1.1 --- a/src/HOL/Library/Topology_Euclidean_Space.thy Fri Oct 23 14:33:07 2009 +0200
1.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,6027 +0,0 @@
1.4 -(* Title: HOL/Library/Topology_Euclidian_Space.thy
1.5 - Author: Amine Chaieb, University of Cambridge
1.6 - Author: Robert Himmelmann, TU Muenchen
1.7 -*)
1.8 -
1.9 -header {* Elementary topology in Euclidean space. *}
1.10 -
1.11 -theory Topology_Euclidean_Space
1.12 -imports SEQ Euclidean_Space Product_Vector
1.13 -begin
1.14 -
1.15 -declare fstcart_pastecart[simp] sndcart_pastecart[simp]
1.16 -
1.17 -subsection{* General notion of a topology *}
1.18 -
1.19 -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)"
1.20 -typedef (open) 'a topology = "{L::('a set) set. istopology L}"
1.21 - morphisms "openin" "topology"
1.22 - unfolding istopology_def by blast
1.23 -
1.24 -lemma istopology_open_in[intro]: "istopology(openin U)"
1.25 - using openin[of U] by blast
1.26 -
1.27 -lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
1.28 - using topology_inverse[unfolded mem_def Collect_def] .
1.29 -
1.30 -lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
1.31 - using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
1.32 -
1.33 -lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
1.34 -proof-
1.35 - {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
1.36 - moreover
1.37 - {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
1.38 - hence "openin T1 = openin T2" by (metis mem_def set_ext)
1.39 - hence "topology (openin T1) = topology (openin T2)" by simp
1.40 - hence "T1 = T2" unfolding openin_inverse .}
1.41 - ultimately show ?thesis by blast
1.42 -qed
1.43 -
1.44 -text{* Infer the "universe" from union of all sets in the topology. *}
1.45 -
1.46 -definition "topspace T = \<Union>{S. openin T S}"
1.47 -
1.48 -subsection{* Main properties of open sets *}
1.49 -
1.50 -lemma openin_clauses:
1.51 - fixes U :: "'a topology"
1.52 - shows "openin U {}"
1.53 - "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
1.54 - "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
1.55 - using openin[of U] unfolding istopology_def Collect_def mem_def
1.56 - by (metis mem_def subset_eq)+
1.57 -
1.58 -lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
1.59 - unfolding topspace_def by blast
1.60 -lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
1.61 -
1.62 -lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
1.63 - by (simp add: openin_clauses)
1.64 -
1.65 -lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
1.66 -
1.67 -lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
1.68 - using openin_Union[of "{S,T}" U] by auto
1.69 -
1.70 -lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
1.71 -
1.72 -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")
1.73 -proof-
1.74 - {assume ?lhs then have ?rhs by auto }
1.75 - moreover
1.76 - {assume H: ?rhs
1.77 - then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"
1.78 - unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast
1.79 - from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
1.80 - have "\<Union> t`S = S" using t by auto
1.81 - with openin_Union[OF th0] have "openin U S" by simp }
1.82 - ultimately show ?thesis by blast
1.83 -qed
1.84 -
1.85 -subsection{* Closed sets *}
1.86 -
1.87 -definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
1.88 -
1.89 -lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
1.90 -lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
1.91 -lemma closedin_topspace[intro,simp]:
1.92 - "closedin U (topspace U)" by (simp add: closedin_def)
1.93 -lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
1.94 - by (auto simp add: Diff_Un closedin_def)
1.95 -
1.96 -lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
1.97 -lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
1.98 - shows "closedin U (\<Inter> K)" using Ke Kc unfolding closedin_def Diff_Inter by auto
1.99 -
1.100 -lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
1.101 - using closedin_Inter[of "{S,T}" U] by auto
1.102 -
1.103 -lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
1.104 -lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
1.105 - apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
1.106 - apply (metis openin_subset subset_eq)
1.107 - done
1.108 -
1.109 -lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
1.110 - by (simp add: openin_closedin_eq)
1.111 -
1.112 -lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
1.113 -proof-
1.114 - have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
1.115 - by (auto simp add: topspace_def openin_subset)
1.116 - then show ?thesis using oS cT by (auto simp add: closedin_def)
1.117 -qed
1.118 -
1.119 -lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
1.120 -proof-
1.121 - have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S] oS cT
1.122 - by (auto simp add: topspace_def )
1.123 - then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
1.124 -qed
1.125 -
1.126 -subsection{* Subspace topology. *}
1.127 -
1.128 -definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
1.129 -
1.130 -lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
1.131 -proof-
1.132 - have "{} \<in> ?L" by blast
1.133 - {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
1.134 - 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
1.135 - have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)" using Sa Sb by blast+
1.136 - then have "A \<inter> B \<in> ?L" by blast}
1.137 - moreover
1.138 - {fix K assume K: "K \<subseteq> ?L"
1.139 - have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
1.140 - apply (rule set_ext)
1.141 - apply (simp add: Ball_def image_iff)
1.142 - by (metis mem_def)
1.143 - from K[unfolded th0 subset_image_iff]
1.144 - obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
1.145 - have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
1.146 - moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
1.147 - ultimately have "\<Union>K \<in> ?L" by blast}
1.148 - ultimately show ?thesis unfolding istopology_def by blast
1.149 -qed
1.150 -
1.151 -lemma openin_subtopology:
1.152 - "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
1.153 - unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
1.154 - by (auto simp add: Collect_def)
1.155 -
1.156 -lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
1.157 - by (auto simp add: topspace_def openin_subtopology)
1.158 -
1.159 -lemma closedin_subtopology:
1.160 - "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
1.161 - unfolding closedin_def topspace_subtopology
1.162 - apply (simp add: openin_subtopology)
1.163 - apply (rule iffI)
1.164 - apply clarify
1.165 - apply (rule_tac x="topspace U - T" in exI)
1.166 - by auto
1.167 -
1.168 -lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
1.169 - unfolding openin_subtopology
1.170 - apply (rule iffI, clarify)
1.171 - apply (frule openin_subset[of U]) apply blast
1.172 - apply (rule exI[where x="topspace U"])
1.173 - by auto
1.174 -
1.175 -lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
1.176 - shows "subtopology U V = U"
1.177 -proof-
1.178 - {fix S
1.179 - {fix T assume T: "openin U T" "S = T \<inter> V"
1.180 - from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
1.181 - have "openin U S" unfolding eq using T by blast}
1.182 - moreover
1.183 - {assume S: "openin U S"
1.184 - hence "\<exists>T. openin U T \<and> S = T \<inter> V"
1.185 - using openin_subset[OF S] UV by auto}
1.186 - ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
1.187 - then show ?thesis unfolding topology_eq openin_subtopology by blast
1.188 -qed
1.189 -
1.190 -
1.191 -lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
1.192 - by (simp add: subtopology_superset)
1.193 -
1.194 -lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
1.195 - by (simp add: subtopology_superset)
1.196 -
1.197 -subsection{* The universal Euclidean versions are what we use most of the time *}
1.198 -
1.199 -definition
1.200 - euclidean :: "'a::topological_space topology" where
1.201 - "euclidean = topology open"
1.202 -
1.203 -lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
1.204 - unfolding euclidean_def
1.205 - apply (rule cong[where x=S and y=S])
1.206 - apply (rule topology_inverse[symmetric])
1.207 - apply (auto simp add: istopology_def)
1.208 - by (auto simp add: mem_def subset_eq)
1.209 -
1.210 -lemma topspace_euclidean: "topspace euclidean = UNIV"
1.211 - apply (simp add: topspace_def)
1.212 - apply (rule set_ext)
1.213 - by (auto simp add: open_openin[symmetric])
1.214 -
1.215 -lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
1.216 - by (simp add: topspace_euclidean topspace_subtopology)
1.217 -
1.218 -lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
1.219 - by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
1.220 -
1.221 -lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
1.222 - by (simp add: open_openin openin_subopen[symmetric])
1.223 -
1.224 -subsection{* Open and closed balls. *}
1.225 -
1.226 -definition
1.227 - ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
1.228 - "ball x e = {y. dist x y < e}"
1.229 -
1.230 -definition
1.231 - cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
1.232 - "cball x e = {y. dist x y \<le> e}"
1.233 -
1.234 -lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
1.235 -lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
1.236 -
1.237 -lemma mem_ball_0 [simp]:
1.238 - fixes x :: "'a::real_normed_vector"
1.239 - shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
1.240 - by (simp add: dist_norm)
1.241 -
1.242 -lemma mem_cball_0 [simp]:
1.243 - fixes x :: "'a::real_normed_vector"
1.244 - shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
1.245 - by (simp add: dist_norm)
1.246 -
1.247 -lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e" by simp
1.248 -lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
1.249 -lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
1.250 -lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
1.251 -lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
1.252 - by (simp add: expand_set_eq) arith
1.253 -
1.254 -lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
1.255 - by (simp add: expand_set_eq)
1.256 -
1.257 -subsection{* Topological properties of open balls *}
1.258 -
1.259 -lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
1.260 - "(a::real) - b < 0 \<longleftrightarrow> a < b"
1.261 - "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
1.262 -lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
1.263 - "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b" by arith+
1.264 -
1.265 -lemma open_ball[intro, simp]: "open (ball x e)"
1.266 - unfolding open_dist ball_def Collect_def Ball_def mem_def
1.267 - unfolding dist_commute
1.268 - apply clarify
1.269 - apply (rule_tac x="e - dist xa x" in exI)
1.270 - using dist_triangle_alt[where z=x]
1.271 - apply (clarsimp simp add: diff_less_iff)
1.272 - apply atomize
1.273 - apply (erule_tac x="y" in allE)
1.274 - apply (erule_tac x="xa" in allE)
1.275 - by arith
1.276 -
1.277 -lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
1.278 -lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
1.279 - unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
1.280 -
1.281 -lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
1.282 - by (metis open_contains_ball subset_eq centre_in_ball)
1.283 -
1.284 -lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
1.285 - unfolding mem_ball expand_set_eq
1.286 - apply (simp add: not_less)
1.287 - by (metis zero_le_dist order_trans dist_self)
1.288 -
1.289 -lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
1.290 -
1.291 -subsection{* Basic "localization" results are handy for connectedness. *}
1.292 -
1.293 -lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
1.294 - by (auto simp add: openin_subtopology open_openin[symmetric])
1.295 -
1.296 -lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
1.297 - by (auto simp add: openin_open)
1.298 -
1.299 -lemma open_openin_trans[trans]:
1.300 - "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
1.301 - by (metis Int_absorb1 openin_open_Int)
1.302 -
1.303 -lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
1.304 - by (auto simp add: openin_open)
1.305 -
1.306 -lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
1.307 - by (simp add: closedin_subtopology closed_closedin Int_ac)
1.308 -
1.309 -lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
1.310 - by (metis closedin_closed)
1.311 -
1.312 -lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
1.313 - apply (subgoal_tac "S \<inter> T = T" )
1.314 - apply auto
1.315 - apply (frule closedin_closed_Int[of T S])
1.316 - by simp
1.317 -
1.318 -lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
1.319 - by (auto simp add: closedin_closed)
1.320 -
1.321 -lemma openin_euclidean_subtopology_iff:
1.322 - fixes S U :: "'a::metric_space set"
1.323 - shows "openin (subtopology euclidean U) S
1.324 - \<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")
1.325 -proof-
1.326 - {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
1.327 - by (simp add: open_dist) blast}
1.328 - moreover
1.329 - {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"
1.330 - 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)"
1.331 - by metis
1.332 - let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
1.333 - have oT: "open ?T" by auto
1.334 - { fix x assume "x\<in>S"
1.335 - hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
1.336 - apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
1.337 - by (rule d [THEN conjunct1])
1.338 - hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto }
1.339 - moreover
1.340 - { fix y assume "y\<in>?T"
1.341 - then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
1.342 - then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
1.343 - assume "y\<in>U"
1.344 - hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
1.345 - ultimately have "S = ?T \<inter> U" by blast
1.346 - with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
1.347 - ultimately show ?thesis by blast
1.348 -qed
1.349 -
1.350 -text{* These "transitivity" results are handy too. *}
1.351 -
1.352 -lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
1.353 - \<Longrightarrow> openin (subtopology euclidean U) S"
1.354 - unfolding open_openin openin_open by blast
1.355 -
1.356 -lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
1.357 - by (auto simp add: openin_open intro: openin_trans)
1.358 -
1.359 -lemma closedin_trans[trans]:
1.360 - "closedin (subtopology euclidean T) S \<Longrightarrow>
1.361 - closedin (subtopology euclidean U) T
1.362 - ==> closedin (subtopology euclidean U) S"
1.363 - by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
1.364 -
1.365 -lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
1.366 - by (auto simp add: closedin_closed intro: closedin_trans)
1.367 -
1.368 -subsection{* Connectedness *}
1.369 -
1.370 -definition "connected S \<longleftrightarrow>
1.371 - ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
1.372 - \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
1.373 -
1.374 -lemma connected_local:
1.375 - "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
1.376 - openin (subtopology euclidean S) e1 \<and>
1.377 - openin (subtopology euclidean S) e2 \<and>
1.378 - S \<subseteq> e1 \<union> e2 \<and>
1.379 - e1 \<inter> e2 = {} \<and>
1.380 - ~(e1 = {}) \<and>
1.381 - ~(e2 = {}))"
1.382 -unfolding connected_def openin_open by (safe, blast+)
1.383 -
1.384 -lemma exists_diff: "(\<exists>S. P(UNIV - S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
1.385 -proof-
1.386 -
1.387 - {assume "?lhs" hence ?rhs by blast }
1.388 - moreover
1.389 - {fix S assume H: "P S"
1.390 - have "S = UNIV - (UNIV - S)" by auto
1.391 - with H have "P (UNIV - (UNIV - S))" by metis }
1.392 - ultimately show ?thesis by metis
1.393 -qed
1.394 -
1.395 -lemma connected_clopen: "connected S \<longleftrightarrow>
1.396 - (\<forall>T. openin (subtopology euclidean S) T \<and>
1.397 - closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
1.398 -proof-
1.399 - 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> {})"
1.400 - unfolding connected_def openin_open closedin_closed
1.401 - apply (subst exists_diff) by blast
1.402 - 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> {})"
1.403 - (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def Compl_eq_Diff_UNIV) by metis
1.404 -
1.405 - have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
1.406 - (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
1.407 - unfolding connected_def openin_open closedin_closed by auto
1.408 - {fix e2
1.409 - {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
1.410 - by auto}
1.411 - then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
1.412 - then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
1.413 - then show ?thesis unfolding th0 th1 by simp
1.414 -qed
1.415 -
1.416 -lemma connected_empty[simp, intro]: "connected {}"
1.417 - by (simp add: connected_def)
1.418 -
1.419 -subsection{* Hausdorff and other separation properties *}
1.420 -
1.421 -class t0_space =
1.422 - assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
1.423 -
1.424 -class t1_space =
1.425 - 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"
1.426 -begin
1.427 -
1.428 -subclass t0_space
1.429 -proof
1.430 -qed (fast dest: t1_space)
1.431 -
1.432 -end
1.433 -
1.434 -text {* T2 spaces are also known as Hausdorff spaces. *}
1.435 -
1.436 -class t2_space =
1.437 - 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 = {}"
1.438 -begin
1.439 -
1.440 -subclass t1_space
1.441 -proof
1.442 -qed (fast dest: hausdorff)
1.443 -
1.444 -end
1.445 -
1.446 -instance metric_space \<subseteq> t2_space
1.447 -proof
1.448 - fix x y :: "'a::metric_space"
1.449 - assume xy: "x \<noteq> y"
1.450 - let ?U = "ball x (dist x y / 2)"
1.451 - let ?V = "ball y (dist x y / 2)"
1.452 - have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
1.453 - ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
1.454 - have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
1.455 - using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
1.456 - by (auto simp add: expand_set_eq)
1.457 - then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
1.458 - by blast
1.459 -qed
1.460 -
1.461 -lemma separation_t2:
1.462 - fixes x y :: "'a::t2_space"
1.463 - 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 = {})"
1.464 - using hausdorff[of x y] by blast
1.465 -
1.466 -lemma separation_t1:
1.467 - fixes x y :: "'a::t1_space"
1.468 - 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)"
1.469 - using t1_space[of x y] by blast
1.470 -
1.471 -lemma separation_t0:
1.472 - fixes x y :: "'a::t0_space"
1.473 - shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
1.474 - using t0_space[of x y] by blast
1.475 -
1.476 -subsection{* Limit points *}
1.477 -
1.478 -definition
1.479 - islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
1.480 - (infixr "islimpt" 60) where
1.481 - "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
1.482 -
1.483 -lemma islimptI:
1.484 - assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
1.485 - shows "x islimpt S"
1.486 - using assms unfolding islimpt_def by auto
1.487 -
1.488 -lemma islimptE:
1.489 - assumes "x islimpt S" and "x \<in> T" and "open T"
1.490 - obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
1.491 - using assms unfolding islimpt_def by auto
1.492 -
