src/HOL/Library/Topology_Euclidean_Space.thy
 author huffman Wed Jun 10 15:29:05 2009 -0700 (2009-06-10) changeset 31560 88347c12e267 parent 31559 ca9e56897403 child 31565 da5a5589418e permissions -rw-r--r--
heine_borel instance for products
```     1 (* Title:      Topology
```
```     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)
```
```   103   apply (metis openin_subset subset_eq)
```
```   104   apply (auto simp add: Diff_Diff_Int)
```
```   105   apply (subgoal_tac "topspace U \<inter> S = S")
```
```   106   by auto
```
```   107
```
```   108 lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
```
```   109   by (simp add: openin_closedin_eq)
```
```   110
```
```   111 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
```
```   112 proof-
```
```   113   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
```
```   114     by (auto simp add: topspace_def openin_subset)
```
```   115   then show ?thesis using oS cT by (auto simp add: closedin_def)
```
```   116 qed
```
```   117
```
```   118 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
```
```   119 proof-
```
```   120   have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
```
```   121     by (auto simp add: topspace_def )
```
```   122   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
```
```   123 qed
```
```   124
```
```   125 subsection{* Subspace topology. *}
```
```   126
```
```   127 definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
```
```   128
```
```   129 lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
```
```   130 proof-
```
```   131   have "{} \<in> ?L" by blast
```
```   132   {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
```
```   133     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
```
```   134     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
```
```   135     then have "A \<inter> B \<in> ?L" by blast}
```
```   136   moreover
```
```   137   {fix K assume K: "K \<subseteq> ?L"
```
```   138     have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
```
```   139       apply (rule set_ext)
```
```   140       apply (simp add: Ball_def image_iff)
```
```   141       by (metis mem_def)
```
```   142     from K[unfolded th0 subset_image_iff]
```
```   143     obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
```
```   144     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
```
```   145     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
```
```   146     ultimately have "\<Union>K \<in> ?L" by blast}
```
```   147   ultimately show ?thesis unfolding istopology_def by blast
```
```   148 qed
```
```   149
```
```   150 lemma openin_subtopology:
```
```   151   "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
```
```   152   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
```
```   153   by (auto simp add: Collect_def)
```
```   154
```
```   155 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
```
```   156   by (auto simp add: topspace_def openin_subtopology)
```
```   157
```
```   158 lemma closedin_subtopology:
```
```   159   "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
```
```   160   unfolding closedin_def topspace_subtopology
```
```   161   apply (simp add: openin_subtopology)
```
```   162   apply (rule iffI)
```
```   163   apply clarify
```
```   164   apply (rule_tac x="topspace U - T" in exI)
```
```   165   by auto
```
```   166
```
```   167 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
```
```   168   unfolding openin_subtopology
```
```   169   apply (rule iffI, clarify)
```
```   170   apply (frule openin_subset[of U])  apply blast
```
```   171   apply (rule exI[where x="topspace U"])
```
```   172   by auto
```
```   173
```
```   174 lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
```
```   175   shows "subtopology U V = U"
```
```   176 proof-
```
```   177   {fix S
```
```   178     {fix T assume T: "openin U T" "S = T \<inter> V"
```
```   179       from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
```
```   180       have "openin U S" unfolding eq using T by blast}
```
```   181     moreover
```
```   182     {assume S: "openin U S"
```
```   183       hence "\<exists>T. openin U T \<and> S = T \<inter> V"
```
```   184 	using openin_subset[OF S] UV by auto}
```
```   185     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
```
```   186   then show ?thesis unfolding topology_eq openin_subtopology by blast
```
```   187 qed
```
```   188
```
```   189
```
```   190 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
```
```   191   by (simp add: subtopology_superset)
```
```   192
```
```   193 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
```
```   194   by (simp add: subtopology_superset)
```
```   195
```
```   196 subsection{* The universal Euclidean versions are what we use most of the time *}
```
```   197
```
```   198 definition
```
```   199   euclidean :: "'a::topological_space topology" where
```
```   200   "euclidean = topology open"
```
```   201
```
```   202 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
```
```   203   unfolding euclidean_def
```
```   204   apply (rule cong[where x=S and y=S])
```
```   205   apply (rule topology_inverse[symmetric])
```
```   206   apply (auto simp add: istopology_def)
```
```   207   by (auto simp add: mem_def subset_eq)
```
```   208
```
```   209 lemma topspace_euclidean: "topspace euclidean = UNIV"
```
```   210   apply (simp add: topspace_def)
```
```   211   apply (rule set_ext)
```
```   212   by (auto simp add: open_openin[symmetric])
```
```   213
```
```   214 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
```
```   215   by (simp add: topspace_euclidean topspace_subtopology)
```
```   216
```
```   217 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
```
```   218   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
```
```   219
```
```   220 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
```
```   221   by (simp add: open_openin openin_subopen[symmetric])
```
```   222
```
```   223 subsection{* Open and closed balls. *}
```
```   224
```
```   225 definition
```
```   226   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
```
```   227   "ball x e = {y. dist x y < e}"
```
```   228
```
```   229 definition
```
```   230   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
```
```   231   "cball x e = {y. dist x y \<le> e}"
```
```   232
```
```   233 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
```
```   234 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
```
```   235
```
```   236 lemma mem_ball_0 [simp]:
```
```   237   fixes x :: "'a::real_normed_vector"
```
```   238   shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
```
```   239   by (simp add: dist_norm)
```
```   240
```
```   241 lemma mem_cball_0 [simp]:
```
```   242   fixes x :: "'a::real_normed_vector"
```
```   243   shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
```
```   244   by (simp add: dist_norm)
```
```   245
```
```   246 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
```
```   247 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
```
```   248 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
```
```   249 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
```
```   250 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
```
```   251   by (simp add: expand_set_eq) arith
```
```   252
```
```   253 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
```
```   254   by (simp add: expand_set_eq)
```
```   255
```
```   256 subsection{* Topological properties of open balls *}
```
```   257
```
```   258 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
```
```   259   "(a::real) - b < 0 \<longleftrightarrow> a < b"
```
```   260   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
```
```   261 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
```
```   262   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
```
```   263
```
```   264 lemma open_ball[intro, simp]: "open (ball x e)"
```
```   265   unfolding open_dist ball_def Collect_def Ball_def mem_def
```
```   266   unfolding dist_commute
```
```   267   apply clarify
```
```   268   apply (rule_tac x="e - dist xa x" in exI)
```
```   269   using dist_triangle_alt[where z=x]
```
```   270   apply (clarsimp simp add: diff_less_iff)
```
```   271   apply atomize
```
```   272   apply (erule_tac x="y" in allE)
```
```   273   apply (erule_tac x="xa" in allE)
```
```   274   by arith
```
```   275
```
```   276 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
```
```   277 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
```
```   278   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
```
```   279
```
```   280 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
```
```   281   by (metis open_contains_ball subset_eq centre_in_ball)
```
```   282
```
```   283 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
```
```   284   unfolding mem_ball expand_set_eq
```
```   285   apply (simp add: not_less)
```
```   286   by (metis zero_le_dist order_trans dist_self)
```
```   287
```
```   288 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
```
```   289
```
```   290 subsection{* Basic "localization" results are handy for connectedness. *}
```
```   291
```
```   292 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
```
```   293   by (auto simp add: openin_subtopology open_openin[symmetric])
```
```   294
```
```   295 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
```
```   296   by (auto simp add: openin_open)
```
```   297
```
```   298 lemma open_openin_trans[trans]:
```
```   299  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
```
```   300   by (metis Int_absorb1  openin_open_Int)
```
```   301
```
```   302 lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
```
```   303   by (auto simp add: openin_open)
```
```   304
```
```   305 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
```
```   306   by (simp add: closedin_subtopology closed_closedin Int_ac)
```
```   307
```
```   308 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
```
```   309   by (metis closedin_closed)
```
```   310
```
```   311 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
```
```   312   apply (subgoal_tac "S \<inter> T = T" )
```
```   313   apply auto
```
```   314   apply (frule closedin_closed_Int[of T S])
```
```   315   by simp
```
```   316
```
```   317 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
```
```   318   by (auto simp add: closedin_closed)
```
```   319
```
```   320 lemma openin_euclidean_subtopology_iff:
```
```   321   fixes S U :: "'a::metric_space set"
```
```   322   shows "openin (subtopology euclidean U) S
```
```   323   \<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")
```
```   324 proof-
```
```   325   {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
```
```   326       by (simp add: open_dist) blast}
```
```   327   moreover
```
```   328   {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"
```
```   329     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)"
```
```   330       by metis
```
```   331     let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
```
```   332     have oT: "open ?T" by auto
```
```   333     { fix x assume "x\<in>S"
```
```   334       hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
```
```   335 	apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
```
```   336         by (rule d [THEN conjunct1])
```
```   337       hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }
```
```   338     moreover
```
```   339     { fix y assume "y\<in>?T"
```
```   340       then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
```
```   341       then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
```
```   342       assume "y\<in>U"
```
```   343       hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
```
```   344     ultimately have "S = ?T \<inter> U" by blast
```
```   345     with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
```
```   346   ultimately show ?thesis by blast
```
```   347 qed
```
```   348
```
```   349 text{* These "transitivity" results are handy too. *}
```
```   350
```
```   351 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
```
```   352   \<Longrightarrow> openin (subtopology euclidean U) S"
```
```   353   unfolding open_openin openin_open by blast
```
```   354
```
```   355 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
```
```   356   by (auto simp add: openin_open intro: openin_trans)
```
```   357
```
```   358 lemma closedin_trans[trans]:
```
```   359  "closedin (subtopology euclidean T) S \<Longrightarrow>
```
```   360            closedin (subtopology euclidean U) T
```
```   361            ==> closedin (subtopology euclidean U) S"
```
```   362   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
```
```   363
```
```   364 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
```
```   365   by (auto simp add: closedin_closed intro: closedin_trans)
```
```   366
```
```   367 subsection{* Connectedness *}
```
```   368
```
```   369 definition "connected S \<longleftrightarrow>
```
```   370   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
```
```   371   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
```
```   372
```
```   373 lemma connected_local:
```
```   374  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
```
```   375                  openin (subtopology euclidean S) e1 \<and>
```
```   376                  openin (subtopology euclidean S) e2 \<and>
```
```   377                  S \<subseteq> e1 \<union> e2 \<and>
```
```   378                  e1 \<inter> e2 = {} \<and>
```
```   379                  ~(e1 = {}) \<and>
```
```   380                  ~(e2 = {}))"
```
```   381 unfolding connected_def openin_open by (safe, blast+)
```
```   382
```
```   383 lemma exists_diff: "(\<exists>S. P(UNIV - S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   384 proof-
```
```   385
```
```   386   {assume "?lhs" hence ?rhs by blast }
```
```   387   moreover
```
```   388   {fix S assume H: "P S"
```
```   389     have "S = UNIV - (UNIV - S)" by auto
```
```   390     with H have "P (UNIV - (UNIV - S))" by metis }
```
```   391   ultimately show ?thesis by metis
```
```   392 qed
```
```   393
```
```   394 lemma connected_clopen: "connected S \<longleftrightarrow>
```
```   395         (\<forall>T. openin (subtopology euclidean S) T \<and>
```
```   396             closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   397 proof-
```
```   398   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> {})"
```
```   399     unfolding connected_def openin_open closedin_closed
```
```   400     apply (subst exists_diff) by blast
```
```   401   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> {})"
```
```   402     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def Compl_eq_Diff_UNIV) by metis
```
```   403
```
```   404   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'))"
```
```   405     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
```
```   406     unfolding connected_def openin_open closedin_closed by auto
```
```   407   {fix e2
```
```   408     {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)"
```
```   409 	by auto}
```
```   410     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
```
```   411   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
```
```   412   then show ?thesis unfolding th0 th1 by simp
```
```   413 qed
```
```   414
```
```   415 lemma connected_empty[simp, intro]: "connected {}"
```
```   416   by (simp add: connected_def)
```
```   417
```
```   418 subsection{* Hausdorff and other separation properties *}
```
```   419
```
```   420 class t0_space =
```
```   421   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
```
```   422
```
```   423 class t1_space =
```
```   424   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"
```
```   425 begin
```
```   426
```
```   427 subclass t0_space
```
```   428 proof
```
```   429 qed (fast dest: t1_space)
```
```   430
```
```   431 end
```
```   432
```
```   433 text {* T2 spaces are also known as Hausdorff spaces. *}
```
```   434
```
```   435 class t2_space =
```
```   436   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 = {}"
```
```   437 begin
```
```   438
```
```   439 subclass t1_space
```
```   440 proof
```
```   441 qed (fast dest: hausdorff)
```
```   442
```
```   443 end
```
```   444
```
```   445 instance metric_space \<subseteq> t2_space
```
```   446 proof
```
```   447   fix x y :: "'a::metric_space"
```
```   448   assume xy: "x \<noteq> y"
```
```   449   let ?U = "ball x (dist x y / 2)"
```
```   450   let ?V = "ball y (dist x y / 2)"
```
```   451   have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
```
```   452                ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
```
```   453   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
```
```   454     using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
```
```   455     by (auto simp add: expand_set_eq)
```
```   456   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```   457     by blast
```
```   458 qed
```
```   459
```
```   460 lemma separation_t2:
```
```   461   fixes x y :: "'a::t2_space"
```
```   462   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 = {})"
```
```   463   using hausdorff[of x y] by blast
```
```   464
```
```   465 lemma separation_t1:
```
```   466   fixes x y :: "'a::t1_space"
```
```   467   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)"
```
```   468   using t1_space[of x y] by blast
```
```   469
```
```   470 lemma separation_t0:
```
```   471   fixes x y :: "'a::t0_space"
```
```   472   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
```
```   473   using t0_space[of x y] by blast
```
```   474
```
```   475 subsection{* Limit points *}
```
```   476
```
```   477 definition
```
```   478   islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   479     (infixr "islimpt" 60) where
```
```   480   "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
```
```   481
```
```   482 lemma islimptI:
```
```   483   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
```
```   484   shows "x islimpt S"
```
```   485   using assms unfolding islimpt_def by auto
```
```   486
```
```   487 lemma islimptE:
```
```   488   assumes "x islimpt S" and "x \<in> T" and "open T"
```
```   489   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
```
```   490   using assms unfolding islimpt_def by auto
```
```   491
```
```   492 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
```
```   493
```
```   494 lemma islimpt_approachable:
```
```   495   fixes x :: "'a::metric_space"
```
```   496   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
```
```   497   unfolding islimpt_def
```
```   498   apply auto
```
```   499   apply(erule_tac x="ball x e" in allE)
```
```   500   apply auto
```
```   501   apply(rule_tac x=y in bexI)
```
```   502   apply (auto simp add: dist_commute)
```
```   503   apply (simp add: open_dist, drule (1) bspec)
```
```   504   apply (clarify, drule spec, drule (1) mp, auto)
```
```   505   done
```
```   506
```
```   507 lemma islimpt_approachable_le:
```
```   508   fixes x :: "'a::metric_space"
```
```   509   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
```
```   510   unfolding islimpt_approachable
```
```   511   using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
```
```   512   by metis (* FIXME: VERY slow! *)
```
```   513
```
```   514 class perfect_space =
```
```   515   (* FIXME: perfect_space should inherit from topological_space *)
```
```   516   assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"
```
```   517
```
```   518 lemma perfect_choose_dist:
```
```   519   fixes x :: "'a::perfect_space"
```
```   520   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
```
```   521 using islimpt_UNIV [of x]
```
```   522 by (simp add: islimpt_approachable)
```
```   523
```
```   524 instance real :: perfect_space
```
```   525 apply default
```
```   526 apply (rule islimpt_approachable [THEN iffD2])
```
```   527 apply (clarify, rule_tac x="x + e/2" in bexI)
```
```   528 apply (auto simp add: dist_norm)
```
```   529 done
```
```   530
```
```   531 instance "^" :: (perfect_space, finite) perfect_space
```
```   532 proof
```
```   533   fix x :: "'a ^ 'b"
```
```   534   {
```
```   535     fix e :: real assume "0 < e"
```
```   536     def a \<equiv> "x \$ arbitrary"
```
```   537     have "a islimpt UNIV" by (rule islimpt_UNIV)
```
```   538     with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
```
```   539       unfolding islimpt_approachable by auto
```
```   540     def y \<equiv> "Cart_lambda ((Cart_nth x)(arbitrary := b))"
```
```   541     from `b \<noteq> a` have "y \<noteq> x"
```
```   542       unfolding a_def y_def by (simp add: Cart_eq)
```
```   543     from `dist b a < e` have "dist y x < e"
```
```   544       unfolding dist_vector_def a_def y_def
```
```   545       apply simp
```
```   546       apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
```
```   547       apply (subst setsum_diff1' [where a=arbitrary], simp, simp, simp)
```
```   548       done
```
```   549     from `y \<noteq> x` and `dist y x < e`
```
```   550     have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
```
```   551   }
```
```   552   then show "x islimpt UNIV" unfolding islimpt_approachable by blast
```
```   553 qed
```
```   554
```
```   555 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
```
```   556   unfolding closed_def
```
```   557   apply (subst open_subopen)
```
```   558   apply (simp add: islimpt_def subset_eq Compl_eq_Diff_UNIV)
```
```   559   by (metis DiffE DiffI UNIV_I insertCI insert_absorb mem_def)
```
```   560
```
```   561 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
```
```   562   unfolding islimpt_def by auto
```
```   563
```
```   564 lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x\$i}"
```
```   565 proof-
```
```   566   let ?U = "UNIV :: 'n set"
```
```   567   let ?O = "{x::real^'n. \<forall>i. x\$i\<ge>0}"
```
```   568   {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
```
```   569     and xi: "x\$i < 0"
```
```   570     from xi have th0: "-x\$i > 0" by arith
```
```   571     from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x \$ i" by blast
```
```   572       have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
```
```   573       have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
```
```   574       have th1: "\<bar>x\$i\<bar> \<le> \<bar>(x' - x)\$i\<bar>" using x'(1) xi
```
```   575 	apply (simp only: vector_component)
```
```   576 	by (rule th') auto
```
```   577       have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)\$i\<bar>" using  component_le_norm[of "x'-x" i]
```
```   578 	apply (simp add: dist_norm) by norm
```
```   579       from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
```
```   580   then show ?thesis unfolding closed_limpt islimpt_approachable
```
```   581     unfolding not_le[symmetric] by blast
```
```   582 qed
```
```   583
```
```   584 lemma finite_set_avoid:
```
```   585   fixes a :: "'a::metric_space"
```
```   586   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
```
```   587 proof(induct rule: finite_induct[OF fS])
```
```   588   case 1 thus ?case apply auto by ferrack
```
```   589 next
```
```   590   case (2 x F)
```
```   591   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
```
```   592   {assume "x = a" hence ?case using d by auto  }
```
```   593   moreover
```
```   594   {assume xa: "x\<noteq>a"
```
```   595     let ?d = "min d (dist a x)"
```
```   596     have dp: "?d > 0" using xa d(1) using dist_nz by auto
```
```   597     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
```
```   598     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
```
```   599   ultimately show ?case by blast
```
```   600 qed
```
```   601
```
```   602 lemma islimpt_finite:
```
```   603   fixes S :: "'a::metric_space set"
```
```   604   assumes fS: "finite S" shows "\<not> a islimpt S"
```
```   605   unfolding islimpt_approachable
```
```   606   using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
```
```   607
```
```   608 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
```
```   609   apply (rule iffI)
```
```   610   defer
```
```   611   apply (metis Un_upper1 Un_upper2 islimpt_subset)
```
```   612   unfolding islimpt_def
```
```   613   apply (rule ccontr, clarsimp, rename_tac A B)
```
```   614   apply (drule_tac x="A \<inter> B" in spec)
```
```   615   apply (auto simp add: open_Int)
```
```   616   done
```
```   617
```
```   618 lemma discrete_imp_closed:
```
```   619   fixes S :: "'a::metric_space set"
```
```   620   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
```
```   621   shows "closed S"
```
```   622 proof-
```
```   623   {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
```
```   624     from e have e2: "e/2 > 0" by arith
```
```   625     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
```
```   626     let ?m = "min (e/2) (dist x y) "
```
```   627     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
```
```   628     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
```
```   629     have th: "dist z y < e" using z y
```
```   630       by (intro dist_triangle_lt [where z=x], simp)
```
```   631     from d[rule_format, OF y(1) z(1) th] y z
```
```   632     have False by (auto simp add: dist_commute)}
```
```   633   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
```
```   634 qed
```
```   635
```
```   636 subsection{* Interior of a Set *}
```
```   637 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
```
```   638
```
```   639 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
```
```   640   apply (simp add: expand_set_eq interior_def)
```
```   641   apply (subst (2) open_subopen) by (safe, blast+)
```
```   642
```
```   643 lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
```
```   644
```
```   645 lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
```
```   646
```
```   647 lemma open_interior[simp, intro]: "open(interior S)"
```
```   648   apply (simp add: interior_def)
```
```   649   apply (subst open_subopen) by blast
```
```   650
```
```   651 lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
```
```   652 lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
```
```   653 lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
```
```   654 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
```
```   655 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"
```
```   656   by (metis equalityI interior_maximal interior_subset open_interior)
```
```   657 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
```
```   658   apply (simp add: interior_def)
```
```   659   by (metis open_contains_ball centre_in_ball open_ball subset_trans)
```
```   660
```
```   661 lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
```
```   662   by (metis interior_maximal interior_subset subset_trans)
```
```   663
```
```   664 lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
```
```   665   apply (rule equalityI, simp)
```
```   666   apply (metis Int_lower1 Int_lower2 subset_interior)
```
```   667   by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
```
```   668
```
```   669 lemma interior_limit_point [intro]:
```
```   670   fixes x :: "'a::perfect_space"
```
```   671   assumes x: "x \<in> interior S" shows "x islimpt S"
```
```   672 proof-
```
```   673   from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
```
```   674     unfolding mem_interior subset_eq Ball_def mem_ball by blast
```
```   675   {
```
```   676     fix d::real assume d: "d>0"
```
```   677     let ?m = "min d e"
```
```   678     have mde2: "0 < ?m" using e(1) d(1) by simp
```
```   679     from perfect_choose_dist [OF mde2, of x]
```
```   680     obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
```
```   681     then have "dist y x < e" "dist y x < d" by simp_all
```
```   682     from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
```
```   683     have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
```
```   684       using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
```
```   685   }
```
```   686   then show ?thesis unfolding islimpt_approachable by blast
```
```   687 qed
```
```   688
```
```   689 lemma interior_closed_Un_empty_interior:
```
```   690   assumes cS: "closed S" and iT: "interior T = {}"
```
```   691   shows "interior(S \<union> T) = interior S"
```
```   692 proof
```
```   693   show "interior S \<subseteq> interior (S\<union>T)"
```
```   694     by (rule subset_interior, blast)
```
```   695 next
```
```   696   show "interior (S \<union> T) \<subseteq> interior S"
```
```   697   proof
```
```   698     fix x assume "x \<in> interior (S \<union> T)"
```
```   699     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
```
```   700       unfolding interior_def by fast
```
```   701     show "x \<in> interior S"
```
```   702     proof (rule ccontr)
```
```   703       assume "x \<notin> interior S"
```
```   704       with `x \<in> R` `open R` obtain y where "y \<in> R - S"
```
```   705         unfolding interior_def expand_set_eq by fast
```
```   706       from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
```
```   707       from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
```
```   708       from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
```
```   709       show "False" unfolding interior_def by fast
```
```   710     qed
```
```   711   qed
```
```   712 qed
```
```   713
```
```   714
```
```   715 subsection{* Closure of a Set *}
```
```   716
```
```   717 definition "closure S = S \<union> {x | x. x islimpt S}"
```
```   718
```
```   719 lemma closure_interior: "closure S = UNIV - interior (UNIV - S)"
```
```   720 proof-
```
```   721   { fix x
```
```   722     have "x\<in>UNIV - interior (UNIV - S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
```
```   723     proof
```
```   724       let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> UNIV - S)"
```
```   725       assume "?lhs"
```
```   726       hence *:"\<not> ?exT x"
```
```   727 	unfolding interior_def
```
```   728 	by simp
```
```   729       { assume "\<not> ?rhs"
```
```   730 	hence False using *
```
```   731 	  unfolding closure_def islimpt_def
```
```   732 	  by blast
```
```   733       }
```
```   734       thus "?rhs"
```
```   735 	by blast
```
```   736     next
```
```   737       assume "?rhs" thus "?lhs"
```
```   738 	unfolding closure_def interior_def islimpt_def
```
```   739 	by blast
```
```   740     qed
```
```   741   }
```
```   742   thus ?thesis
```
```   743     by blast
```
```   744 qed
```
```   745
```
```   746 lemma interior_closure: "interior S = UNIV - (closure (UNIV - S))"
```
```   747 proof-
```
```   748   { fix x
```
```   749     have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
```
```   750       unfolding interior_def closure_def islimpt_def
```
```   751       by blast
```
```   752   }
```
```   753   thus ?thesis
```
```   754     by blast
```
```   755 qed
```
```   756
```
```   757 lemma closed_closure[simp, intro]: "closed (closure S)"
```
```   758 proof-
```
```   759   have "closed (UNIV - interior (UNIV -S))" by blast
```
```   760   thus ?thesis using closure_interior[of S] by simp
```
```   761 qed
```
```   762
```
```   763 lemma closure_hull: "closure S = closed hull S"
```
```   764 proof-
```
```   765   have "S \<subseteq> closure S"
```
```   766     unfolding closure_def
```
```   767     by blast
```
```   768   moreover
```
```   769   have "closed (closure S)"
```
```   770     using closed_closure[of S]
```
```   771     by assumption
```
```   772   moreover
```
```   773   { fix t
```
```   774     assume *:"S \<subseteq> t" "closed t"
```
```   775     { fix x
```
```   776       assume "x islimpt S"
```
```   777       hence "x islimpt t" using *(1)
```
```   778 	using islimpt_subset[of x, of S, of t]
```
```   779 	by blast
```
```   780     }
```
```   781     with * have "closure S \<subseteq> t"
```
```   782       unfolding closure_def
```
```   783       using closed_limpt[of t]
```
```   784       by auto
```
```   785   }
```
```   786   ultimately show ?thesis
```
```   787     using hull_unique[of S, of "closure S", of closed]
```
```   788     unfolding mem_def
```
```   789     by simp
```
```   790 qed
```
```   791
```
```   792 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
```
```   793   unfolding closure_hull
```
```   794   using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]
```
```   795   by (metis mem_def subset_eq)
```
```   796
```
```   797 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
```
```   798   using closure_eq[of S]
```
```   799   by simp
```
```   800
```
```   801 lemma closure_closure[simp]: "closure (closure S) = closure S"
```
```   802   unfolding closure_hull
```
```   803   using hull_hull[of closed S]
```
```   804   by assumption
```
```   805
```
```   806 lemma closure_subset: "S \<subseteq> closure S"
```
```   807   unfolding closure_hull
```
```   808   using hull_subset[of S closed]
```
```   809   by assumption
```
```   810
```
```   811 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
```
```   812   unfolding closure_hull
```
```   813   using hull_mono[of S T closed]
```
```   814   by assumption
```
```   815
```
```   816 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
```
```   817   using hull_minimal[of S T closed]
```
```   818   unfolding closure_hull mem_def
```
```   819   by simp
```
```   820
```
```   821 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"
```
```   822   using hull_unique[of S T closed]
```
```   823   unfolding closure_hull mem_def
```
```   824   by simp
```
```   825
```
```   826 lemma closure_empty[simp]: "closure {} = {}"
```
```   827   using closed_empty closure_closed[of "{}"]
```
```   828   by simp
```
```   829
```
```   830 lemma closure_univ[simp]: "closure UNIV = UNIV"
```
```   831   using closure_closed[of UNIV]
```
```   832   by simp
```
```   833
```
```   834 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
```
```   835   using closure_empty closure_subset[of S]
```
```   836   by blast
```
```   837
```
```   838 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
```
```   839   using closure_eq[of S] closure_subset[of S]
```
```   840   by simp
```
```   841
```
```   842 lemma open_inter_closure_eq_empty:
```
```   843   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
```
```   844   using open_subset_interior[of S "UNIV - T"]
```
```   845   using interior_subset[of "UNIV - T"]
```
```   846   unfolding closure_interior
```
```   847   by auto
```
```   848
```
```   849 lemma open_inter_closure_subset:
```
```   850   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
```
```   851 proof
```
```   852   fix x
```
```   853   assume as: "open S" "x \<in> S \<inter> closure T"
```
```   854   { assume *:"x islimpt T"
```
```   855     have "x islimpt (S \<inter> T)"
```
```   856     proof (rule islimptI)
```
```   857       fix A
```
```   858       assume "x \<in> A" "open A"
```
```   859       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
```
```   860         by (simp_all add: open_Int)
```
```   861       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
```
```   862         by (rule islimptE)
```
```   863       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
```
```   864         by simp_all
```
```   865       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
```
```   866     qed
```
```   867   }
```
```   868   then show "x \<in> closure (S \<inter> T)" using as
```
```   869     unfolding closure_def
```
```   870     by blast
```
```   871 qed
```
```   872
```
```   873 lemma closure_complement: "closure(UNIV - S) = UNIV - interior(S)"
```
```   874 proof-
```
```   875   have "S = UNIV - (UNIV - S)"
```
```   876     by auto
```
```   877   thus ?thesis
```
```   878     unfolding closure_interior
```
```   879     by auto
```
```   880 qed
```
```   881
```
```   882 lemma interior_complement: "interior(UNIV - S) = UNIV - closure(S)"
```
```   883   unfolding closure_interior
```
```   884   by blast
```
```   885
```
```   886 subsection{* Frontier (aka boundary) *}
```
```   887
```
```   888 definition "frontier S = closure S - interior S"
```
```   889
```
```   890 lemma frontier_closed: "closed(frontier S)"
```
```   891   by (simp add: frontier_def closed_Diff)
```
```   892
```
```   893 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(UNIV - S))"
```
```   894   by (auto simp add: frontier_def interior_closure)
```
```   895
```
```   896 lemma frontier_straddle:
```
```   897   fixes a :: "'a::metric_space"
```
```   898   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")
```
```   899 proof
```
```   900   assume "?lhs"
```
```   901   { fix e::real
```
```   902     assume "e > 0"
```
```   903     let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
```
```   904     { assume "a\<in>S"
```
```   905       have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
```
```   906       moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
```
```   907 	unfolding frontier_closures closure_def islimpt_def using `e>0`
```
```   908 	by (auto, erule_tac x="ball a e" in allE, auto)
```
```   909       ultimately have ?rhse by auto
```
```   910     }
```
```   911     moreover
```
```   912     { assume "a\<notin>S"
```
```   913       hence ?rhse using `?lhs`
```
```   914 	unfolding frontier_closures closure_def islimpt_def
```
```   915 	using open_ball[of a e] `e > 0`
```
```   916 	by (auto, erule_tac x = "ball a e" in allE, auto) (* FIXME: VERY slow! *)
```
```   917     }
```
```   918     ultimately have ?rhse by auto
```
```   919   }
```
```   920   thus ?rhs by auto
```
```   921 next
```
```   922   assume ?rhs
```
```   923   moreover
```
```   924   { fix T assume "a\<notin>S" and
```
```   925     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"
```
```   926     from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
```
```   927     then obtain e where "e>0" "ball a e \<subseteq> T" by auto
```
```   928     then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
```
```   929     have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
```
```   930       using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
```
```   931   }
```
```   932   hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
```
```   933   moreover
```
```   934   { fix T assume "a \<in> T"  "open T" "a\<in>S"
```
```   935     then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
```
```   936     obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
```
```   937     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
```
```   938   }
```
```   939   hence "a islimpt (UNIV - S) \<or> a\<notin>S" unfolding islimpt_def by auto
```
```   940   ultimately show ?lhs unfolding frontier_closures using closure_def[of "UNIV - S"] by auto
```
```   941 qed
```
```   942
```
```   943 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
```
```   944   by (metis frontier_def closure_closed Diff_subset)
```
```   945
```
```   946 lemma frontier_empty: "frontier {} = {}"
```
```   947   by (simp add: frontier_def closure_empty)
```
```   948
```
```   949 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
```
```   950 proof-
```
```   951   { assume "frontier S \<subseteq> S"
```
```   952     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
```
```   953     hence "closed S" using closure_subset_eq by auto
```
```   954   }
```
```   955   thus ?thesis using frontier_subset_closed[of S] by auto
```
```   956 qed
```
```   957
```
```   958 lemma frontier_complement: "frontier(UNIV - S) = frontier S"
```
```   959   by (auto simp add: frontier_def closure_complement interior_complement)
```
```   960
```
```   961 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
```
```   962   using frontier_complement frontier_subset_eq[of "UNIV - S"]
```
```   963   unfolding open_closed Compl_eq_Diff_UNIV by auto
```
```   964
```
```   965 subsection{* Common nets and The "within" modifier for nets. *}
```
```   966
```
```   967 definition
```
```   968   at_infinity :: "'a::real_normed_vector net" where
```
```   969   "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
```
```   970
```
```   971 definition
```
```   972   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
```
```   973   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
```
```   974
```
```   975 text{* Prove That They are all nets. *}
```
```   976
```
```   977 lemma Rep_net_at_infinity:
```
```   978   "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
```
```   979 unfolding at_infinity_def
```
```   980 apply (rule Abs_net_inverse')
```
```   981 apply (rule image_nonempty, simp)
```
```   982 apply (clarsimp, rename_tac r s)
```
```   983 apply (rule_tac x="max r s" in exI, auto)
```
```   984 done
```
```   985
```
```   986 lemma within_UNIV: "net within UNIV = net"
```
```   987   by (simp add: Rep_net_inject [symmetric] Rep_net_within)
```
```   988
```
```   989 subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
```
```   990
```
```   991 definition
```
```   992   trivial_limit :: "'a net \<Rightarrow> bool" where
```
```   993   "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
```
```   994
```
```   995 lemma trivial_limit_within:
```
```   996   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
```
```   997 proof
```
```   998   assume "trivial_limit (at a within S)"
```
```   999   thus "\<not> a islimpt S"
```
```  1000     unfolding trivial_limit_def
```
```  1001     unfolding Rep_net_within Rep_net_at
```
```  1002     unfolding islimpt_def
```
```  1003     apply (clarsimp simp add: expand_set_eq)
```
```  1004     apply (rename_tac T, rule_tac x=T in exI)
```
```  1005     apply (clarsimp, drule_tac x=y in spec, simp)
```
```  1006     done
```
```  1007 next
```
```  1008   assume "\<not> a islimpt S"
```
```  1009   thus "trivial_limit (at a within S)"
```
```  1010     unfolding trivial_limit_def
```
```  1011     unfolding Rep_net_within Rep_net_at
```
```  1012     unfolding islimpt_def
```
```  1013     apply (clarsimp simp add: image_image)
```
```  1014     apply (rule_tac x=T in image_eqI)
```
```  1015     apply (auto simp add: expand_set_eq)
```
```  1016     done
```
```  1017 qed
```
```  1018
```
```  1019 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
```
```  1020   using trivial_limit_within [of a UNIV]
```
```  1021   by (simp add: within_UNIV)
```
```  1022
```
```  1023 lemma trivial_limit_at:
```
```  1024   fixes a :: "'a::perfect_space"
```
```  1025   shows "\<not> trivial_limit (at a)"
```
```  1026   by (simp add: trivial_limit_at_iff)
```
```  1027
```
```  1028 lemma trivial_limit_at_infinity:
```
```  1029   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
```
```  1030   (* FIXME: find a more appropriate type class *)
```
```  1031   unfolding trivial_limit_def Rep_net_at_infinity
```
```  1032   apply (clarsimp simp add: expand_set_eq)
```
```  1033   apply (drule_tac x="scaleR r (sgn 1)" in spec)
```
```  1034   apply (simp add: norm_scaleR norm_sgn)
```
```  1035   done
```
```  1036
```
```  1037 lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
```
```  1038   by (auto simp add: trivial_limit_def Rep_net_sequentially)
```
```  1039
```
```  1040 subsection{* Some property holds "sufficiently close" to the limit point. *}
```
```  1041
```
```  1042 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
```
```  1043   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
```
```  1044 unfolding eventually_at dist_nz by auto
```
```  1045
```
```  1046 lemma eventually_at_infinity:
```
```  1047   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
```
```  1048 unfolding eventually_def Rep_net_at_infinity by auto
```
```  1049
```
```  1050 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
```
```  1051         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
```
```  1052 unfolding eventually_within eventually_at dist_nz by auto
```
```  1053
```
```  1054 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
```
```  1055         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
```
```  1056 unfolding eventually_within
```
```  1057 apply safe
```
```  1058 apply (rule_tac x="d/2" in exI, simp)
```
```  1059 apply (rule_tac x="d" in exI, simp)
```
```  1060 done
```
```  1061
```
```  1062 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
```
```  1063   unfolding eventually_def trivial_limit_def
```
```  1064   using Rep_net_nonempty [of net] by auto
```
```  1065
```
```  1066 lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
```
```  1067   unfolding eventually_def trivial_limit_def
```
```  1068   using Rep_net_nonempty [of net] by auto
```
```  1069
```
```  1070 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
```
```  1071   unfolding trivial_limit_def eventually_def by auto
```
```  1072
```
```  1073 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
```
```  1074   unfolding trivial_limit_def eventually_def by auto
```
```  1075
```
```  1076 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
```
```  1077   apply (safe elim!: trivial_limit_eventually)
```
```  1078   apply (simp add: eventually_False [symmetric])
```
```  1079   done
```
```  1080
```
```  1081 text{* Combining theorems for "eventually" *}
```
```  1082
```
```  1083 lemma eventually_conjI:
```
```  1084   "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
```
```  1085     \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
```
```  1086 by (rule eventually_conj)
```
```  1087
```
```  1088 lemma eventually_rev_mono:
```
```  1089   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
```
```  1090 using eventually_mono [of P Q] by fast
```
```  1091
```
```  1092 lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
```
```  1093   by (auto intro!: eventually_conjI elim: eventually_rev_mono)
```
```  1094
```
```  1095 lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
```
```  1096   by (auto simp add: eventually_False)
```
```  1097
```
```  1098 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
```
```  1099   by (simp add: eventually_False)
```
```  1100
```
```  1101 subsection{* Limits, defined as vacuously true when the limit is trivial. *}
```
```  1102
```
```  1103   text{* Notation Lim to avoid collition with lim defined in analysis *}
```
```  1104 definition
```
```  1105   Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'b" where
```
```  1106   "Lim net f = (THE l. (f ---> l) net)"
```
```  1107
```
```  1108 lemma Lim:
```
```  1109  "(f ---> l) net \<longleftrightarrow>
```
```  1110         trivial_limit net \<or>
```
```  1111         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
```
```  1112   unfolding tendsto_iff trivial_limit_eq by auto
```
```  1113
```
```  1114
```
```  1115 text{* Show that they yield usual definitions in the various cases. *}
```
```  1116
```
```  1117 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
```
```  1118            (\<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)"
```
```  1119   by (auto simp add: tendsto_iff eventually_within_le)
```
```  1120
```
```  1121 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
```
```  1122         (\<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)"
```
```  1123   by (auto simp add: tendsto_iff eventually_within)
```
```  1124
```
```  1125 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
```
```  1126         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
```
```  1127   by (auto simp add: tendsto_iff eventually_at)
```
```  1128
```
```  1129 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
```
```  1130   unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
```
```  1131
```
```  1132 lemma Lim_at_infinity:
```
```  1133   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
```
```  1134   by (auto simp add: tendsto_iff eventually_at_infinity)
```
```  1135
```
```  1136 lemma Lim_sequentially:
```
```  1137  "(S ---> l) sequentially \<longleftrightarrow>
```
```  1138           (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
```
```  1139   by (auto simp add: tendsto_iff eventually_sequentially)
```
```  1140
```
```  1141 lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
```
```  1142   unfolding Lim_sequentially LIMSEQ_def ..
