transformation theorems. *}
  3420 
  3421 lemma continuous_transform_within:
  3422   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3423   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
  3424           "continuous (at x within s) f"
  3425   shows "continuous (at x within s) g"
  3426 unfolding continuous_within
  3427 proof (rule Lim_transform_within)
  3428   show "0 < d" by fact
  3429   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  3430     using assms(3) by auto
  3431   have "f x = g x"
  3432     using assms(1,2,3) by auto
  3433   thus "(f ---> g x) (at x within s)"
  3434     using assms(4) unfolding continuous_within by simp
  3435 qed
  3436 
  3437 lemma continuous_transform_at:
  3438   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3439   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
  3440           "continuous (at x) f"
  3441   shows "continuous (at x) g"
  3442   using continuous_transform_within [of d x UNIV f g] assms
  3443   by (simp add: within_UNIV)
  3444 
  3445 text{* Combination results for pointwise continuity. *}
  3446 
  3447 lemma continuous_const: "continuous net (\<lambda>x. c)"
  3448   by (auto simp add: continuous_def tendsto_const)
  3449 
  3450 lemma continuous_cmul:
  3451   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3452   shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
  3453   by (auto simp add: continuous_def intro: tendsto_intros)
  3454 
  3455 lemma continuous_neg:
  3456   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3457   shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
  3458   by (auto simp add: continuous_def tendsto_minus)
  3459 
  3460 lemma continuous_add:
  3461   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3462   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
  3463   by (auto simp add: continuous_def tendsto_add)
  3464 
  3465 lemma continuous_sub:
  3466   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3467   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
  3468   by (auto simp add: continuous_def tendsto_diff)
  3469 
  3470 
  3471 text{* Same thing for setwise continuity. *}
  3472 
  3473 lemma continuous_on_const:
  3474  "continuous_on s (\<lambda>x. c)"
  3475   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3476 
  3477 lemma continuous_on_cmul:
  3478   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3479   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
  3480   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3481 
  3482 lemma continuous_on_neg:
  3483   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3484   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
  3485   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3486 
  3487 lemma continuous_on_add:
  3488   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3489   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3490            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
  3491   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3492 
  3493 lemma continuous_on_sub:
  3494   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  3495   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3496            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
  3497   unfolding continuous_on_def by (auto intro: tendsto_intros)
  3498 
  3499 text{* Same thing for uniform continuity, using sequential formulations. *}
  3500 
  3501 lemma uniformly_continuous_on_const:
  3502  "uniformly_continuous_on s (\<lambda>x. c)"
  3503   unfolding uniformly_continuous_on_def by simp
  3504 
  3505 lemma uniformly_continuous_on_cmul:
  3506   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3507   assumes "uniformly_continuous_on s f"
  3508   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
  3509 proof-
  3510   { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
  3511     hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
  3512       using tendsto_scaleR [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
  3513       unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
  3514   }
  3515   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
  3516     unfolding dist_norm tendsto_norm_zero_iff by auto
  3517 qed
  3518 
  3519 lemma dist_minus:
  3520   fixes x y :: "'a::real_normed_vector"
  3521   shows "dist (- x) (- y) = dist x y"
  3522   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
  3523 
  3524 lemma uniformly_continuous_on_neg:
  3525   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3526   shows "uniformly_continuous_on s f
  3527          ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
  3528   unfolding uniformly_continuous_on_def dist_minus .
  3529 
  3530 lemma uniformly_continuous_on_add:
  3531   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3532   assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
  3533   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
  3534 proof-
  3535   {  fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
  3536                     "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
  3537     hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
  3538       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
  3539     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  }
  3540   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
  3541     unfolding dist_norm tendsto_norm_zero_iff by auto
  3542 qed
  3543 
  3544 lemma uniformly_continuous_on_sub:
  3545   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3546   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
  3547            ==> uniformly_continuous_on s  (\<lambda>x. f x - g x)"
  3548   unfolding ab_diff_minus
  3549   using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
  3550   using uniformly_continuous_on_neg[of s g] by auto
  3551 
  3552 text{* Identity function is continuous in every sense. *}
  3553 
  3554 lemma continuous_within_id:
  3555  "continuous (at a within s) (\<lambda>x. x)"
  3556   unfolding continuous_within by (rule Lim_at_within [OF LIM_ident])
  3557 
  3558 lemma continuous_at_id:
  3559  "continuous (at a) (\<lambda>x. x)"
  3560   unfolding continuous_at by (rule LIM_ident)
  3561 
  3562 lemma continuous_on_id:
  3563  "continuous_on s (\<lambda>x. x)"
  3564   unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)
  3565 
  3566 lemma uniformly_continuous_on_id:
  3567  "uniformly_continuous_on s (\<lambda>x. x)"
  3568   unfolding uniformly_continuous_on_def by auto
  3569 
  3570 text{* Continuity of all kinds is preserved under composition. *}
  3571 
  3572 lemma continuous_within_topological:
  3573   "continuous (at x within s) f \<longleftrightarrow>
  3574     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  3575       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  3576 unfolding continuous_within
  3577 unfolding tendsto_def Limits.eventually_within eventually_at_topological
  3578 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  3579 
  3580 lemma continuous_within_compose:
  3581   assumes "continuous (at x within s) f"
  3582   assumes "continuous (at (f x) within f ` s) g"
  3583   shows "continuous (at x within s) (g o f)"
  3584 using assms unfolding continuous_within_topological by simp metis
  3585 
  3586 lemma continuous_at_compose:
  3587   assumes "continuous (at x) f"  "continuous (at (f x)) g"
  3588   shows "continuous (at x) (g o f)"
  3589 proof-
  3590   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
  3591   thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
  3592 qed
  3593 
  3594 lemma continuous_on_compose:
  3595   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
  3596   unfolding continuous_on_topological by simp metis
  3597 
  3598 lemma uniformly_continuous_on_compose:
  3599   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f ` s) g"
  3600   shows "uniformly_continuous_on s (g o f)"
  3601 proof-
  3602   { fix e::real assume "e>0"
  3603     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
  3604     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
  3605     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  }
  3606   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
  3607 qed
  3608 
  3609 text{* Continuity in terms of open preimages. *}
  3610 
  3611 lemma continuous_at_open:
  3612   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)))"
  3613 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
  3614 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
  3615 
  3616 lemma continuous_on_open:
  3617   shows "continuous_on s f \<longleftrightarrow>
  3618         (\<forall>t. openin (subtopology euclidean (f ` s)) t
  3619             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  3620 proof (safe)
  3621   fix t :: "'b set"
  3622   assume 1: "continuous_on s f"
  3623   assume 2: "openin (subtopology euclidean (f ` s)) t"
  3624   from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
  3625     unfolding openin_open by auto
  3626   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
  3627   have "open U" unfolding U_def by (simp add: open_Union)
  3628   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
  3629   proof (intro ballI iffI)
  3630     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
  3631       unfolding U_def t by auto
  3632   next
  3633     fix x assume "x \<in> s" and "f x \<in> t"
  3634     hence "x \<in> s" and "f x \<in> B"
  3635       unfolding t by auto
  3636     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
  3637       unfolding t continuous_on_topological by metis
  3638     then show "x \<in> U"
  3639       unfolding U_def by auto
  3640   qed
  3641   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
  3642   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3643     unfolding openin_open by fast
  3644 next
  3645   assume "?rhs" show "continuous_on s f"
  3646   unfolding continuous_on_topological
  3647   proof (clarify)
  3648     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
  3649     have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
  3650       unfolding openin_open using `open B` by auto
  3651     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
  3652       using `?rhs` by fast
  3653     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
  3654       unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
  3655   qed
  3656 qed
  3657 
  3658 text {* Similarly in terms of closed sets. *}
  3659 
  3660 lemma continuous_on_closed:
  3661   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")
  3662 proof
  3663   assume ?lhs
  3664   { fix t
  3665     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  3666     have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
  3667     assume as:"closedin (subtopology euclidean (f ` s)) t"
  3668     hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
  3669     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"]]
  3670       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }
  3671   thus ?rhs by auto
  3672 next
  3673   assume ?rhs
  3674   { fix t
  3675     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  3676     assume as:"openin (subtopology euclidean (f ` s)) t"
  3677     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
  3678       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
  3679   thus ?lhs unfolding continuous_on_open by auto
  3680 qed
  3681 
  3682 text {* Half-global and completely global cases. *}
  3683 
  3684 lemma continuous_open_in_preimage:
  3685   assumes "continuous_on s f"  "open t"
  3686   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3687 proof-
  3688   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
  3689   have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  3690     using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
  3691   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  3692 qed
  3693 
  3694 lemma continuous_closed_in_preimage:
  3695   assumes "continuous_on s f"  "closed t"
  3696   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3697 proof-
  3698   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
  3699   have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  3700     using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
  3701   thus ?thesis
  3702     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  3703 qed
  3704 
  3705 lemma continuous_open_preimage:
  3706   assumes "continuous_on s f" "open s" "open t"
  3707   shows "open {x \<in> s. f x \<in> t}"
  3708 proof-
  3709   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  3710     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
  3711   thus ?thesis using open_Int[of s T, OF assms(2)] by auto
  3712 qed
  3713 
  3714 lemma continuous_closed_preimage:
  3715   assumes "continuous_on s f" "closed s" "closed t"
  3716   shows "closed {x \<in> s. f x \<in> t}"
  3717 proof-
  3718   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  3719     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
  3720   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
  3721 qed
  3722 
  3723 lemma continuous_open_preimage_univ:
  3724   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
  3725   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
  3726 
  3727 lemma continuous_closed_preimage_univ:
  3728   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
  3729   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
  3730 
  3731 lemma continuous_open_vimage:
  3732   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
  3733   unfolding vimage_def by (rule continuous_open_preimage_univ)
  3734 
  3735 lemma continuous_closed_vimage:
  3736   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
  3737   unfolding vimage_def by (rule continuous_closed_preimage_univ)
  3738 
  3739 lemma interior_image_subset:
  3740   assumes "\<forall>x. continuous (at x) f" "inj f"
  3741   shows "interior (f ` s) \<subseteq> f ` (interior s)"
  3742 proof
  3743   fix x assume "x \<in> interior (f ` s)"
  3744   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` s" ..
