src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author huffman Wed Aug 31 13:28:29 2011 -0700 (2011-08-31) changeset 44632 076a45f65e12 parent 44628 bd17b7543af1 child 44647 e4de7750cdeb permissions -rw-r--r--
simplify/generalize 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

    11 begin

    12

    13 subsection {* General notion of a topology as a value *}

    14

    15 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))"

    16 typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"

    17   morphisms "openin" "topology"

    18   unfolding istopology_def by blast

    19

    20 lemma istopology_open_in[intro]: "istopology(openin U)"

    21   using openin[of U] by blast

    22

    23 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"

    24   using topology_inverse[unfolded mem_Collect_eq] .

    25

    26 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"

    27   using topology_inverse[of U] istopology_open_in[of "topology U"] by auto

    28

    29 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"

    30 proof-

    31   {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}

    32   moreover

    33   {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"

    34     hence "openin T1 = openin T2" by (simp add: fun_eq_iff)

    35     hence "topology (openin T1) = topology (openin T2)" by simp

    36     hence "T1 = T2" unfolding openin_inverse .}

    37   ultimately show ?thesis by blast

    38 qed

    39

    40 text{* Infer the "universe" from union of all sets in the topology. *}

    41

    42 definition "topspace T =  \<Union>{S. openin T S}"

    43

    44 subsubsection {* Main properties of open sets *}

    45

    46 lemma openin_clauses:

    47   fixes U :: "'a topology"

    48   shows "openin U {}"

    49   "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"

    50   "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"

    51   using openin[of U] unfolding istopology_def mem_Collect_eq

    52   by fast+

    53

    54 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"

    55   unfolding topspace_def by blast

    56 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)

    57

    58 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"

    59   using openin_clauses by simp

    60

    61 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"

    62   using openin_clauses by simp

    63

    64 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"

    65   using openin_Union[of "{S,T}" U] by auto

    66

    67 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)

    68

    69 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")

    70 proof

    71   assume ?lhs then show ?rhs by auto

    72 next

    73   assume H: ?rhs

    74   let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"

    75   have "openin U ?t" by (simp add: openin_Union)

    76   also have "?t = S" using H by auto

    77   finally show "openin U S" .

    78 qed

    79

    80 subsubsection {* Closed sets *}

    81

    82 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"

    83

    84 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)

    85 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)

    86 lemma closedin_topspace[intro,simp]:

    87   "closedin U (topspace U)" by (simp add: closedin_def)

    88 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"

    89   by (auto simp add: Diff_Un closedin_def)

    90

    91 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto

    92 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"

    93   shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto

    94

    95 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"

    96   using closedin_Inter[of "{S,T}" U] by auto

    97

    98 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast

    99 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"

   100   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   101   apply (metis openin_subset subset_eq)

   102   done

   103

   104 lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"

   105   by (simp add: openin_closedin_eq)

   106

   107 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"

   108 proof-

   109   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT

   110     by (auto simp add: topspace_def openin_subset)

   111   then show ?thesis using oS cT by (auto simp add: closedin_def)

   112 qed

   113

   114 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"

   115 proof-

   116   have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT

   117     by (auto simp add: topspace_def )

   118   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)

   119 qed

   120

   121 subsubsection {* Subspace topology *}

   122

   123 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

   124

   125 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

   126   (is "istopology ?L")

   127 proof-

   128   have "?L {}" by blast

   129   {fix A B assume A: "?L A" and B: "?L B"

   130     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

   131     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+

   132     then have "?L (A \<inter> B)" by blast}

   133   moreover

   134   {fix K assume K: "K \<subseteq> Collect ?L"

   135     have th0: "Collect ?L = (\<lambda>S. S \<inter> V)  Collect (openin U)"

   136       apply (rule set_eqI)

   137       apply (simp add: Ball_def image_iff)

   138       by metis

   139     from K[unfolded th0 subset_image_iff]

   140     obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V)  Sk" by blast

   141     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto

   142     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)

   143     ultimately have "?L (\<Union>K)" by blast}

   144   ultimately show ?thesis

   145     unfolding subset_eq mem_Collect_eq istopology_def by blast

   146 qed

   147

   148 lemma openin_subtopology:

   149   "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"

   150   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   151   by auto

   152

   153 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"

   154   by (auto simp add: topspace_def openin_subtopology)

   155

   156 lemma closedin_subtopology:

   157   "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"

   158   unfolding closedin_def topspace_subtopology

   159   apply (simp add: openin_subtopology)

   160   apply (rule iffI)

   161   apply clarify

   162   apply (rule_tac x="topspace U - T" in exI)

   163   by auto

   164

   165 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"

   166   unfolding openin_subtopology

   167   apply (rule iffI, clarify)

   168   apply (frule openin_subset[of U])  apply blast

   169   apply (rule exI[where x="topspace U"])

   170   by auto

   171

   172 lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"

   173   shows "subtopology U V = U"

   174 proof-

   175   {fix S

   176     {fix T assume T: "openin U T" "S = T \<inter> V"

   177       from T openin_subset[OF T(1)] UV have eq: "S = T" by blast

   178       have "openin U S" unfolding eq using T by blast}

   179     moreover

   180     {assume S: "openin U S"

   181       hence "\<exists>T. openin U T \<and> S = T \<inter> V"

   182         using openin_subset[OF S] UV by auto}

   183     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}

   184   then show ?thesis unfolding topology_eq openin_subtopology by blast

   185 qed

   186

   187 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"

   188   by (simp add: subtopology_superset)

   189

   190 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"

   191   by (simp add: subtopology_superset)

   192

   193 subsubsection {* The standard Euclidean topology *}

   194

   195 definition

   196   euclidean :: "'a::topological_space topology" where

   197   "euclidean = topology open"

   198

   199 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"

   200   unfolding euclidean_def

   201   apply (rule cong[where x=S and y=S])

   202   apply (rule topology_inverse[symmetric])

   203   apply (auto simp add: istopology_def)

   204   done

   205

   206 lemma topspace_euclidean: "topspace euclidean = UNIV"

   207   apply (simp add: topspace_def)

   208   apply (rule set_eqI)

   209   by (auto simp add: open_openin[symmetric])

   210

   211 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"

   212   by (simp add: topspace_euclidean topspace_subtopology)

   213

   214 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"

   215   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)

   216

   217 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"

   218   by (simp add: open_openin openin_subopen[symmetric])

   219

   220 text {* Basic "localization" results are handy for connectedness. *}

   221

   222 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"

   223   by (auto simp add: openin_subtopology open_openin[symmetric])

   224

   225 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"

   226   by (auto simp add: openin_open)

   227

   228 lemma open_openin_trans[trans]:

   229  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"

   230   by (metis Int_absorb1  openin_open_Int)

   231

   232 lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"

   233   by (auto simp add: openin_open)

   234

   235 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"

   236   by (simp add: closedin_subtopology closed_closedin Int_ac)

   237

   238 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"

   239   by (metis closedin_closed)

   240

   241 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"

   242   apply (subgoal_tac "S \<inter> T = T" )

   243   apply auto

   244   apply (frule closedin_closed_Int[of T S])

   245   by simp

   246

   247 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"

   248   by (auto simp add: closedin_closed)

   249

   250 lemma openin_euclidean_subtopology_iff:

   251   fixes S U :: "'a::metric_space set"

   252   shows "openin (subtopology euclidean U) S

   253   \<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")

   254 proof

   255   assume ?lhs thus ?rhs unfolding openin_open open_dist by blast

   256 next

   257   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}"

   258   have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"

   259     unfolding T_def

   260     apply clarsimp

   261     apply (rule_tac x="d - dist x a" in exI)

   262     apply (clarsimp simp add: less_diff_eq)

   263     apply (erule rev_bexI)

   264     apply (rule_tac x=d in exI, clarify)

   265     apply (erule le_less_trans [OF dist_triangle])

   266     done

   267   assume ?rhs hence 2: "S = U \<inter> T"

   268     unfolding T_def

   269     apply auto

   270     apply (drule (1) bspec, erule rev_bexI)

   271     apply auto

   272     done

   273   from 1 2 show ?lhs

   274     unfolding openin_open open_dist by fast

   275 qed

   276

   277 text {* These "transitivity" results are handy too *}

   278

   279 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T

   280   \<Longrightarrow> openin (subtopology euclidean U) S"

   281   unfolding open_openin openin_open by blast

   282

   283 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"

   284   by (auto simp add: openin_open intro: openin_trans)

   285

   286 lemma closedin_trans[trans]:

   287  "closedin (subtopology euclidean T) S \<Longrightarrow>

   288            closedin (subtopology euclidean U) T

   289            ==> closedin (subtopology euclidean U) S"

   290   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)

   291

   292 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"

   293   by (auto simp add: closedin_closed intro: closedin_trans)

   294

   295

   296 subsection {* Open and closed balls *}

   297

   298 definition

   299   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

   300   "ball x e = {y. dist x y < e}"

   301

   302 definition

   303   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

   304   "cball x e = {y. dist x y \<le> e}"

   305

   306 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)

   307 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)

   308

   309 lemma mem_ball_0 [simp]:

   310   fixes x :: "'a::real_normed_vector"

   311   shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"

   312   by (simp add: dist_norm)

   313

   314 lemma mem_cball_0 [simp]:

   315   fixes x :: "'a::real_normed_vector"

   316   shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"

   317   by (simp add: dist_norm)

   318

   319 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp

   320 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)

   321 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)

   322 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)

   323 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"

   324   by (simp add: set_eq_iff) arith

   325

   326 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"

   327   by (simp add: set_eq_iff)

   328

   329 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"

   330   "(a::real) - b < 0 \<longleftrightarrow> a < b"

   331   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+

   332 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"

   333   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+

   334

   335 lemma open_ball[intro, simp]: "open (ball x e)"

   336   unfolding open_dist ball_def mem_Collect_eq Ball_def

   337   unfolding dist_commute

   338   apply clarify

   339   apply (rule_tac x="e - dist xa x" in exI)

   340   using dist_triangle_alt[where z=x]

   341   apply (clarsimp simp add: diff_less_iff)

   342   apply atomize

   343   apply (erule_tac x="y" in allE)

   344   apply (erule_tac x="xa" in allE)

   345   by arith

   346

   347 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)

   348 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"

   349   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   350

   351 lemma openE[elim?]:

   352   assumes "open S" "x\<in>S"

   353   obtains e where "e>0" "ball x e \<subseteq> S"

   354   using assms unfolding open_contains_ball by auto

   355

   356 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

   357   by (metis open_contains_ball subset_eq centre_in_ball)

   358

   359 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"

   360   unfolding mem_ball set_eq_iff

   361   apply (simp add: not_less)

   362   by (metis zero_le_dist order_trans dist_self)

   363

   364 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp

   365

   366

   367 subsection{* Connectedness *}

   368

   369 definition "connected S \<longleftrightarrow>

   370   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})

   371   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"

   372

   373 lemma connected_local:

   374  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.

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

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

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

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

   379                  ~(e1 = {}) \<and>

   380                  ~(e2 = {}))"

   381 unfolding connected_def openin_open by (safe, blast+)

   382

   383 lemma exists_diff:

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

   385   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")

   386 proof-

   387   {assume "?lhs" hence ?rhs by blast }

   388   moreover

   389   {fix S assume H: "P S"

   390     have "S = - (- S)" by auto

   391     with H have "P (- (- S))" by metis }

   392   ultimately show ?thesis by metis

   393 qed

   394

   395 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

   398 proof-

   399   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> {})"

   400     unfolding connected_def openin_open closedin_closed

   401     apply (subst exists_diff) by blast

   402   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> {})"

   403     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis

   404

   405   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'))"

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

   407     unfolding connected_def openin_open closedin_closed by auto

   408   {fix e2

   409     {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)"

   410         by auto}

   411     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}

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

   413   then show ?thesis unfolding th0 th1 by simp

   414 qed

   415

   416 lemma connected_empty[simp, intro]: "connected {}"

   417   by (simp add: connected_def)

   418

   419

   420 subsection{* Limit points *}

   421

   422 definition (in topological_space)

   423   islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where

   424   "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"

   425

   426 lemma islimptI:

   427   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

   428   shows "x islimpt S"

   429   using assms unfolding islimpt_def by auto

   430

   431 lemma islimptE:

   432   assumes "x islimpt S" and "x \<in> T" and "open T"

   433   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"

   434   using assms unfolding islimpt_def by auto

   435

   436 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"

   437   unfolding islimpt_def eventually_at_topological by auto

   438

   439 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"

   440   unfolding islimpt_def by fast

   441

   442 lemma islimpt_approachable:

   443   fixes x :: "'a::metric_space"

   444   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

   445   unfolding islimpt_iff_eventually eventually_at by fast

   446

   447 lemma islimpt_approachable_le:

   448   fixes x :: "'a::metric_space"

   449   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"

   450   unfolding islimpt_approachable

   451   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

   452     THEN arg_cong [where f=Not]]

   453   by (simp add: Bex_def conj_commute conj_left_commute)

   454

   455 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

   456   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

   457

   458 text {* A perfect space has no isolated points. *}

   459

   460 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

   461   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

   462

   463 lemma perfect_choose_dist:

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

   465   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

   466 using islimpt_UNIV [of x]

   467 by (simp add: islimpt_approachable)

   468

   469 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

   470   unfolding closed_def

   471   apply (subst open_subopen)

   472   apply (simp add: islimpt_def subset_eq)

   473   by (metis ComplE ComplI)

   474

   475 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

   476   unfolding islimpt_def by auto

   477

   478 lemma finite_set_avoid:

   479   fixes a :: "'a::metric_space"

   480   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"

   481 proof(induct rule: finite_induct[OF fS])

   482   case 1 thus ?case by (auto intro: zero_less_one)

   483 next

   484   case (2 x F)

   485   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast

   486   {assume "x = a" hence ?case using d by auto  }

   487   moreover

   488   {assume xa: "x\<noteq>a"

   489     let ?d = "min d (dist a x)"

   490     have dp: "?d > 0" using xa d(1) using dist_nz by auto

   491     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto

   492     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }

   493   ultimately show ?case by blast

   494 qed

   495

   496 lemma islimpt_finite:

   497   fixes S :: "'a::metric_space set"

   498   assumes fS: "finite S" shows "\<not> a islimpt S"

   499   unfolding islimpt_approachable

   500   using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)

   501

   502 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

   503   apply (rule iffI)

   504   defer

   505   apply (metis Un_upper1 Un_upper2 islimpt_subset)

   506   unfolding islimpt_def

   507   apply (rule ccontr, clarsimp, rename_tac A B)

   508   apply (drule_tac x="A \<inter> B" in spec)

   509   apply (auto simp add: open_Int)

   510   done

   511

   512 lemma discrete_imp_closed:

   513   fixes S :: "'a::metric_space set"

   514   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

   515   shows "closed S"

   516 proof-

   517   {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

   518     from e have e2: "e/2 > 0" by arith

   519     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast

   520     let ?m = "min (e/2) (dist x y) "

   521     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])

   522     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast

   523     have th: "dist z y < e" using z y

   524       by (intro dist_triangle_lt [where z=x], simp)

   525     from d[rule_format, OF y(1) z(1) th] y z

   526     have False by (auto simp add: dist_commute)}

   527   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])

   528 qed

   529

   530

   531 subsection {* Interior of a Set *}

   532

   533 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

   534

   535 lemma interiorI [intro?]:

   536   assumes "open T" and "x \<in> T" and "T \<subseteq> S"

   537   shows "x \<in> interior S"

   538   using assms unfolding interior_def by fast

   539

   540 lemma interiorE [elim?]:

   541   assumes "x \<in> interior S"

   542   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

   543   using assms unfolding interior_def by fast

   544

   545 lemma open_interior [simp, intro]: "open (interior S)"

   546   by (simp add: interior_def open_Union)

   547

   548 lemma interior_subset: "interior S \<subseteq> S"

   549   by (auto simp add: interior_def)

   550

   551 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

   552   by (auto simp add: interior_def)

   553

   554 lemma interior_open: "open S \<Longrightarrow> interior S = S"

   555   by (intro equalityI interior_subset interior_maximal subset_refl)

   556

   557 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

   558   by (metis open_interior interior_open)

   559

   560 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

   561   by (metis interior_maximal interior_subset subset_trans)

   562

   563 lemma interior_empty [simp]: "interior {} = {}"

   564   using open_empty by (rule interior_open)

   565

   566 lemma interior_UNIV [simp]: "interior UNIV = UNIV"

   567   using open_UNIV by (rule interior_open)

   568

   569 lemma interior_interior [simp]: "interior (interior S) = interior S"

   570   using open_interior by (rule interior_open)

   571

   572 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

   573   by (auto simp add: interior_def)

   574

   575 lemma interior_unique:

   576   assumes "T \<subseteq> S" and "open T"

   577   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"

   578   shows "interior S = T"

   579   by (intro equalityI assms interior_subset open_interior interior_maximal)

   580

   581 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"

   582   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

   583     Int_lower2 interior_maximal interior_subset open_Int open_interior)

   584

   585 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

   586   using open_contains_ball_eq [where S="interior S"]

   587   by (simp add: open_subset_interior)

   588

   589 lemma interior_limit_point [intro]:

   590   fixes x :: "'a::perfect_space"

   591   assumes x: "x \<in> interior S" shows "x islimpt S"

   592   using x islimpt_UNIV [of x]

   593   unfolding interior_def islimpt_def

   594   apply (clarsimp, rename_tac T T')

   595   apply (drule_tac x="T \<inter> T'" in spec)

   596   apply (auto simp add: open_Int)

   597   done

   598

   599 lemma interior_closed_Un_empty_interior:

   600   assumes cS: "closed S" and iT: "interior T = {}"

   601   shows "interior (S \<union> T) = interior S"

   602 proof

   603   show "interior S \<subseteq> interior (S \<union> T)"

   604     by (rule interior_mono, rule Un_upper1)

   605 next

   606   show "interior (S \<union> T) \<subseteq> interior S"

   607   proof

   608     fix x assume "x \<in> interior (S \<union> T)"

   609     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

   610     show "x \<in> interior S"

   611     proof (rule ccontr)

   612       assume "x \<notin> interior S"

   613       with x \<in> R open R obtain y where "y \<in> R - S"

   614         unfolding interior_def by fast

   615       from open R closed S have "open (R - S)" by (rule open_Diff)

   616       from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast

   617       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}

   618       show "False" unfolding interior_def by fast

   619     qed

   620   qed

   621 qed

   622

   623 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

   624 proof (rule interior_unique)

   625   show "interior A \<times> interior B \<subseteq> A \<times> B"

   626     by (intro Sigma_mono interior_subset)

   627   show "open (interior A \<times> interior B)"

   628     by (intro open_Times open_interior)

   629   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"

   630   proof (safe)

   631     fix x y assume "(x, y) \<in> T"

   632     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

   633       using open T unfolding open_prod_def by fast

   634     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

   635       using T \<subseteq> A \<times> B by auto

   636     thus "x \<in> interior A" and "y \<in> interior B"

   637       by (auto intro: interiorI)

   638   qed

   639 qed

   640

   641

   642 subsection {* Closure of a Set *}

   643

   644 definition "closure S = S \<union> {x | x. x islimpt S}"

   645

   646 lemma interior_closure: "interior S = - (closure (- S))"

   647   unfolding interior_def closure_def islimpt_def by auto

   648

   649 lemma closure_interior: "closure S = - interior (- S)"

   650   unfolding interior_closure by simp

   651

   652 lemma closed_closure[simp, intro]: "closed (closure S)"

   653   unfolding closure_interior by (simp add: closed_Compl)

   654

   655 lemma closure_subset: "S \<subseteq> closure S"

   656   unfolding closure_def by simp

   657

   658 lemma closure_hull: "closure S = closed hull S"

   659   unfolding hull_def closure_interior interior_def by auto

   660

   661 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

   662   unfolding closure_hull using closed_Inter by (rule hull_eq)

   663

   664 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

   665   unfolding closure_eq .

   666

   667 lemma closure_closure [simp]: "closure (closure S) = closure S"

   668   unfolding closure_hull by (rule hull_hull)

   669

   670 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

   671   unfolding closure_hull by (rule hull_mono)

   672

   673 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

   674   unfolding closure_hull by (rule hull_minimal)

   675

   676 lemma closure_unique:

   677   assumes "S \<subseteq> T" and "closed T"

   678   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

   679   shows "closure S = T"

   680   using assms unfolding closure_hull by (rule hull_unique)

   681

   682 lemma closure_empty [simp]: "closure {} = {}"

   683   using closed_empty by (rule closure_closed)

   684

   685 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

   686   using closed_UNIV by (rule closure_closed)

   687

   688 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

   689   unfolding closure_interior by simp

   690

   691 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

   692   using closure_empty closure_subset[of S]

   693   by blast

   694

   695 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

   696   using closure_eq[of S] closure_subset[of S]

   697   by simp

   698

   699 lemma open_inter_closure_eq_empty:

   700   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

   701   using open_subset_interior[of S "- T"]

   702   using interior_subset[of "- T"]

   703   unfolding closure_interior

   704   by auto

   705

   706 lemma open_inter_closure_subset:

   707   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

   708 proof

   709   fix x

   710   assume as: "open S" "x \<in> S \<inter> closure T"

   711   { assume *:"x islimpt T"

   712     have "x islimpt (S \<inter> T)"

   713     proof (rule islimptI)

   714       fix A

   715       assume "x \<in> A" "open A"

   716       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

   717         by (simp_all add: open_Int)

   718       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

   719         by (rule islimptE)

   720       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

   721         by simp_all

   722       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

   723     qed

   724   }

   725   then show "x \<in> closure (S \<inter> T)" using as

   726     unfolding closure_def

   727     by blast

   728 qed

   729

   730 lemma closure_complement: "closure (- S) = - interior S"

   731   unfolding closure_interior by simp

   732

   733 lemma interior_complement: "interior (- S) = - closure S"

   734   unfolding closure_interior by simp

   735

   736 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

   737 proof (rule closure_unique)

   738   show "A \<times> B \<subseteq> closure A \<times> closure B"

   739     by (intro Sigma_mono closure_subset)

   740   show "closed (closure A \<times> closure B)"

   741     by (intro closed_Times closed_closure)

   742   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"

   743     apply (simp add: closed_def open_prod_def, clarify)

   744     apply (rule ccontr)

   745     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

   746     apply (simp add: closure_interior interior_def)

   747     apply (drule_tac x=C in spec)

   748     apply (drule_tac x=D in spec)

   749     apply auto

   750     done

   751 qed

   752

   753

   754 subsection {* Frontier (aka boundary) *}

   755

   756 definition "frontier S = closure S - interior S"

   757

   758 lemma frontier_closed: "closed(frontier S)"

   759   by (simp add: frontier_def closed_Diff)

   760

   761 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

   762   by (auto simp add: frontier_def interior_closure)

   763

   764 lemma frontier_straddle:

   765   fixes a :: "'a::metric_space"

   766   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")

   767 proof

   768   assume "?lhs"

   769   { fix e::real

   770     assume "e > 0"

   771     let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"

   772     { assume "a\<in>S"

   773       have "\<exists>x\<in>S. dist a x < e" using e>0 a\<in>S by(rule_tac x=a in bexI) auto

   774       moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using ?lhs a\<in>S

   775         unfolding frontier_closures closure_def islimpt_def using e>0

   776         by (auto, erule_tac x="ball a e" in allE, auto)

   777       ultimately have ?rhse by auto

   778     }

   779     moreover

   780     { assume "a\<notin>S"

   781       hence ?rhse using ?lhs

   782         unfolding frontier_closures closure_def islimpt_def

   783         using open_ball[of a e] e > 0

   784           by simp (metis centre_in_ball mem_ball open_ball)

   785     }

   786     ultimately have ?rhse by auto

   787   }

   788   thus ?rhs by auto

   789 next

   790   assume ?rhs

   791   moreover

   792   { fix T assume "a\<notin>S" and

   793     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"

   794     from open T a \<in> T have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto

   795     then obtain e where "e>0" "ball a e \<subseteq> T" by auto

   796     then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto

   797     have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"

   798       using dist a y < e ball a e \<subseteq> T unfolding ball_def using y\<in>S a\<notin>S by auto

   799   }

   800   hence "a \<in> closure S" unfolding closure_def islimpt_def using ?rhs by auto

   801   moreover

   802   { fix T assume "a \<in> T"  "open T" "a\<in>S"

   803     then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using ?rhs by auto

   804     obtain x where "x \<notin> S" "dist a x < e" using ?rhs using e>0 by auto

   805     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

   806   }

   807   hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto

   808   ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto

   809 qed

   810

   811 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

   812   by (metis frontier_def closure_closed Diff_subset)

   813

   814 lemma frontier_empty[simp]: "frontier {} = {}"

   815   by (simp add: frontier_def)

   816

   817 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

   818 proof-

   819   { assume "frontier S \<subseteq> S"

   820     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto

   821     hence "closed S" using closure_subset_eq by auto

   822   }

   823   thus ?thesis using frontier_subset_closed[of S] ..

