src/HOL/Library/Topology_Euclidean_Space.thy
author paulson
Tue Oct 27 12:59:57 2009 +0000 (2009-10-27)
changeset 33269 3b7e2dbbd684
parent 32960 69916a850301
permissions -rw-r--r--
New theory SupInf of the supremum and infimum operators for sets of reals.
     1 (*  Title:      HOL/Library/Topology_Euclidian_Space.thy
     2     Author:     Amine Chaieb, University of Cambridge
     3     Author:     Robert Himmelmann, TU Muenchen
     4 *)
     5 
     6 header {* Elementary topology in Euclidean space. *}
     7 
     8 theory Topology_Euclidean_Space
     9 imports SEQ Euclidean_Space Product_Vector
    10 begin
    11 
    12 declare fstcart_pastecart[simp] sndcart_pastecart[simp]
    13 
    14 subsection{* General notion of a topology *}
    15 
    16 definition "istopology L \<longleftrightarrow> {} \<in> L \<and> (\<forall>S \<in>L. \<forall>T \<in>L. S \<inter> T \<in> L) \<and> (\<forall>K. K \<subseteq>L \<longrightarrow> \<Union> K \<in> L)"
    17 typedef (open) 'a topology = "{L::('a set) set. istopology L}"
    18   morphisms "openin" "topology"
    19   unfolding istopology_def by blast
    20 
    21 lemma istopology_open_in[intro]: "istopology(openin U)"
    22   using openin[of U] by blast
    23 
    24 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
    25   using topology_inverse[unfolded mem_def Collect_def] .
    26 
    27 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
    28   using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
    29 
    30 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
    31 proof-
    32   {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
    33   moreover
    34   {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
    35     hence "openin T1 = openin T2" by (metis mem_def set_ext)
    36     hence "topology (openin T1) = topology (openin T2)" by simp
    37     hence "T1 = T2" unfolding openin_inverse .}
    38   ultimately show ?thesis by blast
    39 qed
    40 
    41 text{* Infer the "universe" from union of all sets in the topology. *}
    42 
    43 definition "topspace T =  \<Union>{S. openin T S}"
    44 
    45 subsection{* Main properties of open sets *}
    46 
    47 lemma openin_clauses:
    48   fixes U :: "'a topology"
    49   shows "openin U {}"
    50   "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
    51   "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
    52   using openin[of U] unfolding istopology_def Collect_def mem_def
    53   by (metis mem_def subset_eq)+
    54 
    55 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
    56   unfolding topspace_def by blast
    57 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
    58 
    59 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
    60   by (simp add: openin_clauses)
    61 
    62 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
    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 have ?rhs by auto }
    72   moreover
    73   {assume H: ?rhs
    74     then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"
    75       unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast
    76     from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
    77     have "\<Union> t`S = S" using t by auto
    78     with openin_Union[OF th0] have "openin U S" by simp }
    79   ultimately show ?thesis by blast
    80 qed
    81 
    82 subsection{* Closed sets *}
    83 
    84 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
    85 
    86 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
    87 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
    88 lemma closedin_topspace[intro,simp]:
    89   "closedin U (topspace U)" by (simp add: closedin_def)
    90 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
    91   by (auto simp add: Diff_Un closedin_def)
    92 
    93 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
    94 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
    95   shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
    96 
    97 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
    98   using closedin_Inter[of "{S,T}" U] by auto
    99 
   100 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
   101 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
   102   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
   103   apply (metis openin_subset subset_eq)
   104   done
   105 
   106 lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
   107   by (simp add: openin_closedin_eq)
   108 
   109 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
   110 proof-
   111   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
   112     by (auto simp add: topspace_def openin_subset)
   113   then show ?thesis using oS cT by (auto simp add: closedin_def)
   114 qed
   115 
   116 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
   117 proof-
   118   have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
   119     by (auto simp add: topspace_def )
   120   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
   121 qed
   122 
   123 subsection{* Subspace topology. *}
   124 
   125 definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
   126 
   127 lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
   128 proof-
   129   have "{} \<in> ?L" by blast
   130   {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
   131     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
   132     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
   133     then have "A \<inter> B \<in> ?L" by blast}
   134   moreover
   135   {fix K assume K: "K \<subseteq> ?L"
   136     have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
   137       apply (rule set_ext)
   138       apply (simp add: Ball_def image_iff)
   139       by (metis mem_def)
   140     from K[unfolded th0 subset_image_iff]
   141     obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
   142     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
   143     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
   144     ultimately have "\<Union>K \<in> ?L" by blast}
   145   ultimately show ?thesis unfolding 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 simp add: Collect_def)
   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 
   188 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
   189   by (simp add: subtopology_superset)
   190 
   191 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
   192   by (simp add: subtopology_superset)
   193 
   194 subsection{* The universal Euclidean versions are what we use most of the time *}
   195 
   196 definition
   197   euclidean :: "'a::topological_space topology" where
   198   "euclidean = topology open"
   199 
   200 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
   201   unfolding euclidean_def
   202   apply (rule cong[where x=S and y=S])
   203   apply (rule topology_inverse[symmetric])
   204   apply (auto simp add: istopology_def)
   205   by (auto simp add: mem_def subset_eq)
   206 
   207 lemma topspace_euclidean: "topspace euclidean = UNIV"
   208   apply (simp add: topspace_def)
   209   apply (rule set_ext)
   210   by (auto simp add: open_openin[symmetric])
   211 
   212 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
   213   by (simp add: topspace_euclidean topspace_subtopology)
   214 
   215 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
   216   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
   217 
   218 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
   219   by (simp add: open_openin openin_subopen[symmetric])
   220 
   221 subsection{* Open and closed balls. *}
   222 
   223 definition
   224   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   225   "ball x e = {y. dist x y < e}"
   226 
   227 definition
   228   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   229   "cball x e = {y. dist x y \<le> e}"
   230 
   231 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
   232 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
   233 
   234 lemma mem_ball_0 [simp]:
   235   fixes x :: "'a::real_normed_vector"
   236   shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
   237   by (simp add: dist_norm)
   238 
   239 lemma mem_cball_0 [simp]:
   240   fixes x :: "'a::real_normed_vector"
   241   shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
   242   by (simp add: dist_norm)
   243 
   244 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
   245 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
   246 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
   247 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
   248 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
   249   by (simp add: expand_set_eq) arith
   250 
   251 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
   252   by (simp add: expand_set_eq)
   253 
   254 subsection{* Topological properties of open balls *}
   255 
   256 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
   257   "(a::real) - b < 0 \<longleftrightarrow> a < b"
   258   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
   259 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
   260   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
   261 
   262 lemma open_ball[intro, simp]: "open (ball x e)"
   263   unfolding open_dist ball_def Collect_def Ball_def mem_def
   264   unfolding dist_commute
   265   apply clarify
   266   apply (rule_tac x="e - dist xa x" in exI)
   267   using dist_triangle_alt[where z=x]
   268   apply (clarsimp simp add: diff_less_iff)
   269   apply atomize
   270   apply (erule_tac x="y" in allE)
   271   apply (erule_tac x="xa" in allE)
   272   by arith
   273 
   274 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
   275 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
   276   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
   277 
   278 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
   279   by (metis open_contains_ball subset_eq centre_in_ball)
   280 
   281 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
   282   unfolding mem_ball expand_set_eq
   283   apply (simp add: not_less)
   284   by (metis zero_le_dist order_trans dist_self)
   285 
   286 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
   287 
   288 subsection{* Basic "localization" results are handy for connectedness. *}
   289 
   290 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
   291   by (auto simp add: openin_subtopology open_openin[symmetric])
   292 
   293 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
   294   by (auto simp add: openin_open)
   295 
   296 lemma open_openin_trans[trans]:
   297  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
   298   by (metis Int_absorb1  openin_open_Int)
   299 
   300 lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
   301   by (auto simp add: openin_open)
   302 
   303 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
   304   by (simp add: closedin_subtopology closed_closedin Int_ac)
   305 
   306 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
   307   by (metis closedin_closed)
   308 
   309 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
   310   apply (subgoal_tac "S \<inter> T = T" )
   311   apply auto
   312   apply (frule closedin_closed_Int[of T S])
   313   by simp
   314 
   315 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
   316   by (auto simp add: closedin_closed)
   317 
   318 lemma openin_euclidean_subtopology_iff:
   319   fixes S U :: "'a::metric_space set"
   320   shows "openin (subtopology euclidean U) S
   321   \<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")
   322 proof-
   323   {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
   324       by (simp add: open_dist) blast}
   325   moreover
   326   {assume SU: "S \<subseteq> U" and H: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S"
   327     from H obtain d where d: "\<And>x . x\<in> S \<Longrightarrow> d x > 0 \<and> (\<forall>x' \<in> U. dist x' x < d x \<longrightarrow> x' \<in> S)"
   328       by metis
   329     let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
   330     have oT: "open ?T" by auto
   331     { fix x assume "x\<in>S"
   332       hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
   333         apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
   334         by (rule d [THEN conjunct1])
   335       hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }
   336     moreover
   337     { fix y assume "y\<in>?T"
   338       then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
   339       then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
   340       assume "y\<in>U"
   341       hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
   342     ultimately have "S = ?T \<inter> U" by blast
   343     with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
   344   ultimately show ?thesis by blast
   345 qed
   346 
   347 text{* These "transitivity" results are handy too. *}
   348 
   349 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
   350   \<Longrightarrow> openin (subtopology euclidean U) S"
   351   unfolding open_openin openin_open by blast
   352 
   353 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
   354   by (auto simp add: openin_open intro: openin_trans)
   355 
   356 lemma closedin_trans[trans]:
   357  "closedin (subtopology euclidean T) S \<Longrightarrow>
   358            closedin (subtopology euclidean U) T
   359            ==> closedin (subtopology euclidean U) S"
   360   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
   361 
   362 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
   363   by (auto simp add: closedin_closed intro: closedin_trans)
   364 
   365 subsection{* Connectedness *}
   366 
   367 definition "connected S \<longleftrightarrow>
   368   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
   369   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
   370 
   371 lemma connected_local:
   372  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
   373                  openin (subtopology euclidean S) e1 \<and>
   374                  openin (subtopology euclidean S) e2 \<and>
   375                  S \<subseteq> e1 \<union> e2 \<and>
   376                  e1 \<inter> e2 = {} \<and>
   377                  ~(e1 = {}) \<and>
   378                  ~(e2 = {}))"
   379 unfolding connected_def openin_open by (safe, blast+)
   380 
   381 lemma exists_diff: "(\<exists>S. P(UNIV - S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
   382 proof-
   383 
   384   {assume "?lhs" hence ?rhs by blast }
   385   moreover
   386   {fix S assume H: "P S"
   387     have "S = UNIV - (UNIV - S)" by auto
   388     with H have "P (UNIV - (UNIV - S))" by metis }
   389   ultimately show ?thesis by metis
   390 qed
   391 
   392 lemma connected_clopen: "connected S \<longleftrightarrow>
   393         (\<forall>T. openin (subtopology euclidean S) T \<and>
   394             closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
   395 proof-
   396   have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (UNIV - e2) \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
   397     unfolding connected_def openin_open closedin_closed
   398     apply (subst exists_diff) by blast
   399   hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
   400     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def Compl_eq_Diff_UNIV) by metis
   401 
   402   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'))"
   403     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
   404     unfolding connected_def openin_open closedin_closed by auto
   405   {fix e2
   406     {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)"
   407         by auto}
   408     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
   409   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
   410   then show ?thesis unfolding th0 th1 by simp
   411 qed
   412 
   413 lemma connected_empty[simp, intro]: "connected {}"
   414   by (simp add: connected_def)
   415 
   416 subsection{* Hausdorff and other separation properties *}
   417 
   418 class t0_space =
   419   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
   420 
   421 class t1_space =
   422   assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<notin> U \<and> x \<notin> V \<and> y \<in> V"
   423 begin
   424 
   425 subclass t0_space
   426 proof
   427 qed (fast dest: t1_space)
   428 
   429 end
   430 
   431 text {* T2 spaces are also known as Hausdorff spaces. *}
   432 
   433 class t2_space =
   434   assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
   435 begin
   436 
   437 subclass t1_space
   438 proof
   439 qed (fast dest: hausdorff)
   440 
   441 end
   442 
   443 instance metric_space \<subseteq> t2_space
   444 proof
   445   fix x y :: "'a::metric_space"
   446   assume xy: "x \<noteq> y"
   447   let ?U = "ball x (dist x y / 2)"
   448   let ?V = "ball y (dist x y / 2)"
   449   have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
   450                ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
   451   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
   452     using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
   453     by (auto simp add: expand_set_eq)
   454   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
   455     by blast
   456 qed
   457 
   458 lemma separation_t2:
   459   fixes x y :: "'a::t2_space"
   460   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
   461   using hausdorff[of x y] by blast
   462 
   463 lemma separation_t1:
   464   fixes x y :: "'a::t1_space"
   465   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in>U \<and> y\<notin> U \<and> x\<notin>V \<and> y\<in>V)"
   466   using t1_space[of x y] by blast
   467 
   468 lemma separation_t0:
   469   fixes x y :: "'a::t0_space"
   470   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
   471   using t0_space[of x y] by blast
   472 
   473 subsection{* Limit points *}
   474 
   475 definition
   476   islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
   477     (infixr "islimpt" 60) where
   478   "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
   479 
   480 lemma islimptI:
   481   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
   482   shows "x islimpt S"
   483   using assms unfolding islimpt_def by auto
   484 
   485 lemma islimptE:
   486   assumes "x islimpt S" and "x \<in> T" and "open T"
   487   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
   488   using assms unfolding islimpt_def by auto
   489 
   490 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
   491 
   492 lemma islimpt_approachable:
   493   fixes x :: "'a::metric_space"
   494   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
   495   unfolding islimpt_def
   496   apply auto
   497   apply(erule_tac x="ball x e" in allE)
   498   apply auto
   499   apply(rule_tac x=y in bexI)
   500   apply (auto simp add: dist_commute)
   501   apply (simp add: open_dist, drule (1) bspec)
   502   apply (clarify, drule spec, drule (1) mp, auto)
   503   done
   504 
   505 lemma islimpt_approachable_le:
   506   fixes x :: "'a::metric_space"
   507   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
   508   unfolding islimpt_approachable
   509   using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
   510   by metis (* FIXME: VERY slow! *)
   511 
   512 class perfect_space =
   513   (* FIXME: perfect_space should inherit from topological_space *)
   514   assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"
   515 
   516 lemma perfect_choose_dist:
   517   fixes x :: "'a::perfect_space"
   518   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
   519 using islimpt_UNIV [of x]
   520 by (simp add: islimpt_approachable)
   521 
   522 instance real :: perfect_space
   523 apply default
   524 apply (rule islimpt_approachable [THEN iffD2])
   525 apply (clarify, rule_tac x="x + e/2" in bexI)
   526 apply (auto simp add: dist_norm)
   527 done
   528 
   529 instance "^" :: (perfect_space, finite) perfect_space
   530 proof
   531   fix x :: "'a ^ 'b"
   532   {
   533     fix e :: real assume "0 < e"
   534     def a \<equiv> "x $ undefined"
   535     have "a islimpt UNIV" by (rule islimpt_UNIV)
   536     with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
   537       unfolding islimpt_approachable by auto
   538     def y \<equiv> "Cart_lambda ((Cart_nth x)(undefined := b))"
   539     from `b \<noteq> a` have "y \<noteq> x"
   540       unfolding a_def y_def by (simp add: Cart_eq)
   541     from `dist b a < e` have "dist y x < e"
   542       unfolding dist_vector_def a_def y_def
   543       apply simp
   544       apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
   545       apply (subst setsum_diff1' [where a=undefined], simp, simp, simp)
   546       done
   547     from `y \<noteq> x` and `dist y x < e`
   548     have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
   549   }
   550   then show "x islimpt UNIV" unfolding islimpt_approachable by blast
   551 qed
   552 
   553 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
   554   unfolding closed_def
   555   apply (subst open_subopen)
   556   apply (simp add: islimpt_def subset_eq Compl_eq_Diff_UNIV)
   557   by (metis DiffE DiffI UNIV_I insertCI insert_absorb mem_def)
   558 
   559 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
   560   unfolding islimpt_def by auto
   561 
   562 lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x$i}"
   563 proof-
   564   let ?U = "UNIV :: 'n set"
   565   let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
   566   {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
   567     and xi: "x$i < 0"
   568     from xi have th0: "-x$i > 0" by arith
   569     from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
   570       have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
   571       have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
   572       have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
   573         apply (simp only: vector_component)
   574         by (rule th') auto
   575       have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[of "x'-x" i]
   576         apply (simp add: dist_norm) by norm
   577       from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
   578   then show ?thesis unfolding closed_limpt islimpt_approachable
   579     unfolding not_le[symmetric] by blast
   580 qed
   581 
   582 lemma finite_set_avoid:
   583   fixes a :: "'a::metric_space"
   584   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
   585 proof(induct rule: finite_induct[OF fS])
   586   case 1 thus ?case apply auto by ferrack
   587 next
   588   case (2 x F)
   589   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
   590   {assume "x = a" hence ?case using d by auto  }
   591   moreover
   592   {assume xa: "x\<noteq>a"
   593     let ?d = "min d (dist a x)"
   594     have dp: "?d > 0" using xa d(1) using dist_nz by auto
   595     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
   596     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
   597   ultimately show ?case by blast
   598 qed
   599 
   600 lemma islimpt_finite:
   601   fixes S :: "'a::metric_space set"
   602   assumes fS: "finite S" shows "\<not> a islimpt S"
   603   unfolding islimpt_approachable
   604   using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
   605 
   606 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
   607   apply (rule iffI)
   608   defer
   609   apply (metis Un_upper1 Un_upper2 islimpt_subset)
   610   unfolding islimpt_def
   611   apply (rule ccontr, clarsimp, rename_tac A B)
   612   apply (drule_tac x="A \<inter> B" in spec)
   613   apply (auto simp add: open_Int)
   614   done
   615 
   616 lemma discrete_imp_closed:
   617   fixes S :: "'a::metric_space set"
   618   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
   619   shows "closed S"
   620 proof-
   621   {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
   622     from e have e2: "e/2 > 0" by arith
   623     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
   624     let ?m = "min (e/2) (dist x y) "
   625     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
   626     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
   627     have th: "dist z y < e" using z y
   628       by (intro dist_triangle_lt [where z=x], simp)
   629     from d[rule_format, OF y(1) z(1) th] y z
   630     have False by (auto simp add: dist_commute)}
   631   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
   632 qed
   633 
   634 subsection{* Interior of a Set *}
   635 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
   636 
   637 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
   638   apply (simp add: expand_set_eq interior_def)
   639   apply (subst (2) open_subopen) by (safe, blast+)
   640 
   641 lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
   642 
   643 lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
   644 
   645 lemma open_interior[simp, intro]: "open(interior S)"
   646   apply (simp add: interior_def)
   647   apply (subst open_subopen) by blast
   648 
   649 lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
   650 lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
   651 lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
   652 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
   653 lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T  \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"
   654   by (metis equalityI interior_maximal interior_subset open_interior)
   655 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
   656   apply (simp add: interior_def)
   657   by (metis open_contains_ball centre_in_ball open_ball subset_trans)
   658 
   659 lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
   660   by (metis interior_maximal interior_subset subset_trans)
   661 
   662 lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
   663   apply (rule equalityI, simp)
   664   apply (metis Int_lower1 Int_lower2 subset_interior)
   665   by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
   666 
   667 lemma interior_limit_point [intro]:
   668   fixes x :: "'a::perfect_space"
   669   assumes x: "x \<in> interior S" shows "x islimpt S"
   670 proof-
   671   from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
   672     unfolding mem_interior subset_eq Ball_def mem_ball by blast
   673   {
   674     fix d::real assume d: "d>0"
   675     let ?m = "min d e"
   676     have mde2: "0 < ?m" using e(1) d(1) by simp
   677     from perfect_choose_dist [OF mde2, of x]
   678     obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
   679     then have "dist y x < e" "dist y x < d" by simp_all
   680     from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
   681     have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
   682       using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
   683   }
   684   then show ?thesis unfolding islimpt_approachable by blast
   685 qed
   686 
   687 lemma interior_closed_Un_empty_interior:
   688   assumes cS: "closed S" and iT: "interior T = {}"
   689   shows "interior(S \<union> T) = interior S"
   690 proof
   691   show "interior S \<subseteq> interior (S\<union>T)"
   692     by (rule subset_interior, blast)
   693 next
   694   show "interior (S \<union> T) \<subseteq> interior S"
   695   proof
   696     fix x assume "x \<in> interior (S \<union> T)"
   697     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
   698       unfolding interior_def by fast
   699     show "x \<in> interior S"
   700     proof (rule ccontr)
   701       assume "x \<notin> interior S"
   702       with `x \<in> R` `open R` obtain y where "y \<in> R - S"
   703         unfolding interior_def expand_set_eq by fast
   704       from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
   705       from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
   706       from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
   707       show "False" unfolding interior_def by fast
   708     qed
   709   qed
   710 qed
   711 
   712 
   713 subsection{* Closure of a Set *}
   714 
   715 definition "closure S = S \<union> {x | x. x islimpt S}"
   716 
   717 lemma closure_interior: "closure S = UNIV - interior (UNIV - S)"
   718 proof-
   719   { fix x
   720     have "x\<in>UNIV - interior (UNIV - S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
   721     proof
   722       let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> UNIV - S)"
   723       assume "?lhs"
   724       hence *:"\<not> ?exT x"
   725         unfolding interior_def
   726         by simp
   727       { assume "\<not> ?rhs"
   728         hence False using *
   729           unfolding closure_def islimpt_def
   730           by blast
   731       }
   732       thus "?rhs"
   733         by blast
   734     next
   735       assume "?rhs" thus "?lhs"
   736         unfolding closure_def interior_def islimpt_def
   737         by blast
   738     qed
   739   }
   740   thus ?thesis
   741     by blast
   742 qed
   743 