  3745   hence "x \<in> f ` s" by auto
  3746   then obtain y where y: "y \<in> s" "x = f y" by auto
  3747   have "open (vimage f T)"
  3748     using assms(1) `open T` by (rule continuous_open_vimage)
  3749   moreover have "y \<in> vimage f T"
  3750     using `x = f y` `x \<in> T` by simp
  3751   moreover have "vimage f T \<subseteq> s"
  3752     using `T \<subseteq> image f s` `inj f` unfolding inj_on_def subset_eq by auto
  3753   ultimately have "y \<in> interior s" ..
  3754   with `x = f y` show "x \<in> f ` interior s" ..
  3755 qed
  3756 
  3757 text {* Equality of continuous functions on closure and related results. *}
  3758 
  3759 lemma continuous_closed_in_preimage_constant:
  3760   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3761   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
  3762   using continuous_closed_in_preimage[of s f "{a}"] by auto
  3763 
  3764 lemma continuous_closed_preimage_constant:
  3765   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3766   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
  3767   using continuous_closed_preimage[of s f "{a}"] by auto
  3768 
  3769 lemma continuous_constant_on_closure:
  3770   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3771   assumes "continuous_on (closure s) f"
  3772           "\<forall>x \<in> s. f x = a"
  3773   shows "\<forall>x \<in> (closure s). f x = a"
  3774     using continuous_closed_preimage_constant[of "closure s" f a]
  3775     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
  3776 
  3777 lemma image_closure_subset:
  3778   assumes "continuous_on (closure s) f"  "closed t"  "(f ` s) \<subseteq> t"
  3779   shows "f ` (closure s) \<subseteq> t"
  3780 proof-
  3781   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
  3782   moreover have "closed {x \<in> closure s. f x \<in> t}"
  3783     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
  3784   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
  3785     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
  3786   thus ?thesis by auto
  3787 qed
  3788 
  3789 lemma continuous_on_closure_norm_le:
  3790   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3791   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"
  3792   shows "norm(f x) \<le> b"
  3793 proof-
  3794   have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
  3795   show ?thesis
  3796     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
  3797     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
  3798 qed
  3799 
  3800 text {* Making a continuous function avoid some value in a neighbourhood. *}
  3801 
  3802 lemma continuous_within_avoid:
  3803   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3804   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"
  3805   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
  3806 proof-
  3807   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"
  3808     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
  3809   { fix y assume " y\<in>s"  "dist x y < d"
  3810     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
  3811       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
  3812   thus ?thesis using `d>0` by auto
  3813 qed
  3814 
  3815 lemma continuous_at_avoid:
  3816   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3817   assumes "continuous (at x) f"  "f x \<noteq> a"
  3818   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  3819 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
  3820 
  3821 lemma continuous_on_avoid:
  3822   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
  3823   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"
  3824   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
  3825 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
  3826 
  3827 lemma continuous_on_open_avoid:
  3828   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
  3829   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"
  3830   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  3831 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
  3832 
  3833 text {* Proving a function is constant by proving open-ness of level set. *}
  3834 
  3835 lemma continuous_levelset_open_in_cases:
  3836   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3837   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  3838         openin (subtopology euclidean s) {x \<in> s. f x = a}
  3839         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
  3840 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
  3841 
  3842 lemma continuous_levelset_open_in:
  3843   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3844   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  3845         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
  3846         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"
  3847 using continuous_levelset_open_in_cases[of s f ]
  3848 by meson
  3849 
  3850 lemma continuous_levelset_open:
  3851   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  3852   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
  3853   shows "\<forall>x \<in> s. f x = a"
  3854 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
  3855 
  3856 text {* Some arithmetical combinations (more to prove). *}
  3857 
  3858 lemma open_scaling[intro]:
  3859   fixes s :: "'a::real_normed_vector set"
  3860   assumes "c \<noteq> 0"  "open s"
  3861   shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
  3862 proof-
  3863   { fix x assume "x \<in> s"
  3864     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
  3865     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
  3866     moreover
  3867     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
  3868       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
  3869         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
  3870           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
  3871       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  }
  3872     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  }
  3873   thus ?thesis unfolding open_dist by auto
  3874 qed
  3875 
  3876 lemma minus_image_eq_vimage:
  3877   fixes A :: "'a::ab_group_add set"
  3878   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
  3879   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
  3880 
  3881 lemma open_negations:
  3882   fixes s :: "'a::real_normed_vector set"
  3883   shows "open s ==> open ((\<lambda> x. -x) ` s)"
  3884   unfolding scaleR_minus1_left [symmetric]
  3885   by (rule open_scaling, auto)
  3886 
  3887 lemma open_translation:
  3888   fixes s :: "'a::real_normed_vector set"
  3889   assumes "open s"  shows "open((\<lambda>x. a + x) ` s)"
  3890 proof-
  3891   { 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  }
  3892   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
  3893   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
  3894 qed
  3895 
  3896 lemma open_affinity:
  3897   fixes s :: "'a::real_normed_vector set"
  3898   assumes "open s"  "c \<noteq> 0"
  3899   shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  3900 proof-
  3901   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
  3902   have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
  3903   thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
  3904 qed
  3905 
  3906 lemma interior_translation:
  3907   fixes s :: "'a::real_normed_vector set"
  3908   shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
  3909 proof (rule set_eqI, rule)
  3910   fix x assume "x \<in> interior (op + a ` s)"
  3911   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
  3912   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
  3913   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
  3914 next
  3915   fix x assume "x \<in> op + a ` interior s"
  3916   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
  3917   { fix z have *:"a + y - z = y + a - z" by auto
  3918     assume "z\<in>ball x e"
  3919     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
  3920     hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }
  3921   hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
  3922   thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
  3923 qed
  3924 
  3925 text {* We can now extend limit compositions to consider the scalar multiplier. *}
  3926 
  3927 lemma continuous_vmul:
  3928   fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
  3929   shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
  3930   unfolding continuous_def by (intro tendsto_intros)
  3931 
  3932 lemma continuous_mul:
  3933   fixes c :: "'a::metric_space \<Rightarrow> real"
  3934   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3935   shows "continuous net c \<Longrightarrow> continuous net f
  3936              ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
  3937   unfolding continuous_def by (intro tendsto_intros)
  3938 
  3939 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul
  3940   continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
  3941 
  3942 lemma continuous_on_vmul:
  3943   fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
  3944   shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
  3945   unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
  3946 
  3947 lemma continuous_on_mul:
  3948   fixes c :: "'a::metric_space \<Rightarrow> real"
  3949   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3950   shows "continuous_on s c \<Longrightarrow> continuous_on s f
  3951              ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
  3952   unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
  3953 
  3954 lemma continuous_on_mul_real:
  3955   fixes f :: "'a::metric_space \<Rightarrow> real"
  3956   fixes g :: "'a::metric_space \<Rightarrow> real"
  3957   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3958              ==> continuous_on s (\<lambda>x. f x * g x)"
  3959   using continuous_on_mul[of s f g] unfolding real_scaleR_def .
  3960 
  3961 lemmas continuous_on_intros = continuous_on_add continuous_on_const
  3962   continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg
  3963   continuous_on_sub continuous_on_mul continuous_on_vmul continuous_on_mul_real
  3964   uniformly_continuous_on_add uniformly_continuous_on_const
  3965   uniformly_continuous_on_id uniformly_continuous_on_compose
  3966   uniformly_continuous_on_cmul uniformly_continuous_on_neg
  3967   uniformly_continuous_on_sub
  3968 
  3969 text {* And so we have continuity of inverse. *}
  3970 
  3971 lemma continuous_inv:
  3972   fixes f :: "'a::metric_space \<Rightarrow> real"
  3973   shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
  3974            ==> continuous net (inverse o f)"
  3975   unfolding continuous_def using Lim_inv by auto
  3976 
  3977 lemma continuous_at_within_inv:
  3978   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
  3979   assumes "continuous (at a within s) f" "f a \<noteq> 0"
  3980   shows "continuous (at a within s) (inverse o f)"
  3981   using assms unfolding continuous_within o_def
  3982   by (intro tendsto_intros)
  3983 
  3984 lemma continuous_at_inv:
  3985   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
  3986   shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
  3987          ==> continuous (at a) (inverse o f) "
  3988   using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
  3989 
  3990 text {* Topological properties of linear functions. *}
  3991 
  3992 lemma linear_lim_0:
  3993   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
  3994 proof-
  3995   interpret f: bounded_linear f by fact
  3996   have "(f ---> f 0) (at 0)"
  3997     using tendsto_ident_at by (rule f.tendsto)
  3998   thus ?thesis unfolding f.zero .