```
```  1143
```
```  1144 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
```
```  1145   by (rule topological_tendstoI, auto elim: eventually_rev_mono)
```
```  1146
```
```  1147 text{* The expected monotonicity property. *}
```
```  1148
```
```  1149 lemma Lim_within_empty: "(f ---> l) (net within {})"
```
```  1150   unfolding tendsto_def Limits.eventually_within by simp
```
```  1151
```
```  1152 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
```
```  1153   unfolding tendsto_def Limits.eventually_within
```
```  1154   by (auto elim!: eventually_elim1)
```
```  1155
```
```  1156 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
```
```  1157   shows "(f ---> l) (net within (S \<union> T))"
```
```  1158   using assms unfolding tendsto_def Limits.eventually_within
```
```  1159   apply clarify
```
```  1160   apply (drule spec, drule (1) mp, drule (1) mp)
```
```  1161   apply (drule spec, drule (1) mp, drule (1) mp)
```
```  1162   apply (auto elim: eventually_elim2)
```
```  1163   done
```
```  1164
```
```  1165 lemma Lim_Un_univ:
```
```  1166  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
```
```  1167         ==> (f ---> l) net"
```
```  1168   by (metis Lim_Un within_UNIV)
```
```  1169
```
```  1170 text{* Interrelations between restricted and unrestricted limits. *}
```
```  1171
```
```  1172 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
```
```  1173   (* FIXME: rename *)
```
```  1174   unfolding tendsto_def Limits.eventually_within
```
```  1175   apply (clarify, drule spec, drule (1) mp, drule (1) mp)
```
```  1176   by (auto elim!: eventually_elim1)
```
```  1177
```
```  1178 lemma Lim_within_open:
```
```  1179   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```  1180   assumes"a \<in> S" "open S"
```
```  1181   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1182 proof
```
```  1183   assume ?lhs
```
```  1184   { fix A assume "open A" "l \<in> A"
```
```  1185     with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
```
```  1186       by (rule topological_tendstoD)
```
```  1187     hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
```
```  1188       unfolding Limits.eventually_within .
```
```  1189     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"
```
```  1190       unfolding eventually_at_topological by fast
```
```  1191     hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
```
```  1192       using assms by auto
```
```  1193     hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
```
```  1194       by fast
```
```  1195     hence "eventually (\<lambda>x. f x \<in> A) (at a)"
```
```  1196       unfolding eventually_at_topological .
```
```  1197   }
```
```  1198   thus ?rhs by (rule topological_tendstoI)
```
```  1199 next
```
```  1200   assume ?rhs
```
```  1201   thus ?lhs by (rule Lim_at_within)
```
```  1202 qed
```
```  1203
```
```  1204 text{* Another limit point characterization. *}
```
```  1205
```
```  1206 lemma islimpt_sequential:
```
```  1207   fixes x :: "'a::metric_space" (* FIXME: generalize to topological_space *)
```
```  1208   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
```
```  1209     (is "?lhs = ?rhs")
```
```  1210 proof
```
```  1211   assume ?lhs
```
```  1212   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"
```
```  1213     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
```
```  1214   { fix n::nat
```
```  1215     have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
```
```  1216   }
```
```  1217   moreover
```
```  1218   { fix e::real assume "e>0"
```
```  1219     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
```
```  1220     then obtain N::nat where "inverse (real (N + 1)) < e" by auto
```
```  1221     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)
```
```  1222     moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
```
```  1223     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
```
```  1224   }
```
```  1225   hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
```
```  1226     unfolding Lim_sequentially using f by auto
```
```  1227   ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
```
```  1228 next
```
```  1229   assume ?rhs
```
```  1230   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
```
```  1231   { fix e::real assume "e>0"
```
```  1232     then obtain N where "dist (f N) x < e" using f(2) by auto
```
```  1233     moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
```
```  1234     ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
```
```  1235   }
```
```  1236   thus ?lhs unfolding islimpt_approachable by auto
```
```  1237 qed
```
```  1238
```
```  1239 text{* Basic arithmetical combining theorems for limits. *}
```
```  1240
```
```  1241 lemma Lim_linear: fixes f :: "('a \<Rightarrow> real^'n::finite)" and h :: "(real^'n \<Rightarrow> real^'m::finite)"
```
```  1242   assumes "(f ---> l) net" "linear h"
```
```  1243   shows "((\<lambda>x. h (f x)) ---> h l) net"
```
```  1244 using `linear h` `(f ---> l) net`
```
```  1245 unfolding linear_conv_bounded_linear
```
```  1246 by (rule bounded_linear.tendsto)
```
```  1247
```
```  1248 lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
```
```  1249   unfolding tendsto_def Limits.eventually_at_topological by fast
```
```  1250
```
```  1251 lemma Lim_const: "((\<lambda>x. a) ---> a) net"
```
```  1252   by (rule tendsto_const)
```
```  1253
```
```  1254 lemma Lim_cmul:
```
```  1255   fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
```
```  1256   shows "(f ---> l) net ==> ((\<lambda>x. c *s f x) ---> c *s l) net"
```
```  1257   apply (rule Lim_linear[where f = f])
```
```  1258   apply simp
```
```  1259   apply (rule linear_compose_cmul)
```
```  1260   apply (rule linear_id[unfolded id_def])
```
```  1261   done
```
```  1262
```
```  1263 lemma Lim_neg:
```
```  1264   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1265   shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
```
```  1266   by (rule tendsto_minus)
```
```  1267
```
```  1268 lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
```
```  1269  "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
```
```  1270   by (rule tendsto_add)
```
```  1271
```
```  1272 lemma Lim_sub:
```
```  1273   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1274   shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
```
```  1275   by (rule tendsto_diff)
```
```  1276
```
```  1277 lemma dist_triangle3: (* TODO: move *)
```
```  1278   fixes x y :: "'a::metric_space"
```
```  1279   shows "dist x y \<le> dist a x + dist a y"
```
```  1280 using dist_triangle2 [of x y a]
```
```  1281 by (simp add: dist_commute)
```
```  1282
```
```  1283 lemma tendsto_dist: (* TODO: move *)
```
```  1284   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
```
```  1285   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
```
```  1286 proof (rule tendstoI)
```
```  1287   fix e :: real assume "0 < e"
```
```  1288   hence e2: "0 < e/2" by simp
```
```  1289   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```  1290   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
```
```  1291   proof (rule eventually_elim2)
```
```  1292     fix x assume x: "dist (f x) l < e/2" "dist (g x) m < e/2"
```
```  1293     have "dist (f x) (g x) - dist l m \<le> dist (f x) l + dist (g x) m"
```
```  1294       using dist_triangle2 [of "f x" "g x" "l"]
```
```  1295       using dist_triangle2 [of "g x" "l" "m"]
```
```  1296       by arith
```
```  1297     moreover
```
```  1298     have "dist l m - dist (f x) (g x) \<le> dist (f x) l + dist (g x) m"
```
```  1299       using dist_triangle3 [of "l" "m" "f x"]
```
```  1300       using dist_triangle [of "f x" "m" "g x"]
```
```  1301       by arith
```
```  1302     ultimately
```
```  1303     have "dist (dist (f x) (g x)) (dist l m) \<le> dist (f x) l + dist (g x) m"
```
```  1304       unfolding dist_norm real_norm_def by arith
```
```  1305     with x show "dist (dist (f x) (g x)) (dist l m) < e"
```
```  1306       by arith
```
```  1307   qed
```
```  1308 qed
```
```  1309
```
```  1310 lemma Lim_null:
```
```  1311   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1312   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
```
```  1313
```
```  1314 lemma Lim_null_norm:
```
```  1315   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1316   shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
```
```  1317   by (simp add: Lim dist_norm)
```
```  1318
```
```  1319 lemma Lim_null_comparison:
```
```  1320   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1321   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
```
```  1322   shows "(f ---> 0) net"
```
```  1323 proof(simp add: tendsto_iff, rule+)
```
```  1324   fix e::real assume "0<e"
```
```  1325   { fix x
```
```  1326     assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
```
```  1327     hence "dist (f x) 0 < e" by (simp add: dist_norm)
```
```  1328   }
```
```  1329   thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
```
```  1330     using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
```
```  1331     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]
```
```  1332     using assms `e>0` unfolding tendsto_iff by auto
```
```  1333 qed
```
```  1334
```
```  1335 lemma Lim_component:
```
```  1336   fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
```
```  1337   shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a \$i) ---> l\$i) net"
```
```  1338   unfolding tendsto_iff
```
```  1339   apply (clarify)
```
```  1340   apply (drule spec, drule (1) mp)
```
```  1341   apply (erule eventually_elim1)
```
```  1342   apply (erule le_less_trans [OF dist_nth_le])
```
```  1343   done
```
```  1344
```
```  1345 lemma Lim_transform_bound:
```
```  1346   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1347   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
```
```  1348   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
```
```  1349   shows "(f ---> 0) net"
```
```  1350 proof (rule tendstoI)
```
```  1351   fix e::real assume "e>0"
```
```  1352   { fix x
```
```  1353     assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
```
```  1354     hence "dist (f x) 0 < e" by (simp add: dist_norm)}
```
```  1355   thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
```
```  1356     using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
```
```  1357     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]
```
```  1358     using assms `e>0` unfolding tendsto_iff by blast
```
```  1359 qed
```
```  1360
```
```  1361 text{* Deducing things about the limit from the elements. *}
```
```  1362
```
```  1363 lemma Lim_in_closed_set:
```
```  1364   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
```
```  1365   shows "l \<in> S"
```
```  1366 proof (rule ccontr)
```
```  1367   assume "l \<notin> S"
```
```  1368   with `closed S` have "open (- S)" "l \<in> - S"
```
```  1369     by (simp_all add: open_Compl)
```
```  1370   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
```
```  1371     by (rule topological_tendstoD)
```
```  1372   with assms(2) have "eventually (\<lambda>x. False) net"
```
```  1373     by (rule eventually_elim2) simp
```
```  1374   with assms(3) show "False"
```
```  1375     by (simp add: eventually_False)
```
```  1376 qed
```
```  1377
```
```  1378 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
```
```  1379
```
```  1380 lemma Lim_dist_ubound:
```
```  1381   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
```
```  1382   shows "dist a l <= e"
```
```  1383 proof (rule ccontr)
```
```  1384   assume "\<not> dist a l \<le> e"
```
```  1385   then have "0 < dist a l - e" by simp
```
```  1386   with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
```
```  1387     by (rule tendstoD)
```
```  1388   with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
```
```  1389     by (rule eventually_conjI)
```
```  1390   then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
```
```  1391     using assms(1) eventually_happens by auto
```
```  1392   hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
```
```  1393     by (rule add_le_less_mono)
```
```  1394   hence "dist a (f w) + dist (f w) l < dist a l"
```
```  1395     by simp
```
```  1396   also have "\<dots> \<le> dist a (f w) + dist (f w) l"
```
```  1397     by (rule dist_triangle)
```
```  1398   finally show False by simp
```
```  1399 qed
```
```  1400
```
```  1401 lemma Lim_norm_ubound:
```
```  1402   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1403   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
```
```  1404   shows "norm(l) <= e"
```
```  1405 proof (rule ccontr)
```
```  1406   assume "\<not> norm l \<le> e"
```
```  1407   then have "0 < norm l - e" by simp
```
```  1408   with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
```
```  1409     by (rule tendstoD)
```
```  1410   with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
```
```  1411     by (rule eventually_conjI)
```
```  1412   then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
```
```  1413     using assms(1) eventually_happens by auto
```
```  1414   hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
```
```  1415   hence "norm (f w - l) + norm (f w) < norm l" by simp
```
```  1416   hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
```
```  1417   thus False using `\<not> norm l \<le> e` by simp
```
```  1418 qed
```
```  1419
```
```  1420 lemma Lim_norm_lbound:
```
```  1421   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```  1422   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
```
```  1423   shows "e \<le> norm l"
```
```  1424 proof (rule ccontr)
```
```  1425   assume "\<not> e \<le> norm l"
```
```  1426   then have "0 < e - norm l" by simp
```
```  1427   with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
```
```  1428     by (rule tendstoD)
```
```  1429   with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
```
```  1430     by (rule eventually_conjI)
```
```  1431   then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
```
```  1432     using assms(1) eventually_happens by auto
```
```  1433   hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
```
```  1434   hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
```
```  1435   hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
```
```  1436   thus False by simp
```
```  1437 qed
```
```  1438
```
```  1439 text{* Uniqueness of the limit, when nontrivial. *}
```
```  1440
```
```  1441 lemma Lim_unique:
```
```  1442   fixes f :: "'a \<Rightarrow> 'b::metric_space"
```
```  1443   assumes "\<not> trivial_limit net"  "(f ---> l) net"  "(f ---> l') net"
```
```  1444   shows "l = l'"
```
```  1445 proof (rule ccontr)
```
```  1446   let ?d = "dist l l' / 2"
```
```  1447   assume "l \<noteq> l'"
```
```  1448   then have "0 < ?d" by (simp add: dist_nz)
```
```  1449   have "eventually (\<lambda>x. dist (f x) l < ?d) net"
```
```  1450     using `(f ---> l) net` `0 < ?d` by (rule tendstoD)
```
```  1451   moreover
```
```  1452   have "eventually (\<lambda>x. dist (f x) l' < ?d) net"
```
```  1453     using `(f ---> l') net` `0 < ?d` by (rule tendstoD)
```
```  1454   ultimately
```
```  1455   have "eventually (\<lambda>x. False) net"
```
```  1456   proof (rule eventually_elim2)
```
```  1457     fix x
```
```  1458     assume *: "dist (f x) l < ?d" "dist (f x) l' < ?d"
```
```  1459     have "dist l l' \<le> dist (f x) l + dist (f x) l'"
```
```  1460       by (rule dist_triangle_alt)
```
```  1461     also from * have "\<dots> < ?d + ?d"
```
```  1462       by (rule add_strict_mono)
```
```  1463     also have "\<dots> = dist l l'" by simp
```
```  1464     finally show "False" by simp
```
```  1465   qed
```
```  1466   with `\<not> trivial_limit net` show "False"
```
```  1467     by (simp add: eventually_False)
```
```  1468 qed
```
```  1469
```
```  1470 lemma tendsto_Lim:
```
```  1471   fixes f :: "'a \<Rightarrow> 'b::metric_space"
```
```  1472   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
```
```  1473   unfolding Lim_def using Lim_unique[of net f] by auto
```
```  1474
```
```  1475 text{* Limit under bilinear function *}
```
```  1476
```
```  1477 lemma Lim_bilinear:
```
```  1478   fixes net :: "'a net" and h:: "real ^'m::finite \<Rightarrow> real ^'n::finite \<Rightarrow> real ^'p::finite"
```
```  1479   assumes "(f ---> l) net" and "(g ---> m) net" and "bilinear h"
```
```  1480   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
```
```  1481 using `bilinear h` `(f ---> l) net` `(g ---> m) net`
```
```  1482 unfolding bilinear_conv_bounded_bilinear
```
```  1483 by (rule bounded_bilinear.tendsto)
```
```  1484
```
```  1485 text{* These are special for limits out of the same vector space. *}
```
```  1486
```
```  1487 lemma Lim_within_id: "(id ---> a) (at a within s)"
```
```  1488   unfolding tendsto_def Limits.eventually_within eventually_at_topological
```
```  1489   by auto
```
```  1490
```
```  1491 lemma Lim_at_id: "(id ---> a) (at a)"
```
```  1492 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
```
```  1493
```
```  1494 lemma Lim_at_zero:
```
```  1495   fixes a :: "'a::real_normed_vector"
```
```  1496   fixes l :: "'b::topological_space"
```
```  1497   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
```
```  1498 proof
```
```  1499   assume "?lhs"
```
```  1500   { fix S assume "open S" "l \<in> S"
```
```  1501     with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
```
```  1502       by (rule topological_tendstoD)
```
```  1503     then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
```
```  1504       unfolding Limits.eventually_at by fast
```
```  1505     { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
```
```  1506       hence "f (a + x) \<in> S" using d
```
```  1507       apply(erule_tac x="x+a" in allE)
```
```  1508       by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
```
```  1509     }
```
```  1510     hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
```
```  1511       using d(1) by auto
```
```  1512     hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
```
```  1513       unfolding Limits.eventually_at .