   824 qed

   825

   826 lemma frontier_complement: "frontier(- S) = frontier S"

   827   by (auto simp add: frontier_def closure_complement interior_complement)

   828

   829 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

   830   using frontier_complement frontier_subset_eq[of "- S"]

   831   unfolding open_closed by auto

   832

   833

   834 subsection {* Filters and the eventually true'' quantifier *}

   835

   836 definition

   837   at_infinity :: "'a::real_normed_vector filter" where

   838   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"

   839

   840 definition

   841   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

   842     (infixr "indirection" 70) where

   843   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

   844

   845 text{* Prove That They are all filters. *}

   846

   847 lemma eventually_at_infinity:

   848   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"

   849 unfolding at_infinity_def

   850 proof (rule eventually_Abs_filter, rule is_filter.intro)

   851   fix P Q :: "'a \<Rightarrow> bool"

   852   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"

   853   then obtain r s where

   854     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto

   855   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp

   856   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..

   857 qed auto

   858

   859 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

   860

   861 lemma trivial_limit_within:

   862   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

   863 proof

   864   assume "trivial_limit (at a within S)"

   865   thus "\<not> a islimpt S"

   866     unfolding trivial_limit_def

   867     unfolding eventually_within eventually_at_topological

   868     unfolding islimpt_def

   869     apply (clarsimp simp add: set_eq_iff)

   870     apply (rename_tac T, rule_tac x=T in exI)

   871     apply (clarsimp, drule_tac x=y in bspec, simp_all)

   872     done

   873 next

   874   assume "\<not> a islimpt S"

   875   thus "trivial_limit (at a within S)"

   876     unfolding trivial_limit_def

   877     unfolding eventually_within eventually_at_topological

   878     unfolding islimpt_def

   879     apply clarsimp

   880     apply (rule_tac x=T in exI)

   881     apply auto

   882     done

   883 qed

   884

   885 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

   886   using trivial_limit_within [of a UNIV]

   887   by (simp add: within_UNIV)

   888

   889 lemma trivial_limit_at:

   890   fixes a :: "'a::perfect_space"

   891   shows "\<not> trivial_limit (at a)"

   892   by (rule at_neq_bot)

   893

   894 lemma trivial_limit_at_infinity:

   895   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

   896   unfolding trivial_limit_def eventually_at_infinity

   897   apply clarsimp

   898   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

   899    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

   900   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

   901   apply (drule_tac x=UNIV in spec, simp)

   902   done

   903

   904 text {* Some property holds "sufficiently close" to the limit point. *}

   905

   906 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)

   907   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

   908 unfolding eventually_at dist_nz by auto

   909

   910 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>

   911         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

   912 unfolding eventually_within eventually_at dist_nz by auto

   913

   914 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>

   915         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")

   916 unfolding eventually_within

   917 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl)

   918

   919 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"

   920   unfolding trivial_limit_def

   921   by (auto elim: eventually_rev_mp)

   922

   923 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

   924   unfolding trivial_limit_def by (auto elim: eventually_rev_mp)

   925

   926 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

   927   by (simp add: filter_eq_iff)

   928

   929 text{* Combining theorems for "eventually" *}

   930

   931 lemma eventually_rev_mono:

   932   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

   933 using eventually_mono [of P Q] by fast

   934

   935 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"

   936   by (simp add: eventually_False)

   937

   938

   939 subsection {* Limits *}

   940

   941 text{* Notation Lim to avoid collition with lim defined in analysis *}

   942

   943 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"

   944   where "Lim A f = (THE l. (f ---> l) A)"

   945

   946 lemma Lim:

   947  "(f ---> l) net \<longleftrightarrow>

   948         trivial_limit net \<or>

   949         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

   950   unfolding tendsto_iff trivial_limit_eq by auto

   951

   952 text{* Show that they yield usual definitions in the various cases. *}

   953

   954 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

   955            (\<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)"

   956   by (auto simp add: tendsto_iff eventually_within_le)

   957

   958 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

   959         (\<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)"

   960   by (auto simp add: tendsto_iff eventually_within)

   961

   962 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

   963         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

   964   by (auto simp add: tendsto_iff eventually_at)

   965

   966 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"

   967   unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)

   968

   969 lemma Lim_at_infinity:

   970   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"

   971   by (auto simp add: tendsto_iff eventually_at_infinity)

   972

   973 lemma Lim_sequentially:

   974  "(S ---> l) sequentially \<longleftrightarrow>

   975           (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"

   976   by (rule LIMSEQ_def) (* FIXME: redundant *)

   977

   978 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

   979   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

   980

   981 text{* The expected monotonicity property. *}

   982

   983 lemma Lim_within_empty: "(f ---> l) (net within {})"

   984   unfolding tendsto_def Limits.eventually_within by simp

   985

   986 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"

   987   unfolding tendsto_def Limits.eventually_within

   988   by (auto elim!: eventually_elim1)

   989

   990 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"

   991   shows "(f ---> l) (net within (S \<union> T))"

   992   using assms unfolding tendsto_def Limits.eventually_within

   993   apply clarify

   994   apply (drule spec, drule (1) mp, drule (1) mp)

   995   apply (drule spec, drule (1) mp, drule (1) mp)

   996   apply (auto elim: eventually_elim2)

   997   done

   998

   999 lemma Lim_Un_univ:

  1000  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV

  1001         ==> (f ---> l) net"

  1002   by (metis Lim_Un within_UNIV)

  1003

  1004 text{* Interrelations between restricted and unrestricted limits. *}

  1005

  1006 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"

  1007   (* FIXME: rename *)

  1008   unfolding tendsto_def Limits.eventually_within

  1009   apply (clarify, drule spec, drule (1) mp, drule (1) mp)

  1010   by (auto elim!: eventually_elim1)

  1011

  1012 lemma eventually_within_interior:

  1013   assumes "x \<in> interior S"

  1014   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")

  1015 proof-

  1016   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1017   { assume "?lhs"

  1018     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1019       unfolding Limits.eventually_within Limits.eventually_at_topological

  1020       by auto

  1021     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"

  1022       by auto

  1023     then have "?rhs"

  1024       unfolding Limits.eventually_at_topological by auto

  1025   } moreover

  1026   { assume "?rhs" hence "?lhs"

  1027       unfolding Limits.eventually_within

  1028       by (auto elim: eventually_elim1)

  1029   } ultimately

  1030   show "?thesis" ..

  1031 qed

  1032

  1033 lemma at_within_interior:

  1034   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1035   by (simp add: filter_eq_iff eventually_within_interior)

  1036

  1037 lemma at_within_open:

  1038   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"

  1039   by (simp only: at_within_interior interior_open)

  1040

  1041 lemma Lim_within_open:

  1042   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"

  1043   assumes"a \<in> S" "open S"

  1044   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"

  1045   using assms by (simp only: at_within_open)

  1046

  1047 lemma Lim_within_LIMSEQ:

  1048   fixes a :: "'a::metric_space"

  1049   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1050   shows "(X ---> L) (at a within T)"

  1051   using assms unfolding tendsto_def [where l=L]

  1052   by (simp add: sequentially_imp_eventually_within)

  1053

  1054 lemma Lim_right_bound:

  1055   fixes f :: "real \<Rightarrow> real"

  1056   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"

  1057   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1058   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1059 proof cases

  1060   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)

  1061 next

  1062   assume [simp]: "{x<..} \<inter> I \<noteq> {}"

  1063   show ?thesis

  1064   proof (rule Lim_within_LIMSEQ, safe)

  1065     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"

  1066

  1067     show "(\<lambda>n. f (S n)) ----> Inf (f  ({x<..} \<inter> I))"

  1068     proof (rule LIMSEQ_I, rule ccontr)

  1069       fix r :: real assume "0 < r"

  1070       with Inf_close[of "f  ({x<..} \<inter> I)" r]

  1071       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f  ({x <..} \<inter> I)) + r" by auto

  1072       from x < y have "0 < y - x" by auto

  1073       from S(2)[THEN LIMSEQ_D, OF this]

  1074       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto

  1075

  1076       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f  ({x<..} \<inter> I))) < r)"

  1077       moreover have "\<And>n. Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1078         using S bnd by (intro Inf_lower[where z=K]) auto

  1079       ultimately obtain n where n: "N \<le> n" "r + Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1080         by (auto simp: not_less field_simps)

  1081       with N[OF n(1)] mono[OF _ y \<in> I, of "S n"] S(1)[THEN spec, of n] y

  1082       show False by auto

  1083     qed

  1084   qed

  1085 qed

  1086

  1087 text{* Another limit point characterization. *}

  1088

  1089 lemma islimpt_sequential:

  1090   fixes x :: "'a::metric_space"

  1091   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"

  1092     (is "?lhs = ?rhs")

  1093 proof

  1094   assume ?lhs

  1095   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"

  1096     unfolding islimpt_approachable

  1097     using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto

  1098   let ?I = "\<lambda>n. inverse (real (Suc n))"

  1099   have "\<forall>n. f (?I n) \<in> S - {x}" using f by simp

  1100   moreover have "(\<lambda>n. f (?I n)) ----> x"

  1101   proof (rule metric_tendsto_imp_tendsto)

  1102     show "?I ----> 0"

  1103       by (rule LIMSEQ_inverse_real_of_nat)

  1104     show "eventually (\<lambda>n. dist (f (?I n)) x \<le> dist (?I n) 0) sequentially"

  1105       by (simp add: norm_conv_dist [symmetric] less_imp_le f)

  1106   qed

  1107   ultimately show ?rhs by fast

  1108 next

  1109   assume ?rhs

  1110   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

  1111   { fix e::real assume "e>0"

  1112     then obtain N where "dist (f N) x < e" using f(2) by auto

  1113     moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto

  1114     ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto

  1115   }

  1116   thus ?lhs unfolding islimpt_approachable by auto

  1117 qed

  1118

  1119 lemma Lim_inv: (* TODO: delete *)

  1120   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"

  1121   assumes "(f ---> l) A" and "l \<noteq> 0"

  1122   shows "((inverse o f) ---> inverse l) A"

  1123   unfolding o_def using assms by (rule tendsto_inverse)

  1124

  1125 lemma Lim_null:

  1126   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1127   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1128   by (simp add: Lim dist_norm)

  1129

  1130 lemma Lim_null_comparison:

  1131   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1132   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1133   shows "(f ---> 0) net"

  1134 proof (rule metric_tendsto_imp_tendsto)

  1135   show "(g ---> 0) net" by fact

  1136   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1137     using assms(1) by (rule eventually_elim1, simp add: dist_norm)

  1138 qed

  1139

  1140 lemma Lim_transform_bound:

  1141   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1142   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1143   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"

  1144   shows "(f ---> 0) net"

  1145   using assms(1) tendsto_norm_zero [OF assms(2)]

  1146   by (rule Lim_null_comparison)

  1147

  1148 text{* Deducing things about the limit from the elements. *}

  1149

  1150 lemma Lim_in_closed_set:

  1151   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"

  1152   shows "l \<in> S"

  1153 proof (rule ccontr)

  1154   assume "l \<notin> S"

  1155   with closed S have "open (- S)" "l \<in> - S"

  1156     by (simp_all add: open_Compl)

  1157   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1158     by (rule topological_tendstoD)

  1159   with assms(2) have "eventually (\<lambda>x. False) net"

  1160     by (rule eventually_elim2) simp

  1161   with assms(3) show "False"

  1162     by (simp add: eventually_False)

  1163 qed

  1164

  1165 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1166

  1167 lemma Lim_dist_ubound:

  1168   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"

  1169   shows "dist a l <= e"

  1170 proof-

  1171   have "dist a l \<in> {..e}"

  1172   proof (rule Lim_in_closed_set)

  1173     show "closed {..e}" by simp

  1174     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)

  1175     show "\<not> trivial_limit net" by fact

  1176     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)

  1177   qed

  1178   thus ?thesis by simp

  1179 qed

  1180

  1181 lemma Lim_norm_ubound:

  1182   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1183   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"

  1184   shows "norm(l) <= e"

  1185 proof-

  1186   have "norm l \<in> {..e}"

  1187   proof (rule Lim_in_closed_set)

  1188     show "closed {..e}" by simp

  1189     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)

  1190     show "\<not> trivial_limit net" by fact

  1191     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1192   qed

  1193   thus ?thesis by simp

  1194 qed

  1195

  1196 lemma Lim_norm_lbound:

  1197   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1198   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"

  1199   shows "e \<le> norm l"

  1200 proof-

  1201   have "norm l \<in> {e..}"

  1202   proof (rule Lim_in_closed_set)

  1203     show "closed {e..}" by simp

  1204     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)

  1205     show "\<not> trivial_limit net" by fact

  1206     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1207   qed

  1208   thus ?thesis by simp

  1209 qed

  1210

  1211 text{* Uniqueness of the limit, when nontrivial. *}

  1212

  1213 lemma tendsto_Lim:

  1214   fixes f :: "'a \<Rightarrow> 'b::t2_space"

  1215   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"

  1216   unfolding Lim_def using tendsto_unique[of net f] by auto

  1217

  1218 text{* Limit under bilinear function *}

  1219

  1220 lemma Lim_bilinear:

  1221   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"

  1222   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1223 using bounded_bilinear h (f ---> l) net (g ---> m) net

  1224 by (rule bounded_bilinear.tendsto)

  1225

  1226 text{* These are special for limits out of the same vector space. *}

  1227

  1228 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1229   unfolding tendsto_def Limits.eventually_within eventually_at_topological

  1230   by auto

  1231

  1232 lemma Lim_at_id: "(id ---> a) (at a)"

  1233 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)

  1234

  1235 lemma Lim_at_zero:

  1236   fixes a :: "'a::real_normed_vector"

  1237   fixes l :: "'b::topological_space"

  1238   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")

  1239   using LIM_offset_zero LIM_offset_zero_cancel ..

  1240

  1241 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1242

  1243 definition

  1244   netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where

  1245   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"

  1246

  1247 lemma netlimit_within:

  1248   assumes "\<not> trivial_limit (at a within S)"

  1249   shows "netlimit (at a within S) = a"

  1250 unfolding netlimit_def

  1251 apply (rule some_equality)

  1252 apply (rule Lim_at_within)

  1253 apply (rule tendsto_ident_at)

  1254 apply (erule tendsto_unique [OF assms])

  1255 apply (rule Lim_at_within)

  1256 apply (rule tendsto_ident_at)

  1257 done

  1258

  1259 lemma netlimit_at:

  1260   fixes a :: "'a::{perfect_space,t2_space}"

  1261   shows "netlimit (at a) = a"

  1262   apply (subst within_UNIV[symmetric])

  1263   using netlimit_within[of a UNIV]

  1264   by (simp add: within_UNIV)

  1265

  1266 lemma lim_within_interior:

  1267   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1268   by (simp add: at_within_interior)

  1269

  1270 lemma netlimit_within_interior:

  1271   fixes x :: "'a::{t2_space,perfect_space}"

  1272   assumes "x \<in> interior S"

  1273   shows "netlimit (at x within S) = x"

  1274 using assms by (simp add: at_within_interior netlimit_at)

  1275

  1276 text{* Transformation of limit. *}

  1277

  1278 lemma Lim_transform:

  1279   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  1280   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1281   shows "(g ---> l) net"

  1282   using tendsto_diff [OF assms(2) assms(1)] by simp

  1283

  1284 lemma Lim_transform_eventually:

  1285   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1286   apply (rule topological_tendstoI)

  1287   apply (drule (2) topological_tendstoD)

  1288   apply (erule (1) eventually_elim2, simp)

  1289   done

  1290

  1291 lemma Lim_transform_within:

  1292   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1293   and "(f ---> l) (at x within S)"

  1294   shows "(g ---> l) (at x within S)"

  1295 proof (rule Lim_transform_eventually)

  1296   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1297     unfolding eventually_within

  1298     using assms(1,2) by auto

  1299   show "(f ---> l) (at x within S)" by fact

  1300 qed

  1301

  1302 lemma Lim_transform_at:

  1303   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1304   and "(f ---> l) (at x)"

  1305   shows "(g ---> l) (at x)"

  1306 proof (rule Lim_transform_eventually)

  1307   show "eventually (\<lambda>x. f x = g x) (at x)"

  1308     unfolding eventually_at

  1309     using assms(1,2) by auto

  1310   show "(f ---> l) (at x)" by fact

  1311 qed

  1312

  1313 text{* Common case assuming being away from some crucial point like 0. *}

  1314

  1315 lemma Lim_transform_away_within:

  1316   fixes a b :: "'a::t1_space"

  1317   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1318   and "(f ---> l) (at a within S)"

  1319   shows "(g ---> l) (at a within S)"

  1320 proof (rule Lim_transform_eventually)

  1321   show "(f ---> l) (at a within S)" by fact

  1322   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1323     unfolding Limits.eventually_within eventually_at_topological

  1324     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1325 qed

  1326

  1327 lemma Lim_transform_away_at:

  1328   fixes a b :: "'a::t1_space"

  1329   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1330   and fl: "(f ---> l) (at a)"

  1331   shows "(g ---> l) (at a)"

  1332   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl

  1333   by (auto simp add: within_UNIV)

  1334

  1335 text{* Alternatively, within an open set. *}

  1336

  1337 lemma Lim_transform_within_open:

  1338   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1339   and "(f ---> l) (at a)"

  1340   shows "(g ---> l) (at a)"

  1341 proof (rule Lim_transform_eventually)

  1342   show "eventually (\<lambda>x. f x = g x) (at a)"

  1343     unfolding eventually_at_topological

  1344     using assms(1,2,3) by auto

  1345   show "(f ---> l) (at a)" by fact

  1346 qed

  1347

  1348 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1349

  1350 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1351

  1352 lemma Lim_cong_within(*[cong add]*):

  1353   assumes "a = b" "x = y" "S = T"

  1354   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1355   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1356   unfolding tendsto_def Limits.eventually_within eventually_at_topological

  1357   using assms by simp

  1358

  1359 lemma Lim_cong_at(*[cong add]*):

  1360   assumes "a = b" "x = y"

  1361   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1362   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1363   unfolding tendsto_def eventually_at_topological

  1364   using assms by simp

  1365

  1366 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1367

  1368 lemma closure_sequential:

  1369   fixes l :: "'a::metric_space"

  1370   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")

  1371 proof

  1372   assume "?lhs" moreover

  1373   { assume "l \<in> S"

  1374     hence "?rhs" using tendsto_const[of l sequentially] by auto

  1375   } moreover

  1376   { assume "l islimpt S"

  1377     hence "?rhs" unfolding islimpt_sequential by auto

  1378   } ultimately

  1379   show "?rhs" unfolding closure_def by auto

  1380 next

  1381   assume "?rhs"

  1382   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto

  1383 qed

  1384

  1385 lemma closed_sequential_limits:

  1386   fixes S :: "'a::metric_space set"

  1387   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1388   unfolding closed_limpt

  1389   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]

  1390   by metis

  1391

  1392 lemma closure_approachable:

  1393   fixes S :: "'a::metric_space set"

  1394   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1395   apply (auto simp add: closure_def islimpt_approachable)

  1396   by (metis dist_self)

  1397

  1398 lemma closed_approachable:

  1399   fixes S :: "'a::metric_space set"

  1400   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1401   by (metis closure_closed closure_approachable)

  1402

  1403 text{* Some other lemmas about sequences. *}

  1404

  1405 lemma sequentially_offset:

  1406   assumes "eventually (\<lambda>i. P i) sequentially"

  1407   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  1408   using assms unfolding eventually_sequentially by (metis trans_le_add1)

  1409

  1410 lemma seq_offset:

  1411   assumes "(f ---> l) sequentially"

  1412   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"

  1413   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

  1414

  1415 lemma seq_offset_neg:

  1416   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  1417   apply (rule topological_tendstoI)

  1418   apply (drule (2) topological_tendstoD)

  1419   apply (simp only: eventually_sequentially)

  1420   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")

  1421   apply metis

  1422   by arith

  1423

  1424 lemma seq_offset_rev:

  1425   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"

  1426   by (rule LIMSEQ_offset) (* FIXME: redundant *)

  1427

  1428 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  1429   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

  1430

  1431 subsection {* More properties of closed balls *}

  1432

  1433 lemma closed_cball: "closed (cball x e)"

  1434 unfolding cball_def closed_def

  1435 unfolding Collect_neg_eq [symmetric] not_le

  1436 apply (clarsimp simp add: open_dist, rename_tac y)

  1437 apply (rule_tac x="dist x y - e" in exI, clarsimp)

  1438 apply (rename_tac x')

  1439 apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  1440 apply simp

  1441 done

  1442

  1443 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  1444 proof-

  1445   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  1446     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  1447   } moreover

  1448   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  1449     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto

  1450   } ultimately

  1451   show ?thesis unfolding open_contains_ball by auto

  1452 qed

  1453

  1454 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  1455   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  1456

  1457 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  1458   apply (simp add: interior_def, safe)

  1459   apply (force simp add: open_contains_cball)

  1460   apply (rule_tac x="ball x e" in exI)

  1461   apply (simp add: subset_trans [OF ball_subset_cball])

  1462   done

  1463

  1464 lemma islimpt_ball:

  1465   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  1466   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")

  1467 proof

  1468   assume "?lhs"

  1469   { assume "e \<le> 0"

  1470     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto

  1471     have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto

  1472   }

  1473   hence "e > 0" by (metis not_less)

  1474   moreover

  1475   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

  1476   ultimately show "?rhs" by auto

  1477 next

  1478   assume "?rhs" hence "e>0"  by auto

  1479   { fix d::real assume "d>0"

  1480     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1481     proof(cases "d \<le> dist x y")

  1482       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1483       proof(cases "x=y")

  1484         case True hence False using d \<le> dist x y d>0 by auto

  1485         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto

  1486       next

  1487         case False

  1488

  1489         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))

  1490               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1491           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto

  1492         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  1493           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]

  1494           unfolding scaleR_minus_left scaleR_one

  1495           by (auto simp add: norm_minus_commute)

  1496         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  1497           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  1498           unfolding left_distrib using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto

  1499         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)

  1500         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto

  1501

  1502         moreover

  1503

  1504         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  1505           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)

  1506         moreover

  1507         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel

  1508           using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  1509           unfolding dist_norm by auto