  3999 qed
  4000 
  4001 lemma linear_continuous_at:
  4002   assumes "bounded_linear f"  shows "continuous (at a) f"
  4003   unfolding continuous_at using assms
  4004   apply (rule bounded_linear.tendsto)
  4005   apply (rule tendsto_ident_at)
  4006   done
  4007 
  4008 lemma linear_continuous_within:
  4009   shows "bounded_linear f ==> continuous (at x within s) f"
  4010   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
  4011 
  4012 lemma linear_continuous_on:
  4013   shows "bounded_linear f ==> continuous_on s f"
  4014   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
  4015 
  4016 text {* Also bilinear functions, in composition form. *}
  4017 
  4018 lemma bilinear_continuous_at_compose:
  4019   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
  4020         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
  4021   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
  4022 
  4023 lemma bilinear_continuous_within_compose:
  4024   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
  4025         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
  4026   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
  4027 
  4028 lemma bilinear_continuous_on_compose:
  4029   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
  4030              ==> continuous_on s (\<lambda>x. h (f x) (g x))"
  4031   unfolding continuous_on_def
  4032   by (fast elim: bounded_bilinear.tendsto)
  4033 
  4034 text {* Preservation of compactness and connectedness under continuous function. *}
  4035 
  4036 lemma compact_continuous_image:
  4037   assumes "continuous_on s f"  "compact s"
  4038   shows "compact(f ` s)"
  4039 proof-
  4040   { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
  4041     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
  4042     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
  4043     { fix e::real assume "e>0"
  4044       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
  4045       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
  4046       { 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  }
  4047       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }
  4048     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  }
  4049   thus ?thesis unfolding compact_def by auto
  4050 qed
  4051 
  4052 lemma connected_continuous_image:
  4053   assumes "continuous_on s f"  "connected s"
  4054   shows "connected(f ` s)"
  4055 proof-
  4056   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f ` s"  "openin (subtopology euclidean (f ` s)) T"  "closedin (subtopology euclidean (f ` s)) T"
  4057     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
  4058       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
  4059       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
  4060       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
  4061     hence False using as(1,2)
  4062       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
  4063   thus ?thesis unfolding connected_clopen by auto
  4064 qed
  4065 
  4066 text {* Continuity implies uniform continuity on a compact domain. *}
  4067 
  4068 lemma compact_uniformly_continuous:
  4069   assumes "continuous_on s f"  "compact s"
  4070   shows "uniformly_continuous_on s f"
  4071 proof-
  4072     { fix x assume x:"x\<in>s"
  4073       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
  4074       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  }
  4075     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
  4076     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)"
  4077       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
  4078 
  4079   { fix e::real assume "e>0"
  4080 
  4081     { 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  }
  4082     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
  4083     moreover
  4084     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }
  4085     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
  4086 
  4087     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
  4088       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
  4089       hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
  4090       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`
  4091         by (auto  simp add: dist_commute)
  4092       moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
  4093         by (auto simp add: dist_commute)
  4094       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`
  4095         by (auto  simp add: dist_commute)
  4096       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
  4097         by (auto simp add: dist_commute)  }
  4098     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  }
  4099   thus ?thesis unfolding uniformly_continuous_on_def by auto
  4100 qed
  4101 
  4102 text{* Continuity of inverse function on compact domain. *}
  4103 
  4104 lemma continuous_on_inverse:
  4105   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  4106     (* TODO: can this be generalized more? *)
  4107   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"
  4108   shows "continuous_on (f ` s) g"
  4109 proof-
  4110   have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
  4111   { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
  4112     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
  4113     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
  4114       unfolding T(2) and Int_left_absorb by auto
  4115     moreover have "compact (s \<inter> T)"
  4116       using assms(2) unfolding compact_eq_bounded_closed
  4117       using bounded_subset[of s "s \<inter> T"] and T(1) by auto
  4118     ultimately have "closed (f ` t)" using T(1) unfolding T(2)
  4119       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
  4120     moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
  4121     ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
  4122       unfolding closedin_closed by auto  }
  4123   thus ?thesis unfolding continuous_on_closed by auto
  4124 qed
  4125 
  4126 text {* A uniformly convergent limit of continuous functions is continuous. *}
  4127 
  4128 lemma continuous_uniform_limit:
  4129   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
  4130   assumes "\<not> trivial_limit F"
  4131   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
  4132   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
  4133   shows "continuous_on s g"
  4134 proof-
  4135   { fix x and e::real assume "x\<in>s" "e>0"
  4136     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
  4137       using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
  4138     from eventually_happens [OF eventually_conj [OF this assms(2)]]
  4139     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"
  4140       using assms(1) by blast
  4141     have "e / 3 > 0" using `e>0` by auto
  4142     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"
  4143       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
  4144     { fix y assume "y \<in> s" and "dist y x < d"
  4145       hence "dist (f n y) (f n x) < e / 3"
  4146         by (rule d [rule_format])
  4147       hence "dist (f n y) (g x) < 2 * e / 3"
  4148         using dist_triangle [of "f n y" "g x" "f n x"]
  4149         using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
  4150         by auto
  4151       hence "dist (g y) (g x) < e"
  4152         using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
  4153         using dist_triangle3 [of "g y" "g x" "f n y"]
  4154         by auto }
  4155     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
  4156       using `d>0` by auto }
  4157   thus ?thesis unfolding continuous_on_iff by auto
  4158 qed
  4159 
  4160 
  4161 subsection {* Topological stuff lifted from and dropped to R *}
  4162 
  4163 lemma open_real:
  4164   fixes s :: "real set" shows
  4165  "open s \<longleftrightarrow>
  4166         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
  4167   unfolding open_dist dist_norm by simp
  4168 
  4169 lemma islimpt_approachable_real:
  4170   fixes s :: "real set"
  4171   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
  4172   unfolding islimpt_approachable dist_norm by simp
  4173 
  4174 lemma closed_real:
  4175   fixes s :: "real set"
  4176   shows "closed s \<longleftrightarrow>
  4177         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
  4178             --> x \<in> s)"
  4179   unfolding closed_limpt islimpt_approachable dist_norm by simp
  4180 
  4181 lemma continuous_at_real_range:
  4182   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4183   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  4184         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
  4185   unfolding continuous_at unfolding Lim_at
  4186   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
  4187   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
  4188   apply(erule_tac x=e in allE) by auto
  4189 
  4190 lemma continuous_on_real_range:
  4191   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4192   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))"
  4193   unfolding continuous_on_iff dist_norm by simp
  4194 
  4195 lemma continuous_at_norm: "continuous (at x) norm"
  4196   unfolding continuous_at by (intro tendsto_intros)
  4197 
  4198 lemma continuous_on_norm: "continuous_on s norm"
  4199 unfolding continuous_on by (intro ballI tendsto_intros)
  4200 
  4201 lemma continuous_at_infnorm: "continuous (at x) infnorm"
  4202   unfolding continuous_at Lim_at o_def unfolding dist_norm
  4203   apply auto apply (rule_tac x=e in exI) apply auto
  4204   using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
  4205 
  4206 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
  4207 
  4208 lemma compact_attains_sup:
  4209   fixes s :: "real set"
  4210   assumes "compact s"  "s \<noteq> {}"
  4211   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
  4212 proof-
  4213   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4214   { 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"
  4215     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
  4216     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
  4217     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto  }
  4218   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
  4219     apply(rule_tac x="Sup s" in bexI) by auto
  4220 qed
  4221 
  4222 lemma Inf:
  4223   fixes S :: "real set"
  4224   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
  4225 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def) 
  4226 
  4227 lemma compact_attains_inf:
  4228   fixes s :: "real set"
  4229   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
  4230 proof-
  4231   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4232   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"
  4233       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
  4234     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
  4235     moreover
  4236     { fix x assume "x \<in> s"
  4237       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
  4238       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
  4239     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
  4240     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto  }
  4241   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
  4242     apply(rule_tac x="Inf s" in bexI) by auto
  4243 qed
  4244 
  4245 lemma continuous_attains_sup:
  4246   fixes f :: "'a::metric_space \<Rightarrow> real"
  4247   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4248         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"
  4249   using compact_attains_sup[of "f ` s"]
  4250   using compact_continuous_image[of s f] by auto
  4251 
  4252 lemma continuous_attains_inf:
  4253   fixes f :: "'a::metric_space \<Rightarrow> real"
  4254   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4255         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
  4256   using compact_attains_inf[of "f ` s"]
  4257   using compact_continuous_image[of s f] by auto
  4258 
  4259 lemma distance_attains_sup:
  4260   assumes "compact s" "s \<noteq> {}"
  4261   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
  4262 proof (rule continuous_attains_sup [OF assms])
  4263   { fix x assume "x\<in>s"
  4264     have "(dist a ---> dist a x) (at x within s)"
  4265       by (intro tendsto_dist tendsto_const Lim_at_within LIM_ident)
  4266   }
  4267   thus "continuous_on s (dist a)"
  4268     unfolding continuous_on ..