1.493 -lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
1.494 -
1.495 -lemma islimpt_approachable:
1.496 - fixes x :: "'a::metric_space"
1.497 - shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
1.498 - unfolding islimpt_def
1.499 - apply auto
1.500 - apply(erule_tac x="ball x e" in allE)
1.501 - apply auto
1.502 - apply(rule_tac x=y in bexI)
1.503 - apply (auto simp add: dist_commute)
1.504 - apply (simp add: open_dist, drule (1) bspec)
1.505 - apply (clarify, drule spec, drule (1) mp, auto)
1.506 - done
1.507 -
1.508 -lemma islimpt_approachable_le:
1.509 - fixes x :: "'a::metric_space"
1.510 - shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
1.511 - unfolding islimpt_approachable
1.512 - using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
1.513 - by metis (* FIXME: VERY slow! *)
1.514 -
1.515 -class perfect_space =
1.516 - (* FIXME: perfect_space should inherit from topological_space *)
1.517 - assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"
1.518 -
1.519 -lemma perfect_choose_dist:
1.520 - fixes x :: "'a::perfect_space"
1.521 - shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
1.522 -using islimpt_UNIV [of x]
1.523 -by (simp add: islimpt_approachable)
1.524 -
1.525 -instance real :: perfect_space
1.526 -apply default
1.527 -apply (rule islimpt_approachable [THEN iffD2])
1.528 -apply (clarify, rule_tac x="x + e/2" in bexI)
1.529 -apply (auto simp add: dist_norm)
1.530 -done
1.531 -
1.532 -instance "^" :: (perfect_space, finite) perfect_space
1.533 -proof
1.534 - fix x :: "'a ^ 'b"
1.535 - {
1.536 - fix e :: real assume "0 < e"
1.537 - def a \<equiv> "x $ undefined"
1.538 - have "a islimpt UNIV" by (rule islimpt_UNIV)
1.539 - with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
1.540 - unfolding islimpt_approachable by auto
1.541 - def y \<equiv> "Cart_lambda ((Cart_nth x)(undefined := b))"
1.542 - from `b \<noteq> a` have "y \<noteq> x"
1.543 - unfolding a_def y_def by (simp add: Cart_eq)
1.544 - from `dist b a < e` have "dist y x < e"
1.545 - unfolding dist_vector_def a_def y_def
1.546 - apply simp
1.547 - apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
1.548 - apply (subst setsum_diff1' [where a=undefined], simp, simp, simp)
1.549 - done
1.550 - from `y \<noteq> x` and `dist y x < e`
1.551 - have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
1.552 - }
1.553 - then show "x islimpt UNIV" unfolding islimpt_approachable by blast
1.554 -qed
1.555 -
1.556 -lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
1.557 - unfolding closed_def
1.558 - apply (subst open_subopen)
1.559 - apply (simp add: islimpt_def subset_eq Compl_eq_Diff_UNIV)
1.560 - by (metis DiffE DiffI UNIV_I insertCI insert_absorb mem_def)
1.561 -
1.562 -lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
1.563 - unfolding islimpt_def by auto
1.564 -
1.565 -lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x$i}"
1.566 -proof-
1.567 - let ?U = "UNIV :: 'n set"
1.568 - let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
1.569 - {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
1.570 - and xi: "x$i < 0"
1.571 - from xi have th0: "-x$i > 0" by arith
1.572 - from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
1.573 - have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
1.574 - have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
1.575 - have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
1.576 - apply (simp only: vector_component)
1.577 - by (rule th') auto
1.578 - have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using component_le_norm[of "x'-x" i]
1.579 - apply (simp add: dist_norm) by norm
1.580 - from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
1.581 - then show ?thesis unfolding closed_limpt islimpt_approachable
1.582 - unfolding not_le[symmetric] by blast
1.583 -qed
1.584 -
1.585 -lemma finite_set_avoid:
1.586 - fixes a :: "'a::metric_space"
1.587 - assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
1.588 -proof(induct rule: finite_induct[OF fS])
1.589 - case 1 thus ?case apply auto by ferrack
1.590 -next
1.591 - case (2 x F)
1.592 - from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
1.593 - {assume "x = a" hence ?case using d by auto }
1.594 - moreover
1.595 - {assume xa: "x\<noteq>a"
1.596 - let ?d = "min d (dist a x)"
1.597 - have dp: "?d > 0" using xa d(1) using dist_nz by auto
1.598 - from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
1.599 - with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
1.600 - ultimately show ?case by blast
1.601 -qed
1.602 -
1.603 -lemma islimpt_finite:
1.604 - fixes S :: "'a::metric_space set"
1.605 - assumes fS: "finite S" shows "\<not> a islimpt S"
1.606 - unfolding islimpt_approachable
1.607 - using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)
1.608 -
1.609 -lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
1.610 - apply (rule iffI)
1.611 - defer
1.612 - apply (metis Un_upper1 Un_upper2 islimpt_subset)
1.613 - unfolding islimpt_def
1.614 - apply (rule ccontr, clarsimp, rename_tac A B)
1.615 - apply (drule_tac x="A \<inter> B" in spec)
1.616 - apply (auto simp add: open_Int)
1.617 - done
1.618 -
1.619 -lemma discrete_imp_closed:
1.620 - fixes S :: "'a::metric_space set"
1.621 - assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
1.622 - shows "closed S"
1.623 -proof-
1.624 - {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
1.625 - from e have e2: "e/2 > 0" by arith
1.626 - from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
1.627 - let ?m = "min (e/2) (dist x y) "
1.628 - from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
1.629 - from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
1.630 - have th: "dist z y < e" using z y
1.631 - by (intro dist_triangle_lt [where z=x], simp)
1.632 - from d[rule_format, OF y(1) z(1) th] y z
1.633 - have False by (auto simp add: dist_commute)}
1.634 - then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
1.635 -qed
1.636 -
1.637 -subsection{* Interior of a Set *}
1.638 -definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
1.639 -
1.640 -lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
1.641 - apply (simp add: expand_set_eq interior_def)
1.642 - apply (subst (2) open_subopen) by (safe, blast+)
1.643 -
1.644 -lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
1.645 -
1.646 -lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
1.647 -
1.648 -lemma open_interior[simp, intro]: "open(interior S)"
1.649 - apply (simp add: interior_def)
1.650 - apply (subst open_subopen) by blast
1.651 -
1.652 -lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
1.653 -lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
1.654 -lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
1.655 -lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
1.656 -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"
1.657 - by (metis equalityI interior_maximal interior_subset open_interior)
1.658 -lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
1.659 - apply (simp add: interior_def)
1.660 - by (metis open_contains_ball centre_in_ball open_ball subset_trans)
1.661 -
1.662 -lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
1.663 - by (metis interior_maximal interior_subset subset_trans)
1.664 -
1.665 -lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
1.666 - apply (rule equalityI, simp)
1.667 - apply (metis Int_lower1 Int_lower2 subset_interior)
1.668 - by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
1.669 -
1.670 -lemma interior_limit_point [intro]:
1.671 - fixes x :: "'a::perfect_space"
1.672 - assumes x: "x \<in> interior S" shows "x islimpt S"
1.673 -proof-
1.674 - from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
1.675 - unfolding mem_interior subset_eq Ball_def mem_ball by blast
1.676 - {
1.677 - fix d::real assume d: "d>0"
1.678 - let ?m = "min d e"
1.679 - have mde2: "0 < ?m" using e(1) d(1) by simp
1.680 - from perfect_choose_dist [OF mde2, of x]
1.681 - obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
1.682 - then have "dist y x < e" "dist y x < d" by simp_all
1.683 - from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
1.684 - have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
1.685 - using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
1.686 - }
1.687 - then show ?thesis unfolding islimpt_approachable by blast
1.688 -qed
1.689 -
1.690 -lemma interior_closed_Un_empty_interior:
1.691 - assumes cS: "closed S" and iT: "interior T = {}"
1.692 - shows "interior(S \<union> T) = interior S"
1.693 -proof
1.694 - show "interior S \<subseteq> interior (S\<union>T)"
1.695 - by (rule subset_interior, blast)
1.696 -next
1.697 - show "interior (S \<union> T) \<subseteq> interior S"
1.698 - proof
1.699 - fix x assume "x \<in> interior (S \<union> T)"
1.700 - then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
1.701 - unfolding interior_def by fast
1.702 - show "x \<in> interior S"
1.703 - proof (rule ccontr)
1.704 - assume "x \<notin> interior S"
1.705 - with `x \<in> R` `open R` obtain y where "y \<in> R - S"
1.706 - unfolding interior_def expand_set_eq by fast
1.707 - from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
1.708 - from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
1.709 - from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
1.710 - show "False" unfolding interior_def by fast
1.711 - qed
1.712 - qed
1.713 -qed
1.714 -
1.715 -
1.716 -subsection{* Closure of a Set *}
1.717 -
1.718 -definition "closure S = S \<union> {x | x. x islimpt S}"
1.719 -
1.720 -lemma closure_interior: "closure S = UNIV - interior (UNIV - S)"
1.721 -proof-
1.722 - { fix x
1.723 - have "x\<in>UNIV - interior (UNIV - S) \<longleftrightarrow> x \<in> closure S" (is "?lhs = ?rhs")
1.724 - proof
1.725 - let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> UNIV - S)"
1.726 - assume "?lhs"
1.727 - hence *:"\<not> ?exT x"
1.728 - unfolding interior_def
1.729 - by simp
1.730 - { assume "\<not> ?rhs"
1.731 - hence False using *
1.732 - unfolding closure_def islimpt_def
1.733 - by blast
1.734 - }
1.735 - thus "?rhs"
1.736 - by blast
1.737 - next
1.738 - assume "?rhs" thus "?lhs"
1.739 - unfolding closure_def interior_def islimpt_def
1.740 - by blast
1.741 - qed
1.742 - }
1.743 - thus ?thesis
1.744 - by blast
1.745 -qed
1.746 -
1.747 -lemma interior_closure: "interior S = UNIV - (closure (UNIV - S))"
1.748 -proof-
1.749 - { fix x
1.750 - have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
1.751 - unfolding interior_def closure_def islimpt_def
1.752 - by blast (* FIXME: VERY slow! *)
1.753 - }
1.754 - thus ?thesis
1.755 - by blast
1.756 -qed
1.757 -
1.758 -lemma closed_closure[simp, intro]: "closed (closure S)"
1.759 -proof-
1.760 - have "closed (UNIV - interior (UNIV -S))" by blast
1.761 - thus ?thesis using closure_interior[of S] by simp
1.762 -qed
1.763 -
1.764 -lemma closure_hull: "closure S = closed hull S"
1.765 -proof-
1.766 - have "S \<subseteq> closure S"
1.767 - unfolding closure_def
1.768 - by blast
1.769 - moreover
1.770 - have "closed (closure S)"
1.771 - using closed_closure[of S]
1.772 - by assumption
1.773 - moreover
1.774 - { fix t
1.775 - assume *:"S \<subseteq> t" "closed t"
1.776 - { fix x
1.777 - assume "x islimpt S"
1.778 - hence "x islimpt t" using *(1)
1.779 - using islimpt_subset[of x, of S, of t]
1.780 - by blast
1.781 - }
1.782 - with * have "closure S \<subseteq> t"
1.783 - unfolding closure_def
1.784 - using closed_limpt[of t]
1.785 - by auto
1.786 - }
1.787 - ultimately show ?thesis
1.788 - using hull_unique[of S, of "closure S", of closed]
1.789 - unfolding mem_def
1.790 - by simp
1.791 -qed
1.792 -
1.793 -lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
1.794 - unfolding closure_hull
1.795 - using hull_eq[of closed, unfolded mem_def, OF closed_Inter, of S]
1.796 - by (metis mem_def subset_eq)
1.797 -
1.798 -lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
1.799 - using closure_eq[of S]
1.800 - by simp
1.801 -
1.802 -lemma closure_closure[simp]: "closure (closure S) = closure S"
1.803 - unfolding closure_hull
1.804 - using hull_hull[of closed S]
1.805 - by assumption
1.806 -
1.807 -lemma closure_subset: "S \<subseteq> closure S"
1.808 - unfolding closure_hull
1.809 - using hull_subset[of S closed]
1.810 - by assumption
1.811 -
1.812 -lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
1.813 - unfolding closure_hull
1.814 - using hull_mono[of S T closed]
1.815 - by assumption
1.816 -
1.817 -lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
1.818 - using hull_minimal[of S T closed]
1.819 - unfolding closure_hull mem_def
1.820 - by simp
1.821 -
1.822 -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"
1.823 - using hull_unique[of S T closed]
1.824 - unfolding closure_hull mem_def
1.825 - by simp
1.826 -
1.827 -lemma closure_empty[simp]: "closure {} = {}"
1.828 - using closed_empty closure_closed[of "{}"]
1.829 - by simp
1.830 -
1.831 -lemma closure_univ[simp]: "closure UNIV = UNIV"
1.832 - using closure_closed[of UNIV]
1.833 - by simp
1.834 -
1.835 -lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
1.836 - using closure_empty closure_subset[of S]
1.837 - by blast
1.838 -
1.839 -lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
1.840 - using closure_eq[of S] closure_subset[of S]
1.841 - by simp
1.842 -
1.843 -lemma open_inter_closure_eq_empty:
1.844 - "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
1.845 - using open_subset_interior[of S "UNIV - T"]
1.846 - using interior_subset[of "UNIV - T"]
1.847 - unfolding closure_interior
1.848 - by auto
1.849 -
1.850 -lemma open_inter_closure_subset:
1.851 - "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
1.852 -proof
1.853 - fix x
1.854 - assume as: "open S" "x \<in> S \<inter> closure T"
1.855 - { assume *:"x islimpt T"
1.856 - have "x islimpt (S \<inter> T)"
1.857 - proof (rule islimptI)
1.858 - fix A
1.859 - assume "x \<in> A" "open A"
1.860 - with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
1.861 - by (simp_all add: open_Int)
1.862 - with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
1.863 - by (rule islimptE)
1.864 - hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
1.865 - by simp_all
1.866 - thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
1.867 - qed
1.868 - }
1.869 - then show "x \<in> closure (S \<inter> T)" using as
1.870 - unfolding closure_def
1.871 - by blast
1.872 -qed
1.873 -
1.874 -lemma closure_complement: "closure(UNIV - S) = UNIV - interior(S)"
1.875 -proof-
1.876 - have "S = UNIV - (UNIV - S)"
1.877 - by auto
1.878 - thus ?thesis
1.879 - unfolding closure_interior
1.880 - by auto
1.881 -qed
1.882 -
1.883 -lemma interior_complement: "interior(UNIV - S) = UNIV - closure(S)"
1.884 - unfolding closure_interior
1.885 - by blast
1.886 -
1.887 -subsection{* Frontier (aka boundary) *}
1.888 -
1.889 -definition "frontier S = closure S - interior S"
1.890 -
1.891 -lemma frontier_closed: "closed(frontier S)"
1.892 - by (simp add: frontier_def closed_Diff)
1.893 -
1.894 -lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(UNIV - S))"
1.895 - by (auto simp add: frontier_def interior_closure)
1.896 -
1.897 -lemma frontier_straddle:
1.898 - fixes a :: "'a::metric_space"
1.899 - 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")
1.900 -proof
1.901 - assume "?lhs"
1.902 - { fix e::real
1.903 - assume "e > 0"
1.904 - let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
1.905 - { assume "a\<in>S"
1.906 - have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
1.907 - moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
1.908 - unfolding frontier_closures closure_def islimpt_def using `e>0`
1.909 - by (auto, erule_tac x="ball a e" in allE, auto)
1.910 - ultimately have ?rhse by auto
1.911 - }
1.912 - moreover
1.913 - { assume "a\<notin>S"
1.914 - hence ?rhse using `?lhs`
1.915 - unfolding frontier_closures closure_def islimpt_def
1.916 - using open_ball[of a e] `e > 0`
1.917 - by (auto, erule_tac x = "ball a e" in allE, auto) (* FIXME: VERY slow! *)
1.918 - }
1.919 - ultimately have ?rhse by auto
1.920 - }
1.921 - thus ?rhs by auto
1.922 -next
1.923 - assume ?rhs
1.924 - moreover
1.925 - { fix T assume "a\<notin>S" and
1.926 - 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"
1.927 - from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
1.928 - then obtain e where "e>0" "ball a e \<subseteq> T" by auto
1.929 - then obtain y where y:"y\<in>S" "dist a y < e" using as(1) by auto
1.930 - have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
1.931 - using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
1.932 - }
1.933 - hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
1.934 - moreover
1.935 - { fix T assume "a \<in> T" "open T" "a\<in>S"
1.936 - then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
1.937 - obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
1.938 - 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
1.939 - }
1.940 - hence "a islimpt (UNIV - S) \<or> a\<notin>S" unfolding islimpt_def by auto
1.941 - ultimately show ?lhs unfolding frontier_closures using closure_def[of "UNIV - S"] by auto
1.942 -qed
1.943 -
1.944 -lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
1.945 - by (metis frontier_def closure_closed Diff_subset)
1.946 -
1.947 -lemma frontier_empty: "frontier {} = {}"
1.948 - by (simp add: frontier_def closure_empty)
1.949 -
1.950 -lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
1.951 -proof-
1.952 - { assume "frontier S \<subseteq> S"
1.953 - hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
1.954 - hence "closed S" using closure_subset_eq by auto
1.955 - }
1.956 - thus ?thesis using frontier_subset_closed[of S] by auto
1.957 -qed
1.958 -
1.959 -lemma frontier_complement: "frontier(UNIV - S) = frontier S"
1.960 - by (auto simp add: frontier_def closure_complement interior_complement)
1.961 -
1.962 -lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
1.963 - using frontier_complement frontier_subset_eq[of "UNIV - S"]
1.964 - unfolding open_closed Compl_eq_Diff_UNIV by auto
1.965 -
1.966 -subsection{* Common nets and The "within" modifier for nets. *}
1.967 -
1.968 -definition
1.969 - at_infinity :: "'a::real_normed_vector net" where
1.970 - "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
1.971 -
1.972 -definition
1.973 - indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
1.974 - "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
1.975 -
1.976 -text{* Prove That They are all nets. *}
1.977 -
1.978 -lemma Rep_net_at_infinity:
1.979 - "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
1.980 -unfolding at_infinity_def
1.981 -apply (rule Abs_net_inverse')
1.982 -apply (rule image_nonempty, simp)
1.983 -apply (clarsimp, rename_tac r s)
1.984 -apply (rule_tac x="max r s" in exI, auto)
1.985 -done
1.986 -
1.987 -lemma within_UNIV: "net within UNIV = net"
1.988 - by (simp add: Rep_net_inject [symmetric] Rep_net_within)
1.989 -
1.990 -subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
1.991 -
1.992 -definition
1.993 - trivial_limit :: "'a net \<Rightarrow> bool" where
1.994 - "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
1.995 -
1.996 -lemma trivial_limit_within:
1.997 - shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
1.998 -proof
1.999 - assume "trivial_limit (at a within S)"
1.1000 - thus "\<not> a islimpt S"
1.1001 - unfolding trivial_limit_def
1.1002 - unfolding Rep_net_within Rep_net_at
1.1003 - unfolding islimpt_def
1.1004 - apply (clarsimp simp add: expand_set_eq)