  1510         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

  1511       qed

  1512     next

  1513       case False hence "d > dist x y" by auto

  1514       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1515       proof(cases "x=y")

  1516         case True

  1517         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  1518           using perfect_choose_dist[of "min e d" y]

  1519           using d > 0 e>0 by auto

  1520         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1521           unfolding x = y

  1522           using z \<noteq> y **

  1523           by (rule_tac x=z in bexI, auto simp add: dist_commute)

  1524       next

  1525         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1526           using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)

  1527       qed

  1528     qed  }

  1529   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto

  1530 qed

  1531

  1532 lemma closure_ball_lemma:

  1533   fixes x y :: "'a::real_normed_vector"

  1534   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"

  1535 proof (rule islimptI)

  1536   fix T assume "y \<in> T" "open T"

  1537   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  1538     unfolding open_dist by fast

  1539   (* choose point between x and y, within distance r of y. *)

  1540   def k \<equiv> "min 1 (r / (2 * dist x y))"

  1541   def z \<equiv> "y + scaleR k (x - y)"

  1542   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  1543     unfolding z_def by (simp add: algebra_simps)

  1544   have "dist z y < r"

  1545     unfolding z_def k_def using 0 < r

  1546     by (simp add: dist_norm min_def)

  1547   hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  1548   have "dist x z < dist x y"

  1549     unfolding z_def2 dist_norm

  1550     apply (simp add: norm_minus_commute)

  1551     apply (simp only: dist_norm [symmetric])

  1552     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  1553     apply (rule mult_strict_right_mono)

  1554     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  1555     apply (simp add: zero_less_dist_iff x \<noteq> y)

  1556     done

  1557   hence "z \<in> ball x (dist x y)" by simp

  1558   have "z \<noteq> y"

  1559     unfolding z_def k_def using x \<noteq> y 0 < r

  1560     by (simp add: min_def)

  1561   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  1562     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  1563     by fast

  1564 qed

  1565

  1566 lemma closure_ball:

  1567   fixes x :: "'a::real_normed_vector"

  1568   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  1569 apply (rule equalityI)

  1570 apply (rule closure_minimal)

  1571 apply (rule ball_subset_cball)

  1572 apply (rule closed_cball)

  1573 apply (rule subsetI, rename_tac y)

  1574 apply (simp add: le_less [where 'a=real])

  1575 apply (erule disjE)

  1576 apply (rule subsetD [OF closure_subset], simp)

  1577 apply (simp add: closure_def)

  1578 apply clarify

  1579 apply (rule closure_ball_lemma)

  1580 apply (simp add: zero_less_dist_iff)

  1581 done

  1582

  1583 (* In a trivial vector space, this fails for e = 0. *)

  1584 lemma interior_cball:

  1585   fixes x :: "'a::{real_normed_vector, perfect_space}"

  1586   shows "interior (cball x e) = ball x e"

  1587 proof(cases "e\<ge>0")

  1588   case False note cs = this

  1589   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover

  1590   { fix y assume "y \<in> cball x e"

  1591     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }

  1592   hence "cball x e = {}" by auto

  1593   hence "interior (cball x e) = {}" using interior_empty by auto

  1594   ultimately show ?thesis by blast

  1595 next

  1596   case True note cs = this

  1597   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover

  1598   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  1599     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast

  1600

  1601     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  1602       using perfect_choose_dist [of d] by auto

  1603     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)

  1604     hence xa_cball:"xa \<in> cball x e" using as(1) by auto

  1605

  1606     hence "y \<in> ball x e" proof(cases "x = y")

  1607       case True

  1608       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)

  1609       thus "y \<in> ball x e" using x = y  by simp

  1610     next

  1611       case False

  1612       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm

  1613         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  1614       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast

  1615       have "y - x \<noteq> 0" using x \<noteq> y by auto

  1616       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]

  1617         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  1618

  1619       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)"

  1620         by (auto simp add: dist_norm algebra_simps)

  1621       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1622         by (auto simp add: algebra_simps)

  1623       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  1624         using ** by auto

  1625       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)

  1626       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)

  1627       thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto

  1628     qed  }

  1629   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto

  1630   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

  1631 qed

  1632

  1633 lemma frontier_ball:

  1634   fixes a :: "'a::real_normed_vector"

  1635   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

  1636   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  1637   apply (simp add: set_eq_iff)

  1638   by arith

  1639

  1640 lemma frontier_cball:

  1641   fixes a :: "'a::{real_normed_vector, perfect_space}"

  1642   shows "frontier(cball a e) = {x. dist a x = e}"

  1643   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  1644   apply (simp add: set_eq_iff)

  1645   by arith

  1646

  1647 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"

  1648   apply (simp add: set_eq_iff not_le)

  1649   by (metis zero_le_dist dist_self order_less_le_trans)

  1650 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)

  1651

  1652 lemma cball_eq_sing:

  1653   fixes x :: "'a::{metric_space,perfect_space}"

  1654   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"

  1655 proof (rule linorder_cases)

  1656   assume e: "0 < e"

  1657   obtain a where "a \<noteq> x" "dist a x < e"

  1658     using perfect_choose_dist [OF e] by auto

  1659   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)

  1660   with e show ?thesis by (auto simp add: set_eq_iff)

  1661 qed auto

  1662

  1663 lemma cball_sing:

  1664   fixes x :: "'a::metric_space"

  1665   shows "e = 0 ==> cball x e = {x}"

  1666   by (auto simp add: set_eq_iff)

  1667

  1668

  1669 subsection {* Boundedness *}

  1670

  1671   (* FIXME: This has to be unified with BSEQ!! *)

  1672 definition (in metric_space)

  1673   bounded :: "'a set \<Rightarrow> bool" where

  1674   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  1675

  1676 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  1677 unfolding bounded_def

  1678 apply safe

  1679 apply (rule_tac x="dist a x + e" in exI, clarify)

  1680 apply (drule (1) bspec)

  1681 apply (erule order_trans [OF dist_triangle add_left_mono])

  1682 apply auto

  1683 done

  1684

  1685 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  1686 unfolding bounded_any_center [where a=0]

  1687 by (simp add: dist_norm)

  1688

  1689 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)

  1690 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"

  1691   by (metis bounded_def subset_eq)

  1692

  1693 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

  1694   by (metis bounded_subset interior_subset)

  1695

  1696 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"

  1697 proof-

  1698   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto

  1699   { fix y assume "y \<in> closure S"

  1700     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  1701       unfolding closure_sequential by auto

  1702     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  1703     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  1704       by (rule eventually_mono, simp add: f(1))

  1705     have "dist x y \<le> a"

  1706       apply (rule Lim_dist_ubound [of sequentially f])

  1707       apply (rule trivial_limit_sequentially)

  1708       apply (rule f(2))

  1709       apply fact

  1710       done

  1711   }

  1712   thus ?thesis unfolding bounded_def by auto

  1713 qed

  1714

  1715 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  1716   apply (simp add: bounded_def)

  1717   apply (rule_tac x=x in exI)

  1718   apply (rule_tac x=e in exI)

  1719   apply auto

  1720   done

  1721

  1722 lemma bounded_ball[simp,intro]: "bounded(ball x e)"

  1723   by (metis ball_subset_cball bounded_cball bounded_subset)

  1724

  1725 lemma finite_imp_bounded[intro]:

  1726   fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"

  1727 proof-

  1728   { fix a and F :: "'a set" assume as:"bounded F"

  1729     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto

  1730     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto

  1731     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)

  1732   }

  1733   thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto

  1734 qed

  1735

  1736 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  1737   apply (auto simp add: bounded_def)

  1738   apply (rename_tac x y r s)

  1739   apply (rule_tac x=x in exI)

  1740   apply (rule_tac x="max r (dist x y + s)" in exI)

  1741   apply (rule ballI, rename_tac z, safe)

  1742   apply (drule (1) bspec, simp)

  1743   apply (drule (1) bspec)

  1744   apply (rule min_max.le_supI2)

  1745   apply (erule order_trans [OF dist_triangle add_left_mono])

  1746   done

  1747

  1748 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"

  1749   by (induct rule: finite_induct[of F], auto)

  1750

  1751 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"

  1752   apply (simp add: bounded_iff)

  1753   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")

  1754   by metis arith

  1755

  1756 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  1757   by (metis Int_lower1 Int_lower2 bounded_subset)

  1758

  1759 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

  1760 apply (metis Diff_subset bounded_subset)

  1761 done

  1762

  1763 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"

  1764   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)

  1765

  1766 lemma not_bounded_UNIV[simp, intro]:

  1767   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  1768 proof(auto simp add: bounded_pos not_le)

  1769   obtain x :: 'a where "x \<noteq> 0"

  1770     using perfect_choose_dist [OF zero_less_one] by fast

  1771   fix b::real  assume b: "b >0"

  1772   have b1: "b +1 \<ge> 0" using b by simp

  1773   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  1774     by (simp add: norm_sgn)

  1775   then show "\<exists>x::'a. b < norm x" ..

  1776 qed

  1777

  1778 lemma bounded_linear_image:

  1779   assumes "bounded S" "bounded_linear f"

  1780   shows "bounded(f  S)"

  1781 proof-

  1782   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  1783   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)

  1784   { fix x assume "x\<in>S"

  1785     hence "norm x \<le> b" using b by auto

  1786     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)

  1787       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  1788   }

  1789   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)

  1790     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)

  1791 qed

  1792

  1793 lemma bounded_scaling:

  1794   fixes S :: "'a::real_normed_vector set"

  1795   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  1796   apply (rule bounded_linear_image, assumption)

  1797   apply (rule bounded_linear_scaleR_right)

  1798   done

  1799

  1800 lemma bounded_translation:

  1801   fixes S :: "'a::real_normed_vector set"

  1802   assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"

  1803 proof-

  1804   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  1805   { fix x assume "x\<in>S"

  1806     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto

  1807   }

  1808   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]

  1809     by (auto intro!: add exI[of _ "b + norm a"])

  1810 qed

  1811

  1812

  1813 text{* Some theorems on sups and infs using the notion "bounded". *}

  1814

  1815 lemma bounded_real:

  1816   fixes S :: "real set"

  1817   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

  1818   by (simp add: bounded_iff)

  1819

  1820 lemma bounded_has_Sup:

  1821   fixes S :: "real set"

  1822   assumes "bounded S" "S \<noteq> {}"

  1823   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"

  1824 proof

  1825   fix x assume "x\<in>S"

  1826   thus "x \<le> Sup S"

  1827     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)

  1828 next

  1829   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms

  1830     by (metis SupInf.Sup_least)

  1831 qed

  1832

  1833 lemma Sup_insert:

  1834   fixes S :: "real set"

  1835   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  1836 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)

  1837

  1838 lemma Sup_insert_finite:

  1839   fixes S :: "real set"

  1840   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  1841   apply (rule Sup_insert)

  1842   apply (rule finite_imp_bounded)

  1843   by simp

  1844

  1845 lemma bounded_has_Inf:

  1846   fixes S :: "real set"

  1847   assumes "bounded S"  "S \<noteq> {}"

  1848   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"

  1849 proof

  1850   fix x assume "x\<in>S"

  1851   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto

  1852   thus "x \<ge> Inf S" using x\<in>S

  1853     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)

  1854 next

  1855   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms

  1856     by (metis SupInf.Inf_greatest)

  1857 qed

  1858

  1859 lemma Inf_insert:

  1860   fixes S :: "real set"

  1861   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  1862 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)

  1863 lemma Inf_insert_finite:

  1864   fixes S :: "real set"

  1865   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  1866   by (rule Inf_insert, rule finite_imp_bounded, simp)

  1867

  1868 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)

  1869 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"

  1870   apply (frule isGlb_isLb)

  1871   apply (frule_tac x = y in isGlb_isLb)

  1872   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)

  1873   done

  1874

  1875

  1876 subsection {* Equivalent versions of compactness *}

  1877

  1878 subsubsection{* Sequential compactness *}

  1879

  1880 definition

  1881   compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)

  1882   "compact S \<longleftrightarrow>

  1883    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>

  1884        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

  1885

  1886 lemma compactI:

  1887   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

  1888   shows "compact S"

  1889   unfolding compact_def using assms by fast

  1890

  1891 lemma compactE:

  1892   assumes "compact S" "\<forall>n. f n \<in> S"

  1893   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  1894   using assms unfolding compact_def by fast

  1895

  1896 text {*

  1897   A metric space (or topological vector space) is said to have the

  1898   Heine-Borel property if every closed and bounded subset is compact.

  1899 *}

  1900

  1901 class heine_borel = metric_space +

  1902   assumes bounded_imp_convergent_subsequence:

  1903     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s

  1904       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  1905

  1906 lemma bounded_closed_imp_compact:

  1907   fixes s::"'a::heine_borel set"

  1908   assumes "bounded s" and "closed s" shows "compact s"

  1909 proof (unfold compact_def, clarify)

  1910   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  1911   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  1912     using bounded_imp_convergent_subsequence [OF bounded s \<forall>n. f n \<in> s] by auto

  1913   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp

  1914   have "l \<in> s" using closed s fr l

  1915     unfolding closed_sequential_limits by blast

  1916   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  1917     using l \<in> s r l by blast

  1918 qed

  1919

  1920 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"

  1921 proof(induct n)

  1922   show "0 \<le> r 0" by auto

  1923 next

  1924   fix n assume "n \<le> r n"

  1925   moreover have "r n < r (Suc n)"

  1926     using assms [unfolded subseq_def] by auto

  1927   ultimately show "Suc n \<le> r (Suc n)" by auto

  1928 qed

  1929

  1930 lemma eventually_subseq:

  1931   assumes r: "subseq r"

  1932   shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"

  1933 unfolding eventually_sequentially

  1934 by (metis subseq_bigger [OF r] le_trans)

  1935

  1936 lemma lim_subseq:

  1937   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"

  1938 unfolding tendsto_def eventually_sequentially o_def

  1939 by (metis subseq_bigger le_trans)

  1940

  1941 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"

  1942   unfolding Ex1_def

  1943   apply (rule_tac x="nat_rec e f" in exI)

  1944   apply (rule conjI)+

  1945 apply (rule def_nat_rec_0, simp)

  1946 apply (rule allI, rule def_nat_rec_Suc, simp)

  1947 apply (rule allI, rule impI, rule ext)

  1948 apply (erule conjE)

  1949 apply (induct_tac x)

  1950 apply simp

  1951 apply (erule_tac x="n" in allE)

  1952 apply (simp)

  1953 done

  1954

  1955 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"

  1956   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"

  1957   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"

  1958 proof-

  1959   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto

  1960   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto

  1961   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"

  1962     { fix n::nat

  1963       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto

  1964       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto

  1965       with n have "s N \<le> t - e" using e>0 by auto

  1966       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  }

  1967     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto

  1968     hence False using isLub_le_isUb[OF t, of "t - e"] and e>0 by auto  }

  1969   thus ?thesis by blast

  1970 qed

  1971

  1972 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"

  1973   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"

  1974   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"

  1975   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]

  1976   unfolding monoseq_def incseq_def

  1977   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]

  1978   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto

  1979

  1980 (* TODO: merge this lemma with the ones above *)

  1981 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"

  1982   assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"

  1983   shows "\<exists>l. (s ---> l) sequentially"

  1984 proof-

  1985   obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto

  1986   { fix m::nat

  1987     have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"

  1988       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)

  1989       apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }

  1990   hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto

  1991   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]

  1992     unfolding monoseq_def by auto

  1993   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)

  1994     unfolding dist_norm  by auto

  1995 qed

  1996

  1997 lemma compact_real_lemma:

  1998   assumes "\<forall>n::nat. abs(s n) \<le> b"

  1999   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"

  2000 proof-

  2001   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"

  2002     using seq_monosub[of s] by auto

  2003   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms

  2004     unfolding tendsto_iff dist_norm eventually_sequentially by auto

  2005 qed

  2006

  2007 instance real :: heine_borel

  2008 proof

  2009   fix s :: "real set" and f :: "nat \<Rightarrow> real"

  2010   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2011   then obtain b where b: "\<forall>n. abs (f n) \<le> b"

  2012     unfolding bounded_iff by auto

  2013   obtain l :: real and r :: "nat \<Rightarrow> nat" where

  2014     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  2015     using compact_real_lemma [OF b] by auto

  2016   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2017     by auto

  2018 qed

  2019

  2020 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$i)  s)"   2021 apply (erule bounded_linear_image)   2022 apply (rule bounded_linear_euclidean_component)   2023 done   2024   2025 lemma compact_lemma:   2026 fixes f :: "nat \<Rightarrow> 'a::euclidean_space"   2027 assumes "bounded s" and "\<forall>n. f n \<in> s"   2028 shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>   2029 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n)$$ i) (l $$i) < e) sequentially)"   2030 proof   2031 fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"   2032 have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto   2033 hence "\<exists>l::'a. \<exists>r. subseq r \<and>   2034 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n)$$ i) (l $$i) < e) sequentially)"   2035 proof(induct d) case empty thus ?case unfolding subseq_def by auto   2036 next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto   2037 have s': "bounded ((\<lambda>x. x$$ k)  s)" using bounded s by (rule bounded_component)

  2038     obtain l1::"'a" and r1 where r1:"subseq r1" and

  2039       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$i) (l1$$ i) < e) sequentially"

  2040       using insert(3) using insert(4) by auto

  2041     have f': "\<forall>n. f (r1 n) $$k \<in> (\<lambda>x. x$$ k)  s" using \<forall>n. f n \<in> s by simp

  2042     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$k) ---> l2) sequentially"   2043 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto   2044 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"   2045 using r1 and r2 unfolding r_def o_def subseq_def by auto   2046 moreover   2047 def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"

  2048     { fix e::real assume "e>0"

  2049       from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$i) (l1$$ i) < e) sequentially" by blast