  4269 qed
  4270 
  4271 text {* For \emph{minimal} distance, we only need closure, not compactness. *}
  4272 
  4273 lemma distance_attains_inf:
  4274   fixes a :: "'a::heine_borel"
  4275   assumes "closed s"  "s \<noteq> {}"
  4276   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
  4277 proof-
  4278   from assms(2) obtain b where "b\<in>s" by auto
  4279   let ?B = "cball a (dist b a) \<inter> s"
  4280   have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
  4281   hence "?B \<noteq> {}" by auto
  4282   moreover
  4283   { fix x assume "x\<in>?B"
  4284     fix e::real assume "e>0"
  4285     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
  4286       from as have "\<bar>dist a x' - dist a x\<bar> < e"
  4287         unfolding abs_less_iff minus_diff_eq
  4288         using dist_triangle2 [of a x' x]
  4289         using dist_triangle [of a x x']
  4290         by arith
  4291     }
  4292     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
  4293       using `e>0` by auto
  4294   }
  4295   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
  4296     unfolding continuous_on Lim_within dist_norm real_norm_def
  4297     by fast
  4298   moreover have "compact ?B"
  4299     using compact_cball[of a "dist b a"]
  4300     unfolding compact_eq_bounded_closed
  4301     using bounded_Int and closed_Int and assms(1) by auto
  4302   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"
  4303     using continuous_attains_inf[of ?B "dist a"] by fastsimp
  4304   thus ?thesis by fastsimp
  4305 qed
  4306 
  4307 
  4308 subsection {* Pasted sets *}
  4309 
  4310 lemma bounded_Times:
  4311   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
  4312 proof-
  4313   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
  4314     using assms [unfolded bounded_def] by auto
  4315   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
  4316     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
  4317   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
  4318 qed
  4319 
  4320 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
  4321 by (induct x) simp
  4322 
  4323 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
  4324 unfolding compact_def
  4325 apply clarify
  4326 apply (drule_tac x="fst \<circ> f" in spec)
  4327 apply (drule mp, simp add: mem_Times_iff)
  4328 apply (clarify, rename_tac l1 r1)
  4329 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
  4330 apply (drule mp, simp add: mem_Times_iff)
  4331 apply (clarify, rename_tac l2 r2)
  4332 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
  4333 apply (rule_tac x="r1 \<circ> r2" in exI)
  4334 apply (rule conjI, simp add: subseq_def)
  4335 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
  4336 apply (drule (1) tendsto_Pair) back
  4337 apply (simp add: o_def)
  4338 done
  4339 
  4340 text{* Hence some useful properties follow quite easily. *}
  4341 
  4342 lemma compact_scaling:
  4343   fixes s :: "'a::real_normed_vector set"
  4344   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
  4345 proof-
  4346   let ?f = "\<lambda>x. scaleR c x"
  4347   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)
  4348   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
  4349     using linear_continuous_at[OF *] assms by auto
  4350 qed
  4351 
  4352 lemma compact_negations:
  4353   fixes s :: "'a::real_normed_vector set"
  4354   assumes "compact s"  shows "compact ((\<lambda>x. -x) ` s)"
  4355   using compact_scaling [OF assms, of "- 1"] by auto
  4356 
  4357 lemma compact_sums:
  4358   fixes s t :: "'a::real_normed_vector set"
  4359   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
  4360 proof-
  4361   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
  4362     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
  4363   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
  4364     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  4365   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
  4366 qed
  4367 
  4368 lemma compact_differences:
  4369   fixes s t :: "'a::real_normed_vector set"
  4370   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
  4371 proof-
  4372   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
  4373     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4374   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
  4375 qed
  4376 
  4377 lemma compact_translation:
  4378   fixes s :: "'a::real_normed_vector set"
  4379   assumes "compact s"  shows "compact ((\<lambda>x. a + x) ` s)"
  4380 proof-
  4381   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
  4382   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
  4383 qed
  4384 
  4385 lemma compact_affinity:
  4386   fixes s :: "'a::real_normed_vector set"
  4387   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  4388 proof-
  4389   have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  4390   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
  4391 qed
  4392 
  4393 text {* Hence we get the following. *}
  4394 
  4395 lemma compact_sup_maxdistance:
  4396   fixes s :: "'a::real_normed_vector set"
  4397   assumes "compact s"  "s \<noteq> {}"
  4398   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
  4399 proof-
  4400   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
  4401   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"
  4402     using compact_differences[OF assms(1) assms(1)]
  4403     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
  4404   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
  4405   thus ?thesis using x(2)[unfolded `x = a - b`] by blast
  4406 qed
  4407 
  4408 text {* We can state this in terms of diameter of a set. *}
  4409 
  4410 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
  4411   (* TODO: generalize to class metric_space *)
  4412 
  4413 lemma diameter_bounded:
  4414   assumes "bounded s"
  4415   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
  4416         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
  4417 proof-
  4418   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
  4419   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
  4420   { fix x y assume "x \<in> s" "y \<in> s"
  4421     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)  }
  4422   note * = this
  4423   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto
  4424     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`
  4425       by simp (blast intro!: Sup_upper *) }
  4426   moreover
  4427   { fix d::real assume "d>0" "d < diameter s"
  4428     hence "s\<noteq>{}" unfolding diameter_def by auto
  4429     have "\<exists>d' \<in> ?D. d' > d"
  4430     proof(rule ccontr)
  4431       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
  4432       hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE) 
  4433       thus False using `d < diameter s` `s\<noteq>{}` 
  4434         apply (auto simp add: diameter_def) 
  4435         apply (drule Sup_real_iff [THEN [2] rev_iffD2])
  4436         apply (auto, force) 
  4437         done
  4438     qed
  4439     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }
  4440   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
  4441         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
  4442 qed
  4443 
  4444 lemma diameter_bounded_bound:
  4445  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
  4446   using diameter_bounded by blast
  4447 
  4448 lemma diameter_compact_attained:
  4449   fixes s :: "'a::real_normed_vector set"
  4450   assumes "compact s"  "s \<noteq> {}"
  4451   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
  4452 proof-
  4453   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
  4454   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
  4455   hence "diameter s \<le> norm (x - y)"
  4456     unfolding diameter_def by clarsimp (rule Sup_least, fast+)
  4457   thus ?thesis
  4458     by (metis b diameter_bounded_bound order_antisym xys)
  4459 qed
  4460 
  4461 text {* Related results with closure as the conclusion. *}
  4462 
  4463 lemma closed_scaling:
  4464   fixes s :: "'a::real_normed_vector set"
  4465   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
  4466 proof(cases "s={}")
  4467   case True thus ?thesis by auto
  4468 next
  4469   case False
  4470   show ?thesis
  4471   proof(cases "c=0")
  4472     have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
  4473     case True thus ?thesis apply auto unfolding * by auto
  4474   next
  4475     case False
  4476     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s"  "(x ---> l) sequentially"
  4477       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
  4478           using as(1)[THEN spec[where x=n]]
  4479           using `c\<noteq>0` by auto
  4480       }
  4481       moreover
  4482       { fix e::real assume "e>0"
  4483         hence "0 < e *\<bar>c\<bar>"  using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
  4484         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
  4485           using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
  4486         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
  4487           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
  4488           using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto  }
  4489       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
  4490       ultimately have "l \<in> scaleR c ` s"
  4491         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"]]
  4492         unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }
  4493     thus ?thesis unfolding closed_sequential_limits by fast
  4494   qed
  4495 qed
  4496 
  4497 lemma closed_negations:
  4498   fixes s :: "'a::real_normed_vector set"
  4499   assumes "closed s"  shows "closed ((\<lambda>x. -x) ` s)"
  4500   using closed_scaling[OF assms, of "- 1"] by simp
  4501 
  4502 lemma compact_closed_sums:
  4503   fixes s :: "'a::real_normed_vector set"
  4504   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  4505 proof-
  4506   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
  4507   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"
  4508     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"
  4509       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
  4510     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
  4511       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
  4512     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
  4513       using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
  4514     hence "l - l' \<in> t"
  4515       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
  4516       using f(3) by auto
  4517     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
  4518   }
  4519   thus ?thesis unfolding closed_sequential_limits by fast
  4520 qed
  4521 
  4522 lemma closed_compact_sums:
  4523   fixes s t :: "'a::real_normed_vector set"
  4524   assumes "closed s"  "compact t"
  4525   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  4526 proof-
  4527   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
  4528     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
  4529   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
  4530 qed
  4531 
  4532 lemma compact_closed_differences:
  4533   fixes s t :: "'a::real_normed_vector set"
  4534   assumes "compact s"  "closed t"
  4535   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  4536 proof-
  4537   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"
  4538     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4539   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
  4540 qed
  4541 
  4542 lemma closed_compact_differences:
  4543   fixes s t :: "'a::real_normed_vector set"
  4544   assumes "closed s" "compact t"
  4545   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  4546 proof-
  4547   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
  4548     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4549  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
  4550 qed
  4551 
  4552 lemma closed_translation:
  4553   fixes a :: "'a::real_normed_vector"
  4554   assumes "closed s"  shows "closed ((\<lambda>x. a + x) ` s)"
  4555 proof-
  4556   have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
  4557   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
  4558 qed
  4559 
  4560 lemma translation_Compl:
  4561   fixes a :: "'a::ab_group_add"
  4562   shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
  4563   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
  4564 
  4565 lemma translation_UNIV:
  4566   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
  4567   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
  4568 
  4569 lemma translation_diff:
  4570   fixes a :: "'a::ab_group_add"
  4571   shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
  4572   by auto
  4573 
  4574 lemma closure_translation:
  4575   fixes a :: "'a::real_normed_vector"
  4576   shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
  4577 proof-