1.1005 - apply (rename_tac T, rule_tac x=T in exI)
1.1006 - apply (clarsimp, drule_tac x=y in spec, simp)
1.1007 - done
1.1008 -next
1.1009 - assume "\<not> a islimpt S"
1.1010 - thus "trivial_limit (at a within S)"
1.1011 - unfolding trivial_limit_def
1.1012 - unfolding Rep_net_within Rep_net_at
1.1013 - unfolding islimpt_def
1.1014 - apply (clarsimp simp add: image_image)
1.1015 - apply (rule_tac x=T in image_eqI)
1.1016 - apply (auto simp add: expand_set_eq)
1.1017 - done
1.1018 -qed
1.1019 -
1.1020 -lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
1.1021 - using trivial_limit_within [of a UNIV]
1.1022 - by (simp add: within_UNIV)
1.1023 -
1.1024 -lemma trivial_limit_at:
1.1025 - fixes a :: "'a::perfect_space"
1.1026 - shows "\<not> trivial_limit (at a)"
1.1027 - by (simp add: trivial_limit_at_iff)
1.1028 -
1.1029 -lemma trivial_limit_at_infinity:
1.1030 - "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
1.1031 - (* FIXME: find a more appropriate type class *)
1.1032 - unfolding trivial_limit_def Rep_net_at_infinity
1.1033 - apply (clarsimp simp add: expand_set_eq)
1.1034 - apply (drule_tac x="scaleR r (sgn 1)" in spec)
1.1035 - apply (simp add: norm_sgn)
1.1036 - done
1.1037 -
1.1038 -lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
1.1039 - by (auto simp add: trivial_limit_def Rep_net_sequentially)
1.1040 -
1.1041 -subsection{* Some property holds "sufficiently close" to the limit point. *}
1.1042 -
1.1043 -lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
1.1044 - "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
1.1045 -unfolding eventually_at dist_nz by auto
1.1046 -
1.1047 -lemma eventually_at_infinity:
1.1048 - "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
1.1049 -unfolding eventually_def Rep_net_at_infinity by auto
1.1050 -
1.1051 -lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
1.1052 - (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
1.1053 -unfolding eventually_within eventually_at dist_nz by auto
1.1054 -
1.1055 -lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
1.1056 - (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
1.1057 -unfolding eventually_within
1.1058 -apply safe
1.1059 -apply (rule_tac x="d/2" in exI, simp)
1.1060 -apply (rule_tac x="d" in exI, simp)
1.1061 -done
1.1062 -
1.1063 -lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
1.1064 - unfolding eventually_def trivial_limit_def
1.1065 - using Rep_net_nonempty [of net] by auto
1.1066 -
1.1067 -lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
1.1068 - unfolding eventually_def trivial_limit_def
1.1069 - using Rep_net_nonempty [of net] by auto
1.1070 -
1.1071 -lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
1.1072 - unfolding trivial_limit_def eventually_def by auto
1.1073 -
1.1074 -lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1.1075 - unfolding trivial_limit_def eventually_def by auto
1.1076 -
1.1077 -lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
1.1078 - apply (safe elim!: trivial_limit_eventually)
1.1079 - apply (simp add: eventually_False [symmetric])
1.1080 - done
1.1081 -
1.1082 -text{* Combining theorems for "eventually" *}
1.1083 -
1.1084 -lemma eventually_conjI:
1.1085 - "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
1.1086 - \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
1.1087 -by (rule eventually_conj)
1.1088 -
1.1089 -lemma eventually_rev_mono:
1.1090 - "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
1.1091 -using eventually_mono [of P Q] by fast
1.1092 -
1.1093 -lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
1.1094 - by (auto intro!: eventually_conjI elim: eventually_rev_mono)
1.1095 -
1.1096 -lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1.1097 - by (auto simp add: eventually_False)
1.1098 -
1.1099 -lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
1.1100 - by (simp add: eventually_False)
1.1101 -
1.1102 -subsection{* Limits, defined as vacuously true when the limit is trivial. *}
1.1103 -
1.1104 - text{* Notation Lim to avoid collition with lim defined in analysis *}
1.1105 -definition
1.1106 - Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b" where
1.1107 - "Lim net f = (THE l. (f ---> l) net)"
1.1108 -
1.1109 -lemma Lim:
1.1110 - "(f ---> l) net \<longleftrightarrow>
1.1111 - trivial_limit net \<or>
1.1112 - (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
1.1113 - unfolding tendsto_iff trivial_limit_eq by auto
1.1114 -
1.1115 -
1.1116 -text{* Show that they yield usual definitions in the various cases. *}
1.1117 -
1.1118 -lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
1.1119 - (\<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)"
1.1120 - by (auto simp add: tendsto_iff eventually_within_le)
1.1121 -
1.1122 -lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
1.1123 - (\<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)"
1.1124 - by (auto simp add: tendsto_iff eventually_within)
1.1125 -
1.1126 -lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
1.1127 - (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
1.1128 - by (auto simp add: tendsto_iff eventually_at)
1.1129 -
1.1130 -lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
1.1131 - unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
1.1132 -
1.1133 -lemma Lim_at_infinity:
1.1134 - "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
1.1135 - by (auto simp add: tendsto_iff eventually_at_infinity)
1.1136 -
1.1137 -lemma Lim_sequentially:
1.1138 - "(S ---> l) sequentially \<longleftrightarrow>
1.1139 - (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
1.1140 - by (auto simp add: tendsto_iff eventually_sequentially)
1.1141 -
1.1142 -lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
1.1143 - unfolding Lim_sequentially LIMSEQ_def ..
1.1144 -
1.1145 -lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1.1146 - by (rule topological_tendstoI, auto elim: eventually_rev_mono)
1.1147 -
1.1148 -text{* The expected monotonicity property. *}
1.1149 -
1.1150 -lemma Lim_within_empty: "(f ---> l) (net within {})"
1.1151 - unfolding tendsto_def Limits.eventually_within by simp
1.1152 -
1.1153 -lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
1.1154 - unfolding tendsto_def Limits.eventually_within
1.1155 - by (auto elim!: eventually_elim1)
1.1156 -
1.1157 -lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
1.1158 - shows "(f ---> l) (net within (S \<union> T))"
1.1159 - using assms unfolding tendsto_def Limits.eventually_within
1.1160 - apply clarify
1.1161 - apply (drule spec, drule (1) mp, drule (1) mp)
1.1162 - apply (drule spec, drule (1) mp, drule (1) mp)
1.1163 - apply (auto elim: eventually_elim2)
1.1164 - done
1.1165 -
1.1166 -lemma Lim_Un_univ:
1.1167 - "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
1.1168 - ==> (f ---> l) net"
1.1169 - by (metis Lim_Un within_UNIV)
1.1170 -
1.1171 -text{* Interrelations between restricted and unrestricted limits. *}
1.1172 -
1.1173 -lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
1.1174 - (* FIXME: rename *)
1.1175 - unfolding tendsto_def Limits.eventually_within
1.1176 - apply (clarify, drule spec, drule (1) mp, drule (1) mp)
1.1177 - by (auto elim!: eventually_elim1)
1.1178 -
1.1179 -lemma Lim_within_open:
1.1180 - fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
1.1181 - assumes"a \<in> S" "open S"
1.1182 - shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
1.1183 -proof
1.1184 - assume ?lhs
1.1185 - { fix A assume "open A" "l \<in> A"
1.1186 - with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
1.1187 - by (rule topological_tendstoD)
1.1188 - hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
1.1189 - unfolding Limits.eventually_within .
1.1190 - 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"
1.1191 - unfolding eventually_at_topological by fast
1.1192 - hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
1.1193 - using assms by auto
1.1194 - hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
1.1195 - by fast
1.1196 - hence "eventually (\<lambda>x. f x \<in> A) (at a)"
1.1197 - unfolding eventually_at_topological .
1.1198 - }
1.1199 - thus ?rhs by (rule topological_tendstoI)
1.1200 -next
1.1201 - assume ?rhs
1.1202 - thus ?lhs by (rule Lim_at_within)
1.1203 -qed
1.1204 -
1.1205 -text{* Another limit point characterization. *}
1.1206 -
1.1207 -lemma islimpt_sequential:
1.1208 - fixes x :: "'a::metric_space" (* FIXME: generalize to topological_space *)
1.1209 - shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1.1210 - (is "?lhs = ?rhs")
1.1211 -proof
1.1212 - assume ?lhs
1.1213 - 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"
1.1214 - 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
1.1215 - { fix n::nat
1.1216 - have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
1.1217 - }
1.1218 - moreover
1.1219 - { fix e::real assume "e>0"
1.1220 - 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
1.1221 - then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1.1222 - 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)
1.1223 - moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
1.1224 - 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
1.1225 - }
1.1226 - hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
1.1227 - unfolding Lim_sequentially using f by auto
1.1228 - ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
1.1229 -next
1.1230 - assume ?rhs
1.1231 - 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
1.1232 - { fix e::real assume "e>0"
1.1233 - then obtain N where "dist (f N) x < e" using f(2) by auto
1.1234 - moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1.1235 - ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1.1236 - }
1.1237 - thus ?lhs unfolding islimpt_approachable by auto
1.1238 -qed
1.1239 -
1.1240 -text{* Basic arithmetical combining theorems for limits. *}
1.1241 -
1.1242 -lemma Lim_linear:
1.1243 - assumes "(f ---> l) net" "bounded_linear h"
1.1244 - shows "((\<lambda>x. h (f x)) ---> h l) net"
1.1245 -using `bounded_linear h` `(f ---> l) net`
1.1246 -by (rule bounded_linear.tendsto)
1.1247 -
1.1248 -lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
1.1249 - unfolding tendsto_def Limits.eventually_at_topological by fast
1.1250 -
1.1251 -lemma Lim_const: "((\<lambda>x. a) ---> a) net"
1.1252 - by (rule tendsto_const)
1.1253 -
1.1254 -lemma Lim_cmul:
1.1255 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1256 - shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
1.1257 - by (intro tendsto_intros)
1.1258 -
1.1259 -lemma Lim_neg:
1.1260 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1261 - shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
1.1262 - by (rule tendsto_minus)
1.1263 -
1.1264 -lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
1.1265 - "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
1.1266 - by (rule tendsto_add)
1.1267 -
1.1268 -lemma Lim_sub:
1.1269 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1270 - shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
1.1271 - by (rule tendsto_diff)
1.1272 -
1.1273 -lemma Lim_null:
1.1274 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1275 - shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
1.1276 -
1.1277 -lemma Lim_null_norm:
1.1278 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1279 - shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
1.1280 - by (simp add: Lim dist_norm)
1.1281 -
1.1282 -lemma Lim_null_comparison:
1.1283 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1284 - assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1.1285 - shows "(f ---> 0) net"
1.1286 -proof(simp add: tendsto_iff, rule+)
1.1287 - fix e::real assume "0<e"
1.1288 - { fix x
1.1289 - assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
1.1290 - hence "dist (f x) 0 < e" by (simp add: dist_norm)
1.1291 - }
1.1292 - thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1.1293 - using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
1.1294 - 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]
1.1295 - using assms `e>0` unfolding tendsto_iff by auto
1.1296 -qed
1.1297 -
1.1298 -lemma Lim_component:
1.1299 - fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
1.1300 - shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
1.1301 - unfolding tendsto_iff
1.1302 - apply (clarify)
1.1303 - apply (drule spec, drule (1) mp)
1.1304 - apply (erule eventually_elim1)
1.1305 - apply (erule le_less_trans [OF dist_nth_le])
1.1306 - done
1.1307 -
1.1308 -lemma Lim_transform_bound:
1.1309 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1310 - fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1.1311 - assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1.1312 - shows "(f ---> 0) net"
1.1313 -proof (rule tendstoI)
1.1314 - fix e::real assume "e>0"
1.1315 - { fix x
1.1316 - assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
1.1317 - hence "dist (f x) 0 < e" by (simp add: dist_norm)}
1.1318 - thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1.1319 - using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
1.1320 - 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]
1.1321 - using assms `e>0` unfolding tendsto_iff by blast
1.1322 -qed
1.1323 -
1.1324 -text{* Deducing things about the limit from the elements. *}
1.1325 -
1.1326 -lemma Lim_in_closed_set:
1.1327 - assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1.1328 - shows "l \<in> S"
1.1329 -proof (rule ccontr)
1.1330 - assume "l \<notin> S"
1.1331 - with `closed S` have "open (- S)" "l \<in> - S"
1.1332 - by (simp_all add: open_Compl)
1.1333 - with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1.1334 - by (rule topological_tendstoD)
1.1335 - with assms(2) have "eventually (\<lambda>x. False) net"
1.1336 - by (rule eventually_elim2) simp
1.1337 - with assms(3) show "False"
1.1338 - by (simp add: eventually_False)
1.1339 -qed
1.1340 -
1.1341 -text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1.1342 -
1.1343 -lemma Lim_dist_ubound:
1.1344 - assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1.1345 - shows "dist a l <= e"
1.1346 -proof (rule ccontr)
1.1347 - assume "\<not> dist a l \<le> e"
1.1348 - then have "0 < dist a l - e" by simp
1.1349 - with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
1.1350 - by (rule tendstoD)
1.1351 - with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
1.1352 - by (rule eventually_conjI)
1.1353 - then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
1.1354 - using assms(1) eventually_happens by auto
1.1355 - hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
1.1356 - by (rule add_le_less_mono)
1.1357 - hence "dist a (f w) + dist (f w) l < dist a l"
1.1358 - by simp
1.1359 - also have "\<dots> \<le> dist a (f w) + dist (f w) l"
1.1360 - by (rule dist_triangle)
1.1361 - finally show False by simp
1.1362 -qed
1.1363 -
1.1364 -lemma Lim_norm_ubound:
1.1365 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1366 - assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1.1367 - shows "norm(l) <= e"
1.1368 -proof (rule ccontr)
1.1369 - assume "\<not> norm l \<le> e"
1.1370 - then have "0 < norm l - e" by simp
1.1371 - with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
1.1372 - by (rule tendstoD)
1.1373 - with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
1.1374 - by (rule eventually_conjI)
1.1375 - then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
1.1376 - using assms(1) eventually_happens by auto
1.1377 - hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
1.1378 - hence "norm (f w - l) + norm (f w) < norm l" by simp
1.1379 - hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
1.1380 - thus False using `\<not> norm l \<le> e` by simp
1.1381 -qed
1.1382 -
1.1383 -lemma Lim_norm_lbound:
1.1384 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.1385 - assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1.1386 - shows "e \<le> norm l"
1.1387 -proof (rule ccontr)
1.1388 - assume "\<not> e \<le> norm l"
1.1389 - then have "0 < e - norm l" by simp
1.1390 - with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
1.1391 - by (rule tendstoD)
1.1392 - with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
1.1393 - by (rule eventually_conjI)
1.1394 - then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
1.1395 - using assms(1) eventually_happens by auto
1.1396 - hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
1.1397 - hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
1.1398 - hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
1.1399 - thus False by simp
1.1400 -qed
1.1401 -
1.1402 -text{* Uniqueness of the limit, when nontrivial. *}
1.1403 -
1.1404 -lemma Lim_unique:
1.1405 - fixes f :: "'a \<Rightarrow> 'b::t2_space"
1.1406 - assumes "\<not> trivial_limit net" "(f ---> l) net" "(f ---> l') net"
1.1407 - shows "l = l'"
1.1408 -proof (rule ccontr)
1.1409 - assume "l \<noteq> l'"
1.1410 - obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
1.1411 - using hausdorff [OF `l \<noteq> l'`] by fast
1.1412 - have "eventually (\<lambda>x. f x \<in> U) net"
1.1413 - using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
1.1414 - moreover
1.1415 - have "eventually (\<lambda>x. f x \<in> V) net"
1.1416 - using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
1.1417 - ultimately
1.1418 - have "eventually (\<lambda>x. False) net"
1.1419 - proof (rule eventually_elim2)
1.1420 - fix x
1.1421 - assume "f x \<in> U" "f x \<in> V"
1.1422 - hence "f x \<in> U \<inter> V" by simp
1.1423 - with `U \<inter> V = {}` show "False" by simp
1.1424 - qed
1.1425 - with `\<not> trivial_limit net` show "False"
1.1426 - by (simp add: eventually_False)
1.1427 -qed
1.1428 -
1.1429 -lemma tendsto_Lim:
1.1430 - fixes f :: "'a \<Rightarrow> 'b::t2_space"
1.1431 - shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1.1432 - unfolding Lim_def using Lim_unique[of net f] by auto
1.1433 -
1.1434 -text{* Limit under bilinear function *}
1.1435 -
1.1436 -lemma Lim_bilinear:
1.1437 - assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1.1438 - shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1.1439 -using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1.1440 -by (rule bounded_bilinear.tendsto)
1.1441 -
1.1442 -text{* These are special for limits out of the same vector space. *}
1.1443 -
1.1444 -lemma Lim_within_id: "(id ---> a) (at a within s)"
1.1445 - unfolding tendsto_def Limits.eventually_within eventually_at_topological
1.1446 - by auto
1.1447 -
1.1448 -lemma Lim_at_id: "(id ---> a) (at a)"
1.1449 -apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
1.1450 -
1.1451 -lemma Lim_at_zero:
1.1452 - fixes a :: "'a::real_normed_vector"
1.1453 - fixes l :: "'b::topological_space"
1.1454 - shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1.1455 -proof
1.1456 - assume "?lhs"
1.1457 - { fix S assume "open S" "l \<in> S"
1.1458 - with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
1.1459 - by (rule topological_tendstoD)
1.1460 - then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
1.1461 - unfolding Limits.eventually_at by fast
1.1462 - { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
1.1463 - hence "f (a + x) \<in> S" using d
1.1464 - apply(erule_tac x="x+a" in allE)
1.1465 - by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
1.1466 - }
1.1467 - hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1.1468 - using d(1) by auto
1.1469 - hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1.1470 - unfolding Limits.eventually_at .