```
```  1514   }
```
```  1515   thus "?rhs" by (rule topological_tendstoI)
```
```  1516 next
```
```  1517   assume "?rhs"
```
```  1518   { fix S assume "open S" "l \<in> S"
```
```  1519     with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
```
```  1520       by (rule topological_tendstoD)
```
```  1521     then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
```
```  1522       unfolding Limits.eventually_at by fast
```
```  1523     { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
```
```  1524       hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
```
```  1525 	by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
```
```  1526     }
```
```  1527     hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
```
```  1528     hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
```
```  1529   }
```
```  1530   thus "?lhs" by (rule topological_tendstoI)
```
```  1531 qed
```
```  1532
```
```  1533 text{* It's also sometimes useful to extract the limit point from the net.  *}
```
```  1534
```
```  1535 definition
```
```  1536   netlimit :: "'a::metric_space net \<Rightarrow> 'a" where
```
```  1537   "netlimit net = (SOME a. \<forall>r>0. eventually (\<lambda>x. dist x a < r) net)"
```
```  1538
```
```  1539 lemma netlimit_within:
```
```  1540   assumes "\<not> trivial_limit (at a within S)"
```
```  1541   shows "netlimit (at a within S) = a"
```
```  1542 using assms
```
```  1543 apply (simp add: trivial_limit_within)
```
```  1544 apply (simp add: netlimit_def eventually_within zero_less_dist_iff)
```
```  1545 apply (rule some_equality, fast)
```
```  1546 apply (rename_tac b)
```
```  1547 apply (rule ccontr)
```
```  1548 apply (drule_tac x="dist b a / 2" in spec, drule mp, simp add: dist_nz)
```
```  1549 apply (clarify, rename_tac r)
```
```  1550 apply (simp only: islimpt_approachable)
```
```  1551 apply (drule_tac x="min r (dist b a / 2)" in spec, drule mp, simp add: dist_nz)
```
```  1552 apply (clarify)
```
```  1553 apply (drule_tac x=x' in bspec, simp)
```
```  1554 apply (drule mp, simp)
```
```  1555 apply (subgoal_tac "dist b a < dist b a / 2 + dist b a / 2", simp)
```
```  1556 apply (rule le_less_trans [OF dist_triangle3])
```
```  1557 apply (erule add_strict_mono)
```
```  1558 apply simp
```
```  1559 done
```
```  1560
```
```  1561 lemma netlimit_at:
```
```  1562   fixes a :: "'a::perfect_space"
```
```  1563   shows "netlimit (at a) = a"
```
```  1564   apply (subst within_UNIV[symmetric])
```
```  1565   using netlimit_within[of a UNIV]
```
```  1566   by (simp add: trivial_limit_at within_UNIV)
```
```  1567
```
```  1568 text{* Transformation of limit. *}
```
```  1569
```
```  1570 lemma Lim_transform:
```
```  1571   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
```
```  1572   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
```
```  1573   shows "(g ---> l) net"
```
```  1574 proof-
```
```  1575   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
```
```  1576   thus "?thesis" using Lim_neg [of "\<lambda> x. - g x" "-l" net] by auto
```
```  1577 qed
```
```  1578
```
```  1579 lemma Lim_transform_eventually:
```
```  1580   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net ==> (g ---> l) net"
```
```  1581   apply (rule topological_tendstoI)
```
```  1582   apply (drule (2) topological_tendstoD)
```
```  1583   apply (erule (1) eventually_elim2, simp)
```
```  1584   done
```
```  1585
```
```  1586 lemma Lim_transform_within:
```
```  1587   fixes l :: "'b::metric_space" (* TODO: generalize *)
```
```  1588   assumes "0 < d" "(\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x')"
```
```  1589           "(f ---> l) (at x within S)"
```
```  1590   shows   "(g ---> l) (at x within S)"
```
```  1591   using assms(1,3) unfolding Lim_within
```
```  1592   apply -
```
```  1593   apply (clarify, rename_tac e)
```
```  1594   apply (drule_tac x=e in spec, clarsimp, rename_tac r)
```
```  1595   apply (rule_tac x="min d r" in exI, clarsimp, rename_tac y)
```
```  1596   apply (drule_tac x=y in bspec, assumption, clarsimp)
```
```  1597   apply (simp add: assms(2))
```
```  1598   done
```
```  1599
```
```  1600 lemma Lim_transform_at:
```
```  1601   fixes l :: "'b::metric_space" (* TODO: generalize *)
```
```  1602   shows "0 < d \<Longrightarrow> (\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x') \<Longrightarrow>
```
```  1603   (f ---> l) (at x) ==> (g ---> l) (at x)"
```
```  1604   apply (subst within_UNIV[symmetric])
```
```  1605   using Lim_transform_within[of d UNIV x f g l]
```
```  1606   by (auto simp add: within_UNIV)
```
```  1607
```
```  1608 text{* Common case assuming being away from some crucial point like 0. *}
```
```  1609
```
```  1610 lemma Lim_transform_away_within:
```
```  1611   fixes a b :: "'a::metric_space"
```
```  1612   fixes l :: "'b::metric_space" (* TODO: generalize *)
```
```  1613   assumes "a\<noteq>b" "\<forall>x\<in> S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
```
```  1614   and "(f ---> l) (at a within S)"
```
```  1615   shows "(g ---> l) (at a within S)"
```
```  1616 proof-
```
```  1617   have "\<forall>x'\<in>S. 0 < dist x' a \<and> dist x' a < dist a b \<longrightarrow> f x' = g x'" using assms(2)
```
```  1618     apply auto apply(erule_tac x=x' in ballE) by (auto simp add: dist_commute)
```
```  1619   thus ?thesis using Lim_transform_within[of "dist a b" S a f g l] using assms(1,3) unfolding dist_nz by auto
```
```  1620 qed
```
```  1621
```
```  1622 lemma Lim_transform_away_at:
```
```  1623   fixes a b :: "'a::metric_space"
```
```  1624   fixes l :: "'b::metric_space" (* TODO: generalize *)
```
```  1625   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
```
```  1626   and fl: "(f ---> l) (at a)"
```
```  1627   shows "(g ---> l) (at a)"
```
```  1628   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
```
```  1629   by (auto simp add: within_UNIV)
```
```  1630
```
```  1631 text{* Alternatively, within an open set. *}
```
```  1632
```
```  1633 lemma Lim_transform_within_open:
```
```  1634   fixes a :: "'a::metric_space"
```
```  1635   fixes l :: "'b::metric_space" (* TODO: generalize *)
```
```  1636   assumes "open S"  "a \<in> S"  "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"  "(f ---> l) (at a)"
```
```  1637   shows "(g ---> l) (at a)"
```
```  1638 proof-
```
```  1639   from assms(1,2) obtain e::real where "e>0" and e:"ball a e \<subseteq> S" unfolding open_contains_ball by auto
```
```  1640   hence "\<forall>x'. 0 < dist x' a \<and> dist x' a < e \<longrightarrow> f x' = g x'" using assms(3)
```
```  1641     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)
```
```  1642   thus ?thesis using Lim_transform_at[of e a f g l] `e>0` assms(4) by auto
```
```  1643 qed
```
```  1644
```
```  1645 text{* A congruence rule allowing us to transform limits assuming not at point. *}
```
```  1646
```
```  1647 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
```
```  1648
```
```  1649 lemma Lim_cong_within[cong add]:
```
```  1650   fixes a :: "'a::metric_space"
```
```  1651   fixes l :: "'b::metric_space" (* TODO: generalize *)
```
```  1652   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))"
```
```  1653   by (simp add: Lim_within dist_nz[symmetric])
```
```  1654
```
```  1655 lemma Lim_cong_at[cong add]:
```
```  1656   fixes a :: "'a::metric_space"
```
```  1657   fixes l :: "'b::metric_space" (* TODO: generalize *)
```
```  1658   shows "(\<And>x. x \<noteq> a ==> f x = g x) ==> (((\<lambda>x. f x) ---> l) (at a) \<longleftrightarrow> ((g ---> l) (at a)))"
```
```  1659   by (simp add: Lim_at dist_nz[symmetric])
```
```  1660
```
```  1661 text{* Useful lemmas on closure and set of possible sequential limits.*}
```
```  1662
```
```  1663 lemma closure_sequential:
```
```  1664   fixes l :: "'a::metric_space" (* TODO: generalize *)
```
```  1665   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
```
```  1666 proof
```
```  1667   assume "?lhs" moreover
```
```  1668   { assume "l \<in> S"
```
```  1669     hence "?rhs" using Lim_const[of l sequentially] by auto
```
```  1670   } moreover
```
```  1671   { assume "l islimpt S"
```
```  1672     hence "?rhs" unfolding islimpt_sequential by auto
```
```  1673   } ultimately
```
```  1674   show "?rhs" unfolding closure_def by auto
```
```  1675 next
```
```  1676   assume "?rhs"
```
```  1677   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
```
```  1678 qed
```
```  1679
```
```  1680 lemma closed_sequential_limits:
```
```  1681   fixes S :: "'a::metric_space set"
```
```  1682   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
```
```  1683   unfolding closed_limpt
```
```  1684   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]
```
```  1685   by metis
```
```  1686
```
```  1687 lemma closure_approachable:
```
```  1688   fixes S :: "'a::metric_space set"
```
```  1689   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
```
```  1690   apply (auto simp add: closure_def islimpt_approachable)
```
```  1691   by (metis dist_self)
```
```  1692
```
```  1693 lemma closed_approachable:
```
```  1694   fixes S :: "'a::metric_space set"
```
```  1695   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
```
```  1696   by (metis closure_closed closure_approachable)
```
```  1697
```
```  1698 text{* Some other lemmas about sequences. *}
```
```  1699
```
```  1700 lemma seq_offset:
```
```  1701   fixes l :: "'a::metric_space" (* TODO: generalize *)
```
```  1702   shows "(f ---> l) sequentially ==> ((\<lambda>i. f( i + k)) ---> l) sequentially"
```
```  1703   apply (auto simp add: Lim_sequentially)
```
```  1704   by (metis trans_le_add1 )
```
```  1705
```
```  1706 lemma seq_offset_neg:
```
```  1707   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
```
```  1708   apply (rule topological_tendstoI)
```
```  1709   apply (drule (2) topological_tendstoD)
```
```  1710   apply (simp only: eventually_sequentially)
```
```  1711   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
```
```  1712   apply metis
```
```  1713   by arith
```
```  1714
```
```  1715 lemma seq_offset_rev:
```
```  1716   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
```
```  1717   apply (rule topological_tendstoI)
```
```  1718   apply (drule (2) topological_tendstoD)
```
```  1719   apply (simp only: eventually_sequentially)
```
```  1720   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
```
```  1721   by metis arith
```
```  1722
```
```  1723 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
```
```  1724 proof-
```
```  1725   { fix e::real assume "e>0"
```
```  1726     hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
```
```  1727       using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
```
```  1728       by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
```
```  1729   }
```
```  1730   thus ?thesis unfolding Lim_sequentially dist_norm by simp
```
```  1731 qed
```
```  1732
```
```  1733 text{* More properties of closed balls. *}
```
```  1734
```
```  1735 lemma closed_cball: "closed (cball x e)"
```
```  1736 unfolding cball_def closed_def
```
```  1737 unfolding Collect_neg_eq [symmetric] not_le
```
```  1738 apply (clarsimp simp add: open_dist, rename_tac y)
```
```  1739 apply (rule_tac x="dist x y - e" in exI, clarsimp)
```
```  1740 apply (rename_tac x')
```
```  1741 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
```
```  1742 apply simp
```
```  1743 done
```
```  1744
```
```  1745 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
```
```  1746 proof-
```
```  1747   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
```
```  1748     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
```
```  1749   } moreover
```
```  1750   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
```
```  1751     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
```
```  1752   } ultimately
```
```  1753   show ?thesis unfolding open_contains_ball by auto
```
```  1754 qed
```
```  1755
```
```  1756 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
```
```  1757   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
```
```  1758
```
```  1759 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
```
```  1760   apply (simp add: interior_def, safe)
```
```  1761   apply (force simp add: open_contains_cball)
```
```  1762   apply (rule_tac x="ball x e" in exI)
```
```  1763   apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
```
```  1764   done
```
```  1765
```
```  1766 lemma islimpt_ball:
```
```  1767   fixes x y :: "'a::{real_normed_vector,perfect_space}"
```
```  1768   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
```
```  1769 proof
```
```  1770   assume "?lhs"
```
```  1771   { assume "e \<le> 0"
```
```  1772     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
```
```  1773     have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
```
```  1774   }
```
```  1775   hence "e > 0" by (metis not_less)
```
```  1776   moreover
```
```  1777   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
```
```  1778   ultimately show "?rhs" by auto
```
```  1779 next
```
```  1780   assume "?rhs" hence "e>0"  by auto
```
```  1781   { fix d::real assume "d>0"
```
```  1782     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  1783     proof(cases "d \<le> dist x y")
```
```  1784       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  1785       proof(cases "x=y")
```
```  1786 	case True hence False using `d \<le> dist x y` `d>0` by auto
```
```  1787 	thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
```
```  1788       next
```
```  1789 	case False
```
```  1790
```
```  1791 	have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
```
```  1792 	      = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
```
```  1793 	  unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
```
```  1794 	also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
```
```  1795 	  using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
```
```  1796 	  unfolding scaleR_minus_left scaleR_one
```
```  1797 	  by (auto simp add: norm_minus_commute norm_scaleR)
```
```  1798 	also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
```
```  1799 	  unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
```
```  1800 	  unfolding real_add_mult_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
```
```  1801 	also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
```
```  1802 	finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
```
```  1803
```
```  1804 	moreover
```
```  1805
```
```  1806 	have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
```
```  1807 	  using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
```
```  1808 	moreover
```
```  1809 	have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel norm_scaleR
```
```  1810 	  using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
```
```  1811 	  unfolding dist_norm by auto
```
```  1812 	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
```
```  1813       qed
```
```  1814     next
```
```  1815       case False hence "d > dist x y" by auto
```
```  1816       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  1817       proof(cases "x=y")
```
```  1818 	case True
```
```  1819 	obtain z where **: "z \<noteq> y" "dist z y < min e d"
```
```  1820           using perfect_choose_dist[of "min e d" y]
```
```  1821 	  using `d > 0` `e>0` by auto
```
```  1822 	show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  1823           unfolding `x = y`
```
```  1824           using `z \<noteq> y` **
```
```  1825           by (rule_tac x=z in bexI, auto simp add: dist_commute)
```
```  1826       next
```
```  1827 	case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
```
```  1828 	  using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
```
```  1829       qed
```
```  1830     qed  }
```
```  1831   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
```
```  1832 qed
```
```  1833
```
```  1834 lemma closure_ball_lemma:
```
```  1835   fixes x y :: "'a::real_normed_vector"
```
```  1836   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
```
```  1837 proof (rule islimptI)
```
```  1838   fix T assume "y \<in> T" "open T"
```
```  1839   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
```
```  1840     unfolding open_dist by fast
```
```  1841   (* choose point between x and y, within distance r of y. *)
```
```  1842   def k \<equiv> "min 1 (r / (2 * dist x y))"
```
```  1843   def z \<equiv> "y + scaleR k (x - y)"
```
```  1844   have z_def2: "z = x + scaleR (1 - k) (y - x)"
```
```  1845     unfolding z_def by (simp add: algebra_simps)
```
```  1846   have "dist z y < r"
```
```  1847     unfolding z_def k_def using `0 < r`
```
```  1848     by (simp add: dist_norm norm_scaleR min_def)
```
```  1849   hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
```
```  1850   have "dist x z < dist x y"
```
```  1851     unfolding z_def2 dist_norm
```
```  1852     apply (simp add: norm_scaleR norm_minus_commute)
```
```  1853     apply (simp only: dist_norm [symmetric])
```
```  1854     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
```
```  1855     apply (rule mult_strict_right_mono)
```
```  1856     apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
```
```  1857     apply (simp add: zero_less_dist_iff `x \<noteq> y`)
```
```  1858     done
```
```  1859   hence "z \<in> ball x (dist x y)" by simp
```
```  1860   have "z \<noteq> y"
```
```  1861     unfolding z_def k_def using `x \<noteq> y` `0 < r`
```
```  1862     by (simp add: min_def)
```
```  1863   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
```
```  1864     using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
```
```  1865     by fast
```
```  1866 qed
```
```  1867
```
```  1868 lemma closure_ball:
```
```  1869   fixes x :: "'a::real_normed_vector"
```
```  1870   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
```
```  1871 apply (rule equalityI)
```
```  1872 apply (rule closure_minimal)
```
```  1873 apply (rule ball_subset_cball)
```
```  1874 apply (rule closed_cball)
```
```  1875 apply (rule subsetI, rename_tac y)
```
```  1876 apply (simp add: le_less [where 'a=real])
```
```  1877 apply (erule disjE)
```
```  1878 apply (rule subsetD [OF closure_subset], simp)
```
```  1879 apply (simp add: closure_def)
```
```  1880 apply clarify
```
```  1881 apply (rule closure_ball_lemma)
```
```  1882 apply (simp add: zero_less_dist_iff)
```
```  1883 done
```
```  1884
```
```  1885 lemma interior_cball:
```
```  1886   fixes x :: "real ^ _" (* FIXME: generalize *)
```
```  1887   shows "interior(cball x e) = ball x e"
```
```  1888 proof(cases "e\<ge>0")
```
```  1889   case False note cs = this
```
```  1890   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
```
```  1891   { fix y assume "y \<in> cball x e"
```
```  1892     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
```
```  1893   hence "cball x e = {}" by auto
```
```  1894   hence "interior (cball x e) = {}" using interior_empty by auto
```
```  1895   ultimately show ?thesis by blast
```
```  1896 next
```
```  1897   case True note cs = this
```
```  1898   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
```
```  1899   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
```
```  1900     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
```
```  1901
```
```  1902     then obtain xa where xa:"dist y xa = d / 2" using vector_choose_dist[of "d/2" y] by auto
```
```  1903     hence xa_y:"xa \<noteq> y" using dist_nz[of y xa] using `d>0` by auto
```
```  1904     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa apply(auto simp add: dist_commute) unfolding dist_nz[THEN sym] using xa_y by auto
```
```  1905     hence xa_cball:"xa \<in> cball x e" using as(1) by auto
```
```  1906
```
```  1907     hence "y \<in> ball x e" proof(cases "x = y")
```
```  1908       case True
```
```  1909       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
```
```  1910       thus "y \<in> ball x e" using `x = y ` by simp
```
```  1911     next
```
```  1912       case False
```
```  1913       have "dist (y + (d / 2 / dist y x) *s (y - x)) y < d" unfolding dist_norm
```
```  1914 	using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
```
```  1915       hence *:"y + (d / 2 / dist y x) *s (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
```
```  1916       have "y - x \<noteq> 0" using `x \<noteq> y` by auto
```
```  1917       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
```
```  1918 	using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
```
```  1919
```
```  1920       have "dist (y + (d / 2 / dist y x) *s (y - x)) x = norm (y + (d / (2 * norm (y - x))) *s y - (d / (2 * norm (y - x))) *s x - x)"
```
```  1921 	by (auto simp add: dist_norm vector_ssub_ldistrib add_diff_eq)
```
```  1922       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *s (y - x))"
```
```  1923 	by (auto simp add: vector_sadd_rdistrib vector_smult_lid ring_simps vector_sadd_rdistrib vector_ssub_ldistrib)
```
```  1924       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)" using ** by auto
```
```  1925       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
```
```  1926       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
```
```  1927       thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
```
```  1928     qed  }
```
```  1929   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
```
```  1930   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
```
```  1931 qed
```
```  1932
```
```  1933 lemma frontier_ball:
```
```  1934   fixes a :: "real ^ _" (* FIXME: generalize *)
```
```  1935   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
```
```  1936   apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
```
```  1937   apply (simp add: expand_set_eq)
```
```  1938   by arith
```
```  1939
```
```  1940 lemma frontier_cball:
```
```  1941   fixes a :: "real ^ _" (* FIXME: generalize *)
```
```  1942   shows "frontier(cball a e) = {x. dist a x = e}"
```
```  1943   apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
```
```  1944   apply (simp add: expand_set_eq)
```
```  1945   by arith
```
```  1946
```
```  1947 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
```
```  1948   apply (simp add: expand_set_eq not_le)
```
```  1949   by (metis zero_le_dist dist_self order_less_le_trans)
```
```  1950 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
```
```  1951
```
```  1952 lemma cball_eq_sing:
```
```  1953   fixes x :: "real ^ _" (* FIXME: generalize *)
```
```  1954   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
```
```  1955 proof-
```
```  1956   { assume as:"\<forall>xa. (dist x xa \<le> e) = (xa = x)"
```
```  1957     hence "e \<ge> 0" apply (erule_tac x=x in allE) by auto
```
```  1958     then obtain y where y:"dist x y = e" using vector_choose_dist[of e] by auto
```
```  1959     hence "e = 0" using as apply(erule_tac x=y in allE) by auto
```
```  1960   }
```
```  1961   thus ?thesis unfolding expand_set_eq mem_cball by (auto simp add: dist_nz)
```
```  1962 qed
```
```  1963
```
```  1964 lemma cball_sing:
```
```  1965   fixes x :: "real ^ _" (* FIXME: generalize *)
```
```  1966   shows "e = 0 ==> cball x e = {x}" by (simp add: cball_eq_sing)
```
```  1967
```
```  1968 text{* For points in the interior, localization of limits makes no difference.   *}
```
```  1969
```
```  1970 lemma eventually_within_interior:
```
```  1971   assumes "x \<in> interior S"
```
```  1972   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
```
```  1973 proof-
```
```  1974   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
```
```  1975     unfolding interior_def by fast
```
```  1976   { assume "?lhs"
```
```  1977     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
```
```  1978       unfolding Limits.eventually_within Limits.eventually_at_topological
```
```  1979       by auto
```
```  1980     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
```
```  1981       by auto
```
```  1982     then have "?rhs"
```
```  1983       unfolding Limits.eventually_at_topological by auto
```
```  1984   } moreover
```
```  1985   { assume "?rhs" hence "?lhs"
```
```  1986       unfolding Limits.eventually_within
```
```  1987       by (auto elim: eventually_elim1)
```
```  1988   } ultimately
```
```  1989   show "?thesis" ..
```
```  1990 qed
```
```  1991
```
```  1992 lemma lim_within_interior:
```
```  1993   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
```
```  1994   unfolding tendsto_def by (simp add: eventually_within_interior)
```
```  1995
```
```  1996 lemma netlimit_within_interior:
```
```  1997   fixes x :: "'a::{perfect_space, real_normed_vector}"
```
```  1998     (* FIXME: generalize to perfect_space *)
```
```  1999   assumes "x \<in> interior S"
```
```  2000   shows "netlimit(at x within S) = x" (is "?lhs = ?rhs")
```
```  2001 proof-
```
```  2002   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
```
```  2003   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
```
```  2004   thus ?thesis using netlimit_within by auto
```
```  2005 qed
```
```  2006
```
```  2007 subsection{* Boundedness. *}
```
```  2008
```
```  2009   (* FIXME: This has to be unified with BSEQ!! *)
```
```  2010 definition
```
```  2011   bounded :: "'a::metric_space set \<Rightarrow> bool" where
```
```  2012   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
```
```  2013
```
```  2014 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
```
```  2015 unfolding bounded_def
```
```  2016 apply safe
```
```  2017 apply (rule_tac x="dist a x + e" in exI, clarify)
```
```  2018 apply (drule (1) bspec)
```
```  2019 apply (erule order_trans [OF dist_triangle add_left_mono])
```
```  2020 apply auto
```
```  2021 done
```
```  2022
```
```  2023 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
```
```  2024 unfolding bounded_any_center [where a=0]
```
```  2025 by (simp add: dist_norm)
```
```  2026
```
```  2027 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
```
```  2028 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
```
```  2029   by (metis bounded_def subset_eq)
```
```  2030
```
```  2031 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
```
```  2032   by (metis bounded_subset interior_subset)
```
```  2033
```
```  2034 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
```
```  2035 proof-
```
```  2036   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
```
```  2037   { fix y assume "y \<in> closure S"
```
```  2038     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
```
```  2039       unfolding closure_sequential by auto
```
```  2040     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
```
```  2041     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
```
```  2042       by (rule eventually_mono, simp add: f(1))
```
```  2043     have "dist x y \<le> a"
```
```  2044       apply (rule Lim_dist_ubound [of sequentially f])
```
```  2045       apply (rule trivial_limit_sequentially)
```
```  2046       apply (rule f(2))
```
```  2047       apply fact
```
```  2048       done
```
```  2049   }
```
```  2050   thus ?thesis unfolding bounded_def by auto
```
```  2051 qed
```
```  2052
```
```  2053 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
```
```  2054   apply (simp add: bounded_def)
```
```  2055   apply (rule_tac x=x in exI)
```
```  2056   apply (rule_tac x=e in exI)
```
```  2057   apply auto
```
```  2058   done
```
```  2059
```
```  2060 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
```
```  2061   by (metis ball_subset_cball bounded_cball bounded_subset)
```
```  2062
```
```  2063 lemma finite_imp_bounded[intro]: assumes "finite S" shows "bounded S"
```
```  2064 proof-
```
```  2065   { fix a F assume as:"bounded F"
```
```  2066     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
```
```  2067     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
```
```  2068     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
```
```  2069   }
```
```  2070   thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto
```
```  2071 qed
```
```  2072
```
```  2073 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
```
```  2074   apply (auto simp add: bounded_def)
```
```  2075   apply (rename_tac x y r s)
```
```  2076   apply (rule_tac x=x in exI)
```
```  2077   apply (rule_tac x="max r (dist x y + s)" in exI)
```
```  2078   apply (rule ballI, rename_tac z, safe)
```
```  2079   apply (drule (1) bspec, simp)
```
```  2080   apply (drule (1) bspec)
```
```  2081   apply (rule min_max.le_supI2)
```
```  2082   apply (erule order_trans [OF dist_triangle add_left_mono])
```
```  2083   done
```
```  2084
```
```  2085 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
```
```  2086   by (induct rule: finite_induct[of F], auto)
```
```  2087
```
```  2088 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
```
```  2089   apply (simp add: bounded_iff)
```
```  2090   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
```
```  2091   by metis arith
```
```  2092
```
```  2093 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
```
```  2094   by (metis Int_lower1 Int_lower2 bounded_subset)
```
```  2095
```
```  2096 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
```
```  2097 apply (metis Diff_subset bounded_subset)
```
```  2098 done
```
```  2099
```
```  2100 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
```
```  2101   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
```
```  2102
```
```  2103 lemma not_bounded_UNIV[simp, intro]:
```
```  2104   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
```
```  2105 proof(auto simp add: bounded_pos not_le)
```
```  2106   obtain x :: 'a where "x \<noteq> 0"
```
```  2107     using perfect_choose_dist [OF zero_less_one] by fast
```
```  2108   fix b::real  assume b: "b >0"
```
```  2109   have b1: "b +1 \<ge> 0" using b by simp
```
```  2110   with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
```
```  2111     by (simp add: norm_scaleR norm_sgn)
```
```  2112   then show "\<exists>x::'a. b < norm x" ..
```
```  2113 qed
```
```  2114
```
```  2115 lemma bounded_linear_image:
```
```  2116   fixes f :: "real^'m::finite \<Rightarrow> real^'n::finite"
```
```  2117   assumes "bounded S" "linear f"
```
```  2118   shows "bounded(f ` S)"
```
```  2119 proof-
```
```  2120   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
```
```  2121   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x"  using linear_bounded_pos by auto
```
```  2122   { fix x assume "x\<in>S"
```
```  2123     hence "norm x \<le> b" using b by auto
```
```  2124     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
```
```  2125       by (metis B(1) B(2) real_le_trans real_mult_le_cancel_iff2)
```
```  2126   }
```
```  2127   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
```
```  2128     using b B real_mult_order[of b B] by (auto simp add: real_mult_commute)
```
```  2129 qed
```
```  2130
```
```  2131 lemma bounded_scaling:
```
```  2132   fixes S :: "(real ^ 'n::finite) set"
```
```  2133   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *s x) ` S)"
```
```  2134   apply (rule bounded_linear_image, assumption)
```
```  2135   by (rule linear_compose_cmul, rule linear_id[unfolded id_def])
```
```  2136
```
```  2137 lemma bounded_translation:
```
```  2138   fixes S :: "'a::real_normed_vector set"
```
```  2139   assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
```
```  2140 proof-
```
```  2141   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
```
```  2142   { fix x assume "x\<in>S"
```
```  2143     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
```
```  2144   }
```
```  2145   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
```
```  2146     by (auto intro!: add exI[of _ "b + norm a"])
```
```  2147 qed
```
```  2148
```
```  2149
```
```  2150 text{* Some theorems on sups and infs using the notion "bounded". *}
```
```  2151
```
```  2152 lemma bounded_real:
```
```  2153   fixes S :: "real set"
```
```  2154   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
```
```  2155   by (simp add: bounded_iff)
```
```  2156
```
```  2157 lemma bounded_has_rsup: assumes "bounded S" "S \<noteq> {}"
```
```  2158   shows "\<forall>x\<in>S. x <= rsup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> rsup S <= b"
```
```  2159 proof
```
```  2160   fix x assume "x\<in>S"
```
```  2161   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
```
```  2162   hence *:"S *<= a" using setleI[of S a] by (metis abs_le_interval_iff mem_def)
```
```  2163   thus "x \<le> rsup S" using rsup[OF `S\<noteq>{}`] using assms(1)[unfolded bounded_real] using isLubD2[of UNIV S "rsup S" x] using `x\<in>S` by auto
```
```  2164 next
```
```  2165   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> rsup S \<le> b" using assms
```
```  2166   using rsup[of S, unfolded isLub_def isUb_def leastP_def setle_def setge_def]
```
```  2167   apply (auto simp add: bounded_real)
```
```  2168   by (auto simp add: isLub_def isUb_def leastP_def setle_def setge_def)
```
```  2169 qed
```
```  2170
```
```  2171 lemma rsup_insert: assumes "bounded S"
```
```  2172   shows "rsup(insert x S) = (if S = {} then x else max x (rsup S))"
```
```  2173 proof(cases "S={}")
```
```  2174   case True thus ?thesis using rsup_finite_in[of "{x}"] by auto
```
```  2175 next
```
```  2176   let ?S = "insert x S"
```
```  2177   case False
```
```  2178   hence *:"\<forall>x\<in>S. x \<le> rsup S" using bounded_has_rsup(1)[of S] using assms by auto
```
```  2179   hence "insert x S *<= max x (rsup S)" unfolding setle_def by auto
```
```  2180   hence "isLub UNIV ?S (rsup ?S)" using rsup[of ?S] by auto
```
```  2181   moreover
```
```  2182   have **:"isUb UNIV ?S (max x (rsup S))" unfolding isUb_def setle_def using * by auto
```
```  2183   { fix y assume as:"isUb UNIV (insert x S) y"
```
```  2184     hence "max x (rsup S) \<le> y" unfolding isUb_def using rsup_le[OF `S\<noteq>{}`]
```
```  2185       unfolding setle_def by auto  }
```
```  2186   hence "max x (rsup S) <=* isUb UNIV (insert x S)" unfolding setge_def Ball_def mem_def by auto
```
```  2187   hence "isLub UNIV ?S (max x (rsup S))" using ** isLubI2[of UNIV ?S "max x (rsup S)"] unfolding Collect_def by auto
```
```  2188   ultimately show ?thesis using real_isLub_unique[of UNIV ?S] using `S\<noteq>{}` by auto
```
```  2189 qed
```
```  2190
```
```  2191 lemma sup_insert_finite: "finite S \<Longrightarrow> rsup(insert x S) = (if S = {} then x else max x (rsup S))"
```
```  2192   apply (rule rsup_insert)
```
```  2193   apply (rule finite_imp_bounded)
```
```  2194   by simp
```
```  2195
```
```  2196 lemma bounded_has_rinf:
```
```  2197   assumes "bounded S"  "S \<noteq> {}"
```
```  2198   shows "\<forall>x\<in>S. x >= rinf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S >= b"
```
```  2199 proof
```
```  2200   fix x assume "x\<in>S"
```
```  2201   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
```
```  2202   hence *:"- a <=* S" using setgeI[of S "-a"] unfolding abs_le_interval_iff by auto
```
```  2203   thus "x \<ge> rinf S" using rinf[OF `S\<noteq>{}`] using isGlbD2[of UNIV S "rinf S" x] using `x\<in>S` by auto
```
```  2204 next
```
```  2205   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S \<ge> b" using assms
```
```  2206   using rinf[of S, unfolded isGlb_def isLb_def greatestP_def setle_def setge_def]
```
```  2207   apply (auto simp add: bounded_real)
```
```  2208   by (auto simp add: isGlb_def isLb_def greatestP_def setle_def setge_def)
```
```  2209 qed
```
```  2210
```
```  2211 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
```
```  2212 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
```
```  2213   apply (frule isGlb_isLb)
```
```  2214   apply (frule_tac x = y in isGlb_isLb)
```
```  2215   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
```
```  2216   done
```
```  2217
```
```  2218 lemma rinf_insert: assumes "bounded S"
```
```  2219   shows "rinf(insert x S) = (if S = {} then x else min x (rinf S))" (is "?lhs = ?rhs")
```
```  2220 proof(cases "S={}")
```
```  2221   case True thus ?thesis using rinf_finite_in[of "{x}"] by auto
```
```  2222 next
```
```  2223   let ?S = "insert x S"
```
```  2224   case False
```
```  2225   hence *:"\<forall>x\<in>S. x \<ge> rinf S" using bounded_has_rinf(1)[of S] using assms by auto
```
```  2226   hence "min x (rinf S) <=* insert x S" unfolding setge_def by auto
```
```  2227   hence "isGlb UNIV ?S (rinf ?S)" using rinf[of ?S] by auto
```
```  2228   moreover
```
```  2229   have **:"isLb UNIV ?S (min x (rinf S))" unfolding isLb_def setge_def using * by auto
```
```  2230   { fix y assume as:"isLb UNIV (insert x S) y"
```
```  2231     hence "min x (rinf S) \<ge> y" unfolding isLb_def using rinf_ge[OF `S\<noteq>{}`]
```
```  2232       unfolding setge_def by auto  }
```
```  2233   hence "isLb UNIV (insert x S) *<= min x (rinf S)" unfolding setle_def Ball_def mem_def by auto
```
```  2234   hence "isGlb UNIV ?S (min x (rinf S))" using ** isGlbI2[of UNIV ?S "min x (rinf S)"] unfolding Collect_def by auto
```
```  2235   ultimately show ?thesis using real_isGlb_unique[of UNIV ?S] using `S\<noteq>{}` by auto
```
```  2236 qed
```
```  2237
```
```  2238 lemma inf_insert_finite: "finite S ==> rinf(insert x S) = (if S = {} then x else min x (rinf S))"
```
```  2239   by (rule rinf_insert, rule finite_imp_bounded, simp)
```
```  2240
```
```  2241 subsection{* Compactness (the definition is the one based on convegent subsequences). *}
```
```  2242
```
```  2243 definition
```
```  2244   compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
```
```  2245   "compact S \<longleftrightarrow>
```
```  2246    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
```
```  2247        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
```
```  2248
```
```  2249 text {*
```
```  2250   A metric space (or topological vector space) is said to have the
```
```  2251   Heine-Borel property if every closed and bounded subset is compact.
```
```  2252 *}
```
```  2253
```
```  2254 class heine_borel =
```
```  2255   assumes bounded_imp_convergent_subsequence:
```
```  2256     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
```
```  2257       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
```
```  2258
```
```  2259 lemma bounded_closed_imp_compact:
```
```  2260   fixes s::"'a::heine_borel set"
```
```  2261   assumes "bounded s" and "closed s" shows "compact s"
```
```  2262 proof (unfold compact_def, clarify)
```
```  2263   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
```
```  2264   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
```
```  2265     using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
```
```  2266   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
```
```  2267   have "l \<in> s" using `closed s` fr l
```
```  2268     unfolding closed_sequential_limits by blast
```
```  2269   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
```
```  2270     using `l \<in> s` r l by blast
```
```  2271 qed
```
```  2272
```
```  2273 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
```
```  2274 proof(induct n)
```
```  2275   show "0 \<le> r 0" by auto
```
```  2276 next
```
```  2277   fix n assume "n \<le> r n"
```
```  2278   moreover have "r n < r (Suc n)"
```
```  2279     using assms [unfolded subseq_def] by auto
```
```  2280   ultimately show "Suc n \<le> r (Suc n)" by auto
```
```  2281 qed
```
```  2282
```
```  2283 lemma eventually_subseq:
```
```  2284   assumes r: "subseq r"
```
```  2285   shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
```
```  2286 unfolding eventually_sequentially
```
```  2287 by (metis subseq_bigger [OF r] le_trans)
```
```  2288
```
```  2289 lemma lim_subseq:
```
```  2290   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
```
```  2291 unfolding tendsto_def eventually_sequentially o_def
```
```  2292 by (metis subseq_bigger le_trans)
```
```  2293
```
```  2294 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
```
```  2295   unfolding Ex1_def
```
```  2296   apply (rule_tac x="nat_rec e f" in exI)
```
```  2297   apply (rule conjI)+
```
```  2298 apply (rule def_nat_rec_0, simp)
```
```  2299 apply (rule allI, rule def_nat_rec_Suc, simp)
```
```  2300 apply (rule allI, rule impI, rule ext)
```
```  2301 apply (erule conjE)
```
```  2302 apply (induct_tac x)
```
```  2303 apply (simp add: nat_rec_0)
```
```  2304 apply (erule_tac x="n" in allE)
```
```  2305 apply (simp)
```
```  2306 done
```
```  2307
```
```  2308 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
```
```  2309   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
```
```  2310   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"
```
```  2311 proof-
```
```  2312   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
```
```  2313   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
```
```  2314   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
```
```  2315     { fix n::nat
```
```  2316       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
```
```  2317       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
```
```  2318       with n have "s N \<le> t - e" using `e>0` by auto
```
```  2319       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  }
```
```  2320     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
```
```  2321     hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto  }
```
```  2322   thus ?thesis by blast
```
```  2323 qed
```
```  2324
```
```  2325 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
```
```  2326   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
```
```  2327   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
```
```  2328   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
```
```  2329   unfolding monoseq_def incseq_def
```
```  2330   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
```
```  2331   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
```
```  2332
```
```  2333 lemma compact_real_lemma:
```
```  2334   assumes "\<forall>n::nat. abs(s n) \<le> b"
```
```  2335   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
```
```  2336 proof-
```
```  2337   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
```
```  2338     using seq_monosub[of s] by auto
```
```  2339   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
```
```  2340     unfolding tendsto_iff dist_norm eventually_sequentially by auto
```
```  2341 qed
```
```  2342
```
```  2343 instance real :: heine_borel
```
```  2344 proof
```
```  2345   fix s :: "real set" and f :: "nat \<Rightarrow> real"
```
```  2346   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
```
```  2347   then obtain b where b: "\<forall>n. abs (f n) \<le> b"
```
```  2348     unfolding bounded_iff by auto
```
```  2349   obtain l :: real and r :: "nat \<Rightarrow> nat" where
```
```  2350     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
```
```  2351     using compact_real_lemma [OF b] by auto
```
```  2352   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
```
```  2353     by auto
```
```  2354 qed
```
```  2355
```
```  2356 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x \$ i) ` s)"
```
```  2357 unfolding bounded_def
```
```  2358 apply clarify
```
```  2359 apply (rule_tac x="x \$ i" in exI)
```
```  2360 apply (rule_tac x="e" in exI)
```
```  2361 apply clarify
```
```  2362 apply (rule order_trans [OF dist_nth_le], simp)
```
```  2363 done
```
```  2364
```
```  2365 lemma compact_lemma:
```
```  2366   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n::finite"
```
```  2367   assumes "bounded s" and "\<forall>n. f n \<in> s"
```
```  2368   shows "\<forall>d.