  2050       from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$k) l2 < e) sequentially" by (rule tendstoD)   2051 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n))$$ i) (l1 $$i) < e) sequentially"   2052 by (rule eventually_subseq)   2053 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n)$$ i) (l $$i) < e) sequentially"   2054 using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def   2055 using insert.prems by auto   2056 }   2057 ultimately show ?case by auto   2058 qed   2059 thus "\<exists>l::'a. \<exists>r. subseq r \<and>   2060 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n)$$ i) (l $$i) < e) sequentially)"   2061 apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe   2062 apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe   2063 apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)   2064 apply(erule_tac x=i in ballE)   2065 proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a   2066 assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"   2067 hence *:"i\<ge>DIM('a)" by auto   2068 thus "dist (f (r n)$$ i) (l $$i) < e" using e by auto   2069 qed   2070 qed   2071   2072 instance euclidean_space \<subseteq> heine_borel   2073 proof   2074 fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"   2075 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"   2076 then obtain l::'a and r where r: "subseq r"   2077 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n)$$ i) (l $$i) < e) sequentially"   2078 using compact_lemma [OF s f] by blast   2079 let ?d = "{..<DIM('a)}"   2080 { fix e::real assume "e>0"   2081 hence "0 < e / (real_of_nat (card ?d))"   2082 using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto   2083 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n)$$ i) (l $$i) < e / (real_of_nat (card ?d))) sequentially"   2084 by simp   2085 moreover   2086 { fix n assume n: "\<forall>i. dist (f (r n)$$ i) (l $$i) < e / (real_of_nat (card ?d))"   2087 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n)$$ i) (l $$i))"   2088 apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)   2089 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"   2090 apply(rule setsum_strict_mono) using n by auto   2091 finally have "dist (f (r n)) l < e" unfolding setsum_constant   2092 using DIM_positive[where 'a='a] by auto   2093 }   2094 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"   2095 by (rule eventually_elim1)   2096 }   2097 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp   2098 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto   2099 qed   2100   2101 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"   2102 unfolding bounded_def   2103 apply clarify   2104 apply (rule_tac x="a" in exI)   2105 apply (rule_tac x="e" in exI)   2106 apply clarsimp   2107 apply (drule (1) bspec)   2108 apply (simp add: dist_Pair_Pair)   2109 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])   2110 done   2111   2112 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"   2113 unfolding bounded_def   2114 apply clarify   2115 apply (rule_tac x="b" in exI)   2116 apply (rule_tac x="e" in exI)   2117 apply clarsimp   2118 apply (drule (1) bspec)   2119 apply (simp add: dist_Pair_Pair)   2120 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])   2121 done   2122   2123 instance prod :: (heine_borel, heine_borel) heine_borel   2124 proof   2125 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"   2126 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"   2127 from s have s1: "bounded (fst  s)" by (rule bounded_fst)   2128 from f have f1: "\<forall>n. fst (f n) \<in> fst  s" by simp   2129 obtain l1 r1 where r1: "subseq r1"   2130 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"   2131 using bounded_imp_convergent_subsequence [OF s1 f1]   2132 unfolding o_def by fast   2133 from s have s2: "bounded (snd  s)" by (rule bounded_snd)   2134 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd  s" by simp   2135 obtain l2 r2 where r2: "subseq r2"   2136 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"   2137 using bounded_imp_convergent_subsequence [OF s2 f2]   2138 unfolding o_def by fast   2139 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"   2140 using lim_subseq [OF r2 l1] unfolding o_def .   2141 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"   2142 using tendsto_Pair [OF l1' l2] unfolding o_def by simp   2143 have r: "subseq (r1 \<circ> r2)"   2144 using r1 r2 unfolding subseq_def by simp   2145 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2146 using l r by fast   2147 qed   2148   2149 subsubsection{* Completeness *}   2150   2151 lemma cauchy_def:   2152 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"   2153 unfolding Cauchy_def by blast   2154   2155 definition   2156 complete :: "'a::metric_space set \<Rightarrow> bool" where   2157 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f   2158 --> (\<exists>l \<in> s. (f ---> l) sequentially))"   2159   2160 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")   2161 proof-   2162 { assume ?rhs   2163 { fix e::real   2164 assume "e>0"   2165 with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"   2166 by (erule_tac x="e/2" in allE) auto   2167 { fix n m   2168 assume nm:"N \<le> m \<and> N \<le> n"   2169 hence "dist (s m) (s n) < e" using N   2170 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]   2171 by blast   2172 }   2173 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"   2174 by blast   2175 }   2176 hence ?lhs   2177 unfolding cauchy_def   2178 by blast   2179 }   2180 thus ?thesis   2181 unfolding cauchy_def   2182 using dist_triangle_half_l   2183 by blast   2184 qed   2185   2186 lemma convergent_imp_cauchy:   2187 "(s ---> l) sequentially ==> Cauchy s"   2188 proof(simp only: cauchy_def, rule, rule)   2189 fix e::real assume "e>0" "(s ---> l) sequentially"   2190 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   2191 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   2192 qed   2193   2194 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"   2195 proof-   2196 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   2197 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto   2198 moreover   2199 have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto   2200 then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"   2201 unfolding bounded_any_center [where a="s N"] by auto   2202 ultimately show "?thesis"   2203 unfolding bounded_any_center [where a="s N"]   2204 apply(rule_tac x="max a 1" in exI) apply auto   2205 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto   2206 qed   2207   2208 lemma compact_imp_complete: assumes "compact s" shows "complete s"   2209 proof-   2210 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"   2211 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   2212   2213 note lr' = subseq_bigger [OF lr(2)]   2214   2215 { fix e::real assume "e>0"   2216 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   2217 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   2218 { fix n::nat assume n:"n \<ge> max N M"   2219 have "dist ((f \<circ> r) n) l < e/2" using n M by auto   2220 moreover have "r n \<ge> N" using lr'[of n] n by auto   2221 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto   2222 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) }   2223 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }   2224 hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding Lim_sequentially by auto }   2225 thus ?thesis unfolding complete_def by auto   2226 qed   2227   2228 instance heine_borel < complete_space   2229 proof   2230 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"   2231 hence "bounded (range f)"   2232 by (rule cauchy_imp_bounded)   2233 hence "compact (closure (range f))"   2234 using bounded_closed_imp_compact [of "closure (range f)"] by auto   2235 hence "complete (closure (range f))"   2236 by (rule compact_imp_complete)   2237 moreover have "\<forall>n. f n \<in> closure (range f)"   2238 using closure_subset [of "range f"] by auto   2239 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"   2240 using Cauchy f unfolding complete_def by auto   2241 then show "convergent f"   2242 unfolding convergent_def by auto   2243 qed   2244   2245 instance euclidean_space \<subseteq> banach ..   2246   2247 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"   2248 proof(simp add: complete_def, rule, rule)   2249 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"   2250 hence "convergent f" by (rule Cauchy_convergent)   2251 thus "\<exists>l. f ----> l" unfolding convergent_def .   2252 qed   2253   2254 lemma complete_imp_closed: assumes "complete s" shows "closed s"   2255 proof -   2256 { fix x assume "x islimpt s"   2257 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"   2258 unfolding islimpt_sequential by auto   2259 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"   2260 using complete s[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto   2261 hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto   2262 }   2263 thus "closed s" unfolding closed_limpt by auto   2264 qed   2265   2266 lemma complete_eq_closed:   2267 fixes s :: "'a::complete_space set"   2268 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")   2269 proof   2270 assume ?lhs thus ?rhs by (rule complete_imp_closed)   2271 next   2272 assume ?rhs   2273 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"   2274 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto   2275 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 }   2276 thus ?lhs unfolding complete_def by auto   2277 qed   2278   2279 lemma convergent_eq_cauchy:   2280 fixes s :: "nat \<Rightarrow> 'a::complete_space"   2281 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"   2282 unfolding Cauchy_convergent_iff convergent_def ..   2283   2284 lemma convergent_imp_bounded:   2285 fixes s :: "nat \<Rightarrow> 'a::metric_space"   2286 shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"   2287 by (intro cauchy_imp_bounded convergent_imp_cauchy)   2288   2289 subsubsection{* Total boundedness *}   2290   2291 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where   2292 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"   2293 declare helper_1.simps[simp del]   2294   2295 lemma compact_imp_totally_bounded:   2296 assumes "compact s"   2297 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"   2298 proof(rule, rule, rule ccontr)   2299 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)"   2300 def x \<equiv> "helper_1 s e"   2301 { fix n   2302 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"   2303 proof(induct_tac rule:nat_less_induct)   2304 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"   2305 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"   2306 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   2307 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto   2308 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]   2309 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto   2310 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto   2311 qed }   2312 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+   2313 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   2314 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto   2315 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   2316 show False   2317 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]   2318 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]   2319 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto   2320 qed   2321   2322 subsubsection{* Heine-Borel theorem *}   2323   2324 text {* Following Burkill \& Burkill vol. 2. *}   2325   2326 lemma heine_borel_lemma: fixes s::"'a::metric_space set"   2327 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"   2328 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"   2329 proof(rule ccontr)   2330 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"   2331 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto   2332 { fix n::nat   2333 have "1 / real (n + 1) > 0" by auto   2334 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 }   2335 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   2336 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)"   2337 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   2338   2339 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"   2340 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto   2341   2342 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto   2343 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"   2344 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto   2345   2346 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"   2347 using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto   2348   2349 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and e>0 by auto   2350 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"   2351 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2   2352 using subseq_bigger[OF r, of "N1 + N2"] by auto   2353   2354 def x \<equiv> "(f (r (N1 + N2)))"   2355 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def   2356 using f[THEN spec[where x="r (N1 + N2)"]] using b\<in>t by auto   2357 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto   2358 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto   2359   2360 have "dist x l < e/2" using N1 unfolding x_def o_def by auto   2361 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)   2362   2363 thus False using e and y\<notin>b by auto   2364 qed   2365   2366 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)   2367 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"   2368 proof clarify   2369 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"   2370 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   2371 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto   2372 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   2373 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast   2374   2375 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   2376 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" by auto   2377   2378 have "finite (bb  k)" using k(1) by auto   2379 moreover   2380 { fix x assume "x\<in>s"   2381 hence "x\<in>\<Union>(\<lambda>x. ball x e)  k" using k(3) unfolding subset_eq by auto   2382 hence "\<exists>X\<in>bb  k. x \<in> X" using bb k(2) by blast   2383 hence "x \<in> \<Union>(bb  k)" using Union_iff[of x "bb  k"] by auto   2384 }   2385 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   2386 qed   2387   2388 subsubsection {* Bolzano-Weierstrass property *}   2389   2390 lemma heine_borel_imp_bolzano_weierstrass:   2391 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'))"   2392 "infinite t" "t \<subseteq> s"   2393 shows "\<exists>x \<in> s. x islimpt t"   2394 proof(rule ccontr)   2395 assume "\<not> (\<exists>x \<in> s. x islimpt t)"   2396 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   2397 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   2398 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"   2399 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto   2400 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto   2401 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"   2402 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   2403 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 }   2404 hence "inj_on f t" unfolding inj_on_def by simp   2405 hence "infinite (f  t)" using assms(2) using finite_imageD by auto   2406 moreover   2407 { fix x assume "x\<in>t" "f x \<notin> g"   2408 from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto   2409 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto   2410 hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto   2411 hence False using f x \<notin> g h\<in>g unfolding h = f y by auto }   2412 hence "f  t \<subseteq> g" by auto   2413 ultimately show False using g(2) using finite_subset by auto   2414 qed   2415   2416 subsubsection {* Complete the chain of compactness variants *}   2417   2418 lemma islimpt_range_imp_convergent_subsequence:   2419 fixes f :: "nat \<Rightarrow> 'a::metric_space"   2420 assumes "l islimpt (range f)"   2421 shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2422 proof (intro exI conjI)   2423 have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"   2424 using assms unfolding islimpt_def   2425 by (drule_tac x="ball l e" in spec)   2426 (auto simp add: zero_less_dist_iff dist_commute)   2427   2428 def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"   2429 have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"   2430 unfolding t_def by (rule LeastI2_ex [OF * conjunct1])   2431 have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"   2432 unfolding t_def by (rule LeastI2_ex [OF * conjunct2])   2433 have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"   2434 unfolding t_def by (simp add: Least_le)   2435 have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"   2436 unfolding t_def by (drule not_less_Least) simp   2437 have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"   2438 apply (rule t_le)   2439 apply (erule f_t_neq)   2440 apply (erule (1) less_le_trans [OF f_t_closer])   2441 done   2442 have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"   2443 by (drule f_t_closer) auto   2444 have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"   2445 apply (subst less_le)   2446 apply (rule conjI)   2447 apply (rule t_antimono)   2448 apply (erule f_t_neq)   2449 apply (erule f_t_closer [THEN less_imp_le])   2450 apply (rule t_dist_f_neq [symmetric])   2451 apply (erule f_t_neq)   2452 done   2453 have dist_f_t_less':   2454 "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"   2455 apply (simp add: le_less)   2456 apply (erule disjE)   2457 apply (rule less_trans)   2458 apply (erule f_t_closer)   2459 apply (rule le_less_trans)   2460 apply (erule less_tD)   2461 apply (erule f_t_neq)   2462 apply (erule f_t_closer)   2463 apply (erule subst)   2464 apply (erule f_t_closer)   2465 done   2466   2467 def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"   2468 have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"   2469 unfolding r_def by simp_all   2470 have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"   2471 by (induct_tac n) (simp_all add: r_simps f_t_neq)   2472   2473 show "subseq r"   2474 unfolding subseq_Suc_iff   2475 apply (rule allI)   2476 apply (case_tac n)   2477 apply (simp_all add: r_simps)   2478 apply (rule t_less, rule zero_less_one)   2479 apply (rule t_less, rule f_r_neq)   2480 done   2481 show "((f \<circ> r) ---> l) sequentially"   2482 unfolding Lim_sequentially o_def   2483 apply (clarify, rule_tac x="t e" in exI, clarify)   2484 apply (drule le_trans, rule seq_suble [OF subseq r])   2485 apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)   2486 done   2487 qed   2488   2489 lemma finite_range_imp_infinite_repeats:   2490 fixes f :: "nat \<Rightarrow> 'a"   2491 assumes "finite (range f)"   2492 shows "\<exists>k. infinite {n. f n = k}"   2493 proof -   2494 { fix A :: "'a set" assume "finite A"   2495 hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"   2496 proof (induct)   2497 case empty thus ?case by simp   2498 next   2499 case (insert x A)   2500 show ?case   2501 proof (cases "finite {n. f n = x}")   2502 case True   2503 with infinite {n. f n \<in> insert x A}   2504 have "infinite {n. f n \<in> A}" by simp   2505 thus "\<exists>k. infinite {n. f n = k}" by (rule insert)   2506 next   2507 case False thus "\<exists>k. infinite {n. f n = k}" ..   2508 qed   2509 qed   2510 } note H = this   2511 from assms show "\<exists>k. infinite {n. f n = k}"   2512 by (rule H) simp   2513 qed   2514   2515 lemma bolzano_weierstrass_imp_compact:   2516 fixes s :: "'a::metric_space set"   2517 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"   2518 shows "compact s"   2519 proof -   2520 { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"   2521 have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2522 proof (cases "finite (range f)")   2523 case True   2524 hence "\<exists>l. infinite {n. f n = l}"   2525 by (rule finite_range_imp_infinite_repeats)   2526 then obtain l where "infinite {n. f n = l}" ..   2527 hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"   2528 by (rule infinite_enumerate)   2529 then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto   2530 hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2531 unfolding o_def by (simp add: fr tendsto_const)   2532 hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2533 by - (rule exI)   2534 from f have "\<forall>n. f (r n) \<in> s" by simp   2535 hence "l \<in> s" by (simp add: fr)   2536 thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2537 by (rule rev_bexI) fact   2538 next   2539 case False   2540 with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto   2541 then obtain l where "l \<in> s" "l islimpt (range f)" ..   2542 have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2543 using l islimpt (range f)   2544 by (rule islimpt_range_imp_convergent_subsequence)   2545 with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..   2546 qed   2547 }   2548 thus ?thesis unfolding compact_def by auto   2549 qed   2550   2551 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where   2552 "helper_2 beyond 0 = beyond 0" |   2553 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"   2554   2555 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"   2556 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"   2557 shows "bounded s"   2558 proof(rule ccontr)   2559 assume "\<not> bounded s"   2560 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"   2561 unfolding bounded_any_center [where a=undefined]   2562 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto   2563 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"   2564 unfolding linorder_not_le by auto   2565 def x \<equiv> "helper_2 beyond"   2566   2567 { fix m n ::nat assume "m<n"   2568 hence "dist undefined (x m) + 1 < dist undefined (x n)"   2569 proof(induct n)   2570 case 0 thus ?case by auto   2571 next   2572 case (Suc n)   2573 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"   2574 unfolding x_def and helper_2.simps   2575 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto   2576 thus ?case proof(cases "m < n")   2577 case True thus ?thesis using Suc and * by auto   2578 next   2579 case False hence "m = n" using Suc(2) by auto   2580 thus ?thesis using * by auto   2581 qed   2582 qed } note * = this   2583 { fix m n ::nat assume "m\<noteq>n"   2584 have "1 < dist (x m) (x n)"   2585 proof(cases "m<n")   2586 case True   2587 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto   2588 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith   2589 next   2590 case False hence "n<m" using m\<noteq>n by auto   2591 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto   2592 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith   2593 qed } note ** = this   2594 { fix a b assume "x a = x b" "a \<noteq> b"   2595 hence False using **[of a b] by auto }   2596 hence "inj x" unfolding inj_on_def by auto   2597 moreover   2598 { fix n::nat   2599 have "x n \<in> s"   2600 proof(cases "n = 0")   2601 case True thus ?thesis unfolding x_def using beyond by auto   2602 next   2603 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto   2604 thus ?thesis unfolding x_def using beyond by auto   2605 qed }   2606 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   2607   2608 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto   2609 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   2610 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"]]   2611 unfolding dist_nz by auto   2612 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto   2613 qed   2614   2615 lemma sequence_infinite_lemma:   2616 fixes f :: "nat \<Rightarrow> 'a::t1_space"   2617 assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"   2618 shows "infinite (range f)"   2619 proof   2620 assume "finite (range f)"   2621 hence "closed (range f)" by (rule finite_imp_closed)   2622 hence "open (- range f)" by (rule open_Compl)   2623 from assms(1) have "l \<in> - range f" by auto   2624 from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"   2625 using open (- range f) l \<in> - range f by (rule topological_tendstoD)   2626 thus False unfolding eventually_sequentially by auto   2627 qed   2628   2629 lemma closure_insert:   2630 fixes x :: "'a::t1_space"   2631 shows "closure (insert x s) = insert x (closure s)"   2632 apply (rule closure_unique)   2633 apply (rule insert_mono [OF closure_subset])   2634 apply (rule closed_insert [OF closed_closure])   2635 apply (simp add: closure_minimal)   2636 done   2637   2638 lemma islimpt_insert:   2639 fixes x :: "'a::t1_space"   2640 shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"   2641 proof   2642 assume *: "x islimpt (insert a s)"   2643 show "x islimpt s"   2644 proof (rule islimptI)   2645 fix t assume t: "x \<in> t" "open t"   2646 show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"   2647 proof (cases "x = a")   2648 case True   2649 obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"   2650 using * t by (rule islimptE)   2651 with x = a show ?thesis by auto   2652 next   2653 case False   2654 with t have t': "x \<in> t - {a}" "open (t - {a})"   2655 by (simp_all add: open_Diff)   2656 obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"   2657 using * t' by (rule islimptE)   2658 thus ?thesis by auto   2659 qed   2660 qed   2661 next   2662 assume "x islimpt s" thus "x islimpt (insert a s)"   2663 by (rule islimpt_subset) auto   2664 qed   2665   2666 lemma islimpt_union_finite:   2667 fixes x :: "'a::t1_space"   2668 shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"   2669 by (induct set: finite, simp_all add: islimpt_insert)   2670   2671 lemma sequence_unique_limpt:   2672 fixes f :: "nat \<Rightarrow> 'a::t2_space"   2673 assumes "(f ---> l) sequentially" and "l' islimpt (range f)"   2674 shows "l' = l"   2675 proof (rule ccontr)   2676 assume "l' \<noteq> l"   2677 obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"   2678 using hausdorff [OF l' \<noteq> l] by auto   2679 have "eventually (\<lambda>n. f n \<in> t) sequentially"   2680 using assms(1) open t l \<in> t by (rule topological_tendstoD)   2681 then obtain N where "\<forall>n\<ge>N. f n \<in> t"   2682 unfolding eventually_sequentially by auto   2683   2684 have "UNIV = {..<N} \<union> {N..}" by auto   2685 hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp   2686 hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)   2687 hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)   2688 then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"   2689 using l' \<in> s open s by (rule islimptE)   2690 then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto   2691 with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp   2692 with s \<inter> t = {} show False by simp   2693 qed   2694   2695 lemma bolzano_weierstrass_imp_closed:   2696 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)   2697 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"   2698 shows "closed s"   2699 proof-   2700 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"   2701 hence "l \<in> s"   2702 proof(cases "\<forall>n. x n \<noteq> l")   2703 case False thus "l\<in>s" using as(1) by auto   2704 next   2705 case True note cas = this   2706 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto   2707 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto   2708 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto   2709 qed }   2710 thus ?thesis unfolding closed_sequential_limits by fast   2711 qed   2712   2713 text {* Hence express everything as an equivalence. *}   2714   2715 lemma compact_eq_heine_borel:   2716 fixes s :: "'a::metric_space set"   2717 shows "compact s \<longleftrightarrow>   2718 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)   2719 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")   2720 proof   2721 assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)   2722 next   2723 assume ?rhs   2724 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"   2725 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])   2726 thus ?lhs by (rule bolzano_weierstrass_imp_compact)   2727 qed   2728   2729 lemma compact_eq_bolzano_weierstrass:   2730 fixes s :: "'a::metric_space set"   2731 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")   2732 proof   2733 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto   2734 next   2735 assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)   2736 qed   2737   2738 lemma compact_eq_bounded_closed:   2739 fixes s :: "'a::heine_borel set"   2740 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")   2741 proof   2742 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto   2743 next   2744 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto   2745 qed   2746   2747 lemma compact_imp_bounded:   2748 fixes s :: "'a::metric_space set"   2749 shows "compact s ==> bounded s"   2750 proof -   2751 assume "compact s"   2752 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')"   2753 by (rule compact_imp_heine_borel)   2754 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"   2755 using heine_borel_imp_bolzano_weierstrass[of s] by auto   2756 thus "bounded s"   2757 by (rule bolzano_weierstrass_imp_bounded)   2758 qed   2759   2760 lemma compact_imp_closed:   2761 fixes s :: "'a::metric_space set"   2762 shows "compact s ==> closed s"   2763 proof -   2764 assume "compact s"   2765 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')"   2766 by (rule compact_imp_heine_borel)   2767 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"   2768 using heine_borel_imp_bolzano_weierstrass[of s] by auto   2769 thus "closed s"   2770 by (rule bolzano_weierstrass_imp_closed)   2771 qed   2772   2773 text{* In particular, some common special cases. *}   2774   2775 lemma compact_empty[simp]:   2776 "compact {}"   2777 unfolding compact_def   2778 by simp   2779   2780 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"   2781 unfolding subseq_def by simp (* TODO: move somewhere else *)   2782   2783 lemma compact_union [intro]:   2784 assumes "compact s" and "compact t"   2785 shows "compact (s \<union> t)"   2786 proof (rule compactI)   2787 fix f :: "nat \<Rightarrow> 'a"   2788 assume "\<forall>n. f n \<in> s \<union> t"   2789 hence "infinite {n. f n \<in> s \<union> t}" by simp   2790 hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp   2791 thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2792 proof   2793 assume "infinite {n. f n \<in> s}"   2794 from infinite_enumerate [OF this]   2795 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto   2796 obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"   2797 using compact s \<forall>n. (f \<circ> q) n \<in> s by (rule compactE)   2798 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"   2799 using subseq q by (simp_all add: subseq_o o_assoc)   2800 thus ?thesis by auto   2801 next   2802 assume "infinite {n. f n \<in> t}"   2803 from infinite_enumerate [OF this]   2804 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto   2805 obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"   2806 using compact t \<forall>n. (f \<circ> q) n \<in> t by (rule compactE)   2807 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"   2808 using subseq q by (simp_all add: subseq_o o_assoc)   2809 thus ?thesis by auto   2810 qed   2811 qed   2812   2813 lemma compact_inter_closed [intro]:   2814 assumes "compact s" and "closed t"   2815 shows "compact (s \<inter> t)"   2816 proof (rule compactI)   2817 fix f :: "nat \<Rightarrow> 'a"   2818 assume "\<forall>n. f n \<in> s \<inter> t"   2819 hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all   2820 obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"   2821 using compact s \<forall>n. f n \<in> s by (rule compactE)   2822 moreover   2823 from closed t \<forall>n. f n \<in> t ((f \<circ> r) ---> l) sequentially have "l \<in> t"   2824 unfolding closed_sequential_limits o_def by fast   2825 ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2826 by auto   2827 qed   2828   2829 lemma closed_inter_compact [intro]:   2830 assumes "closed s" and "compact t"   2831 shows "compact (s \<inter> t)"   2832 using compact_inter_closed [of t s] assms   2833 by (simp add: Int_commute)   2834   2835 lemma compact_inter [intro]:   2836 assumes "compact s" and "compact t"   2837 shows "compact (s \<inter> t)"   2838 using assms by (intro compact_inter_closed compact_imp_closed)   2839   2840 lemma compact_sing [simp]: "compact {a}"   2841 unfolding compact_def o_def subseq_def   2842 by (auto simp add: tendsto_const)   2843   2844 lemma compact_insert [simp]:   2845 assumes "compact s" shows "compact (insert x s)"   2846 proof -   2847 have "compact ({x} \<union> s)"   2848 using compact_sing assms by (rule compact_union)   2849 thus ?thesis by simp   2850 qed   2851   2852 lemma finite_imp_compact:   2853 shows "finite s \<Longrightarrow> compact s"   2854 by (induct set: finite) simp_all   2855   2856 lemma compact_cball[simp]:   2857 fixes x :: "'a::heine_borel"   2858 shows "compact(cball x e)"   2859 using compact_eq_bounded_closed bounded_cball closed_cball   2860 by blast   2861   2862 lemma compact_frontier_bounded[intro]:   2863 fixes s :: "'a::heine_borel set"   2864 shows "bounded s ==> compact(frontier s)"   2865 unfolding frontier_def   2866 using compact_eq_bounded_closed   2867 by blast   2868   2869 lemma compact_frontier[intro]:   2870 fixes s :: "'a::heine_borel set"   2871 shows "compact s ==> compact (frontier s)"   2872 using compact_eq_bounded_closed compact_frontier_bounded   2873 by blast   2874   2875 lemma frontier_subset_compact:   2876 fixes s :: "'a::heine_borel set"   2877 shows "compact s ==> frontier s \<subseteq> s"   2878 using frontier_subset_closed compact_eq_bounded_closed   2879 by blast   2880   2881 lemma open_delete:   2882 fixes s :: "'a::t1_space set"   2883 shows "open s \<Longrightarrow> open (s - {x})"   2884 by (simp add: open_Diff)   2885   2886 text{* Finite intersection property. I could make it an equivalence in fact. *}   2887   2888 lemma compact_imp_fip:   2889 assumes "compact s" "\<forall>t \<in> f. closed t"   2890 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"   2891 shows "s \<inter> (\<Inter> f) \<noteq> {}"   2892 proof   2893 assume as:"s \<inter> (\<Inter> f) = {}"   2894 hence "s \<subseteq> \<Union> uminus  f" by auto   2895 moreover have "Ball (uminus  f) open" using open_Diff closed_Diff using assms(2) by auto   2896 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   2897 hence "finite (uminus  f') \<and> uminus  f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)   2898 hence "s \<inter> \<Inter>uminus  f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus  f'"]] by auto   2899 thus False using f'(3) unfolding subset_eq and Union_iff by blast   2900 qed   2901   2902   2903 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}   2904   2905 lemma bounded_closed_nest:   2906 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"   2907 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"   2908 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"   2909 proof-   2910 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   2911 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto   2912   2913 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"   2914 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast   2915   2916 { fix n::nat   2917 { fix e::real assume "e>0"   2918 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto   2919 hence "dist ((x \<circ> r) (max N n)) l < e" by auto   2920 moreover   2921 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto   2922 hence "(x \<circ> r) (max N n) \<in> s n"   2923 using x apply(erule_tac x=n in allE)   2924 using x apply(erule_tac x="r (max N n)" in allE)   2925 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto   2926 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto   2927 }   2928 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast   2929 }   2930 thus ?thesis by auto   2931 qed   2932   2933 text {* Decreasing case does not even need compactness, just completeness. *}   2934   2935 lemma decreasing_closed_nest:   2936 assumes "\<forall>n. closed(s n)"   2937 "\<forall>n. (s n \<noteq> {})"   2938 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"   2939 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"   2940 shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"   2941 proof-   2942 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto   2943 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto   2944 then obtain t where t: "\<forall>n. t n \<in> s n" by auto   2945 { fix e::real assume "e>0"   2946 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto   2947 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"   2948 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+   2949 hence "dist (t m) (t n) < e" using N by auto   2950 }   2951 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto   2952 }   2953 hence "Cauchy t" unfolding cauchy_def by auto   2954 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto   2955 { fix n::nat   2956 { fix e::real assume "e>0"   2957 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto   2958 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   2959 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto   2960 }   2961 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto   2962 }   2963 then show ?thesis by auto   2964 qed   2965   2966 text {* Strengthen it to the intersection actually being a singleton. *}   2967   2968 lemma decreasing_closed_nest_sing:   2969 fixes s :: "nat \<Rightarrow> 'a::complete_space set"   2970 assumes "\<forall>n. closed(s n)"   2971 "\<forall>n. s n \<noteq> {}"   2972 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"   2973 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"   2974 shows "\<exists>a. \<Inter>(range s) = {a}"   2975 proof-   2976 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto   2977 { fix b assume b:"b \<in> \<Inter>(range s)"   2978 { fix e::real assume "e>0"   2979 hence "dist a b < e" using assms(4 )using b using a by blast   2980 }   2981 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)   2982 }   2983 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto   2984 thus ?thesis ..   2985 qed   2986   2987 text{* Cauchy-type criteria for uniform convergence. *}   2988   2989 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows   2990 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>   2991 (\<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")   2992 proof(rule)   2993 assume ?lhs   2994 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   2995 { fix e::real assume "e>0"   2996 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   2997 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"   2998 hence "dist (s m x) (s n x) < e"   2999 using N[THEN spec[where x=m], THEN spec[where x=x]]   3000 using N[THEN spec[where x=n], THEN spec[where x=x]]   3001 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }   3002 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 }   3003 thus ?rhs by auto   3004 next   3005 assume ?rhs   3006 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   3007 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]   3008 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto   3009 { fix e::real assume "e>0"   3010 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"   3011 using ?rhs[THEN spec[where x="e/2"]] by auto   3012 { fix x assume "P x"   3013 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"   3014 using l[THEN spec[where x=x], unfolded Lim_sequentially] using e>0 by(auto elim!: allE[where x="e/2"])   3015 fix n::nat assume "n\<ge>N"   3016 hence "dist(s n x)(l x) < e" using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]   3017 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) }   3018 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }   3019 thus ?lhs by auto   3020 qed   3021   3022 lemma uniformly_cauchy_imp_uniformly_convergent:   3023 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"   3024 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"   3025 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"   3026 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"   3027 proof-   3028 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"   3029 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto   3030 moreover   3031 { fix x assume "P x"   3032 hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]   3033 using l and assms(2) unfolding Lim_sequentially by blast }   3034 ultimately show ?thesis by auto   3035 qed   3036   3037   3038 subsection {* Continuity *}   3039   3040 text {* Define continuity over a net to take in restrictions of the set. *}   3041   3042 definition   3043 continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"   3044 where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"   3045   3046 lemma continuous_trivial_limit:   3047 "trivial_limit net ==> continuous net f"   3048 unfolding continuous_def tendsto_def trivial_limit_eq by auto   3049   3050 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"   3051 unfolding continuous_def   3052 unfolding tendsto_def   3053 using netlimit_within[of x s]   3054 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)   3055   3056 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"   3057 using continuous_within [of x UNIV f] by (simp add: within_UNIV)   3058   3059 lemma continuous_at_within:   3060 assumes "continuous (at x) f" shows "continuous (at x within s) f"   3061 using assms unfolding continuous_at continuous_within   3062 by (rule Lim_at_within)   3063   3064 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}   3065   3066 lemma continuous_within_eps_delta:   3067 "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)"   3068 unfolding continuous_within and Lim_within   3069 apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto   3070   3071 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.   3072 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"   3073 using continuous_within_eps_delta[of x UNIV f]   3074 unfolding within_UNIV by blast   3075   3076 text{* Versions in terms of open balls. *}   3077   3078 lemma continuous_within_ball:   3079 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.   3080 f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")   3081 proof   3082 assume ?lhs   3083 { fix e::real assume "e>0"   3084 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"   3085 using ?lhs[unfolded continuous_within Lim_within] by auto   3086 { fix y assume "y\<in>f  (ball x d \<inter> s)"   3087 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]   3088 apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto   3089 }   3090 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) }   3091 thus ?rhs by auto   3092 next   3093 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq   3094 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto   3095 qed   3096   3097 lemma continuous_at_ball:   3098 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")   3099 proof   3100 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball   3101 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)   3102 unfolding dist_nz[THEN sym] by auto   3103 next   3104 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball   3105 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)   3106 qed   3107   3108 text{* Define setwise continuity in terms of limits within the set. *}   3109   3110 definition   3111 continuous_on ::   3112 "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"   3113 where   3114 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"   3115   3116 lemma continuous_on_topological:   3117 "continuous_on s f \<longleftrightarrow>   3118 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>   3119 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"   3120 unfolding continuous_on_def tendsto_def   3121 unfolding Limits.eventually_within eventually_at_topological   3122 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto   3123   3124 lemma continuous_on_iff:   3125 "continuous_on s f \<longleftrightarrow>   3126 (\<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)"   3127 unfolding continuous_on_def Lim_within   3128 apply (intro ball_cong [OF refl] all_cong ex_cong)   3129 apply (rename_tac y, case_tac "y = x", simp)   3130 apply (simp add: dist_nz)   3131 done   3132   3133 definition   3134 uniformly_continuous_on ::   3135 "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"   3136 where   3137 "uniformly_continuous_on s f \<longleftrightarrow>   3138 (\<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)"   3139   3140 text{* Some simple consequential lemmas. *}   3141   3142 lemma uniformly_continuous_imp_continuous:   3143 " uniformly_continuous_on s f ==> continuous_on s f"   3144 unfolding uniformly_continuous_on_def continuous_on_iff by blast   3145   3146 lemma continuous_at_imp_continuous_within:   3147 "continuous (at x) f ==> continuous (at x within s) f"   3148 unfolding continuous_within continuous_at using Lim_at_within by auto   3149   3150 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"   3151 unfolding tendsto_def by (simp add: trivial_limit_eq)   3152   3153 lemma continuous_at_imp_continuous_on:   3154 assumes "\<forall>x\<in>s. continuous (at x) f"   3155 shows "continuous_on s f"   3156 unfolding continuous_on_def   3157 proof   3158 fix x assume "x \<in> s"   3159 with assms have *: "(f ---> f (netlimit (at x))) (at x)"   3160 unfolding continuous_def by simp   3161 have "(f ---> f x) (at x)"   3162 proof (cases "trivial_limit (at x)")   3163 case True thus ?thesis   3164 by (rule Lim_trivial_limit)   3165 next   3166 case False   3167 hence 1: "netlimit (at x) = x"   3168 using netlimit_within [of x UNIV]   3169 by (simp add: within_UNIV)   3170 with * show ?thesis by simp   3171 qed   3172 thus "(f ---> f x) (at x within s)"   3173 by (rule Lim_at_within)   3174 qed   3175   3176 lemma continuous_on_eq_continuous_within:   3177 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"   3178 unfolding continuous_on_def continuous_def   3179 apply (rule ball_cong [OF refl])   3180 apply (case_tac "trivial_limit (at x within s)")   3181 apply (simp add: Lim_trivial_limit)   3182 apply (simp add: netlimit_within)   3183 done   3184   3185 lemmas continuous_on = continuous_on_def -- "legacy theorem name"   3186   3187 lemma continuous_on_eq_continuous_at:   3188 shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"   3189 by (auto simp add: continuous_on continuous_at Lim_within_open)   3190   3191 lemma continuous_within_subset:   3192 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s   3193 ==> continuous (at x within t) f"   3194 unfolding continuous_within by(metis Lim_within_subset)   3195   3196 lemma continuous_on_subset:   3197 shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"   3198 unfolding continuous_on by (metis subset_eq Lim_within_subset)   3199   3200 lemma continuous_on_interior:   3201 shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"   3202 by (erule interiorE, drule (1) continuous_on_subset,   3203 simp add: continuous_on_eq_continuous_at)   3204   3205 lemma continuous_on_eq:   3206 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"   3207 unfolding continuous_on_def tendsto_def Limits.eventually_within   3208 by simp   3209   3210 text {* Characterization of various kinds of continuity in terms of sequences. *}   3211   3212 lemma continuous_within_sequentially:   3213 fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"   3214 shows "continuous (at a within s) f \<longleftrightarrow>   3215 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially   3216 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")   3217 proof   3218 assume ?lhs   3219 { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"   3220 fix T::"'b set" assume "open T" and "f a \<in> T"   3221 with ?lhs obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> f x \<in> T"   3222 unfolding continuous_within tendsto_def eventually_within by auto   3223 have "eventually (\<lambda>n. dist (x n) a < d) sequentially"   3224 using x(2) d>0 by simp   3225 hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"   3226 proof (rule eventually_elim1)   3227 fix n assume "dist (x n) a < d" thus "(f \<circ> x) n \<in> T"   3228 using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto   3229 qed   3230 }   3231 thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp   3232 next   3233 assume ?rhs thus ?lhs   3234 unfolding continuous_within tendsto_def [where l="f a"]   3235 by (simp add: sequentially_imp_eventually_within)   3236 qed   3237   3238 lemma continuous_at_sequentially:   3239 fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"   3240 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially   3241 --> ((f o x) ---> f a) sequentially)"   3242 using continuous_within_sequentially[of a UNIV f]   3243 unfolding within_UNIV by auto   3244   3245 lemma continuous_on_sequentially:   3246 fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"   3247 shows "continuous_on s f \<longleftrightarrow>   3248 (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially   3249 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")   3250 proof   3251 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto   3252 next   3253 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto   3254 qed   3255   3256 lemma uniformly_continuous_on_sequentially':   3257 "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>   3258 ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially   3259 \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")   3260 proof   3261 assume ?lhs   3262 { 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"   3263 { fix e::real assume "e>0"   3264 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"   3265 using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto   3266 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   3267 { fix n assume "n\<ge>N"   3268 hence "dist (f (x n)) (f (y n)) < e"   3269 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   3270 unfolding dist_commute by simp }   3271 hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }   3272 hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto }   3273 thus ?rhs by auto   3274 next   3275 assume ?rhs   3276 { assume "\<not> ?lhs"   3277 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   3278 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"   3279 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   3280 by (auto simp add: dist_commute)   3281 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"   3282 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"   3283 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"   3284 unfolding x_def and y_def using fa by auto   3285 { fix e::real assume "e>0"   3286 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   3287 { fix n::nat assume "n\<ge>N"   3288 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto   3289 also have "\<dots> < e" using N by auto   3290 finally have "inverse (real n + 1) < e" by auto   3291 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }   3292 hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }   3293 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   3294 hence False using fxy and e>0 by auto }   3295 thus ?lhs unfolding uniformly_continuous_on_def by blast   3296 qed   3297   3298 lemma uniformly_continuous_on_sequentially:   3299 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"   3300 shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>   3301 ((\<lambda>n. x n - y n) ---> 0) sequentially   3302 \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")   3303 (* BH: maybe the previous lemma should replace this one? *)   3304 unfolding uniformly_continuous_on_sequentially'   3305 unfolding dist_norm tendsto_norm_zero_iff ..   3306   3307 text{* The usual transformation theorems. *}   3308   3309 lemma continuous_transform_within:   3310 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"   3311 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"   3312 "continuous (at x within s) f"   3313 shows "continuous (at x within s) g"   3314 unfolding continuous_within   3315 proof (rule Lim_transform_within)   3316 show "0 < d" by fact   3317 show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"   3318 using assms(3) by auto   3319 have "f x = g x"   3320 using assms(1,2,3) by auto   3321 thus "(f ---> g x) (at x within s)"   3322 using assms(4) unfolding continuous_within by simp   3323 qed   3324   3325 lemma continuous_transform_at:   3326 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"   3327 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"   3328 "continuous (at x) f"   3329 shows "continuous (at x) g"   3330 using continuous_transform_within [of d x UNIV f g] assms   3331 by (simp add: within_UNIV)   3332   3333 text{* Combination results for pointwise continuity. *}   3334   3335 lemma continuous_const: "continuous net (\<lambda>x. c)"   3336 by (auto simp add: continuous_def tendsto_const)   3337   3338 lemma continuous_cmul:   3339 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"   3340 shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"   3341 by (auto simp add: continuous_def intro: tendsto_intros)   3342   3343 lemma continuous_neg:   3344 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"   3345 shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"   3346 by (auto simp add: continuous_def tendsto_minus)   3347   3348 lemma continuous_add:   3349 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"   3350 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"   3351 by (auto simp add: continuous_def tendsto_add)   3352   3353 lemma continuous_sub:   3354 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"   3355 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"   3356 by (auto simp add: continuous_def tendsto_diff)   3357   3358   3359 text{* Same thing for setwise continuity. *}   3360   3361 lemma continuous_on_const: "continuous_on s (\<lambda>x. c)"   3362 unfolding continuous_on_def by (auto intro: tendsto_intros)   3363   3364 lemma continuous_on_minus:   3365 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3366 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"   3367 unfolding continuous_on_def by (auto intro: tendsto_intros)   3368   3369 lemma continuous_on_add:   3370 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3371 shows "continuous_on s f \<Longrightarrow> continuous_on s g   3372 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"   3373 unfolding continuous_on_def by (auto intro: tendsto_intros)   3374   3375 lemma continuous_on_diff:   3376 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3377 shows "continuous_on s f \<Longrightarrow> continuous_on s g   3378 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"   3379 unfolding continuous_on_def by (auto intro: tendsto_intros)   3380   3381 lemma (in bounded_linear) continuous_on:   3382 "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"   3383 unfolding continuous_on_def by (fast intro: tendsto)   3384   3385 lemma (in bounded_bilinear) continuous_on:   3386 "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"   3387 unfolding continuous_on_def by (fast intro: tendsto)   3388   3389 lemma continuous_on_scaleR:   3390 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3391 assumes "continuous_on s f" and "continuous_on s g"   3392 shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"   3393 using bounded_bilinear_scaleR assms   3394 by (rule bounded_bilinear.continuous_on)   3395   3396 lemma continuous_on_mult:   3397 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"   3398 assumes "continuous_on s f" and "continuous_on s g"   3399 shows "continuous_on s (\<lambda>x. f x * g x)"   3400 using bounded_bilinear_mult assms   3401 by (rule bounded_bilinear.continuous_on)   3402   3403 lemma continuous_on_inner:   3404 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"   3405 assumes "continuous_on s f" and "continuous_on s g"   3406 shows "continuous_on s (\<lambda>x. inner (f x) (g x))"   3407 using bounded_bilinear_inner assms   3408 by (rule bounded_bilinear.continuous_on)   3409   3410 lemma continuous_on_euclidean_component:   3411 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x$$ i)"