  4578   have *:"op + a ` (- s) = - op + a ` s"
  4579     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
  4580   show ?thesis unfolding closure_interior translation_Compl
  4581     using interior_translation[of a "- s"] unfolding * by auto
  4582 qed
  4583 
  4584 lemma frontier_translation:
  4585   fixes a :: "'a::real_normed_vector"
  4586   shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
  4587   unfolding frontier_def translation_diff interior_translation closure_translation by auto
  4588 
  4589 
  4590 subsection {* Separation between points and sets *}
  4591 
  4592 lemma separate_point_closed:
  4593   fixes s :: "'a::heine_borel set"
  4594   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
  4595 proof(cases "s = {}")
  4596   case True
  4597   thus ?thesis by(auto intro!: exI[where x=1])
  4598 next
  4599   case False
  4600   assume "closed s" "a \<notin> s"
  4601   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
  4602   with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
  4603 qed
  4604 
  4605 lemma separate_compact_closed:
  4606   fixes s t :: "'a::{heine_borel, real_normed_vector} set"
  4607     (* TODO: does this generalize to heine_borel? *)
  4608   assumes "compact s" and "closed t" and "s \<inter> t = {}"
  4609   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  4610 proof-
  4611   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
  4612   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"
  4613     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
  4614   { fix x y assume "x\<in>s" "y\<in>t"
  4615     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
  4616     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
  4617       by (auto  simp add: dist_commute)
  4618     hence "d \<le> dist x y" unfolding dist_norm by auto  }
  4619   thus ?thesis using `d>0` by auto
  4620 qed
  4621 
  4622 lemma separate_closed_compact:
  4623   fixes s t :: "'a::{heine_borel, real_normed_vector} set"
  4624   assumes "closed s" and "compact t" and "s \<inter> t = {}"
  4625   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  4626 proof-
  4627   have *:"t \<inter> s = {}" using assms(3) by auto
  4628   show ?thesis using separate_compact_closed[OF assms(2,1) *]
  4629     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
  4630     by (auto simp add: dist_commute)
  4631 qed
  4632 
  4633 
  4634 subsection {* Intervals *}
  4635   
  4636 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
  4637   "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
  4638   "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
  4639   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  4640 
  4641 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4642   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
  4643   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
  4644   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  4645 
  4646 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
  4647  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
  4648  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
  4649 proof-
  4650   { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
  4651     hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
  4652     hence "a$$i < b$$i" by auto
  4653     hence False using as by auto  }
  4654   moreover
  4655   { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
  4656     let ?x = "(1/2) *\<^sub>R (a + b)"
  4657     { fix i assume i:"i<DIM('a)" 
  4658       have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
  4659       hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
  4660         unfolding euclidean_simps by auto }
  4661     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
  4662   ultimately show ?th1 by blast
  4663 
  4664   { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
  4665     hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
  4666     hence "a$$i \<le> b$$i" by auto
  4667     hence False using as by auto  }
  4668   moreover
  4669   { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
  4670     let ?x = "(1/2) *\<^sub>R (a + b)"
  4671     { fix i assume i:"i<DIM('a)"
  4672       have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
  4673       hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
  4674         unfolding euclidean_simps by auto }
  4675     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
  4676   ultimately show ?th2 by blast
  4677 qed
  4678 
  4679 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
  4680   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
  4681   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
  4682   unfolding interval_eq_empty[of a b] by fastsimp+
  4683 
  4684 lemma interval_sing: fixes a :: "'a::ordered_euclidean_space" shows
  4685  "{a .. a} = {a}" "{a<..<a} = {}"
  4686   apply(auto simp add: set_eq_iff euclidean_eq[where 'a='a] eucl_less[where 'a='a] eucl_le[where 'a='a])
  4687   apply (simp add: order_eq_iff) apply(rule_tac x=0 in exI) by (auto simp add: not_less less_imp_le)
  4688 
  4689 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
  4690  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
  4691  "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
  4692  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
  4693  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
  4694   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval 
  4695   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
  4696 
  4697 lemma interval_open_subset_closed:  fixes a :: "'a::ordered_euclidean_space" shows
  4698  "{a<..<b} \<subseteq> {a .. b}"
  4699 proof(simp add: subset_eq, rule)
  4700   fix x
  4701   assume x:"x \<in>{a<..<b}"
  4702   { fix i assume "i<DIM('a)"
  4703     hence "a $$ i \<le> x $$ i"
  4704       using x order_less_imp_le[of "a$$i" "x$$i"] 
  4705       by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
  4706   }
  4707   moreover
  4708   { fix i assume "i<DIM('a)"
  4709     hence "x $$ i \<le> b $$ i"
  4710       using x order_less_imp_le[of "x$$i" "b$$i"]
  4711       by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
  4712   }
  4713   ultimately
  4714   show "a \<le> x \<and> x \<le> b"
  4715     by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
  4716 qed
  4717 
  4718 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4719  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and
  4720  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and
  4721  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and
  4722  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)
  4723 proof-
  4724   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
  4725   show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
  4726   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
  4727     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
  4728     fix i assume i:"i<DIM('a)"
  4729     (** TODO combine the following two parts as done in the HOL_light version. **)
  4730     { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
  4731       assume as2: "a$$i > c$$i"
  4732       { fix j assume j:"j<DIM('a)"
  4733         hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
  4734           apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
  4735           by (auto simp add: as2)  }
  4736       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
  4737       moreover
  4738       have "?x\<notin>{a .. b}"
  4739         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
  4740         using as(2)[THEN spec[where x=i]] and as2 i
  4741         by auto
  4742       ultimately have False using as by auto  }
  4743     hence "a$$i \<le> c$$i" by(rule ccontr)auto
  4744     moreover
  4745     { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
  4746       assume as2: "b$$i < d$$i"
  4747       { fix j assume "j<DIM('a)"
  4748         hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j" 
  4749           apply(cases "j=i") using as(2)[THEN spec[where x=j]]
  4750           by (auto simp add: as2)  }
  4751       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
  4752       moreover
  4753       have "?x\<notin>{a .. b}"
  4754         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
  4755         using as(2)[THEN spec[where x=i]] and as2 using i
  4756         by auto
  4757       ultimately have False using as by auto  }
  4758     hence "b$$i \<ge> d$$i" by(rule ccontr)auto
  4759     ultimately
  4760     have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
  4761   } note part1 = this
  4762   show ?th3 unfolding subset_eq and Ball_def and mem_interval 
  4763     apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
  4764     prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+ 
  4765   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
  4766     fix i assume i:"i<DIM('a)"
  4767     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
  4768     hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto  } note * = this
  4769   show ?th4 unfolding subset_eq and Ball_def and mem_interval 
  4770     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
  4771     apply auto by(erule_tac x=i in allE, simp)+ 
  4772 qed
  4773 
  4774 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
  4775   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and
  4776   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and
  4777   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and
  4778   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)
  4779 proof-
  4780   let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
  4781   note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
  4782   show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4783     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4784   show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4785     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4786   show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4787     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4788   show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
  4789     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
  4790 qed
  4791 
  4792 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4793  "{a .. b} \<inter> {c .. d} =  {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
  4794   unfolding set_eq_iff and Int_iff and mem_interval
  4795   by auto
  4796 
  4797 (* Moved interval_open_subset_closed a bit upwards *)
  4798 
  4799 lemma open_interval[intro]:
  4800   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
  4801 proof-
  4802   have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i})"
  4803     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
  4804       linear_continuous_at bounded_linear_euclidean_component
  4805       open_real_greaterThanLessThan)
  4806   also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i}) = {a<..<b}"
  4807     by (auto simp add: eucl_less [where 'a='a])
  4808   finally show "open {a<..<b}" .
  4809 qed
  4810 
  4811 lemma closed_interval[intro]:
  4812   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
  4813 proof-
  4814   have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i})"
  4815     by (intro closed_INT ballI continuous_closed_vimage allI
  4816       linear_continuous_at bounded_linear_euclidean_component
  4817       closed_real_atLeastAtMost)
  4818   also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i}) = {a .. b}"
  4819     by (auto simp add: eucl_le [where 'a='a])
  4820   finally show "closed {a .. b}" .
  4821 qed
  4822 
  4823 lemma interior_closed_interval [intro]:
  4824   fixes a b :: "'a::ordered_euclidean_space"
  4825   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")
  4826 proof(rule subset_antisym)
  4827   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval
  4828     by (rule interior_maximal)
  4829 next
  4830   { fix x assume "x \<in> interior {a..b}"
  4831     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..
  4832     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
  4833     { fix i assume i:"i<DIM('a)"
  4834       have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
  4835            "dist (x + (e / 2) *\<^sub>R basis i) x < e"
  4836         unfolding dist_norm apply auto
  4837         unfolding norm_minus_cancel using norm_basis and `e>0` by auto
  4838       hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
  4839                      "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
  4840         using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
  4841         and   e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
  4842         unfolding mem_interval by (auto elim!: allE[where x=i])
  4843       hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
  4844         unfolding basis_component using `e>0` i by auto  }
  4845     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
  4846   thus "?L \<subseteq> ?R" ..