1.1471 - }
1.1472 - thus "?rhs" by (rule topological_tendstoI)
1.1473 -next
1.1474 - assume "?rhs"
1.1475 - { fix S assume "open S" "l \<in> S"
1.1476 - with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1.1477 - by (rule topological_tendstoD)
1.1478 - then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1.1479 - unfolding Limits.eventually_at by fast
1.1480 - { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
1.1481 - hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
1.1482 - by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
1.1483 - }
1.1484 - hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
1.1485 - hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
1.1486 - }
1.1487 - thus "?lhs" by (rule topological_tendstoI)
1.1488 -qed
1.1489 -
1.1490 -text{* It's also sometimes useful to extract the limit point from the net. *}
1.1491 -
1.1492 -definition
1.1493 - netlimit :: "'a::t2_space net \<Rightarrow> 'a" where
1.1494 - "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1.1495 -
1.1496 -lemma netlimit_within:
1.1497 - assumes "\<not> trivial_limit (at a within S)"
1.1498 - shows "netlimit (at a within S) = a"
1.1499 -unfolding netlimit_def
1.1500 -apply (rule some_equality)
1.1501 -apply (rule Lim_at_within)
1.1502 -apply (rule Lim_ident_at)
1.1503 -apply (erule Lim_unique [OF assms])
1.1504 -apply (rule Lim_at_within)
1.1505 -apply (rule Lim_ident_at)
1.1506 -done
1.1507 -
1.1508 -lemma netlimit_at:
1.1509 - fixes a :: "'a::perfect_space"
1.1510 - shows "netlimit (at a) = a"
1.1511 - apply (subst within_UNIV[symmetric])
1.1512 - using netlimit_within[of a UNIV]
1.1513 - by (simp add: trivial_limit_at within_UNIV)
1.1514 -
1.1515 -text{* Transformation of limit. *}
1.1516 -
1.1517 -lemma Lim_transform:
1.1518 - fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1.1519 - assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1.1520 - shows "(g ---> l) net"
1.1521 -proof-
1.1522 - 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
1.1523 - thus "?thesis" using Lim_neg [of "\<lambda> x. - g x" "-l" net] by auto
1.1524 -qed
1.1525 -
1.1526 -lemma Lim_transform_eventually:
1.1527 - "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net ==> (g ---> l) net"
1.1528 - apply (rule topological_tendstoI)
1.1529 - apply (drule (2) topological_tendstoD)
1.1530 - apply (erule (1) eventually_elim2, simp)
1.1531 - done
1.1532 -
1.1533 -lemma Lim_transform_within:
1.1534 - fixes l :: "'b::metric_space" (* TODO: generalize *)
1.1535 - assumes "0 < d" "(\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x')"
1.1536 - "(f ---> l) (at x within S)"
1.1537 - shows "(g ---> l) (at x within S)"
1.1538 - using assms(1,3) unfolding Lim_within
1.1539 - apply -
1.1540 - apply (clarify, rename_tac e)
1.1541 - apply (drule_tac x=e in spec, clarsimp, rename_tac r)
1.1542 - apply (rule_tac x="min d r" in exI, clarsimp, rename_tac y)
1.1543 - apply (drule_tac x=y in bspec, assumption, clarsimp)
1.1544 - apply (simp add: assms(2))
1.1545 - done
1.1546 -
1.1547 -lemma Lim_transform_at:
1.1548 - fixes l :: "'b::metric_space" (* TODO: generalize *)
1.1549 - shows "0 < d \<Longrightarrow> (\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x') \<Longrightarrow>
1.1550 - (f ---> l) (at x) ==> (g ---> l) (at x)"
1.1551 - apply (subst within_UNIV[symmetric])
1.1552 - using Lim_transform_within[of d UNIV x f g l]
1.1553 - by (auto simp add: within_UNIV)
1.1554 -
1.1555 -text{* Common case assuming being away from some crucial point like 0. *}
1.1556 -
1.1557 -lemma Lim_transform_away_within:
1.1558 - fixes a b :: "'a::metric_space"
1.1559 - fixes l :: "'b::metric_space" (* TODO: generalize *)
1.1560 - assumes "a\<noteq>b" "\<forall>x\<in> S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1.1561 - and "(f ---> l) (at a within S)"
1.1562 - shows "(g ---> l) (at a within S)"
1.1563 -proof-
1.1564 - have "\<forall>x'\<in>S. 0 < dist x' a \<and> dist x' a < dist a b \<longrightarrow> f x' = g x'" using assms(2)
1.1565 - apply auto apply(erule_tac x=x' in ballE) by (auto simp add: dist_commute)
1.1566 - thus ?thesis using Lim_transform_within[of "dist a b" S a f g l] using assms(1,3) unfolding dist_nz by auto
1.1567 -qed
1.1568 -
1.1569 -lemma Lim_transform_away_at:
1.1570 - fixes a b :: "'a::metric_space"
1.1571 - fixes l :: "'b::metric_space" (* TODO: generalize *)
1.1572 - assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1.1573 - and fl: "(f ---> l) (at a)"
1.1574 - shows "(g ---> l) (at a)"
1.1575 - using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
1.1576 - by (auto simp add: within_UNIV)
1.1577 -
1.1578 -text{* Alternatively, within an open set. *}
1.1579 -
1.1580 -lemma Lim_transform_within_open:
1.1581 - fixes a :: "'a::metric_space"
1.1582 - fixes l :: "'b::metric_space" (* TODO: generalize *)
1.1583 - assumes "open S" "a \<in> S" "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x" "(f ---> l) (at a)"
1.1584 - shows "(g ---> l) (at a)"
1.1585 -proof-
1.1586 - from assms(1,2) obtain e::real where "e>0" and e:"ball a e \<subseteq> S" unfolding open_contains_ball by auto
1.1587 - hence "\<forall>x'. 0 < dist x' a \<and> dist x' a < e \<longrightarrow> f x' = g x'" using assms(3)
1.1588 - 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)
1.1589 - thus ?thesis using Lim_transform_at[of e a f g l] `e>0` assms(4) by auto
1.1590 -qed
1.1591 -
1.1592 -text{* A congruence rule allowing us to transform limits assuming not at point. *}
1.1593 -
1.1594 -(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1.1595 -
1.1596 -lemma Lim_cong_within[cong add]:
1.1597 - fixes a :: "'a::metric_space"
1.1598 - fixes l :: "'b::metric_space" (* TODO: generalize *)
1.1599 - 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))"
1.1600 - by (simp add: Lim_within dist_nz[symmetric])
1.1601 -
1.1602 -lemma Lim_cong_at[cong add]:
1.1603 - fixes a :: "'a::metric_space"
1.1604 - fixes l :: "'b::metric_space" (* TODO: generalize *)
1.1605 - shows "(\<And>x. x \<noteq> a ==> f x = g x) ==> (((\<lambda>x. f x) ---> l) (at a) \<longleftrightarrow> ((g ---> l) (at a)))"
1.1606 - by (simp add: Lim_at dist_nz[symmetric])
1.1607 -
1.1608 -text{* Useful lemmas on closure and set of possible sequential limits.*}
1.1609 -
1.1610 -lemma closure_sequential:
1.1611 - fixes l :: "'a::metric_space" (* TODO: generalize *)
1.1612 - shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1.1613 -proof
1.1614 - assume "?lhs" moreover
1.1615 - { assume "l \<in> S"
1.1616 - hence "?rhs" using Lim_const[of l sequentially] by auto
1.1617 - } moreover
1.1618 - { assume "l islimpt S"
1.1619 - hence "?rhs" unfolding islimpt_sequential by auto
1.1620 - } ultimately
1.1621 - show "?rhs" unfolding closure_def by auto
1.1622 -next
1.1623 - assume "?rhs"
1.1624 - thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1.1625 -qed
1.1626 -
1.1627 -lemma closed_sequential_limits:
1.1628 - fixes S :: "'a::metric_space set"
1.1629 - shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1.1630 - unfolding closed_limpt
1.1631 - 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]
1.1632 - by metis
1.1633 -
1.1634 -lemma closure_approachable:
1.1635 - fixes S :: "'a::metric_space set"
1.1636 - shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1.1637 - apply (auto simp add: closure_def islimpt_approachable)
1.1638 - by (metis dist_self)
1.1639 -
1.1640 -lemma closed_approachable:
1.1641 - fixes S :: "'a::metric_space set"
1.1642 - shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1.1643 - by (metis closure_closed closure_approachable)
1.1644 -
1.1645 -text{* Some other lemmas about sequences. *}
1.1646 -
1.1647 -lemma seq_offset:
1.1648 - fixes l :: "'a::metric_space" (* TODO: generalize *)
1.1649 - shows "(f ---> l) sequentially ==> ((\<lambda>i. f( i + k)) ---> l) sequentially"
1.1650 - apply (auto simp add: Lim_sequentially)
1.1651 - by (metis trans_le_add1 )
1.1652 -
1.1653 -lemma seq_offset_neg:
1.1654 - "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1.1655 - apply (rule topological_tendstoI)
1.1656 - apply (drule (2) topological_tendstoD)
1.1657 - apply (simp only: eventually_sequentially)
1.1658 - apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1.1659 - apply metis
1.1660 - by arith
1.1661 -
1.1662 -lemma seq_offset_rev:
1.1663 - "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1.1664 - apply (rule topological_tendstoI)
1.1665 - apply (drule (2) topological_tendstoD)
1.1666 - apply (simp only: eventually_sequentially)
1.1667 - apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
1.1668 - by metis arith
1.1669 -
1.1670 -lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1.1671 -proof-
1.1672 - { fix e::real assume "e>0"
1.1673 - hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
1.1674 - using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
1.1675 - by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
1.1676 - }
1.1677 - thus ?thesis unfolding Lim_sequentially dist_norm by simp
1.1678 -qed
1.1679 -
1.1680 -text{* More properties of closed balls. *}
1.1681 -
1.1682 -lemma closed_cball: "closed (cball x e)"
1.1683 -unfolding cball_def closed_def
1.1684 -unfolding Collect_neg_eq [symmetric] not_le
1.1685 -apply (clarsimp simp add: open_dist, rename_tac y)
1.1686 -apply (rule_tac x="dist x y - e" in exI, clarsimp)
1.1687 -apply (rename_tac x')
1.1688 -apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1.1689 -apply simp
1.1690 -done
1.1691 -
1.1692 -lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1.1693 -proof-
1.1694 - { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1.1695 - hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1.1696 - } moreover
1.1697 - { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1.1698 - hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1.1699 - } ultimately
1.1700 - show ?thesis unfolding open_contains_ball by auto
1.1701 -qed
1.1702 -
1.1703 -lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1.1704 - by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
1.1705 -
1.1706 -lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1.1707 - apply (simp add: interior_def, safe)
1.1708 - apply (force simp add: open_contains_cball)
1.1709 - apply (rule_tac x="ball x e" in exI)
1.1710 - apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
1.1711 - done
1.1712 -
1.1713 -lemma islimpt_ball:
1.1714 - fixes x y :: "'a::{real_normed_vector,perfect_space}"
1.1715 - shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1.1716 -proof
1.1717 - assume "?lhs"
1.1718 - { assume "e \<le> 0"
1.1719 - hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1.1720 - have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1.1721 - }
1.1722 - hence "e > 0" by (metis not_less)
1.1723 - moreover
1.1724 - 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
1.1725 - ultimately show "?rhs" by auto
1.1726 -next
1.1727 - assume "?rhs" hence "e>0" by auto
1.1728 - { fix d::real assume "d>0"
1.1729 - have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1.1730 - proof(cases "d \<le> dist x y")
1.1731 - case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1.1732 - proof(cases "x=y")
1.1733 - case True hence False using `d \<le> dist x y` `d>0` by auto
1.1734 - thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1.1735 - next
1.1736 - case False
1.1737 -
1.1738 - have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1.1739 - = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1.1740 - unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1.1741 - also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1.1742 - using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1.1743 - unfolding scaleR_minus_left scaleR_one
1.1744 - by (auto simp add: norm_minus_commute)
1.1745 - also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1.1746 - unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1.1747 - unfolding real_add_mult_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1.1748 - also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1.1749 - finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1.1750 -
1.1751 - moreover
1.1752 -
1.1753 - have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1.1754 - using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1.1755 - moreover
1.1756 - have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1.1757 - using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1.1758 - unfolding dist_norm by auto
1.1759 - 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
1.1760 - qed
1.1761 - next
1.1762 - case False hence "d > dist x y" by auto
1.1763 - show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1.1764 - proof(cases "x=y")
1.1765 - case True
1.1766 - obtain z where **: "z \<noteq> y" "dist z y < min e d"
1.1767 - using perfect_choose_dist[of "min e d" y]
1.1768 - using `d > 0` `e>0` by auto
1.1769 - show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1.1770 - unfolding `x = y`
1.1771 - using `z \<noteq> y` **
1.1772 - by (rule_tac x=z in bexI, auto simp add: dist_commute)
1.1773 - next
1.1774 - case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1.1775 - using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1.1776 - qed
1.1777 - qed }
1.1778 - thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1.1779 -qed
1.1780 -
1.1781 -lemma closure_ball_lemma:
1.1782 - fixes x y :: "'a::real_normed_vector"
1.1783 - assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1.1784 -proof (rule islimptI)
1.1785 - fix T assume "y \<in> T" "open T"
1.1786 - then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1.1787 - unfolding open_dist by fast
1.1788 - (* choose point between x and y, within distance r of y. *)
1.1789 - def k \<equiv> "min 1 (r / (2 * dist x y))"
1.1790 - def z \<equiv> "y + scaleR k (x - y)"
1.1791 - have z_def2: "z = x + scaleR (1 - k) (y - x)"
1.1792 - unfolding z_def by (simp add: algebra_simps)
1.1793 - have "dist z y < r"
1.1794 - unfolding z_def k_def using `0 < r`
1.1795 - by (simp add: dist_norm min_def)
1.1796 - hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1.1797 - have "dist x z < dist x y"
1.1798 - unfolding z_def2 dist_norm
1.1799 - apply (simp add: norm_minus_commute)
1.1800 - apply (simp only: dist_norm [symmetric])
1.1801 - apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1.1802 - apply (rule mult_strict_right_mono)
1.1803 - apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1.1804 - apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1.1805 - done
1.1806 - hence "z \<in> ball x (dist x y)" by simp
1.1807 - have "z \<noteq> y"
1.1808 - unfolding z_def k_def using `x \<noteq> y` `0 < r`
1.1809 - by (simp add: min_def)
1.1810 - show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1.1811 - using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1.1812 - by fast
1.1813 -qed
1.1814 -
1.1815 -lemma closure_ball:
1.1816 - fixes x :: "'a::real_normed_vector"
1.1817 - shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1.1818 -apply (rule equalityI)
1.1819 -apply (rule closure_minimal)
1.1820 -apply (rule ball_subset_cball)
1.1821 -apply (rule closed_cball)
1.1822 -apply (rule subsetI, rename_tac y)
1.1823 -apply (simp add: le_less [where 'a=real])
1.1824 -apply (erule disjE)
1.1825 -apply (rule subsetD [OF closure_subset], simp)
1.1826 -apply (simp add: closure_def)
1.1827 -apply clarify
1.1828 -apply (rule closure_ball_lemma)
1.1829 -apply (simp add: zero_less_dist_iff)
1.1830 -done
1.1831 -
1.1832 -(* In a trivial vector space, this fails for e = 0. *)
1.1833 -lemma interior_cball:
1.1834 - fixes x :: "'a::{real_normed_vector, perfect_space}"
1.1835 - shows "interior (cball x e) = ball x e"
1.1836 -proof(cases "e\<ge>0")
1.1837 - case False note cs = this
1.1838 - from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1.1839 - { fix y assume "y \<in> cball x e"
1.1840 - hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1.1841 - hence "cball x e = {}" by auto
1.1842 - hence "interior (cball x e) = {}" using interior_empty by auto
1.1843 - ultimately show ?thesis by blast
1.1844 -next
1.1845 - case True note cs = this
1.1846 - have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1.1847 - { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1.1848 - then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1.1849 -
1.1850 - then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1.1851 - using perfect_choose_dist [of d] by auto
1.1852 - have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1.1853 - hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1.1854 -
1.1855 - hence "y \<in> ball x e" proof(cases "x = y")
1.1856 - case True
1.1857 - hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1.1858 - thus "y \<in> ball x e" using `x = y ` by simp
1.1859 - next
1.1860 - case False
1.1861 - have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1.1862 - using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1.1863 - hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1.1864 - have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1.1865 - hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1.1866 - using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1.1867 -
1.1868 - have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
1.1869 - by (auto simp add: dist_norm algebra_simps)
1.1870 - also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1.1871 - by (auto simp add: algebra_simps)
1.1872 - also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1.1873 - using ** by auto
1.1874 - also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1.1875 - finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1.1876 - thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1.1877 - qed }
1.1878 - hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1.1879 - ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1.1880 -qed
1.1881 -
1.1882 -lemma frontier_ball:
1.1883 - fixes a :: "'a::real_normed_vector"
1.1884 - shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1.1885 - apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
1.1886 - apply (simp add: expand_set_eq)
1.1887 - by arith
1.1888 -
1.1889 -lemma frontier_cball:
1.1890 - fixes a :: "'a::{real_normed_vector, perfect_space}"
1.1891 - shows "frontier(cball a e) = {x. dist a x = e}"
1.1892 - apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
1.1893 - apply (simp add: expand_set_eq)
1.1894 - by arith
1.1895 -
1.1896 -lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1.1897 - apply (simp add: expand_set_eq not_le)
1.1898 - by (metis zero_le_dist dist_self order_less_le_trans)
1.1899 -lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1.1900 -
1.1901 -lemma cball_eq_sing:
1.1902 - fixes x :: "'a::perfect_space"
1.1903 - shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1.1904 -proof (rule linorder_cases)
1.1905 - assume e: "0 < e"
1.1906 - obtain a where "a \<noteq> x" "dist a x < e"
1.1907 - using perfect_choose_dist [OF e] by auto
1.1908 - hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1.1909 - with e show ?thesis by (auto simp add: expand_set_eq)
1.1910 -qed auto
1.1911 -
1.1912 -lemma cball_sing:
1.1913 - fixes x :: "'a::metric_space"
1.1914 - shows "e = 0 ==> cball x e = {x}"
1.1915 - by (auto simp add: expand_set_eq)
1.1916 -
1.1917 -text{* For points in the interior, localization of limits makes no difference. *}
1.1918 -
1.1919 -lemma eventually_within_interior:
1.1920 - assumes "x \<in> interior S"
1.1921 - shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
1.1922 -proof-
1.1923 - from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
1.1924 - unfolding interior_def by fast
1.1925 - { assume "?lhs"
1.1926 - then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1.1927 - unfolding Limits.eventually_within Limits.eventually_at_topological
1.1928 - by auto
1.1929 - with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1.1930 - by auto
1.1931 - then have "?rhs"
1.1932 - unfolding Limits.eventually_at_topological by auto
1.1933 - } moreover
1.1934 - { assume "?rhs" hence "?lhs"
1.1935 - unfolding Limits.eventually_within
1.1936 - by (auto elim: eventually_elim1)
1.1937 - } ultimately
1.1938 - show "?thesis" ..