  3412   using bounded_linear_euclidean_component

  3413   by (rule bounded_linear.continuous_on)

  3414

  3415 text{* Same thing for uniform continuity, using sequential formulations. *}

  3416

  3417 lemma uniformly_continuous_on_const:

  3418  "uniformly_continuous_on s (\<lambda>x. c)"

  3419   unfolding uniformly_continuous_on_def by simp

  3420

  3421 lemma uniformly_continuous_on_cmul:

  3422   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3423   assumes "uniformly_continuous_on s f"

  3424   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  3425 proof-

  3426   { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"

  3427     hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"

  3428       using tendsto_scaleR [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]

  3429       unfolding scaleR_zero_right scaleR_right_diff_distrib by auto

  3430   }

  3431   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'

  3432     unfolding dist_norm tendsto_norm_zero_iff by auto

  3433 qed

  3434

  3435 lemma dist_minus:

  3436   fixes x y :: "'a::real_normed_vector"

  3437   shows "dist (- x) (- y) = dist x y"

  3438   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  3439

  3440 lemma uniformly_continuous_on_neg:

  3441   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3442   shows "uniformly_continuous_on s f

  3443          ==> uniformly_continuous_on s (\<lambda>x. -(f x))"

  3444   unfolding uniformly_continuous_on_def dist_minus .

  3445

  3446 lemma uniformly_continuous_on_add:

  3447   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3448   assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"

  3449   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  3450 proof-

  3451   {  fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"

  3452                     "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"

  3453     hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"

  3454       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

  3455     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  }

  3456   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'

  3457     unfolding dist_norm tendsto_norm_zero_iff by auto

  3458 qed

  3459

  3460 lemma uniformly_continuous_on_sub:

  3461   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3462   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g

  3463            ==> uniformly_continuous_on s  (\<lambda>x. f x - g x)"

  3464   unfolding ab_diff_minus

  3465   using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]

  3466   using uniformly_continuous_on_neg[of s g] by auto

  3467

  3468 text{* Identity function is continuous in every sense. *}

  3469

  3470 lemma continuous_within_id:

  3471  "continuous (at a within s) (\<lambda>x. x)"

  3472   unfolding continuous_within by (rule Lim_at_within [OF tendsto_ident_at])

  3473

  3474 lemma continuous_at_id:

  3475  "continuous (at a) (\<lambda>x. x)"

  3476   unfolding continuous_at by (rule tendsto_ident_at)

  3477

  3478 lemma continuous_on_id:

  3479  "continuous_on s (\<lambda>x. x)"

  3480   unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)

  3481

  3482 lemma uniformly_continuous_on_id:

  3483  "uniformly_continuous_on s (\<lambda>x. x)"

  3484   unfolding uniformly_continuous_on_def by auto

  3485

  3486 text{* Continuity of all kinds is preserved under composition. *}

  3487

  3488 lemma continuous_within_topological:

  3489   "continuous (at x within s) f \<longleftrightarrow>

  3490     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  3491       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  3492 unfolding continuous_within

  3493 unfolding tendsto_def Limits.eventually_within eventually_at_topological

  3494 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  3495

  3496 lemma continuous_within_compose:

  3497   assumes "continuous (at x within s) f"

  3498   assumes "continuous (at (f x) within f  s) g"

  3499   shows "continuous (at x within s) (g o f)"

  3500 using assms unfolding continuous_within_topological by simp metis

  3501

  3502 lemma continuous_at_compose:

  3503   assumes "continuous (at x) f"  "continuous (at (f x)) g"

  3504   shows "continuous (at x) (g o f)"

  3505 proof-

  3506   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

  3507   thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto

  3508 qed

  3509

  3510 lemma continuous_on_compose:

  3511   "continuous_on s f \<Longrightarrow> continuous_on (f  s) g \<Longrightarrow> continuous_on s (g o f)"

  3512   unfolding continuous_on_topological by simp metis

  3513

  3514 lemma uniformly_continuous_on_compose:

  3515   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  3516   shows "uniformly_continuous_on s (g o f)"

  3517 proof-

  3518   { fix e::real assume "e>0"

  3519     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

  3520     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

  3521     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  }

  3522   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  3523 qed

  3524

  3525 text{* Continuity in terms of open preimages. *}

  3526

  3527 lemma continuous_at_open:

  3528   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)))"

  3529 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]

  3530 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  3531

  3532 lemma continuous_on_open:

  3533   shows "continuous_on s f \<longleftrightarrow>

  3534         (\<forall>t. openin (subtopology euclidean (f  s)) t

  3535             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  3536 proof (safe)

  3537   fix t :: "'b set"

  3538   assume 1: "continuous_on s f"

  3539   assume 2: "openin (subtopology euclidean (f  s)) t"

  3540   from 2 obtain B where B: "open B" and t: "t = f  s \<inter> B"

  3541     unfolding openin_open by auto

  3542   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"

  3543   have "open U" unfolding U_def by (simp add: open_Union)

  3544   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"

  3545   proof (intro ballI iffI)

  3546     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"

  3547       unfolding U_def t by auto

  3548   next

  3549     fix x assume "x \<in> s" and "f x \<in> t"

  3550     hence "x \<in> s" and "f x \<in> B"

  3551       unfolding t by auto

  3552     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"

  3553       unfolding t continuous_on_topological by metis

  3554     then show "x \<in> U"

  3555       unfolding U_def by auto

  3556   qed

  3557   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto

  3558   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  3559     unfolding openin_open by fast

  3560 next

  3561   assume "?rhs" show "continuous_on s f"

  3562   unfolding continuous_on_topological

  3563   proof (clarify)

  3564     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"

  3565     have "openin (subtopology euclidean (f  s)) (f  s \<inter> B)"

  3566       unfolding openin_open using open B by auto

  3567     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f  s \<inter> B}"

  3568       using ?rhs by fast

  3569     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"

  3570       unfolding openin_open using x \<in> s and f x \<in> B by auto

  3571   qed

  3572 qed

  3573

  3574 text {* Similarly in terms of closed sets. *}

  3575

  3576 lemma continuous_on_closed:

  3577   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")

  3578 proof

  3579   assume ?lhs

  3580   { fix t

  3581     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  3582     have **:"f  s - (f  s - (f  s - t)) = f  s - t" by auto

  3583     assume as:"closedin (subtopology euclidean (f  s)) t"

  3584     hence "closedin (subtopology euclidean (f  s)) (f  s - (f  s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto

  3585     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"]]

  3586       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }

  3587   thus ?rhs by auto

  3588 next

  3589   assume ?rhs

  3590   { fix t

  3591     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  3592     assume as:"openin (subtopology euclidean (f  s)) t"

  3593     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?rhs[THEN spec[where x="(f  s) - t"]]

  3594       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }

  3595   thus ?lhs unfolding continuous_on_open by auto

  3596 qed

  3597

  3598 text {* Half-global and completely global cases. *}

  3599

  3600 lemma continuous_open_in_preimage:

  3601   assumes "continuous_on s f"  "open t"

  3602   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  3603 proof-

  3604   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

  3605   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  3606     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  3607   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  3608 qed

  3609

  3610 lemma continuous_closed_in_preimage:

  3611   assumes "continuous_on s f"  "closed t"

  3612   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  3613 proof-

  3614   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

  3615   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  3616     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  3617   thus ?thesis

  3618     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  3619 qed

  3620

  3621 lemma continuous_open_preimage:

  3622   assumes "continuous_on s f" "open s" "open t"

  3623   shows "open {x \<in> s. f x \<in> t}"

  3624 proof-

  3625   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  3626     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  3627   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  3628 qed

  3629

  3630 lemma continuous_closed_preimage:

  3631   assumes "continuous_on s f" "closed s" "closed t"

  3632   shows "closed {x \<in> s. f x \<in> t}"

  3633 proof-

  3634   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  3635     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  3636   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  3637 qed

  3638

  3639 lemma continuous_open_preimage_univ:

  3640   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  3641   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  3642

  3643 lemma continuous_closed_preimage_univ:

  3644   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  3645   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  3646

  3647 lemma continuous_open_vimage:

  3648   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  3649   unfolding vimage_def by (rule continuous_open_preimage_univ)

  3650

  3651 lemma continuous_closed_vimage:

  3652   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  3653   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  3654

  3655 lemma interior_image_subset:

  3656   assumes "\<forall>x. continuous (at x) f" "inj f"

  3657   shows "interior (f  s) \<subseteq> f  (interior s)"

  3658 proof

  3659   fix x assume "x \<in> interior (f  s)"

  3660   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  3661   hence "x \<in> f  s" by auto

  3662   then obtain y where y: "y \<in> s" "x = f y" by auto

  3663   have "open (vimage f T)"

  3664     using assms(1) open T by (rule continuous_open_vimage)

  3665   moreover have "y \<in> vimage f T"

  3666     using x = f y x \<in> T by simp

  3667   moreover have "vimage f T \<subseteq> s"

  3668     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  3669   ultimately have "y \<in> interior s" ..

  3670   with x = f y show "x \<in> f  interior s" ..