   744 lemma interior_closure: "interior S = UNIV - (closure (UNIV - S))"
   745 proof-
   746   { fix x
   747     have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
   748       unfolding interior_def closure_def islimpt_def
   749       by blast (* FIXME: VERY slow! *)
   750   }
   751   thus ?thesis
   752     by blast
   753 qed
   754 
   755 lemma closed_closure[simp, intro]: "closed (closure S)"
   756 proof-
   757   have "closed (UNIV - interior (UNIV -S))" by blast
   758   thus ?thesis using closure_interior[of S] by simp
   759 qed
   760 
   761 lemma closure_hull: "closure S = closed hull S"
   762 proof-
   763   have "S \<subseteq> closure S"
   764     unfolding closure_def
   765     by blast
   766   moreover
   767   have "closed (closure S)"
   768     using closed_closure[of S]
   769     by assumption
   770   moreover
   771   { fix t
   772     assume *:"S \<subseteq> t" "closed t"
   773     { fix x
   774       assume "x islimpt S"
   775       hence "x islimpt t" using *(1)
   776         using islimpt_subset[of x, of S, of t]
   777         by blast
   778     }
   779     with * have "closure S \<subseteq> t"
   780       unfolding closure_def
   781       using closed_limpt[of t]
   782       by auto
   783   }
   784   ultimately show ?thesis
   785     using hull_unique[of S, of "closure S", of closed]
   786     unfolding mem_def
   787     by simp
   788 qed
   789 
   790 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
   791   unfolding closure_hull
   792   using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]
   793   by (metis mem_def subset_eq)
   794 
   795 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
   796   using closure_eq[of S]
   797   by simp
   798 
   799 lemma closure_closure[simp]: "closure (closure S) = closure S"
   800   unfolding closure_hull
   801   using hull_hull[of closed S]
   802   by assumption
   803 
   804 lemma closure_subset: "S \<subseteq> closure S"
   805   unfolding closure_hull
   806   using hull_subset[of S closed]
   807   by assumption
   808 
   809 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
   810   unfolding closure_hull
   811   using hull_mono[of S T closed]
   812   by assumption
   813 
   814 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
   815   using hull_minimal[of S T closed]
   816   unfolding closure_hull mem_def
   817   by simp
   818 
   819 lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"
   820   using hull_unique[of S T closed]
   821   unfolding closure_hull mem_def
   822   by simp
   823 
   824 lemma closure_empty[simp]: "closure {} = {}"
   825   using closed_empty closure_closed[of "{}"]
   826   by simp
   827 
   828 lemma closure_univ[simp]: "closure UNIV = UNIV"
   829   using closure_closed[of UNIV]
   830   by simp
   831 
   832 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
   833   using closure_empty closure_subset[of S]
   834   by blast
   835 
   836 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
   837   using closure_eq[of S] closure_subset[of S]
   838   by simp
   839 
   840 lemma open_inter_closure_eq_empty:
   841   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
   842   using open_subset_interior[of S "UNIV - T"]
   843   using interior_subset[of "UNIV - T"]
   844   unfolding closure_interior
   845   by auto
   846 
   847 lemma open_inter_closure_subset:
   848   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
   849 proof
   850   fix x
   851   assume as: "open S" "x \<in> S \<inter> closure T"
   852   { assume *:"x islimpt T"
   853     have "x islimpt (S \<inter> T)"
   854     proof (rule islimptI)
   855       fix A
   856       assume "x \<in> A" "open A"
   857       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
   858         by (simp_all add: open_Int)
   859       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
   860         by (rule islimptE)
   861       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
   862         by simp_all
   863       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
   864     qed
   865   }
   866   then show "x \<in> closure (S \<inter> T)" using as
   867     unfolding closure_def
   868     by blast
   869 qed
   870 
   871 lemma closure_complement: "closure(UNIV - S) = UNIV - interior(S)"
   872 proof-
   873   have "S = UNIV - (UNIV - S)"
   874     by auto
   875   thus ?thesis
   876     unfolding closure_interior
   877     by auto
   878 qed
   879 
   880 lemma interior_complement: "interior(UNIV - S) = UNIV - closure(S)"
   881   unfolding closure_interior
   882   by blast
   883 
   884 subsection{* Frontier (aka boundary) *}
   885 
   886 definition "frontier S = closure S - interior S"
   887 
   888 lemma frontier_closed: "closed(frontier S)"
   889   by (simp add: frontier_def closed_Diff)
   890 
   891 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(UNIV - S))"
   892   by (auto simp add: frontier_def interior_closure)
   893 
   894 lemma frontier_straddle:
   895   fixes a :: "'a::metric_space"
   896   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")
   897 proof
   898   assume "?lhs"
   899   { fix e::real
   900     assume "e > 0"
   901     let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
   902     { assume "a\<in>S"
   903       have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
   904       moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
   905         unfolding frontier_closures closure_def islimpt_def using `e>0`
   906         by (auto, erule_tac x="ball a e" in allE, auto)
   907       ultimately have ?rhse by auto
   908     }
   909     moreover
   910     { assume "a\<notin>S"
   911       hence ?rhse using `?lhs`
   912         unfolding frontier_closures closure_def islimpt_def
   913         using open_ball[of a e] `e > 0`
   914         by (auto, erule_tac x = "ball a e" in allE, auto) (* FIXME: VERY slow! *)
   915     }
   916     ultimately have ?rhse by auto
   917   }
   918   thus ?rhs by auto
   919 next
   920   assume ?rhs
   921   moreover
   922   { fix T assume "a\<notin>S" and
   923     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"
   924     from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
   925     then obtain e where "e>0" "ball a e \<subseteq> T" by auto
   926     then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
   927     have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
   928       using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
   929   }
   930   hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
   931   moreover
   932   { fix T assume "a \<in> T"  "open T" "a\<in>S"
   933     then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
   934     obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
   935     hence "\<exists>y\<in>UNIV - S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
   936   }
   937   hence "a islimpt (UNIV - S) \<or> a\<notin>S" unfolding islimpt_def by auto
   938   ultimately show ?lhs unfolding frontier_closures using closure_def[of "UNIV - S"] by auto
   939 qed
   940 
   941 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
   942   by (metis frontier_def closure_closed Diff_subset)
   943 
   944 lemma frontier_empty: "frontier {} = {}"
   945   by (simp add: frontier_def closure_empty)
   946 
   947 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
   948 proof-
   949   { assume "frontier S \<subseteq> S"
   950     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
   951     hence "closed S" using closure_subset_eq by auto
   952   }
   953   thus ?thesis using frontier_subset_closed[of S] by auto
   954 qed
   955 
   956 lemma frontier_complement: "frontier(UNIV - S) = frontier S"
   957   by (auto simp add: frontier_def closure_complement interior_complement)
   958 
   959 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
   960   using frontier_complement frontier_subset_eq[of "UNIV - S"]
   961   unfolding open_closed Compl_eq_Diff_UNIV by auto
   962 
   963 subsection{* Common nets and The "within" modifier for nets. *}
   964 
   965 definition
   966   at_infinity :: "'a::real_normed_vector net" where
   967   "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
   968 
   969 definition
   970   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
   971   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
   972 
   973 text{* Prove That They are all nets. *}
   974 
   975 lemma Rep_net_at_infinity:
   976   "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
   977 unfolding at_infinity_def
   978 apply (rule Abs_net_inverse')
   979 apply (rule image_nonempty, simp)
   980 apply (clarsimp, rename_tac r s)
   981 apply (rule_tac x="max r s" in exI, auto)
   982 done
   983 
   984 lemma within_UNIV: "net within UNIV = net"
   985   by (simp add: Rep_net_inject [symmetric] Rep_net_within)
   986 
   987 subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
   988 
   989 definition
   990   trivial_limit :: "'a net \<Rightarrow> bool" where
   991   "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
   992 
   993 lemma trivial_limit_within:
   994   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
   995 proof
   996   assume "trivial_limit (at a within S)"
   997   thus "\<not> a islimpt S"
   998     unfolding trivial_limit_def
   999     unfolding Rep_net_within Rep_net_at
  1000     unfolding islimpt_def
  1001     apply (clarsimp simp add: expand_set_eq)
  1002     apply (rename_tac T, rule_tac x=T in exI)
  1003     apply (clarsimp, drule_tac x=y in spec, simp)
  1004     done
  1005 next
  1006   assume "\<not> a islimpt S"
  1007   thus "trivial_limit (at a within S)"
  1008     unfolding trivial_limit_def
  1009     unfolding Rep_net_within Rep_net_at
  1010     unfolding islimpt_def
  1011     apply (clarsimp simp add: image_image)
  1012     apply (rule_tac x=T in image_eqI)
  1013     apply (auto simp add: expand_set_eq)
  1014     done
  1015 qed
  1016 
  1017 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
  1018   using trivial_limit_within [of a UNIV]
  1019   by (simp add: within_UNIV)
  1020 
  1021 lemma trivial_limit_at:
  1022   fixes a :: "'a::perfect_space"
  1023   shows "\<not> trivial_limit (at a)"
  1024   by (simp add: trivial_limit_at_iff)
  1025 
  1026 lemma trivial_limit_at_infinity:
  1027   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
  1028   (* FIXME: find a more appropriate type class *)
  1029   unfolding trivial_limit_def Rep_net_at_infinity
  1030   apply (clarsimp simp add: expand_set_eq)
  1031   apply (drule_tac x="scaleR r (sgn 1)" in spec)
  1032   apply (simp add: norm_sgn)
  1033   done
  1034 
  1035 lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
  1036   by (auto simp add: trivial_limit_def Rep_net_sequentially)
  1037 
  1038 subsection{* Some property holds "sufficiently close" to the limit point. *}
  1039 
  1040 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
  1041   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
  1042 unfolding eventually_at dist_nz by auto
  1043 
  1044 lemma eventually_at_infinity:
  1045   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
  1046 unfolding eventually_def Rep_net_at_infinity by auto
  1047 
  1048 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
  1049         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
  1050 unfolding eventually_within eventually_at dist_nz by auto
  1051 
  1052 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
  1053         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
  1054 unfolding eventually_within
  1055 apply safe
  1056 apply (rule_tac x="d/2" in exI, simp)
  1057 apply (rule_tac x="d" in exI, simp)
  1058 done
  1059 
  1060 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
  1061   unfolding eventually_def trivial_limit_def
  1062   using Rep_net_nonempty [of net] by auto
  1063 
  1064 lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
  1065   unfolding eventually_def trivial_limit_def
  1066   using Rep_net_nonempty [of net] by auto
  1067 
  1068 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
  1069   unfolding trivial_limit_def eventually_def by auto
  1070 
  1071 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
  1072   unfolding trivial_limit_def eventually_def by auto
  1073 
  1074 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
  1075   apply (safe elim!: trivial_limit_eventually)
  1076   apply (simp add: eventually_False [symmetric])
  1077   done
  1078 
  1079 text{* Combining theorems for "eventually" *}
  1080 
  1081 lemma eventually_conjI:
  1082   "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
  1083     \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
  1084 by (rule eventually_conj)
  1085 
  1086 lemma eventually_rev_mono:
  1087   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
  1088 using eventually_mono [of P Q] by fast
  1089 
  1090 lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
  1091   by (auto intro!: eventually_conjI elim: eventually_rev_mono)
  1092 
  1093 lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
  1094   by (auto simp add: eventually_False)
  1095 
  1096 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
  1097   by (simp add: eventually_False)
  1098 
  1099 subsection{* Limits, defined as vacuously true when the limit is trivial. *}
  1100 
  1101   text{* Notation Lim to avoid collition with lim defined in analysis *}
  1102 definition
  1103   Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b" where
  1104   "Lim net f = (THE l. (f ---> l) net)"
  1105 
  1106 lemma Lim:
  1107  "(f ---> l) net \<longleftrightarrow>
  1108         trivial_limit net \<or>
  1109         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
  1110   unfolding tendsto_iff trivial_limit_eq by auto
  1111 
  1112 
  1113 text{* Show that they yield usual definitions in the various cases. *}
  1114 
  1115 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
  1116            (\<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)"
  1117   by (auto simp add: tendsto_iff eventually_within_le)
  1118 
  1119 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
  1120         (\<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)"
  1121   by (auto simp add: tendsto_iff eventually_within)
  1122 
  1123 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
  1124         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
  1125   by (auto simp add: tendsto_iff eventually_at)
  1126 
  1127 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
  1128   unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
  1129 
  1130 lemma Lim_at_infinity:
  1131   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
  1132   by (auto simp add: tendsto_iff eventually_at_infinity)
  1133 
  1134 lemma Lim_sequentially:
  1135  "(S ---> l) sequentially \<longleftrightarrow>
  1136           (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
  1137   by (auto simp add: tendsto_iff eventually_sequentially)
  1138 
  1139 lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
  1140   unfolding Lim_sequentially LIMSEQ_def ..
  1141 
  1142 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
  1143   by (rule topological_tendstoI, auto elim: eventually_rev_mono)
  1144 
  1145 text{* The expected monotonicity property. *}
  1146 
  1147 lemma Lim_within_empty: "(f ---> l) (net within {})"
  1148   unfolding tendsto_def Limits.eventually_within by simp
  1149 
  1150 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
  1151   unfolding tendsto_def Limits.eventually_within
  1152   by (auto elim!: eventually_elim1)
  1153 
  1154 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
  1155   shows "(f ---> l) (net within (S \<union> T))"
  1156   using assms unfolding tendsto_def Limits.eventually_within
  1157   apply clarify
  1158   apply (drule spec, drule (1) mp, drule (1) mp)
  1159   apply (drule spec, drule (1) mp, drule (1) mp)
  1160   apply (auto elim: eventually_elim2)
  1161   done
  1162 
  1163 lemma Lim_Un_univ:
  1164  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
  1165         ==> (f ---> l) net"
  1166   by (metis Lim_Un within_UNIV)
  1167 
  1168 text{* Interrelations between restricted and unrestricted limits. *}
  1169 
  1170 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
  1171   (* FIXME: rename *)
  1172   unfolding tendsto_def Limits.eventually_within
  1173   apply (clarify, drule spec, drule (1) mp, drule (1) mp)
  1174   by (auto elim!: eventually_elim1)
  1175 
  1176 lemma Lim_within_open:
  1177   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
  1178   assumes"a \<in> S" "open S"
  1179   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
  1180 proof
  1181   assume ?lhs
  1182   { fix A assume "open A" "l \<in> A"
  1183     with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
  1184       by (rule topological_tendstoD)
  1185     hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
  1186       unfolding Limits.eventually_within .
  1187     then obtain T where "open T" "a \<in> T" "\<forall>x\<in>T. x \<noteq> a \<longrightarrow> x \<in> S \<longrightarrow> f x \<in> A"
  1188       unfolding eventually_at_topological by fast
  1189     hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
  1190       using assms by auto
  1191     hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
  1192       by fast
  1193     hence "eventually (\<lambda>x. f x \<in> A) (at a)"
  1194       unfolding eventually_at_topological .
  1195   }
  1196   thus ?rhs by (rule topological_tendstoI)
  1197 next
  1198   assume ?rhs
  1199   thus ?lhs by (rule Lim_at_within)
  1200 qed
  1201 
  1202 text{* Another limit point characterization. *}
  1203 
  1204 lemma islimpt_sequential:
  1205   fixes x :: "'a::metric_space" (* FIXME: generalize to topological_space *)
  1206   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
  1207     (is "?lhs = ?rhs")
  1208 proof
  1209   assume ?lhs
  1210   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"
  1211     unfolding islimpt_approachable using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto
  1212   { fix n::nat
  1213     have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
  1214   }
  1215   moreover
  1216   { fix e::real assume "e>0"
  1217     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
  1218     then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  1219     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)
  1220     moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
  1221     ultimately have "\<exists>N::nat. \<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < e" apply(rule_tac x=N in exI) apply auto apply(erule_tac x=n in allE)+ by auto
  1222   }
  1223   hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
  1224     unfolding Lim_sequentially using f by auto
  1225   ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
  1226 next
  1227   assume ?rhs
  1228   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
  1229   { fix e::real assume "e>0"
  1230     then obtain N where "dist (f N) x < e" using f(2) by auto
  1231     moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
  1232     ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
  1233   }
  1234   thus ?lhs unfolding islimpt_approachable by auto
  1235 qed
  1236 
  1237 text{* Basic arithmetical combining theorems for limits. *}
  1238 
  1239 lemma Lim_linear:
  1240   assumes "(f ---> l) net" "bounded_linear h"
  1241   shows "((\<lambda>x. h (f x)) ---> h l) net"
  1242 using `bounded_linear h` `(f ---> l) net`
  1243 by (rule bounded_linear.tendsto)
  1244 
  1245 lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
  1246   unfolding tendsto_def Limits.eventually_at_topological by fast
  1247 
  1248 lemma Lim_const: "((\<lambda>x. a) ---> a) net"
  1249   by (rule tendsto_const)
  1250 
  1251 lemma Lim_cmul:
  1252   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1253   shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
  1254   by (intro tendsto_intros)
  1255 
  1256 lemma Lim_neg:
  1257   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1258   shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
  1259   by (rule tendsto_minus)
  1260 
  1261 lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
  1262  "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
  1263   by (rule tendsto_add)
  1264 
  1265 lemma Lim_sub:
  1266   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1267   shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
  1268   by (rule tendsto_diff)
  1269 
  1270 lemma Lim_null:
  1271   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1272   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
  1273 
  1274 lemma Lim_null_norm:
  1275   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1276   shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
  1277   by (simp add: Lim dist_norm)
  1278 
  1279 lemma Lim_null_comparison:
  1280   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1281   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
  1282   shows "(f ---> 0) net"
  1283 proof(simp add: tendsto_iff, rule+)
  1284   fix e::real assume "0<e"
  1285   { fix x
  1286     assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
  1287     hence "dist (f x) 0 < e" by (simp add: dist_norm)
  1288   }
  1289   thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
  1290     using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
  1291     using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (g x) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
  1292     using assms `e>0` unfolding tendsto_iff by auto
  1293 qed
  1294 
  1295 lemma Lim_component:
  1296   fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
  1297   shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
  1298   unfolding tendsto_iff
  1299   apply (clarify)
  1300   apply (drule spec, drule (1) mp)
  1301   apply (erule eventually_elim1)
  1302   apply (erule le_less_trans [OF dist_nth_le])
  1303   done
  1304 
  1305 lemma Lim_transform_bound:
  1306   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1307   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
  1308   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
  1309   shows "(f ---> 0) net"
  1310 proof (rule tendstoI)
  1311   fix e::real assume "e>0"
  1312   { fix x
  1313     assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
  1314     hence "dist (f x) 0 < e" by (simp add: dist_norm)}
  1315   thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
  1316     using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
  1317     using eventually_mono[of "\<lambda>x. norm (f x) \<le> norm (g x) \<and> dist (g x) 0 < e" "\<lambda>x. dist (f x) 0 < e" net]
  1318     using assms `e>0` unfolding tendsto_iff by blast
  1319 qed
  1320 
  1321 text{* Deducing things about the limit from the elements. *}
  1322 
  1323 lemma Lim_in_closed_set:
  1324   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
  1325   shows "l \<in> S"
  1326 proof (rule ccontr)
  1327   assume "l \<notin> S"
  1328   with `closed S` have "open (- S)" "l \<in> - S"
  1329     by (simp_all add: open_Compl)
  1330   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
  1331     by (rule topological_tendstoD)
  1332   with assms(2) have "eventually (\<lambda>x. False) net"
  1333     by (rule eventually_elim2) simp
  1334   with assms(3) show "False"
  1335     by (simp add: eventually_False)
  1336 qed
  1337 
  1338 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
  1339 
  1340 lemma Lim_dist_ubound:
  1341   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
  1342   shows "dist a l <= e"
  1343 proof (rule ccontr)
  1344   assume "\<not> dist a l \<le> e"
  1345   then have "0 < dist a l - e" by simp
  1346   with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
  1347     by (rule tendstoD)
  1348   with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
  1349     by (rule eventually_conjI)
  1350   then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
  1351     using assms(1) eventually_happens by auto
  1352   hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
  1353     by (rule add_le_less_mono)
  1354   hence "dist a (f w) + dist (f w) l < dist a l"
  1355     by simp
  1356   also have "\<dots> \<le> dist a (f w) + dist (f w) l"
  1357     by (rule dist_triangle)
  1358   finally show False by simp
  1359 qed
  1360 
  1361 lemma Lim_norm_ubound:
  1362   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1363   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
  1364   shows "norm(l) <= e"
  1365 proof (rule ccontr)
  1366   assume "\<not> norm l \<le> e"
  1367   then have "0 < norm l - e" by simp
  1368   with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
  1369     by (rule tendstoD)
  1370   with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
  1371     by (rule eventually_conjI)
  1372   then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
  1373     using assms(1) eventually_happens by auto
  1374   hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
  1375   hence "norm (f w - l) + norm (f w) < norm l" by simp
  1376   hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
  1377   thus False using `\<not> norm l \<le> e` by simp
  1378 qed
  1379 
  1380 lemma Lim_norm_lbound:
  1381   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1382   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
  1383   shows "e \<le> norm l"
  1384 proof (rule ccontr)
  1385   assume "\<not> e \<le> norm l"
  1386   then have "0 < e - norm l" by simp
  1387   with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
  1388     by (rule tendstoD)
  1389   with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
  1390     by (rule eventually_conjI)
  1391   then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
  1392     using assms(1) eventually_happens by auto
  1393   hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
  1394   hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
  1395   hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
  1396   thus False by simp
  1397 qed
  1398 
  1399 text{* Uniqueness of the limit, when nontrivial. *}
  1400 
  1401 lemma Lim_unique:
  1402   fixes f :: "'a \<Rightarrow> 'b::t2_space"
  1403   assumes "\<not> trivial_limit net"  "(f ---> l) net"  "(f ---> l') net"
  1404   shows "l = l'"
  1405 proof (rule ccontr)
  1406   assume "l \<noteq> l'"
  1407   obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
  1408     using hausdorff [OF `l \<noteq> l'`] by fast
  1409   have "eventually (\<lambda>x. f x \<in> U) net"
  1410     using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
  1411   moreover
  1412   have "eventually (\<lambda>x. f x \<in> V) net"
  1413     using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
  1414   ultimately
  1415   have "eventually (\<lambda>x. False) net"
  1416   proof (rule eventually_elim2)
  1417     fix x
  1418     assume "f x \<in> U" "f x \<in> V"
  1419     hence "f x \<in> U \<inter> V" by simp
  1420     with `U \<inter> V = {}` show "False" by simp
  1421   qed
  1422   with `\<not> trivial_limit net` show "False"
  1423     by (simp add: eventually_False)
  1424 qed
  1425 
  1426 lemma tendsto_Lim:
  1427   fixes f :: "'a \<Rightarrow> 'b::t2_space"
  1428   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
  1429   unfolding Lim_def using Lim_unique[of net f] by auto
  1430 
  1431 text{* Limit under bilinear function *}
  1432 
  1433 lemma Lim_bilinear:
  1434   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
  1435   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
  1436 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
  1437 by (rule bounded_bilinear.tendsto)
  1438 
  1439 text{* These are special for limits out of the same vector space. *}
  1440 
  1441 lemma Lim_within_id: "(id ---> a) (at a within s)"
  1442   unfolding tendsto_def Limits.eventually_within eventually_at_topological
  1443   by auto
  1444 
  1445 lemma Lim_at_id: "(id ---> a) (at a)"
  1446 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
  1447 
  1448 lemma Lim_at_zero:
  1449   fixes a :: "'a::real_normed_vector"
  1450   fixes l :: "'b::topological_space"
  1451   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
  1452 proof
  1453   assume "?lhs"
  1454   { fix S assume "open S" "l \<in> S"
  1455     with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
  1456       by (rule topological_tendstoD)
  1457     then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
  1458       unfolding Limits.eventually_at by fast
  1459     { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
  1460       hence "f (a + x) \<in> S" using d
  1461       apply(erule_tac x="x+a" in allE)
  1462       by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
  1463     }
  1464     hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
  1465       using d(1) by auto
  1466     hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
  1467       unfolding Limits.eventually_at .