1.1939 -qed
1.1940 -
1.1941 -lemma lim_within_interior:
1.1942 - "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1.1943 - unfolding tendsto_def by (simp add: eventually_within_interior)
1.1944 -
1.1945 -lemma netlimit_within_interior:
1.1946 - fixes x :: "'a::{perfect_space, real_normed_vector}"
1.1947 - (* FIXME: generalize to perfect_space *)
1.1948 - assumes "x \<in> interior S"
1.1949 - shows "netlimit(at x within S) = x" (is "?lhs = ?rhs")
1.1950 -proof-
1.1951 - 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
1.1952 - 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
1.1953 - thus ?thesis using netlimit_within by auto
1.1954 -qed
1.1955 -
1.1956 -subsection{* Boundedness. *}
1.1957 -
1.1958 - (* FIXME: This has to be unified with BSEQ!! *)
1.1959 -definition
1.1960 - bounded :: "'a::metric_space set \<Rightarrow> bool" where
1.1961 - "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1.1962 -
1.1963 -lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1.1964 -unfolding bounded_def
1.1965 -apply safe
1.1966 -apply (rule_tac x="dist a x + e" in exI, clarify)
1.1967 -apply (drule (1) bspec)
1.1968 -apply (erule order_trans [OF dist_triangle add_left_mono])
1.1969 -apply auto
1.1970 -done
1.1971 -
1.1972 -lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1.1973 -unfolding bounded_any_center [where a=0]
1.1974 -by (simp add: dist_norm)
1.1975 -
1.1976 -lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1.1977 -lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1.1978 - by (metis bounded_def subset_eq)
1.1979 -
1.1980 -lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1.1981 - by (metis bounded_subset interior_subset)
1.1982 -
1.1983 -lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1.1984 -proof-
1.1985 - from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1.1986 - { fix y assume "y \<in> closure S"
1.1987 - then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1.1988 - unfolding closure_sequential by auto
1.1989 - have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1.1990 - hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1.1991 - by (rule eventually_mono, simp add: f(1))
1.1992 - have "dist x y \<le> a"
1.1993 - apply (rule Lim_dist_ubound [of sequentially f])
1.1994 - apply (rule trivial_limit_sequentially)
1.1995 - apply (rule f(2))
1.1996 - apply fact
1.1997 - done
1.1998 - }
1.1999 - thus ?thesis unfolding bounded_def by auto
1.2000 -qed
1.2001 -
1.2002 -lemma bounded_cball[simp,intro]: "bounded (cball x e)"
1.2003 - apply (simp add: bounded_def)
1.2004 - apply (rule_tac x=x in exI)
1.2005 - apply (rule_tac x=e in exI)
1.2006 - apply auto
1.2007 - done
1.2008 -
1.2009 -lemma bounded_ball[simp,intro]: "bounded(ball x e)"
1.2010 - by (metis ball_subset_cball bounded_cball bounded_subset)
1.2011 -
1.2012 -lemma finite_imp_bounded[intro]: assumes "finite S" shows "bounded S"
1.2013 -proof-
1.2014 - { fix a F assume as:"bounded F"
1.2015 - then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
1.2016 - hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
1.2017 - hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
1.2018 - }
1.2019 - thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
1.2020 -qed
1.2021 -
1.2022 -lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
1.2023 - apply (auto simp add: bounded_def)
1.2024 - apply (rename_tac x y r s)
1.2025 - apply (rule_tac x=x in exI)
1.2026 - apply (rule_tac x="max r (dist x y + s)" in exI)
1.2027 - apply (rule ballI, rename_tac z, safe)
1.2028 - apply (drule (1) bspec, simp)
1.2029 - apply (drule (1) bspec)
1.2030 - apply (rule min_max.le_supI2)
1.2031 - apply (erule order_trans [OF dist_triangle add_left_mono])
1.2032 - done
1.2033 -
1.2034 -lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1.2035 - by (induct rule: finite_induct[of F], auto)
1.2036 -
1.2037 -lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
1.2038 - apply (simp add: bounded_iff)
1.2039 - apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
1.2040 - by metis arith
1.2041 -
1.2042 -lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
1.2043 - by (metis Int_lower1 Int_lower2 bounded_subset)
1.2044 -
1.2045 -lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
1.2046 -apply (metis Diff_subset bounded_subset)
1.2047 -done
1.2048 -
1.2049 -lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
1.2050 - by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
1.2051 -
1.2052 -lemma not_bounded_UNIV[simp, intro]:
1.2053 - "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
1.2054 -proof(auto simp add: bounded_pos not_le)
1.2055 - obtain x :: 'a where "x \<noteq> 0"
1.2056 - using perfect_choose_dist [OF zero_less_one] by fast
1.2057 - fix b::real assume b: "b >0"
1.2058 - have b1: "b +1 \<ge> 0" using b by simp
1.2059 - with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
1.2060 - by (simp add: norm_sgn)
1.2061 - then show "\<exists>x::'a. b < norm x" ..
1.2062 -qed
1.2063 -
1.2064 -lemma bounded_linear_image:
1.2065 - assumes "bounded S" "bounded_linear f"
1.2066 - shows "bounded(f ` S)"
1.2067 -proof-
1.2068 - from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1.2069 - from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)
1.2070 - { fix x assume "x\<in>S"
1.2071 - hence "norm x \<le> b" using b by auto
1.2072 - hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
1.2073 - by (metis B(1) B(2) real_le_trans real_mult_le_cancel_iff2)
1.2074 - }
1.2075 - thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
1.2076 - using b B real_mult_order[of b B] by (auto simp add: real_mult_commute)
1.2077 -qed
1.2078 -
1.2079 -lemma bounded_scaling:
1.2080 - fixes S :: "'a::real_normed_vector set"
1.2081 - shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
1.2082 - apply (rule bounded_linear_image, assumption)
1.2083 - apply (rule scaleR.bounded_linear_right)
1.2084 - done
1.2085 -
1.2086 -lemma bounded_translation:
1.2087 - fixes S :: "'a::real_normed_vector set"
1.2088 - assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
1.2089 -proof-
1.2090 - from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1.2091 - { fix x assume "x\<in>S"
1.2092 - hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
1.2093 - }
1.2094 - thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
1.2095 - by (auto intro!: add exI[of _ "b + norm a"])
1.2096 -qed
1.2097 -
1.2098 -
1.2099 -text{* Some theorems on sups and infs using the notion "bounded". *}
1.2100 -
1.2101 -lemma bounded_real:
1.2102 - fixes S :: "real set"
1.2103 - shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
1.2104 - by (simp add: bounded_iff)
1.2105 -
1.2106 -lemma bounded_has_rsup: assumes "bounded S" "S \<noteq> {}"
1.2107 - shows "\<forall>x\<in>S. x <= rsup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> rsup S <= b"
1.2108 -proof
1.2109 - fix x assume "x\<in>S"
1.2110 - from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
1.2111 - hence *:"S *<= a" using setleI[of S a] by (metis abs_le_interval_iff mem_def)
1.2112 - 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
1.2113 -next
1.2114 - show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> rsup S \<le> b" using assms
1.2115 - using rsup[of S, unfolded isLub_def isUb_def leastP_def setle_def setge_def]
1.2116 - apply (auto simp add: bounded_real)
1.2117 - by (auto simp add: isLub_def isUb_def leastP_def setle_def setge_def)
1.2118 -qed
1.2119 -
1.2120 -lemma rsup_insert: assumes "bounded S"
1.2121 - shows "rsup(insert x S) = (if S = {} then x else max x (rsup S))"
1.2122 -proof(cases "S={}")
1.2123 - case True thus ?thesis using rsup_finite_in[of "{x}"] by auto
1.2124 -next
1.2125 - let ?S = "insert x S"
1.2126 - case False
1.2127 - hence *:"\<forall>x\<in>S. x \<le> rsup S" using bounded_has_rsup(1)[of S] using assms by auto
1.2128 - hence "insert x S *<= max x (rsup S)" unfolding setle_def by auto
1.2129 - hence "isLub UNIV ?S (rsup ?S)" using rsup[of ?S] by auto
1.2130 - moreover
1.2131 - have **:"isUb UNIV ?S (max x (rsup S))" unfolding isUb_def setle_def using * by auto
1.2132 - { fix y assume as:"isUb UNIV (insert x S) y"
1.2133 - hence "max x (rsup S) \<le> y" unfolding isUb_def using rsup_le[OF `S\<noteq>{}`]
1.2134 - unfolding setle_def by auto }
1.2135 - hence "max x (rsup S) <=* isUb UNIV (insert x S)" unfolding setge_def Ball_def mem_def by auto
1.2136 - hence "isLub UNIV ?S (max x (rsup S))" using ** isLubI2[of UNIV ?S "max x (rsup S)"] unfolding Collect_def by auto
1.2137 - ultimately show ?thesis using real_isLub_unique[of UNIV ?S] using `S\<noteq>{}` by auto
1.2138 -qed
1.2139 -
1.2140 -lemma sup_insert_finite: "finite S \<Longrightarrow> rsup(insert x S) = (if S = {} then x else max x (rsup S))"
1.2141 - apply (rule rsup_insert)
1.2142 - apply (rule finite_imp_bounded)
1.2143 - by simp
1.2144 -
1.2145 -lemma bounded_has_rinf:
1.2146 - assumes "bounded S" "S \<noteq> {}"
1.2147 - shows "\<forall>x\<in>S. x >= rinf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S >= b"
1.2148 -proof
1.2149 - fix x assume "x\<in>S"
1.2150 - from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
1.2151 - hence *:"- a <=* S" using setgeI[of S "-a"] unfolding abs_le_interval_iff by auto
1.2152 - thus "x \<ge> rinf S" using rinf[OF `S\<noteq>{}`] using isGlbD2[of UNIV S "rinf S" x] using `x\<in>S` by auto
1.2153 -next
1.2154 - show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S \<ge> b" using assms
1.2155 - using rinf[of S, unfolded isGlb_def isLb_def greatestP_def setle_def setge_def]
1.2156 - apply (auto simp add: bounded_real)
1.2157 - by (auto simp add: isGlb_def isLb_def greatestP_def setle_def setge_def)
1.2158 -qed
1.2159 -
1.2160 -(* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
1.2161 -lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
1.2162 - apply (frule isGlb_isLb)
1.2163 - apply (frule_tac x = y in isGlb_isLb)
1.2164 - apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
1.2165 - done
1.2166 -
1.2167 -lemma rinf_insert: assumes "bounded S"
1.2168 - shows "rinf(insert x S) = (if S = {} then x else min x (rinf S))" (is "?lhs = ?rhs")
1.2169 -proof(cases "S={}")
1.2170 - case True thus ?thesis using rinf_finite_in[of "{x}"] by auto
1.2171 -next
1.2172 - let ?S = "insert x S"
1.2173 - case False
1.2174 - hence *:"\<forall>x\<in>S. x \<ge> rinf S" using bounded_has_rinf(1)[of S] using assms by auto
1.2175 - hence "min x (rinf S) <=* insert x S" unfolding setge_def by auto
1.2176 - hence "isGlb UNIV ?S (rinf ?S)" using rinf[of ?S] by auto
1.2177 - moreover
1.2178 - have **:"isLb UNIV ?S (min x (rinf S))" unfolding isLb_def setge_def using * by auto
1.2179 - { fix y assume as:"isLb UNIV (insert x S) y"
1.2180 - hence "min x (rinf S) \<ge> y" unfolding isLb_def using rinf_ge[OF `S\<noteq>{}`]
1.2181 - unfolding setge_def by auto }
1.2182 - hence "isLb UNIV (insert x S) *<= min x (rinf S)" unfolding setle_def Ball_def mem_def by auto
1.2183 - hence "isGlb UNIV ?S (min x (rinf S))" using ** isGlbI2[of UNIV ?S "min x (rinf S)"] unfolding Collect_def by auto
1.2184 - ultimately show ?thesis using real_isGlb_unique[of UNIV ?S] using `S\<noteq>{}` by auto
1.2185 -qed
1.2186 -
1.2187 -lemma inf_insert_finite: "finite S ==> rinf(insert x S) = (if S = {} then x else min x (rinf S))"
1.2188 - by (rule rinf_insert, rule finite_imp_bounded, simp)
1.2189 -
1.2190 -subsection{* Compactness (the definition is the one based on convegent subsequences). *}
1.2191 -
1.2192 -definition
1.2193 - compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
1.2194 - "compact S \<longleftrightarrow>
1.2195 - (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
1.2196 - (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
1.2197 -
1.2198 -text {*
1.2199 - A metric space (or topological vector space) is said to have the
1.2200 - Heine-Borel property if every closed and bounded subset is compact.
1.2201 -*}
1.2202 -
1.2203 -class heine_borel =
1.2204 - assumes bounded_imp_convergent_subsequence:
1.2205 - "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
1.2206 - \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1.2207 -
1.2208 -lemma bounded_closed_imp_compact:
1.2209 - fixes s::"'a::heine_borel set"
1.2210 - assumes "bounded s" and "closed s" shows "compact s"
1.2211 -proof (unfold compact_def, clarify)
1.2212 - fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
1.2213 - obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
1.2214 - using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
1.2215 - from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
1.2216 - have "l \<in> s" using `closed s` fr l
1.2217 - unfolding closed_sequential_limits by blast
1.2218 - show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1.2219 - using `l \<in> s` r l by blast
1.2220 -qed
1.2221 -
1.2222 -lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
1.2223 -proof(induct n)
1.2224 - show "0 \<le> r 0" by auto
1.2225 -next
1.2226 - fix n assume "n \<le> r n"
1.2227 - moreover have "r n < r (Suc n)"
1.2228 - using assms [unfolded subseq_def] by auto
1.2229 - ultimately show "Suc n \<le> r (Suc n)" by auto
1.2230 -qed
1.2231 -
1.2232 -lemma eventually_subseq:
1.2233 - assumes r: "subseq r"
1.2234 - shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
1.2235 -unfolding eventually_sequentially
1.2236 -by (metis subseq_bigger [OF r] le_trans)
1.2237 -
1.2238 -lemma lim_subseq:
1.2239 - "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
1.2240 -unfolding tendsto_def eventually_sequentially o_def
1.2241 -by (metis subseq_bigger le_trans)
1.2242 -
1.2243 -lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
1.2244 - unfolding Ex1_def
1.2245 - apply (rule_tac x="nat_rec e f" in exI)
1.2246 - apply (rule conjI)+
1.2247 -apply (rule def_nat_rec_0, simp)
1.2248 -apply (rule allI, rule def_nat_rec_Suc, simp)
1.2249 -apply (rule allI, rule impI, rule ext)
1.2250 -apply (erule conjE)
1.2251 -apply (induct_tac x)
1.2252 -apply (simp add: nat_rec_0)
1.2253 -apply (erule_tac x="n" in allE)
1.2254 -apply (simp)
1.2255 -done
1.2256 -
1.2257 -lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
1.2258 - assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
1.2259 - shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
1.2260 -proof-
1.2261 - have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
1.2262 - then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
1.2263 - { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
1.2264 - { fix n::nat
1.2265 - obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
1.2266 - have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
1.2267 - with n have "s N \<le> t - e" using `e>0` by auto
1.2268 - 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 }
1.2269 - hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
1.2270 - hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
1.2271 - thus ?thesis by blast
1.2272 -qed
1.2273 -
1.2274 -lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
1.2275 - assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
1.2276 - shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
1.2277 - using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
1.2278 - unfolding monoseq_def incseq_def
1.2279 - apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
1.2280 - unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
1.2281 -
1.2282 -lemma compact_real_lemma:
1.2283 - assumes "\<forall>n::nat. abs(s n) \<le> b"
1.2284 - shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
1.2285 -proof-
1.2286 - obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
1.2287 - using seq_monosub[of s] by auto
1.2288 - thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
1.2289 - unfolding tendsto_iff dist_norm eventually_sequentially by auto
1.2290 -qed
1.2291 -
1.2292 -instance real :: heine_borel
1.2293 -proof
1.2294 - fix s :: "real set" and f :: "nat \<Rightarrow> real"
1.2295 - assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
1.2296 - then obtain b where b: "\<forall>n. abs (f n) \<le> b"
1.2297 - unfolding bounded_iff by auto
1.2298 - obtain l :: real and r :: "nat \<Rightarrow> nat" where
1.2299 - r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
1.2300 - using compact_real_lemma [OF b] by auto
1.2301 - thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1.2302 - by auto
1.2303 -qed
1.2304 -
1.2305 -lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
1.2306 -unfolding bounded_def
1.2307 -apply clarify
1.2308 -apply (rule_tac x="x $ i" in exI)
1.2309 -apply (rule_tac x="e" in exI)
1.2310 -apply clarify
1.2311 -apply (rule order_trans [OF dist_nth_le], simp)
1.2312 -done
1.2313 -
1.2314 -lemma compact_lemma:
1.2315 - fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n::finite"
1.2316 - assumes "bounded s" and "\<forall>n. f n \<in> s"
1.2317 - shows "\<forall>d.