  4847 qed
  4848 
  4849 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
  4850 proof-
  4851   let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
  4852   { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
  4853     { fix i assume "i<DIM('a)"
  4854       hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto  }
  4855     hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
  4856     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
  4857   thus ?thesis unfolding interval and bounded_iff by auto
  4858 qed
  4859 
  4860 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
  4861  "bounded {a .. b} \<and> bounded {a<..<b}"
  4862   using bounded_closed_interval[of a b]
  4863   using interval_open_subset_closed[of a b]
  4864   using bounded_subset[of "{a..b}" "{a<..<b}"]
  4865   by simp
  4866 
  4867 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
  4868  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
  4869   using bounded_interval[of a b] by auto
  4870 
  4871 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
  4872   using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
  4873   by auto
  4874 
  4875 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
  4876   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
  4877 proof-
  4878   { fix i assume "i<DIM('a)"
  4879     hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
  4880       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
  4881       unfolding euclidean_simps by auto  }
  4882   thus ?thesis unfolding mem_interval by auto
  4883 qed
  4884 
  4885 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
  4886   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
  4887   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
  4888 proof-
  4889   { fix i assume i:"i<DIM('a)"
  4890     have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
  4891     also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
  4892       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  4893       using x unfolding mem_interval using i apply simp
  4894       using y unfolding mem_interval using i apply simp
  4895       done
  4896     finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
  4897     moreover {
  4898     have "b $$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp
  4899     also have "\<dots> > e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
  4900       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  4901       using x unfolding mem_interval using i apply simp
  4902       using y unfolding mem_interval using i apply simp
  4903       done
  4904     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
  4905     } ultimately have "a $$ 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 }
  4906   thus ?thesis unfolding mem_interval by auto
  4907 qed
  4908 
  4909 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
  4910   assumes "{a<..<b} \<noteq> {}"
  4911   shows "closure {a<..<b} = {a .. b}"
  4912 proof-
  4913   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
  4914   let ?c = "(1 / 2) *\<^sub>R (a + b)"
  4915   { fix x assume as:"x \<in> {a .. b}"
  4916     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
  4917     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
  4918       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
  4919       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
  4920         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
  4921         by (auto simp add: algebra_simps)
  4922       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
  4923       hence False using fn unfolding f_def using xc by auto  }
  4924     moreover
  4925     { assume "\<not> (f ---> x) sequentially"
  4926       { fix e::real assume "e>0"
  4927         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
  4928         then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  4929         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)
  4930         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }
  4931       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
  4932         unfolding Lim_sequentially by(auto simp add: dist_norm)
  4933       hence "(f ---> x) sequentially" unfolding f_def
  4934         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]
  4935         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  }
  4936     ultimately have "x \<in> closure {a<..<b}"
  4937       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }
  4938   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
  4939 qed
  4940 
  4941 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
  4942   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
  4943 proof-
  4944   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
  4945   def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
  4946   { fix x assume "x\<in>s"
  4947     fix i assume i:"i<DIM('a)"
  4948     hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
  4949       and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto  }
  4950   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
  4951 qed
  4952 
  4953 lemma bounded_subset_open_interval:
  4954   fixes s :: "('a::ordered_euclidean_space) set"
  4955   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
  4956   by (auto dest!: bounded_subset_open_interval_symmetric)
  4957 
  4958 lemma bounded_subset_closed_interval_symmetric:
  4959   fixes s :: "('a::ordered_euclidean_space) set"
  4960   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
  4961 proof-
  4962   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
  4963   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
  4964 qed
  4965 
  4966 lemma bounded_subset_closed_interval:
  4967   fixes s :: "('a::ordered_euclidean_space) set"
  4968   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
  4969   using bounded_subset_closed_interval_symmetric[of s] by auto
  4970 
  4971 lemma frontier_closed_interval:
  4972   fixes a b :: "'a::ordered_euclidean_space"
  4973   shows "frontier {a .. b} = {a .. b} - {a<..<b}"
  4974   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
  4975 
  4976 lemma frontier_open_interval:
  4977   fixes a b :: "'a::ordered_euclidean_space"
  4978   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
  4979 proof(cases "{a<..<b} = {}")
  4980   case True thus ?thesis using frontier_empty by auto
  4981 next
  4982   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
  4983 qed
  4984 
  4985 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
  4986   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
  4987   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
  4988 
  4989 
  4990 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)
  4991 
  4992 lemma closed_interval_left: fixes b::"'a::euclidean_space"
  4993   shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
  4994 proof-
  4995   { fix i assume i:"i<DIM('a)"
  4996     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$ i \<le> b $$ i}. x' \<noteq> x \<and> dist x' x < e"
  4997     { assume "x$$i > b$$i"
  4998       then obtain y where "y $$ i \<le> b $$ i"  "y \<noteq> x"  "dist y x < x$$i - b$$i"
  4999         using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
  5000       hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i 
  5001         by auto   }
  5002     hence "x$$i \<le> b$$i" by(rule ccontr)auto  }
  5003   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  5004 qed
  5005 
  5006 lemma closed_interval_right: fixes a::"'a::euclidean_space"
  5007   shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
  5008 proof-
  5009   { fix i assume i:"i<DIM('a)"
  5010     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$ i \<le> x $$ i}. x' \<noteq> x \<and> dist x' x < e"
  5011     { assume "a$$i > x$$i"
  5012       then obtain y where "a $$ i \<le> y $$ i"  "y \<noteq> x"  "dist y x < a$$i - x$$i"
  5013         using x[THEN spec[where x="a$$i - x$$i"]] i by auto
  5014       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto   }
  5015     hence "a$$i \<le> x$$i" by(rule ccontr)auto  }
  5016   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  5017 qed
  5018 
  5019 text {* Intervals in general, including infinite and mixtures of open and closed. *}
  5020 
  5021 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
  5022   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$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)"
  5023 
  5024 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
  5025   "is_interval {a<..<b}" (is ?th2) proof -
  5026   have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
  5027   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
  5028     by(meson order_trans le_less_trans less_le_trans *)+ qed
  5029 
  5030 lemma is_interval_empty:
  5031  "is_interval {}"
  5032   unfolding is_interval_def
  5033   by simp
  5034 
  5035 lemma is_interval_univ:
  5036  "is_interval UNIV"
  5037   unfolding is_interval_def
  5038   by simp
  5039 
  5040 
  5041 subsection {* Closure of halfspaces and hyperplanes *}
  5042 
  5043 lemma isCont_open_vimage:
  5044   assumes "\<And>x. isCont f x" and "open s" shows "open (f -` s)"
  5045 proof -
  5046   from assms(1) have "continuous_on UNIV f"
  5047     unfolding isCont_def continuous_on_def within_UNIV by simp
  5048   hence "open {x \<in> UNIV. f x \<in> s}"
  5049     using open_UNIV `open s` by (rule continuous_open_preimage)
  5050   thus "open (f -` s)"
  5051     by (simp add: vimage_def)
  5052 qed
  5053 
  5054 lemma isCont_closed_vimage:
  5055   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f -` s)"
  5056   using assms unfolding closed_def vimage_Compl [symmetric]
  5057   by (rule isCont_open_vimage)
  5058 
  5059 lemma open_Collect_less:
  5060   fixes f g :: "'a::topological_space \<Rightarrow> real"
  5061   assumes f: "\<And>x. isCont f x"
  5062   assumes g: "\<And>x. isCont g x"
  5063   shows "open {x. f x < g x}"
  5064 proof -
  5065   have "open ((\<lambda>x. g x - f x) -` {0<..})"
  5066     using isCont_diff [OF g f] open_real_greaterThan
  5067     by (rule isCont_open_vimage)
  5068   also have "((\<lambda>x. g x - f x) -` {0<..}) = {x. f x < g x}"
  5069     by auto
  5070   finally show ?thesis .
  5071 qed
  5072 
  5073 lemma closed_Collect_le:
  5074   fixes f g :: "'a::topological_space \<Rightarrow> real"
  5075   assumes f: "\<And>x. isCont f x"
  5076   assumes g: "\<And>x. isCont g x"
  5077   shows "closed {x. f x \<le> g x}"
  5078 proof -
  5079   have "closed ((\<lambda>x. g x - f x) -` {0..})"
  5080     using isCont_diff [OF g f] closed_real_atLeast
  5081     by (rule isCont_closed_vimage)
  5082   also have "((\<lambda>x. g x - f x) -` {0..}) = {x. f x \<le> g x}"
  5083     by auto
  5084   finally show ?thesis .
  5085 qed
  5086 
  5087 lemma closed_Collect_eq:
  5088   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
  5089   assumes f: "\<And>x. isCont f x"
  5090   assumes g: "\<And>x. isCont g x"
  5091   shows "closed {x. f x = g x}"
  5092 proof -
  5093   have "open {(x::'b, y::'b). x \<noteq> y}"
  5094     unfolding open_prod_def by (auto dest!: hausdorff)
  5095   hence "closed {(x::'b, y::'b). x = y}"
  5096     unfolding closed_def split_def Collect_neg_eq .
  5097   with isCont_Pair [OF f g]
  5098   have "closed ((\<lambda>x. (f x, g x)) -` {(x, y). x = y})"
  5099     by (rule isCont_closed_vimage)
  5100   also have "\<dots> = {x. f x = g x}" by auto
  5101   finally show ?thesis .