  1468   }
  1469   thus "?rhs" by (rule topological_tendstoI)
  1470 next
  1471   assume "?rhs"
  1472   { fix S assume "open S" "l \<in> S"
  1473     with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
  1474       by (rule topological_tendstoD)
  1475     then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
  1476       unfolding Limits.eventually_at by fast
  1477     { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
  1478       hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
  1479         by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
  1480     }
  1481     hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
  1482     hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
  1483   }
  1484   thus "?lhs" by (rule topological_tendstoI)
  1485 qed
  1486 
  1487 text{* It's also sometimes useful to extract the limit point from the net.  *}
  1488 
  1489 definition
  1490   netlimit :: "'a::t2_space net \<Rightarrow> 'a" where
  1491   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
  1492 
  1493 lemma netlimit_within:
  1494   assumes "\<not> trivial_limit (at a within S)"
  1495   shows "netlimit (at a within S) = a"
  1496 unfolding netlimit_def
  1497 apply (rule some_equality)
  1498 apply (rule Lim_at_within)
  1499 apply (rule Lim_ident_at)
  1500 apply (erule Lim_unique [OF assms])
  1501 apply (rule Lim_at_within)
  1502 apply (rule Lim_ident_at)
  1503 done
  1504 
  1505 lemma netlimit_at:
  1506   fixes a :: "'a::perfect_space"
  1507   shows "netlimit (at a) = a"
  1508   apply (subst within_UNIV[symmetric])
  1509   using netlimit_within[of a UNIV]
  1510   by (simp add: trivial_limit_at within_UNIV)
  1511 
  1512 text{* Transformation of limit. *}
  1513 
  1514 lemma Lim_transform:
  1515   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
  1516   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
  1517   shows "(g ---> l) net"
  1518 proof-
  1519   from assms have "((\<lambda>x. f x - g x - f x) ---> 0 - l) net" using Lim_sub[of "\<lambda>x. f x - g x" 0 net f l] by auto
  1520   thus "?thesis" using Lim_neg [of "\<lambda> x. - g x" "-l" net] by auto
  1521 qed
  1522 
  1523 lemma Lim_transform_eventually:
  1524   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net ==> (g ---> l) net"
  1525   apply (rule topological_tendstoI)
  1526   apply (drule (2) topological_tendstoD)
  1527   apply (erule (1) eventually_elim2, simp)
  1528   done
  1529 
  1530 lemma Lim_transform_within:
  1531   fixes l :: "'b::metric_space" (* TODO: generalize *)
  1532   assumes "0 < d" "(\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x')"
  1533           "(f ---> l) (at x within S)"
  1534   shows   "(g ---> l) (at x within S)"
  1535   using assms(1,3) unfolding Lim_within
  1536   apply -
  1537   apply (clarify, rename_tac e)
  1538   apply (drule_tac x=e in spec, clarsimp, rename_tac r)
  1539   apply (rule_tac x="min d r" in exI, clarsimp, rename_tac y)
  1540   apply (drule_tac x=y in bspec, assumption, clarsimp)
  1541   apply (simp add: assms(2))
  1542   done
  1543 
  1544 lemma Lim_transform_at:
  1545   fixes l :: "'b::metric_space" (* TODO: generalize *)
  1546   shows "0 < d \<Longrightarrow> (\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x') \<Longrightarrow>
  1547   (f ---> l) (at x) ==> (g ---> l) (at x)"
  1548   apply (subst within_UNIV[symmetric])
  1549   using Lim_transform_within[of d UNIV x f g l]
  1550   by (auto simp add: within_UNIV)
  1551 
  1552 text{* Common case assuming being away from some crucial point like 0. *}
  1553 
  1554 lemma Lim_transform_away_within:
  1555   fixes a b :: "'a::metric_space"
  1556   fixes l :: "'b::metric_space" (* TODO: generalize *)
  1557   assumes "a\<noteq>b" "\<forall>x\<in> S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1558   and "(f ---> l) (at a within S)"
  1559   shows "(g ---> l) (at a within S)"
  1560 proof-
  1561   have "\<forall>x'\<in>S. 0 < dist x' a \<and> dist x' a < dist a b \<longrightarrow> f x' = g x'" using assms(2)
  1562     apply auto apply(erule_tac x=x' in ballE) by (auto simp add: dist_commute)
  1563   thus ?thesis using Lim_transform_within[of "dist a b" S a f g l] using assms(1,3) unfolding dist_nz by auto
  1564 qed
  1565 
  1566 lemma Lim_transform_away_at:
  1567   fixes a b :: "'a::metric_space"
  1568   fixes l :: "'b::metric_space" (* TODO: generalize *)
  1569   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1570   and fl: "(f ---> l) (at a)"
  1571   shows "(g ---> l) (at a)"
  1572   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
  1573   by (auto simp add: within_UNIV)
  1574 
  1575 text{* Alternatively, within an open set. *}
  1576 
  1577 lemma Lim_transform_within_open:
  1578   fixes a :: "'a::metric_space"
  1579   fixes l :: "'b::metric_space" (* TODO: generalize *)
  1580   assumes "open S"  "a \<in> S"  "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"  "(f ---> l) (at a)"
  1581   shows "(g ---> l) (at a)"
  1582 proof-
  1583   from assms(1,2) obtain e::real where "e>0" and e:"ball a e \<subseteq> S" unfolding open_contains_ball by auto
  1584   hence "\<forall>x'. 0 < dist x' a \<and> dist x' a < e \<longrightarrow> f x' = g x'" using assms(3)
  1585     unfolding ball_def subset_eq apply auto apply(erule_tac x=x' in allE) apply(erule_tac x=x' in ballE) by(auto simp add: dist_commute)
  1586   thus ?thesis using Lim_transform_at[of e a f g l] `e>0` assms(4) by auto
  1587 qed
  1588 
  1589 text{* A congruence rule allowing us to transform limits assuming not at point. *}
  1590 
  1591 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
  1592 
  1593 lemma Lim_cong_within[cong add]:
  1594   fixes a :: "'a::metric_space"
  1595   fixes l :: "'b::metric_space" (* TODO: generalize *)
  1596   shows "(\<And>x. x \<noteq> a \<Longrightarrow> f x = g x) ==> ((\<lambda>x. f x) ---> l) (at a within S) \<longleftrightarrow> ((g ---> l) (at a within S))"
  1597   by (simp add: Lim_within dist_nz[symmetric])
  1598 
  1599 lemma Lim_cong_at[cong add]:
  1600   fixes a :: "'a::metric_space"
  1601   fixes l :: "'b::metric_space" (* TODO: generalize *)
  1602   shows "(\<And>x. x \<noteq> a ==> f x = g x) ==> (((\<lambda>x. f x) ---> l) (at a) \<longleftrightarrow> ((g ---> l) (at a)))"
  1603   by (simp add: Lim_at dist_nz[symmetric])
  1604 
  1605 text{* Useful lemmas on closure and set of possible sequential limits.*}
  1606 
  1607 lemma closure_sequential:
  1608   fixes l :: "'a::metric_space" (* TODO: generalize *)
  1609   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
  1610 proof
  1611   assume "?lhs" moreover
  1612   { assume "l \<in> S"
  1613     hence "?rhs" using Lim_const[of l sequentially] by auto
  1614   } moreover
  1615   { assume "l islimpt S"
  1616     hence "?rhs" unfolding islimpt_sequential by auto
  1617   } ultimately
  1618   show "?rhs" unfolding closure_def by auto
  1619 next
  1620   assume "?rhs"
  1621   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
  1622 qed
  1623 
  1624 lemma closed_sequential_limits:
  1625   fixes S :: "'a::metric_space set"
  1626   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
  1627   unfolding closed_limpt
  1628   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]
  1629   by metis
  1630 
  1631 lemma closure_approachable:
  1632   fixes S :: "'a::metric_space set"
  1633   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
  1634   apply (auto simp add: closure_def islimpt_approachable)
  1635   by (metis dist_self)
  1636 
  1637 lemma closed_approachable:
  1638   fixes S :: "'a::metric_space set"
  1639   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
  1640   by (metis closure_closed closure_approachable)
  1641 
  1642 text{* Some other lemmas about sequences. *}
  1643 
  1644 lemma seq_offset:
  1645   fixes l :: "'a::metric_space" (* TODO: generalize *)
  1646   shows "(f ---> l) sequentially ==> ((\<lambda>i. f( i + k)) ---> l) sequentially"
  1647   apply (auto simp add: Lim_sequentially)
  1648   by (metis trans_le_add1 )
  1649 
  1650 lemma seq_offset_neg:
  1651   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
  1652   apply (rule topological_tendstoI)
  1653   apply (drule (2) topological_tendstoD)
  1654   apply (simp only: eventually_sequentially)
  1655   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
  1656   apply metis
  1657   by arith
  1658 
  1659 lemma seq_offset_rev:
  1660   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
  1661   apply (rule topological_tendstoI)
  1662   apply (drule (2) topological_tendstoD)
  1663   apply (simp only: eventually_sequentially)
  1664   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
  1665   by metis arith
  1666 
  1667 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
  1668 proof-
  1669   { fix e::real assume "e>0"
  1670     hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
  1671       using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
  1672       by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
  1673   }
  1674   thus ?thesis unfolding Lim_sequentially dist_norm by simp
  1675 qed
  1676 
  1677 text{* More properties of closed balls. *}
  1678 
  1679 lemma closed_cball: "closed (cball x e)"
  1680 unfolding cball_def closed_def
  1681 unfolding Collect_neg_eq [symmetric] not_le
  1682 apply (clarsimp simp add: open_dist, rename_tac y)
  1683 apply (rule_tac x="dist x y - e" in exI, clarsimp)
  1684 apply (rename_tac x')
  1685 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
  1686 apply simp
  1687 done
  1688 
  1689 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
  1690 proof-
  1691   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
  1692     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
  1693   } moreover
  1694   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
  1695     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
  1696   } ultimately
  1697   show ?thesis unfolding open_contains_ball by auto
  1698 qed
  1699 
  1700 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
  1701   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
  1702 
  1703 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
  1704   apply (simp add: interior_def, safe)
  1705   apply (force simp add: open_contains_cball)
  1706   apply (rule_tac x="ball x e" in exI)
  1707   apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
  1708   done
  1709 
  1710 lemma islimpt_ball:
  1711   fixes x y :: "'a::{real_normed_vector,perfect_space}"
  1712   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
  1713 proof
  1714   assume "?lhs"
  1715   { assume "e \<le> 0"
  1716     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
  1717     have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
  1718   }
  1719   hence "e > 0" by (metis not_less)
  1720   moreover
  1721   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
  1722   ultimately show "?rhs" by auto
  1723 next
  1724   assume "?rhs" hence "e>0"  by auto
  1725   { fix d::real assume "d>0"
  1726     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1727     proof(cases "d \<le> dist x y")
  1728       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1729       proof(cases "x=y")
  1730         case True hence False using `d \<le> dist x y` `d>0` by auto
  1731         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
  1732       next
  1733         case False
  1734 
  1735         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
  1736               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1737           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
  1738         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
  1739           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
  1740           unfolding scaleR_minus_left scaleR_one
  1741           by (auto simp add: norm_minus_commute)
  1742         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
  1743           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
  1744           unfolding real_add_mult_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
  1745         also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
  1746         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
  1747 
  1748         moreover
  1749 
  1750         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
  1751           using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
  1752         moreover
  1753         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
  1754           using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
  1755           unfolding dist_norm by auto
  1756         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
  1757       qed
  1758     next
  1759       case False hence "d > dist x y" by auto
  1760       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1761       proof(cases "x=y")
  1762         case True
  1763         obtain z where **: "z \<noteq> y" "dist z y < min e d"
  1764           using perfect_choose_dist[of "min e d" y]
  1765           using `d > 0` `e>0` by auto
  1766         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1767           unfolding `x = y`
  1768           using `z \<noteq> y` **
  1769           by (rule_tac x=z in bexI, auto simp add: dist_commute)
  1770       next
  1771         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1772           using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
  1773       qed
  1774     qed  }
  1775   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
  1776 qed
  1777 
  1778 lemma closure_ball_lemma:
  1779   fixes x y :: "'a::real_normed_vector"
  1780   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
  1781 proof (rule islimptI)
  1782   fix T assume "y \<in> T" "open T"
  1783   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
  1784     unfolding open_dist by fast
  1785   (* choose point between x and y, within distance r of y. *)
  1786   def k \<equiv> "min 1 (r / (2 * dist x y))"
  1787   def z \<equiv> "y + scaleR k (x - y)"
  1788   have z_def2: "z = x + scaleR (1 - k) (y - x)"
  1789     unfolding z_def by (simp add: algebra_simps)
  1790   have "dist z y < r"
  1791     unfolding z_def k_def using `0 < r`
  1792     by (simp add: dist_norm min_def)
  1793   hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
  1794   have "dist x z < dist x y"
  1795     unfolding z_def2 dist_norm
  1796     apply (simp add: norm_minus_commute)
  1797     apply (simp only: dist_norm [symmetric])
  1798     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
  1799     apply (rule mult_strict_right_mono)
  1800     apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
  1801     apply (simp add: zero_less_dist_iff `x \<noteq> y`)
  1802     done
  1803   hence "z \<in> ball x (dist x y)" by simp
  1804   have "z \<noteq> y"
  1805     unfolding z_def k_def using `x \<noteq> y` `0 < r`
  1806     by (simp add: min_def)
  1807   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
  1808     using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
  1809     by fast
  1810 qed
  1811 
  1812 lemma closure_ball:
  1813   fixes x :: "'a::real_normed_vector"
  1814   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
  1815 apply (rule equalityI)
  1816 apply (rule closure_minimal)
  1817 apply (rule ball_subset_cball)
  1818 apply (rule closed_cball)
  1819 apply (rule subsetI, rename_tac y)
  1820 apply (simp add: le_less [where 'a=real])
  1821 apply (erule disjE)
  1822 apply (rule subsetD [OF closure_subset], simp)
  1823 apply (simp add: closure_def)
  1824 apply clarify
  1825 apply (rule closure_ball_lemma)
  1826 apply (simp add: zero_less_dist_iff)
  1827 done
  1828 
  1829 (* In a trivial vector space, this fails for e = 0. *)
  1830 lemma interior_cball:
  1831   fixes x :: "'a::{real_normed_vector, perfect_space}"
  1832   shows "interior (cball x e) = ball x e"
  1833 proof(cases "e\<ge>0")
  1834   case False note cs = this
  1835   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
  1836   { fix y assume "y \<in> cball x e"
  1837     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
  1838   hence "cball x e = {}" by auto
  1839   hence "interior (cball x e) = {}" using interior_empty by auto
  1840   ultimately show ?thesis by blast
  1841 next
  1842   case True note cs = this
  1843   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
  1844   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
  1845     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
  1846 
  1847     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
  1848       using perfect_choose_dist [of d] by auto
  1849     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
  1850     hence xa_cball:"xa \<in> cball x e" using as(1) by auto
  1851 
  1852     hence "y \<in> ball x e" proof(cases "x = y")
  1853       case True
  1854       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
  1855       thus "y \<in> ball x e" using `x = y ` by simp
  1856     next
  1857       case False
  1858       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
  1859         using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
  1860       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
  1861       have "y - x \<noteq> 0" using `x \<noteq> y` by auto
  1862       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
  1863         using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
  1864 
  1865       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)"
  1866         by (auto simp add: dist_norm algebra_simps)
  1867       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1868         by (auto simp add: algebra_simps)
  1869       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
  1870         using ** by auto
  1871       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
  1872       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
  1873       thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
  1874     qed  }
  1875   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
  1876   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
  1877 qed
  1878 
  1879 lemma frontier_ball:
  1880   fixes a :: "'a::real_normed_vector"
  1881   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
  1882   apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
  1883   apply (simp add: expand_set_eq)
  1884   by arith
  1885 
  1886 lemma frontier_cball:
  1887   fixes a :: "'a::{real_normed_vector, perfect_space}"
  1888   shows "frontier(cball a e) = {x. dist a x = e}"
  1889   apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
  1890   apply (simp add: expand_set_eq)
  1891   by arith
  1892 
  1893 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
  1894   apply (simp add: expand_set_eq not_le)
  1895   by (metis zero_le_dist dist_self order_less_le_trans)
  1896 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
  1897 
  1898 lemma cball_eq_sing:
  1899   fixes x :: "'a::perfect_space"
  1900   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
  1901 proof (rule linorder_cases)
  1902   assume e: "0 < e"
  1903   obtain a where "a \<noteq> x" "dist a x < e"
  1904     using perfect_choose_dist [OF e] by auto
  1905   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
  1906   with e show ?thesis by (auto simp add: expand_set_eq)
  1907 qed auto
  1908 
  1909 lemma cball_sing:
  1910   fixes x :: "'a::metric_space"
  1911   shows "e = 0 ==> cball x e = {x}"
  1912   by (auto simp add: expand_set_eq)
  1913 
  1914 text{* For points in the interior, localization of limits makes no difference.   *}
  1915 
  1916 lemma eventually_within_interior:
  1917   assumes "x \<in> interior S"
  1918   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
  1919 proof-
  1920   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
  1921     unfolding interior_def by fast
  1922   { assume "?lhs"
  1923     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
  1924       unfolding Limits.eventually_within Limits.eventually_at_topological
  1925       by auto
  1926     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
  1927       by auto
  1928     then have "?rhs"
  1929       unfolding Limits.eventually_at_topological by auto
  1930   } moreover
  1931   { assume "?rhs" hence "?lhs"
  1932       unfolding Limits.eventually_within
  1933       by (auto elim: eventually_elim1)
  1934   } ultimately
  1935   show "?thesis" ..
  1936 qed
  1937 
  1938 lemma lim_within_interior:
  1939   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
  1940   unfolding tendsto_def by (simp add: eventually_within_interior)
  1941 
  1942 lemma netlimit_within_interior:
  1943   fixes x :: "'a::{perfect_space, real_normed_vector}"
  1944     (* FIXME: generalize to perfect_space *)
  1945   assumes "x \<in> interior S"
  1946   shows "netlimit(at x within S) = x" (is "?lhs = ?rhs")
  1947 proof-
  1948   from assms obtain e::real where e:"e>0" "ball x e \<subseteq> S" using open_interior[of S] unfolding open_contains_ball using interior_subset[of S] by auto
  1949   hence "\<not> trivial_limit (at x within S)" using islimpt_subset[of x "ball x e" S] unfolding trivial_limit_within islimpt_ball centre_in_cball by auto
  1950   thus ?thesis using netlimit_within by auto
  1951 qed
  1952 
  1953 subsection{* Boundedness. *}
  1954 
  1955   (* FIXME: This has to be unified with BSEQ!! *)
  1956 definition
  1957   bounded :: "'a::metric_space set \<Rightarrow> bool" where
  1958   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
  1959 
  1960 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
  1961 unfolding bounded_def
  1962 apply safe
  1963 apply (rule_tac x="dist a x + e" in exI, clarify)
  1964 apply (drule (1) bspec)
  1965 apply (erule order_trans [OF dist_triangle add_left_mono])
  1966 apply auto
  1967 done
  1968 
  1969 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
  1970 unfolding bounded_any_center [where a=0]
  1971 by (simp add: dist_norm)
  1972 
  1973 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
  1974 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
  1975   by (metis bounded_def subset_eq)
  1976 
  1977 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
  1978   by (metis bounded_subset interior_subset)
  1979 
  1980 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
  1981 proof-
  1982   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
  1983   { fix y assume "y \<in> closure S"
  1984     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
  1985       unfolding closure_sequential by auto
  1986     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
  1987     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
  1988       by (rule eventually_mono, simp add: f(1))
  1989     have "dist x y \<le> a"
  1990       apply (rule Lim_dist_ubound [of sequentially f])
  1991       apply (rule trivial_limit_sequentially)
  1992       apply (rule f(2))
  1993       apply fact
  1994       done
  1995   }
  1996   thus ?thesis unfolding bounded_def by auto
  1997 qed
  1998 
  1999 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
  2000   apply (simp add: bounded_def)
  2001   apply (rule_tac x=x in exI)
  2002   apply (rule_tac x=e in exI)
  2003   apply auto
  2004   done
  2005 
  2006 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
  2007   by (metis ball_subset_cball bounded_cball bounded_subset)
  2008 
  2009 lemma finite_imp_bounded[intro]: assumes "finite S" shows "bounded S"
  2010 proof-
  2011   { fix a F assume as:"bounded F"
  2012     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
  2013     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
  2014     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
  2015   }
  2016   thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto
  2017 qed
  2018 
  2019 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
  2020   apply (auto simp add: bounded_def)
  2021   apply (rename_tac x y r s)
  2022   apply (rule_tac x=x in exI)
  2023   apply (rule_tac x="max r (dist x y + s)" in exI)
  2024   apply (rule ballI, rename_tac z, safe)
  2025   apply (drule (1) bspec, simp)
  2026   apply (drule (1) bspec)
  2027   apply (rule min_max.le_supI2)
  2028   apply (erule order_trans [OF dist_triangle add_left_mono])
  2029   done
  2030 
  2031 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
  2032   by (induct rule: finite_induct[of F], auto)
  2033 
  2034 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
  2035   apply (simp add: bounded_iff)
  2036   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
  2037   by metis arith
  2038 
  2039 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
  2040   by (metis Int_lower1 Int_lower2 bounded_subset)
  2041 
  2042 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
  2043 apply (metis Diff_subset bounded_subset)
  2044 done
  2045 
  2046 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
  2047   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
  2048 
  2049 lemma not_bounded_UNIV[simp, intro]:
  2050   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
  2051 proof(auto simp add: bounded_pos not_le)
  2052   obtain x :: 'a where "x \<noteq> 0"
  2053     using perfect_choose_dist [OF zero_less_one] by fast
  2054   fix b::real  assume b: "b >0"
  2055   have b1: "b +1 \<ge> 0" using b by simp
  2056   with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
  2057     by (simp add: norm_sgn)
  2058   then show "\<exists>x::'a. b < norm x" ..
  2059 qed
  2060 
  2061 lemma bounded_linear_image:
  2062   assumes "bounded S" "bounded_linear f"
  2063   shows "bounded(f ` S)"
  2064 proof-
  2065   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  2066   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)
  2067   { fix x assume "x\<in>S"
  2068     hence "norm x \<le> b" using b by auto
  2069     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
  2070       by (metis B(1) B(2) real_le_trans real_mult_le_cancel_iff2)
  2071   }
  2072   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
  2073     using b B real_mult_order[of b B] by (auto simp add: real_mult_commute)
  2074 qed
  2075 
  2076 lemma bounded_scaling:
  2077   fixes S :: "'a::real_normed_vector set"
  2078   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
  2079   apply (rule bounded_linear_image, assumption)
  2080   apply (rule scaleR.bounded_linear_right)
  2081   done
  2082 
  2083 lemma bounded_translation:
  2084   fixes S :: "'a::real_normed_vector set"
  2085   assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
  2086 proof-
  2087   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  2088   { fix x assume "x\<in>S"
  2089     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
  2090   }
  2091   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
  2092     by (auto intro!: add exI[of _ "b + norm a"])
  2093 qed
  2094 
  2095 
  2096 text{* Some theorems on sups and infs using the notion "bounded". *}
  2097 
  2098 lemma bounded_real:
  2099   fixes S :: "real set"
  2100   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
  2101   by (simp add: bounded_iff)
  2102 
  2103 lemma bounded_has_Sup:
  2104   fixes S :: "real set"
  2105   assumes "bounded S" "S \<noteq> {}"
  2106   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
  2107 proof
  2108   fix x assume "x\<in>S"
  2109   thus "x \<le> Sup S"
  2110     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
  2111 next
  2112   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
  2113     by (metis SupInf.Sup_least)
  2114 qed
  2115 
  2116 lemma Sup_insert:
  2117   fixes S :: "real set"
  2118   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" 
  2119 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) 
  2120 
  2121 lemma Sup_insert_finite:
  2122   fixes S :: "real set"
  2123   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
  2124   apply (rule Sup_insert)
  2125   apply (rule finite_imp_bounded)
  2126   by simp
  2127 
  2128 lemma bounded_has_Inf:
  2129   fixes S :: "real set"
  2130   assumes "bounded S"  "S \<noteq> {}"
  2131   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
  2132 proof
  2133   fix x assume "x\<in>S"
  2134   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
  2135   thus "x \<ge> Inf S" using `x\<in>S`
  2136     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
  2137 next
  2138   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
  2139     by (metis SupInf.Inf_greatest)
  2140 qed
  2141 
  2142 lemma Inf_insert:
  2143   fixes S :: "real set"
  2144   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" 
  2145 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal) 
  2146 lemma Inf_insert_finite:
  2147   fixes S :: "real set"
  2148   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
  2149   by (rule Inf_insert, rule finite_imp_bounded, simp)
  2150 
  2151 
  2152 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
  2153 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
  2154   apply (frule isGlb_isLb)
  2155   apply (frule_tac x = y in isGlb_isLb)
  2156   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
  2157   done
  2158 
  2159 subsection{* Compactness (the definition is the one based on convegent subsequences). *}
  2160 
  2161 definition
  2162   compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
  2163   "compact S \<longleftrightarrow>
  2164    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
  2165        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
  2166 
  2167 text {*
  2168   A metric space (or topological vector space) is said to have the
  2169   Heine-Borel property if every closed and bounded subset is compact.
  2170 *}
  2171 
  2172 class heine_borel =
  2173   assumes bounded_imp_convergent_subsequence:
  2174     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
  2175       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2176 
  2177 lemma bounded_closed_imp_compact:
  2178   fixes s::"'a::heine_borel set"
  2179   assumes "bounded s" and "closed s" shows "compact s"
  2180 proof (unfold compact_def, clarify)
  2181   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  2182   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  2183     using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
  2184   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
  2185   have "l \<in> s" using `closed s` fr l
  2186     unfolding closed_sequential_limits by blast
  2187   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2188     using `l \<in> s` r l by blast
  2189 qed
  2190 
  2191 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
  2192 proof(induct n)
  2193   show "0 \<le> r 0" by auto
  2194 next
  2195   fix n assume "n \<le> r n"
  2196   moreover have "r n < r (Suc n)"
  2197     using assms [unfolded subseq_def] by auto
  2198   ultimately show "Suc n \<le> r (Suc n)" by auto
  2199 qed
  2200 
  2201 lemma eventually_subseq:
  2202   assumes r: "subseq r"
  2203   shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
  2204 unfolding eventually_sequentially
  2205 by (metis subseq_bigger [OF r] le_trans)
  2206 
  2207 lemma lim_subseq:
  2208   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
  2209 unfolding tendsto_def eventually_sequentially o_def
  2210 by (metis subseq_bigger le_trans)
  2211 
  2212 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
  2213   unfolding Ex1_def
  2214   apply (rule_tac x="nat_rec e f" in exI)
  2215   apply (rule conjI)+
  2216 apply (rule def_nat_rec_0, simp)
  2217 apply (rule allI, rule def_nat_rec_Suc, simp)
  2218 apply (rule allI, rule impI, rule ext)
  2219 apply (erule conjE)
  2220 apply (induct_tac x)
  2221 apply (simp add: nat_rec_0)
  2222 apply (erule_tac x="n" in allE)
  2223 apply (simp)
  2224 done
  2225 
  2226 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
  2227   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
  2228   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"
  2229 proof-
  2230   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
  2231   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
  2232   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
  2233     { fix n::nat
  2234       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
  2235       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
  2236       with n have "s N \<le> t - e" using `e>0` by auto
  2237       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  }
  2238     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
  2239     hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto  }
  2240   thus ?thesis by blast
  2241 qed
  2242 
  2243 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
  2244   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
  2245   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
  2246   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
  2247   unfolding monoseq_def incseq_def
  2248   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
  2249   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
  2250 
  2251 lemma compact_real_lemma:
  2252   assumes "\<forall>n::nat. abs(s n) \<le> b"
  2253   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
  2254 proof-
  2255   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
  2256     using seq_monosub[of s] by auto
  2257   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
  2258     unfolding tendsto_iff dist_norm eventually_sequentially by auto
  2259 qed
  2260 
  2261 instance real :: heine_borel
  2262 proof
  2263   fix s :: "real set" and f :: "nat \<Rightarrow> real"
  2264   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2265   then obtain b where b: "\<forall>n. abs (f n) \<le> b"
  2266     unfolding bounded_iff by auto
  2267   obtain l :: real and r :: "nat \<Rightarrow> nat" where
  2268     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  2269     using compact_real_lemma [OF b] by auto
  2270   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2271     by auto
  2272 qed
  2273 
  2274 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
  2275 unfolding bounded_def
  2276 apply clarify
  2277 apply (rule_tac x="x $ i" in exI)
  2278 apply (rule_tac x="e" in exI)
  2279 apply clarify
  2280 apply (rule order_trans [OF dist_nth_le], simp)
  2281 done
  2282 
  2283 lemma compact_lemma:
  2284   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n::finite"
  2285   assumes "bounded s" and "\<forall>n. f n \<in> s"
  2286   shows "\<forall>d.
  2287         \<exists>l r. subseq r \<and>
  2288         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
  2289 proof
  2290   fix d::"'n set" have "finite d" by simp
  2291   thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
  2292       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
  2293   proof(induct d) case empty thus ?case unfolding subseq_def by auto
  2294   next case (insert k d)
  2295     have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component)
  2296     obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
  2297       using insert(3) by auto
  2298     have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
  2299     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
  2300       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
  2301     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
  2302       using r1 and r2 unfolding r_def o_def subseq_def by auto
  2303     moreover
  2304     def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
  2305     { fix e::real assume "e>0"
  2306       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
  2307       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
  2308       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
  2309         by (rule eventually_subseq)
  2310       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
  2311         using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
  2312     }
  2313     ultimately show ?case by auto
  2314   qed
  2315 qed
  2316 
  2317 instance "^" :: (heine_borel, finite) heine_borel
  2318 proof
  2319   fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
  2320   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2321   then obtain l r where r: "subseq r"
  2322     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
  2323     using compact_lemma [OF s f] by blast
  2324   let ?d = "UNIV::'b set"
  2325   { fix e::real assume "e>0"
  2326     hence "0 < e / (real_of_nat (card ?d))"
  2327       using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
  2328     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
  2329       by simp
  2330     moreover
  2331     { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
  2332       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
  2333         unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)
  2334       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
  2335         by (rule setsum_strict_mono) (simp_all add: n)
  2336       finally have "dist (f (r n)) l < e" by simp
  2337     }
  2338     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
  2339       by (rule eventually_elim1)
  2340   }
  2341   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
  2342   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
  2343 qed
  2344 
  2345 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
  2346 unfolding bounded_def
  2347 apply clarify
  2348 apply (rule_tac x="a" in exI)
  2349 apply (rule_tac x="e" in exI)
  2350 apply clarsimp
  2351 apply (drule (1) bspec)
  2352 apply (simp add: dist_Pair_Pair)
  2353 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
  2354 done
  2355 
  2356 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
  2357 unfolding bounded_def
  2358 apply clarify
  2359 apply (rule_tac x="b" in exI)
  2360 apply (rule_tac x="e" in exI)
  2361 apply clarsimp
  2362 apply (drule (1) bspec)
  2363 apply (simp add: dist_Pair_Pair)
  2364 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
  2365 done
  2366 
  2367 instance "*" :: (heine_borel, heine_borel) heine_borel
  2368 proof
  2369   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
  2370   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
  2371   from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
  2372   from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
  2373   obtain l1 r1 where r1: "subseq r1"
  2374     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
  2375     using bounded_imp_convergent_subsequence [OF s1 f1]
  2376     unfolding o_def by fast
  2377   from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
  2378   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
  2379   obtain l2 r2 where r2: "subseq r2"
  2380     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
  2381     using bounded_imp_convergent_subsequence [OF s2 f2]
  2382     unfolding o_def by fast
  2383   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
  2384     using lim_subseq [OF r2 l1] unfolding o_def .