1.2318 - \<exists>l r. subseq r \<and>
1.2319 - (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
1.2320 -proof
1.2321 - fix d::"'n set" have "finite d" by simp
1.2322 - thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
1.2323 - (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
1.2324 - proof(induct d) case empty thus ?case unfolding subseq_def by auto
1.2325 - next case (insert k d)
1.2326 - have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component)
1.2327 - 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"
1.2328 - using insert(3) by auto
1.2329 - have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
1.2330 - obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
1.2331 - using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
1.2332 - def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
1.2333 - using r1 and r2 unfolding r_def o_def subseq_def by auto
1.2334 - moreover
1.2335 - def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
1.2336 - { fix e::real assume "e>0"
1.2337 - from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
1.2338 - from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
1.2339 - from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
1.2340 - by (rule eventually_subseq)
1.2341 - have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
1.2342 - using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
1.2343 - }
1.2344 - ultimately show ?case by auto
1.2345 - qed
1.2346 -qed
1.2347 -
1.2348 -instance "^" :: (heine_borel, finite) heine_borel
1.2349 -proof
1.2350 - fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
1.2351 - assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
1.2352 - then obtain l r where r: "subseq r"
1.2353 - and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
1.2354 - using compact_lemma [OF s f] by blast
1.2355 - let ?d = "UNIV::'b set"
1.2356 - { fix e::real assume "e>0"
1.2357 - hence "0 < e / (real_of_nat (card ?d))"
1.2358 - using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
1.2359 - with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
1.2360 - by simp
1.2361 - moreover
1.2362 - { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
1.2363 - have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
1.2364 - unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)
1.2365 - also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
1.2366 - by (rule setsum_strict_mono) (simp_all add: n)
1.2367 - finally have "dist (f (r n)) l < e" by simp
1.2368 - }
1.2369 - ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
1.2370 - by (rule eventually_elim1)
1.2371 - }
1.2372 - hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
1.2373 - with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
1.2374 -qed
1.2375 -
1.2376 -lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
1.2377 -unfolding bounded_def
1.2378 -apply clarify
1.2379 -apply (rule_tac x="a" in exI)
1.2380 -apply (rule_tac x="e" in exI)
1.2381 -apply clarsimp
1.2382 -apply (drule (1) bspec)
1.2383 -apply (simp add: dist_Pair_Pair)
1.2384 -apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
1.2385 -done
1.2386 -
1.2387 -lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
1.2388 -unfolding bounded_def
1.2389 -apply clarify
1.2390 -apply (rule_tac x="b" in exI)
1.2391 -apply (rule_tac x="e" in exI)
1.2392 -apply clarsimp
1.2393 -apply (drule (1) bspec)
1.2394 -apply (simp add: dist_Pair_Pair)
1.2395 -apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
1.2396 -done
1.2397 -
1.2398 -instance "*" :: (heine_borel, heine_borel) heine_borel
1.2399 -proof
1.2400 - fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
1.2401 - assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
1.2402 - from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
1.2403 - from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
1.2404 - obtain l1 r1 where r1: "subseq r1"
1.2405 - and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
1.2406 - using bounded_imp_convergent_subsequence [OF s1 f1]
1.2407 - unfolding o_def by fast
1.2408 - from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
1.2409 - from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
1.2410 - obtain l2 r2 where r2: "subseq r2"
1.2411 - and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
1.2412 - using bounded_imp_convergent_subsequence [OF s2 f2]
1.2413 - unfolding o_def by fast
1.2414 - have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
1.2415 - using lim_subseq [OF r2 l1] unfolding o_def .
1.2416 - have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
1.2417 - using tendsto_Pair [OF l1' l2] unfolding o_def by simp
1.2418 - have r: "subseq (r1 \<circ> r2)"
1.2419 - using r1 r2 unfolding subseq_def by simp
1.2420 - show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1.2421 - using l r by fast
1.2422 -qed
1.2423 -
1.2424 -subsection{* Completeness. *}
1.2425 -
1.2426 -lemma cauchy_def:
1.2427 - "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
1.2428 -unfolding Cauchy_def by blast
1.2429 -
1.2430 -definition
1.2431 - complete :: "'a::metric_space set \<Rightarrow> bool" where
1.2432 - "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
1.2433 - --> (\<exists>l \<in> s. (f ---> l) sequentially))"
1.2434 -
1.2435 -lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
1.2436 -proof-
1.2437 - { assume ?rhs
1.2438 - { fix e::real
1.2439 - assume "e>0"
1.2440 - with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
1.2441 - by (erule_tac x="e/2" in allE) auto
1.2442 - { fix n m
1.2443 - assume nm:"N \<le> m \<and> N \<le> n"
1.2444 - hence "dist (s m) (s n) < e" using N
1.2445 - using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
1.2446 - by blast
1.2447 - }
1.2448 - hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
1.2449 - by blast
1.2450 - }
1.2451 - hence ?lhs
1.2452 - unfolding cauchy_def
1.2453 - by blast
1.2454 - }
1.2455 - thus ?thesis
1.2456 - unfolding cauchy_def
1.2457 - using dist_triangle_half_l
1.2458 - by blast
1.2459 -qed
1.2460 -
1.2461 -lemma convergent_imp_cauchy:
1.2462 - "(s ---> l) sequentially ==> Cauchy s"
1.2463 -proof(simp only: cauchy_def, rule, rule)
1.2464 - fix e::real assume "e>0" "(s ---> l) sequentially"
1.2465 - 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
1.2466 - 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
1.2467 -qed
1.2468 -
1.2469 -lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded {y. (\<exists>n::nat. y = s n)}"
1.2470 -proof-
1.2471 - 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
1.2472 - hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
1.2473 - moreover
1.2474 - have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
1.2475 - then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
1.2476 - unfolding bounded_any_center [where a="s N"] by auto
1.2477 - ultimately show "?thesis"
1.2478 - unfolding bounded_any_center [where a="s N"]
1.2479 - apply(rule_tac x="max a 1" in exI) apply auto
1.2480 - apply(erule_tac x=n in allE) apply(erule_tac x=n in ballE) by auto
1.2481 -qed
1.2482 -
1.2483 -lemma compact_imp_complete: assumes "compact s" shows "complete s"
1.2484 -proof-
1.2485 - { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
1.2486 - 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
1.2487 -
1.2488 - note lr' = subseq_bigger [OF lr(2)]
1.2489 -
1.2490 - { fix e::real assume "e>0"
1.2491 - 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
1.2492 - 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
1.2493 - { fix n::nat assume n:"n \<ge> max N M"
1.2494 - have "dist ((f \<circ> r) n) l < e/2" using n M by auto
1.2495 - moreover have "r n \<ge> N" using lr'[of n] n by auto
1.2496 - hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
1.2497 - 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) }
1.2498 - hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
1.2499 - hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto }
1.2500 - thus ?thesis unfolding complete_def by auto
1.2501 -qed
1.2502 -
1.2503 -instance heine_borel < complete_space
1.2504 -proof
1.2505 - fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
1.2506 - hence "bounded (range f)" unfolding image_def
1.2507 - using cauchy_imp_bounded [of f] by auto
1.2508 - hence "compact (closure (range f))"
1.2509 - using bounded_closed_imp_compact [of "closure (range f)"] by auto
1.2510 - hence "complete (closure (range f))"
1.2511 - using compact_imp_complete by auto
1.2512 - moreover have "\<forall>n. f n \<in> closure (range f)"
1.2513 - using closure_subset [of "range f"] by auto
1.2514 - ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
1.2515 - using `Cauchy f` unfolding complete_def by auto
1.2516 - then show "convergent f"
1.2517 - unfolding convergent_def LIMSEQ_conv_tendsto [symmetric] by auto
1.2518 -qed
1.2519 -
1.2520 -lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
1.2521 -proof(simp add: complete_def, rule, rule)
1.2522 - fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
1.2523 - hence "convergent f" by (rule Cauchy_convergent)
1.2524 - hence "\<exists>l. f ----> l" unfolding convergent_def .
1.2525 - thus "\<exists>l. (f ---> l) sequentially" unfolding LIMSEQ_conv_tendsto .
1.2526 -qed
1.2527 -
1.2528 -lemma complete_imp_closed: assumes "complete s" shows "closed s"
1.2529 -proof -
1.2530 - { fix x assume "x islimpt s"
1.2531 - then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
1.2532 - unfolding islimpt_sequential by auto
1.2533 - then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
1.2534 - using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
1.2535 - hence "x \<in> s" using Lim_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
1.2536 - }
1.2537 - thus "closed s" unfolding closed_limpt by auto
1.2538 -qed
1.2539 -
1.2540 -lemma complete_eq_closed:
1.2541 - fixes s :: "'a::complete_space set"
1.2542 - shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
1.2543 -proof
1.2544 - assume ?lhs thus ?rhs by (rule complete_imp_closed)
1.2545 -next
1.2546 - assume ?rhs
1.2547 - { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
1.2548 - then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
1.2549 - 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 }
1.2550 - thus ?lhs unfolding complete_def by auto
1.2551 -qed
1.2552 -
1.2553 -lemma convergent_eq_cauchy:
1.2554 - fixes s :: "nat \<Rightarrow> 'a::complete_space"
1.2555 - shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
1.2556 -proof
1.2557 - assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
1.2558 - thus ?rhs using convergent_imp_cauchy by auto
1.2559 -next
1.2560 - assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
1.2561 -qed
1.2562 -
1.2563 -lemma convergent_imp_bounded:
1.2564 - fixes s :: "nat \<Rightarrow> 'a::metric_space"
1.2565 - shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
1.2566 - using convergent_imp_cauchy[of s]
1.2567 - using cauchy_imp_bounded[of s]
1.2568 - unfolding image_def
1.2569 - by auto
1.2570 -
1.2571 -subsection{* Total boundedness. *}
1.2572 -
1.2573 -fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
1.2574 - "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
1.2575 -declare helper_1.simps[simp del]
1.2576 -
1.2577 -lemma compact_imp_totally_bounded:
1.2578 - assumes "compact s"
1.2579 - shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
1.2580 -proof(rule, rule, rule ccontr)
1.2581 - 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)"
1.2582 - def x \<equiv> "helper_1 s e"
1.2583 - { fix n
1.2584 - have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
1.2585 - proof(induct_tac rule:nat_less_induct)
1.2586 - fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
1.2587 - assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
1.2588 - 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
1.2589 - then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
1.2590 - have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
1.2591 - apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
1.2592 - thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
1.2593 - qed }
1.2594 - hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
1.2595 - 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
1.2596 - from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
1.2597 - 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
1.2598 - show False
1.2599 - using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
1.2600 - using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
1.2601 - using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
1.2602 -qed
1.2603 -
1.2604 -subsection{* Heine-Borel theorem (following Burkill \& Burkill vol. 2) *}
1.2605 -
1.2606 -lemma heine_borel_lemma: fixes s::"'a::metric_space set"
1.2607 - assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
1.2608 - shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
1.2609 -proof(rule ccontr)
1.2610 - assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
1.2611 - hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
1.2612 - { fix n::nat
1.2613 - have "1 / real (n + 1) > 0" by auto
1.2614 - 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 }
1.2615 - 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
1.2616 - 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)"
1.2617 - 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
1.2618 -
1.2619 - then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
1.2620 - using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
1.2621 -
1.2622 - obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
1.2623 - then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
1.2624 - using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
1.2625 -
1.2626 - then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
1.2627 - using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
1.2628 -
1.2629 - obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
1.2630 - have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
1.2631 - apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
1.2632 - using subseq_bigger[OF r, of "N1 + N2"] by auto
1.2633 -
1.2634 - def x \<equiv> "(f (r (N1 + N2)))"
1.2635 - have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
1.2636 - using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
1.2637 - have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
1.2638 - then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
1.2639 -
1.2640 - have "dist x l < e/2" using N1 unfolding x_def o_def by auto
1.2641 - hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
1.2642 -
1.2643 - thus False using e and `y\<notin>b` by auto
1.2644 -qed
1.2645 -
1.2646 -lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
1.2647 - \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
1.2648 -proof clarify
1.2649 - fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
1.2650 - 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
1.2651 - hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
1.2652 - 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
1.2653 - then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
1.2654 -
1.2655 - 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
1.2656 - then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
1.2657 -
1.2658 - have "finite (bb ` k)" using k(1) by auto
1.2659 - moreover
1.2660 - { fix x assume "x\<in>s"
1.2661 - hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
1.2662 - hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
1.2663 - hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
1.2664 - }
1.2665 - 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
1.2666 -qed
1.2667 -
1.2668 -subsection{* Bolzano-Weierstrass property. *}
1.2669 -
1.2670 -lemma heine_borel_imp_bolzano_weierstrass:
1.2671 - 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'))"
1.2672 - "infinite t" "t \<subseteq> s"
1.2673 - shows "\<exists>x \<in> s. x islimpt t"
1.2674 -proof(rule ccontr)
1.2675 - assume "\<not> (\<exists>x \<in> s. x islimpt t)"
1.2676 - 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
1.2677 - 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
1.2678 - obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
1.2679 - using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
1.2680 - from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
1.2681 - { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
1.2682 - 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
1.2683 - 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 }
1.2684 - hence "infinite (f ` t)" using assms(2) using finite_imageD[unfolded inj_on_def, of f t] by auto
1.2685 - moreover
1.2686 - { fix x assume "x\<in>t" "f x \<notin> g"
1.2687 - from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
1.2688 - then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
1.2689 - hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
1.2690 - hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
1.2691 - hence "f ` t \<subseteq> g" by auto
1.2692 - ultimately show False using g(2) using finite_subset by auto
1.2693 -qed
1.2694 -
1.2695 -subsection{* Complete the chain of compactness variants. *}
1.2696 -
1.2697 -primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
1.2698 - "helper_2 beyond 0 = beyond 0" |
1.2699 - "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
1.2700 -
1.2701 -lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
1.2702 - assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
1.2703 - shows "bounded s"
1.2704 -proof(rule ccontr)
1.2705 - assume "\<not> bounded s"
1.2706 - then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
1.2707 - unfolding bounded_any_center [where a=undefined]
1.2708 - apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
1.2709 - hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
1.2710 - unfolding linorder_not_le by auto
1.2711 - def x \<equiv> "helper_2 beyond"
1.2712 -
1.2713 - { fix m n ::nat assume "m<n"
1.2714 - hence "dist undefined (x m) + 1 < dist undefined (x n)"
1.2715 - proof(induct n)
1.2716 - case 0 thus ?case by auto
1.2717 - next
1.2718 - case (Suc n)
1.2719 - have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
1.2720 - unfolding x_def and helper_2.simps
1.2721 - using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
1.2722 - thus ?case proof(cases "m < n")
1.2723 - case True thus ?thesis using Suc and * by auto
1.2724 - next
1.2725 - case False hence "m = n" using Suc(2) by auto
1.2726 - thus ?thesis using * by auto
1.2727 - qed
1.2728 - qed } note * = this
1.2729 - { fix m n ::nat assume "m\<noteq>n"
1.2730 - have "1 < dist (x m) (x n)"
1.2731 - proof(cases "m<n")
1.2732 - case True
1.2733 - hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
1.2734 - thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
1.2735 - next
1.2736 - case False hence "n<m" using `m\<noteq>n` by auto
1.2737 - hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
1.2738 - thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
1.2739 - qed } note ** = this
1.2740 - { fix a b assume "x a = x b" "a \<noteq> b"
1.2741 - hence False using **[of a b] by auto }
1.2742 - hence "inj x" unfolding inj_on_def by auto
1.2743 - moreover
1.2744 - { fix n::nat
1.2745 - have "x n \<in> s"
1.2746 - proof(cases "n = 0")
1.2747 - case True thus ?thesis unfolding x_def using beyond by auto
1.2748 - next
1.2749 - case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
1.2750 - thus ?thesis unfolding x_def using beyond by auto
1.2751 - qed }
1.2752 - 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
1.2753 -
1.2754 - then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
1.2755 - 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