  5102 qed
  5103 
  5104 lemma Lim_inner:
  5105   assumes "(f ---> l) net"  shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
  5106   by (intro tendsto_intros assms)
  5107 
  5108 lemma continuous_at_inner: "continuous (at x) (inner a)"
  5109   unfolding continuous_at by (intro tendsto_intros)
  5110 
  5111 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
  5112   unfolding euclidean_component_def by (rule continuous_at_inner)
  5113 
  5114 lemma continuous_on_inner:
  5115   fixes s :: "'a::real_inner set"
  5116   shows "continuous_on s (inner a)"
  5117   unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  5118 
  5119 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
  5120   by (simp add: closed_Collect_le)
  5121 
  5122 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
  5123   by (simp add: closed_Collect_le)
  5124 
  5125 lemma closed_hyperplane: "closed {x. inner a x = b}"
  5126   by (simp add: closed_Collect_eq)
  5127 
  5128 lemma closed_halfspace_component_le:
  5129   shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
  5130   by (simp add: closed_Collect_le)
  5131 
  5132 lemma closed_halfspace_component_ge:
  5133   shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
  5134   by (simp add: closed_Collect_le)
  5135 
  5136 text {* Openness of halfspaces. *}
  5137 
  5138 lemma open_halfspace_lt: "open {x. inner a x < b}"
  5139   by (simp add: open_Collect_less)
  5140 
  5141 lemma open_halfspace_gt: "open {x. inner a x > b}"
  5142   by (simp add: open_Collect_less)
  5143 
  5144 lemma open_halfspace_component_lt:
  5145   shows "open {x::'a::euclidean_space. x$$i < a}"
  5146   by (simp add: open_Collect_less)
  5147 
  5148 lemma open_halfspace_component_gt:
  5149   shows "open {x::'a::euclidean_space. x$$i > a}"
  5150   by (simp add: open_Collect_less)
  5151 
  5152 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
  5153 
  5154 lemma eucl_lessThan_eq_halfspaces:
  5155   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5156   shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
  5157  by (auto simp: eucl_less[where 'a='a])
  5158 
  5159 lemma eucl_greaterThan_eq_halfspaces:
  5160   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5161   shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
  5162  by (auto simp: eucl_less[where 'a='a])
  5163 
  5164 lemma eucl_atMost_eq_halfspaces:
  5165   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5166   shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
  5167  by (auto simp: eucl_le[where 'a='a])
  5168 
  5169 lemma eucl_atLeast_eq_halfspaces:
  5170   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5171   shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
  5172  by (auto simp: eucl_le[where 'a='a])
  5173 
  5174 lemma open_eucl_lessThan[simp, intro]:
  5175   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5176   shows "open {..< a}"
  5177   by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
  5178 
  5179 lemma open_eucl_greaterThan[simp, intro]:
  5180   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5181   shows "open {a <..}"
  5182   by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
  5183 
  5184 lemma closed_eucl_atMost[simp, intro]:
  5185   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5186   shows "closed {.. a}"
  5187   unfolding eucl_atMost_eq_halfspaces
  5188   by (simp add: closed_INT closed_Collect_le)
  5189 
  5190 lemma closed_eucl_atLeast[simp, intro]:
  5191   fixes a :: "'a\<Colon>ordered_euclidean_space"
  5192   shows "closed {a ..}"
  5193   unfolding eucl_atLeast_eq_halfspaces
  5194   by (simp add: closed_INT closed_Collect_le)
  5195 
  5196 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
  5197   by (auto intro!: continuous_open_vimage)
  5198 
  5199 text {* This gives a simple derivation of limit component bounds. *}
  5200 
  5201 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
  5202   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$$i \<le> b) net"
  5203   shows "l$$i \<le> b"
  5204 proof-
  5205   { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
  5206       unfolding euclidean_component_def by auto  } note * = this
  5207   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
  5208     using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
  5209 qed
  5210 
  5211 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
  5212   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$$i) net"
  5213   shows "b \<le> l$$i"
  5214 proof-
  5215   { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
  5216       unfolding euclidean_component_def by auto  } note * = this
  5217   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
  5218     using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
  5219 qed
  5220 
  5221 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
  5222   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
  5223   shows "l$$i = b"
  5224   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
  5225 text{* Limits relative to a union.                                               *}
  5226 
  5227 lemma eventually_within_Un:
  5228   "eventually P (net within (s \<union> t)) \<longleftrightarrow>
  5229     eventually P (net within s) \<and> eventually P (net within t)"
  5230   unfolding Limits.eventually_within
  5231   by (auto elim!: eventually_rev_mp)
  5232 
  5233 lemma Lim_within_union:
  5234  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
  5235   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
  5236   unfolding tendsto_def
  5237   by (auto simp add: eventually_within_Un)
  5238 
  5239 lemma Lim_topological:
  5240  "(f ---> l) net \<longleftrightarrow>
  5241         trivial_limit net \<or>
  5242         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
  5243   unfolding tendsto_def trivial_limit_eq by auto
  5244 
  5245 lemma continuous_on_union:
  5246   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
  5247   shows "continuous_on (s \<union> t) f"
  5248   using assms unfolding continuous_on Lim_within_union
  5249   unfolding Lim_topological trivial_limit_within closed_limpt by auto
  5250 
  5251 lemma continuous_on_cases:
  5252   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
  5253           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
  5254   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
  5255 proof-
  5256   let ?h = "(\<lambda>x. if P x then f x else g x)"
  5257   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
  5258   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
  5259   moreover
  5260   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
  5261   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
  5262   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
  5263 qed
  5264 
  5265 
  5266 text{* Some more convenient intermediate-value theorem formulations.             *}
  5267 
  5268 lemma connected_ivt_hyperplane:
  5269   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
  5270   shows "\<exists>z \<in> s. inner a z = b"
  5271 proof(rule ccontr)
  5272   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
  5273   let ?A = "{x. inner a x < b}"
  5274   let ?B = "{x. inner a x > b}"
  5275   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
  5276   moreover have "?A \<inter> ?B = {}" by auto
  5277   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
  5278   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
  5279 qed
  5280 
  5281 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
  5282  "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)"
  5283   using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
  5284   unfolding euclidean_component_def by auto
  5285 
  5286 
  5287 subsection {* Homeomorphisms *}
  5288 
  5289 definition "homeomorphism s t f g \<equiv>
  5290      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
  5291      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
  5292 
  5293 definition
  5294   homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
  5295     (infixr "homeomorphic" 60) where
  5296   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
  5297 
  5298 lemma homeomorphic_refl: "s homeomorphic s"
  5299   unfolding homeomorphic_def
  5300   unfolding homeomorphism_def
  5301   using continuous_on_id
  5302   apply(rule_tac x = "(\<lambda>x. x)" in exI)
  5303   apply(rule_tac x = "(\<lambda>x. x)" in exI)
  5304   by blast
  5305 
  5306 lemma homeomorphic_sym:
  5307  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
  5308 unfolding homeomorphic_def
  5309 unfolding homeomorphism_def
  5310 by blast 
  5311 
  5312 lemma homeomorphic_trans:
  5313   assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
  5314 proof-
  5315   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"
  5316     using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
  5317   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"
  5318     using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
  5319 
  5320   { 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 }
  5321   moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
  5322   moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
  5323   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 }
  5324   moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
  5325   moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6)  unfolding fg2(5) by auto
  5326   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
  5327 qed
  5328 
  5329 lemma homeomorphic_minimal:
  5330  "s homeomorphic t \<longleftrightarrow>
  5331     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
  5332            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
  5333            continuous_on s f \<and> continuous_on t g)"
  5334 unfolding homeomorphic_def homeomorphism_def
  5335 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
  5336 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
  5337 unfolding image_iff
  5338 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
  5339 apply auto apply(rule_tac x="g x" in bexI) apply auto
  5340 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
  5341 apply auto apply(rule_tac x="f x" in bexI) by auto
  5342 
  5343 text {* Relatively weak hypotheses if a set is compact. *}
  5344 
  5345 lemma homeomorphism_compact:
  5346   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  5347     (* class constraint due to continuous_on_inverse *)
  5348   assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
  5349   shows "\<exists>g. homeomorphism s t f g"
  5350 proof-
  5351   def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
  5352   have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
  5353   { fix y assume "y\<in>t"
  5354     then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
  5355     hence "g (f x) = x" using g by auto
  5356     hence "f (g y) = y" unfolding x(1)[THEN sym] by auto  }
  5357   hence g':"\<forall>x\<in>t. f (g x) = x" by auto
  5358   moreover
  5359   { fix x
  5360     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"])
  5361     moreover
  5362     { assume "x\<in>g ` t"
  5363       then obtain y where y:"y\<in>t" "g y = x" by auto
  5364       then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
  5365       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 }
  5366     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..  }
  5367   hence "g ` t = s" by auto
  5368   ultimately
  5369   show ?thesis unfolding homeomorphism_def homeomorphic_def
  5370     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
  5371 qed
  5372 
  5373 lemma homeomorphic_compact:
  5374   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  5375     (* class constraint due to continuous_on_inverse *)
  5376   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
  5377           \<Longrightarrow> s homeomorphic t"
  5378   unfolding homeomorphic_def by (metis homeomorphism_compact)
  5379 
  5380 text{* Preservation of topological properties.                                   *}
  5381 
  5382 lemma homeomorphic_compactness:
  5383  "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
  5384 unfolding homeomorphic_def homeomorphism_def
  5385 by (metis compact_continuous_image)
  5386 
  5387 text{* Results on translation, scaling etc.                                      *}
  5388 
  5389 lemma homeomorphic_scaling:
  5390   fixes s :: "'a::real_normed_vector set"
  5391   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
  5392   unfolding homeomorphic_minimal
  5393   apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
  5394   apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
  5395   using assms apply auto
  5396   using continuous_on_cmul[OF continuous_on_id] by auto
  5397 
  5398 lemma homeomorphic_translation:
  5399   fixes s :: "'a::real_normed_vector set"
  5400   shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
  5401   unfolding homeomorphic_minimal
  5402   apply(rule_tac x="\<lambda>x. a + x" in exI)
  5403   apply(rule_tac x="\<lambda>x. -a + x" in exI)
  5404   using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
  5405 
  5406 lemma homeomorphic_affinity:
  5407   fixes s :: "'a::real_normed_vector set"
  5408   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  5409 proof-
  5410   have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  5411   show ?thesis
  5412     using homeomorphic_trans
  5413     using homeomorphic_scaling[OF assms, of s]
  5414     using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
  5415 qed
  5416 
  5417 lemma homeomorphic_balls:
  5418   fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
  5419   assumes "0 < d"  "0 < e"
  5420   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
  5421         "(cball a d) homeomorphic (cball b e)" (is ?cth)
  5422 proof-
  5423   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
  5424   show ?th unfolding homeomorphic_minimal