  2385   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
  2386     using tendsto_Pair [OF l1' l2] unfolding o_def by simp
  2387   have r: "subseq (r1 \<circ> r2)"
  2388     using r1 r2 unfolding subseq_def by simp
  2389   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  2390     using l r by fast
  2391 qed
  2392 
  2393 subsection{* Completeness. *}
  2394 
  2395 lemma cauchy_def:
  2396   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
  2397 unfolding Cauchy_def by blast
  2398 
  2399 definition
  2400   complete :: "'a::metric_space set \<Rightarrow> bool" where
  2401   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
  2402                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"
  2403 
  2404 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
  2405 proof-
  2406   { assume ?rhs
  2407     { fix e::real
  2408       assume "e>0"
  2409       with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
  2410         by (erule_tac x="e/2" in allE) auto
  2411       { fix n m
  2412         assume nm:"N \<le> m \<and> N \<le> n"
  2413         hence "dist (s m) (s n) < e" using N
  2414           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
  2415           by blast
  2416       }
  2417       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
  2418         by blast
  2419     }
  2420     hence ?lhs
  2421       unfolding cauchy_def
  2422       by blast
  2423   }
  2424   thus ?thesis
  2425     unfolding cauchy_def
  2426     using dist_triangle_half_l
  2427     by blast
  2428 qed
  2429 
  2430 lemma convergent_imp_cauchy:
  2431  "(s ---> l) sequentially ==> Cauchy s"
  2432 proof(simp only: cauchy_def, rule, rule)
  2433   fix e::real assume "e>0" "(s ---> l) sequentially"
  2434   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
  2435   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
  2436 qed
  2437 
  2438 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded {y. (\<exists>n::nat. y = s n)}"
  2439 proof-
  2440   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
  2441   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
  2442   moreover
  2443   have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
  2444   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
  2445     unfolding bounded_any_center [where a="s N"] by auto
  2446   ultimately show "?thesis"
  2447     unfolding bounded_any_center [where a="s N"]
  2448     apply(rule_tac x="max a 1" in exI) apply auto
  2449     apply(erule_tac x=n in allE) apply(erule_tac x=n in ballE) by auto
  2450 qed
  2451 
  2452 lemma compact_imp_complete: assumes "compact s" shows "complete s"
  2453 proof-
  2454   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
  2455     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
  2456 
  2457     note lr' = subseq_bigger [OF lr(2)]
  2458 
  2459     { fix e::real assume "e>0"
  2460       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
  2461       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
  2462       { fix n::nat assume n:"n \<ge> max N M"
  2463         have "dist ((f \<circ> r) n) l < e/2" using n M by auto
  2464         moreover have "r n \<ge> N" using lr'[of n] n by auto
  2465         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
  2466         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)  }
  2467       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }
  2468     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto  }
  2469   thus ?thesis unfolding complete_def by auto
  2470 qed
  2471 
  2472 instance heine_borel < complete_space
  2473 proof
  2474   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  2475   hence "bounded (range f)" unfolding image_def
  2476     using cauchy_imp_bounded [of f] by auto
  2477   hence "compact (closure (range f))"
  2478     using bounded_closed_imp_compact [of "closure (range f)"] by auto
  2479   hence "complete (closure (range f))"
  2480     using compact_imp_complete by auto
  2481   moreover have "\<forall>n. f n \<in> closure (range f)"
  2482     using closure_subset [of "range f"] by auto
  2483   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
  2484     using `Cauchy f` unfolding complete_def by auto
  2485   then show "convergent f"
  2486     unfolding convergent_def LIMSEQ_conv_tendsto [symmetric] by auto
  2487 qed
  2488 
  2489 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
  2490 proof(simp add: complete_def, rule, rule)
  2491   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  2492   hence "convergent f" by (rule Cauchy_convergent)
  2493   hence "\<exists>l. f ----> l" unfolding convergent_def .  
  2494   thus "\<exists>l. (f ---> l) sequentially" unfolding LIMSEQ_conv_tendsto .
  2495 qed
  2496 
  2497 lemma complete_imp_closed: assumes "complete s" shows "closed s"
  2498 proof -
  2499   { fix x assume "x islimpt s"
  2500     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
  2501       unfolding islimpt_sequential by auto
  2502     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
  2503       using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
  2504     hence "x \<in> s"  using Lim_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
  2505   }
  2506   thus "closed s" unfolding closed_limpt by auto
  2507 qed
  2508 
  2509 lemma complete_eq_closed:
  2510   fixes s :: "'a::complete_space set"
  2511   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
  2512 proof
  2513   assume ?lhs thus ?rhs by (rule complete_imp_closed)
  2514 next
  2515   assume ?rhs
  2516   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
  2517     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
  2518     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  }
  2519   thus ?lhs unfolding complete_def by auto
  2520 qed
  2521 
  2522 lemma convergent_eq_cauchy:
  2523   fixes s :: "nat \<Rightarrow> 'a::complete_space"
  2524   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
  2525 proof
  2526   assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
  2527   thus ?rhs using convergent_imp_cauchy by auto
  2528 next
  2529   assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
  2530 qed
  2531 
  2532 lemma convergent_imp_bounded:
  2533   fixes s :: "nat \<Rightarrow> 'a::metric_space"
  2534   shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
  2535   using convergent_imp_cauchy[of s]
  2536   using cauchy_imp_bounded[of s]
  2537   unfolding image_def
  2538   by auto
  2539 
  2540 subsection{* Total boundedness. *}
  2541 
  2542 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
  2543   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
  2544 declare helper_1.simps[simp del]
  2545 
  2546 lemma compact_imp_totally_bounded:
  2547   assumes "compact s"
  2548   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
  2549 proof(rule, rule, rule ccontr)
  2550   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)"
  2551   def x \<equiv> "helper_1 s e"
  2552   { fix n
  2553     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
  2554     proof(induct_tac rule:nat_less_induct)
  2555       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
  2556       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
  2557       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
  2558       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
  2559       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
  2560         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
  2561       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
  2562     qed }
  2563   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
  2564   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
  2565   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
  2566   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
  2567   show False
  2568     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
  2569     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
  2570     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
  2571 qed
  2572 
  2573 subsection{* Heine-Borel theorem (following Burkill \& Burkill vol. 2) *}
  2574 
  2575 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
  2576   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"
  2577   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
  2578 proof(rule ccontr)
  2579   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
  2580   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
  2581   { fix n::nat
  2582     have "1 / real (n + 1) > 0" by auto
  2583     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 }
  2584   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
  2585   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)"
  2586     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
  2587 
  2588   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
  2589     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
  2590 
  2591   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
  2592   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
  2593     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
  2594 
  2595   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
  2596     using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
  2597 
  2598   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
  2599   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
  2600     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
  2601     using subseq_bigger[OF r, of "N1 + N2"] by auto
  2602 
  2603   def x \<equiv> "(f (r (N1 + N2)))"
  2604   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
  2605     using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
  2606   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
  2607   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
  2608 
  2609   have "dist x l < e/2" using N1 unfolding x_def o_def by auto
  2610   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
  2611 
  2612   thus False using e and `y\<notin>b` by auto
  2613 qed
  2614 
  2615 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
  2616                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
  2617 proof clarify
  2618   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
  2619   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
  2620   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
  2621   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
  2622   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
  2623 
  2624   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
  2625   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
  2626 
  2627   have "finite (bb ` k)" using k(1) by auto
  2628   moreover
  2629   { fix x assume "x\<in>s"
  2630     hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3)  unfolding subset_eq by auto
  2631     hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
  2632     hence "x \<in> \<Union>(bb ` k)" using  Union_iff[of x "bb ` k"] by auto
  2633   }
  2634   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
  2635 qed
  2636 
  2637 subsection{* Bolzano-Weierstrass property. *}
  2638 
  2639 lemma heine_borel_imp_bolzano_weierstrass:
  2640   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'))"
  2641           "infinite t"  "t \<subseteq> s"
  2642   shows "\<exists>x \<in> s. x islimpt t"
  2643 proof(rule ccontr)
  2644   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
  2645   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
  2646     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
  2647   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
  2648     using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
  2649   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
  2650   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
  2651     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
  2652     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  }
  2653   hence "infinite (f ` t)" using assms(2) using finite_imageD[unfolded inj_on_def, of f t] by auto
  2654   moreover
  2655   { fix x assume "x\<in>t" "f x \<notin> g"
  2656     from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
  2657     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
  2658     hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
  2659     hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto  }
  2660   hence "f ` t \<subseteq> g" by auto
  2661   ultimately show False using g(2) using finite_subset by auto
  2662 qed
  2663 
  2664 subsection{* Complete the chain of compactness variants. *}
  2665 
  2666 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
  2667   "helper_2 beyond 0 = beyond 0" |
  2668   "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
  2669 
  2670 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
  2671   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2672   shows "bounded s"
  2673 proof(rule ccontr)
  2674   assume "\<not> bounded s"
  2675   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
  2676     unfolding bounded_any_center [where a=undefined]
  2677     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
  2678   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
  2679     unfolding linorder_not_le by auto
  2680   def x \<equiv> "helper_2 beyond"
  2681 
  2682   { fix m n ::nat assume "m<n"
  2683     hence "dist undefined (x m) + 1 < dist undefined (x n)"
  2684     proof(induct n)
  2685       case 0 thus ?case by auto
  2686     next
  2687       case (Suc n)
  2688       have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
  2689         unfolding x_def and helper_2.simps
  2690         using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
  2691       thus ?case proof(cases "m < n")
  2692         case True thus ?thesis using Suc and * by auto
  2693       next
  2694         case False hence "m = n" using Suc(2) by auto
  2695         thus ?thesis using * by auto
  2696       qed
  2697     qed  } note * = this
  2698   { fix m n ::nat assume "m\<noteq>n"
  2699     have "1 < dist (x m) (x n)"
  2700     proof(cases "m<n")
  2701       case True
  2702       hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
  2703       thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
  2704     next
  2705       case False hence "n<m" using `m\<noteq>n` by auto
  2706       hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
  2707       thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
  2708     qed  } note ** = this
  2709   { fix a b assume "x a = x b" "a \<noteq> b"
  2710     hence False using **[of a b] by auto  }
  2711   hence "inj x" unfolding inj_on_def by auto
  2712   moreover
  2713   { fix n::nat
  2714     have "x n \<in> s"
  2715     proof(cases "n = 0")
  2716       case True thus ?thesis unfolding x_def using beyond by auto
  2717     next
  2718       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
  2719       thus ?thesis unfolding x_def using beyond by auto
  2720     qed  }
  2721   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
  2722 
  2723   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
  2724   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
  2725   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"]]
  2726     unfolding dist_nz by auto
  2727   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
  2728 qed
  2729 
  2730 lemma sequence_infinite_lemma:
  2731   fixes l :: "'a::metric_space" (* TODO: generalize *)
  2732   assumes "\<forall>n::nat. (f n  \<noteq> l)"  "(f ---> l) sequentially"
  2733   shows "infinite {y. (\<exists> n. y = f n)}"
  2734 proof(rule ccontr)
  2735   let ?A = "(\<lambda>x. dist x l) ` {y. \<exists>n. y = f n}"
  2736   assume "\<not> infinite {y. \<exists>n. y = f n}"
  2737   hence **:"finite ?A" "?A \<noteq> {}" by auto
  2738   obtain k where k:"dist (f k) l = Min ?A" using Min_in[OF **] by auto
  2739   have "0 < Min ?A" using assms(1) unfolding dist_nz unfolding Min_gr_iff[OF **] by auto
  2740   then obtain N where "dist (f N) l < Min ?A" using assms(2)[unfolded Lim_sequentially, THEN spec[where x="Min ?A"]] by auto
  2741   moreover have "dist (f N) l \<in> ?A" by auto
  2742   ultimately show False using Min_le[OF **(1), of "dist (f N) l"] by auto
  2743 qed
  2744 
  2745 lemma sequence_unique_limpt:
  2746   fixes l :: "'a::metric_space" (* TODO: generalize *)
  2747   assumes "\<forall>n::nat. (f n \<noteq> l)"  "(f ---> l) sequentially"  "l' islimpt {y.  (\<exists>n. y = f n)}"
  2748   shows "l' = l"
  2749 proof(rule ccontr)
  2750   def e \<equiv> "dist l' l"
  2751   assume "l' \<noteq> l" hence "e>0" unfolding dist_nz e_def by auto
  2752   then obtain N::nat where N:"\<forall>n\<ge>N. dist (f n) l < e / 2"
  2753     using assms(2)[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
  2754   def d \<equiv> "Min (insert (e/2) ((\<lambda>n. if dist (f n) l' = 0 then e/2 else dist (f n) l') ` {0 .. N}))"
  2755   have "d>0" using `e>0` unfolding d_def e_def using zero_le_dist[of _ l', unfolded order_le_less] by auto
  2756   obtain k where k:"f k \<noteq> l'"  "dist (f k) l' < d" using `d>0` and assms(3)[unfolded islimpt_approachable, THEN spec[where x="d"]] by auto
  2757   have "k\<ge>N" using k(1)[unfolded dist_nz] using k(2)[unfolded d_def]
  2758     by force
  2759   hence "dist l' l < e" using N[THEN spec[where x=k]] using k(2)[unfolded d_def] and dist_triangle_half_r[of "f k" l' e l] by auto
  2760   thus False unfolding e_def by auto
  2761 qed
  2762 
  2763 lemma bolzano_weierstrass_imp_closed:
  2764   fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
  2765   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2766   shows "closed s"
  2767 proof-
  2768   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
  2769     hence "l \<in> s"
  2770     proof(cases "\<forall>n. x n \<noteq> l")
  2771       case False thus "l\<in>s" using as(1) by auto
  2772     next
  2773       case True note cas = this
  2774       with as(2) have "infinite {y. \<exists>n. y = x n}" using sequence_infinite_lemma[of x l] by auto
  2775       then obtain l' where "l'\<in>s" "l' islimpt {y. \<exists>n. y = x n}" using assms[THEN spec[where x="{y. \<exists>n. y = x n}"]] as(1) by auto
  2776       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
  2777     qed  }
  2778   thus ?thesis unfolding closed_sequential_limits by fast
  2779 qed
  2780 
  2781 text{* Hence express everything as an equivalence.   *}
  2782 
  2783 lemma compact_eq_heine_borel:
  2784   fixes s :: "'a::heine_borel set"
  2785   shows "compact s \<longleftrightarrow>
  2786            (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
  2787                --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
  2788 proof
  2789   assume ?lhs thus ?rhs using compact_imp_heine_borel[of s] by blast
  2790 next
  2791   assume ?rhs
  2792   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
  2793     by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
  2794   thus ?lhs using bolzano_weierstrass_imp_bounded[of s] bolzano_weierstrass_imp_closed[of s] bounded_closed_imp_compact[of s] by blast
  2795 qed
  2796 
  2797 lemma compact_eq_bolzano_weierstrass:
  2798   fixes s :: "'a::heine_borel set"
  2799   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
  2800 proof
  2801   assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2802 next
  2803   assume ?rhs thus ?lhs using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed bounded_closed_imp_compact by auto
  2804 qed
  2805 
  2806 lemma compact_eq_bounded_closed:
  2807   fixes s :: "'a::heine_borel set"
  2808   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")
  2809 proof
  2810   assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
  2811 next
  2812   assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
  2813 qed
  2814 
  2815 lemma compact_imp_bounded:
  2816   fixes s :: "'a::metric_space set"
  2817   shows "compact s ==> bounded s"
  2818 proof -
  2819   assume "compact s"
  2820   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')"
  2821     by (rule compact_imp_heine_borel)
  2822   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
  2823     using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2824   thus "bounded s"
  2825     by (rule bolzano_weierstrass_imp_bounded)
  2826 qed
  2827 
  2828 lemma compact_imp_closed:
  2829   fixes s :: "'a::metric_space set"
  2830   shows "compact s ==> closed s"
  2831 proof -
  2832   assume "compact s"
  2833   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')"
  2834     by (rule compact_imp_heine_borel)
  2835   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
  2836     using heine_borel_imp_bolzano_weierstrass[of s] by auto
  2837   thus "closed s"
  2838     by (rule bolzano_weierstrass_imp_closed)
  2839 qed
  2840 
  2841 text{* In particular, some common special cases. *}
  2842 
  2843 lemma compact_empty[simp]:
  2844  "compact {}"
  2845   unfolding compact_def
  2846   by simp
  2847 
  2848 (* TODO: can any of the next 3 lemmas be generalized to metric spaces? *)
  2849 
  2850   (* FIXME : Rename *)
  2851 lemma compact_union[intro]:
  2852   fixes s t :: "'a::heine_borel set"
  2853   shows "compact s \<Longrightarrow> compact t ==> compact (s \<union> t)"
  2854   unfolding compact_eq_bounded_closed
  2855   using bounded_Un[of s t]
  2856   using closed_Un[of s t]
  2857   by simp
  2858 
  2859 lemma compact_inter[intro]:
  2860   fixes s t :: "'a::heine_borel set"
  2861   shows "compact s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
  2862   unfolding compact_eq_bounded_closed
  2863   using bounded_Int[of s t]
  2864   using closed_Int[of s t]
  2865   by simp
  2866 
  2867 lemma compact_inter_closed[intro]:
  2868   fixes s t :: "'a::heine_borel set"
  2869   shows "compact s \<Longrightarrow> closed t ==> compact (s \<inter> t)"
  2870   unfolding compact_eq_bounded_closed
  2871   using closed_Int[of s t]
  2872   using bounded_subset[of "s \<inter> t" s]
  2873   by blast
  2874 
  2875 lemma closed_inter_compact[intro]:
  2876   fixes s t :: "'a::heine_borel set"
  2877   shows "closed s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
  2878 proof-
  2879   assume "closed s" "compact t"
  2880   moreover
  2881   have "s \<inter> t = t \<inter> s" by auto ultimately
  2882   show ?thesis
  2883     using compact_inter_closed[of t s]
  2884     by auto
  2885 qed
  2886 
  2887 lemma closed_sing [simp]:
  2888   fixes a :: "'a::metric_space"
  2889   shows "closed {a}"
  2890   apply (clarsimp simp add: closed_def open_dist)
  2891   apply (rule ccontr)
  2892   apply (drule_tac x="dist x a" in spec)
  2893   apply (simp add: dist_nz dist_commute)
  2894   done
  2895 
  2896 lemma finite_imp_closed:
  2897   fixes s :: "'a::metric_space set"
  2898   shows "finite s ==> closed s"
  2899 proof (induct set: finite)
  2900   case empty show "closed {}" by simp
  2901 next
  2902   case (insert x F)
  2903   hence "closed ({x} \<union> F)" by (simp only: closed_Un closed_sing)
  2904   thus "closed (insert x F)" by simp
  2905 qed
  2906 
  2907 lemma finite_imp_compact:
  2908   fixes s :: "'a::heine_borel set"
  2909   shows "finite s ==> compact s"
  2910   unfolding compact_eq_bounded_closed
  2911   using finite_imp_closed finite_imp_bounded
  2912   by blast
  2913 
  2914 lemma compact_sing [simp]: "compact {a}"
  2915   unfolding compact_def o_def subseq_def
  2916   by (auto simp add: tendsto_const)
  2917 
  2918 lemma compact_cball[simp]:
  2919   fixes x :: "'a::heine_borel"
  2920   shows "compact(cball x e)"
  2921   using compact_eq_bounded_closed bounded_cball closed_cball
  2922   by blast
  2923 
  2924 lemma compact_frontier_bounded[intro]:
  2925   fixes s :: "'a::heine_borel set"
  2926   shows "bounded s ==> compact(frontier s)"
  2927   unfolding frontier_def
  2928   using compact_eq_bounded_closed
  2929   by blast
  2930 
  2931 lemma compact_frontier[intro]:
  2932   fixes s :: "'a::heine_borel set"
  2933   shows "compact s ==> compact (frontier s)"
  2934   using compact_eq_bounded_closed compact_frontier_bounded
  2935   by blast
  2936 
  2937 lemma frontier_subset_compact:
  2938   fixes s :: "'a::heine_borel set"
  2939   shows "compact s ==> frontier s \<subseteq> s"
  2940   using frontier_subset_closed compact_eq_bounded_closed
  2941   by blast
  2942 
  2943 lemma open_delete:
  2944   fixes s :: "'a::metric_space set"
  2945   shows "open s ==> open(s - {x})"
  2946   using open_Diff[of s "{x}"] closed_sing
  2947   by blast
  2948 
  2949 text{* Finite intersection property. I could make it an equivalence in fact. *}
  2950 
  2951 lemma compact_imp_fip:
  2952   fixes s :: "'a::heine_borel set"
  2953   assumes "compact s"  "\<forall>t \<in> f. closed t"
  2954         "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
  2955   shows "s \<inter> (\<Inter> f) \<noteq> {}"
  2956 proof
  2957   assume as:"s \<inter> (\<Inter> f) = {}"
  2958   hence "s \<subseteq> \<Union>op - UNIV ` f" by auto
  2959   moreover have "Ball (op - UNIV ` f) open" using open_Diff closed_Diff using assms(2) by auto
  2960   ultimately obtain f' where f':"f' \<subseteq> op - UNIV ` f"  "finite f'"  "s \<subseteq> \<Union>f'" using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="(\<lambda>t. UNIV - t) ` f"]] by auto
  2961   hence "finite (op - UNIV ` f') \<and> op - UNIV ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
  2962   hence "s \<inter> \<Inter>op - UNIV ` f' \<noteq> {}" using assms(3)[THEN spec[where x="op - UNIV ` f'"]] by auto
  2963   thus False using f'(3) unfolding subset_eq and Union_iff by blast
  2964 qed
  2965 
  2966 subsection{* Bounded closed nest property (proof does not use Heine-Borel).            *}
  2967 
  2968 lemma bounded_closed_nest:
  2969   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
  2970   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
  2971   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
  2972 proof-
  2973   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
  2974   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
  2975 
  2976   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
  2977     unfolding compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast
  2978 
  2979   { fix n::nat
  2980     { fix e::real assume "e>0"
  2981       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
  2982       hence "dist ((x \<circ> r) (max N n)) l < e" by auto
  2983       moreover
  2984       have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
  2985       hence "(x \<circ> r) (max N n) \<in> s n"
  2986         using x apply(erule_tac x=n in allE)
  2987         using x apply(erule_tac x="r (max N n)" in allE)
  2988         using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
  2989       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
  2990     }
  2991     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
  2992   }
  2993   thus ?thesis by auto
  2994 qed
  2995 
  2996 text{* Decreasing case does not even need compactness, just completeness.        *}
  2997 
  2998 lemma decreasing_closed_nest:
  2999   assumes "\<forall>n. closed(s n)"
  3000           "\<forall>n. (s n \<noteq> {})"
  3001           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3002           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
  3003   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
  3004 proof-
  3005   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
  3006   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
  3007   then obtain t where t: "\<forall>n. t n \<in> s n" by auto
  3008   { fix e::real assume "e>0"
  3009     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
  3010     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
  3011       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+
  3012       hence "dist (t m) (t n) < e" using N by auto
  3013     }
  3014     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
  3015   }
  3016   hence  "Cauchy t" unfolding cauchy_def by auto
  3017   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
  3018   { fix n::nat
  3019     { fix e::real assume "e>0"
  3020       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
  3021       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
  3022       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
  3023     }
  3024     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
  3025   }
  3026   then show ?thesis by auto
  3027 qed
  3028 
  3029 text{* Strengthen it to the intersection actually being a singleton.             *}
  3030 
  3031 lemma decreasing_closed_nest_sing:
  3032   assumes "\<forall>n. closed(s n)"
  3033           "\<forall>n. s n \<noteq> {}"
  3034           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3035           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
  3036   shows "\<exists>a::'a::heine_borel. \<Inter> {t. (\<exists>n::nat. t = s n)} = {a}"
  3037 proof-
  3038   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
  3039   { fix b assume b:"b \<in> \<Inter>{t. \<exists>n. t = s n}"
  3040     { fix e::real assume "e>0"
  3041       hence "dist a b < e" using assms(4 )using b using a by blast
  3042     }
  3043     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz real_less_def)
  3044   }
  3045   with a have "\<Inter>{t. \<exists>n. t = s n} = {a}"  by auto
  3046   thus ?thesis by auto
  3047 qed
  3048 
  3049 text{* Cauchy-type criteria for uniform convergence. *}
  3050 
  3051 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
  3052  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
  3053   (\<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")
  3054 proof(rule)
  3055   assume ?lhs
  3056   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
  3057   { fix e::real assume "e>0"
  3058     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
  3059     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
  3060       hence "dist (s m x) (s n x) < e"
  3061         using N[THEN spec[where x=m], THEN spec[where x=x]]
  3062         using N[THEN spec[where x=n], THEN spec[where x=x]]
  3063         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }
  3064     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  }
  3065   thus ?rhs by auto
  3066 next
  3067   assume ?rhs
  3068   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
  3069   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
  3070     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
  3071   { fix e::real assume "e>0"
  3072     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"
  3073       using `?rhs`[THEN spec[where x="e/2"]] by auto
  3074     { fix x assume "P x"
  3075       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
  3076         using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
  3077       fix n::nat assume "n\<ge>N"
  3078       hence "dist(s n x)(l x) < e"  using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
  3079         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)  }
  3080     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
  3081   thus ?lhs by auto
  3082 qed
  3083 
  3084 lemma uniformly_cauchy_imp_uniformly_convergent:
  3085   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
  3086   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"
  3087           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
  3088   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
  3089 proof-
  3090   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"
  3091     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
  3092   moreover
  3093   { fix x assume "P x"
  3094     hence "l x = l' x" using Lim_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
  3095       using l and assms(2) unfolding Lim_sequentially by blast  }
  3096   ultimately show ?thesis by auto
  3097 qed
  3098 
  3099 subsection{* Define continuity over a net to take in restrictions of the set. *}
  3100 
  3101 definition
  3102   continuous :: "'a::t2_space net \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
  3103   "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
  3104 
  3105 lemma continuous_trivial_limit:
  3106  "trivial_limit net ==> continuous net f"
  3107   unfolding continuous_def tendsto_def trivial_limit_eq by auto
  3108 
  3109 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
  3110   unfolding continuous_def
  3111   unfolding tendsto_def
  3112   using netlimit_within[of x s]
  3113   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
  3114 
  3115 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
  3116   using continuous_within [of x UNIV f] by (simp add: within_UNIV)
  3117 
  3118 lemma continuous_at_within:
  3119   assumes "continuous (at x) f"  shows "continuous (at x within s) f"
  3120   using assms unfolding continuous_at continuous_within
  3121   by (rule Lim_at_within)
  3122 
  3123 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
  3124 
  3125 lemma continuous_within_eps_delta:
  3126   "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)"
  3127   unfolding continuous_within and Lim_within
  3128   apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
  3129 
  3130 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.