  3671 qed

  3672

  3673 text {* Equality of continuous functions on closure and related results. *}

  3674

  3675 lemma continuous_closed_in_preimage_constant:

  3676   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  3677   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  3678   using continuous_closed_in_preimage[of s f "{a}"] by auto

  3679

  3680 lemma continuous_closed_preimage_constant:

  3681   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  3682   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  3683   using continuous_closed_preimage[of s f "{a}"] by auto

  3684

  3685 lemma continuous_constant_on_closure:

  3686   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  3687   assumes "continuous_on (closure s) f"

  3688           "\<forall>x \<in> s. f x = a"

  3689   shows "\<forall>x \<in> (closure s). f x = a"

  3690     using continuous_closed_preimage_constant[of "closure s" f a]

  3691     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  3692

  3693 lemma image_closure_subset:

  3694   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  3695   shows "f  (closure s) \<subseteq> t"

  3696 proof-

  3697   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  3698   moreover have "closed {x \<in> closure s. f x \<in> t}"

  3699     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  3700   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  3701     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  3702   thus ?thesis by auto

  3703 qed

  3704

  3705 lemma continuous_on_closure_norm_le:

  3706   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3707   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  3708   shows "norm(f x) \<le> b"

  3709 proof-

  3710   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  3711   show ?thesis

  3712     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  3713     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  3714 qed

  3715

  3716 text {* Making a continuous function avoid some value in a neighbourhood. *}

  3717

  3718 lemma continuous_within_avoid:

  3719   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  3720   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"

  3721   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  3722 proof-

  3723   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"

  3724     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto

  3725   { fix y assume " y\<in>s"  "dist x y < d"

  3726     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]

  3727       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }

  3728   thus ?thesis using d>0 by auto

  3729 qed

  3730

  3731 lemma continuous_at_avoid:

  3732   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  3733   assumes "continuous (at x) f"  "f x \<noteq> a"

  3734   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  3735 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto

  3736

  3737 lemma continuous_on_avoid:

  3738   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  3739   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  3740   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  3741 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

  3742

  3743 lemma continuous_on_open_avoid:

  3744   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  3745   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  3746   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  3747 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

  3748

  3749 text {* Proving a function is constant by proving open-ness of level set. *}

  3750

  3751 lemma continuous_levelset_open_in_cases:

  3752   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  3753   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  3754         openin (subtopology euclidean s) {x \<in> s. f x = a}

  3755         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  3756 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  3757

  3758 lemma continuous_levelset_open_in:

  3759   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  3760   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  3761         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  3762         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  3763 using continuous_levelset_open_in_cases[of s f ]

  3764 by meson

  3765

  3766 lemma continuous_levelset_open:

  3767   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  3768   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  3769   shows "\<forall>x \<in> s. f x = a"

  3770 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  3771

  3772 text {* Some arithmetical combinations (more to prove). *}

  3773

  3774 lemma open_scaling[intro]:

  3775   fixes s :: "'a::real_normed_vector set"

  3776   assumes "c \<noteq> 0"  "open s"

  3777   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  3778 proof-

  3779   { fix x assume "x \<in> s"

  3780     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

  3781     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  3782     moreover

  3783     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  3784       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  3785         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  3786           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  3787       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  }

  3788     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  }

  3789   thus ?thesis unfolding open_dist by auto

  3790 qed

  3791

  3792 lemma minus_image_eq_vimage:

  3793   fixes A :: "'a::ab_group_add set"

  3794   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  3795   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  3796

  3797 lemma open_negations:

  3798   fixes s :: "'a::real_normed_vector set"

  3799   shows "open s ==> open ((\<lambda> x. -x)  s)"

  3800   unfolding scaleR_minus1_left [symmetric]

  3801   by (rule open_scaling, auto)

  3802

  3803 lemma open_translation:

  3804   fixes s :: "'a::real_normed_vector set"

  3805   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  3806 proof-

  3807   { 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  }

  3808   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

  3809   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  3810 qed

  3811

  3812 lemma open_affinity:

  3813   fixes s :: "'a::real_normed_vector set"

  3814   assumes "open s"  "c \<noteq> 0"

  3815   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  3816 proof-

  3817   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  3818   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  3819   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  3820 qed

  3821

  3822 lemma interior_translation:

  3823   fixes s :: "'a::real_normed_vector set"

  3824   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  3825 proof (rule set_eqI, rule)

  3826   fix x assume "x \<in> interior (op + a  s)"

  3827   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  3828   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

  3829   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

  3830 next

  3831   fix x assume "x \<in> op + a  interior s"

  3832   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

  3833   { fix z have *:"a + y - z = y + a - z" by auto

  3834     assume "z\<in>ball x e"

  3835     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

  3836     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  3837   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  3838   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  3839 qed

  3840

  3841 text {* We can now extend limit compositions to consider the scalar multiplier. *}

  3842

  3843 lemma continuous_vmul:

  3844   fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"

  3845   shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"

  3846   unfolding continuous_def by (intro tendsto_intros)

  3847

  3848 lemma continuous_mul:

  3849   fixes c :: "'a::metric_space \<Rightarrow> real"

  3850   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3851   shows "continuous net c \<Longrightarrow> continuous net f

  3852              ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "

  3853   unfolding continuous_def by (intro tendsto_intros)

  3854

  3855 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul

  3856   continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul

  3857

  3858 lemmas continuous_on_intros = continuous_on_add continuous_on_const

  3859   continuous_on_id continuous_on_compose continuous_on_minus

  3860   continuous_on_diff continuous_on_scaleR continuous_on_mult

  3861   continuous_on_inner continuous_on_euclidean_component

  3862   uniformly_continuous_on_add uniformly_continuous_on_const

  3863   uniformly_continuous_on_id uniformly_continuous_on_compose

  3864   uniformly_continuous_on_cmul uniformly_continuous_on_neg

  3865   uniformly_continuous_on_sub

  3866

  3867 text {* And so we have continuity of inverse. *}

  3868

  3869 lemma continuous_inv:

  3870   fixes f :: "'a::metric_space \<Rightarrow> real"

  3871   shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0

  3872            ==> continuous net (inverse o f)"

  3873   unfolding continuous_def using Lim_inv by auto

  3874

  3875 lemma continuous_at_within_inv:

  3876   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"

  3877   assumes "continuous (at a within s) f" "f a \<noteq> 0"

  3878   shows "continuous (at a within s) (inverse o f)"

  3879   using assms unfolding continuous_within o_def

  3880   by (intro tendsto_intros)

  3881

  3882 lemma continuous_at_inv:

  3883   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"

  3884   shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0

  3885          ==> continuous (at a) (inverse o f) "

  3886   using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto

  3887

  3888 text {* Topological properties of linear functions. *}

  3889

  3890 lemma linear_lim_0:

  3891   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  3892 proof-

  3893   interpret f: bounded_linear f by fact

  3894   have "(f ---> f 0) (at 0)"

  3895     using tendsto_ident_at by (rule f.tendsto)

  3896   thus ?thesis unfolding f.zero .

  3897 qed

  3898

  3899 lemma linear_continuous_at:

  3900   assumes "bounded_linear f"  shows "continuous (at a) f"

  3901   unfolding continuous_at using assms

  3902   apply (rule bounded_linear.tendsto)

  3903   apply (rule tendsto_ident_at)

  3904   done

  3905

  3906 lemma linear_continuous_within:

  3907   shows "bounded_linear f ==> continuous (at x within s) f"

  3908   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  3909

  3910 lemma linear_continuous_on:

  3911   shows "bounded_linear f ==> continuous_on s f"

  3912   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  3913

  3914 text {* Also bilinear functions, in composition form. *}

  3915

  3916 lemma bilinear_continuous_at_compose:

  3917   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  3918         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  3919   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  3920

  3921 lemma bilinear_continuous_within_compose:

  3922   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  3923         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  3924   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  3925

  3926 lemma bilinear_continuous_on_compose:

  3927   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  3928              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  3929   unfolding continuous_on_def

  3930   by (fast elim: bounded_bilinear.tendsto)

  3931

  3932 text {* Preservation of compactness and connectedness under continuous function. *}

  3933

  3934 lemma compact_continuous_image:

  3935   assumes "continuous_on s f"  "compact s"

  3936   shows "compact(f  s)"

  3937 proof-

  3938   { fix x assume x:"\<forall>n::nat. x n \<in> f  s"

  3939     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

  3940     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

  3941     { fix e::real assume "e>0"

  3942       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

  3943       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

  3944       { 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  }

  3945       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }

  3946     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  }

  3947   thus ?thesis unfolding compact_def by auto

  3948 qed

  3949

  3950 lemma connected_continuous_image:

  3951   assumes "continuous_on s f"  "connected s"

  3952   shows "connected(f  s)"

  3953 proof-

  3954   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  3955     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  3956       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  3957       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  3958       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  3959     hence False using as(1,2)

  3960       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  3961   thus ?thesis unfolding connected_clopen by auto

  3962 qed

  3963

  3964 text {* Continuity implies uniform continuity on a compact domain. *}

  3965

  3966 lemma compact_uniformly_continuous:

  3967   assumes "continuous_on s f"  "compact s"

  3968   shows "uniformly_continuous_on s f"

  3969 proof-

  3970     { fix x assume x:"x\<in>s"

  3971       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

  3972       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  }

  3973     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

  3974     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)"

  3975       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

  3976

  3977   { fix e::real assume "e>0"

  3978

  3979     { 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  }

  3980     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto

  3981     moreover

  3982     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }

  3983     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

  3984

  3985     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"

  3986       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

  3987       hence "x\<in>ball z (d z (e / 2))" using ea>0 unfolding subset_eq by auto

  3988       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

  3989         by (auto  simp add: dist_commute)

  3990       moreover have "y\<in>ball z (d z (e / 2))" using as and ea>0 and z[unfolded subset_eq]

  3991         by (auto simp add: dist_commute)

  3992       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

  3993         by (auto  simp add: dist_commute)

  3994       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]

  3995         by (auto simp add: dist_commute)  }

  3996     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  }

  3997   thus ?thesis unfolding uniformly_continuous_on_def by auto

  3998 qed

  3999

  4000 text{* Continuity of inverse function on compact domain. *}

  4001

  4002 lemma continuous_on_inverse:

  4003   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"

  4004     (* TODO: can this be generalized more? *)

  4005   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"

  4006   shows "continuous_on (f  s) g"

  4007 proof-

  4008   have *:"g  f  s = s" using assms(3) by (auto simp add: image_iff)

  4009   { fix t assume t:"closedin (subtopology euclidean (g  f  s)) t"

  4010     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto

  4011     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]

  4012       unfolding T(2) and Int_left_absorb by auto

  4013     moreover have "compact (s \<inter> T)"

  4014       using assms(2) unfolding compact_eq_bounded_closed

  4015       using bounded_subset[of s "s \<inter> T"] and T(1) by auto

  4016     ultimately have "closed (f  t)" using T(1) unfolding T(2)

  4017       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto

  4018     moreover have "{x \<in> f  s. g x \<in> t} = f  s \<inter> f  t" using assms(3) unfolding T(2) by auto

  4019     ultimately have "closedin (subtopology euclidean (f  s)) {x \<in> f  s. g x \<in> t}"

  4020       unfolding closedin_closed by auto  }

  4021   thus ?thesis unfolding continuous_on_closed by auto

  4022 qed

  4023

  4024 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4025

  4026 lemma continuous_uniform_limit:

  4027   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4028   assumes "\<not> trivial_limit F"

  4029   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4030   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4031   shows "continuous_on s g"

  4032 proof-

  4033   { fix x and e::real assume "x\<in>s" "e>0"

  4034     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4035       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4036     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4037     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4038       using assms(1) by blast

  4039     have "e / 3 > 0" using e>0 by auto

  4040     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"

  4041       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4042     { fix y assume "y \<in> s" and "dist y x < d"

  4043       hence "dist (f n y) (f n x) < e / 3"

  4044         by (rule d [rule_format])

  4045       hence "dist (f n y) (g x) < 2 * e / 3"

  4046         using dist_triangle [of "f n y" "g x" "f n x"]

  4047         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4048         by auto

  4049       hence "dist (g y) (g x) < e"

  4050         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4051         using dist_triangle3 [of "g y" "g x" "f n y"]

  4052         by auto }

  4053     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4054       using d>0 by auto }

  4055   thus ?thesis unfolding continuous_on_iff by auto

  4056 qed

  4057

  4058

  4059 subsection {* Topological stuff lifted from and dropped to R *}

  4060

  4061 lemma open_real:

  4062   fixes s :: "real set" shows

  4063  "open s \<longleftrightarrow>

  4064         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4065   unfolding open_dist dist_norm by simp

  4066

  4067 lemma islimpt_approachable_real:

  4068   fixes s :: "real set"

  4069   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4070   unfolding islimpt_approachable dist_norm by simp

  4071

  4072 lemma closed_real:

  4073   fixes s :: "real set"

  4074   shows "closed s \<longleftrightarrow>

  4075         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4076             --> x \<in> s)"

  4077   unfolding closed_limpt islimpt_approachable dist_norm by simp

  4078

  4079 lemma continuous_at_real_range:

  4080   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4081   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  4082         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  4083   unfolding continuous_at unfolding Lim_at

  4084   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  4085   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

  4086   apply(erule_tac x=e in allE) by auto

  4087

  4088 lemma continuous_on_real_range:

  4089   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4090   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))"

  4091   unfolding continuous_on_iff dist_norm by simp

  4092

  4093 lemma continuous_at_norm: "continuous (at x) norm"

  4094   unfolding continuous_at by (intro tendsto_intros)

  4095

  4096 lemma continuous_on_norm: "continuous_on s norm"

  4097 unfolding continuous_on by (intro ballI tendsto_intros)

  4098

  4099 lemma continuous_at_infnorm: "continuous (at x) infnorm"

  4100   unfolding continuous_at Lim_at o_def unfolding dist_norm

  4101   apply auto apply (rule_tac x=e in exI) apply auto

  4102   using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))

  4103

  4104 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  4105

  4106 lemma compact_attains_sup:

  4107   fixes s :: "real set"

  4108   assumes "compact s"  "s \<noteq> {}"

  4109   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"

  4110 proof-

  4111   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4112   { 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"

  4113     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto

  4114     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto

  4115     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using e>0 by auto  }

  4116   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]

  4117     apply(rule_tac x="Sup s" in bexI) by auto

  4118 qed

  4119

  4120 lemma Inf:

  4121   fixes S :: "real set"

  4122   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"

  4123 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)

  4124

  4125 lemma compact_attains_inf:

  4126   fixes s :: "real set"

  4127   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"

  4128 proof-

  4129   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4130   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"

  4131       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"

  4132     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto

  4133     moreover

  4134     { fix x assume "x \<in> s"

  4135       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto

  4136       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) x\<in>s unfolding * by auto }

  4137     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto

  4138     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using e>0 by auto  }

  4139   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]

  4140     apply(rule_tac x="Inf s" in bexI) by auto

  4141 qed

  4142

  4143 lemma continuous_attains_sup:

  4144   fixes f :: "'a::metric_space \<Rightarrow> real"

  4145   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4146         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"

  4147   using compact_attains_sup[of "f  s"]

  4148   using compact_continuous_image[of s f] by auto

  4149

  4150 lemma continuous_attains_inf:

  4151   fixes f :: "'a::metric_space \<Rightarrow> real"

  4152   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4153         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"

  4154   using compact_attains_inf[of "f  s"]

  4155   using compact_continuous_image[of s f] by auto

  4156

  4157 lemma distance_attains_sup:

  4158   assumes "compact s" "s \<noteq> {}"

  4159   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"

  4160 proof (rule continuous_attains_sup [OF assms])

  4161   { fix x assume "x\<in>s"

  4162     have "(dist a ---> dist a x) (at x within s)"

  4163       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)

  4164   }

  4165   thus "continuous_on s (dist a)"

  4166     unfolding continuous_on ..

  4167 qed

  4168

  4169 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  4170

  4171 lemma distance_attains_inf:

  4172   fixes a :: "'a::heine_borel"

  4173   assumes "closed s"  "s \<noteq> {}"

  4174   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"

  4175 proof-

  4176   from assms(2) obtain b where "b\<in>s" by auto

  4177   let ?B = "cball a (dist b a) \<inter> s"

  4178   have "b \<in> ?B" using b\<in>s by (simp add: dist_commute)

  4179   hence "?B \<noteq> {}" by auto

  4180   moreover

  4181   { fix x assume "x\<in>?B"

  4182     fix e::real assume "e>0"

  4183     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"

  4184       from as have "\<bar>dist a x' - dist a x\<bar> < e"

  4185         unfolding abs_less_iff minus_diff_eq

  4186         using dist_triangle2 [of a x' x]

  4187         using dist_triangle [of a x x']

  4188         by arith

  4189     }

  4190     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"

  4191       using e>0 by auto

  4192   }

  4193   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"

  4194     unfolding continuous_on Lim_within dist_norm real_norm_def

  4195     by fast

  4196   moreover have "compact ?B"

  4197     using compact_cball[of a "dist b a"]

  4198     unfolding compact_eq_bounded_closed

  4199     using bounded_Int and closed_Int and assms(1) by auto

  4200   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"

  4201     using continuous_attains_inf[of ?B "dist a"] by fastsimp

  4202   thus ?thesis by fastsimp

  4203 qed

  4204

  4205

  4206 subsection {* Pasted sets *}

  4207

  4208 lemma bounded_Times:

  4209   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  4210 proof-

  4211   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  4212     using assms [unfolded bounded_def] by auto

  4213   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"

  4214     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  4215   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  4216 qed

  4217

  4218 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  4219 by (induct x) simp

  4220

  4221 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"

  4222 unfolding compact_def

  4223 apply clarify

  4224 apply (drule_tac x="fst \<circ> f" in spec)

  4225 apply (drule mp, simp add: mem_Times_iff)

  4226 apply (clarify, rename_tac l1 r1)

  4227 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  4228 apply (drule mp, simp add: mem_Times_iff)

  4229 apply (clarify, rename_tac l2 r2)

  4230 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  4231 apply (rule_tac x="r1 \<circ> r2" in exI)

  4232 apply (rule conjI, simp add: subseq_def)

  4233 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)

  4234 apply (drule (1) tendsto_Pair) back

  4235 apply (simp add: o_def)

  4236 done

  4237

  4238 text{* Hence some useful properties follow quite easily. *}

  4239

  4240 lemma compact_scaling:

  4241   fixes s :: "'a::real_normed_vector set"

  4242   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  4243 proof-

  4244   let ?f = "\<lambda>x. scaleR c x"

  4245   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  4246   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  4247     using linear_continuous_at[OF *] assms by auto

  4248 qed

  4249

  4250 lemma compact_negations:

  4251   fixes s :: "'a::real_normed_vector set"

  4252   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  4253   using compact_scaling [OF assms, of "- 1"] by auto

  4254

  4255 lemma compact_sums:

  4256   fixes s t :: "'a::real_normed_vector set"

  4257   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  4258 proof-

  4259   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  4260     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  4261   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  4262     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  4263   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  4264 qed

  4265

  4266 lemma compact_differences:

  4267   fixes s t :: "'a::real_normed_vector set"

  4268   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  4269 proof-

  4270   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  4271     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4272   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  4273 qed

  4274

  4275 lemma compact_translation:

  4276   fixes s :: "'a::real_normed_vector set"

  4277   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  4278 proof-

  4279   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  4280   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  4281 qed

  4282

  4283 lemma compact_affinity:

  4284   fixes s :: "'a::real_normed_vector set"

  4285   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4286 proof-

  4287   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  4288   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  4289 qed

  4290

  4291 text {* Hence we get the following. *}

  4292

  4293 lemma compact_sup_maxdistance:

  4294   fixes s :: "'a::real_normed_vector set"

  4295   assumes "compact s"  "s \<noteq> {}"

  4296   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"

  4297 proof-

  4298   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using s \<noteq> {} by auto

  4299   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"

  4300     using compact_differences[OF assms(1) assms(1)]

  4301     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

  4302   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto

  4303   thus ?thesis using x(2)[unfolded x = a - b] by blast

  4304 qed

  4305

  4306 text {* We can state this in terms of diameter of a set. *}

  4307

  4308 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"

  4309   (* TODO: generalize to class metric_space *)

  4310

  4311 lemma diameter_bounded:

  4312   assumes "bounded s"

  4313   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4314         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"

  4315 proof-

  4316   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"

  4317   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto

  4318   { fix x y assume "x \<in> s" "y \<in> s"

  4319     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)  }

  4320   note * = this

  4321   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto

  4322     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

  4323       by simp (blast del: Sup_upper intro!: * Sup_upper) }

  4324   moreover

  4325   { fix d::real assume "d>0" "d < diameter s"

  4326     hence "s\<noteq>{}" unfolding diameter_def by auto

  4327     have "\<exists>d' \<in> ?D. d' > d"

  4328     proof(rule ccontr)

  4329       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"

  4330       hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)

  4331       thus False using d < diameter s s\<noteq>{}

  4332         apply (auto simp add: diameter_def)

  4333         apply (drule Sup_real_iff [THEN [2] rev_iffD2])

  4334         apply (auto, force)

  4335         done

  4336     qed

  4337     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }

  4338   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4339         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto

  4340 qed

  4341

  4342 lemma diameter_bounded_bound:

  4343  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"

  4344   using diameter_bounded by blast

  4345

  4346 lemma diameter_compact_attained:

  4347   fixes s :: "'a::real_normed_vector set"

  4348   assumes "compact s"  "s \<noteq> {}"

  4349   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"

  4350 proof-

  4351   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  4352   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

  4353   hence "diameter s \<le> norm (x - y)"

  4354     unfolding diameter_def by clarsimp (rule Sup_least, fast+)

  4355   thus ?thesis

  4356     by (metis b diameter_bounded_bound order_antisym xys)

  4357 qed

  4358

  4359 text {* Related results with closure as the conclusion. *}

  4360

  4361 lemma closed_scaling:

  4362   fixes s :: "'a::real_normed_vector set"

  4363   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  4364 proof(cases "s={}")

  4365   case True thus ?thesis by auto

  4366 next

  4367   case False

  4368   show ?thesis

  4369   proof(cases "c=0")

  4370     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  4371     case True thus ?thesis apply auto unfolding * by auto

  4372   next

  4373     case False

  4374     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  4375       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  4376           using as(1)[THEN spec[where x=n]]

  4377           using c\<noteq>0 by auto

  4378       }

  4379       moreover

  4380       { fix e::real assume "e>0"

  4381         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  4382         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  4383           using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto

  4384         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  4385           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  4386           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  4387       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto

  4388       ultimately have "l \<in> scaleR c  s"

  4389         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"]]

  4390         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  4391     thus ?thesis unfolding closed_sequential_limits by fast

  4392   qed

  4393 qed

  4394

  4395 lemma closed_negations:

  4396   fixes s :: "'a::real_normed_vector set"

  4397   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  4398   using closed_scaling[OF assms, of "- 1"] by simp

  4399

  4400 lemma compact_closed_sums:

  4401   fixes s :: "'a::real_normed_vector set"

  4402   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4403 proof-

  4404   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  4405   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  4406     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"

  4407       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  4408     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  4409       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  4410     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  4411       using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto

  4412     hence "l - l' \<in> t"

  4413       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  4414       using f(3) by auto

  4415     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

  4416   }

  4417   thus ?thesis unfolding closed_sequential_limits by fast

  4418 qed

  4419

  4420 lemma closed_compact_sums:

  4421   fixes s t :: "'a::real_normed_vector set"

  4422   assumes "closed s"  "compact t"