```
```  2369         \<exists>l r. subseq r \<and>
```
```  2370         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```  2371 proof
```
```  2372   fix d::"'n set" have "finite d" by simp
```
```  2373   thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
```
```  2374       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
```
```  2375   proof(induct d) case empty thus ?case unfolding subseq_def by auto
```
```  2376   next case (insert k d)
```
```  2377     have s': "bounded ((\<lambda>x. x \$ k) ` s)" using `bounded s` by (rule bounded_component)
```
```  2378     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"
```
```  2379       using insert(3) by auto
```
```  2380     have f': "\<forall>n. f (r1 n) \$ k \<in> (\<lambda>x. x \$ k) ` s" using `\<forall>n. f n \<in> s` by simp
```
```  2381     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \$ k) ---> l2) sequentially"
```
```  2382       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
```
```  2383     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
```
```  2384       using r1 and r2 unfolding r_def o_def subseq_def by auto
```
```  2385     moreover
```
```  2386     def l \<equiv> "(\<chi> i. if i = k then l2 else l1\$i)::'a^'n"
```
```  2387     { fix e::real assume "e>0"
```
```  2388       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially" by blast
```
```  2389       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \$ k) l2 < e) sequentially" by (rule tendstoD)
```
```  2390       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \$ i) (l1 \$ i) < e) sequentially"
```
```  2391         by (rule eventually_subseq)
```
```  2392       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```  2393         using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
```
```  2394     }
```
```  2395     ultimately show ?case by auto
```
```  2396   qed
```
```  2397 qed
```
```  2398
```
```  2399 instance "^" :: (heine_borel, finite) heine_borel
```
```  2400 proof
```
```  2401   fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
```
```  2402   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
```
```  2403   then obtain l r where r: "subseq r"
```
```  2404     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially"
```
```  2405     using compact_lemma [OF s f] by blast
```
```  2406   let ?d = "UNIV::'b set"
```
```  2407   { fix e::real assume "e>0"
```
```  2408     hence "0 < e / (real_of_nat (card ?d))"
```
```  2409       using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
```
```  2410     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
```
```  2411       by simp
```
```  2412     moreover
```
```  2413     { fix n assume n: "\<forall>i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
```
```  2414       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) \$ i) (l \$ i))"
```
```  2415         unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)
```
```  2416       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
```
```  2417         by (rule setsum_strict_mono) (simp_all add: n)
```
```  2418       finally have "dist (f (r n)) l < e" by simp
```
```  2419     }
```
```  2420     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```  2421       by (rule eventually_elim1)
```
```  2422   }
```
```  2423   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
```
```  2424   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
```
```  2425 qed
```
```  2426
```
```  2427 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
```
```  2428 unfolding bounded_def
```
```  2429 apply clarify
```
```  2430 apply (rule_tac x="a" in exI)
```
```  2431 apply (rule_tac x="e" in exI)
```
```  2432 apply clarsimp
```
```  2433 apply (drule (1) bspec)
```
```  2434 apply (simp add: dist_Pair_Pair)
```
```  2435 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
```
```  2436 done
```
```  2437
```
```  2438 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
```
```  2439 unfolding bounded_def
```
```  2440 apply clarify
```
```  2441 apply (rule_tac x="b" in exI)
```
```  2442 apply (rule_tac x="e" in exI)
```
```  2443 apply clarsimp
```
```  2444 apply (drule (1) bspec)
```
```  2445 apply (simp add: dist_Pair_Pair)
```
```  2446 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
```
```  2447 done
```
```  2448
```
```  2449 instance "*" :: (heine_borel, heine_borel) heine_borel
```
```  2450 proof
```
```  2451   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
```
```  2452   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
```
```  2453   from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
```
```  2454   from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
```
```  2455   obtain l1 r1 where r1: "subseq r1"
```
```  2456     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
```
```  2457     using bounded_imp_convergent_subsequence [OF s1 f1]
```
```  2458     unfolding o_def by fast
```
```  2459   from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
```
```  2460   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
```
```  2461   obtain l2 r2 where r2: "subseq r2"
```
```  2462     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
```
```  2463     using bounded_imp_convergent_subsequence [OF s2 f2]
```
```  2464     unfolding o_def by fast
```
```  2465   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
```
```  2466     using lim_subseq [OF r2 l1] unfolding o_def .
```
```  2467   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
```
```  2468     using tendsto_Pair [OF l1' l2] unfolding o_def by simp
```
```  2469   have r: "subseq (r1 \<circ> r2)"
```
```  2470     using r1 r2 unfolding subseq_def by simp
```
```  2471   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
```
```  2472     using l r by fast
```
```  2473 qed
```
```  2474
```
```  2475 subsection{* Completeness. *}
```
```  2476
```
```  2477 lemma cauchy_def:
```
```  2478   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
```
```  2479 unfolding Cauchy_def by blast
```
```  2480
```
```  2481 definition
```
```  2482   complete :: "'a::metric_space set \<Rightarrow> bool" where
```
```  2483   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
```
```  2484                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"
```
```  2485
```
```  2486 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
```
```  2487 proof-
```
```  2488   { assume ?rhs
```
```  2489     { fix e::real
```
```  2490       assume "e>0"
```
```  2491       with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
```
```  2492 	by (erule_tac x="e/2" in allE) auto
```
```  2493       { fix n m
```
```  2494 	assume nm:"N \<le> m \<and> N \<le> n"
```
```  2495 	hence "dist (s m) (s n) < e" using N
```
```  2496 	  using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
```
```  2497 	  by blast
```
```  2498       }
```
```  2499       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
```
```  2500 	by blast
```
```  2501     }
```
```  2502     hence ?lhs
```
```  2503       unfolding cauchy_def
```
```  2504       by blast
```
```  2505   }
```
```  2506   thus ?thesis
```
```  2507     unfolding cauchy_def
```
```  2508     using dist_triangle_half_l
```
```  2509     by blast
```
```  2510 qed
```
```  2511
```
```  2512 lemma convergent_imp_cauchy:
```
```  2513  "(s ---> l) sequentially ==> Cauchy s"
```
```  2514 proof(simp only: cauchy_def, rule, rule)
```
```  2515   fix e::real assume "e>0" "(s ---> l) sequentially"
```
```  2516   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
```
```  2517   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
```
```  2518 qed
```
```  2519
```
```  2520 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded {y. (\<exists>n::nat. y = s n)}"
```
```  2521 proof-
```
```  2522   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
```
```  2523   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
```
```  2524   moreover
```
```  2525   have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
```
```  2526   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
```
```  2527     unfolding bounded_any_center [where a="s N"] by auto
```
```  2528   ultimately show "?thesis"
```
```  2529     unfolding bounded_any_center [where a="s N"]
```
```  2530     apply(rule_tac x="max a 1" in exI) apply auto
```
```  2531     apply(erule_tac x=n in allE) apply(erule_tac x=n in ballE) by auto
```
```  2532 qed
```
```  2533
```
```  2534 lemma compact_imp_complete: assumes "compact s" shows "complete s"
```
```  2535 proof-
```
```  2536   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
```
```  2537     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
```
```  2538
```
```  2539     note lr' = subseq_bigger [OF lr(2)]
```
```  2540
```
```  2541     { fix e::real assume "e>0"
```
```  2542       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
```
```  2543       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
```
```  2544       { fix n::nat assume n:"n \<ge> max N M"
```
```  2545 	have "dist ((f \<circ> r) n) l < e/2" using n M by auto
```
```  2546 	moreover have "r n \<ge> N" using lr'[of n] n by auto
```
```  2547 	hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
```
```  2548 	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)  }
```
```  2549       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }
```
```  2550     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto  }
```
```  2551   thus ?thesis unfolding complete_def by auto
```
```  2552 qed
```
```  2553
```
```  2554 instance heine_borel < complete_space
```
```  2555 proof
```
```  2556   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
```
```  2557   hence "bounded (range f)" unfolding image_def
```
```  2558     using cauchy_imp_bounded [of f] by auto
```
```  2559   hence "compact (closure (range f))"
```
```  2560     using bounded_closed_imp_compact [of "closure (range f)"] by auto
```
```  2561   hence "complete (closure (range f))"
```
```  2562     using compact_imp_complete by auto
```
```  2563   moreover have "\<forall>n. f n \<in> closure (range f)"
```
```  2564     using closure_subset [of "range f"] by auto
```
```  2565   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
```
```  2566     using `Cauchy f` unfolding complete_def by auto
```
```  2567   then show "convergent f"
```
```  2568     unfolding convergent_def LIMSEQ_conv_tendsto [symmetric] by auto
```
```  2569 qed
```
```  2570
```
```  2571 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
```
```  2572 proof(simp add: complete_def, rule, rule)
```
```  2573   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
```
```  2574   hence "convergent f" by (rule Cauchy_convergent)
```
```  2575   hence "\<exists>l. f ----> l" unfolding convergent_def .
```
```  2576   thus "\<exists>l. (f ---> l) sequentially" unfolding LIMSEQ_conv_tendsto .
```
```  2577 qed
```
```  2578
```
```  2579 lemma complete_imp_closed: assumes "complete s" shows "closed s"
```
```  2580 proof -
```
```  2581   { fix x assume "x islimpt s"
```
```  2582     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
```
```  2583       unfolding islimpt_sequential by auto
```
```  2584     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
```
```  2585       using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
```
```  2586     hence "x \<in> s"  using Lim_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
```
```  2587   }
```
```  2588   thus "closed s" unfolding closed_limpt by auto
```
```  2589 qed
```
```  2590
```
```  2591 lemma complete_eq_closed:
```
```  2592   fixes s :: "'a::complete_space set"
```
```  2593   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
```
```  2594 proof
```
```  2595   assume ?lhs thus ?rhs by (rule complete_imp_closed)
```
```  2596 next
```
```  2597   assume ?rhs
```
```  2598   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
```
```  2599     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
```
```  2600     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  }
```
```  2601   thus ?lhs unfolding complete_def by auto
```
```  2602 qed
```
```  2603
```
```  2604 lemma convergent_eq_cauchy:
```
```  2605   fixes s :: "nat \<Rightarrow> 'a::complete_space"
```
```  2606   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
```
```  2607 proof
```
```  2608   assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
```
```  2609   thus ?rhs using convergent_imp_cauchy by auto
```
```  2610 next
```
```  2611   assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
```
```  2612 qed
```
```  2613
```
```  2614 lemma convergent_imp_bounded:
```
```  2615   fixes s :: "nat \<Rightarrow> 'a::metric_space"
```
```  2616   shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
```
```  2617   using convergent_imp_cauchy[of s]
```
```  2618   using cauchy_imp_bounded[of s]
```
```  2619   unfolding image_def
```
```  2620   by auto
```
```  2621
```
```  2622 subsection{* Total boundedness. *}
```
```  2623
```
```  2624 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
```
```  2625   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
```
```  2626 declare helper_1.simps[simp del]
```
```  2627
```
```  2628 lemma compact_imp_totally_bounded:
```
```  2629   assumes "compact s"
```
```  2630   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
```
```  2631 proof(rule, rule, rule ccontr)
```
```  2632   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)"
```
```  2633   def x \<equiv> "helper_1 s e"
```
```  2634   { fix n
```
```  2635     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
```
```  2636     proof(induct_tac rule:nat_less_induct)
```
```  2637       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
```
```  2638       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
```
```  2639       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
```
```  2640       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
```
```  2641       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
```
```  2642 	apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
```
```  2643       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
```
```  2644     qed }
```
```  2645   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
```
```  2646   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
```
```  2647   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
```
```  2648   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
```
```  2649   show False
```
```  2650     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
```
```  2651     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
```
```  2652     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
```
```  2653 qed
```
```  2654
```
```  2655 subsection{* Heine-Borel theorem (following Burkill \& Burkill vol. 2) *}
```
```  2656
```
```  2657 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
```
```  2658   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"
```
```  2659   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
```
```  2660 proof(rule ccontr)
```
```  2661   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
```
```  2662   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
```
```  2663   { fix n::nat
```
```  2664     have "1 / real (n + 1) > 0" by auto
```
```  2665     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 }
```
```  2666   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
```
```  2667   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)"
```
```  2668     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
```
```  2669
```
```  2670   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
```
```  2671     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
```
```  2672
```
```  2673   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
```
```  2674   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
```
```  2675     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
```
```  2676
```
```  2677   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
```
```  2678     using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
```
```  2679
```
```  2680   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
```
```  2681   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
```
```  2682     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
```
```  2683     using subseq_bigger[OF r, of "N1 + N2"] by auto
```
```  2684
```
```  2685   def x \<equiv> "(f (r (N1 + N2)))"
```
```  2686   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
```
```  2687     using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
```
```  2688   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
```
```  2689   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
```
```  2690
```
```  2691   have "dist x l < e/2" using N1 unfolding x_def o_def by auto
```
```  2692   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
```
```  2693
```
```  2694   thus False using e and `y\<notin>b` by auto
```
```  2695 qed
```
```  2696
```
```  2697 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
```
```  2698                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
```
```  2699 proof clarify
```
```  2700   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
```
```  2701   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
```
```  2702   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
```
```  2703   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
```
```  2704   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
```
```  2705
```
```  2706   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
```
```  2707   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
```
```  2708
```
```  2709   have "finite (bb ` k)" using k(1) by auto
```
```  2710   moreover
```
```  2711   { fix x assume "x\<in>s"
```
```  2712     hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3)  unfolding subset_eq by auto
```
```  2713     hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
```
```  2714     hence "x \<in> \<Union>(bb ` k)" using  Union_iff[of x "bb ` k"] by auto
```
```  2715   }
```
```  2716   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
```
```  2717 qed
```
```  2718
```
```  2719 subsection{* Bolzano-Weierstrass property. *}
```
```  2720
```
```  2721 lemma heine_borel_imp_bolzano_weierstrass:
```
```  2722   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'))"
```
```  2723           "infinite t"  "t \<subseteq> s"
```
```  2724   shows "\<exists>x \<in> s. x islimpt t"
```
```  2725 proof(rule ccontr)
```
```  2726   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
```
```  2727   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
```
```  2728     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
```
```  2729   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
```
```  2730     using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
```
```  2731   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
```
```  2732   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
```
```  2733     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
```
```  2734     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  }
```
```  2735   hence "infinite (f ` t)" using assms(2) using finite_imageD[unfolded inj_on_def, of f t] by auto
```
```  2736   moreover
```
```  2737   { fix x assume "x\<in>t" "f x \<notin> g"
```
```  2738     from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
```
```  2739     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
```
```  2740     hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
```
```  2741     hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto  }
```
```  2742   hence "f ` t \<subseteq> g" by auto
```
```  2743   ultimately show False using g(2) using finite_subset by auto
```
```  2744 qed
```
```  2745
```
```  2746 subsection{* Complete the chain of compactness variants. *}
```
```  2747
```
```  2748 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
```
```  2749   "helper_2 beyond 0 = beyond 0" |
```
```  2750   "helper_2 beyond (Suc n) = beyond (dist arbitrary (helper_2 beyond n) + 1 )"
```
```  2751
```
```  2752 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
```
```  2753   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
```
```  2754   shows "bounded s"
```
```  2755 proof(rule ccontr)
```
```  2756   assume "\<not> bounded s"
```
```  2757   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist arbitrary (beyond a) \<le> a"
```
```  2758     unfolding bounded_any_center [where a=arbitrary]
```
```  2759     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist arbitrary x \<le> a"] by auto
```
```  2760   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist arbitrary (beyond a) > a"
```
```  2761     unfolding linorder_not_le by auto
```
```  2762   def x \<equiv> "helper_2 beyond"
```
```  2763
```
```  2764   { fix m n ::nat assume "m<n"
```
```  2765     hence "dist arbitrary (x m) + 1 < dist arbitrary (x n)"
```
```  2766     proof(induct n)
```
```  2767       case 0 thus ?case by auto
```
```  2768     next
```
```  2769       case (Suc n)
```
```  2770       have *:"dist arbitrary (x n) + 1 < dist arbitrary (x (Suc n))"
```
```  2771         unfolding x_def and helper_2.simps
```
```  2772 	using beyond(2)[of "dist arbitrary (helper_2 beyond n) + 1"] by auto
```
```  2773       thus ?case proof(cases "m < n")
```
```  2774 	case True thus ?thesis using Suc and * by auto
```
```  2775       next
```
```  2776 	case False hence "m = n" using Suc(2) by auto
```
```  2777 	thus ?thesis using * by auto
```
```  2778       qed
```
```  2779     qed  } note * = this
```
```  2780   { fix m n ::nat assume "m\<noteq>n"
```
```  2781     have "1 < dist (x m) (x n)"
```
```  2782     proof(cases "m<n")
```
```  2783       case True
```
```  2784       hence "1 < dist arbitrary (x n) - dist arbitrary (x m)" using *[of m n] by auto
```
```  2785       thus ?thesis using dist_triangle [of arbitrary "x n" "x m"] by arith
```
```  2786     next
```
```  2787       case False hence "n<m" using `m\<noteq>n` by auto
```
```  2788       hence "1 < dist arbitrary (x m) - dist arbitrary (x n)" using *[of n m] by auto
```
```  2789       thus ?thesis using dist_triangle2 [of arbitrary "x m" "x n"] by arith
```
```  2790     qed  } note ** = this
```
```  2791   { fix a b assume "x a = x b" "a \<noteq> b"
```
```  2792     hence False using **[of a b] by auto  }
```
```  2793   hence "inj x" unfolding inj_on_def by auto
```
```  2794   moreover
```
```  2795   { fix n::nat
```
```  2796     have "x n \<in> s"
```
```  2797     proof(cases "n = 0")
```
```  2798       case True thus ?thesis unfolding x_def using beyond by auto
```
```  2799     next
```
```  2800       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
```
```  2801       thus ?thesis unfolding x_def using beyond by auto
```
```  2802     qed  }
```
```  2803   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
```
```  2804
```
```  2805   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
```
```  2806   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
```
```  2807   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"]]
```
```  2808     unfolding dist_nz by auto
```
```  2809   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
```
```  2810 qed
```
```  2811
```
```  2812 lemma sequence_infinite_lemma:
```
```  2813   fixes l :: "'a::metric_space" (* TODO: generalize *)
```
```  2814   assumes "\<forall>n::nat. (f n  \<noteq> l)"  "(f ---> l) sequentially"
```
```  2815   shows "infinite {y. (\<exists> n. y = f n)}"
```
```  2816 proof(rule ccontr)
```
```  2817   let ?A = "(\<lambda>x. dist x l) ` {y. \<exists>n. y = f n}"
```
```  2818   assume "\<not> infinite {y. \<exists>n. y = f n}"
```
```  2819   hence **:"finite ?A" "?A \<noteq> {}" by auto
```
```  2820   obtain k where k:"dist (f k) l = Min ?A" using Min_in[OF **] by auto
```
```  2821   have "0 < Min ?A" using assms(1) unfolding dist_nz unfolding Min_gr_iff[OF **] by auto
```
```  2822   then obtain N where "dist (f N) l < Min ?A" using assms(2)[unfolded Lim_sequentially, THEN spec[where x="Min ?A"]] by auto
```
```  2823   moreover have "dist (f N) l \<in> ?A" by auto
```
```  2824   ultimately show False using Min_le[OF **(1), of "dist (f N) l"] by auto
```
```  2825 qed
```
```  2826
```
```  2827 lemma sequence_unique_limpt:
```
```  2828   fixes l :: "'a::metric_space" (* TODO: generalize *)
```
```  2829   assumes "\<forall>n::nat. (f n \<noteq> l)"  "(f ---> l) sequentially"  "l' islimpt {y.  (\<exists>n. y = f n)}"
```
```  2830   shows "l' = l"
```
```  2831 proof(rule ccontr)
```
```  2832   def e \<equiv> "dist l' l"
```
```  2833   assume "l' \<noteq> l" hence "e>0" unfolding dist_nz e_def by auto
```
```  2834   then obtain N::nat where N:"\<forall>n\<ge>N. dist (f n) l < e / 2"
```
```  2835     using assms(2)[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
```
```  2836   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}))"
```
```  2837   have "d>0" using `e>0` unfolding d_def e_def using zero_le_dist[of _ l', unfolded order_le_less] by auto
```
```  2838   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
```
```  2839   have "k\<ge>N" using k(1)[unfolded dist_nz] using k(2)[unfolded d_def]
```
```  2840     by force
```
```  2841   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
```
```  2842   thus False unfolding e_def by auto
```
```  2843 qed
```
```  2844
```
```  2845 lemma bolzano_weierstrass_imp_closed:
```
```  2846   fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
```
```  2847   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
```
```  2848   shows "closed s"
```
```  2849 proof-
```
```  2850   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
```
```  2851     hence "l \<in> s"
```
```  2852     proof(cases "\<forall>n. x n \<noteq> l")
```
```  2853       case False thus "l\<in>s" using as(1) by auto
```
```  2854     next
```
```  2855       case True note cas = this
```
```  2856       with as(2) have "infinite {y. \<exists>n. y = x n}" using sequence_infinite_lemma[of x l] by auto
```
```  2857       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
```
```  2858       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
```
```  2859     qed  }
```
```  2860   thus ?thesis unfolding closed_sequential_limits by fast
```
```  2861 qed
```
```  2862
```
```  2863 text{* Hence express everything as an equivalence.   *}
```
```  2864
```
```  2865 lemma compact_eq_heine_borel:
```
```  2866   fixes s :: "'a::heine_borel set"
```
```  2867   shows "compact s \<longleftrightarrow>
```
```  2868            (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
```
```  2869                --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
```
```  2870 proof
```
```  2871   assume ?lhs thus ?rhs using compact_imp_heine_borel[of s] by blast
```
```  2872 next
```
```  2873   assume ?rhs
```
```  2874   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
```
```  2875     by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
```
```  2876   thus ?lhs using bolzano_weierstrass_imp_bounded[of s] bolzano_weierstrass_imp_closed[of s] bounded_closed_imp_compact[of s] by blast
```
```  2877 qed
```
```  2878
```
```  2879 lemma compact_eq_bolzano_weierstrass:
```
```  2880   fixes s :: "'a::heine_borel set"
```
```  2881   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
```
```  2882 proof
```
```  2883   assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
```
```  2884 next
```
```  2885   assume ?rhs thus ?lhs using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed bounded_closed_imp_compact by auto
```
```  2886 qed
```
```  2887
```
```  2888 lemma compact_eq_bounded_closed:
```
```  2889   fixes s :: "'a::heine_borel set"
```
```  2890   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")
```
```  2891 proof
```
```  2892   assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
```
```  2893 next
```
```  2894   assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
```
```  2895 qed
```
```  2896
```
```  2897 lemma compact_imp_bounded:
```
```  2898   fixes s :: "'a::metric_space set"
```
```  2899   shows "compact s ==> bounded s"
```
```  2900 proof -
```
```  2901   assume "compact s"
```
```  2902   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')"
```
```  2903     by (rule compact_imp_heine_borel)
```
```  2904   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
```
```  2905     using heine_borel_imp_bolzano_weierstrass[of s] by auto
```
```  2906   thus "bounded s"
```
```  2907     by (rule bolzano_weierstrass_imp_bounded)
```
```  2908 qed
```
```  2909
```
```  2910 lemma compact_imp_closed:
```
```  2911   fixes s :: "'a::metric_space set"
```
```  2912   shows "compact s ==> closed s"
```
```  2913 proof -
```
```  2914   assume "compact s"
```
```  2915   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')"
```
```  2916     by (rule compact_imp_heine_borel)
```
```  2917   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
```
```  2918     using heine_borel_imp_bolzano_weierstrass[of s] by auto
```
```  2919   thus "closed s"
```
```  2920     by (rule bolzano_weierstrass_imp_closed)
```
```  2921 qed
```
```  2922
```
```  2923 text{* In particular, some common special cases. *}
```
```  2924
```
```  2925 lemma compact_empty[simp]:
```
```  2926  "compact {}"
```
```  2927   unfolding compact_def
```
```  2928   by simp
```
```  2929
```
```  2930 (* TODO: can any of the next 3 lemmas be generalized to metric spaces? *)
```
```  2931
```
```  2932   (* FIXME : Rename *)
```
```  2933 lemma compact_union[intro]:
```
```  2934   fixes s t :: "'a::heine_borel set"
```
```  2935   shows "compact s \<Longrightarrow> compact t ==> compact (s \<union> t)"
```
```  2936   unfolding compact_eq_bounded_closed
```
```  2937   using bounded_Un[of s t]
```
```  2938   using closed_Un[of s t]
```
```  2939   by simp
```
```  2940
```
```  2941 lemma compact_inter[intro]:
```
```  2942   fixes s t :: "'a::heine_borel set"
```
```  2943   shows "compact s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
```
```  2944   unfolding compact_eq_bounded_closed
```
```  2945   using bounded_Int[of s t]
```
```  2946   using closed_Int[of s t]
```
```  2947   by simp
```
```  2948
```
```  2949 lemma compact_inter_closed[intro]:
```
```  2950   fixes s t :: "'a::heine_borel set"
```
```  2951   shows "compact s \<Longrightarrow> closed t ==> compact (s \<inter> t)"
```
```  2952   unfolding compact_eq_bounded_closed
```
```  2953   using closed_Int[of s t]
```
```  2954   using bounded_subset[of "s \<inter> t" s]
```
```  2955   by blast
```
```  2956
```
```  2957 lemma closed_inter_compact[intro]:
```
```  2958   fixes s t :: "'a::heine_borel set"
```
```  2959   shows "closed s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
```
```  2960 proof-
```
```  2961   assume "closed s" "compact t"
```
```  2962   moreover
```
```  2963   have "s \<inter> t = t \<inter> s" by auto ultimately
```
```  2964   show ?thesis
```
```  2965     using compact_inter_closed[of t s]
```
```  2966     by auto
```
```  2967 qed
```
```  2968
```
```  2969 lemma closed_sing [simp]:
```
```  2970   fixes a :: "'a::metric_space"
```
```  2971   shows "closed {a}"
```
```  2972   apply (clarsimp simp add: closed_def open_dist)
```
```  2973   apply (rule ccontr)
```
```  2974   apply (drule_tac x="dist x a" in spec)
```
```  2975   apply (simp add: dist_nz dist_commute)
```
```  2976   done
```
```  2977
```
```  2978 lemma finite_imp_closed:
```
```  2979   fixes s :: "'a::metric_space set"
```
```  2980   shows "finite s ==> closed s"
```
```  2981 proof (induct set: finite)
```
```  2982   case empty show "closed {}" by simp
```
```  2983 next
```
```  2984   case (insert x F)
```
```  2985   hence "closed ({x} \<union> F)" by (simp only: closed_Un closed_sing)
```
```  2986   thus "closed (insert x F)" by simp
```
```  2987 qed
```
```  2988
```
```  2989 lemma finite_imp_compact:
```
```  2990   fixes s :: "'a::heine_borel set"
```
```  2991   shows "finite s ==> compact s"
```
```  2992   unfolding compact_eq_bounded_closed
```
```  2993   using finite_imp_closed finite_imp_bounded
```
```  2994   by blast
```
```  2995
```
```  2996 lemma compact_sing [simp]: "compact {a}"
```
```  2997   unfolding compact_def o_def subseq_def
```
```  2998   by (auto simp add: tendsto_const)
```
```  2999
```
```  3000 lemma compact_cball[simp]:
```
```  3001   fixes x :: "'a::heine_borel"
```
```  3002   shows "compact(cball x e)"
```
```  3003   using compact_eq_bounded_closed bounded_cball closed_cball
```
```  3004   by blast
```
```  3005
```
```  3006 lemma compact_frontier_bounded[intro]:
```
```  3007   fixes s :: "'a::heine_borel set"
```
```  3008   shows "bounded s ==> compact(frontier s)"
```
```  3009   unfolding frontier_def
```
```  3010   using compact_eq_bounded_closed
```
```  3011   by blast
```
```  3012
```
```  3013 lemma compact_frontier[intro]:
```
```  3014   fixes s :: "'a::heine_borel set"
```
```  3015   shows "compact s ==> compact (frontier s)"
```
```  3016   using compact_eq_bounded_closed compact_frontier_bounded
```
```  3017   by blast
```
```  3018
```
```  3019 lemma frontier_subset_compact:
```
```  3020   fixes s :: "'a::heine_borel set"
```
```  3021   shows "compact s ==> frontier s \<subseteq> s"
```
```  3022   using frontier_subset_closed compact_eq_bounded_closed
```
```  3023   by blast
```
```  3024
```
```  3025 lemma open_delete:
```
```  3026   fixes s :: "'a::metric_space set"
```
```  3027   shows "open s ==> open(s - {x})"
```
```  3028   using open_Diff[of s "{x}"] closed_sing
```
```  3029   by blast
```
```  3030
```
```  3031 text{* Finite intersection property. I could make it an equivalence in fact. *}
```
```  3032
```
```  3033 lemma compact_imp_fip:
```
```  3034   fixes s :: "'a::heine_borel set"
```
```  3035   assumes "compact s"  "\<forall>t \<in> f. closed t"
```
```  3036         "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
```
```  3037   shows "s \<inter> (\<Inter> f) \<noteq> {}"
```
```  3038 proof
```
```  3039   assume as:"s \<inter> (\<Inter> f) = {}"
```
```  3040   hence "s \<subseteq> \<Union>op - UNIV ` f" by auto
```
```  3041   moreover have "Ball (op - UNIV ` f) open" using open_Diff closed_Diff using assms(2) by auto
```
```  3042   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
```
```  3043   hence "finite (op - UNIV ` f') \<and> op - UNIV ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
```
```  3044   hence "s \<inter> \<Inter>op - UNIV ` f' \<noteq> {}" using assms(3)[THEN spec[where x="op - UNIV ` f'"]] by auto
```
```  3045   thus False using f'(3) unfolding subset_eq and Union_iff by blast
```
```  3046 qed
```
```  3047
```
```  3048 subsection{* Bounded closed nest property (proof does not use Heine-Borel).            *}
```
```  3049
```
```  3050 lemma bounded_closed_nest:
```
```  3051   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
```
```  3052   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
```
```  3053   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
```
```  3054 proof-
```
```  3055   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
```
```  3056   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
```
```  3057
```
```  3058   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
```
```  3059     unfolding compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast
```
```  3060
```
```  3061   { fix n::nat
```
```  3062     { fix e::real assume "e>0"
```
```  3063       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
```
```  3064       hence "dist ((x \<circ> r) (max N n)) l < e" by auto
```
```  3065       moreover
```
```  3066       have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
```
```  3067       hence "(x \<circ> r) (max N n) \<in> s n"
```
```  3068 	using x apply(erule_tac x=n in allE)
```
```  3069 	using x apply(erule_tac x="r (max N n)" in allE)
```
```  3070 	using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
```
```  3071       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
```
```  3072     }
```
```  3073     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
```
```  3074   }
```
```  3075   thus ?thesis by auto
```
```  3076 qed
```
```  3077
```
```  3078 text{* Decreasing case does not even need compactness, just completeness.        *}
```
```  3079
```
```  3080 lemma decreasing_closed_nest:
```
```  3081   assumes "\<forall>n. closed(s n)"
```
```  3082           "\<forall>n. (s n \<noteq> {})"
```
```  3083           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
```
```  3084           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
```
```  3085   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
```
```  3086 proof-
```
```  3087   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
```
```  3088   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
```
```  3089   then obtain t where t: "\<forall>n. t n \<in> s n" by auto
```
```  3090   { fix e::real assume "e>0"
```
```  3091     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
```
```  3092     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
```
```  3093       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+
```
```  3094       hence "dist (t m) (t n) < e" using N by auto
```
```  3095     }
```
```  3096     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
```
```  3097   }
```
```  3098   hence  "Cauchy t" unfolding cauchy_def by auto
```
```  3099   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
```
```  3100   { fix n::nat
```
```  3101     { fix e::real assume "e>0"
```
```  3102       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
```
```  3103       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
```
```  3104       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
```
```  3105     }
```
```  3106     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
```
```  3107   }
```
```  3108   then show ?thesis by auto
```
```  3109 qed
```
```  3110
```
```  3111 text{* Strengthen it to the intersection actually being a singleton.             *}
```
```  3112
```
```  3113 lemma decreasing_closed_nest_sing:
```
```  3114   assumes "\<forall>n. closed(s n)"
```
```  3115           "\<forall>n. s n \<noteq> {}"
```
```  3116           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
```
```  3117           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
```
```  3118   shows "\<exists>a::'a::heine_borel. \<Inter> {t. (\<exists>n::nat. t = s n)} = {a}"
```
```  3119 proof-
```
```  3120   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
```
```  3121   { fix b assume b:"b \<in> \<Inter>{t. \<exists>n. t = s n}"
```
```  3122     { fix e::real assume "e>0"
```
```  3123       hence "dist a b < e" using assms(4 )using b using a by blast
```
```  3124     }
```
```  3125     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz real_less_def)
```
```  3126   }
```
```  3127   with a have "\<Inter>{t. \<exists>n. t = s n} = {a}"  by auto
```
```  3128   thus ?thesis by auto
```
```  3129 qed
```
```  3130
```
```  3131 text{* Cauchy-type criteria for uniform convergence. *}
```
```  3132
```
```  3133 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
```
```  3134  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
```
```  3135   (\<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")
```
```  3136 proof(rule)
```
```  3137   assume ?lhs
```
```  3138   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
```
```  3139   { fix e::real assume "e>0"
```
```  3140     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
```
```  3141     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
```
```  3142       hence "dist (s m x) (s n x) < e"
```
```  3143 	using N[THEN spec[where x=m], THEN spec[where x=x]]
```
```  3144 	using N[THEN spec[where x=n], THEN spec[where x=x]]
```
```  3145 	using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }
```
```  3146     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  }
```
```  3147   thus ?rhs by auto
```
```  3148 next
```
```  3149   assume ?rhs
```
```  3150   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
```
```  3151   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
```
```  3152     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
```
```  3153   { fix e::real assume "e>0"
```
```  3154     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"
```
```  3155       using `?rhs`[THEN spec[where x="e/2"]] by auto
```
```  3156     { fix x assume "P x"
```
```  3157       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
```
```  3158 	using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
```
```  3159       fix n::nat assume "n\<ge>N"
```
```  3160       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]]
```
```  3161 	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)  }
```
```  3162     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
```
```  3163   thus ?lhs by auto
```
```  3164 qed
```
```  3165
```
```  3166 lemma uniformly_cauchy_imp_uniformly_convergent:
```
```  3167   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
```
```  3168   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"
```
```  3169           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
```
```  3170   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
```
```  3171 proof-
```
```  3172   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"
```
```  3173     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
```
```  3174   moreover
```
```  3175   { fix x assume "P x"
```
```  3176     hence "l x = l' x" using Lim_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
```
```  3177       using l and assms(2) unfolding Lim_sequentially by blast  }
```
```  3178   ultimately show ?thesis by auto
```
```  3179 qed
```
```  3180
```
```  3181 subsection{* Define continuity over a net to take in restrictions of the set. *}
```
```  3182
```
```  3183 definition
```
```  3184   continuous :: "'a::metric_space net \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
```
```  3185   "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
```
```  3186   (* FIXME: generalize 'b to topological_space *)
```
```  3187
```
```  3188 lemma continuous_trivial_limit:
```
```  3189  "trivial_limit net ==> continuous net f"
```
```  3190   unfolding continuous_def tendsto_iff trivial_limit_eq by auto
```
```  3191
```
```  3192 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
```
```  3193   unfolding continuous_def
```
```  3194   unfolding tendsto_iff
```
```  3195   using netlimit_within[of x s]
```
```  3196   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
```
```  3197
```
```  3198 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
```
```  3199   using continuous_within [of x UNIV f] by (simp add: within_UNIV)
```
```  3200
```
```  3201 lemma continuous_at_within:
```
```  3202   fixes x :: "'a::perfect_space"
```
```  3203   assumes "continuous (at x) f"  shows "continuous (at x within s) f"
```
```  3204   (* FIXME: generalize *)
```
```  3205 proof(cases "x islimpt s")
```
```  3206   case True show ?thesis using assms unfolding continuous_def and netlimit_at
```
```  3207     using Lim_at_within[of f "f x" "at x" s]
```
```  3208     unfolding netlimit_within[unfolded trivial_limit_within not_not, OF True] by blast
```
```  3209 next
```
```  3210   case False thus ?thesis unfolding continuous_def and netlimit_at
```
```  3211     unfolding Lim and trivial_limit_within by auto
```
```  3212 qed
```
```  3213
```
```  3214 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
```
```  3215
```
```  3216 lemma continuous_within_eps_delta:
```
```  3217   "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)"
```
```  3218   unfolding continuous_within and Lim_within
```
```  3219   apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
```
```  3220
```
```  3221 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.