1.2756 - 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"]]
1.2757 - unfolding dist_nz by auto
1.2758 - show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
1.2759 -qed
1.2760 -
1.2761 -lemma sequence_infinite_lemma:
1.2762 - fixes l :: "'a::metric_space" (* TODO: generalize *)
1.2763 - assumes "\<forall>n::nat. (f n \<noteq> l)" "(f ---> l) sequentially"
1.2764 - shows "infinite {y. (\<exists> n. y = f n)}"
1.2765 -proof(rule ccontr)
1.2766 - let ?A = "(\<lambda>x. dist x l) ` {y. \<exists>n. y = f n}"
1.2767 - assume "\<not> infinite {y. \<exists>n. y = f n}"
1.2768 - hence **:"finite ?A" "?A \<noteq> {}" by auto
1.2769 - obtain k where k:"dist (f k) l = Min ?A" using Min_in[OF **] by auto
1.2770 - have "0 < Min ?A" using assms(1) unfolding dist_nz unfolding Min_gr_iff[OF **] by auto
1.2771 - then obtain N where "dist (f N) l < Min ?A" using assms(2)[unfolded Lim_sequentially, THEN spec[where x="Min ?A"]] by auto
1.2772 - moreover have "dist (f N) l \<in> ?A" by auto
1.2773 - ultimately show False using Min_le[OF **(1), of "dist (f N) l"] by auto
1.2774 -qed
1.2775 -
1.2776 -lemma sequence_unique_limpt:
1.2777 - fixes l :: "'a::metric_space" (* TODO: generalize *)
1.2778 - assumes "\<forall>n::nat. (f n \<noteq> l)" "(f ---> l) sequentially" "l' islimpt {y. (\<exists>n. y = f n)}"
1.2779 - shows "l' = l"
1.2780 -proof(rule ccontr)
1.2781 - def e \<equiv> "dist l' l"
1.2782 - assume "l' \<noteq> l" hence "e>0" unfolding dist_nz e_def by auto
1.2783 - then obtain N::nat where N:"\<forall>n\<ge>N. dist (f n) l < e / 2"
1.2784 - using assms(2)[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
1.2785 - 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}))"
1.2786 - have "d>0" using `e>0` unfolding d_def e_def using zero_le_dist[of _ l', unfolded order_le_less] by auto
1.2787 - 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
1.2788 - have "k\<ge>N" using k(1)[unfolded dist_nz] using k(2)[unfolded d_def]
1.2789 - by force
1.2790 - 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
1.2791 - thus False unfolding e_def by auto
1.2792 -qed
1.2793 -
1.2794 -lemma bolzano_weierstrass_imp_closed:
1.2795 - fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
1.2796 - assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
1.2797 - shows "closed s"
1.2798 -proof-
1.2799 - { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
1.2800 - hence "l \<in> s"
1.2801 - proof(cases "\<forall>n. x n \<noteq> l")
1.2802 - case False thus "l\<in>s" using as(1) by auto
1.2803 - next
1.2804 - case True note cas = this
1.2805 - with as(2) have "infinite {y. \<exists>n. y = x n}" using sequence_infinite_lemma[of x l] by auto
1.2806 - 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
1.2807 - thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
1.2808 - qed }
1.2809 - thus ?thesis unfolding closed_sequential_limits by fast
1.2810 -qed
1.2811 -
1.2812 -text{* Hence express everything as an equivalence. *}
1.2813 -
1.2814 -lemma compact_eq_heine_borel:
1.2815 - fixes s :: "'a::heine_borel set"
1.2816 - shows "compact s \<longleftrightarrow>
1.2817 - (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
1.2818 - --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
1.2819 -proof
1.2820 - assume ?lhs thus ?rhs using compact_imp_heine_borel[of s] by blast
1.2821 -next
1.2822 - assume ?rhs
1.2823 - hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
1.2824 - by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
1.2825 - thus ?lhs using bolzano_weierstrass_imp_bounded[of s] bolzano_weierstrass_imp_closed[of s] bounded_closed_imp_compact[of s] by blast
1.2826 -qed
1.2827 -
1.2828 -lemma compact_eq_bolzano_weierstrass:
1.2829 - fixes s :: "'a::heine_borel set"
1.2830 - shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
1.2831 -proof
1.2832 - assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
1.2833 -next
1.2834 - assume ?rhs thus ?lhs using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed bounded_closed_imp_compact by auto
1.2835 -qed
1.2836 -
1.2837 -lemma compact_eq_bounded_closed:
1.2838 - fixes s :: "'a::heine_borel set"
1.2839 - shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
1.2840 -proof
1.2841 - assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
1.2842 -next
1.2843 - assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
1.2844 -qed
1.2845 -
1.2846 -lemma compact_imp_bounded:
1.2847 - fixes s :: "'a::metric_space set"
1.2848 - shows "compact s ==> bounded s"
1.2849 -proof -
1.2850 - assume "compact s"
1.2851 - 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')"
1.2852 - by (rule compact_imp_heine_borel)
1.2853 - hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
1.2854 - using heine_borel_imp_bolzano_weierstrass[of s] by auto
1.2855 - thus "bounded s"
1.2856 - by (rule bolzano_weierstrass_imp_bounded)
1.2857 -qed
1.2858 -
1.2859 -lemma compact_imp_closed:
1.2860 - fixes s :: "'a::metric_space set"
1.2861 - shows "compact s ==> closed s"
1.2862 -proof -
1.2863 - assume "compact s"
1.2864 - 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')"
1.2865 - by (rule compact_imp_heine_borel)
1.2866 - hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
1.2867 - using heine_borel_imp_bolzano_weierstrass[of s] by auto
1.2868 - thus "closed s"
1.2869 - by (rule bolzano_weierstrass_imp_closed)
1.2870 -qed
1.2871 -
1.2872 -text{* In particular, some common special cases. *}
1.2873 -
1.2874 -lemma compact_empty[simp]:
1.2875 - "compact {}"
1.2876 - unfolding compact_def
1.2877 - by simp
1.2878 -
1.2879 -(* TODO: can any of the next 3 lemmas be generalized to metric spaces? *)
1.2880 -
1.2881 - (* FIXME : Rename *)
1.2882 -lemma compact_union[intro]:
1.2883 - fixes s t :: "'a::heine_borel set"
1.2884 - shows "compact s \<Longrightarrow> compact t ==> compact (s \<union> t)"
1.2885 - unfolding compact_eq_bounded_closed
1.2886 - using bounded_Un[of s t]
1.2887 - using closed_Un[of s t]
1.2888 - by simp
1.2889 -
1.2890 -lemma compact_inter[intro]:
1.2891 - fixes s t :: "'a::heine_borel set"
1.2892 - shows "compact s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
1.2893 - unfolding compact_eq_bounded_closed
1.2894 - using bounded_Int[of s t]
1.2895 - using closed_Int[of s t]
1.2896 - by simp
1.2897 -
1.2898 -lemma compact_inter_closed[intro]:
1.2899 - fixes s t :: "'a::heine_borel set"
1.2900 - shows "compact s \<Longrightarrow> closed t ==> compact (s \<inter> t)"
1.2901 - unfolding compact_eq_bounded_closed
1.2902 - using closed_Int[of s t]
1.2903 - using bounded_subset[of "s \<inter> t" s]
1.2904 - by blast
1.2905 -
1.2906 -lemma closed_inter_compact[intro]:
1.2907 - fixes s t :: "'a::heine_borel set"
1.2908 - shows "closed s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
1.2909 -proof-
1.2910 - assume "closed s" "compact t"
1.2911 - moreover
1.2912 - have "s \<inter> t = t \<inter> s" by auto ultimately
1.2913 - show ?thesis
1.2914 - using compact_inter_closed[of t s]
1.2915 - by auto
1.2916 -qed
1.2917 -
1.2918 -lemma closed_sing [simp]:
1.2919 - fixes a :: "'a::metric_space"
1.2920 - shows "closed {a}"
1.2921 - apply (clarsimp simp add: closed_def open_dist)
1.2922 - apply (rule ccontr)
1.2923 - apply (drule_tac x="dist x a" in spec)
1.2924 - apply (simp add: dist_nz dist_commute)
1.2925 - done
1.2926 -
1.2927 -lemma finite_imp_closed:
1.2928 - fixes s :: "'a::metric_space set"
1.2929 - shows "finite s ==> closed s"
1.2930 -proof (induct set: finite)
1.2931 - case empty show "closed {}" by simp
1.2932 -next
1.2933 - case (insert x F)
1.2934 - hence "closed ({x} \<union> F)" by (simp only: closed_Un closed_sing)
1.2935 - thus "closed (insert x F)" by simp
1.2936 -qed
1.2937 -
1.2938 -lemma finite_imp_compact:
1.2939 - fixes s :: "'a::heine_borel set"
1.2940 - shows "finite s ==> compact s"
1.2941 - unfolding compact_eq_bounded_closed
1.2942 - using finite_imp_closed finite_imp_bounded
1.2943 - by blast
1.2944 -
1.2945 -lemma compact_sing [simp]: "compact {a}"
1.2946 - unfolding compact_def o_def subseq_def
1.2947 - by (auto simp add: tendsto_const)
1.2948 -
1.2949 -lemma compact_cball[simp]:
1.2950 - fixes x :: "'a::heine_borel"
1.2951 - shows "compact(cball x e)"
1.2952 - using compact_eq_bounded_closed bounded_cball closed_cball
1.2953 - by blast
1.2954 -
1.2955 -lemma compact_frontier_bounded[intro]:
1.2956 - fixes s :: "'a::heine_borel set"
1.2957 - shows "bounded s ==> compact(frontier s)"
1.2958 - unfolding frontier_def
1.2959 - using compact_eq_bounded_closed
1.2960 - by blast
1.2961 -
1.2962 -lemma compact_frontier[intro]:
1.2963 - fixes s :: "'a::heine_borel set"
1.2964 - shows "compact s ==> compact (frontier s)"
1.2965 - using compact_eq_bounded_closed compact_frontier_bounded
1.2966 - by blast
1.2967 -
1.2968 -lemma frontier_subset_compact:
1.2969 - fixes s :: "'a::heine_borel set"
1.2970 - shows "compact s ==> frontier s \<subseteq> s"
1.2971 - using frontier_subset_closed compact_eq_bounded_closed
1.2972 - by blast
1.2973 -
1.2974 -lemma open_delete:
1.2975 - fixes s :: "'a::metric_space set"
1.2976 - shows "open s ==> open(s - {x})"
1.2977 - using open_Diff[of s "{x}"] closed_sing
1.2978 - by blast
1.2979 -
1.2980 -text{* Finite intersection property. I could make it an equivalence in fact. *}
1.2981 -
1.2982 -lemma compact_imp_fip:
1.2983 - fixes s :: "'a::heine_borel set"
1.2984 - assumes "compact s" "\<forall>t \<in> f. closed t"
1.2985 - "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
1.2986 - shows "s \<inter> (\<Inter> f) \<noteq> {}"
1.2987 -proof
1.2988 - assume as:"s \<inter> (\<Inter> f) = {}"
1.2989 - hence "s \<subseteq> \<Union>op - UNIV ` f" by auto
1.2990 - moreover have "Ball (op - UNIV ` f) open" using open_Diff closed_Diff using assms(2) by auto
1.2991 - 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
1.2992 - hence "finite (op - UNIV ` f') \<and> op - UNIV ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
1.2993 - hence "s \<inter> \<Inter>op - UNIV ` f' \<noteq> {}" using assms(3)[THEN spec[where x="op - UNIV ` f'"]] by auto
1.2994 - thus False using f'(3) unfolding subset_eq and Union_iff by blast
1.2995 -qed
1.2996 -
1.2997 -subsection{* Bounded closed nest property (proof does not use Heine-Borel). *}
1.2998 -
1.2999 -lemma bounded_closed_nest:
1.3000 - assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
1.3001 - "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
1.3002 - shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
1.3003 -proof-
1.3004 - 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
1.3005 - from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
1.3006 -
1.3007 - then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
1.3008 - unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
1.3009 -
1.3010 - { fix n::nat
1.3011 - { fix e::real assume "e>0"
1.3012 - with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
1.3013 - hence "dist ((x \<circ> r) (max N n)) l < e" by auto
1.3014 - moreover
1.3015 - have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
1.3016 - hence "(x \<circ> r) (max N n) \<in> s n"
1.3017 - using x apply(erule_tac x=n in allE)
1.3018 - using x apply(erule_tac x="r (max N n)" in allE)
1.3019 - using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
1.3020 - ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
1.3021 - }
1.3022 - hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
1.3023 - }
1.3024 - thus ?thesis by auto
1.3025 -qed
1.3026 -
1.3027 -text{* Decreasing case does not even need compactness, just completeness. *}
1.3028 -
1.3029 -lemma decreasing_closed_nest:
1.3030 - assumes "\<forall>n. closed(s n)"
1.3031 - "\<forall>n. (s n \<noteq> {})"
1.3032 - "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
1.3033 - "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
1.3034 - shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
1.3035 -proof-
1.3036 - have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
1.3037 - hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
1.3038 - then obtain t where t: "\<forall>n. t n \<in> s n" by auto
1.3039 - { fix e::real assume "e>0"
1.3040 - then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
1.3041 - { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
1.3042 - hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
1.3043 - hence "dist (t m) (t n) < e" using N by auto
1.3044 - }
1.3045 - hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
1.3046 - }
1.3047 - hence "Cauchy t" unfolding cauchy_def by auto
1.3048 - then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
1.3049 - { fix n::nat
1.3050 - { fix e::real assume "e>0"
1.3051 - then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
1.3052 - 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
1.3053 - hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
1.3054 - }
1.3055 - hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
1.3056 - }
1.3057 - then show ?thesis by auto
1.3058 -qed
1.3059 -
1.3060 -text{* Strengthen it to the intersection actually being a singleton. *}
1.3061 -
1.3062 -lemma decreasing_closed_nest_sing:
1.3063 - assumes "\<forall>n. closed(s n)"
1.3064 - "\<forall>n. s n \<noteq> {}"
1.3065 - "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
1.3066 - "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
1.3067 - shows "\<exists>a::'a::heine_borel. \<Inter> {t. (\<exists>n::nat. t = s n)} = {a}"
1.3068 -proof-
1.3069 - obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
1.3070 - { fix b assume b:"b \<in> \<Inter>{t. \<exists>n. t = s n}"
1.3071 - { fix e::real assume "e>0"
1.3072 - hence "dist a b < e" using assms(4 )using b using a by blast
1.3073 - }
1.3074 - hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz real_less_def)
1.3075 - }
1.3076 - with a have "\<Inter>{t. \<exists>n. t = s n} = {a}" by auto
1.3077 - thus ?thesis by auto
1.3078 -qed
1.3079 -
1.3080 -text{* Cauchy-type criteria for uniform convergence. *}
1.3081 -
1.3082 -lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
1.3083 - "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
1.3084 - (\<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")
1.3085 -proof(rule)
1.3086 - assume ?lhs
1.3087 - 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
1.3088 - { fix e::real assume "e>0"
1.3089 - 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
1.3090 - { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
1.3091 - hence "dist (s m x) (s n x) < e"
1.3092 - using N[THEN spec[where x=m], THEN spec[where x=x]]
1.3093 - using N[THEN spec[where x=n], THEN spec[where x=x]]
1.3094 - using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
1.3095 - 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 }
1.3096 - thus ?rhs by auto
1.3097 -next
1.3098 - assume ?rhs
1.3099 - 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
1.3100 - then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
1.3101 - using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
1.3102 - { fix e::real assume "e>0"
1.3103 - 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"
1.3104 - using `?rhs`[THEN spec[where x="e/2"]] by auto
1.3105 - { fix x assume "P x"
1.3106 - then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
1.3107 - using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
1.3108 - fix n::nat assume "n\<ge>N"
1.3109 - 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]]
1.3110 - 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) }
1.3111 - hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
1.3112 - thus ?lhs by auto
1.3113 -qed
1.3114 -
1.3115 -lemma uniformly_cauchy_imp_uniformly_convergent:
1.3116 - fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
1.3117 - 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"
1.3118 - "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
1.3119 - shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
1.3120 -proof-
1.3121 - 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"
1.3122 - using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
1.3123 - moreover
1.3124 - { fix x assume "P x"
1.3125 - hence "l x = l' x" using Lim_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
1.3126 - using l and assms(2) unfolding Lim_sequentially by blast }
1.3127 - ultimately show ?thesis by auto
1.3128 -qed
1.3129 -
1.3130 -subsection{* Define continuity over a net to take in restrictions of the set. *}
1.3131 -
1.3132 -definition
1.3133 - continuous :: "'a::t2_space net \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
1.3134 - "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
1.3135 -
1.3136 -lemma continuous_trivial_limit:
1.3137 - "trivial_limit net ==> continuous net f"
1.3138 - unfolding continuous_def tendsto_def trivial_limit_eq by auto
1.3139 -
1.3140 -lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
1.3141 - unfolding continuous_def
1.3142 - unfolding tendsto_def
1.3143 - using netlimit_within[of x s]
1.3144 - by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
1.3145 -
1.3146 -lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
1.3147 - using continuous_within [of x UNIV f] by (simp add: within_UNIV)
1.3148 -
1.3149 -lemma continuous_at_within:
1.3150 - assumes "continuous (at x) f" shows "continuous (at x within s) f"
1.3151 - using assms unfolding continuous_at continuous_within
1.3152 - by (rule Lim_at_within)
1.3153 -
1.3154 -text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
1.3155 -
1.3156 -lemma continuous_within_eps_delta:
1.3157 - "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)"
1.3158 - unfolding continuous_within and Lim_within
1.3159 - apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
1.3160 -
1.3161 -lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
1.3162 - \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
1.3163 - using continuous_within_eps_delta[of x UNIV f]
1.3164 - unfolding within_UNIV by blast
1.3165 -
1.3166 -text{* Versions in terms of open balls. *}
1.3167 -
1.3168 -lemma continuous_within_ball:
1.3169 - "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
1.3170 - f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
1.3171 -proof
1.3172 - assume ?lhs
1.3173 - { fix e::real assume "e>0"
1.3174 - 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"
1.3175 - using `?lhs`[unfolded continuous_within Lim_within] by auto
1.3176 - { fix y assume "y\<in>f ` (ball x d \<inter> s)"
1.3177 - hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
1.3178 - apply (auto simp add: dist_commute mem_ball) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
1.3179 - }
1.3180 - 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) }
1.3181 - thus ?rhs by auto
1.3182 -next
1.3183 - assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
1.3184 - apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
1.3185 -qed
1.3186 -
1.3187 -lemma continuous_at_ball:
1.3188 - "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
1.3189 -proof
1.3190 - assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
1.3191 - 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)
1.3192 - unfolding dist_nz[THEN sym] by auto
1.3193 -next
1.3194 - assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
1.3195 - 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)
1.3196 -qed
1.3197 -
1.3198 -text{* For setwise continuity, just start from the epsilon-delta definitions. *}
1.3199 -
1.3200 -definition
1.3201 - continuous_on :: "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
1.3202 - "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)"
1.3203 -
1.3204 -
1.3205 -definition
1.3206 - uniformly_continuous_on ::
1.3207 - "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
1.3208 - "uniformly_continuous_on s f \<longleftrightarrow>