  4423   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4424 proof-

  4425   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

  4426     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  4427   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  4428 qed

  4429

  4430 lemma compact_closed_differences:

  4431   fixes s t :: "'a::real_normed_vector set"

  4432   assumes "compact s"  "closed t"

  4433   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4434 proof-

  4435   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  4436     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4437   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  4438 qed

  4439

  4440 lemma closed_compact_differences:

  4441   fixes s t :: "'a::real_normed_vector set"

  4442   assumes "closed s" "compact t"

  4443   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4444 proof-

  4445   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  4446     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4447  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  4448 qed

  4449

  4450 lemma closed_translation:

  4451   fixes a :: "'a::real_normed_vector"

  4452   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  4453 proof-

  4454   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  4455   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  4456 qed

  4457

  4458 lemma translation_Compl:

  4459   fixes a :: "'a::ab_group_add"

  4460   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  4461   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  4462

  4463 lemma translation_UNIV:

  4464   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  4465   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  4466

  4467 lemma translation_diff:

  4468   fixes a :: "'a::ab_group_add"

  4469   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  4470   by auto

  4471

  4472 lemma closure_translation:

  4473   fixes a :: "'a::real_normed_vector"

  4474   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  4475 proof-

  4476   have *:"op + a  (- s) = - op + a  s"

  4477     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  4478   show ?thesis unfolding closure_interior translation_Compl

  4479     using interior_translation[of a "- s"] unfolding * by auto

  4480 qed

  4481

  4482 lemma frontier_translation:

  4483   fixes a :: "'a::real_normed_vector"

  4484   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  4485   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  4486

  4487

  4488 subsection {* Separation between points and sets *}

  4489

  4490 lemma separate_point_closed:

  4491   fixes s :: "'a::heine_borel set"

  4492   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  4493 proof(cases "s = {}")

  4494   case True

  4495   thus ?thesis by(auto intro!: exI[where x=1])

  4496 next

  4497   case False

  4498   assume "closed s" "a \<notin> s"

  4499   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

  4500   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  4501 qed

  4502

  4503 lemma separate_compact_closed:

  4504   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  4505     (* TODO: does this generalize to heine_borel? *)

  4506   assumes "compact s" and "closed t" and "s \<inter> t = {}"

  4507   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  4508 proof-

  4509   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto

  4510   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"

  4511     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto

  4512   { fix x y assume "x\<in>s" "y\<in>t"

  4513     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto

  4514     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute

  4515       by (auto  simp add: dist_commute)

  4516     hence "d \<le> dist x y" unfolding dist_norm by auto  }

  4517   thus ?thesis using d>0 by auto

  4518 qed

  4519

  4520 lemma separate_closed_compact:

  4521   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  4522   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  4523   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  4524 proof-

  4525   have *:"t \<inter> s = {}" using assms(3) by auto

  4526   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  4527     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  4528     by (auto simp add: dist_commute)

  4529 qed

  4530

  4531

  4532 subsection {* Intervals *}

  4533

  4534 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  4535   "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and

  4536   "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"

  4537   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  4538

  4539 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  4540   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"

  4541   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"

  4542   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  4543

  4544 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  4545  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and

  4546  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)

  4547 proof-

  4548   { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"

  4549     hence "a $$i < x$$ i \<and> x $$i < b$$ i" unfolding mem_interval by auto

  4550     hence "a$$i < b$$i" by auto

  4551     hence False using as by auto  }

  4552   moreover

  4553   { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"

  4554     let ?x = "(1/2) *\<^sub>R (a + b)"

  4555     { fix i assume i:"i<DIM('a)"

  4556       have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto

  4557       hence "a$$i < ((1/2) *\<^sub>R (a+b))$$ i" "((1/2) *\<^sub>R (a+b)) $$i < b$$i"

  4558         unfolding euclidean_simps by auto }

  4559     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  4560   ultimately show ?th1 by blast

  4561

  4562   { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"

  4563     hence "a $$i \<le> x$$ i \<and> x $$i \<le> b$$ i" unfolding mem_interval by auto

  4564     hence "a$$i \<le> b$$i" by auto

  4565     hence False using as by auto  }

  4566   moreover

  4567   { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"

  4568     let ?x = "(1/2) *\<^sub>R (a + b)"

  4569     { fix i assume i:"i<DIM('a)"

  4570       have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto

  4571       hence "a$$i \<le> ((1/2) *\<^sub>R (a+b))$$ i" "((1/2) *\<^sub>R (a+b)) $$i \<le> b$$i"

  4572         unfolding euclidean_simps by auto }

  4573     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  4574   ultimately show ?th2 by blast

  4575 qed

  4576

  4577 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  4578   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and

  4579   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"

  4580   unfolding interval_eq_empty[of a b] by fastsimp+

  4581

  4582 lemma interval_sing:

  4583   fixes a :: "'a::ordered_euclidean_space"

  4584   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  4585   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  4586   by (auto simp add: euclidean_eq[where 'a='a] eq_commute

  4587     eucl_less[where 'a='a] eucl_le[where 'a='a])

  4588

  4589 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  4590  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and

  4591  "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  4592  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and

  4593  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"

  4594   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  4595   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  4596

  4597 lemma interval_open_subset_closed:

  4598   fixes a :: "'a::ordered_euclidean_space"

  4599   shows "{a<..<b} \<subseteq> {a .. b}"

  4600   unfolding subset_eq [unfolded Ball_def] mem_interval

  4601   by (fast intro: less_imp_le)

  4602

  4603 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  4604  "{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

  4605  "{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

  4606  "{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

  4607  "{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)

  4608 proof-

  4609   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  4610   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)

  4611   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"

  4612     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  4613     fix i assume i:"i<DIM('a)"

  4614     (** TODO combine the following two parts as done in the HOL_light version. **)

  4615     { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"   4616 assume as2: "a$$i > c$$i"   4617 { fix j assume j:"j<DIM('a)"   4618 hence "c$$ j < ?x $$j \<and> ?x$$ j < d $$j"   4619 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i   4620 by (auto simp add: as2) }   4621 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto   4622 moreover   4623 have "?x\<notin>{a .. b}"   4624 unfolding mem_interval apply auto apply(rule_tac x=i in exI)   4625 using as(2)[THEN spec[where x=i]] and as2 i   4626 by auto   4627 ultimately have False using as by auto }   4628 hence "a$$i \<le> c$$i" by(rule ccontr)auto   4629 moreover   4630 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"

  4631       assume as2: "b$$i < d$$i"

  4632       { fix j assume "j<DIM('a)"

  4633         hence "d $$j > ?x$$ j \<and> ?x $$j > c$$ j"

  4634           apply(cases "j=i") using as(2)[THEN spec[where x=j]]

  4635           by (auto simp add: as2)  }

  4636       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  4637       moreover

  4638       have "?x\<notin>{a .. b}"

  4639         unfolding mem_interval apply auto apply(rule_tac x=i in exI)

  4640         using as(2)[THEN spec[where x=i]] and as2 using i

  4641         by auto

  4642       ultimately have False using as by auto  }

  4643     hence "b$$i \<ge> d$$i" by(rule ccontr)auto

  4644     ultimately

  4645     have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto

  4646   } note part1 = this

  4647   show ?th3 unfolding subset_eq and Ball_def and mem_interval

  4648     apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval

  4649     prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+

  4650   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"

  4651     fix i assume i:"i<DIM('a)"

  4652     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  4653     hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto  } note * = this

  4654   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  4655     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  4656     apply auto by(erule_tac x=i in allE, simp)+

  4657 qed

  4658

  4659 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  4660   "{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

  4661   "{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

  4662   "{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

  4663   "{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)

  4664 proof-

  4665   let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"

  4666   note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False

  4667   show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)

  4668     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  4669   show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)

  4670     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  4671   show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)

  4672     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  4673   show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)

  4674     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  4675 qed

  4676

  4677 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  4678  "{a .. b} \<inter> {c .. d} =  {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"

  4679   unfolding set_eq_iff and Int_iff and mem_interval

  4680   by auto

  4681

  4682 (* Moved interval_open_subset_closed a bit upwards *)

  4683

  4684 lemma open_interval[intro]:

  4685   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  4686 proof-

  4687   have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i<..<b$$i})"   4688 by (intro open_INT finite_lessThan ballI continuous_open_vimage allI   4689 linear_continuous_at bounded_linear_euclidean_component   4690 open_real_greaterThanLessThan)   4691 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i<..<b$$i}) = {a<..<b}"

  4692     by (auto simp add: eucl_less [where 'a='a])

  4693   finally show "open {a<..<b}" .

  4694 qed

  4695

  4696 lemma closed_interval[intro]:

  4697   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  4698 proof-

  4699   have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i .. b$$i})"   4700 by (intro closed_INT ballI continuous_closed_vimage allI   4701 linear_continuous_at bounded_linear_euclidean_component   4702 closed_real_atLeastAtMost)   4703 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i .. b$$i}) = {a .. b}"

  4704     by (auto simp add: eucl_le [where 'a='a])

  4705   finally show "closed {a .. b}" .

  4706 qed

  4707

  4708 lemma interior_closed_interval [intro]:

  4709   fixes a b :: "'a::ordered_euclidean_space"

  4710   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  4711 proof(rule subset_antisym)

  4712   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  4713     by (rule interior_maximal)

  4714 next

  4715   { fix x assume "x \<in> interior {a..b}"

  4716     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  4717     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

  4718     { fix i assume i:"i<DIM('a)"

  4719       have "dist (x - (e / 2) *\<^sub>R basis i) x < e"

  4720            "dist (x + (e / 2) *\<^sub>R basis i) x < e"

  4721         unfolding dist_norm apply auto

  4722         unfolding norm_minus_cancel using norm_basis and e>0 by auto

  4723       hence "a $$i \<le> (x - (e / 2) *\<^sub>R basis i)$$ i"

  4724                      "(x + (e / 2) *\<^sub>R basis i) $$i \<le> b$$ i"

  4725         using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]

  4726         and   e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]

  4727         unfolding mem_interval using i by blast+

  4728       hence "a $$i < x$$ i" and "x $$i < b$$ i" unfolding euclidean_simps

  4729         unfolding basis_component using e>0 i by auto  }

  4730     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  4731   thus "?L \<subseteq> ?R" ..

  4732 qed

  4733

  4734 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  4735 proof-

  4736   let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"

  4737   { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$i \<le> x$$ i \<and> x $$i \<le> b$$ i"

  4738     { fix i assume "i<DIM('a)"

  4739       hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }   4740 hence "(\<Sum>i<DIM('a). \<bar>x$$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  4741     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  4742   thus ?thesis unfolding interval and bounded_iff by auto

  4743 qed

  4744

  4745 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  4746  "bounded {a .. b} \<and> bounded {a<..<b}"

  4747   using bounded_closed_interval[of a b]

  4748   using interval_open_subset_closed[of a b]

  4749   using bounded_subset[of "{a..b}" "{a<..<b}"]

  4750   by simp

  4751

  4752 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  4753  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  4754   using bounded_interval[of a b] by auto

  4755

  4756 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  4757   using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]

  4758   by auto

  4759

  4760 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  4761   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  4762 proof-

  4763   { fix i assume "i<DIM('a)"

  4764     hence "a $$i < ((1 / 2) *\<^sub>R (a + b))$$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$i < b$$ i"

  4765       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]

  4766       unfolding euclidean_simps by auto  }

  4767   thus ?thesis unfolding mem_interval by auto

  4768 qed

  4769

  4770 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  4771   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  4772   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  4773 proof-

  4774   { fix i assume i:"i<DIM('a)"

  4775     have "a $$i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp   4776 also have "\<dots> < e * x$$ i + (1 - e) * y $$i" apply(rule add_less_le_mono)   4777 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all   4778 using x unfolding mem_interval using i apply simp   4779 using y unfolding mem_interval using i apply simp   4780 done   4781 finally have "a$$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$i" unfolding euclidean_simps by auto   4782 moreover {   4783 have "b$$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp

  4784     also have "\<dots> > e * x $$i + (1 - e) * y$$ i" apply(rule add_less_le_mono)

  4785       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  4786       using x unfolding mem_interval using i apply simp

  4787       using y unfolding mem_interval using i apply simp

  4788       done

  4789     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$i < b$$ i" unfolding euclidean_simps by auto

  4790     } 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 }

  4791   thus ?thesis unfolding mem_interval by auto

  4792 qed

  4793

  4794 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"

  4795   assumes "{a<..<b} \<noteq> {}"

  4796   shows "closure {a<..<b} = {a .. b}"

  4797 proof-

  4798   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto

  4799   let ?c = "(1 / 2) *\<^sub>R (a + b)"

  4800   { fix x assume as:"x \<in> {a .. b}"

  4801     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"

  4802     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"

  4803       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto

  4804       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =

  4805         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"

  4806         by (auto simp add: algebra_simps)

  4807       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

  4808       hence False using fn unfolding f_def using xc by auto  }

  4809     moreover

  4810     { assume "\<not> (f ---> x) sequentially"

  4811       { fix e::real assume "e>0"

  4812         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

  4813         then obtain N::nat where "inverse (real (N + 1)) < e" by auto

  4814         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)

  4815         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }

  4816       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"

  4817         unfolding Lim_sequentially by(auto simp add: dist_norm)

  4818       hence "(f ---> x) sequentially" unfolding f_def

  4819         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]

  4820         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  }

  4821     ultimately have "x \<in> closure {a<..<b}"

  4822       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }

  4823   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

  4824 qed

  4825

  4826 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"

  4827   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"

  4828 proof-

  4829   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto

  4830   def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"

  4831   { fix x assume "x\<in>s"

  4832     fix i assume i:"i<DIM('a)"

  4833     hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF x\<in>s]

  4834       and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto  }

  4835   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

  4836 qed

  4837

  4838 lemma bounded_subset_open_interval:

  4839   fixes s :: "('a::ordered_euclidean_space) set"

  4840   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"

  4841   by (auto dest!: bounded_subset_open_interval_symmetric)

  4842

  4843 lemma bounded_subset_closed_interval_symmetric:

  4844   fixes s :: "('a::ordered_euclidean_space) set"

  4845   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"

  4846 proof-

  4847   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto

  4848   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto

  4849 qed

  4850

  4851 lemma bounded_subset_closed_interval:

  4852   fixes s :: "('a::ordered_euclidean_space) set"

  4853   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"

  4854   using bounded_subset_closed_interval_symmetric[of s] by auto

  4855

  4856 lemma frontier_closed_interval:

  4857   fixes a b :: "'a::ordered_euclidean_space"

  4858   shows "frontier {a .. b} = {a .. b} - {a<..<b}"

  4859   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

  4860

  4861 lemma frontier_open_interval:

  4862   fixes a b :: "'a::ordered_euclidean_space"

  4863   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

  4864 proof(cases "{a<..<b} = {}")

  4865   case True thus ?thesis using frontier_empty by auto

  4866 next

  4867   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto

  4868 qed

  4869

  4870 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"

  4871   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"

  4872   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

  4873

  4874

  4875 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)

  4876

  4877 lemma closed_interval_left: fixes b::"'a::euclidean_space"

  4878   shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"

  4879 proof-

  4880   { fix i assume i:"i<DIM('a)"

  4881     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"

  4882     { assume "x$$i > b$$i"

  4883       then obtain y where "y $$i \<le> b$$ i"  "y \<noteq> x"  "dist y x < x$$i - b$$i"

  4884         using x[THEN spec[where x="x$$i - b$$i"]] using i by auto

  4885       hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i

  4886         by auto   }

  4887     hence "x$$i \<le> b$$i" by(rule ccontr)auto  }

  4888   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  4889 qed

  4890

  4891 lemma closed_interval_right: fixes a::"'a::euclidean_space"

  4892   shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"

  4893 proof-

  4894   { fix i assume i:"i<DIM('a)"

  4895     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"

  4896     { assume "a$$i > x$$i"

  4897       then obtain y where "a $$i \<le> y$$ i"  "y \<noteq> x"  "dist y x < a$$i - x$$i"

  4898         using x[THEN spec[where x="a$$i - x$$i"]] i by auto

  4899       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto   }

  4900     hence "a$$i \<le> x$$i" by(rule ccontr)auto  }

  4901   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  4902 qed

  4903

  4904 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  4905

  4906 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  4907   (\<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)"

  4908

  4909 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

  4910   "is_interval {a<..<b}" (is ?th2) proof -

  4911   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

  4912     by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

  4913

  4914 lemma is_interval_empty:

  4915  "is_interval {}"

  4916   unfolding is_interval_def

  4917   by simp

  4918

  4919 lemma is_interval_univ:

  4920  "is_interval UNIV"

  4921   unfolding is_interval_def

  4922   by simp

  4923

  4924

  4925 subsection {* Closure of halfspaces and hyperplanes *}

  4926

  4927 lemma isCont_open_vimage:

  4928   assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"

  4929 proof -

  4930   from assms(1) have "continuous_on UNIV f"

  4931     unfolding isCont_def continuous_on_def within_UNIV by simp

  4932   hence "open {x \<in> UNIV. f x \<in> s}"

  4933     using open_UNIV open s by (rule continuous_open_preimage)

  4934   thus "open (f - s)"

  4935     by (simp add: vimage_def)

  4936 qed

  4937

  4938 lemma isCont_closed_vimage:

  4939   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"

  4940   using assms unfolding closed_def vimage_Compl [symmetric]

  4941   by (rule isCont_open_vimage)

  4942

  4943 lemma open_Collect_less:

  4944   fixes f g :: "'a::topological_space \<Rightarrow> real"

  4945   assumes f: "\<And>x. isCont f x"

  4946   assumes g: "\<And>x. isCont g x"

  4947   shows "open {x. f x < g x}"

  4948 proof -

  4949   have "open ((\<lambda>x. g x - f x) - {0<..})"

  4950     using isCont_diff [OF g f] open_real_greaterThan

  4951     by (rule isCont_open_vimage)

  4952   also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"

  4953     by auto

  4954   finally show ?thesis .

  4955 qed

  4956

  4957 lemma closed_Collect_le:

  4958   fixes f g :: "'a::topological_space \<Rightarrow> real"

  4959   assumes f: "\<And>x. isCont f x"

  4960   assumes g: "\<And>x. isCont g x"

  4961   shows "closed {x. f x \<le> g x}"

  4962 proof -

  4963   have "closed ((\<lambda>x. g x - f x) - {0..})"

  4964     using isCont_diff [OF g f] closed_real_atLeast

  4965     by (rule isCont_closed_vimage)

  4966   also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"

  4967     by auto

  4968   finally show ?thesis .

  4969 qed

  4970

  4971 lemma closed_Collect_eq:

  4972   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  4973   assumes f: "\<And>x. isCont f x"

  4974   assumes g: "\<And>x. isCont g x"

  4975   shows "closed {x. f x = g x}"

  4976 proof -

  4977   have "open {(x::'b, y::'b). x \<noteq> y}"

  4978     unfolding open_prod_def by (auto dest!: hausdorff)

  4979   hence "closed {(x::'b, y::'b). x = y}"

  4980     unfolding closed_def split_def Collect_neg_eq .

  4981   with isCont_Pair [OF f g]

  4982   have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"

  4983     by (rule isCont_closed_vimage)

  4984   also have "\<dots> = {x. f x = g x}" by auto

  4985   finally show ?thesis .

  4986 qed

  4987

  4988 lemma continuous_at_inner: "continuous (at x) (inner a)"

  4989   unfolding continuous_at by (intro tendsto_intros)

  4990

  4991 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$i)"   4992 unfolding euclidean_component_def by (rule continuous_at_inner)   4993   4994 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"   4995 by (simp add: closed_Collect_le)   4996   4997 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"   4998 by (simp add: closed_Collect_le)   4999   5000 lemma closed_hyperplane: "closed {x. inner a x = b}"   5001 by (simp add: closed_Collect_eq)   5002   5003 lemma closed_halfspace_component_le:   5004 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"

  5005   by (simp add: closed_Collect_le)

  5006

  5007 lemma closed_halfspace_component_ge:

  5008   shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"   5009 by (simp add: closed_Collect_le)   5010   5011 text {* Openness of halfspaces. *}   5012   5013 lemma open_halfspace_lt: "open {x. inner a x < b}"   5014 by (simp add: open_Collect_less)   5015   5016 lemma open_halfspace_gt: "open {x. inner a x > b}"   5017 by (simp add: open_Collect_less)   5018   5019 lemma open_halfspace_component_lt:   5020 shows "open {x::'a::euclidean_space. x$$i < a}"

  5021   by (simp add: open_Collect_less)

  5022

  5023 lemma open_halfspace_component_gt:

  5024   shows "open {x::'a::euclidean_space. x$$i > a}"   5025 by (simp add: open_Collect_less)   5026   5027 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}   5028   5029 lemma eucl_lessThan_eq_halfspaces:   5030 fixes a :: "'a\<Colon>ordered_euclidean_space"   5031 shows "{..<a} = (\<Inter>i<DIM('a). {x. x$$ i < a $$i})"   5032 by (auto simp: eucl_less[where 'a='a])   5033   5034 lemma eucl_greaterThan_eq_halfspaces:   5035 fixes a :: "'a\<Colon>ordered_euclidean_space"   5036 shows "{a<..} = (\<Inter>i<DIM('a). {x. a$$ i < x $$i})"   5037 by (auto simp: eucl_less[where 'a='a])   5038   5039 lemma eucl_atMost_eq_halfspaces:   5040 fixes a :: "'a\<Colon>ordered_euclidean_space"   5041 shows "{.. a} = (\<Inter>i<DIM('a). {x. x$$ i \<le> a $$i})"   5042 by (auto simp: eucl_le[where 'a='a])   5043   5044 lemma eucl_atLeast_eq_halfspaces:   5045 fixes a :: "'a\<Colon>ordered_euclidean_space"   5046 shows "{a ..} = (\<Inter>i<DIM('a). {x. a$$ i \<le> x $$i})"   5047 by (auto simp: eucl_le[where 'a='a])   5048   5049 lemma open_eucl_lessThan[simp, intro]:   5050 fixes a :: "'a\<Colon>ordered_euclidean_space"   5051 shows "open {..< a}"   5052 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)   5053   5054 lemma open_eucl_greaterThan[simp, intro]:   5055 fixes a :: "'a\<Colon>ordered_euclidean_space"   5056 shows "open {a <..}"   5057 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)   5058   5059 lemma closed_eucl_atMost[simp, intro]:   5060 fixes a :: "'a\<Colon>ordered_euclidean_space"   5061 shows "closed {.. a}"   5062 unfolding eucl_atMost_eq_halfspaces   5063 by (simp add: closed_INT closed_Collect_le)   5064   5065 lemma closed_eucl_atLeast[simp, intro]:   5066 fixes a :: "'a\<Colon>ordered_euclidean_space"   5067 shows "closed {a ..}"   5068 unfolding eucl_atLeast_eq_halfspaces   5069 by (simp add: closed_INT closed_Collect_le)   5070   5071 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x$$ i) - S)"

  5072   by (auto intro!: continuous_open_vimage)

  5073

  5074 text {* This gives a simple derivation of limit component bounds. *}

  5075

  5076 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5077   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$$i \<le> b) net"   5078 shows "l$$i \<le> b"

  5079 proof-

  5080   { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"   5081 unfolding euclidean_component_def by auto } note * = this   5082 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *   5083 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto   5084 qed   5085   5086 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"   5087 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"

  5088   shows "b \<le> l$$i"   5089 proof-   5090 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"

  5091       unfolding euclidean_component_def by auto  } note * = this

  5092   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *

  5093     using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto

  5094 qed

  5095

  5096 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5097   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"   5098 shows "l$$i = b"