```
```  3222                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
```
```  3223   using continuous_within_eps_delta[of x UNIV f]
```
```  3224   unfolding within_UNIV by blast
```
```  3225
```
```  3226 text{* Versions in terms of open balls. *}
```
```  3227
```
```  3228 lemma continuous_within_ball:
```
```  3229  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
```
```  3230                             f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
```
```  3231 proof
```
```  3232   assume ?lhs
```
```  3233   { fix e::real assume "e>0"
```
```  3234     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"
```
```  3235       using `?lhs`[unfolded continuous_within Lim_within] by auto
```
```  3236     { fix y assume "y\<in>f ` (ball x d \<inter> s)"
```
```  3237       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
```
```  3238 	apply (auto simp add: dist_commute mem_ball) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
```
```  3239     }
```
```  3240     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)  }
```
```  3241   thus ?rhs by auto
```
```  3242 next
```
```  3243   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
```
```  3244     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
```
```  3245 qed
```
```  3246
```
```  3247 lemma continuous_at_ball:
```
```  3248   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
```
```  3249 proof
```
```  3250   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
```
```  3251     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)
```
```  3252     unfolding dist_nz[THEN sym] by auto
```
```  3253 next
```
```  3254   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
```
```  3255     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)
```
```  3256 qed
```
```  3257
```
```  3258 text{* For setwise continuity, just start from the epsilon-delta definitions. *}
```
```  3259
```
```  3260 definition
```
```  3261   continuous_on :: "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
```
```  3262   "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)"
```
```  3263
```
```  3264
```
```  3265 definition
```
```  3266   uniformly_continuous_on ::
```
```  3267     "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
```
```  3268   "uniformly_continuous_on s f \<longleftrightarrow>
```
```  3269         (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall> x'\<in>s. dist x' x < d
```
```  3270                            --> dist (f x') (f x) < e)"
```
```  3271
```
```  3272 text{* Some simple consequential lemmas. *}
```
```  3273
```
```  3274 lemma uniformly_continuous_imp_continuous:
```
```  3275  " uniformly_continuous_on s f ==> continuous_on s f"
```
```  3276   unfolding uniformly_continuous_on_def continuous_on_def by blast
```
```  3277
```
```  3278 lemma continuous_at_imp_continuous_within:
```
```  3279  "continuous (at x) f ==> continuous (at x within s) f"
```
```  3280   unfolding continuous_within continuous_at using Lim_at_within by auto
```
```  3281
```
```  3282 lemma continuous_at_imp_continuous_on: assumes "(\<forall>x \<in> s. continuous (at x) f)"
```
```  3283   shows "continuous_on s f"
```
```  3284 proof(simp add: continuous_at continuous_on_def, rule, rule, rule)
```
```  3285   fix x and e::real assume "x\<in>s" "e>0"
```
```  3286   hence "eventually (\<lambda>xa. dist (f xa) (f x) < e) (at x)" using assms unfolding continuous_at tendsto_iff by auto
```
```  3287   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
```
```  3288   { fix x' assume "\<not> 0 < dist x' x"
```
```  3289     hence "x=x'"
```
```  3290       using dist_nz[of x' x] by auto
```
```  3291     hence "dist (f x') (f x) < e" using `e>0` by auto
```
```  3292   }
```
```  3293   thus "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using d by auto
```
```  3294 qed
```
```  3295
```
```  3296 lemma continuous_on_eq_continuous_within:
```
```  3297  "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)" (is "?lhs = ?rhs")
```
```  3298 proof
```
```  3299   assume ?rhs
```
```  3300   { fix x assume "x\<in>s"
```
```  3301     fix e::real assume "e>0"
```
```  3302     assume "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
```
```  3303     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
```
```  3304     { fix x' assume as:"x'\<in>s" "dist x' x < d"
```
```  3305       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) }
```
```  3306     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using `d>0` by auto
```
```  3307   }
```
```  3308   thus ?lhs using `?rhs` unfolding continuous_on_def continuous_within Lim_within by auto
```
```  3309 next
```
```  3310   assume ?lhs
```
```  3311   thus ?rhs unfolding continuous_on_def continuous_within Lim_within by blast
```
```  3312 qed
```
```  3313
```
```  3314 lemma continuous_on:
```
```  3315  "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. (f ---> f(x)) (at x within s))"
```
```  3316   by (auto simp add: continuous_on_eq_continuous_within continuous_within)
```
```  3317
```
```  3318 lemma continuous_on_eq_continuous_at:
```
```  3319  "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
```
```  3320   by (auto simp add: continuous_on continuous_at Lim_within_open)
```
```  3321
```
```  3322 lemma continuous_within_subset:
```
```  3323  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
```
```  3324              ==> continuous (at x within t) f"
```
```  3325   unfolding continuous_within by(metis Lim_within_subset)
```
```  3326
```
```  3327 lemma continuous_on_subset:
```
```  3328  "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
```
```  3329   unfolding continuous_on by (metis subset_eq Lim_within_subset)
```
```  3330
```
```  3331 lemma continuous_on_interior:
```
```  3332  "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
```
```  3333 unfolding interior_def
```
```  3334 apply simp
```
```  3335 by (meson continuous_on_eq_continuous_at continuous_on_subset)
```
```  3336
```
```  3337 lemma continuous_on_eq:
```
```  3338  "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f
```
```  3339            ==> continuous_on s g"
```
```  3340   by (simp add: continuous_on_def)
```
```  3341
```
```  3342 text{* Characterization of various kinds of continuity in terms of sequences.  *}
```
```  3343
```
```  3344 lemma continuous_within_sequentially:
```
```  3345  "continuous (at a within s) f \<longleftrightarrow>
```
```  3346                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
```
```  3347                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
```
```  3348 proof
```
```  3349   assume ?lhs
```
```  3350   { 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"
```
```  3351     fix e::real assume "e>0"
```
```  3352     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
```
```  3353     from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
```
```  3354     hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
```
```  3355       apply(rule_tac  x=N in exI) using N d  apply auto using x(1)
```
```  3356       apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
```
```  3357       apply(erule_tac x="x n" in ballE)  apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
```
```  3358   }
```
```  3359   thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
```
```  3360 next
```
```  3361   assume ?rhs
```
```  3362   { fix e::real assume "e>0"
```
```  3363     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)"
```
```  3364     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
```
```  3365     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)"
```
```  3366       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
```
```  3367     { fix d::real assume "d>0"
```
```  3368       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
```
```  3369       then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
```
```  3370       { fix n::nat assume n:"n\<ge>N"
```
```  3371 	hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
```
```  3372 	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)
```
```  3373 	ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
```
```  3374       }
```
```  3375       hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
```
```  3376     }
```
```  3377     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
```
```  3378     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
```
```  3379     hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
```
```  3380   }
```
```  3381   thus ?lhs  unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
```
```  3382 qed
```
```  3383
```
```  3384 lemma continuous_at_sequentially:
```
```  3385  "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
```
```  3386                   --> ((f o x) ---> f a) sequentially)"
```
```  3387   using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
```
```  3388
```
```  3389 lemma continuous_on_sequentially:
```
```  3390  "continuous_on s f \<longleftrightarrow>  (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
```
```  3391                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
```
```  3392 proof
```
```  3393   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
```
```  3394 next
```
```  3395   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
```
```  3396 qed
```
```  3397
```
```  3398 lemma uniformly_continuous_on_sequentially:
```
```  3399   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```  3400   shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
```
```  3401                     ((\<lambda>n. x n - y n) ---> 0) sequentially
```
```  3402                     \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
```
```  3403 proof
```
```  3404   assume ?lhs
```
```  3405   { 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"
```
```  3406     { fix e::real assume "e>0"
```
```  3407       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"
```
```  3408 	using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
```
```  3409       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
```
```  3410       { fix n assume "n\<ge>N"
```
```  3411 	hence "norm (f (x n) - f (y n) - 0) < e"
```
```  3412 	  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
```
```  3413 	  unfolding dist_commute and dist_norm by simp  }
```
```  3414       hence "\<exists>N. \<forall>n\<ge>N. norm (f (x n) - f (y n) - 0) < e"  by auto  }
```
```  3415     hence "((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially" unfolding Lim_sequentially and dist_norm by auto  }
```
```  3416   thus ?rhs by auto
```
```  3417 next
```
```  3418   assume ?rhs
```
```  3419   { assume "\<not> ?lhs"
```
```  3420     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
```
```  3421     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"
```
```  3422       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
```
```  3423       by (auto simp add: dist_commute)
```
```  3424     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
```
```  3425     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
```
```  3426     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"
```
```  3427       unfolding x_def and y_def using fa by auto
```
```  3428     have 1:"\<And>(x::'a) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
```
```  3429     have 2:"\<And>(x::'b) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
```
```  3430     { fix e::real assume "e>0"
```
```  3431       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
```
```  3432       { fix n::nat assume "n\<ge>N"
```
```  3433 	hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
```
```  3434 	also have "\<dots> < e" using N by auto
```
```  3435 	finally have "inverse (real n + 1) < e" by auto
```
```  3436 	hence "dist (x n - y n) 0 < e" unfolding 1 using xy0[THEN spec[where x=n]] by auto  }
```
```  3437       hence "\<exists>N. \<forall>n\<ge>N. dist (x n - y n) 0 < e" by auto  }
```
```  3438     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
```
```  3439     hence False unfolding 2 using fxy and `e>0` by auto  }
```
```  3440   thus ?lhs unfolding uniformly_continuous_on_def by blast
```
```  3441 qed
```
```  3442
```
```  3443 text{* The usual transformation theorems. *}
```
```  3444
```
```  3445 lemma continuous_transform_within:
```
```  3446   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
```
```  3447   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
```
```  3448           "continuous (at x within s) f"
```
```  3449   shows "continuous (at x within s) g"
```
```  3450 proof-
```
```  3451   { fix e::real assume "e>0"
```
```  3452     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
```
```  3453     { fix x' assume "x'\<in>s" "0 < dist x' x" "dist x' x < (min d d')"
```
```  3454       hence "dist (f x') (g x) < e" using assms(2,3) apply(erule_tac x=x in ballE) using d' by auto  }
```
```  3455     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
```
```  3456     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  }
```
```  3457   hence "(f ---> g x) (at x within s)" unfolding Lim_within using assms(1) by auto
```
```  3458   thus ?thesis unfolding continuous_within using Lim_transform_within[of d s x f g "g x"] using assms by blast
```
```  3459 qed
```
```  3460
```
```  3461 lemma continuous_transform_at:
```
```  3462   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
```
```  3463   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
```
```  3464           "continuous (at x) f"
```
```  3465   shows "continuous (at x) g"
```
```  3466 proof-
```
```  3467   { fix e::real assume "e>0"
```
```  3468     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
```
```  3469     { fix x' assume "0 < dist x' x" "dist x' x < (min d d')"
```
```  3470       hence "dist (f x') (g x) < e" using assms(2) apply(erule_tac x=x in allE) using d' by auto
```
```  3471     }
```
```  3472     hence "\<forall>xa. 0 < dist xa x \<and> dist xa x < (min d d') \<longrightarrow> dist (f xa) (g x) < e" by blast
```
```  3473     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
```
```  3474   }
```
```  3475   hence "(f ---> g x) (at x)" unfolding Lim_at using assms(1) by auto
```
```  3476   thus ?thesis unfolding continuous_at using Lim_transform_at[of d x f g "g x"] using assms by blast
```
```  3477 qed
```
```  3478
```
```  3479 text{* Combination results for pointwise continuity. *}
```
```  3480
```
```  3481 lemma continuous_const: "continuous net (\<lambda>x. c)"
```
```  3482   by (auto simp add: continuous_def Lim_const)
```
```  3483
```
```  3484 lemma continuous_cmul:
```
```  3485   fixes f :: "'a::metric_space \<Rightarrow> real ^ 'n::finite"
```
```  3486   shows "continuous net f ==> continuous net (\<lambda>x. c *s f x)"
```
```  3487   by (auto simp add: continuous_def Lim_cmul)
```
```  3488
```
```  3489 lemma continuous_neg:
```
```  3490   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3491   shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
```
```  3492   by (auto simp add: continuous_def Lim_neg)
```
```  3493
```
```  3494 lemma continuous_add:
```
```  3495   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3496   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
```
```  3497   by (auto simp add: continuous_def Lim_add)
```
```  3498
```
```  3499 lemma continuous_sub:
```
```  3500   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3501   shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
```
```  3502   by (auto simp add: continuous_def Lim_sub)
```
```  3503
```
```  3504 text{* Same thing for setwise continuity. *}
```
```  3505
```
```  3506 lemma continuous_on_const:
```
```  3507  "continuous_on s (\<lambda>x. c)"
```
```  3508   unfolding continuous_on_eq_continuous_within using continuous_const by blast
```
```  3509
```
```  3510 lemma continuous_on_cmul:
```
```  3511   fixes f :: "'a::metric_space \<Rightarrow> real ^ _"
```
```  3512   shows "continuous_on s f ==>  continuous_on s (\<lambda>x. c *s (f x))"
```
```  3513   unfolding continuous_on_eq_continuous_within using continuous_cmul by blast
```
```  3514
```
```  3515 lemma continuous_on_neg:
```
```  3516   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3517   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
```
```  3518   unfolding continuous_on_eq_continuous_within using continuous_neg by blast
```
```  3519
```
```  3520 lemma continuous_on_add:
```
```  3521   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3522   shows "continuous_on s f \<Longrightarrow> continuous_on s g
```
```  3523            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
```
```  3524   unfolding continuous_on_eq_continuous_within using continuous_add by blast
```
```  3525
```
```  3526 lemma continuous_on_sub:
```
```  3527   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3528   shows "continuous_on s f \<Longrightarrow> continuous_on s g
```
```  3529            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
```
```  3530   unfolding continuous_on_eq_continuous_within using continuous_sub by blast
```
```  3531
```
```  3532 text{* Same thing for uniform continuity, using sequential formulations. *}
```
```  3533
```
```  3534 lemma uniformly_continuous_on_const:
```
```  3535  "uniformly_continuous_on s (\<lambda>x. c)"
```
```  3536   unfolding uniformly_continuous_on_def by simp
```
```  3537
```
```  3538 lemma uniformly_continuous_on_cmul:
```
```  3539   fixes f :: "'a::real_normed_vector \<Rightarrow> real ^ _"
```
```  3540   assumes "uniformly_continuous_on s f"
```
```  3541   shows "uniformly_continuous_on s (\<lambda>x. c *s f(x))"
```
```  3542 proof-
```
```  3543   { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
```
```  3544     hence "((\<lambda>n. c *s f (x n) - c *s f (y n)) ---> 0) sequentially"
```
```  3545       using Lim_cmul[of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
```
```  3546       unfolding  vector_smult_rzero vector_ssub_ldistrib[of c] by auto
```
```  3547   }
```
```  3548   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
```
```  3549 qed
```
```  3550
```
```  3551 lemma dist_minus:
```
```  3552   fixes x y :: "'a::real_normed_vector"
```
```  3553   shows "dist (- x) (- y) = dist x y"
```
```  3554   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
```
```  3555
```
```  3556 lemma uniformly_continuous_on_neg:
```
```  3557   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3558   shows "uniformly_continuous_on s f
```
```  3559          ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
```
```  3560   unfolding uniformly_continuous_on_def dist_minus .
```
```  3561
```
```  3562 lemma uniformly_continuous_on_add:
```
```  3563   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
```
```  3564   assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
```
```  3565   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
```
```  3566 proof-
```
```  3567   {  fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
```
```  3568                     "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
```
```  3569     hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
```
```  3570       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
```
```  3571     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  }
```
```  3572   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
```
```  3573 qed
```
```  3574
```
```  3575 lemma uniformly_continuous_on_sub:
```
```  3576   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
```
```  3577   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
```
```  3578            ==> uniformly_continuous_on s  (\<lambda>x. f x - g x)"
```
```  3579   unfolding ab_diff_minus
```
```  3580   using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
```
```  3581   using uniformly_continuous_on_neg[of s g] by auto
```
```  3582
```
```  3583 text{* Identity function is continuous in every sense. *}
```
```  3584
```
```  3585 lemma continuous_within_id:
```
```  3586  "continuous (at a within s) (\<lambda>x. x)"
```
```  3587   unfolding continuous_within Lim_within by auto
```
```  3588
```
```  3589 lemma continuous_at_id:
```
```  3590  "continuous (at a) (\<lambda>x. x)"
```
```  3591   unfolding continuous_at Lim_at by auto
```
```  3592
```
```  3593 lemma continuous_on_id:
```
```  3594  "continuous_on s (\<lambda>x. x)"
```
```  3595   unfolding continuous_on Lim_within by auto
```
```  3596
```
```  3597 lemma uniformly_continuous_on_id:
```
```  3598  "uniformly_continuous_on s (\<lambda>x. x)"
```
```  3599   unfolding uniformly_continuous_on_def by auto
```
```  3600
```
```  3601 text{* Continuity of all kinds is preserved under composition. *}
```
```  3602
```
```  3603 lemma continuous_within_compose:
```
```  3604   assumes "continuous (at x within s) f"   "continuous (at (f x) within f ` s) g"
```
```  3605   shows "continuous (at x within s) (g o f)"
```
```  3606 proof-
```
```  3607   { fix e::real assume "e>0"
```
```  3608     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
```
```  3609     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
```
```  3610     { fix y assume as:"y\<in>s"  "0 < dist y x"  "dist y x < d'"
```
```  3611       hence "dist (f y) (f x) < d" using d'[THEN bspec[where x=y]] by (auto simp add:dist_commute)
```
```  3612       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   }
```
```  3613     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  }
```
```  3614   thus ?thesis unfolding continuous_within Lim_within by auto
```
```  3615 qed
```
```  3616
```
```  3617 lemma continuous_at_compose:
```
```  3618   assumes "continuous (at x) f"  "continuous (at (f x)) g"
```
```  3619   shows "continuous (at x) (g o f)"
```
```  3620 proof-
```
```  3621   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
```
```  3622   thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
```
```  3623 qed
```
```  3624
```
```  3625 lemma continuous_on_compose:
```
```  3626  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
```
```  3627   unfolding continuous_on_eq_continuous_within using continuous_within_compose[of _ s f g] by auto
```
```  3628
```
```  3629 lemma uniformly_continuous_on_compose:
```
```  3630   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f ` s) g"
```
```  3631   shows "uniformly_continuous_on s (g o f)"
```
```  3632 proof-
```
```  3633   { fix e::real assume "e>0"
```
```  3634     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
```
```  3635     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
```
```  3636     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  }
```
```  3637   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
```
```  3638 qed
```
```  3639
```
```  3640 text{* Continuity in terms of open preimages. *}
```
```  3641
```
```  3642 lemma continuous_at_open:
```
```  3643  "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")
```
```  3644 proof
```
```  3645   assume ?lhs
```
```  3646   { fix t assume as: "open t" "f x \<in> t"
```
```  3647     then obtain e where "e>0" and e:"ball (f x) e \<subseteq> t" unfolding open_contains_ball by auto
```
```  3648
```
```  3649     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
```
```  3650
```
```  3651     have "open (ball x d)" using open_ball by auto
```
```  3652     moreover have "x \<in> ball x d" unfolding centre_in_ball using `d>0` by simp
```
```  3653     moreover
```
```  3654     { fix x' assume "x'\<in>ball x d" hence "f x' \<in> t"
```
```  3655 	using e[unfolded subset_eq Ball_def mem_ball, THEN spec[where x="f x'"]]    d[THEN spec[where x=x']]
```
```  3656 	unfolding mem_ball apply (auto simp add: dist_commute)
```
```  3657 	unfolding dist_nz[THEN sym] using as(2) by auto  }
```
```  3658     hence "\<forall>x'\<in>ball x d. f x' \<in> t" by auto
```
```  3659     ultimately have "\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x'\<in>s. f x' \<in> t)"
```
```  3660       apply(rule_tac x="ball x d" in exI) by simp  }
```
```  3661   thus ?rhs by auto
```
```  3662 next
```
```  3663   assume ?rhs
```
```  3664   { fix e::real assume "e>0"
```
```  3665     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"]]
```
```  3666       unfolding centre_in_ball[of "f x" e, THEN sym] by auto
```
```  3667     then obtain d where "d>0" and d:"ball x d \<subseteq> s" unfolding open_contains_ball by auto
```
```  3668     { fix y assume "0 < dist y x \<and> dist y x < d"
```
```  3669       hence "dist (f y) (f x) < e" using d[unfolded subset_eq Ball_def mem_ball, THEN spec[where x=y]]
```
```  3670 	using s(3)[THEN bspec[where x=y], unfolded mem_ball] by (auto simp add: dist_commute)  }
```
```  3671     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  }
```
```  3672   thus ?lhs unfolding continuous_at Lim_at by auto
```
```  3673 qed
```
```  3674
```
```  3675 lemma continuous_on_open:
```
```  3676  "continuous_on s f \<longleftrightarrow>
```
```  3677         (\<forall>t. openin (subtopology euclidean (f ` s)) t
```
```  3678             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
```
```  3679 proof
```
```  3680   assume ?lhs
```
```  3681   { fix t assume as:"openin (subtopology euclidean (f ` s)) t"
```
```  3682     have "{x \<in> s. f x \<in> t} \<subseteq> s" using as[unfolded openin_euclidean_subtopology_iff] by auto
```
```  3683     moreover
```
```  3684     { fix x assume as':"x\<in>{x \<in> s. f x \<in> t}"
```
```  3685       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
```
```  3686       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
```
```  3687       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)  }
```
```  3688     ultimately have "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" unfolding openin_euclidean_subtopology_iff by auto  }
```
```  3689   thus ?rhs unfolding continuous_on Lim_within using openin by auto
```
```  3690 next
```
```  3691   assume ?rhs
```
```  3692   { fix e::real and x assume "x\<in>s" "e>0"
```
```  3693     { 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)"
```
```  3694       hence "dist (f x') (f x) < e" using dist_triangle[of "f x'" "f x" "f xa"]
```
```  3695 	by (auto simp add: dist_commute)  }
```
```  3696     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
```
```  3697       apply(rule_tac x="e - dist (f xa) (f x)" in exI) using `e>0` by (auto simp add: dist_commute)
```
```  3698     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}"
```
```  3699       using `?rhs`[unfolded openin_euclidean_subtopology_iff, THEN spec[where x="ball (f x) e \<inter> f ` s"]] by auto
```
```  3700     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)  }
```
```  3701   thus ?lhs unfolding continuous_on Lim_within by auto
```
```  3702 qed
```
```  3703
```
```  3704 (* ------------------------------------------------------------------------- *)
```
```  3705 (* Similarly in terms of closed sets.                                        *)
```
```  3706 (* ------------------------------------------------------------------------- *)
```
```  3707
```
```  3708 lemma continuous_on_closed:
```
```  3709  "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")
```
```  3710 proof
```
```  3711   assume ?lhs
```
```  3712   { fix t
```
```  3713     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
```
```  3714     have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
```
```  3715     assume as:"closedin (subtopology euclidean (f ` s)) t"
```
```  3716     hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
```
```  3717     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"]]
```
```  3718       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }
```
```  3719   thus ?rhs by auto
```
```  3720 next
```
```  3721   assume ?rhs
```
```  3722   { fix t
```
```  3723     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
```
```  3724     assume as:"openin (subtopology euclidean (f ` s)) t"
```
```  3725     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
```
```  3726       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
```
```  3727   thus ?lhs unfolding continuous_on_open by auto
```
```  3728 qed
```
```  3729
```
```  3730 text{* Half-global and completely global cases.                                  *}
```
```  3731
```
```  3732 lemma continuous_open_in_preimage:
```
```  3733   assumes "continuous_on s f"  "open t"
```
```  3734   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
```
```  3735 proof-
```
```  3736   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
```
```  3737   have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
```
```  3738     using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
```
```  3739   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
```
```  3740 qed
```
```  3741
```
```  3742 lemma continuous_closed_in_preimage:
```
```  3743   assumes "continuous_on s f"  "closed t"
```
```  3744   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
```
```  3745 proof-
```
```  3746   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
```
```  3747   have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
```
```  3748     using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
```
```  3749   thus ?thesis
```
```  3750     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
```
```  3751 qed
```
```  3752
```
```  3753 lemma continuous_open_preimage:
```
```  3754   assumes "continuous_on s f" "open s" "open t"
```
```  3755   shows "open {x \<in> s. f x \<in> t}"
```
```  3756 proof-
```
```  3757   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
```
```  3758     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
```
```  3759   thus ?thesis using open_Int[of s T, OF assms(2)] by auto
```
```  3760 qed
```
```  3761
```
```  3762 lemma continuous_closed_preimage:
```
```  3763   assumes "continuous_on s f" "closed s" "closed t"
```
```  3764   shows "closed {x \<in> s. f x \<in> t}"
```
```  3765 proof-
```
```  3766   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
```
```  3767     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
```
```  3768   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
```
```  3769 qed
```
```  3770
```
```  3771 lemma continuous_open_preimage_univ:
```
```  3772   fixes f :: "real ^ _ \<Rightarrow> real ^ _" (* FIXME: generalize *)
```
```  3773   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
```
```  3774   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
```
```  3775
```
```  3776 lemma continuous_closed_preimage_univ:
```
```  3777   fixes f :: "real ^ _ \<Rightarrow> real ^ _" (* FIXME: generalize *)
```
```  3778   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
```
```  3779   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
```
```  3780
```
```  3781 text{* Equality of continuous functions on closure and related results.          *}
```
```  3782
```
```  3783 lemma continuous_closed_in_preimage_constant:
```
```  3784  "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
```
```  3785   using continuous_closed_in_preimage[of s f "{a}"] closed_sing by auto
```
```  3786
```
```  3787 lemma continuous_closed_preimage_constant:
```
```  3788  "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
```
```  3789   using continuous_closed_preimage[of s f "{a}"] closed_sing by auto
```
```  3790
```
```  3791 lemma continuous_constant_on_closure:
```
```  3792   assumes "continuous_on (closure s) f"
```
```  3793           "\<forall>x \<in> s. f x = a"
```
```  3794   shows "\<forall>x \<in> (closure s). f x = a"
```
```  3795     using continuous_closed_preimage_constant[of "closure s" f a]
```
```  3796     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
```
```  3797
```
```  3798 lemma image_closure_subset:
```
```  3799   assumes "continuous_on (closure s) f"  "closed t"  "(f ` s) \<subseteq> t"
```
```  3800   shows "f ` (closure s) \<subseteq> t"
```
```  3801 proof-
```
```  3802   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
```
```  3803   moreover have "closed {x \<in> closure s. f x \<in> t}"
```
```  3804     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
```
```  3805   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
```
```  3806     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
```
```  3807   thus ?thesis by auto
```
```  3808 qed
```
```  3809
```
```  3810 lemma continuous_on_closure_norm_le:
```
```  3811   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  3812   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"
```
```  3813   shows "norm(f x) \<le> b"
```
```  3814 proof-
```
```  3815   have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
```
```  3816   show ?thesis
```
```  3817     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
```
```  3818     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
```
```  3819 qed
```
```  3820
```
```  3821 text{* Making a continuous function avoid some value in a neighbourhood.         *}
```
```  3822
```
```  3823 lemma continuous_within_avoid:
```
```  3824   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"
```
```  3825   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
```
```  3826 proof-
```
```  3827   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"
```
```  3828     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
```
```  3829   { fix y assume " y\<in>s"  "dist x y < d"
```
```  3830     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
```
```  3831       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
```
```  3832   thus ?thesis using `d>0` by auto
```
```  3833 qed
```
```  3834
```
```  3835 lemma continuous_at_avoid:
```
```  3836   assumes "continuous (at x) f"  "f x \<noteq> a"
```
```  3837   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
```
```  3838 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
```
```  3839
```
```  3840 lemma continuous_on_avoid:
```
```  3841   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"
```
```  3842   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
```
```  3843 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
```
```  3844
```
```  3845 lemma continuous_on_open_avoid:
```
```  3846   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"
```
```  3847   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
```
```  3848 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
```
```  3849
```
```  3850 text{* Proving a function is constant by proving open-ness of level set.         *}
```
```  3851
```
```  3852 lemma continuous_levelset_open_in_cases:
```
```  3853  "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
```
```  3854         openin (subtopology euclidean s) {x \<in> s. f x = a}
```
```  3855         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
```
```  3856 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
```
```  3857
```
```  3858 lemma continuous_levelset_open_in:
```
```  3859  "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
```
```  3860         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
```
```  3861         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"
```
```  3862 using continuous_levelset_open_in_cases[of s f ]
```
```  3863 by meson
```
```  3864
```
```  3865 lemma continuous_levelset_open:
```
```  3866   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
```
```  3867   shows "\<forall>x \<in> s. f x = a"
```
```  3868 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by auto
```
```  3869
```
```  3870 text{* Some arithmetical combinations (more to prove).                           *}
```
```  3871
```
```  3872 lemma open_scaling[intro]:
```
```  3873   fixes s :: "(real ^ _) set"
```
```  3874   assumes "c \<noteq> 0"  "open s"
```
```  3875   shows "open((\<lambda>x. c *s x) ` s)"
```
```  3876 proof-
```
```  3877   { fix x assume "x \<in> s"
```
```  3878     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
```
```  3879     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using real_mult_order[OF `e>0`] by auto
```
```  3880     moreover
```
```  3881     { fix y assume "dist y (c *s x) < e * \<bar>c\<bar>"
```
```  3882       hence "norm ((1 / c) *s y - x) < e" unfolding dist_norm
```
```  3883 	using norm_mul[of c "(1 / c) *s y - x", unfolded vector_ssub_ldistrib, unfolded vector_smult_assoc] assms(1)
```
```  3884 	  assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
```
```  3885       hence "y \<in> op *s c ` s" using rev_image_eqI[of "(1 / c) *s y" s y "op *s c"]  e[THEN spec[where x="(1 / c) *s y"]]  assms(1) unfolding dist_norm vector_smult_assoc by auto  }
```
```  3886     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *s x) < e \<longrightarrow> x' \<in> op *s c ` s" apply(rule_tac x="e * abs c" in exI) by auto  }
```
```  3887   thus ?thesis unfolding open_dist by auto
```
```  3888 qed
```
```  3889
```
```  3890 lemma minus_image_eq_vimage:
```
```  3891   fixes A :: "'a::ab_group_add set"
```
```  3892   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
```
```  3893   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
```
```  3894
```
```  3895 lemma open_negations:
```
```  3896   fixes s :: "(real ^ _) set" (* FIXME: generalize *)
```
```  3897   shows "open s ==> open ((\<lambda> x. -x) ` s)"
```
```  3898   unfolding vector_sneg_minus1 by auto
```
```  3899
```
```  3900 lemma open_translation:
```
```  3901   fixes s :: "(real ^ _) set" (* FIXME: generalize *)
```
```  3902   assumes "open s"  shows "open((\<lambda>x. a + x) ` s)"
```
```  3903 proof-
```
```  3904   { 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  }
```
```  3905   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
```
```  3906   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
```
```  3907 qed
```
```  3908
```
```  3909 lemma open_affinity:
```
```  3910   fixes s :: "(real ^ _) set"
```
```  3911   assumes "open s"  "c \<noteq> 0"
```
```  3912   shows "open ((\<lambda>x. a + c *s x) ` s)"
```
```  3913 proof-
```
```  3914   have *:"(\<lambda>x. a + c *s x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *s x)" unfolding o_def ..