  3131                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
  3132   using continuous_within_eps_delta[of x UNIV f]
  3133   unfolding within_UNIV by blast
  3134 
  3135 text{* Versions in terms of open balls. *}
  3136 
  3137 lemma continuous_within_ball:
  3138  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  3139                             f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3140 proof
  3141   assume ?lhs
  3142   { fix e::real assume "e>0"
  3143     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"
  3144       using `?lhs`[unfolded continuous_within Lim_within] by auto
  3145     { fix y assume "y\<in>f ` (ball x d \<inter> s)"
  3146       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
  3147         apply (auto simp add: dist_commute mem_ball) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
  3148     }
  3149     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)  }
  3150   thus ?rhs by auto
  3151 next
  3152   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
  3153     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
  3154 qed
  3155 
  3156 lemma continuous_at_ball:
  3157   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3158 proof
  3159   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3160     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)
  3161     unfolding dist_nz[THEN sym] by auto
  3162 next
  3163   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3164     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)
  3165 qed
  3166 
  3167 text{* For setwise continuity, just start from the epsilon-delta definitions. *}
  3168 
  3169 definition
  3170   continuous_on :: "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
  3171   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d::real>0. \<forall>x' \<in> s. dist x' x < d --> dist (f x') (f x) < e)"
  3172 
  3173 
  3174 definition
  3175   uniformly_continuous_on ::
  3176     "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
  3177   "uniformly_continuous_on s f \<longleftrightarrow>
  3178         (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall> x'\<in>s. dist x' x < d
  3179                            --> dist (f x') (f x) < e)"
  3180 
  3181 text{* Some simple consequential lemmas. *}
  3182 
  3183 lemma uniformly_continuous_imp_continuous:
  3184  " uniformly_continuous_on s f ==> continuous_on s f"
  3185   unfolding uniformly_continuous_on_def continuous_on_def by blast
  3186 
  3187 lemma continuous_at_imp_continuous_within:
  3188  "continuous (at x) f ==> continuous (at x within s) f"
  3189   unfolding continuous_within continuous_at using Lim_at_within by auto
  3190 
  3191 lemma continuous_at_imp_continuous_on: assumes "(\<forall>x \<in> s. continuous (at x) f)"
  3192   shows "continuous_on s f"
  3193 proof(simp add: continuous_at continuous_on_def, rule, rule, rule)
  3194   fix x and e::real assume "x\<in>s" "e>0"
  3195   hence "eventually (\<lambda>xa. dist (f xa) (f x) < e) (at x)" using assms unfolding continuous_at tendsto_iff by auto
  3196   then obtain d where d:"d>0" "\<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e" unfolding eventually_at by auto
  3197   { fix x' assume "\<not> 0 < dist x' x"
  3198     hence "x=x'"
  3199       using dist_nz[of x' x] by auto
  3200     hence "dist (f x') (f x) < e" using `e>0` by auto
  3201   }
  3202   thus "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using d by auto
  3203 qed
  3204 
  3205 lemma continuous_on_eq_continuous_within:
  3206  "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)" (is "?lhs = ?rhs")
  3207 proof
  3208   assume ?rhs
  3209   { fix x assume "x\<in>s"
  3210     fix e::real assume "e>0"
  3211     assume "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
  3212     then 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) < e" by auto
  3213     { fix x' assume as:"x'\<in>s" "dist x' x < d"
  3214       hence "dist (f x') (f x) < e" using `e>0` d `x'\<in>s` dist_eq_0_iff[of x' x] zero_le_dist[of x' x] as(2) by (metis dist_eq_0_iff dist_nz) }
  3215     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using `d>0` by auto
  3216   }
  3217   thus ?lhs using `?rhs` unfolding continuous_on_def continuous_within Lim_within by auto
  3218 next
  3219   assume ?lhs
  3220   thus ?rhs unfolding continuous_on_def continuous_within Lim_within by blast
  3221 qed
  3222 
  3223 lemma continuous_on:
  3224  "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. (f ---> f(x)) (at x within s))"
  3225   by (auto simp add: continuous_on_eq_continuous_within continuous_within)
  3226 
  3227 lemma continuous_on_eq_continuous_at:
  3228  "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
  3229   by (auto simp add: continuous_on continuous_at Lim_within_open)
  3230 
  3231 lemma continuous_within_subset:
  3232  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
  3233              ==> continuous (at x within t) f"
  3234   unfolding continuous_within by(metis Lim_within_subset)
  3235 
  3236 lemma continuous_on_subset:
  3237  "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
  3238   unfolding continuous_on by (metis subset_eq Lim_within_subset)
  3239 
  3240 lemma continuous_on_interior:
  3241  "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
  3242 unfolding interior_def
  3243 apply simp
  3244 by (meson continuous_on_eq_continuous_at continuous_on_subset)
  3245 
  3246 lemma continuous_on_eq:
  3247  "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f
  3248            ==> continuous_on s g"
  3249   by (simp add: continuous_on_def)
  3250 
  3251 text{* Characterization of various kinds of continuity in terms of sequences.  *}
  3252 
  3253 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
  3254 lemma continuous_within_sequentially:
  3255   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3256   shows "continuous (at a within s) f \<longleftrightarrow>
  3257                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
  3258                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
  3259 proof
  3260   assume ?lhs
  3261   { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (x n) a < e"
  3262     fix e::real assume "e>0"
  3263     from `?lhs` obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) (f a) < e" unfolding continuous_within Lim_within using `e>0` by auto
  3264     from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
  3265     hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
  3266       apply(rule_tac  x=N in exI) using N d  apply auto using x(1)
  3267       apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
  3268       apply(erule_tac x="x n" in ballE)  apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
  3269   }
  3270   thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
  3271 next
  3272   assume ?rhs
  3273   { fix e::real assume "e>0"
  3274     assume "\<not> (\<exists>d>0. \<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) (f a) < e)"
  3275     hence "\<forall>d. \<exists>x. d>0 \<longrightarrow> x\<in>s \<and> (0 < dist x a \<and> dist x a < d \<and> \<not> dist (f x) (f a) < e)" by blast
  3276     then obtain x where x:"\<forall>d>0. x d \<in> s \<and> (0 < dist (x d) a \<and> dist (x d) a < d \<and> \<not> dist (f (x d)) (f a) < e)"
  3277       using choice[of "\<lambda>d x.0<d \<longrightarrow> x\<in>s \<and> (0 < dist x a \<and> dist x a < d \<and> \<not> dist (f x) (f a) < e)"] by auto
  3278     { fix d::real assume "d>0"
  3279       hence "\<exists>N::nat. inverse (real (N + 1)) < d" using real_arch_inv[of d] by (auto, rule_tac x="n - 1" in exI)auto
  3280       then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
  3281       { fix n::nat assume n:"n\<ge>N"
  3282         hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
  3283         moreover have "inverse (real (n + 1)) < d" using N n by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
  3284         ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
  3285       }
  3286       hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
  3287     }
  3288     hence "(\<forall>n::nat. x (inverse (real (n + 1))) \<in> s) \<and> (\<forall>e>0. \<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < e)" using x by auto
  3289     hence "\<forall>e>0. \<exists>N::nat. \<forall>n\<ge>N. dist (f (x (inverse (real (n + 1))))) (f a) < e"  using `?rhs`[THEN spec[where x="\<lambda>n::nat. x (inverse (real (n+1)))"], unfolded Lim_sequentially] by auto
  3290     hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
  3291   }
  3292   thus ?lhs  unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
  3293 qed
  3294 
  3295 lemma continuous_at_sequentially:
  3296   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3297   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
  3298                   --> ((f o x) ---> f a) sequentially)"
  3299   using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
  3300 
  3301 lemma continuous_on_sequentially:
  3302  "continuous_on s f \<longleftrightarrow>  (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
  3303                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
  3304 proof
  3305   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
  3306 next
  3307   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
  3308 qed
  3309 
  3310 lemma uniformly_continuous_on_sequentially:
  3311   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  3312   shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
  3313                     ((\<lambda>n. x n - y n) ---> 0) sequentially
  3314                     \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
  3315 proof
  3316   assume ?lhs
  3317   { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. x n - y n) ---> 0) sequentially"
  3318     { fix e::real assume "e>0"
  3319       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"
  3320         using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
  3321       obtain N where N:"\<forall>n\<ge>N. norm (x n - y n - 0) < d" using xy[unfolded Lim_sequentially dist_norm] and `d>0` by auto
  3322       { fix n assume "n\<ge>N"
  3323         hence "norm (f (x n) - f (y n) - 0) < e"
  3324           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
  3325           unfolding dist_commute and dist_norm by simp  }
  3326       hence "\<exists>N. \<forall>n\<ge>N. norm (f (x n) - f (y n) - 0) < e"  by auto  }
  3327     hence "((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially" unfolding Lim_sequentially and dist_norm by auto  }
  3328   thus ?rhs by auto
  3329 next
  3330   assume ?rhs
  3331   { assume "\<not> ?lhs"
  3332     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
  3333     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"
  3334       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
  3335       by (auto simp add: dist_commute)
  3336     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
  3337     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
  3338     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"
  3339       unfolding x_def and y_def using fa by auto
  3340     have 1:"\<And>(x::'a) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
  3341     have 2:"\<And>(x::'b) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
  3342     { fix e::real assume "e>0"
  3343       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
  3344       { fix n::nat assume "n\<ge>N"
  3345         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
  3346         also have "\<dots> < e" using N by auto
  3347         finally have "inverse (real n + 1) < e" by auto
  3348         hence "dist (x n - y n) 0 < e" unfolding 1 using xy0[THEN spec[where x=n]] by auto  }
  3349       hence "\<exists>N. \<forall>n\<ge>N. dist (x n - y n) 0 < e" by auto  }
  3350     hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n) - f (y n)) 0 < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding Lim_sequentially by auto
  3351     hence False unfolding 2 using fxy and `e>0` by auto  }
  3352   thus ?lhs unfolding uniformly_continuous_on_def by blast
  3353 qed
  3354 
  3355 text{* The usual transformation theorems. *}
  3356 
  3357 lemma continuous_transform_within:
  3358   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3359   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
  3360           "continuous (at x within s) f"
  3361   shows "continuous (at x within s) g"
  3362 proof-
  3363   { fix e::real assume "e>0"
  3364     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" using assms(4) unfolding continuous_within Lim_within by auto
  3365     { fix x' assume "x'\<in>s" "0 < dist x' x" "dist x' x < (min d d')"
  3366       hence "dist (f x') (g x) < e" using assms(2,3) apply(erule_tac x=x in ballE) using d' by auto  }
  3367     hence "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < (min d d') \<longrightarrow> dist (f xa) (g x) < e" by blast
  3368     hence "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (g x) < e" using `d>0` `d'>0` by(rule_tac x="min d d'" in exI)auto  }
  3369   hence "(f ---> g x) (at x within s)" unfolding Lim_within using assms(1) by auto
  3370   thus ?thesis unfolding continuous_within using Lim_transform_within[of d s x f g "g x"] using assms by blast
  3371 qed
  3372 
  3373 lemma continuous_transform_at:
  3374   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  3375   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
  3376           "continuous (at x) f"
  3377   shows "continuous (at x) g"
  3378 proof-
  3379   { fix e::real assume "e>0"
  3380     then obtain d' where d':"d'>0" "\<forall>xa. 0 < dist xa x \<and> dist xa x < d' \<longrightarrow> dist (f xa) (f x) < e" using assms(3) unfolding continuous_at Lim_at by auto
  3381     { fix x' assume "0 < dist x' x" "dist x' x < (min d d')"
  3382       hence "dist (f x') (g x) < e" using assms(2) apply(erule_tac x=x in allE) using d' by auto
  3383     }
  3384     hence "\<forall>xa. 0 < dist xa x \<and> dist xa x < (min d d') \<longrightarrow> dist (f xa) (g x) < e" by blast
  3385     hence "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (g x) < e" using `d>0` `d'>0` by(rule_tac x="min d d'" in exI)auto
  3386   }
  3387   hence "(f ---> g x) (at x)" unfolding Lim_at using assms(1) by auto
  3388   thus ?thesis unfolding continuous_at using Lim_transform_at[of d x f g "g x"] using assms by blast
  3389 qed
  3390 
  3391 text{* Combination results for pointwise continuity. *}
  3392 
  3393 lemma continuous_const: "continuous net (\<lambda>x. c)"
  3394   by (auto simp add: continuous_def Lim_const)
  3395 
  3396 lemma continuous_cmul:
  3397   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3398   shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
  3399   by (auto simp add: continuous_def Lim_cmul)
  3400 
  3401 lemma continuous_neg:
  3402   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3403   shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
  3404   by (auto simp add: continuous_def Lim_neg)
  3405 
  3406 lemma continuous_add:
  3407   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3408   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
  3409   by (auto simp add: continuous_def Lim_add)
  3410 
  3411 lemma continuous_sub:
  3412   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  3413   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
  3414   by (auto simp add: continuous_def Lim_sub)
  3415 
  3416 text{* Same thing for setwise continuity. *}
  3417 
  3418 lemma continuous_on_const:
  3419  "continuous_on s (\<lambda>x. c)"
  3420   unfolding continuous_on_eq_continuous_within using continuous_const by blast
  3421 
  3422 lemma continuous_on_cmul:
  3423   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3424   shows "continuous_on s f ==>  continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
  3425   unfolding continuous_on_eq_continuous_within using continuous_cmul by blast
  3426 
  3427 lemma continuous_on_neg:
  3428   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3429   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
  3430   unfolding continuous_on_eq_continuous_within using continuous_neg by blast
  3431 
  3432 lemma continuous_on_add:
  3433   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3434   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3435            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
  3436   unfolding continuous_on_eq_continuous_within using continuous_add by blast
  3437 
  3438 lemma continuous_on_sub:
  3439   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3440   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  3441            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
  3442   unfolding continuous_on_eq_continuous_within using continuous_sub by blast
  3443 
  3444 text{* Same thing for uniform continuity, using sequential formulations. *}
  3445 
  3446 lemma uniformly_continuous_on_const:
  3447  "uniformly_continuous_on s (\<lambda>x. c)"
  3448   unfolding uniformly_continuous_on_def by simp
  3449 
  3450 lemma uniformly_continuous_on_cmul:
  3451   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  3452     (* FIXME: generalize 'a to metric_space *)
  3453   assumes "uniformly_continuous_on s f"
  3454   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
  3455 proof-
  3456   { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
  3457     hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
  3458       using Lim_cmul[of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
  3459       unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
  3460   }
  3461   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
  3462 qed
  3463 
  3464 lemma dist_minus:
  3465   fixes x y :: "'a::real_normed_vector"
  3466   shows "dist (- x) (- y) = dist x y"
  3467   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
  3468 
  3469 lemma uniformly_continuous_on_neg:
  3470   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3471   shows "uniformly_continuous_on s f
  3472          ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
  3473   unfolding uniformly_continuous_on_def dist_minus .
  3474 
  3475 lemma uniformly_continuous_on_add:
  3476   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
  3477   assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
  3478   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
  3479 proof-
  3480   {  fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
  3481                     "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
  3482     hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
  3483       using Lim_add[of "\<lambda> n. f (x n) - f (y n)" 0  sequentially "\<lambda> n. g (x n) - g (y n)" 0] by auto
  3484     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  }
  3485   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
  3486 qed
  3487 
  3488 lemma uniformly_continuous_on_sub:
  3489   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
  3490   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
  3491            ==> uniformly_continuous_on s  (\<lambda>x. f x - g x)"
  3492   unfolding ab_diff_minus
  3493   using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
  3494   using uniformly_continuous_on_neg[of s g] by auto
  3495 
  3496 text{* Identity function is continuous in every sense. *}
  3497 
  3498 lemma continuous_within_id:
  3499  "continuous (at a within s) (\<lambda>x. x)"
  3500   unfolding continuous_within by (rule Lim_at_within [OF Lim_ident_at])
  3501 
  3502 lemma continuous_at_id:
  3503  "continuous (at a) (\<lambda>x. x)"
  3504   unfolding continuous_at by (rule Lim_ident_at)
  3505 
  3506 lemma continuous_on_id:
  3507  "continuous_on s (\<lambda>x. x)"
  3508   unfolding continuous_on Lim_within by auto
  3509 
  3510 lemma uniformly_continuous_on_id:
  3511  "uniformly_continuous_on s (\<lambda>x. x)"
  3512   unfolding uniformly_continuous_on_def by auto
  3513 
  3514 text{* Continuity of all kinds is preserved under composition. *}
  3515 
  3516 lemma continuous_within_compose:
  3517   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3518   fixes g :: "'b::metric_space \<Rightarrow> 'c::metric_space"
  3519   assumes "continuous (at x within s) f"   "continuous (at (f x) within f ` s) g"
  3520   shows "continuous (at x within s) (g o f)"
  3521 proof-
  3522   { fix e::real assume "e>0"
  3523     with assms(2)[unfolded continuous_within Lim_within] obtain d  where "d>0" and d:"\<forall>xa\<in>f ` s. 0 < dist xa (f x) \<and> dist xa (f x) < d \<longrightarrow> dist (g xa) (g (f x)) < e" by auto
  3524     from assms(1)[unfolded continuous_within Lim_within] 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) < d" using `d>0` by auto
  3525     { fix y assume as:"y\<in>s"  "0 < dist y x"  "dist y x < d'"
  3526       hence "dist (f y) (f x) < d" using d'[THEN bspec[where x=y]] by (auto simp add:dist_commute)
  3527       hence "dist (g (f y)) (g (f x)) < e" using as(1) d[THEN bspec[where x="f y"]] unfolding dist_nz[THEN sym] using `e>0` by auto   }
  3528     hence "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (g (f xa)) (g (f x)) < e" using `d'>0` by auto  }
  3529   thus ?thesis unfolding continuous_within Lim_within by auto
  3530 qed
  3531 
  3532 lemma continuous_at_compose:
  3533   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3534   fixes g :: "'b::metric_space \<Rightarrow> 'c::metric_space"
  3535   assumes "continuous (at x) f"  "continuous (at (f x)) g"
  3536   shows "continuous (at x) (g o f)"
  3537 proof-
  3538   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
  3539   thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
  3540 qed
  3541 
  3542 lemma continuous_on_compose:
  3543  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
  3544   unfolding continuous_on_eq_continuous_within using continuous_within_compose[of _ s f g] by auto
  3545 
  3546 lemma uniformly_continuous_on_compose:
  3547   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f ` s) g"
  3548   shows "uniformly_continuous_on s (g o f)"
  3549 proof-
  3550   { fix e::real assume "e>0"
  3551     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
  3552     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
  3553     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  }
  3554   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
  3555 qed
  3556 
  3557 text{* Continuity in terms of open preimages. *}
  3558 
  3559 lemma continuous_at_open:
  3560   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3561   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)))" (is "?lhs = ?rhs")
  3562 proof
  3563   assume ?lhs
  3564   { fix t assume as: "open t" "f x \<in> t"
  3565     then obtain e where "e>0" and e:"ball (f x) e \<subseteq> t" unfolding open_contains_ball by auto
  3566 
  3567     obtain d where "d>0" and d:"\<forall>y. 0 < dist y x \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e" using `e>0` using `?lhs`[unfolded continuous_at Lim_at open_dist] by auto
  3568 
  3569     have "open (ball x d)" using open_ball by auto
  3570     moreover have "x \<in> ball x d" unfolding centre_in_ball using `d>0` by simp
  3571     moreover
  3572     { fix x' assume "x'\<in>ball x d" hence "f x' \<in> t"
  3573         using e[unfolded subset_eq Ball_def mem_ball, THEN spec[where x="f x'"]]    d[THEN spec[where x=x']]
  3574         unfolding mem_ball apply (auto simp add: dist_commute)
  3575         unfolding dist_nz[THEN sym] using as(2) by auto  }
  3576     hence "\<forall>x'\<in>ball x d. f x' \<in> t" by auto
  3577     ultimately have "\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x'\<in>s. f x' \<in> t)"
  3578       apply(rule_tac x="ball x d" in exI) by simp  }
  3579   thus ?rhs by auto
  3580 next
  3581   assume ?rhs
  3582   { fix e::real assume "e>0"
  3583     then obtain s where s: "open s"  "x \<in> s"  "\<forall>x'\<in>s. f x' \<in> ball (f x) e" using `?rhs`[unfolded continuous_at Lim_at, THEN spec[where x="ball (f x) e"]]
  3584       unfolding centre_in_ball[of "f x" e, THEN sym] by auto
  3585     then obtain d where "d>0" and d:"ball x d \<subseteq> s" unfolding open_contains_ball by auto
  3586     { fix y assume "0 < dist y x \<and> dist y x < d"
  3587       hence "dist (f y) (f x) < e" using d[unfolded subset_eq Ball_def mem_ball, THEN spec[where x=y]]
  3588         using s(3)[THEN bspec[where x=y], unfolded mem_ball] by (auto simp add: dist_commute)  }
  3589     hence "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e" using `d>0` by auto  }
  3590   thus ?lhs unfolding continuous_at Lim_at by auto
  3591 qed
  3592 
  3593 lemma continuous_on_open:
  3594  "continuous_on s f \<longleftrightarrow>
  3595         (\<forall>t. openin (subtopology euclidean (f ` s)) t
  3596             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  3597 proof
  3598   assume ?lhs
  3599   { fix t assume as:"openin (subtopology euclidean (f ` s)) t"
  3600     have "{x \<in> s. f x \<in> t} \<subseteq> s" using as[unfolded openin_euclidean_subtopology_iff] by auto
  3601     moreover
  3602     { fix x assume as':"x\<in>{x \<in> s. f x \<in> t}"
  3603       then obtain e where e: "e>0" "\<forall>x'\<in>f ` s. dist x' (f x) < e \<longrightarrow> x' \<in> t" using as[unfolded openin_euclidean_subtopology_iff, THEN conjunct2, THEN bspec[where x="f x"]] by auto
  3604       from this(1) 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" using `?lhs`[unfolded continuous_on Lim_within, THEN bspec[where x=x]] using as' by auto
  3605       have "\<exists>e>0. \<forall>x'\<in>s. dist x' x < e \<longrightarrow> x' \<in> {x \<in> s. f x \<in> t}" using d e unfolding dist_nz[THEN sym] by (rule_tac x=d in exI, auto)  }
  3606     ultimately have "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" unfolding openin_euclidean_subtopology_iff by auto  }
  3607   thus ?rhs unfolding continuous_on Lim_within using openin by auto
  3608 next
  3609   assume ?rhs
  3610   { fix e::real and x assume "x\<in>s" "e>0"
  3611     { fix xa x' assume "dist (f xa) (f x) < e" "xa \<in> s" "x' \<in> s" "dist (f xa) (f x') < e - dist (f xa) (f x)"
  3612       hence "dist (f x') (f x) < e" using dist_triangle[of "f x'" "f x" "f xa"]
  3613         by (auto simp add: dist_commute)  }
  3614     hence "ball (f x) e \<inter> f ` s \<subseteq> f ` s \<and> (\<forall>xa\<in>ball (f x) e \<inter> f ` s. \<exists>ea>0. \<forall>x'\<in>f ` s. dist x' xa < ea \<longrightarrow> x' \<in> ball (f x) e \<inter> f ` s)" apply auto
  3615       apply(rule_tac x="e - dist (f xa) (f x)" in exI) using `e>0` by (auto simp add: dist_commute)
  3616     hence "\<forall>xa\<in>{xa \<in> s. f xa \<in> ball (f x) e \<inter> f ` s}. \<exists>ea>0. \<forall>x'\<in>s. dist x' xa < ea \<longrightarrow> x' \<in> {xa \<in> s. f xa \<in> ball (f x) e \<inter> f ` s}"
  3617       using `?rhs`[unfolded openin_euclidean_subtopology_iff, THEN spec[where x="ball (f x) e \<inter> f ` s"]] by auto
  3618     hence "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e" apply(erule_tac x=x in ballE) apply auto using `e>0` `x\<in>s` by (auto simp add: dist_commute)  }
  3619   thus ?lhs unfolding continuous_on Lim_within by auto
  3620 qed
  3621 
  3622 (* ------------------------------------------------------------------------- *)
  3623 (* Similarly in terms of closed sets.                                        *)
  3624 (* ------------------------------------------------------------------------- *)
  3625 
  3626 lemma continuous_on_closed:
  3627  "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")
  3628 proof
  3629   assume ?lhs
  3630   { fix t
  3631     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  3632     have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
  3633     assume as:"closedin (subtopology euclidean (f ` s)) t"
  3634     hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
  3635     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"]]
  3636       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }
  3637   thus ?rhs by auto
  3638 next
  3639   assume ?rhs
  3640   { fix t
  3641     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  3642     assume as:"openin (subtopology euclidean (f ` s)) t"
  3643     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
  3644       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
  3645   thus ?lhs unfolding continuous_on_open by auto
  3646 qed
  3647 
  3648 text{* Half-global and completely global cases.                                  *}
  3649 
  3650 lemma continuous_open_in_preimage:
  3651   assumes "continuous_on s f"  "open t"
  3652   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3653 proof-
  3654   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
  3655   have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  3656     using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
  3657   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  3658 qed
  3659 
  3660 lemma continuous_closed_in_preimage:
  3661   assumes "continuous_on s f"  "closed t"
  3662   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  3663 proof-
  3664   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
  3665   have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  3666     using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
  3667   thus ?thesis
  3668     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  3669 qed
  3670 
  3671 lemma continuous_open_preimage:
  3672   assumes "continuous_on s f" "open s" "open t"
  3673   shows "open {x \<in> s. f x \<in> t}"
  3674 proof-
  3675   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  3676     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
  3677   thus ?thesis using open_Int[of s T, OF assms(2)] by auto
  3678 qed
  3679 
  3680 lemma continuous_closed_preimage:
  3681   assumes "continuous_on s f" "closed s" "closed t"
  3682   shows "closed {x \<in> s. f x \<in> t}"
  3683 proof-
  3684   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  3685     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
  3686   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
  3687 qed
  3688 
  3689 lemma continuous_open_preimage_univ:
  3690   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3691   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
  3692   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
  3693 
  3694 lemma continuous_closed_preimage_univ:
  3695   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3696   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
  3697   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
  3698 
  3699 lemma continuous_open_vimage:
  3700   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3701   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
  3702   unfolding vimage_def by (rule continuous_open_preimage_univ)
  3703 
  3704 lemma continuous_closed_vimage:
  3705   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3706   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
  3707   unfolding vimage_def by (rule continuous_closed_preimage_univ)
  3708 
  3709 text{* Equality of continuous functions on closure and related results.          *}
  3710 
  3711 lemma continuous_closed_in_preimage_constant:
  3712  "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
  3713   using continuous_closed_in_preimage[of s f "{a}"] closed_sing by auto
  3714 
  3715 lemma continuous_closed_preimage_constant:
  3716  "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