  5425     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
  5426     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
  5427     using assms apply (auto simp add: dist_commute)
  5428     unfolding dist_norm
  5429     apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
  5430     unfolding continuous_on
  5431     by (intro ballI tendsto_intros, simp)+
  5432 next
  5433   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
  5434   show ?cth unfolding homeomorphic_minimal
  5435     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
  5436     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
  5437     using assms apply (auto simp add: dist_commute)
  5438     unfolding dist_norm
  5439     apply (auto simp add: pos_divide_le_eq)
  5440     unfolding continuous_on
  5441     by (intro ballI tendsto_intros, simp)+
  5442 qed
  5443 
  5444 text{* "Isometry" (up to constant bounds) of injective linear map etc.           *}
  5445 
  5446 lemma cauchy_isometric:
  5447   fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
  5448   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)"
  5449   shows "Cauchy x"
  5450 proof-
  5451   interpret f: bounded_linear f by fact
  5452   { fix d::real assume "d>0"
  5453     then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
  5454       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
  5455     { fix n assume "n\<ge>N"
  5456       hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
  5457       moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
  5458         using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
  5459         using normf[THEN bspec[where x="x n - x N"]] by auto
  5460       ultimately have "norm (x n - x N) < d" using `e>0`
  5461         using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto   }
  5462     hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
  5463   thus ?thesis unfolding cauchy and dist_norm by auto
  5464 qed
  5465 
  5466 lemma complete_isometric_image:
  5467   fixes f :: "'a::euclidean_space => 'b::euclidean_space"
  5468   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"
  5469   shows "complete(f ` s)"
  5470 proof-
  5471   { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
  5472     then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)" 
  5473       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
  5474     hence x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)" by auto
  5475     hence "f \<circ> x = g" unfolding fun_eq_iff by auto
  5476     then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
  5477       using cs[unfolded complete_def, THEN spec[where x="x"]]
  5478       using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
  5479     hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
  5480       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
  5481       unfolding `f \<circ> x = g` by auto  }
  5482   thus ?thesis unfolding complete_def by auto
  5483 qed
  5484 
  5485 lemma dist_0_norm:
  5486   fixes x :: "'a::real_normed_vector"
  5487   shows "dist 0 x = norm x"
  5488 unfolding dist_norm by simp
  5489 
  5490 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  5491   assumes s:"closed s"  "subspace s"  and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
  5492   shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
  5493 proof(cases "s \<subseteq> {0::'a}")
  5494   case True
  5495   { fix x assume "x \<in> s"
  5496     hence "x = 0" using True by auto
  5497     hence "norm x \<le> norm (f x)" by auto  }
  5498   thus ?thesis by(auto intro!: exI[where x=1])
  5499 next
  5500   interpret f: bounded_linear f by fact
  5501   case False
  5502   then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
  5503   from False have "s \<noteq> {}" by auto
  5504   let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
  5505   let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
  5506   let ?S'' = "{x::'a. norm x = norm a}"
  5507 
  5508   have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
  5509   hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
  5510   moreover have "?S' = s \<inter> ?S''" by auto
  5511   ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
  5512   moreover have *:"f ` ?S' = ?S" by auto
  5513   ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
  5514   hence "closed ?S" using compact_imp_closed by auto
  5515   moreover have "?S \<noteq> {}" using a by auto
  5516   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
  5517   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
  5518 
  5519   let ?e = "norm (f b) / norm b"
  5520   have "norm b > 0" using ba and a and norm_ge_zero by auto
  5521   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
  5522   ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
  5523   moreover
  5524   { fix x assume "x\<in>s"
  5525     hence "norm (f b) / norm b * norm x \<le> norm (f x)"
  5526     proof(cases "x=0")
  5527       case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
  5528     next
  5529       case False
  5530       hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
  5531       have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
  5532       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
  5533       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"]]
  5534         unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
  5535         by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
  5536     qed }
  5537   ultimately
  5538   show ?thesis by auto
  5539 qed
  5540 
  5541 lemma closed_injective_image_subspace:
  5542   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  5543   assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
  5544   shows "closed(f ` s)"
  5545 proof-
  5546   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
  5547   show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
  5548     unfolding complete_eq_closed[THEN sym] by auto
  5549 qed
  5550 
  5551 
  5552 subsection {* Some properties of a canonical subspace *}
  5553 
  5554 lemma subspace_substandard:
  5555   "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
  5556   unfolding subspace_def by auto
  5557 
  5558 lemma closed_substandard:
  5559  "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
  5560 proof-
  5561   let ?D = "{i. P i} \<inter> {..<DIM('a)}"
  5562   have "closed (\<Inter>i\<in>?D. {x::'a. x$$i = 0})"
  5563     by (simp add: closed_INT closed_Collect_eq)
  5564   also have "(\<Inter>i\<in>?D. {x::'a. x$$i = 0}) = ?A"
  5565     by auto
  5566   finally show "closed ?A" .
  5567 qed
  5568 
  5569 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
  5570   shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
  5571 proof-
  5572   let ?D = "{..<DIM('a)}"
  5573   let ?B = "(basis::nat => 'a) ` d"
  5574   let ?bas = "basis::nat \<Rightarrow> 'a"
  5575   have "?B \<subseteq> ?A" by auto
  5576   moreover
  5577   { fix x::"'a" assume "x\<in>?A"
  5578     hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
  5579     hence "x\<in> span ?B"
  5580     proof(induct d arbitrary: x)
  5581       case empty hence "x=0" apply(subst euclidean_eq) by auto
  5582       thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
  5583     next
  5584       case (insert k F)
  5585       hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
  5586       have **:"F \<subseteq> insert k F" by auto
  5587       def y \<equiv> "x - x$$k *\<^sub>R basis k"
  5588       have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
  5589       { fix i assume i':"i \<notin> F"
  5590         hence "y $$ i = 0" unfolding y_def 
  5591           using *[THEN spec[where x=i]] by auto }
  5592       hence "y \<in> span (basis ` F)" using insert(3) by auto
  5593       hence "y \<in> span (basis ` (insert k F))"
  5594         using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
  5595         using image_mono[OF **, of basis] using assms by auto
  5596       moreover
  5597       have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
  5598       hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
  5599         using span_mul by auto
  5600       ultimately
  5601       have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
  5602         using span_add by auto
  5603       thus ?case using y by auto
  5604     qed
  5605   }
  5606   hence "?A \<subseteq> span ?B" by auto
  5607   moreover
  5608   { fix x assume "x \<in> ?B"
  5609     hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto  }
  5610   hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
  5611   moreover
  5612   have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
  5613   hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
  5614   have "card ?B = card d" unfolding card_image[OF *] by auto
  5615   ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
  5616 qed
  5617 
  5618 text{* Hence closure and completeness of all subspaces.                          *}
  5619 
  5620 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
  5621 apply (induct n)
  5622 apply (rule_tac x="{}" in exI, simp)
  5623 apply clarsimp
  5624 apply (subgoal_tac "\<exists>x. x \<notin> A")
  5625 apply (erule exE)
  5626 apply (rule_tac x="insert x A" in exI, simp)
  5627 apply (subgoal_tac "A \<noteq> UNIV", auto)
  5628 done
  5629 
  5630 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
  5631   assumes "subspace s" shows "closed s"
  5632 proof-
  5633   have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
  5634   def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
  5635   let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
  5636   have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
  5637       inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
  5638     apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
  5639     using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
  5640   then guess f apply-by(erule exE conjE)+ note f = this
  5641   interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
  5642   have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
  5643     by(erule_tac x=0 in ballE) auto
  5644   moreover have "closed ?t" using closed_substandard .
  5645   moreover have "subspace ?t" using subspace_substandard .
  5646   ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
  5647     unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
  5648 qed
  5649 
  5650 lemma complete_subspace:
  5651   fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
  5652   using complete_eq_closed closed_subspace
  5653   by auto
  5654 
  5655 lemma dim_closure:
  5656   fixes s :: "('a::euclidean_space) set"
  5657   shows "dim(closure s) = dim s" (is "?dc = ?d")
  5658 proof-
  5659   have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
  5660     using closed_subspace[OF subspace_span, of s]
  5661     using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
  5662   thus ?thesis using dim_subset[OF closure_subset, of s] by auto
  5663 qed
  5664 
  5665 
  5666 subsection {* Affine transformations of intervals *}
  5667 
  5668 lemma real_affinity_le:
  5669  "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
  5670   by (simp add: field_simps inverse_eq_divide)
  5671 
  5672 lemma real_le_affinity:
  5673  "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
  5674   by (simp add: field_simps inverse_eq_divide)
  5675 
  5676 lemma real_affinity_lt:
  5677  "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
  5678   by (simp add: field_simps inverse_eq_divide)
  5679 
  5680 lemma real_lt_affinity:
  5681  "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
  5682   by (simp add: field_simps inverse_eq_divide)
  5683 
  5684 lemma real_affinity_eq:
  5685  "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
  5686   by (simp add: field_simps inverse_eq_divide)
  5687 
  5688 lemma real_eq_affinity:
  5689  "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c  \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
  5690   by (simp add: field_simps inverse_eq_divide)
  5691 
  5692 lemma image_affinity_interval: fixes m::real
  5693   fixes a b c :: "'a::ordered_euclidean_space"
  5694   shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
  5695             (if {a .. b} = {} then {}
  5696             else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
  5697             else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
  5698 proof(cases "m=0")  
  5699   { fix x assume "x \<le> c" "c \<le> x"
  5700     hence "x=c" unfolding eucl_le[where 'a='a] apply-
  5701       apply(subst euclidean_eq) by (auto intro: order_antisym) }
  5702   moreover case True
  5703   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])
  5704   ultimately show ?thesis by auto
  5705 next
  5706   case False
  5707   { fix y assume "a \<le> y" "y \<le> b" "m > 0"
  5708     hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
  5709       unfolding eucl_le[where 'a='a] by auto
  5710   } moreover
  5711   { fix y assume "a \<le> y" "y \<le> b" "m < 0"
  5712     hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
  5713       unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg)
  5714   } moreover
  5715   { fix y assume "m > 0"  "m *\<^sub>R a + c \<le> y"  "y \<le> m *\<^sub>R b + c"
  5716     hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
  5717       unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
  5718       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
  5719       by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff)
  5720   } moreover
  5721   { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
  5722     hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
  5723       unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
  5724       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
  5725       by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff)
  5726   }