```
```  3915   have "op + a ` op *s c ` s = (op + a \<circ> op *s c) ` s" by auto
```
```  3916   thus ?thesis using assms open_translation[of "op *s c ` s" a] unfolding * by auto
```
```  3917 qed
```
```  3918
```
```  3919 lemma interior_translation:
```
```  3920   fixes s :: "'a::real_normed_vector set"
```
```  3921   shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
```
```  3922 proof (rule set_ext, rule)
```
```  3923   fix x assume "x \<in> interior (op + a ` s)"
```
```  3924   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
```
```  3925   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
```
```  3926   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
```
```  3927 next
```
```  3928   fix x assume "x \<in> op + a ` interior s"
```
```  3929   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
```
```  3930   { fix z have *:"a + y - z = y + a - z" by auto
```
```  3931     assume "z\<in>ball x e"
```
```  3932     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
```
```  3933     hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }
```
```  3934   hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
```
```  3935   thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
```
```  3936 qed
```
```  3937
```
```  3938 subsection {* Preservation of compactness and connectedness under continuous function.  *}
```
```  3939
```
```  3940 lemma compact_continuous_image:
```
```  3941   assumes "continuous_on s f"  "compact s"
```
```  3942   shows "compact(f ` s)"
```
```  3943 proof-
```
```  3944   { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
```
```  3945     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
```
```  3946     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
```
```  3947     { fix e::real assume "e>0"
```
```  3948       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
```
```  3949       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
```
```  3950       { 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  }
```
```  3951       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }
```
```  3952     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  }
```
```  3953   thus ?thesis unfolding compact_def by auto
```
```  3954 qed
```
```  3955
```
```  3956 lemma connected_continuous_image:
```
```  3957   assumes "continuous_on s f"  "connected s"
```
```  3958   shows "connected(f ` s)"
```
```  3959 proof-
```
```  3960   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f ` s"  "openin (subtopology euclidean (f ` s)) T"  "closedin (subtopology euclidean (f ` s)) T"
```
```  3961     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
```
```  3962       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
```
```  3963       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
```
```  3964       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
```
```  3965     hence False using as(1,2)
```
```  3966       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
```
```  3967   thus ?thesis unfolding connected_clopen by auto
```
```  3968 qed
```
```  3969
```
```  3970 text{* Continuity implies uniform continuity on a compact domain.                *}
```
```  3971
```
```  3972 lemma compact_uniformly_continuous:
```
```  3973   assumes "continuous_on s f"  "compact s"
```
```  3974   shows "uniformly_continuous_on s f"
```
```  3975 proof-
```
```  3976     { fix x assume x:"x\<in>s"
```
```  3977       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
```
```  3978       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  }
```
```  3979     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
```
```  3980     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)"
```
```  3981       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
```
```  3982
```
```  3983   { fix e::real assume "e>0"
```
```  3984
```
```  3985     { 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  }
```
```  3986     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
```
```  3987     moreover
```
```  3988     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }
```
```  3989     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
```
```  3990
```
```  3991     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
```
```  3992       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
```
```  3993       hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
```
```  3994       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`
```
```  3995 	by (auto  simp add: dist_commute)
```
```  3996       moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
```
```  3997 	by (auto simp add: dist_commute)
```
```  3998       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`
```
```  3999 	by (auto  simp add: dist_commute)
```
```  4000       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
```
```  4001 	by (auto simp add: dist_commute)  }
```
```  4002     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  }
```
```  4003   thus ?thesis unfolding uniformly_continuous_on_def by auto
```
```  4004 qed
```
```  4005
```
```  4006 text{* Continuity of inverse function on compact domain. *}
```
```  4007
```
```  4008 lemma continuous_on_inverse:
```
```  4009   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
```
```  4010   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"
```
```  4011   shows "continuous_on (f ` s) g"
```
```  4012 proof-
```
```  4013   have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
```
```  4014   { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
```
```  4015     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
```
```  4016     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
```
```  4017       unfolding T(2) and Int_left_absorb by auto
```
```  4018     moreover have "compact (s \<inter> T)"
```
```  4019       using assms(2) unfolding compact_eq_bounded_closed
```
```  4020       using bounded_subset[of s "s \<inter> T"] and T(1) by auto
```
```  4021     ultimately have "closed (f ` t)" using T(1) unfolding T(2)
```
```  4022       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
```
```  4023     moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
```
```  4024     ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
```
```  4025       unfolding closedin_closed by auto  }
```
```  4026   thus ?thesis unfolding continuous_on_closed by auto
```
```  4027 qed
```
```  4028
```
```  4029 subsection{* A uniformly convergent limit of continuous functions is continuous.       *}
```
```  4030
```
```  4031 lemma norm_triangle_lt:
```
```  4032   fixes x y :: "'a::real_normed_vector"
```
```  4033   shows "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
```
```  4034 by (rule le_less_trans [OF norm_triangle_ineq])
```
```  4035
```
```  4036 lemma continuous_uniform_limit:
```
```  4037   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
```
```  4038   assumes "\<not> (trivial_limit net)"  "eventually (\<lambda>n. continuous_on s (f n)) net"
```
```  4039   "\<forall>e>0. eventually (\<lambda>n. \<forall>x \<in> s. norm(f n x - g x) < e) net"
```
```  4040   shows "continuous_on s g"
```
```  4041 proof-
```
```  4042   { fix x and e::real assume "x\<in>s" "e>0"
```
```  4043     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
```
```  4044     then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3"  "continuous_on s (f n)"
```
```  4045       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
```
```  4046     have "e / 3 > 0" using `e>0` by auto
```
```  4047     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"
```
```  4048       using n(2)[unfolded continuous_on_def, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
```
```  4049     { fix y assume "y\<in>s" "dist y x < d"
```
```  4050       hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
```
```  4051       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"]
```
```  4052 	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
```
```  4053       hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
```
```  4054 	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)  }
```
```  4055     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto  }
```
```  4056   thus ?thesis unfolding continuous_on_def by auto
```
```  4057 qed
```
```  4058
```
```  4059 subsection{* Topological properties of linear functions.                               *}
```
```  4060
```
```  4061 lemma linear_lim_0: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
```
```  4062   assumes "linear f" shows "(f ---> 0) (at (0))"
```
```  4063 proof-
```
```  4064   obtain B where "B>0" and B:"\<forall>x. norm (f x) \<le> B * norm x" using linear_bounded_pos[OF assms] by auto
```
```  4065   { fix e::real assume "e>0"
```
```  4066     { fix x::"real^'a" assume "norm x < e / B"
```
```  4067       hence "B * norm x < e" using `B>0` using mult_strict_right_mono[of "norm x" " e / B" B] unfolding real_mult_commute by auto
```
```  4068       hence "norm (f x) < e" using B[THEN spec[where x=x]] `B>0` using order_le_less_trans[of "norm (f x)" "B * norm x" e] by auto   }
```
```  4069     moreover have "e / B > 0" using `e>0` `B>0` divide_pos_pos by auto
```
```  4070     ultimately have "\<exists>d>0. \<forall>x. 0 < dist x 0 \<and> dist x 0 < d \<longrightarrow> dist (f x) 0 < e" unfolding dist_norm by auto  }
```
```  4071   thus ?thesis unfolding Lim_at by auto
```
```  4072 qed
```
```  4073
```
```  4074 lemma bounded_linear_continuous_at:
```
```  4075   assumes "bounded_linear f"  shows "continuous (at a) f"
```
```  4076   unfolding continuous_at using assms
```
```  4077   apply (rule bounded_linear.tendsto)
```
```  4078   apply (rule Lim_at_id [unfolded id_def])
```
```  4079   done
```
```  4080
```
```  4081 lemma linear_continuous_at:
```
```  4082   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
```
```  4083   assumes "linear f"  shows "continuous (at a) f"
```
```  4084   using assms unfolding linear_conv_bounded_linear
```
```  4085   by (rule bounded_linear_continuous_at)
```
```  4086
```
```  4087 lemma linear_continuous_within:
```
```  4088   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
```
```  4089   shows "linear f ==> continuous (at x within s) f"
```
```  4090   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
```
```  4091
```
```  4092 lemma linear_continuous_on:
```
```  4093   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
```
```  4094   shows "linear f ==> continuous_on s f"
```
```  4095   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
```
```  4096
```
```  4097 text{* Also bilinear functions, in composition form.                             *}
```
```  4098
```
```  4099 lemma bilinear_continuous_at_compose:
```
```  4100   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
```
```  4101   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bilinear h
```
```  4102         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
```
```  4103   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
```
```  4104
```
```  4105 lemma bilinear_continuous_within_compose:
```
```  4106   fixes h :: "real ^ _ \<Rightarrow> real ^ _ \<Rightarrow> real ^ _"
```
```  4107   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bilinear h
```
```  4108         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
```
```  4109   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
```
```  4110
```
```  4111 lemma bilinear_continuous_on_compose:
```
```  4112   fixes h :: "real ^ _ \<Rightarrow> real ^ _ \<Rightarrow> real ^ _"
```
```  4113   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bilinear h
```
```  4114              ==> continuous_on s (\<lambda>x. h (f x) (g x))"
```
```  4115   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
```
```  4116   using bilinear_continuous_within_compose[of _ s f g h] by auto
```
```  4117
```
```  4118 subsection{* Topological stuff lifted from and dropped to R                            *}
```
```  4119
```
```  4120
```
```  4121 lemma open_real:
```
```  4122   fixes s :: "real set" shows
```
```  4123  "open s \<longleftrightarrow>
```
```  4124         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
```
```  4125   unfolding open_dist dist_norm by simp
```
```  4126
```
```  4127 lemma islimpt_approachable_real:
```
```  4128   fixes s :: "real set"
```
```  4129   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
```
```  4130   unfolding islimpt_approachable dist_norm by simp
```
```  4131
```
```  4132 lemma closed_real:
```
```  4133   fixes s :: "real set"
```
```  4134   shows "closed s \<longleftrightarrow>
```
```  4135         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
```
```  4136             --> x \<in> s)"
```
```  4137   unfolding closed_limpt islimpt_approachable dist_norm by simp
```
```  4138
```
```  4139 lemma continuous_at_real_range:
```
```  4140   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
```
```  4141   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
```
```  4142         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
```
```  4143   unfolding continuous_at unfolding Lim_at
```
```  4144   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
```
```  4145   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
```
```  4146   apply(erule_tac x=e in allE) by auto
```
```  4147
```
```  4148 lemma continuous_on_real_range:
```
```  4149   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
```
```  4150   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))"
```
```  4151   unfolding continuous_on_def dist_norm by simp
```
```  4152
```
```  4153 lemma continuous_at_norm: "continuous (at x) norm"
```
```  4154   unfolding continuous_at by (intro tendsto_norm Lim_ident_at)
```
```  4155
```
```  4156 lemma continuous_on_norm: "continuous_on s norm"
```
```  4157 unfolding continuous_on by (intro ballI tendsto_norm Lim_at_within Lim_ident_at)
```
```  4158
```
```  4159 lemma continuous_at_component: "continuous (at a) (\<lambda>x. x \$ i)"
```
```  4160 unfolding continuous_at by (intro Lim_component Lim_ident_at)
```
```  4161
```
```  4162 lemma continuous_on_component: "continuous_on s (\<lambda>x. x \$ i)"
```
```  4163 unfolding continuous_on by (intro ballI Lim_component Lim_at_within Lim_ident_at)
```
```  4164
```
```  4165 lemma continuous_at_infnorm: "continuous (at x) infnorm"
```
```  4166   unfolding continuous_at Lim_at o_def unfolding dist_norm
```
```  4167   apply auto apply (rule_tac x=e in exI) apply auto
```
```  4168   using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
```
```  4169
```
```  4170 text{* Hence some handy theorems on distance, diameter etc. of/from a set.       *}
```
```  4171
```
```  4172 lemma compact_attains_sup:
```
```  4173   fixes s :: "real set"
```
```  4174   assumes "compact s"  "s \<noteq> {}"
```
```  4175   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
```
```  4176 proof-
```
```  4177   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
```
```  4178   { fix e::real assume as: "\<forall>x\<in>s. x \<le> rsup s" "rsup s \<notin> s"  "0 < e" "\<forall>x'\<in>s. x' = rsup s \<or> \<not> rsup s - x' < e"
```
```  4179     have "isLub UNIV s (rsup s)" using rsup[OF assms(2)] unfolding setle_def using as(1) by auto
```
```  4180     moreover have "isUb UNIV s (rsup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
```
```  4181     ultimately have False using isLub_le_isUb[of UNIV s "rsup s" "rsup s - e"] using `e>0` by auto  }
```
```  4182   thus ?thesis using bounded_has_rsup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rsup s"]]
```
```  4183     apply(rule_tac x="rsup s" in bexI) by auto
```
```  4184 qed
```
```  4185
```
```  4186 lemma compact_attains_inf:
```
```  4187   fixes s :: "real set"
```
```  4188   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
```
```  4189 proof-
```
```  4190   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
```
```  4191   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> rinf s"  "rinf s \<notin> s"  "0 < e"
```
```  4192       "\<forall>x'\<in>s. x' = rinf s \<or> \<not> abs (x' - rinf s) < e"
```
```  4193     have "isGlb UNIV s (rinf s)" using rinf[OF assms(2)] unfolding setge_def using as(1) by auto
```
```  4194     moreover
```
```  4195     { fix x assume "x \<in> s"
```
```  4196       hence *:"abs (x - rinf s) = x - rinf s" using as(1)[THEN bspec[where x=x]] by auto
```
```  4197       have "rinf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
```
```  4198     hence "isLb UNIV s (rinf s + e)" unfolding isLb_def and setge_def by auto
```
```  4199     ultimately have False using isGlb_le_isLb[of UNIV s "rinf s" "rinf s + e"] using `e>0` by auto  }
```
```  4200   thus ?thesis using bounded_has_rinf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rinf s"]]
```
```  4201     apply(rule_tac x="rinf s" in bexI) by auto
```
```  4202 qed
```
```  4203
```
```  4204 lemma continuous_attains_sup:
```
```  4205   fixes f :: "'a::metric_space \<Rightarrow> real"
```
```  4206   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
```
```  4207         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"
```
```  4208   using compact_attains_sup[of "f ` s"]
```
```  4209   using compact_continuous_image[of s f] by auto
```
```  4210
```
```  4211 lemma continuous_attains_inf:
```
```  4212   fixes f :: "'a::metric_space \<Rightarrow> real"
```
```  4213   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
```
```  4214         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
```
```  4215   using compact_attains_inf[of "f ` s"]
```
```  4216   using compact_continuous_image[of s f] by auto
```
```  4217
```
```  4218 lemma distance_attains_sup:
```
```  4219   assumes "compact s" "s \<noteq> {}"
```
```  4220   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
```
```  4221 proof (rule continuous_attains_sup [OF assms])
```
```  4222   { fix x assume "x\<in>s"
```
```  4223     have "(dist a ---> dist a x) (at x within s)"
```
```  4224       by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
```
```  4225   }
```
```  4226   thus "continuous_on s (dist a)"
```
```  4227     unfolding continuous_on ..
```
```  4228 qed
```
```  4229
```
```  4230 text{* For *minimal* distance, we only need closure, not compactness.            *}
```
```  4231
```
```  4232 lemma distance_attains_inf:
```
```  4233   fixes a :: "'a::heine_borel"
```
```  4234   assumes "closed s"  "s \<noteq> {}"
```
```  4235   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
```
```  4236 proof-
```
```  4237   from assms(2) obtain b where "b\<in>s" by auto
```
```  4238   let ?B = "cball a (dist b a) \<inter> s"
```
```  4239   have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
```
```  4240   hence "?B \<noteq> {}" by auto
```
```  4241   moreover
```
```  4242   { fix x assume "x\<in>?B"
```
```  4243     fix e::real assume "e>0"
```
```  4244     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
```
```  4245       from as have "\<bar>dist a x' - dist a x\<bar> < e"
```
```  4246         unfolding abs_less_iff minus_diff_eq
```
```  4247         using dist_triangle2 [of a x' x]
```
```  4248         using dist_triangle [of a x x']
```
```  4249         by arith
```
```  4250     }
```
```  4251     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
```
```  4252       using `e>0` by auto
```
```  4253   }
```
```  4254   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
```
```  4255     unfolding continuous_on Lim_within dist_norm real_norm_def
```
```  4256     by fast
```
```  4257   moreover have "compact ?B"
```
```  4258     using compact_cball[of a "dist b a"]
```
```  4259     unfolding compact_eq_bounded_closed
```
```  4260     using bounded_Int and closed_Int and assms(1) by auto
```
```  4261   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"
```
```  4262     using continuous_attains_inf[of ?B "dist a"] by fastsimp
```
```  4263   thus ?thesis by fastsimp
```
```  4264 qed
```
```  4265
```
```  4266 subsection{* We can now extend limit compositions to consider the scalar multiplier.   *}
```
```  4267
```
```  4268 lemma Lim_mul:
```
```  4269   fixes f :: "'a \<Rightarrow> real ^ _"
```
```  4270   assumes "(c ---> d) net"  "(f ---> l) net"
```
```  4271   shows "((\<lambda>x. c(x) *s f x) ---> (d *s l)) net"
```
```  4272   unfolding smult_conv_scaleR using assms by (rule scaleR.tendsto)
```
```  4273
```
```  4274 lemma Lim_vmul:
```
```  4275   fixes c :: "'a \<Rightarrow> real"
```
```  4276   shows "(c ---> d) net ==> ((\<lambda>x. c(x) *s v) ---> d *s v) net"
```
```  4277   using Lim_mul[of c d net "\<lambda>x. v" v] using Lim_const[of v] by auto
```
```  4278
```
```  4279 lemma continuous_vmul:
```
```  4280   fixes c :: "'a::metric_space \<Rightarrow> real"
```
```  4281   shows "continuous net c ==> continuous net (\<lambda>x. c(x) *s v)"
```
```  4282   unfolding continuous_def using Lim_vmul[of c] by auto
```
```  4283
```
```  4284 lemma continuous_mul:
```
```  4285   fixes c :: "'a::metric_space \<Rightarrow> real"
```
```  4286   shows "continuous net c \<Longrightarrow> continuous net f
```
```  4287              ==> continuous net (\<lambda>x. c(x) *s f x) "
```
```  4288   unfolding continuous_def using Lim_mul[of c] by auto
```
```  4289
```
```  4290 lemma continuous_on_vmul:
```
```  4291   fixes c :: "'a::metric_space \<Rightarrow> real"
```
```  4292   shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *s v)"
```
```  4293   unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
```
```  4294
```
```  4295 lemma continuous_on_mul:
```
```  4296   fixes c :: "'a::metric_space \<Rightarrow> real"
```
```  4297   shows "continuous_on s c \<Longrightarrow> continuous_on s f
```
```  4298              ==> continuous_on s (\<lambda>x. c(x) *s f x)"
```
```  4299   unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
```
```  4300
```
```  4301 text{* And so we have continuity of inverse.                                     *}
```
```  4302
```
```  4303 lemma Lim_inv:
```
```  4304   fixes f :: "'a \<Rightarrow> real"
```
```  4305   assumes "(f ---> l) (net::'a net)"  "l \<noteq> 0"
```
```  4306   shows "((inverse o f) ---> inverse l) net"
```
```  4307   unfolding o_def using assms by (rule tendsto_inverse)
```
```  4308
```
```  4309 lemma continuous_inv:
```
```  4310   fixes f :: "'a::metric_space \<Rightarrow> real"
```
```  4311   shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
```
```  4312            ==> continuous net (inverse o f)"
```
```  4313   unfolding continuous_def using Lim_inv by auto
```
```  4314
```
```  4315 lemma continuous_at_within_inv:
```
```  4316   fixes f :: "real ^ _ \<Rightarrow> real"
```
```  4317   assumes "continuous (at a within s) f" "f a \<noteq> 0"
```
```  4318   shows "continuous (at a within s) (inverse o f)"
```
```  4319   using assms unfolding continuous_within o_apply
```
```  4320   by (rule Lim_inv)
```
```  4321
```
```  4322 lemma continuous_at_inv:
```
```  4323   fixes f :: "real ^ _ \<Rightarrow> real"
```
```  4324   shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
```
```  4325          ==> continuous (at a) (inverse o f) "
```
```  4326   using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
```
```  4327
```
```  4328 subsection{* Preservation properties for pasted sets.                                  *}
```
```  4329
```
```  4330 lemma bounded_pastecart:
```
```  4331   fixes s :: "('a::real_normed_vector ^ _) set" (* FIXME: generalize to metric_space *)
```
```  4332   assumes "bounded s" "bounded t"
```
```  4333   shows "bounded { pastecart x y | x y . (x \<in> s \<and> y \<in> t)}"
```
```  4334 proof-
```
```  4335   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
```
```  4336   { fix x y assume "x\<in>s" "y\<in>t"
```
```  4337     hence "norm x \<le> a" "norm y \<le> b" using ab by auto
```
```  4338     hence "norm (pastecart x y) \<le> a + b" using norm_pastecart[of x y] by auto }
```
```  4339   thus ?thesis unfolding bounded_iff by auto
```
```  4340 qed
```
```  4341
```
```  4342 lemma closed_pastecart:
```
```  4343   fixes s :: "(real ^ 'a::finite) set" (* FIXME: generalize *)
```
```  4344   assumes "closed s"  "closed t"
```
```  4345   shows "closed {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
```
```  4346 proof-
```
```  4347   { 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"
```
```  4348     { 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
```
```  4349     moreover
```
```  4350     { fix e::real assume "e>0"
```
```  4351       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
```
```  4352       { fix n::nat assume "n\<ge>N"
```
```  4353 	hence "dist (fstcart (x n)) (fstcart l) < e" "dist (sndcart (x n)) (sndcart l) < e"
```
```  4354 	  using N[THEN spec[where x=n]] dist_fstcart[of "x n" l] dist_sndcart[of "x n" l] by auto   }
```
```  4355       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  }
```
```  4356     ultimately have "fstcart l \<in> s" "sndcart l \<in> t"
```
```  4357       using assms(1)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. fstcart (x n)"], THEN spec[where x="fstcart l"]]
```
```  4358       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. sndcart (x n)"], THEN spec[where x="sndcart l"]]
```
```  4359       unfolding Lim_sequentially by auto
```
```  4360     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  }
```
```  4361   thus ?thesis unfolding closed_sequential_limits by auto
```
```  4362 qed
```
```  4363
```
```  4364 lemma compact_pastecart:
```
```  4365   fixes s t :: "(real ^ _) set"
```
```  4366   shows "compact s \<Longrightarrow> compact t ==> compact {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
```
```  4367   unfolding compact_eq_bounded_closed using bounded_pastecart[of s t] closed_pastecart[of s t] by auto
```
```  4368
```
```  4369 text{* Hence some useful properties follow quite easily.                         *}
```
```  4370
```
```  4371 lemma compact_scaleR_image:
```
```  4372   fixes s :: "'a::real_normed_vector set"
```
```  4373   assumes "compact s"  shows "compact ((\<lambda>x. scaleR c x) ` s)"
```
```  4374 proof-
```
```  4375   let ?f = "\<lambda>x. scaleR c x"
```
```  4376   have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
```
```  4377   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
```
```  4378     using bounded_linear_continuous_at[OF *] assms by auto
```
```  4379 qed
```
```  4380
```
```  4381 lemma compact_scaling:
```
```  4382   fixes s :: "(real ^ _) set"
```
```  4383   assumes "compact s"  shows "compact ((\<lambda>x. c *s x) ` s)"
```
```  4384   using assms unfolding smult_conv_scaleR by (rule compact_scaleR_image)
```
```  4385
```
```  4386 lemma compact_negations:
```
```  4387   fixes s :: "'a::real_normed_vector set"
```
```  4388   assumes "compact s"  shows "compact ((\<lambda>x. -x) ` s)"
```
```  4389   using compact_scaleR_image [OF assms, of "- 1"] by auto
```
```  4390
```
```  4391 lemma compact_sums:
```
```  4392   fixes s t :: "(real ^ _) set"
```
```  4393   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
```
```  4394 proof-
```
```  4395   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} =(\<lambda>z. fstcart z + sndcart z) ` {pastecart x y | x y.  x \<in> s \<and> y \<in> t}"
```
```  4396     apply auto unfolding image_iff apply(rule_tac x="pastecart xa y" in bexI) unfolding fstcart_pastecart sndcart_pastecart by auto
```
```  4397   have "linear (\<lambda>z::real^('a + 'a). fstcart z + sndcart z)" unfolding linear_def
```
```  4398     unfolding fstcart_add sndcart_add apply auto
```
```  4399     unfolding vector_add_ldistrib fstcart_cmul[THEN sym] sndcart_cmul[THEN sym] by auto
```
```  4400   hence "continuous_on {pastecart x y |x y. x \<in> s \<and> y \<in> t} (\<lambda>z. fstcart z + sndcart z)"
```
```  4401     using continuous_at_imp_continuous_on linear_continuous_at by auto
```
```  4402   thus ?thesis unfolding * using compact_continuous_image compact_pastecart[OF assms] by auto
```
```  4403 qed
```
```  4404
```
```  4405 lemma compact_differences:
```
```  4406   fixes s t :: "(real ^ 'a::finite) set"
```
```  4407   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
```
```  4408 proof-
```
```  4409   have "{x - y | x y::real^'a. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
```
```  4410     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
```
```  4411   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
```
```  4412 qed
```
```  4413
```
```  4414 lemma compact_translation:
```
```  4415   fixes s :: "(real ^ _) set"
```
```  4416   assumes "compact s"  shows "compact ((\<lambda>x. a + x) ` s)"
```
```  4417 proof-
```
```  4418   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
```
```  4419   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
```
```  4420 qed
```
```  4421
```
```  4422 lemma compact_affinity:
```
```  4423   fixes s :: "(real ^ _) set"
```
```  4424   assumes "compact s"  shows "compact ((\<lambda>x. a + c *s x) ` s)"
```
```  4425 proof-
```
```  4426   have "op + a ` op *s c ` s = (\<lambda>x. a + c *s x) ` s" by auto
```
```  4427   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
```
```  4428 qed
```
```  4429
```
```  4430 text{* Hence we get the following.                                               *}
```
```  4431
```
```  4432 lemma compact_sup_maxdistance:
```
```  4433   fixes s :: "(real ^ 'n::finite) set"
```
```  4434   assumes "compact s"  "s \<noteq> {}"
```
```  4435   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
```
```  4436 proof-
```
```  4437   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
```
```  4438   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"
```
```  4439     using compact_differences[OF assms(1) assms(1)]
```
```  4440     using distance_attains_sup[where 'a="real ^ 'n", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
```
```  4441   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
```
```  4442   thus ?thesis using x(2)[unfolded `x = a - b`] by blast
```
```  4443 qed
```
```  4444
```
```  4445 text{* We can state this in terms of diameter of a set.                          *}
```
```  4446
```
```  4447 definition "diameter s = (if s = {} then 0::real else rsup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
```
```  4448   (* TODO: generalize to class metric_space *)
```
```  4449
```
```  4450 lemma diameter_bounded:
```
```  4451   assumes "bounded s"
```
```  4452   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
```
```  4453         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
```
```  4454 proof-
```
```  4455   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
```
```  4456   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
```
```  4457   { fix x y assume "x \<in> s" "y \<in> s"
```
```  4458     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)  }
```
```  4459   note * = this
```
```  4460   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto
```
```  4461     have lub:"isLub UNIV ?D (rsup ?D)" using * rsup[of ?D] using `s\<noteq>{}` unfolding setle_def by auto
```
```  4462     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  }
```
```  4463   moreover
```
```  4464   { fix d::real assume "d>0" "d < diameter s"
```
```  4465     hence "s\<noteq>{}" unfolding diameter_def by auto
```
```  4466     hence lub:"isLub UNIV ?D (rsup ?D)" using * rsup[of ?D] unfolding setle_def by auto
```
```  4467     have "\<exists>d' \<in> ?D. d' > d"
```
```  4468     proof(rule ccontr)
```
```  4469       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
```
```  4470       hence as:"\<forall>d'\<in>?D. d' \<le> d" apply auto apply(erule_tac x="norm (x - y)" in allE) by auto
```
```  4471       hence "isUb UNIV ?D d" unfolding isUb_def unfolding setle_def by auto
```
```  4472       thus False using `d < diameter s` `s\<noteq>{}` isLub_le_isUb[OF lub, of d] unfolding diameter_def  by auto
```
```  4473     qed
```
```  4474     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }
```
```  4475   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
```
```  4476         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
```
```  4477 qed
```
```  4478
```
```  4479 lemma diameter_bounded_bound:
```
```  4480  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
```
```  4481   using diameter_bounded by blast
```
```  4482
```
```  4483 lemma diameter_compact_attained:
```
```  4484   fixes s :: "(real ^ _) set"
```
```  4485   assumes "compact s"  "s \<noteq> {}"
```
```  4486   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
```
```  4487 proof-
```
```  4488   have b:"bounded s" using assms(1) compact_eq_bounded_closed by auto
```
```  4489   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
```
```  4490   hence "diameter s \<le> norm (x - y)" using rsup_le[of "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}" "norm (x - y)"]
```
```  4491     unfolding setle_def and diameter_def by auto
```
```  4492   thus ?thesis using diameter_bounded(1)[OF b, THEN bspec[where x=x], THEN bspec[where x=y], OF xys] and xys by auto