  3717   using continuous_closed_preimage[of s f "{a}"] closed_sing by auto
  3718 
  3719 lemma continuous_constant_on_closure:
  3720   assumes "continuous_on (closure s) f"
  3721           "\<forall>x \<in> s. f x = a"
  3722   shows "\<forall>x \<in> (closure s). f x = a"
  3723     using continuous_closed_preimage_constant[of "closure s" f a]
  3724     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
  3725 
  3726 lemma image_closure_subset:
  3727   assumes "continuous_on (closure s) f"  "closed t"  "(f ` s) \<subseteq> t"
  3728   shows "f ` (closure s) \<subseteq> t"
  3729 proof-
  3730   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
  3731   moreover have "closed {x \<in> closure s. f x \<in> t}"
  3732     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
  3733   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
  3734     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
  3735   thus ?thesis by auto
  3736 qed
  3737 
  3738 lemma continuous_on_closure_norm_le:
  3739   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  3740   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"
  3741   shows "norm(f x) \<le> b"
  3742 proof-
  3743   have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
  3744   show ?thesis
  3745     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
  3746     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
  3747 qed
  3748 
  3749 text{* Making a continuous function avoid some value in a neighbourhood.         *}
  3750 
  3751 lemma continuous_within_avoid:
  3752   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3753   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"
  3754   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
  3755 proof-
  3756   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"
  3757     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
  3758   { fix y assume " y\<in>s"  "dist x y < d"
  3759     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
  3760       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
  3761   thus ?thesis using `d>0` by auto
  3762 qed
  3763 
  3764 lemma continuous_at_avoid:
  3765   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
  3766   assumes "continuous (at x) f"  "f x \<noteq> a"
  3767   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  3768 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
  3769 
  3770 lemma continuous_on_avoid:
  3771   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"
  3772   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
  3773 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
  3774 
  3775 lemma continuous_on_open_avoid:
  3776   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"
  3777   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  3778 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
  3779 
  3780 text{* Proving a function is constant by proving open-ness of level set.         *}
  3781 
  3782 lemma continuous_levelset_open_in_cases:
  3783  "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  3784         openin (subtopology euclidean s) {x \<in> s. f x = a}
  3785         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
  3786 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
  3787 
  3788 lemma continuous_levelset_open_in:
  3789  "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  3790         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
  3791         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"
  3792 using continuous_levelset_open_in_cases[of s f ]
  3793 by meson
  3794 
  3795 lemma continuous_levelset_open:
  3796   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
  3797   shows "\<forall>x \<in> s. f x = a"
  3798 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by auto
  3799 
  3800 text{* Some arithmetical combinations (more to prove).                           *}
  3801 
  3802 lemma open_scaling[intro]:
  3803   fixes s :: "'a::real_normed_vector set"
  3804   assumes "c \<noteq> 0"  "open s"
  3805   shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
  3806 proof-
  3807   { fix x assume "x \<in> s"
  3808     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
  3809     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using real_mult_order[OF `e>0`] by auto
  3810     moreover
  3811     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
  3812       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
  3813         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
  3814           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
  3815       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  }
  3816     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  }
  3817   thus ?thesis unfolding open_dist by auto
  3818 qed
  3819 
  3820 lemma minus_image_eq_vimage:
  3821   fixes A :: "'a::ab_group_add set"
  3822   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
  3823   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
  3824 
  3825 lemma open_negations:
  3826   fixes s :: "'a::real_normed_vector set"
  3827   shows "open s ==> open ((\<lambda> x. -x) ` s)"
  3828   unfolding scaleR_minus1_left [symmetric]
  3829   by (rule open_scaling, auto)
  3830 
  3831 lemma open_translation:
  3832   fixes s :: "'a::real_normed_vector set"
  3833   assumes "open s"  shows "open((\<lambda>x. a + x) ` s)"
  3834 proof-
  3835   { 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  }
  3836   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
  3837   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
  3838 qed
  3839 
  3840 lemma open_affinity:
  3841   fixes s :: "'a::real_normed_vector set"
  3842   assumes "open s"  "c \<noteq> 0"
  3843   shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  3844 proof-
  3845   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
  3846   have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
  3847   thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
  3848 qed
  3849 
  3850 lemma interior_translation:
  3851   fixes s :: "'a::real_normed_vector set"
  3852   shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
  3853 proof (rule set_ext, rule)
  3854   fix x assume "x \<in> interior (op + a ` s)"
  3855   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
  3856   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
  3857   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
  3858 next
  3859   fix x assume "x \<in> op + a ` interior s"
  3860   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
  3861   { fix z have *:"a + y - z = y + a - z" by auto
  3862     assume "z\<in>ball x e"
  3863     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
  3864     hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }
  3865   hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
  3866   thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
  3867 qed
  3868 
  3869 subsection {* Preservation of compactness and connectedness under continuous function.  *}
  3870 
  3871 lemma compact_continuous_image:
  3872   assumes "continuous_on s f"  "compact s"
  3873   shows "compact(f ` s)"
  3874 proof-
  3875   { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
  3876     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
  3877     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
  3878     { fix e::real assume "e>0"
  3879       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_def, THEN bspec[where x=l], OF `l\<in>s`] by auto
  3880       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
  3881       { 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  }
  3882       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }
  3883     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  }
  3884   thus ?thesis unfolding compact_def by auto
  3885 qed
  3886 
  3887 lemma connected_continuous_image:
  3888   assumes "continuous_on s f"  "connected s"
  3889   shows "connected(f ` s)"
  3890 proof-
  3891   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f ` s"  "openin (subtopology euclidean (f ` s)) T"  "closedin (subtopology euclidean (f ` s)) T"
  3892     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
  3893       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
  3894       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
  3895       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
  3896     hence False using as(1,2)
  3897       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
  3898   thus ?thesis unfolding connected_clopen by auto
  3899 qed
  3900 
  3901 text{* Continuity implies uniform continuity on a compact domain.                *}
  3902 
  3903 lemma compact_uniformly_continuous:
  3904   assumes "continuous_on s f"  "compact s"
  3905   shows "uniformly_continuous_on s f"
  3906 proof-
  3907     { fix x assume x:"x\<in>s"
  3908       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_def, THEN bspec[where x=x]] by auto
  3909       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  }
  3910     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
  3911     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)"
  3912       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
  3913 
  3914   { fix e::real assume "e>0"
  3915 
  3916     { 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  }
  3917     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
  3918     moreover
  3919     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }
  3920     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
  3921 
  3922     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
  3923       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
  3924       hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
  3925       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`
  3926         by (auto  simp add: dist_commute)
  3927       moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
  3928         by (auto simp add: dist_commute)
  3929       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`
  3930         by (auto  simp add: dist_commute)
  3931       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
  3932         by (auto simp add: dist_commute)  }
  3933     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  }
  3934   thus ?thesis unfolding uniformly_continuous_on_def by auto
  3935 qed
  3936 
  3937 text{* Continuity of inverse function on compact domain. *}
  3938 
  3939 lemma continuous_on_inverse:
  3940   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  3941     (* TODO: can this be generalized more? *)
  3942   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"
  3943   shows "continuous_on (f ` s) g"
  3944 proof-
  3945   have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
  3946   { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
  3947     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
  3948     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
  3949       unfolding T(2) and Int_left_absorb by auto
  3950     moreover have "compact (s \<inter> T)"
  3951       using assms(2) unfolding compact_eq_bounded_closed
  3952       using bounded_subset[of s "s \<inter> T"] and T(1) by auto
  3953     ultimately have "closed (f ` t)" using T(1) unfolding T(2)
  3954       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
  3955     moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
  3956     ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
  3957       unfolding closedin_closed by auto  }
  3958   thus ?thesis unfolding continuous_on_closed by auto
  3959 qed
  3960 
  3961 subsection{* A uniformly convergent limit of continuous functions is continuous.       *}
  3962 
  3963 lemma norm_triangle_lt:
  3964   fixes x y :: "'a::real_normed_vector"
  3965   shows "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
  3966 by (rule le_less_trans [OF norm_triangle_ineq])
  3967 
  3968 lemma continuous_uniform_limit:
  3969   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
  3970   assumes "\<not> (trivial_limit net)"  "eventually (\<lambda>n. continuous_on s (f n)) net"
  3971   "\<forall>e>0. eventually (\<lambda>n. \<forall>x \<in> s. norm(f n x - g x) < e) net"
  3972   shows "continuous_on s g"
  3973 proof-
  3974   { fix x and e::real assume "x\<in>s" "e>0"
  3975     have "eventually (\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3) net" using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
  3976     then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3"  "continuous_on s (f n)"
  3977       using eventually_and[of "(\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3)" "(\<lambda>n. continuous_on s (f n))" net] assms(1,2) eventually_happens by blast
  3978     have "e / 3 > 0" using `e>0` by auto
  3979     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"
  3980       using n(2)[unfolded continuous_on_def, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
  3981     { fix y assume "y\<in>s" "dist y x < d"
  3982       hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
  3983       hence "norm (f n y - g x) < 2 * e / 3" using norm_triangle_lt[of "f n y - f n x" "f n x - g x" "2*e/3"]
  3984         using n(1)[THEN bspec[where x=x], OF `x\<in>s`] unfolding dist_norm unfolding ab_group_add_class.ab_diff_minus by auto
  3985       hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
  3986         unfolding norm_minus_cancel[of "f n y - g y", THEN sym] using norm_triangle_lt[of "f n y - g x" "g y - f n y" e] by (auto simp add: uminus_add_conv_diff)  }
  3987     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto  }
  3988   thus ?thesis unfolding continuous_on_def by auto
  3989 qed
  3990 
  3991 subsection{* Topological properties of linear functions.                               *}
  3992 
  3993 lemma linear_lim_0:
  3994   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
  3995 proof-
  3996   interpret f: bounded_linear f by fact
  3997   have "(f ---> f 0) (at 0)"
  3998     using tendsto_ident_at by (rule f.tendsto)
  3999   thus ?thesis unfolding f.zero .
  4000 qed
  4001 
  4002 lemma linear_continuous_at:
  4003   assumes "bounded_linear f"  shows "continuous (at a) f"
  4004   unfolding continuous_at using assms
  4005   apply (rule bounded_linear.tendsto)
  4006   apply (rule tendsto_ident_at)
  4007   done
  4008 
  4009 lemma linear_continuous_within:
  4010   shows "bounded_linear f ==> continuous (at x within s) f"
  4011   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
  4012 
  4013 lemma linear_continuous_on:
  4014   shows "bounded_linear f ==> continuous_on s f"
  4015   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
  4016 
  4017 text{* Also bilinear functions, in composition form.                             *}
  4018 
  4019 lemma bilinear_continuous_at_compose:
  4020   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
  4021         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
  4022   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
  4023 
  4024 lemma bilinear_continuous_within_compose:
  4025   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
  4026         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
  4027   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
  4028 
  4029 lemma bilinear_continuous_on_compose:
  4030   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
  4031              ==> continuous_on s (\<lambda>x. h (f x) (g x))"
  4032   unfolding continuous_on_eq_continuous_within apply auto apply(erule_tac x=x in ballE) apply auto apply(erule_tac x=x in ballE) apply auto
  4033   using bilinear_continuous_within_compose[of _ s f g h] by auto
  4034 
  4035 subsection{* Topological stuff lifted from and dropped to R                            *}
  4036 
  4037 
  4038 lemma open_real:
  4039   fixes s :: "real set" shows
  4040  "open s \<longleftrightarrow>
  4041         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
  4042   unfolding open_dist dist_norm by simp
  4043 
  4044 lemma islimpt_approachable_real:
  4045   fixes s :: "real set"
  4046   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
  4047   unfolding islimpt_approachable dist_norm by simp
  4048 
  4049 lemma closed_real:
  4050   fixes s :: "real set"
  4051   shows "closed s \<longleftrightarrow>
  4052         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
  4053             --> x \<in> s)"
  4054   unfolding closed_limpt islimpt_approachable dist_norm by simp
  4055 
  4056 lemma continuous_at_real_range:
  4057   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4058   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  4059         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
  4060   unfolding continuous_at unfolding Lim_at
  4061   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
  4062   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
  4063   apply(erule_tac x=e in allE) by auto
  4064 
  4065 lemma continuous_on_real_range:
  4066   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4067   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))"
  4068   unfolding continuous_on_def dist_norm by simp
  4069 
  4070 lemma continuous_at_norm: "continuous (at x) norm"
  4071   unfolding continuous_at by (intro tendsto_intros)
  4072 
  4073 lemma continuous_on_norm: "continuous_on s norm"
  4074 unfolding continuous_on by (intro ballI tendsto_intros)
  4075 
  4076 lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
  4077 unfolding continuous_at by (intro tendsto_intros)
  4078 
  4079 lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
  4080 unfolding continuous_on by (intro ballI tendsto_intros)
  4081 
  4082 lemma continuous_at_infnorm: "continuous (at x) infnorm"
  4083   unfolding continuous_at Lim_at o_def unfolding dist_norm
  4084   apply auto apply (rule_tac x=e in exI) apply auto
  4085   using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
  4086 
  4087 text{* Hence some handy theorems on distance, diameter etc. of/from a set.       *}
  4088 
  4089 lemma compact_attains_sup:
  4090   fixes s :: "real set"
  4091   assumes "compact s"  "s \<noteq> {}"
  4092   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
  4093 proof-
  4094   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4095   { 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"
  4096     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
  4097     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
  4098     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto  }
  4099   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
  4100     apply(rule_tac x="Sup s" in bexI) by auto
  4101 qed
  4102 
  4103 lemma Inf:
  4104   fixes S :: "real set"
  4105   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
  4106 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def) 
  4107 
  4108 lemma compact_attains_inf:
  4109   fixes s :: "real set"
  4110   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
  4111 proof-
  4112   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4113   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"
  4114       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
  4115     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
  4116     moreover
  4117     { fix x assume "x \<in> s"
  4118       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
  4119       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
  4120     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
  4121     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto  }
  4122   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
  4123     apply(rule_tac x="Inf s" in bexI) by auto
  4124 qed
  4125 
  4126 lemma continuous_attains_sup:
  4127   fixes f :: "'a::metric_space \<Rightarrow> real"
  4128   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4129         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"
  4130   using compact_attains_sup[of "f ` s"]
  4131   using compact_continuous_image[of s f] by auto
  4132 
  4133 lemma continuous_attains_inf:
  4134   fixes f :: "'a::metric_space \<Rightarrow> real"
  4135   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4136         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
  4137   using compact_attains_inf[of "f ` s"]
  4138   using compact_continuous_image[of s f] by auto
  4139 
  4140 lemma distance_attains_sup:
  4141   assumes "compact s" "s \<noteq> {}"
  4142   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
  4143 proof (rule continuous_attains_sup [OF assms])
  4144   { fix x assume "x\<in>s"
  4145     have "(dist a ---> dist a x) (at x within s)"
  4146       by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
  4147   }
  4148   thus "continuous_on s (dist a)"
  4149     unfolding continuous_on ..
  4150 qed
  4151 
  4152 text{* For *minimal* distance, we only need closure, not compactness.            *}
  4153 
  4154 lemma distance_attains_inf:
  4155   fixes a :: "'a::heine_borel"
  4156   assumes "closed s"  "s \<noteq> {}"
  4157   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
  4158 proof-
  4159   from assms(2) obtain b where "b\<in>s" by auto
  4160   let ?B = "cball a (dist b a) \<inter> s"
  4161   have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
  4162   hence "?B \<noteq> {}" by auto
  4163   moreover
  4164   { fix x assume "x\<in>?B"
  4165     fix e::real assume "e>0"
  4166     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
  4167       from as have "\<bar>dist a x' - dist a x\<bar> < e"
  4168         unfolding abs_less_iff minus_diff_eq
  4169         using dist_triangle2 [of a x' x]
  4170         using dist_triangle [of a x x']
  4171         by arith
  4172     }
  4173     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
  4174       using `e>0` by auto
  4175   }
  4176   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
  4177     unfolding continuous_on Lim_within dist_norm real_norm_def
  4178     by fast
  4179   moreover have "compact ?B"
  4180     using compact_cball[of a "dist b a"]
  4181     unfolding compact_eq_bounded_closed
  4182     using bounded_Int and closed_Int and assms(1) by auto
  4183   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"
  4184     using continuous_attains_inf[of ?B "dist a"] by fastsimp
  4185   thus ?thesis by fastsimp
  4186 qed
  4187 
  4188 subsection{* We can now extend limit compositions to consider the scalar multiplier.   *}
  4189 
  4190 lemma Lim_mul:
  4191   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  4192   assumes "(c ---> d) net"  "(f ---> l) net"
  4193   shows "((\<lambda>x. c(x) *\<^sub>R f x) ---> (d *\<^sub>R l)) net"
  4194   using assms by (rule scaleR.tendsto)
  4195 
  4196 lemma Lim_vmul:
  4197   fixes c :: "'a \<Rightarrow> real" and v :: "'b::real_normed_vector"
  4198   shows "(c ---> d) net ==> ((\<lambda>x. c(x) *\<^sub>R v) ---> d *\<^sub>R v) net"
  4199   by (intro tendsto_intros)
  4200 
  4201 lemma continuous_vmul:
  4202   fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
  4203   shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
  4204   unfolding continuous_def using Lim_vmul[of c] by auto
  4205 
  4206 lemma continuous_mul:
  4207   fixes c :: "'a::metric_space \<Rightarrow> real"
  4208   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4209   shows "continuous net c \<Longrightarrow> continuous net f
  4210              ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
  4211   unfolding continuous_def by (intro tendsto_intros)
  4212 
  4213 lemma continuous_on_vmul:
  4214   fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
  4215   shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
  4216   unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
  4217 
  4218 lemma continuous_on_mul:
  4219   fixes c :: "'a::metric_space \<Rightarrow> real"
  4220   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4221   shows "continuous_on s c \<Longrightarrow> continuous_on s f
  4222              ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
  4223   unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
  4224 
  4225 text{* And so we have continuity of inverse.                                     *}
  4226 
  4227 lemma Lim_inv:
  4228   fixes f :: "'a \<Rightarrow> real"
  4229   assumes "(f ---> l) (net::'a net)"  "l \<noteq> 0"
  4230   shows "((inverse o f) ---> inverse l) net"
  4231   unfolding o_def using assms by (rule tendsto_inverse)
  4232 
  4233 lemma continuous_inv:
  4234   fixes f :: "'a::metric_space \<Rightarrow> real"
  4235   shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
  4236            ==> continuous net (inverse o f)"
  4237   unfolding continuous_def using Lim_inv by auto
  4238 
  4239 lemma continuous_at_within_inv:
  4240   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
  4241   assumes "continuous (at a within s) f" "f a \<noteq> 0"
  4242   shows "continuous (at a within s) (inverse o f)"
  4243   using assms unfolding continuous_within o_def
  4244   by (intro tendsto_intros)
  4245 
  4246 lemma continuous_at_inv:
  4247   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
  4248   shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
  4249          ==> continuous (at a) (inverse o f) "
  4250   using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
  4251 
  4252 subsection{* Preservation properties for pasted sets.                                  *}
  4253 
  4254 lemma bounded_pastecart:
  4255   fixes s :: "('a::real_normed_vector ^ _) set" (* FIXME: generalize to metric_space *)
  4256   assumes "bounded s" "bounded t"
  4257   shows "bounded { pastecart x y | x y . (x \<in> s \<and> y \<in> t)}"
  4258 proof-
  4259   obtain a b where ab:"\<forall>x\<in>s. norm x \<le> a" "\<forall>x\<in>t. norm x \<le> b" using assms[unfolded bounded_iff] by auto
  4260   { fix x y assume "x\<in>s" "y\<in>t"
  4261     hence "norm x \<le> a" "norm y \<le> b" using ab by auto
  4262     hence "norm (pastecart x y) \<le> a + b" using norm_pastecart[of x y] by auto }
  4263   thus ?thesis unfolding bounded_iff by auto
  4264 qed
  4265 
  4266 lemma bounded_Times:
  4267   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
  4268 proof-
  4269   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
  4270     using assms [unfolded bounded_def] by auto
  4271   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
  4272     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
  4273   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
  4274 qed
  4275 
  4276 lemma closed_pastecart:
  4277   fixes s :: "(real ^ 'a::finite) set" (* FIXME: generalize *)
  4278   assumes "closed s"  "closed t"
  4279   shows "closed {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
  4280 proof-
  4281   { fix x l assume as:"\<forall>n::nat. x n \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}"  "(x ---> l) sequentially"
  4282     { fix n::nat have "fstcart (x n) \<in> s" "sndcart (x n) \<in> t" using as(1)[THEN spec[where x=n]] by auto } note * = this
  4283     moreover
  4284     { fix e::real assume "e>0"
  4285       then obtain N::nat where N:"\<forall>n\<ge>N. dist (x n) l < e" using as(2)[unfolded Lim_sequentially, THEN spec[where x=e]] by auto
  4286       { fix n::nat assume "n\<ge>N"
  4287         hence "dist (fstcart (x n)) (fstcart l) < e" "dist (sndcart (x n)) (sndcart l) < e"
  4288           using N[THEN spec[where x=n]] dist_fstcart[of "x n" l] dist_sndcart[of "x n" l] by auto   }
  4289       hence "\<exists>N. \<forall>n\<ge>N. dist (fstcart (x n)) (fstcart l) < e" "\<exists>N. \<forall>n\<ge>N. dist (sndcart (x n)) (sndcart l) < e" by auto  }
  4290     ultimately have "fstcart l \<in> s" "sndcart l \<in> t"
  4291       using assms(1)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. fstcart (x n)"], THEN spec[where x="fstcart l"]]
  4292       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. sndcart (x n)"], THEN spec[where x="sndcart l"]]
  4293       unfolding Lim_sequentially by auto
  4294     hence "l \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}" using pastecart_fst_snd[THEN sym, of l] by auto  }
  4295   thus ?thesis unfolding closed_sequential_limits by auto
  4296 qed
  4297 
  4298 lemma compact_pastecart:
  4299   fixes s t :: "(real ^ _) set"
  4300   shows "compact s \<Longrightarrow> compact t ==> compact {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
  4301   unfolding compact_eq_bounded_closed using bounded_pastecart[of s t] closed_pastecart[of s t] by auto
  4302 
  4303 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
  4304 by (induct x) simp
  4305 
  4306 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
  4307 unfolding compact_def
  4308 apply clarify
  4309 apply (drule_tac x="fst \<circ> f" in spec)
  4310 apply (drule mp, simp add: mem_Times_iff)
  4311 apply (clarify, rename_tac l1 r1)
  4312 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
  4313 apply (drule mp, simp add: mem_Times_iff)
  4314 apply (clarify, rename_tac l2 r2)
  4315 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
  4316 apply (rule_tac x="r1 \<circ> r2" in exI)
  4317 apply (rule conjI, simp add: subseq_def)
  4318 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
  4319 apply (drule (1) tendsto_Pair) back
  4320 apply (simp add: o_def)
  4321 done
  4322 
  4323 text{* Hence some useful properties follow quite easily.                         *}
  4324 
  4325 lemma compact_scaling:
  4326   fixes s :: "'a::real_normed_vector set"
  4327   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
  4328 proof-
  4329   let ?f = "\<lambda>x. scaleR c x"
  4330   have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
  4331   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
  4332     using linear_continuous_at[OF *] assms by auto
  4333 qed
  4334 
  4335 lemma compact_negations:
  4336   fixes s :: "'a::real_normed_vector set"
  4337   assumes "compact s"  shows "compact ((\<lambda>x. -x) ` s)"
  4338   using compact_scaling [OF assms, of "- 1"] by auto
  4339 
  4340 lemma compact_sums:
  4341   fixes s t :: "'a::real_normed_vector set"
  4342   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
  4343 proof-
  4344   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
  4345     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
  4346   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
  4347     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  4348   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
  4349 qed
  4350 
  4351 lemma compact_differences:
  4352   fixes s t :: "'a::real_normed_vector set"
  4353   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
  4354 proof-
  4355   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
  4356     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4357   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
  4358 qed
  4359 
  4360 lemma compact_translation:
  4361   fixes s :: "'a::real_normed_vector set"
  4362   assumes "compact s"  shows "compact ((\<lambda>x. a + x) ` s)"
  4363 proof-
  4364   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
  4365   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
  4366 qed
  4367 
  4368 lemma compact_affinity:
  4369   fixes s :: "'a::real_normed_vector set"
  4370   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  4371 proof-
  4372   have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  4373   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
  4374 qed
  4375 