  5099   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

  5100 text{* Limits relative to a union.                                               *}

  5101

  5102 lemma eventually_within_Un:

  5103   "eventually P (net within (s \<union> t)) \<longleftrightarrow>

  5104     eventually P (net within s) \<and> eventually P (net within t)"

  5105   unfolding Limits.eventually_within

  5106   by (auto elim!: eventually_rev_mp)

  5107

  5108 lemma Lim_within_union:

  5109  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>

  5110   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"

  5111   unfolding tendsto_def

  5112   by (auto simp add: eventually_within_Un)

  5113

  5114 lemma Lim_topological:

  5115  "(f ---> l) net \<longleftrightarrow>

  5116         trivial_limit net \<or>

  5117         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  5118   unfolding tendsto_def trivial_limit_eq by auto

  5119

  5120 lemma continuous_on_union:

  5121   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"

  5122   shows "continuous_on (s \<union> t) f"

  5123   using assms unfolding continuous_on Lim_within_union

  5124   unfolding Lim_topological trivial_limit_within closed_limpt by auto

  5125

  5126 lemma continuous_on_cases:

  5127   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"

  5128           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"

  5129   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

  5130 proof-

  5131   let ?h = "(\<lambda>x. if P x then f x else g x)"

  5132   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto

  5133   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto

  5134   moreover

  5135   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto

  5136   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto

  5137   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto

  5138 qed

  5139

  5140

  5141 text{* Some more convenient intermediate-value theorem formulations.             *}

  5142

  5143 lemma connected_ivt_hyperplane:

  5144   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"

  5145   shows "\<exists>z \<in> s. inner a z = b"

  5146 proof(rule ccontr)

  5147   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"

  5148   let ?A = "{x. inner a x < b}"

  5149   let ?B = "{x. inner a x > b}"

  5150   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto

  5151   moreover have "?A \<inter> ?B = {}" by auto

  5152   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto

  5153   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

  5154 qed

  5155

  5156 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows

  5157  "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)"   5158 using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]   5159 unfolding euclidean_component_def by auto   5160   5161   5162 subsection {* Homeomorphisms *}   5163   5164 definition "homeomorphism s t f g \<equiv>   5165 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>   5166 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"   5167   5168 definition   5169 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"   5170 (infixr "homeomorphic" 60) where   5171 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"   5172   5173 lemma homeomorphic_refl: "s homeomorphic s"   5174 unfolding homeomorphic_def   5175 unfolding homeomorphism_def   5176 using continuous_on_id   5177 apply(rule_tac x = "(\<lambda>x. x)" in exI)   5178 apply(rule_tac x = "(\<lambda>x. x)" in exI)   5179 by blast   5180   5181 lemma homeomorphic_sym:   5182 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"   5183 unfolding homeomorphic_def   5184 unfolding homeomorphism_def   5185 by blast   5186   5187 lemma homeomorphic_trans:   5188 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"   5189 proof-   5190 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"   5191 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto   5192 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"   5193 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto   5194   5195 { 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 }   5196 moreover have "(f2 \<circ> f1)  s = u" using fg1(2) fg2(2) by auto   5197 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto   5198 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 }   5199 moreover have "(g1 \<circ> g2)  u = s" using fg1(5) fg2(5) by auto   5200 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto   5201 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   5202 qed   5203   5204 lemma homeomorphic_minimal:   5205 "s homeomorphic t \<longleftrightarrow>   5206 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>   5207 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>   5208 continuous_on s f \<and> continuous_on t g)"   5209 unfolding homeomorphic_def homeomorphism_def   5210 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)   5211 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto   5212 unfolding image_iff   5213 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)   5214 apply auto apply(rule_tac x="g x" in bexI) apply auto   5215 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)   5216 apply auto apply(rule_tac x="f x" in bexI) by auto   5217   5218 text {* Relatively weak hypotheses if a set is compact. *}   5219   5220 lemma homeomorphism_compact:   5221 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"   5222 (* class constraint due to continuous_on_inverse *)   5223 assumes "compact s" "continuous_on s f" "f  s = t" "inj_on f s"   5224 shows "\<exists>g. homeomorphism s t f g"   5225 proof-   5226 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"   5227 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto   5228 { fix y assume "y\<in>t"   5229 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto   5230 hence "g (f x) = x" using g by auto   5231 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }   5232 hence g':"\<forall>x\<in>t. f (g x) = x" by auto   5233 moreover   5234 { fix x   5235 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"])   5236 moreover   5237 { assume "x\<in>g  t"   5238 then obtain y where y:"y\<in>t" "g y = x" by auto   5239 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto   5240 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 }   5241 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g  t" .. }   5242 hence "g  t = s" by auto   5243 ultimately   5244 show ?thesis unfolding homeomorphism_def homeomorphic_def   5245 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   5246 qed   5247   5248 lemma homeomorphic_compact:   5249 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"   5250 (* class constraint due to continuous_on_inverse *)   5251 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f  s = t) \<Longrightarrow> inj_on f s   5252 \<Longrightarrow> s homeomorphic t"   5253 unfolding homeomorphic_def by (metis homeomorphism_compact)   5254   5255 text{* Preservation of topological properties. *}   5256   5257 lemma homeomorphic_compactness:   5258 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"   5259 unfolding homeomorphic_def homeomorphism_def   5260 by (metis compact_continuous_image)   5261   5262 text{* Results on translation, scaling etc. *}   5263   5264 lemma homeomorphic_scaling:   5265 fixes s :: "'a::real_normed_vector set"   5266 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x)  s)"   5267 unfolding homeomorphic_minimal   5268 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)   5269 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)   5270 using assms by (auto simp add: continuous_on_intros)   5271   5272 lemma homeomorphic_translation:   5273 fixes s :: "'a::real_normed_vector set"   5274 shows "s homeomorphic ((\<lambda>x. a + x)  s)"   5275 unfolding homeomorphic_minimal   5276 apply(rule_tac x="\<lambda>x. a + x" in exI)   5277 apply(rule_tac x="\<lambda>x. -a + x" in exI)   5278 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto   5279   5280 lemma homeomorphic_affinity:   5281 fixes s :: "'a::real_normed_vector set"   5282 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x)  s)"   5283 proof-   5284 have *:"op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto   5285 show ?thesis   5286 using homeomorphic_trans   5287 using homeomorphic_scaling[OF assms, of s]   5288 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x)  s" a] unfolding * by auto   5289 qed   5290   5291 lemma homeomorphic_balls:   5292 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)   5293 assumes "0 < d" "0 < e"   5294 shows "(ball a d) homeomorphic (ball b e)" (is ?th)   5295 "(cball a d) homeomorphic (cball b e)" (is ?cth)   5296 proof-   5297 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto   5298 show ?th unfolding homeomorphic_minimal   5299 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)   5300 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)   5301 using assms apply (auto simp add: dist_commute)   5302 unfolding dist_norm   5303 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)   5304 unfolding continuous_on   5305 by (intro ballI tendsto_intros, simp)+   5306 next   5307 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto   5308 show ?cth unfolding homeomorphic_minimal   5309 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)   5310 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)   5311 using assms apply (auto simp add: dist_commute)   5312 unfolding dist_norm   5313 apply (auto simp add: pos_divide_le_eq)   5314 unfolding continuous_on   5315 by (intro ballI tendsto_intros, simp)+   5316 qed   5317   5318 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}   5319   5320 lemma cauchy_isometric:   5321 fixes x :: "nat \<Rightarrow> 'a::euclidean_space"   5322 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)"   5323 shows "Cauchy x"   5324 proof-   5325 interpret f: bounded_linear f by fact   5326 { fix d::real assume "d>0"   5327 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"   5328 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   5329 { fix n assume "n\<ge>N"   5330 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto   5331 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"   5332 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]   5333 using normf[THEN bspec[where x="x n - x N"]] by auto   5334 ultimately have "norm (x n - x N) < d" using e>0   5335 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }   5336 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }   5337 thus ?thesis unfolding cauchy and dist_norm by auto   5338 qed   5339   5340 lemma complete_isometric_image:   5341 fixes f :: "'a::euclidean_space => 'b::euclidean_space"   5342 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"   5343 shows "complete(f  s)"   5344 proof-   5345 { fix g assume as:"\<forall>n::nat. g n \<in> f  s" and cfg:"Cauchy g"   5346 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"   5347 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto   5348 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto   5349 hence "f \<circ> x = g" unfolding fun_eq_iff by auto   5350 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"   5351 using cs[unfolded complete_def, THEN spec[where x="x"]]   5352 using cauchy_isometric[OF 0<e s f normf] and cfg and x(1) by auto   5353 hence "\<exists>l\<in>f  s. (g ---> l) sequentially"   5354 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]   5355 unfolding f \<circ> x = g by auto }   5356 thus ?thesis unfolding complete_def by auto   5357 qed   5358   5359 lemma dist_0_norm:   5360 fixes x :: "'a::real_normed_vector"   5361 shows "dist 0 x = norm x"   5362 unfolding dist_norm by simp   5363   5364 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"   5365 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"   5366 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"   5367 proof(cases "s \<subseteq> {0::'a}")   5368 case True   5369 { fix x assume "x \<in> s"   5370 hence "x = 0" using True by auto   5371 hence "norm x \<le> norm (f x)" by auto }   5372 thus ?thesis by(auto intro!: exI[where x=1])   5373 next   5374 interpret f: bounded_linear f by fact   5375 case False   5376 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto   5377 from False have "s \<noteq> {}" by auto   5378 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"   5379 let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"   5380 let ?S'' = "{x::'a. norm x = norm a}"   5381   5382 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto   5383 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto   5384 moreover have "?S' = s \<inter> ?S''" by auto   5385 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto   5386 moreover have *:"f  ?S' = ?S" by auto   5387 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto   5388 hence "closed ?S" using compact_imp_closed by auto   5389 moreover have "?S \<noteq> {}" using a by auto   5390 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   5391 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   5392   5393 let ?e = "norm (f b) / norm b"   5394 have "norm b > 0" using ba and a and norm_ge_zero by auto   5395 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   5396 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)   5397 moreover   5398 { fix x assume "x\<in>s"   5399 hence "norm (f b) / norm b * norm x \<le> norm (f x)"   5400 proof(cases "x=0")   5401 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto   5402 next   5403 case False   5404 hence *:"0 < norm a / norm x" using a\<noteq>0 unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)   5405 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto   5406 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   5407 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"]]   5408 unfolding f.scaleR and ba using x\<noteq>0 a\<noteq>0   5409 by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)   5410 qed }   5411 ultimately   5412 show ?thesis by auto   5413 qed   5414   5415 lemma closed_injective_image_subspace:   5416 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"   5417 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"   5418 shows "closed(f  s)"   5419 proof-   5420 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   5421 show ?thesis using complete_isometric_image[OF e>0 assms(1,2) e] and assms(4)   5422 unfolding complete_eq_closed[THEN sym] by auto   5423 qed   5424   5425   5426 subsection {* Some properties of a canonical subspace *}   5427   5428 lemma subspace_substandard:   5429 "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"

  5430   unfolding subspace_def by auto

  5431

  5432 lemma closed_substandard:

  5433  "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")   5434 proof-   5435 let ?D = "{i. P i} \<inter> {..<DIM('a)}"   5436 have "closed (\<Inter>i\<in>?D. {x::'a. x$$i = 0})"

  5437     by (simp add: closed_INT closed_Collect_eq)

  5438   also have "(\<Inter>i\<in>?D. {x::'a. x$$i = 0}) = ?A"   5439 by auto   5440 finally show "closed ?A" .   5441 qed   5442   5443 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"   5444 shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")

  5445 proof-

  5446   let ?D = "{..<DIM('a)}"

  5447   let ?B = "(basis::nat => 'a)  d"

  5448   let ?bas = "basis::nat \<Rightarrow> 'a"

  5449   have "?B \<subseteq> ?A" by auto

  5450   moreover

  5451   { fix x::"'a" assume "x\<in>?A"

  5452     hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)

  5453     hence "x\<in> span ?B"

  5454     proof(induct d arbitrary: x)

  5455       case empty hence "x=0" apply(subst euclidean_eq) by auto

  5456       thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto

  5457     next

  5458       case (insert k F)

  5459       hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$i = 0" by auto   5460 have **:"F \<subseteq> insert k F" by auto   5461 def y \<equiv> "x - x$$k *\<^sub>R basis k"

  5462       have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto   5463 { fix i assume i':"i \<notin> F"   5464 hence "y$$ i = 0" unfolding y_def

  5465           using *[THEN spec[where x=i]] by auto }

  5466       hence "y \<in> span (basis  F)" using insert(3) by auto

  5467       hence "y \<in> span (basis  (insert k F))"

  5468         using span_mono[of "?bas  F" "?bas  (insert k F)"]

  5469         using image_mono[OF **, of basis] using assms by auto

  5470       moreover

  5471       have "basis k \<in> span (?bas  (insert k F))" by(rule span_superset, auto)

  5472       hence "x$$k *\<^sub>R basis k \<in> span (?bas  (insert k F))"   5473 using span_mul by auto   5474 ultimately   5475 have "y + x$$k *\<^sub>R basis k \<in> span (?bas  (insert k F))"

  5476         using span_add by auto

  5477       thus ?case using y by auto

  5478     qed

  5479   }

  5480   hence "?A \<subseteq> span ?B" by auto

  5481   moreover

  5482   { fix x assume "x \<in> ?B"

  5483     hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto  }

  5484   hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto

  5485   moreover

  5486   have "d \<subseteq> ?D" unfolding subset_eq using assms by auto

  5487   hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto

  5488   have "card ?B = card d" unfolding card_image[OF *] by auto

  5489   ultimately show ?thesis using dim_unique[of "basis  d" ?A] by auto

  5490 qed

  5491

  5492 text{* Hence closure and completeness of all subspaces.                          *}

  5493

  5494 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"

  5495 apply (induct n)

  5496 apply (rule_tac x="{}" in exI, simp)

  5497 apply clarsimp

  5498 apply (subgoal_tac "\<exists>x. x \<notin> A")

  5499 apply (erule exE)

  5500 apply (rule_tac x="insert x A" in exI, simp)

  5501 apply (subgoal_tac "A \<noteq> UNIV", auto)

  5502 done

  5503

  5504 lemma closed_subspace: fixes s::"('a::euclidean_space) set"

  5505   assumes "subspace s" shows "closed s"

  5506 proof-

  5507   have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto

  5508   def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto

  5509   let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"   5510 have "\<exists>f. linear f \<and> f  {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$ i = 0} = s \<and>

  5511       inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x  i = 0}"

  5512     apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])

  5513     using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto

  5514   then guess f apply-by(erule exE conjE)+ note f = this

  5515   interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto

  5516   have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]

  5517     by(erule_tac x=0 in ballE) auto

  5518   moreover have "closed ?t" using closed_substandard .

  5519   moreover have "subspace ?t" using subspace_substandard .

  5520   ultimately show ?thesis using closed_injective_image_subspace[of ?t f]

  5521     unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto

  5522 qed

  5523

  5524 lemma complete_subspace:

  5525   fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"

  5526   using complete_eq_closed closed_subspace

  5527   by auto

  5528

  5529 lemma dim_closure:

  5530   fixes s :: "('a::euclidean_space) set"

  5531   shows "dim(closure s) = dim s" (is "?dc = ?d")

  5532 proof-

  5533   have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]

  5534     using closed_subspace[OF subspace_span, of s]

  5535     using dim_subset[of "closure s" "span s"] unfolding dim_span by auto

  5536   thus ?thesis using dim_subset[OF closure_subset, of s] by auto

  5537 qed

  5538

  5539

  5540 subsection {* Affine transformations of intervals *}

  5541

  5542 lemma real_affinity_le:

  5543  "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"

  5544   by (simp add: field_simps inverse_eq_divide)

  5545

  5546 lemma real_le_affinity:

  5547  "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"

  5548   by (simp add: field_simps inverse_eq_divide)

  5549

  5550 lemma real_affinity_lt:

  5551  "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"

  5552   by (simp add: field_simps inverse_eq_divide)

  5553

  5554 lemma real_lt_affinity:

  5555  "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"

  5556   by (simp add: field_simps inverse_eq_divide)

  5557

  5558 lemma real_affinity_eq:

  5559  "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"

  5560   by (simp add: field_simps inverse_eq_divide)

  5561

  5562 lemma real_eq_affinity:

  5563  "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c  \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"

  5564   by (simp add: field_simps inverse_eq_divide)

  5565

  5566 lemma image_affinity_interval: fixes m::real

  5567   fixes a b c :: "'a::ordered_euclidean_space"

  5568   shows "(\<lambda>x. m *\<^sub>R x + c)  {a .. b} =

  5569             (if {a .. b} = {} then {}

  5570             else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}

  5571             else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"

  5572 proof(cases "m=0")

  5573   { fix x assume "x \<le> c" "c \<le> x"

  5574     hence "x=c" unfolding eucl_le[where 'a='a] apply-

  5575       apply(subst euclidean_eq) by (auto intro: order_antisym) }

  5576   moreover case True

  5577   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])

  5578   ultimately show ?thesis by auto

  5579 next

  5580   case False

  5581   { fix y assume "a \<le> y" "y \<le> b" "m > 0"

  5582     hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"

  5583       unfolding eucl_le[where 'a='a] by auto

  5584   } moreover

  5585   { fix y assume "a \<le> y" "y \<le> b" "m < 0"

  5586     hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"

  5587       unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg)

  5588   } moreover

  5589   { fix y assume "m > 0"  "m *\<^sub>R a + c \<le> y"  "y \<le> m *\<^sub>R b + c"

  5590     hence "y \<in> (\<lambda>x. m *\<^sub>R x + c)  {a..b}"

  5591       unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]

  5592       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])

  5593       by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff)

  5594   } moreover

  5595   { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"

  5596     hence "y \<in> (\<lambda>x. m *\<^sub>R x + c)  {a..b}"

  5597       unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]

  5598       apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])

  5599       by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff)

  5600   }

  5601   ultimately show ?thesis using False by auto

  5602 qed

  5603

  5604 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space))  {a..b} =

  5605   (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"

  5606   using image_affinity_interval[of m 0 a b] by auto

  5607

  5608

  5609 subsection {* Banach fixed point theorem (not really topological...) *}

  5610

  5611 lemma banach_fix:

  5612   assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f  s) \<subseteq> s" and

  5613           lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"

  5614   shows "\<exists>! x\<in>s. (f x = x)"

  5615 proof-

  5616   have "1 - c > 0" using c by auto

  5617

  5618   from s(2) obtain z0 where "z0 \<in> s" by auto

  5619   def z \<equiv> "\<lambda>n. (f ^^ n) z0"

  5620   { fix n::nat

  5621     have "z n \<in> s" unfolding z_def

  5622     proof(induct n) case 0 thus ?case using z0 \<in>s by auto

  5623     next case Suc thus ?case using f by auto qed }

  5624   note z_in_s = this

  5625

  5626   def d \<equiv> "dist (z 0) (z 1)"

  5627

  5628   have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto

  5629   { fix n::nat

  5630     have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"

  5631     proof(induct n)

  5632       case 0 thus ?case unfolding d_def by auto

  5633     next

  5634       case (Suc m)

  5635       hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"

  5636         using 0 \<le> c using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto

  5637       thus ?case using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]

  5638         unfolding fzn and mult_le_cancel_left by auto

  5639     qed

  5640   } note cf_z = this

  5641

  5642   { fix n m::nat

  5643     have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"

  5644     proof(induct n)

  5645       case 0 show ?case by auto

  5646     next

  5647       case (Suc k)

  5648       have "(1 - c) * dist (z m) (z (m + Suc k)) \<le> (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"

  5649         using dist_triangle and c by(auto simp add: dist_triangle)

  5650       also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"

  5651         using cf_z[of "m + k"] and c by auto

  5652       also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"

  5653         using Suc by (auto simp add: field_simps)

  5654       also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"

  5655         unfolding power_add by (auto simp add: field_simps)

  5656       also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"

  5657         using c by (auto simp add: field_simps)

  5658       finally show ?case by auto

  5659     qed

  5660   } note cf_z2 = this

  5661   { fix e::real assume "e>0"

  5662     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"

  5663     proof(cases "d = 0")

  5664       case True

  5665       have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using 1 - c > 0

  5666         by (metis mult_zero_left real_mult_commute real_mult_le_cancel_iff1)

  5667       from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def

  5668         by (simp add: *)

  5669       thus ?thesis using e>0 by auto

  5670     next

  5671       case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]

  5672         by (metis False d_def less_le)

  5673       hence "0 < e * (1 - c) / d" using e>0 and 1-c>0

  5674         using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto

  5675       then obtain N where N:"c ^ N < e * (1 - c) / d" using real_arch_pow_inv[of "e * (1 - c) / d" c] and c by auto

  5676       { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"

  5677         have *:"c ^ n \<le> c ^ N" using n\<ge>N and c using power_decreasing[OF n\<ge>N, of c] by auto

  5678         have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using m>n by auto

  5679         hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"

  5680           using mult_pos_pos[OF d>0, of "1 - c ^ (m - n)"]

  5681           using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]

  5682           using 0 < 1 - c by auto

  5683

  5684         have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"

  5685           using cf_z2[of n "m - n"] and m>n unfolding pos_le_divide_eq[OF 1-c>0]

  5686           by (auto simp add: mult_commute dist_commute)

  5687         also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"

  5688           using mult_right_mono[OF * order_less_imp_le[OF **]]

  5689           unfolding mult_assoc by auto

  5690         also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"

  5691           using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto

  5692         also have "\<dots> = e * (1 - c ^ (m - n))" using c and d>0 and 1 - c > 0 by auto

  5693         also have "\<dots> \<le> e" using c and 1 - c ^ (m - n) > 0 and e>0 using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto

  5694         finally have  "dist (z m) (z n) < e" by auto

  5695       } note * = this

  5696       { fix m n::nat assume as:"N\<le>m" "N\<le>n"

  5697         hence "dist (z n) (z m) < e"

  5698         proof(cases "n = m")

  5699           case True thus ?thesis using e>0 by auto

  5700         next

  5701           case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)

  5702         qed }

  5703       thus ?thesis by auto

  5704     qed

  5705   }

  5706   hence "Cauchy z" unfolding cauchy_def by auto

  5707   then obtain x where "x\<in>s" and x:"(z ---> x) sequentially" using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto

  5708

  5709   def e \<equiv> "dist (f x) x"

  5710   have "e = 0" proof(rule ccontr)

  5711     assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]

  5712       by (metis dist_eq_0_iff dist_nz e_def)

  5713     then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"

  5714       using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto

  5715     hence N':"dist (z N) x < e / 2" by auto

  5716

  5717     have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2

  5718       using zero_le_dist[of "z N" x] and c

  5719       by (metis dist_eq_0_iff dist_nz order_less_asym less_le)

  5720     have "dist (f (z N)) (f x) \<le> c * dist (z N) x" using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]

  5721       using z_in_s[of N] x\<in>s using c by auto

  5722     also have "\<dots> < e / 2" using N' and c using * by auto

  5723     finally show False unfolding fzn

  5724       using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]

  5725       unfolding e_def by auto

  5726   qed

  5727   hence "f x = x" unfolding e_def by auto

  5728   moreover

  5729   { fix y assume "f y = y" "y\<in>s"

  5730     hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]

  5731       using x\<in>s and f x = x by auto

  5732     hence "dist x y = 0" unfolding mult_le_cancel_right1

  5733       using c and zero_le_dist[of x y] by auto

  5734     hence "y = x" by auto

  5735   }

  5736   ultimately show ?thesis using x\<in>s by blast+

  5737 qed