```
```  4493 qed
```
```  4494
```
```  4495 text{* Related results with closure as the conclusion.                           *}
```
```  4496
```
```  4497 lemma closed_scaleR_image:
```
```  4498   fixes s :: "'a::real_normed_vector set"
```
```  4499   assumes "closed s" shows "closed ((\<lambda>x. scaleR c x) ` s)"
```
```  4500 proof(cases "s={}")
```
```  4501   case True thus ?thesis by auto
```
```  4502 next
```
```  4503   case False
```
```  4504   show ?thesis
```
```  4505   proof(cases "c=0")
```
```  4506     have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
```
```  4507     case True thus ?thesis apply auto unfolding * using closed_sing by auto
```
```  4508   next
```
```  4509     case False
```
```  4510     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s"  "(x ---> l) sequentially"
```
```  4511       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
```
```  4512           using as(1)[THEN spec[where x=n]]
```
```  4513           using `c\<noteq>0` by (auto simp add: vector_smult_assoc)
```
```  4514       }
```
```  4515       moreover
```
```  4516       { fix e::real assume "e>0"
```
```  4517 	hence "0 < e *\<bar>c\<bar>"  using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
```
```  4518 	then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
```
```  4519           using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
```
```  4520 	hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
```
```  4521           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym] norm_scaleR
```
```  4522 	  using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto  }
```
```  4523       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
```
```  4524       ultimately have "l \<in> scaleR c ` s"
```
```  4525         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"]]
```
```  4526 	unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }
```
```  4527     thus ?thesis unfolding closed_sequential_limits by fast
```
```  4528   qed
```
```  4529 qed
```
```  4530
```
```  4531 lemma closed_scaling:
```
```  4532   fixes s :: "(real ^ _) set"
```
```  4533   assumes "closed s" shows "closed ((\<lambda>x. c *s x) ` s)"
```
```  4534   using assms unfolding smult_conv_scaleR by (rule closed_scaleR_image)
```
```  4535
```
```  4536 lemma closed_negations:
```
```  4537   fixes s :: "'a::real_normed_vector set"
```
```  4538   assumes "closed s"  shows "closed ((\<lambda>x. -x) ` s)"
```
```  4539   using closed_scaleR_image[OF assms, of "- 1"] by simp
```
```  4540
```
```  4541 lemma compact_closed_sums:
```
```  4542   fixes s :: "'a::real_normed_vector set"
```
```  4543   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
```
```  4544 proof-
```
```  4545   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
```
```  4546   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"
```
```  4547     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"
```
```  4548       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
```
```  4549     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
```
```  4550       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
```
```  4551     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
```
```  4552       using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
```
```  4553     hence "l - l' \<in> t"
```
```  4554       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
```
```  4555       using f(3) by auto
```
```  4556     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
```
```  4557   }
```
```  4558   thus ?thesis unfolding closed_sequential_limits by fast
```
```  4559 qed
```
```  4560
```
```  4561 lemma closed_compact_sums:
```
```  4562   fixes s t :: "'a::real_normed_vector set"
```
```  4563   assumes "closed s"  "compact t"
```
```  4564   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
```
```  4565 proof-
```
```  4566   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
```
```  4567     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
```
```  4568   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
```
```  4569 qed
```
```  4570
```
```  4571 lemma compact_closed_differences:
```
```  4572   fixes s t :: "'a::real_normed_vector set"
```
```  4573   assumes "compact s"  "closed t"
```
```  4574   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
```
```  4575 proof-
```
```  4576   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"
```
```  4577     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
```
```  4578   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
```
```  4579 qed
```
```  4580
```
```  4581 lemma closed_compact_differences:
```
```  4582   fixes s t :: "'a::real_normed_vector set"
```
```  4583   assumes "closed s" "compact t"
```
```  4584   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
```
```  4585 proof-
```
```  4586   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
```
```  4587     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
```
```  4588  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
```
```  4589 qed
```
```  4590
```
```  4591 lemma closed_translation:
```
```  4592   fixes a :: "'a::real_normed_vector"
```
```  4593   assumes "closed s"  shows "closed ((\<lambda>x. a + x) ` s)"
```
```  4594 proof-
```
```  4595   have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
```
```  4596   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
```
```  4597 qed
```
```  4598
```
```  4599 lemma translation_UNIV:
```
```  4600   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
```
```  4601   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
```
```  4602
```
```  4603 lemma translation_diff:
```
```  4604   fixes a :: "'a::ab_group_add"
```
```  4605   shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
```
```  4606   by auto
```
```  4607
```
```  4608 lemma closure_translation:
```
```  4609   fixes a :: "'a::real_normed_vector"
```
```  4610   shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
```
```  4611 proof-
```
```  4612   have *:"op + a ` (UNIV - s) = UNIV - op + a ` s"
```
```  4613     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
```
```  4614   show ?thesis unfolding closure_interior translation_diff translation_UNIV
```
```  4615     using interior_translation[of a "UNIV - s"] unfolding * by auto
```
```  4616 qed
```
```  4617
```
```  4618 lemma frontier_translation:
```
```  4619   fixes a :: "'a::real_normed_vector"
```
```  4620   shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
```
```  4621   unfolding frontier_def translation_diff interior_translation closure_translation by auto
```
```  4622
```
```  4623 subsection{* Separation between points and sets.                                       *}
```
```  4624
```
```  4625 lemma separate_point_closed:
```
```  4626   fixes s :: "(real ^ _) set" (* FIXME: generalize *)
```
```  4627   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
```
```  4628 proof(cases "s = {}")
```
```  4629   case True
```
```  4630   thus ?thesis by(auto intro!: exI[where x=1])
```
```  4631 next
```
```  4632   case False
```
```  4633   assume "closed s" "a \<notin> s"
```
```  4634   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
```
```  4635   with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
```
```  4636 qed
```
```  4637
```
```  4638 lemma separate_compact_closed:
```
```  4639   fixes s t :: "(real ^ _) set"
```
```  4640   assumes "compact s" and "closed t" and "s \<inter> t = {}"
```
```  4641   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
```
```  4642 proof-
```
```  4643   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
```
```  4644   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"
```
```  4645     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
```
```  4646   { fix x y assume "x\<in>s" "y\<in>t"
```
```  4647     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
```
```  4648     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
```
```  4649       by (auto  simp add: dist_commute)
```
```  4650     hence "d \<le> dist x y" unfolding dist_norm by auto  }
```
```  4651   thus ?thesis using `d>0` by auto
```
```  4652 qed
```
```  4653
```
```  4654 lemma separate_closed_compact:
```
```  4655   fixes s t :: "(real ^ _) set"
```
```  4656   assumes "closed s" and "compact t" and "s \<inter> t = {}"
```
```  4657   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
```
```  4658 proof-
```
```  4659   have *:"t \<inter> s = {}" using assms(3) by auto
```
```  4660   show ?thesis using separate_compact_closed[OF assms(2,1) *]
```
```  4661     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
```
```  4662     by (auto simp add: dist_commute)
```
```  4663 qed
```
```  4664
```
```  4665 (* A cute way of denoting open and closed intervals using overloading.       *)
```
```  4666
```
```  4667 lemma interval: fixes a :: "'a::ord^'n::finite" shows
```
```  4668   "{a <..< b} = {x::'a^'n. \<forall>i. a\$i < x\$i \<and> x\$i < b\$i}" and
```
```  4669   "{a .. b} = {x::'a^'n. \<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i}"
```
```  4670   by (auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
```
```  4671
```
```  4672 lemma mem_interval: fixes a :: "'a::ord^'n::finite" shows
```
```  4673   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a\$i < x\$i \<and> x\$i < b\$i)"
```
```  4674   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a\$i \<le> x\$i \<and> x\$i \<le> b\$i)"
```
```  4675   using interval[of a b] by(auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
```
```  4676
```
```  4677 lemma mem_interval_1: fixes x :: "real^1" shows
```
```  4678  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
```
```  4679  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
```
```  4680 by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def forall_1)
```
```  4681
```
```  4682 lemma interval_eq_empty: fixes a :: "real^'n::finite" shows
```
```  4683  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b\$i \<le> a\$i))" (is ?th1) and
```
```  4684  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b\$i < a\$i))" (is ?th2)
```
```  4685 proof-
```
```  4686   { fix i x assume as:"b\$i \<le> a\$i" and x:"x\<in>{a <..< b}"
```
```  4687     hence "a \$ i < x \$ i \<and> x \$ i < b \$ i" unfolding mem_interval by auto
```
```  4688     hence "a\$i < b\$i" by auto
```
```  4689     hence False using as by auto  }
```
```  4690   moreover
```
```  4691   { assume as:"\<forall>i. \<not> (b\$i \<le> a\$i)"
```
```  4692     let ?x = "(1/2) *s (a + b)"
```
```  4693     { fix i
```
```  4694       have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
```
```  4695       hence "a\$i < ((1/2) *s (a+b)) \$ i" "((1/2) *s (a+b)) \$ i < b\$i"
```
```  4696 	unfolding vector_smult_component and vector_add_component
```
```  4697 	by (auto simp add: less_divide_eq_number_of1)  }
```
```  4698     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
```
```  4699   ultimately show ?th1 by blast
```
```  4700
```
```  4701   { fix i x assume as:"b\$i < a\$i" and x:"x\<in>{a .. b}"
```
```  4702     hence "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i" unfolding mem_interval by auto
```
```  4703     hence "a\$i \<le> b\$i" by auto
```
```  4704     hence False using as by auto  }
```
```  4705   moreover
```
```  4706   { assume as:"\<forall>i. \<not> (b\$i < a\$i)"
```
```  4707     let ?x = "(1/2) *s (a + b)"
```
```  4708     { fix i
```
```  4709       have "a\$i \<le> b\$i" using as[THEN spec[where x=i]] by auto
```
```  4710       hence "a\$i \<le> ((1/2) *s (a+b)) \$ i" "((1/2) *s (a+b)) \$ i \<le> b\$i"
```
```  4711 	unfolding vector_smult_component and vector_add_component
```
```  4712 	by (auto simp add: less_divide_eq_number_of1)  }
```
```  4713     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
```
```  4714   ultimately show ?th2 by blast
```
```  4715 qed
```
```  4716
```
```  4717 lemma interval_ne_empty: fixes a :: "real^'n::finite" shows
```
```  4718   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i \<le> b\$i)" and
```
```  4719   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a\$i < b\$i)"
```
```  4720   unfolding interval_eq_empty[of a b] by (auto simp add: not_less not_le) (* BH: Why doesn't just "auto" work here? *)
```
```  4721
```
```  4722 lemma subset_interval_imp: fixes a :: "real^'n::finite" shows
```
```  4723  "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
```
```  4724  "(\<forall>i. a\$i < c\$i \<and> d\$i < b\$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
```
```  4725  "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
```
```  4726  "(\<forall>i. a\$i \<le> c\$i \<and> d\$i \<le> b\$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
```
```  4727   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
```
```  4728   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
```
```  4729
```
```  4730 lemma interval_sing: fixes a :: "'a::linorder^'n::finite" shows
```
```  4731  "{a .. a} = {a} \<and> {a<..<a} = {}"
```
```  4732 apply(auto simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
```
```  4733 apply (simp add: order_eq_iff)
```
```  4734 apply (auto simp add: not_less less_imp_le)
```
```  4735 done
```
```  4736
```
```  4737 lemma interval_open_subset_closed:  fixes a :: "'a::preorder^'n::finite" shows
```
```  4738  "{a<..<b} \<subseteq> {a .. b}"
```
```  4739 proof(simp add: subset_eq, rule)
```
```  4740   fix x
```
```  4741   assume x:"x \<in>{a<..<b}"
```
```  4742   { fix i
```
```  4743     have "a \$ i \<le> x \$ i"
```
```  4744       using x order_less_imp_le[of "a\$i" "x\$i"]
```
```  4745       by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
```
```  4746   }
```
```  4747   moreover
```
```  4748   { fix i
```
```  4749     have "x \$ i \<le> b \$ i"
```
```  4750       using x order_less_imp_le[of "x\$i" "b\$i"]
```
```  4751       by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
```
```  4752   }
```
```  4753   ultimately
```
```  4754   show "a \<le> x \<and> x \<le> b"
```
```  4755     by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
```
```  4756 qed
```
```  4757
```
```  4758 lemma subset_interval: fixes a :: "real^'n::finite" shows
```
```  4759  "{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
```
```  4760  "{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
```
```  4761  "{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
```
```  4762  "{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)
```
```  4763 proof-
```
```  4764   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
```
```  4765   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)
```
```  4766   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i. c\$i < d\$i"
```
```  4767     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by (auto, drule_tac x=i in spec, simp) (* BH: Why doesn't just "auto" work? *)
```
```  4768     fix i
```
```  4769     (** TODO combine the following two parts as done in the HOL_light version. **)
```
```  4770     { let ?x = "(\<chi> j. (if j=i then ((min (a\$j) (d\$j))+c\$j)/2 else (c\$j+d\$j)/2))::real^'n"
```
```  4771       assume as2: "a\$i > c\$i"
```
```  4772       { fix j
```
```  4773 	have "c \$ j < ?x \$ j \<and> ?x \$ j < d \$ j" unfolding Cart_lambda_beta
```
```  4774 	  apply(cases "j=i") using as(2)[THEN spec[where x=j]]
```
```  4775 	  by (auto simp add: less_divide_eq_number_of1 as2)  }
```
```  4776       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
```
```  4777       moreover
```
```  4778       have "?x\<notin>{a .. b}"
```
```  4779 	unfolding mem_interval apply auto apply(rule_tac x=i in exI)
```
```  4780 	using as(2)[THEN spec[where x=i]] and as2
```
```  4781 	by (auto simp add: less_divide_eq_number_of1)
```
```  4782       ultimately have False using as by auto  }
```
```  4783     hence "a\$i \<le> c\$i" by(rule ccontr)auto
```
```  4784     moreover
```
```  4785     { let ?x = "(\<chi> j. (if j=i then ((max (b\$j) (c\$j))+d\$j)/2 else (c\$j+d\$j)/2))::real^'n"
```
```  4786       assume as2: "b\$i < d\$i"
```
```  4787       { fix j
```
```  4788 	have "d \$ j > ?x \$ j \<and> ?x \$ j > c \$ j" unfolding Cart_lambda_beta
```
```  4789 	  apply(cases "j=i") using as(2)[THEN spec[where x=j]]
```
```  4790 	  by (auto simp add: less_divide_eq_number_of1 as2)  }
```
```  4791       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
```
```  4792       moreover
```
```  4793       have "?x\<notin>{a .. b}"
```
```  4794 	unfolding mem_interval apply auto apply(rule_tac x=i in exI)
```
```  4795 	using as(2)[THEN spec[where x=i]] and as2
```
```  4796 	by (auto simp add: less_divide_eq_number_of1)
```
```  4797       ultimately have False using as by auto  }
```
```  4798     hence "b\$i \<ge> d\$i" by(rule ccontr)auto
```
```  4799     ultimately
```
```  4800     have "a\$i \<le> c\$i \<and> d\$i \<le> b\$i" by auto
```
```  4801   } note part1 = this
```
```  4802   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)+
```
```  4803   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i. c\$i < d\$i"
```
```  4804     fix i
```
```  4805     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
```
```  4806     hence "a\$i \<le> c\$i \<and> d\$i \<le> b\$i" using part1 and as(2) by auto  } note * = this
```
```  4807   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)+
```
```  4808 qed
```
```  4809
```
```  4810 lemma disjoint_interval: fixes a::"real^'n::finite" shows
```
```  4811   "{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
```
```  4812   "{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
```
```  4813   "{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
```
```  4814   "{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)
```
```  4815 proof-
```
```  4816   let ?z = "(\<chi> i. ((max (a\$i) (c\$i)) + (min (b\$i) (d\$i))) / 2)::real^'n"
```
```  4817   show ?th1 ?th2 ?th3 ?th4
```
```  4818   unfolding expand_set_eq and Int_iff and empty_iff and mem_interval and all_conj_distrib[THEN sym] and eq_False
```
```  4819   apply (auto elim!: allE[where x="?z"])
```
```  4820   apply ((rule_tac x=x in exI, force) | (rule_tac x=i in exI, force))+
```
```  4821   done
```
```  4822 qed
```
```  4823
```
```  4824 lemma inter_interval: fixes a :: "'a::linorder^'n::finite" shows
```
```  4825  "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a\$i) (c\$i)) .. (\<chi> i. min (b\$i) (d\$i))}"
```
```  4826   unfolding expand_set_eq and Int_iff and mem_interval
```
```  4827   by (auto simp add: less_divide_eq_number_of1 intro!: bexI)
```
```  4828
```
```  4829 (* Moved interval_open_subset_closed a bit upwards *)
```
```  4830
```
```  4831 lemma open_interval_lemma: fixes x :: "real" shows
```
```  4832  "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
```
```  4833   by(rule_tac x="min (x - a) (b - x)" in exI, auto)
```
```  4834
```
```  4835 lemma open_interval: fixes a :: "real^'n::finite" shows "open {a<..<b}"
```
```  4836 proof-
```
```  4837   { fix x assume x:"x\<in>{a<..<b}"
```
```  4838     { fix i
```
```  4839       have "\<exists>d>0. \<forall>x'. abs (x' - (x\$i)) < d \<longrightarrow> a\$i < x' \<and> x' < b\$i"
```
```  4840 	using x[unfolded mem_interval, THEN spec[where x=i]]
```
```  4841 	using open_interval_lemma[of "a\$i" "x\$i" "b\$i"] by auto  }
```
```  4842
```
```  4843     hence "\<forall>i. \<exists>d>0. \<forall>x'. abs (x' - (x\$i)) < d \<longrightarrow> a\$i < x' \<and> x' < b\$i" by auto
```
```  4844     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)"
```
```  4845       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
```
```  4846
```
```  4847     let ?d = "Min (range d)"
```
```  4848     have **:"finite (range d)" "range d \<noteq> {}" by auto
```
```  4849     have "?d>0" unfolding Min_gr_iff[OF **] using d by auto
```
```  4850     moreover
```
```  4851     { fix x' assume as:"dist x' x < ?d"
```
```  4852       { fix i
```
```  4853 	have "\<bar>x'\$i - x \$ i\<bar> < d i"
```
```  4854 	  using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
```
```  4855 	  unfolding vector_minus_component and Min_gr_iff[OF **] by auto
```
```  4856 	hence "a \$ i < x' \$ i" "x' \$ i < b \$ i" using d[THEN spec[where x=i]] by auto  }
```
```  4857       hence "a < x' \<and> x' < b" unfolding vector_less_def by auto  }
```
```  4858     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)
```
```  4859   }
```
```  4860   thus ?thesis unfolding open_dist using open_interval_lemma by auto
```
```  4861 qed
```
```  4862
```
```  4863 lemma closed_interval: fixes a :: "real^'n::finite" shows "closed {a .. b}"
```
```  4864 proof-
```
```  4865   { 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"*)
```
```  4866     { assume xa:"a\$i > x\$i"
```
```  4867       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
```
```  4868       hence False unfolding mem_interval and dist_norm
```
```  4869 	using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xa by(auto elim!: allE[where x=i])
```
```  4870     } hence "a\$i \<le> x\$i" by(rule ccontr)auto
```
```  4871     moreover
```
```  4872     { assume xb:"b\$i < x\$i"
```
```  4873       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
```
```  4874       hence False unfolding mem_interval and dist_norm
```
```  4875 	using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xb by(auto elim!: allE[where x=i])
```
```  4876     } hence "x\$i \<le> b\$i" by(rule ccontr)auto
```
```  4877     ultimately
```
```  4878     have "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i" by auto }
```
```  4879   thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
```
```  4880 qed
```
```  4881
```
```  4882 lemma interior_closed_interval: fixes a :: "real^'n::finite" shows
```
```  4883  "interior {a .. b} = {a<..<b}" (is "?L = ?R")
```
```  4884 proof(rule subset_antisym)
```
```  4885   show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
```
```  4886 next
```
```  4887   { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
```
```  4888     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
```
```  4889     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
```
```  4890     { fix i
```
```  4891       have "dist (x - (e / 2) *s basis i) x < e"
```
```  4892 	   "dist (x + (e / 2) *s basis i) x < e"
```
```  4893 	unfolding dist_norm apply auto
```
```  4894 	unfolding norm_minus_cancel and norm_mul using norm_basis[of i] and `e>0` by auto
```
```  4895       hence "a \$ i \<le> (x - (e / 2) *s basis i) \$ i"
```
```  4896                     "(x + (e / 2) *s basis i) \$ i \<le> b \$ i"
```
```  4897 	using e[THEN spec[where x="x - (e/2) *s basis i"]]
```
```  4898 	and   e[THEN spec[where x="x + (e/2) *s basis i"]]
```
```  4899 	unfolding mem_interval by (auto elim!: allE[where x=i])
```
```  4900       hence "a \$ i < x \$ i" and "x \$ i < b \$ i"
```
```  4901 	unfolding vector_minus_component and vector_add_component
```
```  4902 	unfolding vector_smult_component and basis_component using `e>0` by auto   }
```
```  4903     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
```
```  4904   thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
```
```  4905 qed
```
```  4906
```
```  4907 lemma bounded_closed_interval: fixes a :: "real^'n::finite" shows
```
```  4908  "bounded {a .. b}"
```
```  4909 proof-
```
```  4910   let ?b = "\<Sum>i\<in>UNIV. \<bar>a\$i\<bar> + \<bar>b\$i\<bar>"
```
```  4911   { fix x::"real^'n" assume x:"\<forall>i. a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i"
```
```  4912     { fix i
```
```  4913       have "\<bar>x\$i\<bar> \<le> \<bar>a\$i\<bar> + \<bar>b\$i\<bar>" using x[THEN spec[where x=i]] by auto  }
```
```  4914     hence "(\<Sum>i\<in>UNIV. \<bar>x \$ i\<bar>) \<le> ?b" by(rule setsum_mono)
```
```  4915     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
```
```  4916   thus ?thesis unfolding interval and bounded_iff by auto
```
```  4917 qed
```
```  4918
```
```  4919 lemma bounded_interval: fixes a :: "real^'n::finite" shows
```
```  4920  "bounded {a .. b} \<and> bounded {a<..<b}"
```
```  4921   using bounded_closed_interval[of a b]
```
```  4922   using interval_open_subset_closed[of a b]
```
```  4923   using bounded_subset[of "{a..b}" "{a<..<b}"]
```
```  4924   by simp
```
```  4925
```
```  4926 lemma not_interval_univ: fixes a :: "real^'n::finite" shows
```
```  4927  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
```
```  4928   using bounded_interval[of a b]
```
```  4929   by auto
```
```  4930
```
```  4931 lemma compact_interval: fixes a :: "real^'n::finite" shows
```
```  4932  "compact {a .. b}"
```
```  4933   using bounded_closed_imp_compact using bounded_interval[of a b] using closed_interval[of a b] by auto
```
```  4934
```
```  4935 lemma open_interval_midpoint: fixes a :: "real^'n::finite"
```
```  4936   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *s (a + b)) \<in> {a<..<b}"
```
```  4937 proof-
```
```  4938   { fix i
```
```  4939     have "a \$ i < ((1 / 2) *s (a + b)) \$ i \<and> ((1 / 2) *s (a + b)) \$ i < b \$ i"
```
```  4940       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
```
```  4941       unfolding vector_smult_component and vector_add_component
```
```  4942       by(auto simp add: less_divide_eq_number_of1)  }
```
```  4943   thus ?thesis unfolding mem_interval by auto
```
```  4944 qed
```
```  4945
```
```  4946 lemma open_closed_interval_convex: fixes x :: "real^'n::finite"
```
```  4947   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
```
```  4948   shows "(e *s x + (1 - e) *s y) \<in> {a<..<b}"
```
```  4949 proof-
```
```  4950   { fix i
```
```  4951     have "a \$ i = e * a\$i + (1 - e) * a\$i" unfolding left_diff_distrib by simp
```
```  4952     also have "\<dots> < e * x \$ i + (1 - e) * y \$ i" apply(rule add_less_le_mono)
```
```  4953       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
```
```  4954       using x unfolding mem_interval  apply simp
```
```  4955       using y unfolding mem_interval  apply simp
```
```  4956       done
```
```  4957     finally have "a \$ i < (e *s x + (1 - e) *s y) \$ i" by auto
```
```  4958     moreover {
```
```  4959     have "b \$ i = e * b\$i + (1 - e) * b\$i" unfolding left_diff_distrib by simp
```
```  4960     also have "\<dots> > e * x \$ i + (1 - e) * y \$ i" apply(rule add_less_le_mono)
```
```  4961       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
```
```  4962       using x unfolding mem_interval  apply simp
```
```  4963       using y unfolding mem_interval  apply simp
```
```  4964       done
```
```  4965     finally have "(e *s x + (1 - e) *s y) \$ i < b \$ i" by auto
```
```  4966     } ultimately have "a \$ i < (e *s x + (1 - e) *s y) \$ i \<and> (e *s x + (1 - e) *s y) \$ i < b \$ i" by auto }
```
```  4967   thus ?thesis unfolding mem_interval by auto
```
```  4968 qed
```
```  4969
```
```  4970 lemma closure_open_interval: fixes a :: "real^'n::finite"
```
```  4971   assumes "{a<..<b} \<noteq> {}"
```
```  4972   shows "closure {a<..<b} = {a .. b}"
```
```  4973 proof-
```
```  4974   have ab:"a < b" using assms[unfolded interval_ne_empty] unfolding vector_less_def by auto
```
```  4975   let ?c = "(1 / 2) *s (a + b)"
```
```  4976   { fix x assume as:"x \<in> {a .. b}"
```
```  4977     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *s (?c - x)"
```
```  4978     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
```
```  4979       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
```
```  4980       have "(inverse (real n + 1)) *s ((1 / 2) *s (a + b)) + (1 - inverse (real n + 1)) *s x =
```
```  4981 	x + (inverse (real n + 1)) *s ((1 / 2 *s (a + b)) - x)" by (auto simp add: vector_ssub_ldistrib vector_add_ldistrib field_simps vector_sadd_rdistrib[THEN sym])
```
```  4982       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
```
```  4983       hence False using fn unfolding f_def using xc by(auto simp add: vector_mul_lcancel vector_ssub_ldistrib)  }
```
```  4984     moreover
```
```  4985     { assume "\<not> (f ---> x) sequentially"
```
```  4986       { fix e::real assume "e>0"
```
```  4987 	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
```
```  4988 	then obtain N::nat where "inverse (real (N + 1)) < e" by auto
```
```  4989 	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)
```
```  4990 	hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }
```
```  4991       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
```
```  4992 	unfolding Lim_sequentially by(auto simp add: dist_norm)
```
```  4993       hence "(f ---> x) sequentially" unfolding f_def
```
```  4994 	using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *s ((1 / 2) *s (a + b) - x)" 0 sequentially x]
```
```  4995 	using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *s (a + b) - x)"] by auto  }
```
```  4996     ultimately have "x \<in> closure {a<..<b}"
```
```  4997       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }
```
```  4998   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
```
```  4999 qed
```
```  5000
```
```  5001 lemma bounded_subset_open_interval_symmetric: fixes s::"(real^'n::finite) set"
```
```  5002   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
```
```  5003 proof-
```
```  5004   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
```
```  5005   def a \<equiv> "(\<chi> i. b+1)::real^'n"
```
```  5006   { fix x assume "x\<in>s"
```
```  5007     fix i
```
```  5008     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]
```
```  5009       unfolding vector_uminus_component and a_def and Cart_lambda_beta by auto
```
```  5010   }
```
```  5011   thus ?thesis by(auto intro: exI[where x=a] simp add: vector_less_def)
```
```  5012 qed
```
```  5013
```
```  5014 lemma bounded_subset_open_interval:
```
```  5015   fixes s :: "(real ^ 'n::finite) set"
```
```  5016   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
```
```  5017   by (auto dest!: bounded_subset_open_interval_symmetric)
```
```  5018
```
```  5019 lemma bounded_subset_closed_interval_symmetric:
```
```  5020   fixes s :: "(real ^ 'n::finite) set"
```
```  5021   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
```
```  5022 proof-
```
```  5023   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
```
```  5024   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
```
```  5025 qed
```
```  5026
```
```  5027 lemma bounded_subset_closed_interval:
```
```  5028   fixes s :: "(real ^ 'n::finite) set"
```
```  5029   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
```
```  5030   using bounded_subset_closed_interval_symmetric[of s] by auto
```
```  5031
```
```  5032 lemma frontier_closed_interval:
```
```  5033   fixes a b :: "real ^ _"
```
```  5034   shows "frontier {a .. b} = {a .. b} - {a<..<b}"
```
```  5035   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
```
```  5036
```
```  5037 lemma frontier_open_interval:
```
```  5038   fixes a b :: "real ^ _"
```
```  5039   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
```
```  5040 proof(cases "{a<..<b} = {}")
```
```  5041   case True thus ?thesis using frontier_empty by auto
```
```  5042 next
```
```  5043   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
```
```  5044 qed
```
```  5045
```
```  5046 lemma inter_interval_mixed_eq_empty: fixes a :: "real^'n::finite"
```
```  5047   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
```
```  5048   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
```
```  5049
```
```  5050
```
```  5051 (* Some special cases for intervals in R^1.                                  *)
```
```  5052
```
```  5053 lemma all_1: "(\<forall>x::1. P x) \<longleftrightarrow> P 1"
```
```  5054   by (metis num1_eq_iff)
```
```  5055
```
```  5056 lemma ex_1: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
```
```  5057   by auto (metis num1_eq_iff)
```
```  5058
```
```  5059 lemma interval_cases_1: fixes x :: "real^1" shows
```
```  5060  "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
```
```  5061   by(simp add:  Cart_eq vector_less_def vector_less_eq_def all_1, auto)
```
```  5062
```
```  5063 lemma in_interval_1: fixes x :: "real^1" shows
```
```  5064  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
```
```  5065   (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
```
```  5066 by(simp add: Cart_eq vector_less_def vector_less_eq_def all_1 dest_vec1_def)
```
```  5067
```
```  5068 lemma interval_eq_empty_1: fixes a :: "real^1" shows
```
```  5069   "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
```
```  5070   "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
```
```  5071   unfolding interval_eq_empty and ex_1 and dest_vec1_def by auto
```
```  5072
```
```  5073 lemma subset_interval_1: fixes a :: "real^1" shows
```
```  5074  "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
```
```  5075                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
```
```  5076  "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
```
```  5077                 dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
```