  4376 text{* Hence we get the following.                                               *}
  4377 
  4378 lemma compact_sup_maxdistance:
  4379   fixes s :: "'a::real_normed_vector set"
  4380   assumes "compact s"  "s \<noteq> {}"
  4381   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
  4382 proof-
  4383   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
  4384   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"
  4385     using compact_differences[OF assms(1) assms(1)]
  4386     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 simp add: norm_minus_cancel)
  4387   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
  4388   thus ?thesis using x(2)[unfolded `x = a - b`] by blast
  4389 qed
  4390 
  4391 text{* We can state this in terms of diameter of a set.                          *}
  4392 
  4393 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
  4394   (* TODO: generalize to class metric_space *)
  4395 
  4396 lemma diameter_bounded:
  4397   assumes "bounded s"
  4398   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
  4399         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
  4400 proof-
  4401   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
  4402   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
  4403   { fix x y assume "x \<in> s" "y \<in> s"
  4404     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: ring_simps)  }
  4405   note * = this
  4406   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto
  4407     have lub:"isLub UNIV ?D (Sup ?D)" using * Sup[of ?D] using `s\<noteq>{}` unfolding setle_def
  4408       apply auto    (*FIXME: something horrible has happened here!*)
  4409       apply atomize
  4410       apply safe
  4411       apply metis +
  4412       done
  4413     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` isLubD1[OF lub] unfolding setle_def by auto  }
  4414   moreover
  4415   { fix d::real assume "d>0" "d < diameter s"
  4416     hence "s\<noteq>{}" unfolding diameter_def by auto
  4417     hence lub:"isLub UNIV ?D (Sup ?D)" using * Sup[of ?D] unfolding setle_def 
  4418       apply auto    (*FIXME: something horrible has happened here!*)
  4419       apply atomize
  4420       apply safe
  4421       apply metis +
  4422       done
  4423     have "\<exists>d' \<in> ?D. d' > d"
  4424     proof(rule ccontr)
  4425       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
  4426       hence as:"\<forall>d'\<in>?D. d' \<le> d" apply auto apply(erule_tac x="norm (x - y)" in allE) by auto
  4427       hence "isUb UNIV ?D d" unfolding isUb_def unfolding setle_def by auto
  4428       thus False using `d < diameter s` `s\<noteq>{}` isLub_le_isUb[OF lub, of d] unfolding diameter_def  by auto
  4429     qed
  4430     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }
  4431   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
  4432         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
  4433 qed
  4434 
  4435 lemma diameter_bounded_bound:
  4436  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
  4437   using diameter_bounded by blast
  4438 atp_minimize [atp=remote_vampire] Collect_def Max_ge add_increasing2 add_le_cancel_left diameter_def_raw equation_minus_iff finite finite_imageI norm_imp_pos_and_ge rangeI
  4439 
  4440 lemma diameter_compact_attained:
  4441   fixes s :: "'a::real_normed_vector set"
  4442   assumes "compact s"  "s \<noteq> {}"
  4443   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
  4444 proof-
  4445   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
  4446   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
  4447   hence "diameter s \<le> norm (x - y)" 
  4448     by (force simp add: diameter_def intro!: Sup_least) 
  4449   thus ?thesis using diameter_bounded(1)[OF b, THEN bspec[where x=x], THEN bspec[where x=y], OF xys] and xys by auto
  4450 qed
  4451 
  4452 text{* Related results with closure as the conclusion.                           *}
  4453 
  4454 lemma closed_scaling:
  4455   fixes s :: "'a::real_normed_vector set"
  4456   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
  4457 proof(cases "s={}")
  4458   case True thus ?thesis by auto
  4459 next
  4460   case False
  4461   show ?thesis
  4462   proof(cases "c=0")
  4463     have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
  4464     case True thus ?thesis apply auto unfolding * using closed_sing by auto
  4465   next
  4466     case False
  4467     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s"  "(x ---> l) sequentially"
  4468       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
  4469           using as(1)[THEN spec[where x=n]]
  4470           using `c\<noteq>0` by (auto simp add: vector_smult_assoc)
  4471       }
  4472       moreover
  4473       { fix e::real assume "e>0"
  4474         hence "0 < e *\<bar>c\<bar>"  using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
  4475         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
  4476           using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
  4477         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
  4478           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
  4479           using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto  }
  4480       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
  4481       ultimately have "l \<in> scaleR c ` s"
  4482         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"]]
  4483         unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }
  4484     thus ?thesis unfolding closed_sequential_limits by fast
  4485   qed
  4486 qed
  4487 
  4488 lemma closed_negations:
  4489   fixes s :: "'a::real_normed_vector set"
  4490   assumes "closed s"  shows "closed ((\<lambda>x. -x) ` s)"
  4491   using closed_scaling[OF assms, of "- 1"] by simp
  4492 
  4493 lemma compact_closed_sums:
  4494   fixes s :: "'a::real_normed_vector set"
  4495   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  4496 proof-
  4497   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
  4498   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"
  4499     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"
  4500       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
  4501     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
  4502       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
  4503     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
  4504       using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
  4505     hence "l - l' \<in> t"
  4506       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
  4507       using f(3) by auto
  4508     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
  4509   }
  4510   thus ?thesis unfolding closed_sequential_limits by fast
  4511 qed
  4512 
  4513 lemma closed_compact_sums:
  4514   fixes s t :: "'a::real_normed_vector set"
  4515   assumes "closed s"  "compact t"
  4516   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  4517 proof-
  4518   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
  4519     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
  4520   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
  4521 qed
  4522 
  4523 lemma compact_closed_differences:
  4524   fixes s t :: "'a::real_normed_vector set"
  4525   assumes "compact s"  "closed t"
  4526   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  4527 proof-
  4528   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"
  4529     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4530   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
  4531 qed
  4532 
  4533 lemma closed_compact_differences:
  4534   fixes s t :: "'a::real_normed_vector set"
  4535   assumes "closed s" "compact t"
  4536   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  4537 proof-
  4538   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
  4539     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  4540  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
  4541 qed
  4542 
  4543 lemma closed_translation:
  4544   fixes a :: "'a::real_normed_vector"
  4545   assumes "closed s"  shows "closed ((\<lambda>x. a + x) ` s)"
  4546 proof-
  4547   have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
  4548   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
  4549 qed
  4550 
  4551 lemma translation_UNIV:
  4552   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
  4553   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
  4554 
  4555 lemma translation_diff:
  4556   fixes a :: "'a::ab_group_add"
  4557   shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
  4558   by auto
  4559 
  4560 lemma closure_translation:
  4561   fixes a :: "'a::real_normed_vector"
  4562   shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
  4563 proof-
  4564   have *:"op + a ` (UNIV - s) = UNIV - op + a ` s"
  4565     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
  4566   show ?thesis unfolding closure_interior translation_diff translation_UNIV
  4567     using interior_translation[of a "UNIV - s"] unfolding * by auto
  4568 qed
  4569 
  4570 lemma frontier_translation:
  4571   fixes a :: "'a::real_normed_vector"
  4572   shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
  4573   unfolding frontier_def translation_diff interior_translation closure_translation by auto
  4574 
  4575 subsection{* Separation between points and sets.                                       *}
  4576 
  4577 lemma separate_point_closed:
  4578   fixes s :: "'a::heine_borel set"
  4579   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
  4580 proof(cases "s = {}")
  4581   case True
  4582   thus ?thesis by(auto intro!: exI[where x=1])
  4583 next
  4584   case False
  4585   assume "closed s" "a \<notin> s"
  4586   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
  4587   with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
  4588 qed
  4589 
  4590 lemma separate_compact_closed:
  4591   fixes s t :: "'a::{heine_borel, real_normed_vector} set"
  4592     (* TODO: does this generalize to heine_borel? *)
  4593   assumes "compact s" and "closed t" and "s \<inter> t = {}"
  4594   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  4595 proof-
  4596   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
  4597   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"
  4598     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
  4599   { fix x y assume "x\<in>s" "y\<in>t"
  4600     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
  4601     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
  4602       by (auto  simp add: dist_commute)
  4603     hence "d \<le> dist x y" unfolding dist_norm by auto  }
  4604   thus ?thesis using `d>0` by auto
  4605 qed
  4606 
  4607 lemma separate_closed_compact:
  4608   fixes s t :: "'a::{heine_borel, real_normed_vector} set"
  4609   assumes "closed s" and "compact t" and "s \<inter> t = {}"
  4610   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  4611 proof-
  4612   have *:"t \<inter> s = {}" using assms(3) by auto
  4613   show ?thesis using separate_compact_closed[OF assms(2,1) *]
  4614     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
  4615     by (auto simp add: dist_commute)
  4616 qed
  4617 
  4618 (* A cute way of denoting open and closed intervals using overloading.       *)
  4619 
  4620 lemma interval: fixes a :: "'a::ord^'n::finite" shows
  4621   "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
  4622   "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
  4623   by (auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
  4624 
  4625 lemma mem_interval: fixes a :: "'a::ord^'n::finite" shows
  4626   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
  4627   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
  4628   using interval[of a b] by(auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
  4629 
  4630 lemma mem_interval_1: fixes x :: "real^1" shows
  4631  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
  4632  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
  4633 by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def forall_1)
  4634 
  4635 lemma interval_eq_empty: fixes a :: "real^'n::finite" shows
  4636  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
  4637  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
  4638 proof-
  4639   { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
  4640     hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval by auto
  4641     hence "a$i < b$i" by auto
  4642     hence False using as by auto  }
  4643   moreover
  4644   { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
  4645     let ?x = "(1/2) *\<^sub>R (a + b)"
  4646     { fix i
  4647       have "a$i < b$i" using as[THEN spec[where x=i]] by auto
  4648       hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
  4649         unfolding vector_smult_component and vector_add_component
  4650         by (auto simp add: less_divide_eq_number_of1)  }
  4651     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
  4652   ultimately show ?th1 by blast
  4653 
  4654   { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
  4655     hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval by auto
  4656     hence "a$i \<le> b$i" by auto
  4657     hence False using as by auto  }
  4658   moreover
  4659   { assume as:"\<forall>i. \<not> (b$i < a$i)"
  4660     let ?x = "(1/2) *\<^sub>R (a + b)"
  4661     { fix i
  4662       have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
  4663       hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
  4664         unfolding vector_smult_component and vector_add_component
  4665         by (auto simp add: less_divide_eq_number_of1)  }
  4666     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
  4667   ultimately show ?th2 by blast
  4668 qed
  4669 
  4670 lemma interval_ne_empty: fixes a :: "real^'n::finite" shows
  4671   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and
  4672   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
  4673   unfolding interval_eq_empty[of a b] by (auto simp add: not_less not_le) (* BH: Why doesn't just "auto" work here? *)
  4674 
  4675 lemma subset_interval_imp: fixes a :: "real^'n::finite" shows
  4676  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
  4677  "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
  4678  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
  4679  "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
  4680   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
  4681   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
  4682 
  4683 lemma interval_sing: fixes a :: "'a::linorder^'n::finite" shows
  4684  "{a .. a} = {a} \<and> {a<..<a} = {}"
  4685 apply(auto simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
  4686 apply (simp add: order_eq_iff)
  4687 apply (auto simp add: not_less less_imp_le)
  4688 done
  4689 
  4690 lemma interval_open_subset_closed:  fixes a :: "'a::preorder^'n::finite" shows
  4691  "{a<..<b} \<subseteq> {a .. b}"
  4692 proof(simp add: subset_eq, rule)
  4693   fix x
  4694   assume x:"x \<in>{a<..<b}"
  4695   { fix i
  4696     have "a $ i \<le> x $ i"
  4697       using x order_less_imp_le[of "a$i" "x$i"]
  4698       by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
  4699   }
  4700   moreover
  4701   { fix i
  4702     have "x $ i \<le> b $ i"
  4703       using x order_less_imp_le[of "x$i" "b$i"]
  4704       by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
  4705   }
  4706   ultimately
  4707   show "a \<le> x \<and> x \<le> b"
  4708     by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
  4709 qed
  4710 
  4711 lemma subset_interval: fixes a :: "real^'n::finite" shows
  4712  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) and
  4713  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2) and
  4714  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) and
  4715  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
  4716 proof-
  4717   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
  4718   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)
  4719   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i. c$i < d$i"
  4720     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by (auto, drule_tac x=i in spec, simp) (* BH: Why doesn't just "auto" work? *)
  4721     fix i
  4722     (** TODO combine the following two parts as done in the HOL_light version. **)
  4723     { let ?x = "(\<chi> j. (if j=i then ((min (a$j) (d$j))+c$j)/2 else (c$j+d$j)/2))::real^'n"
  4724       assume as2: "a$i > c$i"
  4725       { fix j
  4726         have "c $ j < ?x $ j \<and> ?x $ j < d $ j" unfolding Cart_lambda_beta
  4727           apply(cases "j=i") using as(2)[THEN spec[where x=j]]
  4728           by (auto simp add: less_divide_eq_number_of1 as2)  }
  4729       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
  4730       moreover
  4731       have "?x\<notin>{a .. b}"
  4732         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
  4733         using as(2)[THEN spec[where x=i]] and as2
  4734         by (auto simp add: less_divide_eq_number_of1)
  4735       ultimately have False using as by auto  }
  4736     hence "a$i \<le> c$i" by(rule ccontr)auto
  4737     moreover
  4738     { let ?x = "(\<chi> j. (if j=i then ((max (b$j) (c$j))+d$j)/2 else (c$j+d$j)/2))::real^'n"
  4739       assume as2: "b$i < d$i"
  4740       { fix j
  4741         have "d $ j > ?x $ j \<and> ?x $ j > c $ j" unfolding Cart_lambda_beta
  4742           apply(cases "j=i") using as(2)[THEN spec[where x=j]]
  4743           by (auto simp add: less_divide_eq_number_of1 as2)  }
  4744       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
  4745       moreover
  4746       have "?x\<notin>{a .. b}"
  4747         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
  4748         using as(2)[THEN spec[where x=i]] and as2
  4749         by (auto simp add: less_divide_eq_number_of1)
  4750       ultimately have False using as by auto  }
  4751     hence "b$i \<ge> d$i" by(rule ccontr)auto
  4752     ultimately
  4753     have "a$i \<le> c$i \<and> d$i \<le> b$i" by auto
  4754   } note part1 = this
  4755   thus ?th3 unfolding subset_eq and Ball_def and mem_interval apply auto apply (erule_tac x=ia in allE, simp)+ by (erule_tac x=i in allE, erule_tac x=i in allE, simp)+
  4756   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i. c$i < d$i"
  4757     fix i
  4758     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
  4759     hence "a$i \<le> c$i \<and> d$i \<le> b$i" using part1 and as(2) by auto  } note * = this
  4760   thus ?th4 unfolding subset_eq and Ball_def and mem_interval apply auto apply (erule_tac x=ia in allE, simp)+ by (erule_tac x=i in allE, erule_tac x=i in allE, simp)+
  4761 qed
  4762 
  4763 lemma disjoint_interval: fixes a::"real^'n::finite" shows
  4764   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) and
  4765   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (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
  4766   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (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
  4767   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (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)
  4768 proof-
  4769   let ?z = "(\<chi> i. ((max (a$i) (c$i)) + (min (b$i) (d$i))) / 2)::real^'n"
  4770   show ?th1 ?th2 ?th3 ?th4
  4771   unfolding expand_set_eq and Int_iff and empty_iff and mem_interval and all_conj_distrib[THEN sym] and eq_False
  4772   apply (auto elim!: allE[where x="?z"])
  4773   apply ((rule_tac x=x in exI, force) | (rule_tac x=i in exI, force))+
  4774   done
  4775 qed
  4776 
  4777 lemma inter_interval: fixes a :: "'a::linorder^'n::finite" shows
  4778  "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
  4779   unfolding expand_set_eq and Int_iff and mem_interval
  4780   by (auto simp add: less_divide_eq_number_of1 intro!: bexI)
  4781 
  4782 (* Moved interval_open_subset_closed a bit upwards *)
  4783 
  4784 lemma open_interval_lemma: fixes x :: "real" shows
  4785  "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
  4786   by(rule_tac x="min (x - a) (b - x)" in exI, auto)
  4787 
  4788 lemma open_interval: fixes a :: "real^'n::finite" shows "open {a<..<b}"
  4789 proof-
  4790   { fix x assume x:"x\<in>{a<..<b}"
  4791     { fix i
  4792       have "\<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i"
  4793         using x[unfolded mem_interval, THEN spec[where x=i]]
  4794         using open_interval_lemma[of "a$i" "x$i" "b$i"] by auto  }
  4795 
  4796     hence "\<forall>i. \<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i" by auto
  4797     then obtain d where d:"\<forall>i. 0 < d i \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d i \<longrightarrow> a $ i < x' \<and> x' < b $ i)"
  4798       using bchoice[of "UNIV" "\<lambda>i d. d>0 \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d \<longrightarrow> a $ i < x' \<and> x' < b $ i)"] by auto
  4799 
  4800     let ?d = "Min (range d)"
  4801     have **:"finite (range d)" "range d \<noteq> {}" by auto
  4802     have "?d>0" unfolding Min_gr_iff[OF **] using d by auto
  4803     moreover
  4804     { fix x' assume as:"dist x' x < ?d"
  4805       { fix i
  4806         have "\<bar>x'$i - x $ i\<bar> < d i"
  4807           using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
  4808           unfolding vector_minus_component and Min_gr_iff[OF **] by auto
  4809         hence "a $ i < x' $ i" "x' $ i < b $ i" using d[THEN spec[where x=i]] by auto  }
  4810       hence "a < x' \<and> x' < b" unfolding vector_less_def by auto  }
  4811     ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by (auto, rule_tac x="?d" in exI, simp)
  4812   }
  4813   thus ?thesis unfolding open_dist using open_interval_lemma by auto
  4814 qed
  4815 
  4816 lemma closed_interval: fixes a :: "real^'n::finite" shows "closed {a .. b}"
  4817 proof-
  4818   { fix x i assume as:"\<forall>e>0. \<exists>x'\<in>{a..b}. x' \<noteq> x \<and> dist x' x < e"(* and xab:"a$i > x$i \<or> b$i < x$i"*)
  4819     { assume xa:"a$i > x$i"
  4820       with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < a$i - x$i" by(erule_tac x="a$i - x$i" in allE)auto
  4821       hence False unfolding mem_interval and dist_norm
  4822         using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xa by(auto elim!: allE[where x=i])
  4823     } hence "a$i \<le> x$i" by(rule ccontr)auto
  4824     moreover
  4825     { assume xb:"b$i < x$i"
  4826       with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < x$i - b$i" by(erule_tac x="x$i - b$i" in allE)auto
  4827       hence False unfolding mem_interval and dist_norm
  4828         using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xb by(auto elim!: allE[where x=i])
  4829     } hence "x$i \<le> b$i" by(rule ccontr)auto
  4830     ultimately
  4831     have "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" by auto }
  4832   thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
  4833 qed
  4834 
  4835 lemma interior_closed_interval: fixes a :: "real^'n::finite" shows
  4836  "interior {a .. b} = {a<..<b}" (is "?L = ?R")
  4837 proof(rule subset_antisym)
  4838   show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
  4839 next
  4840   { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
  4841     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
  4842     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
  4843     { fix i
  4844       have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
  4845            "dist (x + (e / 2) *\<^sub>R basis i) x < e"
  4846         unfolding dist_norm apply auto
  4847         unfolding norm_minus_cancel using norm_basis[of i] and `e>0` by auto
  4848       hence "a $ i \<le> (x - (e / 2) *\<^sub>R basis i) $ i"
  4849                     "(x + (e / 2) *\<^sub>R basis i) $ i \<le> b $ i"
  4850         using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
  4851         and   e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
  4852         unfolding mem_interval by (auto elim!: allE[where x=i])
  4853       hence "a $ i < x $ i" and "x $ i < b $ i"
  4854         unfolding vector_minus_component and vector_add_component
  4855         unfolding vector_smult_component and basis_component using `e>0` by auto   }
  4856     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
  4857   thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
  4858 qed
  4859 
  4860 lemma bounded_closed_interval: fixes a :: "real^'n::finite" shows
  4861  "bounded {a .. b}"
  4862 proof-
  4863   let ?b = "\<Sum>i\<in>UNIV. \<bar>a$i\<bar> + \<bar>b$i\<bar>"
  4864   { fix x::"real^'n" assume x:"\<forall>i. a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
  4865     { fix i
  4866       have "\<bar>x$i\<bar> \<le> \<bar>a$i\<bar> + \<bar>b$i\<bar>" using x[THEN spec[where x=i]] by auto  }
  4867     hence "(\<Sum>i\<in>UNIV. \<bar>x $ i\<bar>) \<le> ?b" by(rule setsum_mono)
  4868     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
  4869   thus ?thesis unfolding interval and bounded_iff by auto
  4870 qed
  4871 
  4872 lemma bounded_interval: fixes a :: "real^'n::finite" shows
  4873  "bounded {a .. b} \<and> bounded {a<..<b}"
  4874   using bounded_closed_interval[of a b]
  4875   using interval_open_subset_closed[of a b]
  4876   using bounded_subset[of "{a..b}" "{a<..<b}"]
  4877   by simp
  4878 
  4879 lemma not_interval_univ: fixes a :: "real^'n::finite" shows
  4880  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
  4881   using bounded_interval[of a b]
  4882   by auto
  4883 
  4884 lemma compact_interval: fixes a :: "real^'n::finite" shows
  4885  "compact {a .. b}"
  4886   using bounded_closed_imp_compact using bounded_interval[of a b] using closed_interval[of a b] by auto
  4887 
  4888 lemma open_interval_midpoint: fixes a :: "real^'n::finite"
  4889   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
  4890 proof-
  4891   { fix i
  4892     have "a $ i < ((1 / 2) *\<^sub>R (a + b)) $ i \<and> ((1 / 2) *\<^sub>R (a + b)) $ i < b $ i"
  4893       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
  4894       unfolding vector_smult_component and vector_add_component
  4895       by(auto simp add: less_divide_eq_number_of1)  }
  4896   thus ?thesis unfolding mem_interval by auto
  4897 qed
  4898 
  4899 lemma open_closed_interval_convex: fixes x :: "real^'n::finite"
  4900   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
  4901   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
  4902 proof-
  4903   { fix i
  4904     have "a $ i = e * a$i + (1 - e) * a$i" unfolding left_diff_distrib by simp
  4905     also have "\<dots> < e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
  4906       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  4907       using x unfolding mem_interval  apply simp
  4908       using y unfolding mem_interval  apply simp
  4909       done
  4910     finally have "a $ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i" by auto
  4911     moreover {
  4912     have "b $ i = e * b$i + (1 - e) * b$i" unfolding left_diff_distrib by simp
  4913     also have "\<dots> > e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
  4914       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  4915       using x unfolding mem_interval  apply simp
  4916       using y unfolding mem_interval  apply simp
  4917       done
  4918     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i < b $ i" by auto
  4919     } 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 }
  4920   thus ?thesis unfolding mem_interval by auto
  4921 qed
  4922 
  4923 lemma closure_open_interval: fixes a :: "real^'n::finite"
  4924   assumes "{a<..<b} \<noteq> {}"
  4925   shows "closure {a<..<b} = {a .. b}"
  4926 proof-
  4927   have ab:"a < b" using assms[unfolded interval_ne_empty] unfolding vector_less_def by auto
  4928   let ?c = "(1 / 2) *\<^sub>R (a + b)"
  4929   { fix x assume as:"x \<in> {a .. b}"
  4930     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
  4931     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
  4932       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
  4933       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
  4934         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
  4935         by (auto simp add: algebra_simps)
  4936       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
  4937       hence False using fn unfolding f_def using xc by(auto simp add: vector_mul_lcancel vector_ssub_ldistrib)  }
  4938     moreover
  4939     { assume "\<not> (f ---> x) sequentially"
  4940       { fix e::real assume "e>0"
  4941         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
  4942         then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  4943         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)
  4944         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }
  4945       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
  4946         unfolding Lim_sequentially by(auto simp add: dist_norm)
  4947       hence "(f ---> x) sequentially" unfolding f_def
  4948         using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
  4949         using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto  }
  4950     ultimately have "x \<in> closure {a<..<b}"
  4951       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }
  4952   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
  4953 qed
  4954 
  4955 lemma bounded_subset_open_interval_symmetric: fixes s::"(real^'n::finite) set"
  4956   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
  4957 proof-
  4958   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
  4959   def a \<equiv> "(\<chi> i. b+1)::real^'n"
  4960   { fix x assume "x\<in>s"
  4961     fix i
  4962     have "(-a)$i < x$i" and "x$i < a$i" using b[THEN bspec[where x=x], OF `x\<in>s`] and component_le_norm[of x i]
  4963       unfolding vector_uminus_component and a_def and Cart_lambda_beta by auto
  4964   }
  4965   thus ?thesis by(auto intro: exI[where x=a] simp add: vector_less_def)
  4966 qed
  4967 
  4968 lemma bounded_subset_open_interval:
  4969   fixes s :: "(real ^ 'n::finite) set"
  4970   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
  4971   by (auto dest!: bounded_subset_open_interval_symmetric)
  4972 
  4973 lemma bounded_subset_closed_interval_symmetric:
  4974   fixes s :: "(real ^ 'n::finite) set"
  4975   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
  4976 proof-
  4977   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
  4978   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
  4979 qed
  4980 
  4981 lemma bounded_subset_closed_interval:
  4982   fixes s :: "(real ^ 'n::finite) set"
  4983   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
  4984   using bounded_subset_closed_interval_symmetric[of s] by auto
  4985 
  4986 lemma frontier_closed_interval:
  4987   fixes a b :: "real ^ _"
  4988   shows "frontier {a .. b} = {a .. b} - {a<..<b}"
  4989   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
  4990 
  4991 lemma frontier_open_interval:
  4992   fixes a b :: "real ^ _"
  4993   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
  4994 proof(cases "{a<..<b} = {}")
  4995   case True thus ?thesis using frontier_empty by auto
  4996 next
  4997   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
  4998 qed
  4999 
  5000 lemma inter_interval_mixed_eq_empty: fixes a :: "real^'n::finite"
  5001   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
  5002   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