```  5078  "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b \<le> dest_vec1 a \<or>
```
```  5079                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
```
```  5080  "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
```
```  5081                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
```
```  5082   unfolding subset_interval[of a b c d] unfolding all_1 and dest_vec1_def by auto
```
```  5083
```
```  5084 lemma eq_interval_1: fixes a :: "real^1" shows
```
```  5085  "{a .. b} = {c .. d} \<longleftrightarrow>
```
```  5086           dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
```
```  5087           dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
```
```  5088 using set_eq_subset[of "{a .. b}" "{c .. d}"]
```
```  5089 using subset_interval_1(1)[of a b c d]
```
```  5090 using subset_interval_1(1)[of c d a b]
```
```  5091 by auto (* FIXME: slow *)
```
```  5092
```
```  5093 lemma disjoint_interval_1: fixes a :: "real^1" shows
```
```  5094   "{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"
```
```  5095   "{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"
```
```  5096   "{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"
```
```  5097   "{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"
```
```  5098   unfolding disjoint_interval and dest_vec1_def ex_1 by auto
```
```  5099
```
```  5100 lemma open_closed_interval_1: fixes a :: "real^1" shows
```
```  5101  "{a<..<b} = {a .. b} - {a, b}"
```
```  5102   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
```
```  5103
```
```  5104 lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
```
```  5105   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
```
```  5106
```
```  5107 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)
```
```  5108
```
```  5109 lemma closed_interval_left: fixes b::"real^'n::finite"
```
```  5110   shows "closed {x::real^'n. \<forall>i. x\$i \<le> b\$i}"
```
```  5111 proof-
```
```  5112   { fix i
```
```  5113     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"
```
```  5114     { assume "x\$i > b\$i"
```
```  5115       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
```
```  5116       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
```
```  5117     hence "x\$i \<le> b\$i" by(rule ccontr)auto  }
```
```  5118   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
```
```  5119 qed
```
```  5120
```
```  5121 lemma closed_interval_right: fixes a::"real^'n::finite"
```
```  5122   shows "closed {x::real^'n. \<forall>i. a\$i \<le> x\$i}"
```
```  5123 proof-
```
```  5124   { fix i
```
```  5125     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"
```
```  5126     { assume "a\$i > x\$i"
```
```  5127       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
```
```  5128       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
```
```  5129     hence "a\$i \<le> x\$i" by(rule ccontr)auto  }
```
```  5130   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
```
```  5131 qed
```
```  5132
```
```  5133 subsection{* Intervals in general, including infinite and mixtures of open and closed. *}
```
```  5134
```
```  5135 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)"
```
```  5136
```
```  5137 lemma is_interval_interval: "is_interval {a .. b::real^'n::finite}" (is ?th1) "is_interval {a<..<b}" (is ?th2) proof -
```
```  5138   have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
```
```  5139   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
```
```  5140     by(meson real_le_trans le_less_trans less_le_trans *)+ qed
```
```  5141
```
```  5142 lemma is_interval_empty:
```
```  5143  "is_interval {}"
```
```  5144   unfolding is_interval_def
```
```  5145   by simp
```
```  5146
```
```  5147 lemma is_interval_univ:
```
```  5148  "is_interval UNIV"
```
```  5149   unfolding is_interval_def
```
```  5150   by simp
```
```  5151
```
```  5152 subsection{* Closure of halfspaces and hyperplanes.                                    *}
```
```  5153
```
```  5154 lemma Lim_vec1_dot: fixes f :: "real^'m \<Rightarrow> real^'n::finite"
```
```  5155   assumes "(f ---> l) net"  shows "((vec1 o (\<lambda>y. a \<bullet> (f y))) ---> vec1(a \<bullet> l)) net"
```
```  5156 proof(cases "a = vec 0")
```
```  5157   case True thus ?thesis using dot_lzero and Lim_const[of 0 net] unfolding vec1_vec and o_def by auto
```
```  5158 next
```
```  5159   case False
```
```  5160   { fix e::real
```
```  5161     assume "0 < e"
```
```  5162     with `a \<noteq> vec 0` have "0 < e / norm a" by (simp add: divide_pos_pos)
```
```  5163     with assms(1) have "eventually (\<lambda>x. dist (f x) l < e / norm a) net"
```
```  5164       by (rule tendstoD)
```
```  5165     moreover
```
```  5166     { fix z assume "dist (f z) l < e / norm a"
```
```  5167       hence "norm a * norm (f z - l) < e" unfolding dist_norm and
```
```  5168 	pos_less_divide_eq[OF False[unfolded vec_0 zero_less_norm_iff[of a, THEN sym]]] and real_mult_commute by auto
```
```  5169       hence "\<bar>a \<bullet> f z - a \<bullet> l\<bar> < e"
```
```  5170         using order_le_less_trans[OF norm_cauchy_schwarz_abs[of a "f z - l"], of e]
```
```  5171         unfolding dot_rsub[symmetric] by auto  }
```
```  5172     ultimately have "eventually (\<lambda>x. \<bar>a \<bullet> f x - a \<bullet> l\<bar> < e) net"
```
```  5173       by (auto elim: eventually_rev_mono)
```
```  5174   }
```
```  5175   thus ?thesis unfolding tendsto_iff
```
```  5176     by (auto simp add: dist_vec1)
```
```  5177 qed
```
```  5178
```
```  5179 lemma continuous_at_vec1_dot:
```
```  5180   fixes x :: "real ^ _"
```
```  5181   shows "continuous (at x) (vec1 o (\<lambda>y. a \<bullet> y))"
```
```  5182 proof-
```
```  5183   have "((\<lambda>x. x) ---> x) (at x)" unfolding Lim_at by auto
```
```  5184   thus ?thesis unfolding continuous_at and o_def using Lim_vec1_dot[of "\<lambda>x. x" x "at x" a] by auto
```
```  5185 qed
```
```  5186
```
```  5187 lemma continuous_on_vec1_dot:
```
```  5188   fixes s :: "(real ^ _) set"
```
```  5189   shows "continuous_on s (vec1 o (\<lambda>y. a \<bullet> y)) "
```
```  5190   using continuous_at_imp_continuous_on[of s "vec1 o (\<lambda>y. a \<bullet> y)"]
```
```  5191   using continuous_at_vec1_dot
```
```  5192   by auto
```
```  5193
```
```  5194 lemma closed_halfspace_le: fixes a::"real^'n::finite"
```
```  5195   shows "closed {x. a \<bullet> x \<le> b}"
```
```  5196 proof-
```
```  5197   have *:"{x \<in> UNIV. (vec1 \<circ> op \<bullet> a) x \<in> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b}} = {x. a \<bullet> x \<le> b}" by auto
```
```  5198   let ?T = "{x::real^1. (\<forall>i. x\$i \<le> (vec1 b)\$i)}"
```
```  5199   have "closed ?T" using closed_interval_left[of "vec1 b"] by simp
```
```  5200   moreover have "vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b} = range (vec1 \<circ> op \<bullet> a) \<inter> ?T" unfolding all_1
```
```  5201     unfolding image_def by auto
```
```  5202   ultimately have "\<exists>T. closed T \<and> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b} = range (vec1 \<circ> op \<bullet> a) \<inter> T" by auto
```
```  5203   hence "closedin euclidean {x \<in> UNIV. (vec1 \<circ> op \<bullet> a) x \<in> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b}}"
```
```  5204     using continuous_on_vec1_dot[of UNIV a, unfolded continuous_on_closed subtopology_UNIV] unfolding closedin_closed
```
```  5205     by (auto elim!: allE[where x="vec1 ` {r. (\<exists>x. a \<bullet> x = r \<and> r \<le> b)}"])
```
```  5206   thus ?thesis unfolding closed_closedin[THEN sym] and * by auto
```
```  5207 qed
```
```  5208
```
```  5209 lemma closed_halfspace_ge: "closed {x::real^_. a \<bullet> x \<ge> b}"
```
```  5210   using closed_halfspace_le[of "-a" "-b"] unfolding dot_lneg by auto
```
```  5211
```
```  5212 lemma closed_hyperplane: "closed {x::real^_. a \<bullet> x = b}"
```
```  5213 proof-
```
```  5214   have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x \<le> b}" by auto
```
```  5215   thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
```
```  5216 qed
```
```  5217
```
```  5218 lemma closed_halfspace_component_le:
```
```  5219   shows "closed {x::real^'n::finite. x\$i \<le> a}"
```
```  5220   using closed_halfspace_le[of "(basis i)::real^'n" a] unfolding dot_basis[OF assms] by auto
```
```  5221
```
```  5222 lemma closed_halfspace_component_ge:
```
```  5223   shows "closed {x::real^'n::finite. x\$i \<ge> a}"
```
```  5224   using closed_halfspace_ge[of a "(basis i)::real^'n"] unfolding dot_basis[OF assms] by auto
```
```  5225
```
```  5226 text{* Openness of halfspaces.                                                   *}
```
```  5227
```
```  5228 lemma open_halfspace_lt: "open {x::real^_. a \<bullet> x < b}"
```
```  5229 proof-
```
```  5230   have "UNIV - {x. b \<le> a \<bullet> x} = {x. a \<bullet> x < b}" by auto
```
```  5231   thus ?thesis using closed_halfspace_ge[unfolded closed_def Compl_eq_Diff_UNIV, of b a] by auto
```
```  5232 qed
```
```  5233
```
```  5234 lemma open_halfspace_gt: "open {x::real^_. a \<bullet> x > b}"
```
```  5235 proof-
```
```  5236   have "UNIV - {x. b \<ge> a \<bullet> x} = {x. a \<bullet> x > b}" by auto
```
```  5237   thus ?thesis using closed_halfspace_le[unfolded closed_def Compl_eq_Diff_UNIV, of a b] by auto
```
```  5238 qed
```
```  5239
```
```  5240 lemma open_halfspace_component_lt:
```
```  5241   shows "open {x::real^'n::finite. x\$i < a}"
```
```  5242   using open_halfspace_lt[of "(basis i)::real^'n" a] unfolding dot_basis[OF assms] by auto
```
```  5243
```
```  5244 lemma open_halfspace_component_gt:
```
```  5245   shows "open {x::real^'n::finite. x\$i  > a}"
```
```  5246   using open_halfspace_gt[of a "(basis i)::real^'n"] unfolding dot_basis[OF assms] by auto
```
```  5247
```
```  5248 text{* This gives a simple derivation of limit component bounds.                 *}
```
```  5249
```
```  5250 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> real^'n::finite"
```
```  5251   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)\$i \<le> b) net"
```
```  5252   shows "l\$i \<le> b"
```
```  5253 proof-
```
```  5254   { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<le> b} \<longleftrightarrow> x\$i \<le> b" unfolding dot_basis by auto } note * = this
```
```  5255   show ?thesis using Lim_in_closed_set[of "{x. basis i \<bullet> x \<le> b}" f net l] unfolding *
```
```  5256     using closed_halfspace_le[of "(basis i)::real^'n" b] and assms(1,2,3) by auto
```
```  5257 qed
```
```  5258
```
```  5259 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> real^'n::finite"
```
```  5260   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
```
```  5261   shows "b \<le> l\$i"
```
```  5262 proof-
```
```  5263   { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<ge> b} \<longleftrightarrow> x\$i \<ge> b" unfolding dot_basis by auto } note * = this
```
```  5264   show ?thesis using Lim_in_closed_set[of "{x. basis i \<bullet> x \<ge> b}" f net l] unfolding *
```
```  5265     using closed_halfspace_ge[of b "(basis i)::real^'n"] and assms(1,2,3) by auto
```
```  5266 qed
```
```  5267
```
```  5268 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> real^'n::finite"
```
```  5269   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
```
```  5270   shows "l\$i = b"
```
```  5271   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
```
```  5272
```
```  5273 lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
```
```  5274   "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
```
```  5275   using Lim_component_le[of f l net 1 b] unfolding dest_vec1_def by auto
```
```  5276
```
```  5277 lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows
```
```  5278  "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
```
```  5279   using Lim_component_ge[of f l net b 1] unfolding dest_vec1_def by auto
```
```  5280
```
```  5281 text{* Limits relative to a union.                                               *}
```
```  5282
```
```  5283 lemma eventually_within_Un:
```
```  5284   "eventually P (net within (s \<union> t)) \<longleftrightarrow>
```
```  5285     eventually P (net within s) \<and> eventually P (net within t)"
```
```  5286   unfolding Limits.eventually_within
```
```  5287   by (auto elim!: eventually_rev_mp)
```
```  5288
```
```  5289 lemma Lim_within_union:
```
```  5290  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
```
```  5291   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
```
```  5292   unfolding tendsto_def
```
```  5293   by (auto simp add: eventually_within_Un)
```
```  5294
```
```  5295 lemma continuous_on_union:
```
```  5296   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
```
```  5297   shows "continuous_on (s \<union> t) f"
```
```  5298   using assms unfolding continuous_on unfolding Lim_within_union
```
```  5299   unfolding Lim unfolding trivial_limit_within unfolding closed_limpt by auto
```
```  5300
```
```  5301 lemma continuous_on_cases: fixes g :: "real^'m::finite \<Rightarrow> real ^'n::finite"
```
```  5302   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
```
```  5303           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
```
```  5304   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
```
```  5305 proof-
```
```  5306   let ?h = "(\<lambda>x. if P x then f x else g x)"
```
```  5307   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
```
```  5308   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
```
```  5309   moreover
```
```  5310   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
```
```  5311   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
```
```  5312   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
```
```  5313 qed
```
```  5314
```
```  5315
```
```  5316 text{* Some more convenient intermediate-value theorem formulations.             *}
```
```  5317
```
```  5318 lemma connected_ivt_hyperplane: fixes y :: "real^'n::finite"
```
```  5319   assumes "connected s" "x \<in> s" "y \<in> s" "a \<bullet> x \<le> b" "b \<le> a \<bullet> y"
```
```  5320   shows "\<exists>z \<in> s. a \<bullet> z = b"
```
```  5321 proof(rule ccontr)
```
```  5322   assume as:"\<not> (\<exists>z\<in>s. a \<bullet> z = b)"
```
```  5323   let ?A = "{x::real^'n. a \<bullet> x < b}"
```
```  5324   let ?B = "{x::real^'n. a \<bullet> x > b}"
```
```  5325   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
```
```  5326   moreover have "?A \<inter> ?B = {}" by auto
```
```  5327   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
```
```  5328   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
```
```  5329 qed
```
```  5330
```
```  5331 lemma connected_ivt_component: fixes x::"real^'n::finite" shows
```
```  5332  "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)"
```
```  5333   using connected_ivt_hyperplane[of s x y "(basis k)::real^'n" a] by (auto simp add: dot_basis)
```
```  5334
```
```  5335 text{* Also more convenient formulations of monotone convergence.                *}
```
```  5336
```
```  5337 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real^1"
```
```  5338   assumes "bounded {s n| n::nat. True}"  "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
```
```  5339   shows "\<exists>l. (s ---> l) sequentially"
```
```  5340 proof-
```
```  5341   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
```
```  5342   { fix m::nat
```
```  5343     have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
```
```  5344       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq)  }
```
```  5345   hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
```
```  5346   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
```
```  5347   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)
```
```  5348     unfolding dist_norm unfolding abs_dest_vec1 and dest_vec1_sub by auto
```
```  5349 qed
```
```  5350
```
```  5351 subsection{* Basic homeomorphism definitions.                                          *}
```
```  5352
```
```  5353 definition "homeomorphism s t f g \<equiv>
```
```  5354      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
```
```  5355      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
```
```  5356
```
```  5357 definition homeomorphic :: "((real^'a::finite) set) \<Rightarrow> ((real^'b::finite) set) \<Rightarrow> bool" (infixr "homeomorphic" 60) where
```
```  5358   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
```
```  5359
```
```  5360 lemma homeomorphic_refl: "s homeomorphic s"
```
```  5361   unfolding homeomorphic_def
```
```  5362   unfolding homeomorphism_def
```
```  5363   using continuous_on_id
```
```  5364   apply(rule_tac x = "(\<lambda>x::real^'a.x)" in exI)
```
```  5365   apply(rule_tac x = "(\<lambda>x::real^'b.x)" in exI)
```
```  5366   by blast
```
```  5367
```
```  5368 lemma homeomorphic_sym:
```
```  5369  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
```
```  5370 unfolding homeomorphic_def
```
```  5371 unfolding homeomorphism_def
```
```  5372 by blast
```
```  5373
```
```  5374 lemma homeomorphic_trans:
```
```  5375   assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
```
```  5376 proof-
```
```  5377   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"
```
```  5378     using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
```
```  5379   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"
```
```  5380     using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
```
```  5381
```
```  5382   { 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 }
```
```  5383   moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
```
```  5384   moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
```
```  5385   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 }
```
```  5386   moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
```
```  5387   moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6)  unfolding fg2(5) by auto
```
```  5388   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
```
```  5389 qed
```
```  5390
```
```  5391 lemma homeomorphic_minimal:
```
```  5392  "s homeomorphic t \<longleftrightarrow>
```
```  5393     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
```
```  5394            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
```
```  5395            continuous_on s f \<and> continuous_on t g)"
```
```  5396 unfolding homeomorphic_def homeomorphism_def
```
```  5397 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
```
```  5398 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
```
```  5399 unfolding image_iff
```
```  5400 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
```
```  5401 apply auto apply(rule_tac x="g x" in bexI) apply auto
```
```  5402 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
```
```  5403 apply auto apply(rule_tac x="f x" in bexI) by auto
```
```  5404
```
```  5405 subsection{* Relatively weak hypotheses if a set is compact.                           *}
```
```  5406
```
```  5407 definition "inv_on f s = (\<lambda>x. SOME y. y\<in>s \<and> f y = x)"
```
```  5408
```
```  5409 lemma assumes "inj_on f s" "x\<in>s"
```
```  5410   shows "inv_on f s (f x) = x"
```
```  5411  using assms unfolding inj_on_def inv_on_def by auto
```
```  5412
```
```  5413 lemma homeomorphism_compact:
```
```  5414   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
```
```  5415   assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
```
```  5416   shows "\<exists>g. homeomorphism s t f g"
```
```  5417 proof-
```
```  5418   def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
```
```  5419   have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
```
```  5420   { fix y assume "y\<in>t"
```
```  5421     then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
```
```  5422     hence "g (f x) = x" using g by auto
```
```  5423     hence "f (g y) = y" unfolding x(1)[THEN sym] by auto  }
```
```  5424   hence g':"\<forall>x\<in>t. f (g x) = x" by auto
```
```  5425   moreover
```
```  5426   { fix x
```
```  5427     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"])
```
```  5428     moreover
```
```  5429     { assume "x\<in>g ` t"
```
```  5430       then obtain y where y:"y\<in>t" "g y = x" by auto
```
```  5431       then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
```
```  5432       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 }
```
```  5433     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" by auto  }
```
```  5434   hence "g ` t = s" by auto
```
```  5435   ultimately
```
```  5436   show ?thesis unfolding homeomorphism_def homeomorphic_def
```
```  5437     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
```
```  5438 qed
```
```  5439
```
```  5440 lemma homeomorphic_compact:
```
```  5441  "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
```
```  5442           \<Longrightarrow> s homeomorphic t"
```
```  5443   unfolding homeomorphic_def by(metis homeomorphism_compact)
```
```  5444
```
```  5445 text{* Preservation of topological properties.                                   *}
```
```  5446
```
```  5447 lemma homeomorphic_compactness:
```
```  5448  "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
```
```  5449 unfolding homeomorphic_def homeomorphism_def
```
```  5450 by (metis compact_continuous_image)
```
```  5451
```
```  5452 text{* Results on translation, scaling etc.                                      *}
```
```  5453
```
```  5454 lemma homeomorphic_scaling:
```
```  5455   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. c *s x) ` s)"
```
```  5456   unfolding homeomorphic_minimal
```
```  5457   apply(rule_tac x="\<lambda>x. c *s x" in exI)
```
```  5458   apply(rule_tac x="\<lambda>x. (1 / c) *s x" in exI)
```
```  5459   apply auto unfolding vector_smult_assoc using assms apply auto
```
```  5460   using continuous_on_cmul[OF continuous_on_id] by auto
```
```  5461
```
```  5462 lemma homeomorphic_translation:
```
```  5463  "s homeomorphic ((\<lambda>x. a + x) ` s)"
```
```  5464   unfolding homeomorphic_minimal
```
```  5465   apply(rule_tac x="\<lambda>x. a + x" in exI)
```
```  5466   apply(rule_tac x="\<lambda>x. -a + x" in exI)
```
```  5467   using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
```
```  5468
```
```  5469 lemma homeomorphic_affinity:
```
```  5470   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. a + c *s x) ` s)"
```
```  5471 proof-
```
```  5472   have *:"op + a ` op *s c ` s = (\<lambda>x. a + c *s x) ` s" by auto
```
```  5473   show ?thesis
```
```  5474     using homeomorphic_trans
```
```  5475     using homeomorphic_scaling[OF assms, of s]
```
```  5476     using homeomorphic_translation[of "(\<lambda>x. c *s x) ` s" a] unfolding * by auto
```
```  5477 qed
```
```  5478
```
```  5479 lemma homeomorphic_balls: fixes a b ::"real^'a::finite"
```
```  5480   assumes "0 < d"  "0 < e"
```
```  5481   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
```
```  5482         "(cball a d) homeomorphic (cball b e)" (is ?cth)
```
```  5483 proof-
```
```  5484   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
```
```  5485   show ?th unfolding homeomorphic_minimal
```
```  5486     apply(rule_tac x="\<lambda>x. b + (e/d) *s (x - a)" in exI)
```
```  5487     apply(rule_tac x="\<lambda>x. a + (d/e) *s (x - b)" in exI)
```
```  5488     apply (auto simp add: dist_commute) unfolding dist_norm and vector_smult_assoc using assms apply auto
```
```  5489     unfolding norm_minus_cancel and norm_mul
```
```  5490     using continuous_on_add[OF continuous_on_const continuous_on_cmul[OF continuous_on_sub[OF continuous_on_id continuous_on_const]]]
```
```  5491     apply (auto simp add: dist_commute)
```
```  5492     using pos_less_divide_eq[OF *(1), THEN sym] unfolding real_mult_commute[of _ "\<bar>e / d\<bar>"]
```
```  5493     using pos_less_divide_eq[OF *(2), THEN sym] unfolding real_mult_commute[of _ "\<bar>d / e\<bar>"]
```
```  5494     by (auto simp add: dist_commute)
```
```  5495 next
```
```  5496   have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
```
```  5497   show ?cth unfolding homeomorphic_minimal
```
```  5498     apply(rule_tac x="\<lambda>x. b + (e/d) *s (x - a)" in exI)
```
```  5499     apply(rule_tac x="\<lambda>x. a + (d/e) *s (x - b)" in exI)
```
```  5500     apply (auto simp add: dist_commute) unfolding dist_norm and vector_smult_assoc using assms apply auto
```
```  5501     unfolding norm_minus_cancel and norm_mul
```
```  5502     using continuous_on_add[OF continuous_on_const continuous_on_cmul[OF continuous_on_sub[OF continuous_on_id continuous_on_const]]]
```
```  5503     apply auto
```
```  5504     using pos_le_divide_eq[OF *(1), THEN sym] unfolding real_mult_commute[of _ "\<bar>e / d\<bar>"]
```
```  5505     using pos_le_divide_eq[OF *(2), THEN sym] unfolding real_mult_commute[of _ "\<bar>d / e\<bar>"]
```
```  5506     by auto
```
```  5507 qed
```
```  5508
```
```  5509 text{* "Isometry" (up to constant bounds) of injective linear map etc.           *}
```
```  5510
```
```  5511 lemma cauchy_isometric:
```
```  5512   fixes x :: "nat \<Rightarrow> real ^ 'n::finite"
```
```  5513   assumes e:"0 < e" and s:"subspace s" and f:"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)"
```
```  5514   shows "Cauchy x"
```
```  5515 proof-
```
```  5516   { fix d::real assume "d>0"
```
```  5517     then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
```
```  5518       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
```
```  5519     { fix n assume "n\<ge>N"
```
```  5520       hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding linear_sub[OF f, THEN sym] by auto
```
```  5521       moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
```
```  5522 	using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
```
```  5523 	using normf[THEN bspec[where x="x n - x N"]] by auto
```
```  5524       ultimately have "norm (x n - x N) < d" using `e>0`
```
```  5525 	using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto   }
```
```  5526     hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
```
```  5527   thus ?thesis unfolding cauchy and dist_norm by auto
```
```  5528 qed
```
```  5529
```
```  5530 lemma complete_isometric_image:
```
```  5531   fixes f :: "real ^ _ \<Rightarrow> real ^ _"
```
```  5532   assumes "0 < e" and s:"subspace s" and f:"linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and cs:"complete s"
```
```  5533   shows "complete(f ` s)"
```
```  5534 proof-
```
```  5535   { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
```
```  5536     then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)" unfolding image_iff and Bex_def
```
```  5537       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
```
```  5538     hence x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)" by auto
```
```  5539     hence "f \<circ> x = g" unfolding expand_fun_eq by auto
```
```  5540     then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
```
```  5541       using cs[unfolded complete_def, THEN spec[where x="x"]]
```
```  5542       using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
```
```  5543     hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
```
```  5544       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
```
```  5545       unfolding `f \<circ> x = g` by auto  }
```
```  5546   thus ?thesis unfolding complete_def by auto
```
```  5547 qed
```
```  5548
```
```  5549 lemma dist_0_norm:
```
```  5550   fixes x :: "'a::real_normed_vector"
```
```  5551   shows "dist 0 x = norm x"
```
```  5552 unfolding dist_norm by simp
```
```  5553
```
```  5554 lemma injective_imp_isometric: fixes f::"real^'m::finite \<Rightarrow> real^'n::finite"
```
```  5555   assumes s:"closed s"  "subspace s"  and f:"linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
```
```  5556   shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
```
```  5557 proof(cases "s \<subseteq> {0::real^'m}")
```
```  5558   case True
```
```  5559   { fix x assume "x \<in> s"
```
```  5560     hence "x = 0" using True by auto
```
```  5561     hence "norm x \<le> norm (f x)" by auto  }
```
```  5562   thus ?thesis by(auto intro!: exI[where x=1])
```
```  5563 next
```
```  5564   case False
```
```  5565   then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
```
```  5566   from False have "s \<noteq> {}" by auto
```
```  5567   let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
```
```  5568   let ?S' = "{x::real^'m. x\<in>s \<and> norm x = norm a}"
```
```  5569   let ?S'' = "{x::real^'m. norm x = norm a}"
```
```  5570
```
```  5571   have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by (auto simp add: norm_minus_cancel)
```
```  5572   hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
```
```  5573   moreover have "?S' = s \<inter> ?S''" by auto
```
```  5574   ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
```
```  5575   moreover have *:"f ` ?S' = ?S" by auto
```
```  5576   ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
```
```  5577   hence "closed ?S" using compact_imp_closed by auto
```
```  5578   moreover have "?S \<noteq> {}" using a by auto
```
```  5579   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
```
```  5580   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
```
```  5581
```
```  5582   let ?e = "norm (f b) / norm b"
```
```  5583   have "norm b > 0" using ba and a and norm_ge_zero by auto
```
```  5584   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
```
```  5585   ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
```
```  5586   moreover
```
```  5587   { fix x assume "x\<in>s"
```
```  5588     hence "norm (f b) / norm b * norm x \<le> norm (f x)"
```
```  5589     proof(cases "x=0")
```
```  5590       case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
```
```  5591     next
```
```  5592       case False
```
```  5593       hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
```
```  5594       have "\<forall>c. \<forall>x\<in>s. c *s x \<in> s" using s[unfolded subspace_def] by auto
```
```  5595       hence "(norm a / norm x) *s x \<in> {x \<in> s. norm x = norm a}" using `x\<in>s` and `x\<noteq>0` by auto
```
```  5596       thus "norm (f b) / norm b * norm x \<le> norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *s x"]]
```
```  5597 	unfolding linear_cmul[OF f(1)] and norm_mul and ba using `x\<noteq>0` `a\<noteq>0`
```
```  5598 	by (auto simp add: real_mult_commute pos_le_divide_eq pos_divide_le_eq)
```
```  5599     qed }
```
```  5600   ultimately
```
```  5601   show ?thesis by auto
```
```  5602 qed
```
```  5603
```
```  5604 lemma closed_injective_image_subspace:
```
```  5605   fixes s :: "(real ^ _) set"
```
```  5606   assumes "subspace s" "linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
```
```  5607   shows "closed(f ` s)"
```
```  5608 proof-
```
```  5609   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
```
```  5610   show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
```
```  5611     unfolding complete_eq_closed[THEN sym] by auto
```
```  5612 qed
```
```  5613
```
```  5614 subsection{* Some properties of a canonical subspace.                                  *}
```
```  5615
```
```  5616 lemma subspace_substandard:
```
```  5617  "subspace {x::real^'n. (\<forall>i. P i \<longrightarrow> x\$i = 0)}"
```
```  5618   unfolding subspace_def by(auto simp add: vector_add_component vector_smult_component elim!: ballE)
```
```  5619
```
```  5620 lemma closed_substandard:
```
```  5621  "closed {x::real^'n::finite. \<forall>i. P i --> x\$i = 0}" (is "closed ?A")
```
```  5622 proof-
```
```  5623   let ?D = "{i. P i}"
```
```  5624   let ?Bs = "{{x::real^'n. basis i \<bullet> x = 0}| i. i \<in> ?D}"
```
```  5625   { fix x
```
```  5626     { assume "x\<in>?A"
```
```  5627       hence x:"\<forall>i\<in>?D. x \$ i = 0" by auto
```
```  5628       hence "x\<in> \<Inter> ?Bs" by(auto simp add: dot_basis x) }
```
```  5629     moreover
```
```  5630     { assume x:"x\<in>\<Inter>?Bs"
```
```  5631       { fix i assume i:"i \<in> ?D"
```
```  5632 	then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. basis i \<bullet> x = 0}" by auto
```
```  5633 	hence "x \$ i = 0" unfolding B using x unfolding dot_basis by auto  }
```
```  5634       hence "x\<in>?A" by auto }
```
```  5635     ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
```
```  5636   hence "?A = \<Inter> ?Bs" by auto
```
```  5637   thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
```
```  5638 qed
```
```  5639
```
```  5640 lemma dim_substandard:
```
```  5641   shows "dim {x::real^'n::finite. \<forall>i. i \<notin> d \<longrightarrow> x\$i = 0} = card d" (is "dim ?A = _")
```
```  5642 proof-
```
```  5643   let ?D = "UNIV::'n set"
```
```  5644   let ?B = "(basis::'n\<Rightarrow>real^'n) ` d"
```
```  5645
```
```  5646     let ?bas = "basis::'n \<Rightarrow> real^'n"
```
```  5647
```
```  5648   have "?B \<subseteq> ?A" by auto
```
```  5649
```
```  5650   moreover
```
```  5651   { fix x::"real^'n" assume "x\<in>?A"
```
```  5652     with finite[of d]
```
```  5653     have "x\<in> span ?B"
```
```  5654     proof(induct d arbitrary: x)
```