  5003 
  5004 
  5005 (* Some special cases for intervals in R^1.                                  *)
  5006 
  5007 lemma all_1: "(\<forall>x::1. P x) \<longleftrightarrow> P 1"
  5008   by (metis num1_eq_iff)
  5009 
  5010 lemma ex_1: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
  5011   by auto (metis num1_eq_iff)
  5012 
  5013 lemma interval_cases_1: fixes x :: "real^1" shows
  5014  "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
  5015   by(simp add:  Cart_eq vector_less_def vector_less_eq_def all_1, auto)
  5016 
  5017 lemma in_interval_1: fixes x :: "real^1" shows
  5018  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
  5019   (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
  5020 by(simp add: Cart_eq vector_less_def vector_less_eq_def all_1 dest_vec1_def)
  5021 
  5022 lemma interval_eq_empty_1: fixes a :: "real^1" shows
  5023   "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
  5024   "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
  5025   unfolding interval_eq_empty and ex_1 and dest_vec1_def by auto
  5026 
  5027 lemma subset_interval_1: fixes a :: "real^1" shows
  5028  "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
  5029                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
  5030  "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
  5031                 dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
  5032  "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b \<le> dest_vec1 a \<or>
  5033                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
  5034  "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
  5035                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
  5036   unfolding subset_interval[of a b c d] unfolding all_1 and dest_vec1_def by auto
  5037 
  5038 lemma eq_interval_1: fixes a :: "real^1" shows
  5039  "{a .. b} = {c .. d} \<longleftrightarrow>
  5040           dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
  5041           dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
  5042 using set_eq_subset[of "{a .. b}" "{c .. d}"]
  5043 using subset_interval_1(1)[of a b c d]
  5044 using subset_interval_1(1)[of c d a b]
  5045 by auto (* FIXME: slow *)
  5046 
  5047 lemma disjoint_interval_1: fixes a :: "real^1" shows
  5048   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"
  5049   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
  5050   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
  5051   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
  5052   unfolding disjoint_interval and dest_vec1_def ex_1 by auto
  5053 
  5054 lemma open_closed_interval_1: fixes a :: "real^1" shows
  5055  "{a<..<b} = {a .. b} - {a, b}"
  5056   unfolding expand_set_eq apply simp unfolding vector_less_def and vector_less_eq_def and all_1 and dest_vec1_eq[THEN sym] and dest_vec1_def by auto
  5057 
  5058 lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
  5059   unfolding expand_set_eq apply simp unfolding vector_less_def and vector_less_eq_def and all_1 and dest_vec1_eq[THEN sym] and dest_vec1_def by auto
  5060 
  5061 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)
  5062 
  5063 lemma closed_interval_left: fixes b::"real^'n::finite"
  5064   shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
  5065 proof-
  5066   { fix i
  5067     fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. x $ i \<le> b $ i}. x' \<noteq> x \<and> dist x' x < e"
  5068     { assume "x$i > b$i"
  5069       then obtain y where "y $ i \<le> b $ i"  "y \<noteq> x"  "dist y x < x$i - b$i" using x[THEN spec[where x="x$i - b$i"]] by auto
  5070       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
  5071     hence "x$i \<le> b$i" by(rule ccontr)auto  }
  5072   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  5073 qed
  5074 
  5075 lemma closed_interval_right: fixes a::"real^'n::finite"
  5076   shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
  5077 proof-
  5078   { fix i
  5079     fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. a $ i \<le> x $ i}. x' \<noteq> x \<and> dist x' x < e"
  5080     { assume "a$i > x$i"
  5081       then obtain y where "a $ i \<le> y $ i"  "y \<noteq> x"  "dist y x < a$i - x$i" using x[THEN spec[where x="a$i - x$i"]] by auto
  5082       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
  5083     hence "a$i \<le> x$i" by(rule ccontr)auto  }
  5084   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
  5085 qed
  5086 
  5087 subsection{* Intervals in general, including infinite and mixtures of open and closed. *}
  5088 
  5089 definition "is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((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)"
  5090 
  5091 lemma is_interval_interval: "is_interval {a .. b::real^'n::finite}" (is ?th1) "is_interval {a<..<b}" (is ?th2) proof - 
  5092   have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
  5093   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
  5094     by(meson real_le_trans le_less_trans less_le_trans *)+ qed
  5095 
  5096 lemma is_interval_empty:
  5097  "is_interval {}"
  5098   unfolding is_interval_def
  5099   by simp
  5100 
  5101 lemma is_interval_univ:
  5102  "is_interval UNIV"
  5103   unfolding is_interval_def
  5104   by simp
  5105 
  5106 subsection{* Closure of halfspaces and hyperplanes.                                    *}
  5107 
  5108 lemma Lim_inner:
  5109   assumes "(f ---> l) net"  shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
  5110   by (intro tendsto_intros assms)
  5111 
  5112 lemma continuous_at_inner: "continuous (at x) (inner a)"
  5113   unfolding continuous_at by (intro tendsto_intros)
  5114 
  5115 lemma continuous_on_inner:
  5116   fixes s :: "'a::real_inner set"
  5117   shows "continuous_on s (inner a)"
  5118   unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  5119 
  5120 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
  5121 proof-
  5122   have "\<forall>x. continuous (at x) (inner a)"
  5123     unfolding continuous_at by (rule allI) (intro tendsto_intros)
  5124   hence "closed (inner a -` {..b})"
  5125     using closed_real_atMost by (rule continuous_closed_vimage)
  5126   moreover have "{x. inner a x \<le> b} = inner a -` {..b}" by auto
  5127   ultimately show ?thesis by simp
  5128 qed
  5129 
  5130 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
  5131   using closed_halfspace_le[of "-a" "-b"] unfolding inner_minus_left by auto
  5132 
  5133 lemma closed_hyperplane: "closed {x. inner a x = b}"
  5134 proof-
  5135   have "{x. inner a x = b} = {x. inner a x \<ge> b} \<inter> {x. inner a x \<le> b}" by auto
  5136   thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
  5137 qed
  5138 
  5139 lemma closed_halfspace_component_le:
  5140   shows "closed {x::real^'n::finite. x$i \<le> a}"
  5141   using closed_halfspace_le[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
  5142 
  5143 lemma closed_halfspace_component_ge:
  5144   shows "closed {x::real^'n::finite. x$i \<ge> a}"
  5145   using closed_halfspace_ge[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
  5146 
  5147 text{* Openness of halfspaces.                                                   *}
  5148 
  5149 lemma open_halfspace_lt: "open {x. inner a x < b}"
  5150 proof-
  5151   have "UNIV - {x. b \<le> inner a x} = {x. inner a x < b}" by auto
  5152   thus ?thesis using closed_halfspace_ge[unfolded closed_def Compl_eq_Diff_UNIV, of b a] by auto
  5153 qed
  5154 
  5155 lemma open_halfspace_gt: "open {x. inner a x > b}"
  5156 proof-
  5157   have "UNIV - {x. b \<ge> inner a x} = {x. inner a x > b}" by auto
  5158   thus ?thesis using closed_halfspace_le[unfolded closed_def Compl_eq_Diff_UNIV, of a b] by auto
  5159 qed
  5160 
  5161 lemma open_halfspace_component_lt:
  5162   shows "open {x::real^'n::finite. x$i < a}"
  5163   using open_halfspace_lt[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
  5164 
  5165 lemma open_halfspace_component_gt:
  5166   shows "open {x::real^'n::finite. x$i  > a}"
  5167   using open_halfspace_gt[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
  5168 
  5169 text{* This gives a simple derivation of limit component bounds.                 *}
  5170 
  5171 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> real^'n::finite"
  5172   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$i \<le> b) net"
  5173   shows "l$i \<le> b"
  5174 proof-
  5175   { fix x have "x \<in> {x::real^'n. inner (basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this
  5176   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
  5177     using closed_halfspace_le[of "(basis i)::real^'n" b] and assms(1,2,3) by auto
  5178 qed
  5179 
  5180 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> real^'n::finite"
  5181   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
  5182   shows "b \<le> l$i"
  5183 proof-
  5184   { fix x have "x \<in> {x::real^'n. inner (basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this
  5185   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
  5186     using closed_halfspace_ge[of b "(basis i)::real^'n"] and assms(1,2,3) by auto
  5187 qed
  5188 
  5189 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> real^'n::finite"
  5190   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
  5191   shows "l$i = b"
  5192   using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
  5193 
  5194 lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
  5195   "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
  5196   using Lim_component_le[of f l net 1 b] unfolding dest_vec1_def by auto
  5197 
  5198 lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows
  5199  "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
  5200   using Lim_component_ge[of f l net b 1] unfolding dest_vec1_def by auto
  5201 
  5202 text{* Limits relative to a union.                                               *}
  5203 
  5204 lemma eventually_within_Un:
  5205   "eventually P (net within (s \<union> t)) \<longleftrightarrow>
  5206     eventually P (net within s) \<and> eventually P (net within t)"
  5207   unfolding Limits.eventually_within
  5208   by (auto elim!: eventually_rev_mp)
  5209 
  5210 lemma Lim_within_union:
  5211  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
  5212   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
  5213   unfolding tendsto_def
  5214   by (auto simp add: eventually_within_Un)
  5215 
  5216 lemma continuous_on_union:
  5217   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
  5218   shows "continuous_on (s \<union> t) f"
  5219   using assms unfolding continuous_on unfolding Lim_within_union
  5220   unfolding Lim unfolding trivial_limit_within unfolding closed_limpt by auto
  5221 
  5222 lemma continuous_on_cases:
  5223   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
  5224           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
  5225   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
  5226 proof-
  5227   let ?h = "(\<lambda>x. if P x then f x else g x)"
  5228   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
  5229   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
  5230   moreover
  5231   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
  5232   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
  5233   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
  5234 qed
  5235 
  5236 
  5237 text{* Some more convenient intermediate-value theorem formulations.             *}
  5238 
  5239 lemma connected_ivt_hyperplane:
  5240   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
  5241   shows "\<exists>z \<in> s. inner a z = b"
  5242 proof(rule ccontr)
  5243   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
  5244   let ?A = "{x. inner a x < b}"
  5245   let ?B = "{x. inner a x > b}"
  5246   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
  5247   moreover have "?A \<inter> ?B = {}" by auto
  5248   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
  5249   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
  5250 qed
  5251 
  5252 lemma connected_ivt_component: fixes x::"real^'n::finite" shows
  5253  "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)"
  5254   using connected_ivt_hyperplane[of s x y "(basis k)::real^'n" a] by (auto simp add: inner_basis)
  5255 
  5256 text{* Also more convenient formulations of monotone convergence.                *}
  5257 
  5258 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real^1"
  5259   assumes "bounded {s n| n::nat. True}"  "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
  5260   shows "\<exists>l. (s ---> l) sequentially"
  5261 proof-
  5262   obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto
  5263   { fix m::nat
  5264     have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
  5265       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq)  }
  5266   hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
  5267   then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
  5268   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)
  5269     unfolding dist_norm unfolding abs_dest_vec1 and dest_vec1_sub by auto
  5270 qed
  5271 
  5272 subsection{* Basic homeomorphism definitions.                                          *}
  5273 
  5274 definition "homeomorphism s t f g \<equiv>
  5275      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
  5276      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
  5277 
  5278 definition
  5279   homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
  5280     (infixr "homeomorphic" 60) where
  5281   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
  5282 
  5283 lemma homeomorphic_refl: "s homeomorphic s"
  5284   unfolding homeomorphic_def
  5285   unfolding homeomorphism_def
  5286   using continuous_on_id
  5287   apply(rule_tac x = "(\<lambda>x. x)" in exI)
  5288   apply(rule_tac x = "(\<lambda>x. x)" in exI)
  5289   by blast
  5290 
  5291 lemma homeomorphic_sym:
  5292  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
  5293 unfolding homeomorphic_def
  5294 unfolding homeomorphism_def
  5295 by blast (* FIXME: slow *)
  5296 
  5297 lemma homeomorphic_trans:
  5298   assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
  5299 proof-
  5300   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"
  5301     using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
  5302   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"
  5303     using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
  5304 
  5305   { 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 }
  5306   moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
  5307   moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
  5308   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 }
  5309   moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
  5310   moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6)  unfolding fg2(5) by auto
  5311   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
  5312 qed
  5313 
  5314 lemma homeomorphic_minimal:
  5315  "s homeomorphic t \<longleftrightarrow>
  5316     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
  5317            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
  5318            continuous_on s f \<and> continuous_on t g)"
  5319 unfolding homeomorphic_def homeomorphism_def
  5320 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
  5321 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
  5322 unfolding image_iff
  5323 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
  5324 apply auto apply(rule_tac x="g x" in bexI) apply auto
  5325 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
  5326 apply auto apply(rule_tac x="f x" in bexI) by auto
  5327 
  5328 subsection{* Relatively weak hypotheses if a set is compact.                           *}
  5329 
  5330 definition "inv_on f s = (\<lambda>x. SOME y. y\<in>s \<and> f y = x)"
  5331 
  5332 lemma assumes "inj_on f s" "x\<in>s"
  5333   shows "inv_on f s (f x) = x"
  5334  using assms unfolding inj_on_def inv_on_def by auto
  5335 
  5336 lemma homeomorphism_compact:
  5337   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  5338     (* class constraint due to continuous_on_inverse *)
  5339   assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
  5340   shows "\<exists>g. homeomorphism s t f g"
  5341 proof-
  5342   def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
  5343   have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
  5344   { fix y assume "y\<in>t"
  5345     then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
  5346     hence "g (f x) = x" using g by auto
  5347     hence "f (g y) = y" unfolding x(1)[THEN sym] by auto  }
  5348   hence g':"\<forall>x\<in>t. f (g x) = x" by auto
  5349   moreover
  5350   { fix x
  5351     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"])
  5352     moreover
  5353     { assume "x\<in>g ` t"
  5354       then obtain y where y:"y\<in>t" "g y = x" by auto
  5355       then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
  5356       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 }
  5357     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" by auto  }
  5358   hence "g ` t = s" by auto
  5359   ultimately
  5360   show ?thesis unfolding homeomorphism_def homeomorphic_def
  5361     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
  5362 qed
  5363 
  5364 lemma homeomorphic_compact:
  5365   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
  5366     (* class constraint due to continuous_on_inverse *)
  5367   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
  5368           \<Longrightarrow> s homeomorphic t"
  5369   unfolding homeomorphic_def by(metis homeomorphism_compact)
  5370 
  5371 text{* Preservation of topological properties.                                   *}
  5372 
  5373 lemma homeomorphic_compactness:
  5374  "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
  5375 unfolding homeomorphic_def homeomorphism_def
  5376 by (metis compact_continuous_image)
  5377 
  5378 text{* Results on translation, scaling etc.                                      *}
  5379 
  5380 lemma homeomorphic_scaling:
  5381   fixes s :: "'a::real_normed_vector set"
  5382   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
  5383   unfolding homeomorphic_minimal
  5384   apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
  5385   apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
  5386   using assms apply auto
  5387   using continuous_on_cmul[OF continuous_on_id] by auto
  5388 
  5389 lemma homeomorphic_translation:
  5390   fixes s :: "'a::real_normed_vector set"
  5391   shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
  5392   unfolding homeomorphic_minimal
  5393   apply(rule_tac x="\<lambda>x. a + x" in exI)
  5394   apply(rule_tac x="\<lambda>x. -a + x" in exI)
  5395   using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
  5396 
  5397 lemma homeomorphic_affinity:
  5398   fixes s :: "'a::real_normed_vector set"
  5399   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  5400 proof-
  5401   have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  5402   show ?thesis
  5403     using homeomorphic_trans
  5404     using homeomorphic_scaling[OF assms, of s]
  5405     using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
  5406 qed
  5407 
  5408 lemma homeomorphic_balls:
  5409   fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
  5410   assumes "0 < d"  "0 < e"
  5411   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
  5412         "(cball a d) homeomorphic (cball b e)" (is ?cth)
  5413 proof-
  5414   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
  5415   show ?th unfolding homeomorphic_minimal
  5416     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
  5417     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
  5418     using assms apply (auto simp add: dist_commute)
  5419     unfolding dist_norm
  5420     apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
  5421     unfolding continuous_on
  5422     by (intro ballI tendsto_intros, simp, assumption)+
  5423 next
  5424   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
  5425   show ?cth unfolding homeomorphic_minimal
  5426     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
  5427     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
  5428     using assms apply (auto simp add: dist_commute)
  5429     unfolding dist_norm
  5430     apply (auto simp add: pos_divide_le_eq)
  5431     unfolding continuous_on
  5432     by (intro ballI tendsto_intros, simp, assumption)+
  5433 qed
  5434 
  5435 text{* "Isometry" (up to constant bounds) of injective linear map etc.           *}
  5436 
  5437 lemma cauchy_isometric:
  5438   fixes x :: "nat \<Rightarrow> real ^ 'n::finite"
  5439   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)"
  5440   shows "Cauchy x"
  5441 proof-
  5442   interpret f: bounded_linear f by fact
  5443   { fix d::real assume "d>0"
  5444     then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
  5445       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
  5446     { fix n assume "n\<ge>N"
  5447       hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
  5448       moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
  5449         using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
  5450         using normf[THEN bspec[where x="x n - x N"]] by auto
  5451       ultimately have "norm (x n - x N) < d" using `e>0`
  5452         using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto   }
  5453     hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
  5454   thus ?thesis unfolding cauchy and dist_norm by auto
  5455 qed
  5456 
  5457 lemma complete_isometric_image:
  5458   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
  5459   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"
  5460   shows "complete(f ` s)"
  5461 proof-
  5462   { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
  5463     then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)" unfolding image_iff and Bex_def
  5464       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
  5465     hence x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)" by auto
  5466     hence "f \<circ> x = g" unfolding expand_fun_eq by auto
  5467     then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
  5468       using cs[unfolded complete_def, THEN spec[where x="x"]]
  5469       using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
  5470     hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
  5471       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
  5472       unfolding `f \<circ> x = g` by auto  }
  5473   thus ?thesis unfolding complete_def by auto
  5474 qed
  5475 
  5476 lemma dist_0_norm:
  5477   fixes x :: "'a::real_normed_vector"
  5478   shows "dist 0 x = norm x"
  5479 unfolding dist_norm by simp
  5480 
  5481 lemma injective_imp_isometric: fixes f::"real^'m::finite \<Rightarrow> real^'n::finite"
  5482   assumes s:"closed s"  "subspace s"  and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
  5483   shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
  5484 proof(cases "s \<subseteq> {0::real^'m}")
  5485   case True
  5486   { fix x assume "x \<in> s"
  5487     hence "x = 0" using True by auto
  5488     hence "norm x \<le> norm (f x)" by auto  }
  5489   thus ?thesis by(auto intro!: exI[where x=1])
  5490 next
  5491   interpret f: bounded_linear f by fact
  5492   case False
  5493   then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
  5494   from False have "s \<noteq> {}" by auto
  5495   let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
  5496   let ?S' = "{x::real^'m. x\<in>s \<and> norm x = norm a}"
  5497   let ?S'' = "{x::real^'m. norm x = norm a}"
  5498 
  5499   have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by (auto simp add: norm_minus_cancel)
  5500   hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
  5501   moreover have "?S' = s \<inter> ?S''" by auto
  5502   ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
  5503   moreover have *:"f ` ?S' = ?S" by auto
  5504   ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
  5505   hence "closed ?S" using compact_imp_closed by auto
  5506   moreover have "?S \<noteq> {}" using a by auto
  5507   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
  5508   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
  5509 
  5510   let ?e = "norm (f b) / norm b"
  5511   have "norm b > 0" using ba and a and norm_ge_zero by auto
  5512   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
  5513   ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
  5514   moreover
  5515   { fix x assume "x\<in>s"
  5516     hence "norm (f b) / norm b * norm x \<le> norm (f x)"
  5517     proof(cases "x=0")
  5518       case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
  5519     next
  5520       case False
  5521       hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
  5522       have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def smult_conv_scaleR] by auto
  5523       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
  5524       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"]]
  5525         unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
  5526         by (auto simp add: real_mult_commute pos_le_divide_eq pos_divide_le_eq)
  5527     qed }
  5528   ultimately
  5529   show ?thesis by auto
  5530 qed
  5531 
  5532 lemma closed_injective_image_subspace:
  5533   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
  5534   assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
  5535   shows "closed(f ` s)"
  5536 proof-
  5537   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
  5538   show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
  5539     unfolding complete_eq_closed[THEN sym] by auto
  5540 qed
  5541 
  5542 subsection{* Some properties of a canonical subspace.                                  *}
  5543 
  5544 lemma subspace_substandard:
  5545  "subspace {x::real^'n. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
  5546   unfolding subspace_def by(auto simp add: vector_add_component vector_smult_component elim!: ballE)
  5547 
  5548 lemma closed_substandard:
  5549  "closed {x::real^'n::finite. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
  5550 proof-
  5551   let ?D = "{i. P i}"
  5552   let ?Bs = "{{x::real^'n. inner (basis i) x = 0}| i. i \<in> ?D}"
  5553   { fix x
  5554     { assume "x\<in>?A"
  5555       hence x:"\<forall>i\<in>?D. x $ i = 0" by auto
  5556       hence "x\<in> \<Inter> ?Bs" by(auto simp add: inner_basis x) }
  5557     moreover
  5558     { assume x:"x\<in>\<Inter>?Bs"
  5559       { fix i assume i:"i \<in> ?D"
  5560         then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (basis i) x = 0}" by auto
  5561         hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto  }
  5562       hence "x\<in>?A" by auto }
  5563     ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
  5564   hence "?A = \<Inter> ?Bs" by auto
  5565   thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
  5566 qed
  5567 
  5568 lemma dim_substandard:
  5569   shows "dim {x::real^'n::finite. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
  5570 proof-
  5571   let ?D = "UNIV::'n set"
  5572   let ?B = "(basis::'n\<Rightarrow>real^'n) ` d"
  5573 
  5574     let ?bas = "basis::'n \<Rightarrow> real^'n"
  5575 
  5576   have "?B \<subseteq> ?A" by auto
  5577 
  5578   moreover
  5579   { fix x::"real^'n" assume "x\<in>?A"
  5580     with finite[of d]
  5581     have "x\<in> span ?B"
  5582     proof(induct d arbitrary: x)
  5583       case empty hence "x=0" unfolding Cart_eq by auto
  5584       thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
  5585     next
  5586       case (insert k F)
  5587       hence *:"\<forall>i. i \<notin> insert k F \<longrightarrow> x $ i = 0" by auto
  5588       have **:"F \<subseteq> insert k F" by auto
  5589       def y \<equiv> "x - x$k *\<^sub>R basis k"
  5590       have y:"x = y + (x$k) *\<^sub>R basis k" unfolding y_def by auto
  5591       { fix i assume i':"i \<notin> F"
  5592         hence "y $ i = 0" unfolding y_def unfolding vector_minus_component
  5593           and vector_smult_component and basis_component
  5594           using *[THEN spec[where x=i]] by auto }
  5595       hence "y \<in> span (basis ` (insert k F))" using insert(3)
  5596         using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
  5597         using image_mono[OF **, of basis] by auto
  5598       moreover
  5599       have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
  5600       hence "x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
  5601         using span_mul [where 'a=real, unfolded smult_conv_scaleR] by auto
  5602       ultimately
  5603       have "y + x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
  5604         using span_add by auto
  5605       thus ?case using y by auto
  5606     qed
  5607   }
  5608   hence "?A \<subseteq> span ?B" by auto
  5609 
  5610   moreover
  5611   { fix x assume "x \<in> ?B"
  5612     hence "x\<in>{(basis i)::real^'n |i. i \<in> ?D}" using assms by auto  }
  5613   hence "independent ?B" using independent_mono[OF independent_stdbasis, of ?B] and assms by auto
  5614 
  5615   moreover
  5616   have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
  5617   hence *:"inj_on (basis::'n\<Rightarrow>real^'n) d" using subset_inj_on[OF basis_inj, of "d"] by auto
  5618   have "?B hassize (card d)" unfolding hassize_def and card_image[OF *] by auto
  5619 
  5620   ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
  5621 qed
  5622 
  5623 text{* Hence closure and completeness of all subspaces.                          *}
  5624 
  5625 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
  5626 apply (induct n)
  5627 apply (rule_tac x="{}" in exI, simp)
  5628 apply clarsimp
  5629 apply (subgoal_tac "\<exists>x. x \<notin> A")
  5630 apply (erule exE)
  5631 apply (rule_tac x="insert x A" in exI, simp)
  5632 apply (subgoal_tac "A \<noteq> UNIV", auto)
  5633 done
  5634 
  5635 lemma closed_subspace: fixes s::"(real^'n::finite) set"
  5636   assumes "subspace s" shows "closed s"
  5637 proof-
  5638   have "dim s \<le> card (UNIV :: 'n set)" using dim_subset_univ by auto
  5639   then obtain d::"'n set" where t: "card d = dim s"
  5640     using closed_subspace_lemma by auto
  5641   let ?t = "{x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"
  5642   obtain f where f:"bounded_linear f"  "f ` ?t = s" "inj_on f ?t"
  5643     using subspace_isomorphism[unfolded linear_conv_bounded_linear, OF subspace_substandard[of "\<lambda>i. i \<notin> d"] assms]
  5644     using dim_substandard[of d] and t by auto
  5645   interpret f: bounded_linear f by fact
  5646   have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
  5647     by(erule_tac x=0 in ballE) auto
  5648   moreover have "closed ?t" using closed_substandard .
  5649   moreover have "subspace ?t" using subspace_substandard .
  5650   ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
  5651     unfolding f(2) using f(1) by auto
  5652 qed
  5653 
  5654 lemma complete_subspace:
  5655   fixes s :: "(real ^ _) set" shows "subspace s ==> complete s"
  5656   using complete_eq_closed closed_subspace
  5657   by auto
  5658 
  5659 lemma dim_closure:
  5660   fixes s :: "(real ^ _) set"
  5661   shows "dim(closure s) = dim s" (is "?dc = ?d")
  5662 proof-
  5663   have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]