1.3209 - (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall> x'\<in>s. dist x' x < d
1.3210 - --> dist (f x') (f x) < e)"
1.3211 -
1.3212 -text{* Some simple consequential lemmas. *}
1.3213 -
1.3214 -lemma uniformly_continuous_imp_continuous:
1.3215 - " uniformly_continuous_on s f ==> continuous_on s f"
1.3216 - unfolding uniformly_continuous_on_def continuous_on_def by blast
1.3217 -
1.3218 -lemma continuous_at_imp_continuous_within:
1.3219 - "continuous (at x) f ==> continuous (at x within s) f"
1.3220 - unfolding continuous_within continuous_at using Lim_at_within by auto
1.3221 -
1.3222 -lemma continuous_at_imp_continuous_on: assumes "(\<forall>x \<in> s. continuous (at x) f)"
1.3223 - shows "continuous_on s f"
1.3224 -proof(simp add: continuous_at continuous_on_def, rule, rule, rule)
1.3225 - fix x and e::real assume "x\<in>s" "e>0"
1.3226 - hence "eventually (\<lambda>xa. dist (f xa) (f x) < e) (at x)" using assms unfolding continuous_at tendsto_iff by auto
1.3227 - 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
1.3228 - { fix x' assume "\<not> 0 < dist x' x"
1.3229 - hence "x=x'"
1.3230 - using dist_nz[of x' x] by auto
1.3231 - hence "dist (f x') (f x) < e" using `e>0` by auto
1.3232 - }
1.3233 - thus "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using d by auto
1.3234 -qed
1.3235 -
1.3236 -lemma continuous_on_eq_continuous_within:
1.3237 - "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)" (is "?lhs = ?rhs")
1.3238 -proof
1.3239 - assume ?rhs
1.3240 - { fix x assume "x\<in>s"
1.3241 - fix e::real assume "e>0"
1.3242 - assume "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
1.3243 - 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
1.3244 - { fix x' assume as:"x'\<in>s" "dist x' x < d"
1.3245 - 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) }
1.3246 - hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using `d>0` by auto
1.3247 - }
1.3248 - thus ?lhs using `?rhs` unfolding continuous_on_def continuous_within Lim_within by auto
1.3249 -next
1.3250 - assume ?lhs
1.3251 - thus ?rhs unfolding continuous_on_def continuous_within Lim_within by blast
1.3252 -qed
1.3253 -
1.3254 -lemma continuous_on:
1.3255 - "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. (f ---> f(x)) (at x within s))"
1.3256 - by (auto simp add: continuous_on_eq_continuous_within continuous_within)
1.3257 -
1.3258 -lemma continuous_on_eq_continuous_at:
1.3259 - "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
1.3260 - by (auto simp add: continuous_on continuous_at Lim_within_open)
1.3261 -
1.3262 -lemma continuous_within_subset:
1.3263 - "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
1.3264 - ==> continuous (at x within t) f"
1.3265 - unfolding continuous_within by(metis Lim_within_subset)
1.3266 -
1.3267 -lemma continuous_on_subset:
1.3268 - "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
1.3269 - unfolding continuous_on by (metis subset_eq Lim_within_subset)
1.3270 -
1.3271 -lemma continuous_on_interior:
1.3272 - "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
1.3273 -unfolding interior_def
1.3274 -apply simp
1.3275 -by (meson continuous_on_eq_continuous_at continuous_on_subset)
1.3276 -
1.3277 -lemma continuous_on_eq:
1.3278 - "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f
1.3279 - ==> continuous_on s g"
1.3280 - by (simp add: continuous_on_def)
1.3281 -
1.3282 -text{* Characterization of various kinds of continuity in terms of sequences. *}
1.3283 -
1.3284 -(* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
1.3285 -lemma continuous_within_sequentially:
1.3286 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
1.3287 - shows "continuous (at a within s) f \<longleftrightarrow>
1.3288 - (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
1.3289 - --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
1.3290 -proof
1.3291 - assume ?lhs
1.3292 - { 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"
1.3293 - fix e::real assume "e>0"
1.3294 - 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
1.3295 - from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
1.3296 - hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
1.3297 - apply(rule_tac x=N in exI) using N d apply auto using x(1)
1.3298 - apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
1.3299 - apply(erule_tac x="x n" in ballE) apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
1.3300 - }
1.3301 - thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
1.3302 -next
1.3303 - assume ?rhs
1.3304 - { fix e::real assume "e>0"
1.3305 - 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)"
1.3306 - 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
1.3307 - 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)"
1.3308 - 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
1.3309 - { fix d::real assume "d>0"
1.3310 - 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
1.3311 - then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
1.3312 - { fix n::nat assume n:"n\<ge>N"
1.3313 - hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
1.3314 - 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)
1.3315 - ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
1.3316 - }
1.3317 - hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
1.3318 - }
1.3319 - 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
1.3320 - 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
1.3321 - hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
1.3322 - }
1.3323 - thus ?lhs unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
1.3324 -qed
1.3325 -
1.3326 -lemma continuous_at_sequentially:
1.3327 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
1.3328 - shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
1.3329 - --> ((f o x) ---> f a) sequentially)"
1.3330 - using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
1.3331 -
1.3332 -lemma continuous_on_sequentially:
1.3333 - "continuous_on s f \<longleftrightarrow> (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
1.3334 - --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
1.3335 -proof
1.3336 - assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
1.3337 -next
1.3338 - assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
1.3339 -qed
1.3340 -
1.3341 -lemma uniformly_continuous_on_sequentially:
1.3342 - fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
1.3343 - shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
1.3344 - ((\<lambda>n. x n - y n) ---> 0) sequentially
1.3345 - \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
1.3346 -proof
1.3347 - assume ?lhs
1.3348 - { 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"
1.3349 - { fix e::real assume "e>0"
1.3350 - 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"
1.3351 - using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
1.3352 - 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
1.3353 - { fix n assume "n\<ge>N"
1.3354 - hence "norm (f (x n) - f (y n) - 0) < e"
1.3355 - 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
1.3356 - unfolding dist_commute and dist_norm by simp }
1.3357 - hence "\<exists>N. \<forall>n\<ge>N. norm (f (x n) - f (y n) - 0) < e" by auto }
1.3358 - hence "((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially" unfolding Lim_sequentially and dist_norm by auto }
1.3359 - thus ?rhs by auto
1.3360 -next
1.3361 - assume ?rhs
1.3362 - { assume "\<not> ?lhs"
1.3363 - 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
1.3364 - 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"
1.3365 - 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
1.3366 - by (auto simp add: dist_commute)
1.3367 - def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
1.3368 - def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
1.3369 - 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"
1.3370 - unfolding x_def and y_def using fa by auto
1.3371 - have 1:"\<And>(x::'a) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
1.3372 - have 2:"\<And>(x::'b) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
1.3373 - { fix e::real assume "e>0"
1.3374 - 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
1.3375 - { fix n::nat assume "n\<ge>N"
1.3376 - hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
1.3377 - also have "\<dots> < e" using N by auto
1.3378 - finally have "inverse (real n + 1) < e" by auto
1.3379 - hence "dist (x n - y n) 0 < e" unfolding 1 using xy0[THEN spec[where x=n]] by auto }
1.3380 - hence "\<exists>N. \<forall>n\<ge>N. dist (x n - y n) 0 < e" by auto }
1.3381 - 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
1.3382 - hence False unfolding 2 using fxy and `e>0` by auto }
1.3383 - thus ?lhs unfolding uniformly_continuous_on_def by blast
1.3384 -qed
1.3385 -
1.3386 -text{* The usual transformation theorems. *}
1.3387 -
1.3388 -lemma continuous_transform_within:
1.3389 - fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
1.3390 - assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
1.3391 - "continuous (at x within s) f"
1.3392 - shows "continuous (at x within s) g"
1.3393 -proof-
1.3394 - { fix e::real assume "e>0"
1.3395 - 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
1.3396 - { fix x' assume "x'\<in>s" "0 < dist x' x" "dist x' x < (min d d')"
1.3397 - hence "dist (f x') (g x) < e" using assms(2,3) apply(erule_tac x=x in ballE) using d' by auto }
1.3398 - 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
1.3399 - 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 }
1.3400 - hence "(f ---> g x) (at x within s)" unfolding Lim_within using assms(1) by auto
1.3401 - thus ?thesis unfolding continuous_within using Lim_transform_within[of d s x f g "g x"] using assms by blast
1.3402 -qed
1.3403 -
1.3404 -lemma continuous_transform_at:
1.3405 - fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
1.3406 - assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
1.3407 - "continuous (at x) f"
1.3408 - shows "continuous (at x) g"
1.3409 -proof-
1.3410 - { fix e::real assume "e>0"
1.3411 - 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
1.3412 - { fix x' assume "0 < dist x' x" "dist x' x < (min d d')"
1.3413 - hence "dist (f x') (g x) < e" using assms(2) apply(erule_tac x=x in allE) using d' by auto
1.3414 - }
1.3415 - hence "\<forall>xa. 0 < dist xa x \<and> dist xa x < (min d d') \<longrightarrow> dist (f xa) (g x) < e" by blast
1.3416 - 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
1.3417 - }
1.3418 - hence "(f ---> g x) (at x)" unfolding Lim_at using assms(1) by auto
1.3419 - thus ?thesis unfolding continuous_at using Lim_transform_at[of d x f g "g x"] using assms by blast
1.3420 -qed
1.3421 -
1.3422 -text{* Combination results for pointwise continuity. *}
1.3423 -
1.3424 -lemma continuous_const: "continuous net (\<lambda>x. c)"
1.3425 - by (auto simp add: continuous_def Lim_const)
1.3426 -
1.3427 -lemma continuous_cmul:
1.3428 - fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1.3429 - shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
1.3430 - by (auto simp add: continuous_def Lim_cmul)
1.3431 -
1.3432 -lemma continuous_neg:
1.3433 - fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1.3434 - shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
1.3435 - by (auto simp add: continuous_def Lim_neg)
1.3436 -
1.3437 -lemma continuous_add:
1.3438 - fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1.3439 - shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
1.3440 - by (auto simp add: continuous_def Lim_add)
1.3441 -
1.3442 -lemma continuous_sub:
1.3443 - fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
1.3444 - shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
1.3445 - by (auto simp add: continuous_def Lim_sub)
1.3446 -
1.3447 -text{* Same thing for setwise continuity. *}
1.3448 -
1.3449 -lemma continuous_on_const:
1.3450 - "continuous_on s (\<lambda>x. c)"
1.3451 - unfolding continuous_on_eq_continuous_within using continuous_const by blast
1.3452 -
1.3453 -lemma continuous_on_cmul:
1.3454 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.3455 - shows "continuous_on s f ==> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
1.3456 - unfolding continuous_on_eq_continuous_within using continuous_cmul by blast
1.3457 -
1.3458 -lemma continuous_on_neg:
1.3459 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.3460 - shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
1.3461 - unfolding continuous_on_eq_continuous_within using continuous_neg by blast
1.3462 -
1.3463 -lemma continuous_on_add:
1.3464 - fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.3465 - shows "continuous_on s f \<Longrightarrow> continuous_on s g
1.3466 - \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
1.3467 - unfolding continuous_on_eq_continuous_within using continuous_add by blast
1.3468 -
1.3469 -lemma continuous_on_sub:
1.3470 - fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.3471 - shows "continuous_on s f \<Longrightarrow> continuous_on s g
1.3472 - \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
1.3473 - unfolding continuous_on_eq_continuous_within using continuous_sub by blast
1.3474 -
1.3475 -text{* Same thing for uniform continuity, using sequential formulations. *}
1.3476 -
1.3477 -lemma uniformly_continuous_on_const:
1.3478 - "uniformly_continuous_on s (\<lambda>x. c)"
1.3479 - unfolding uniformly_continuous_on_def by simp
1.3480 -
1.3481 -lemma uniformly_continuous_on_cmul:
1.3482 - fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
1.3483 - (* FIXME: generalize 'a to metric_space *)
1.3484 - assumes "uniformly_continuous_on s f"
1.3485 - shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
1.3486 -proof-
1.3487 - { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
1.3488 - hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
1.3489 - using Lim_cmul[of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
1.3490 - unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
1.3491 - }
1.3492 - thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
1.3493 -qed
1.3494 -
1.3495 -lemma dist_minus:
1.3496 - fixes x y :: "'a::real_normed_vector"
1.3497 - shows "dist (- x) (- y) = dist x y"
1.3498 - unfolding dist_norm minus_diff_minus norm_minus_cancel ..
1.3499 -
1.3500 -lemma uniformly_continuous_on_neg:
1.3501 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.3502 - shows "uniformly_continuous_on s f
1.3503 - ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
1.3504 - unfolding uniformly_continuous_on_def dist_minus .
1.3505 -
1.3506 -lemma uniformly_continuous_on_add:
1.3507 - fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
1.3508 - assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
1.3509 - shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
1.3510 -proof-
1.3511 - { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
1.3512 - "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
1.3513 - hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
1.3514 - 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
1.3515 - 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 }
1.3516 - thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
1.3517 -qed
1.3518 -
1.3519 -lemma uniformly_continuous_on_sub:
1.3520 - fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
1.3521 - shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
1.3522 - ==> uniformly_continuous_on s (\<lambda>x. f x - g x)"
1.3523 - unfolding ab_diff_minus
1.3524 - using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
1.3525 - using uniformly_continuous_on_neg[of s g] by auto
1.3526 -
1.3527 -text{* Identity function is continuous in every sense. *}
1.3528 -
1.3529 -lemma continuous_within_id:
1.3530 - "continuous (at a within s) (\<lambda>x. x)"
1.3531 - unfolding continuous_within by (rule Lim_at_within [OF Lim_ident_at])
1.3532 -
1.3533 -lemma continuous_at_id:
1.3534 - "continuous (at a) (\<lambda>x. x)"
1.3535 - unfolding continuous_at by (rule Lim_ident_at)
1.3536 -
1.3537 -lemma continuous_on_id:
1.3538 - "continuous_on s (\<lambda>x. x)"
1.3539 - unfolding continuous_on Lim_within by auto
1.3540 -
1.3541 -lemma uniformly_continuous_on_id:
1.3542 - "uniformly_continuous_on s (\<lambda>x. x)"
1.3543 - unfolding uniformly_continuous_on_def by auto
1.3544 -
1.3545 -text{* Continuity of all kinds is preserved under composition. *}
1.3546 -
1.3547 -lemma continuous_within_compose:
1.3548 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3549 - fixes g :: "'b::metric_space \<Rightarrow> 'c::metric_space"
1.3550 - assumes "continuous (at x within s) f" "continuous (at (f x) within f ` s) g"
1.3551 - shows "continuous (at x within s) (g o f)"
1.3552 -proof-
1.3553 - { fix e::real assume "e>0"
1.3554 - 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
1.3555 - 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
1.3556 - { fix y assume as:"y\<in>s" "0 < dist y x" "dist y x < d'"
1.3557 - hence "dist (f y) (f x) < d" using d'[THEN bspec[where x=y]] by (auto simp add:dist_commute)
1.3558 - 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 }
1.3559 - 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 }
1.3560 - thus ?thesis unfolding continuous_within Lim_within by auto
1.3561 -qed
1.3562 -
1.3563 -lemma continuous_at_compose:
1.3564 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3565 - fixes g :: "'b::metric_space \<Rightarrow> 'c::metric_space"
1.3566 - assumes "continuous (at x) f" "continuous (at (f x)) g"
1.3567 - shows "continuous (at x) (g o f)"
1.3568 -proof-
1.3569 - 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
1.3570 - thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
1.3571 -qed
1.3572 -
1.3573 -lemma continuous_on_compose:
1.3574 - "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
1.3575 - unfolding continuous_on_eq_continuous_within using continuous_within_compose[of _ s f g] by auto
1.3576 -
1.3577 -lemma uniformly_continuous_on_compose:
1.3578 - assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
1.3579 - shows "uniformly_continuous_on s (g o f)"
1.3580 -proof-
1.3581 - { fix e::real assume "e>0"
1.3582 - 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
1.3583 - 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
1.3584 - 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 }
1.3585 - thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
1.3586 -qed
1.3587 -
1.3588 -text{* Continuity in terms of open preimages. *}
1.3589 -
1.3590 -lemma continuous_at_open:
1.3591 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3592 - shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))" (is "?lhs = ?rhs")
1.3593 -proof
1.3594 - assume ?lhs
1.3595 - { fix t assume as: "open t" "f x \<in> t"
1.3596 - then obtain e where "e>0" and e:"ball (f x) e \<subseteq> t" unfolding open_contains_ball by auto
1.3597 -
1.3598 - 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
1.3599 -
1.3600 - have "open (ball x d)" using open_ball by auto
1.3601 - moreover have "x \<in> ball x d" unfolding centre_in_ball using `d>0` by simp
1.3602 - moreover
1.3603 - { fix x' assume "x'\<in>ball x d" hence "f x' \<in> t"
1.3604 - using e[unfolded subset_eq Ball_def mem_ball, THEN spec[where x="f x'"]] d[THEN spec[where x=x']]
1.3605 - unfolding mem_ball apply (auto simp add: dist_commute)
1.3606 - unfolding dist_nz[THEN sym] using as(2) by auto }
1.3607 - hence "\<forall>x'\<in>ball x d. f x' \<in> t" by auto
1.3608 - ultimately have "\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x'\<in>s. f x' \<in> t)"
1.3609 - apply(rule_tac x="ball x d" in exI) by simp }
1.3610 - thus ?rhs by auto
1.3611 -next
1.3612 - assume ?rhs
1.3613 - { fix e::real assume "e>0"
1.3614 - 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"]]
1.3615 - unfolding centre_in_ball[of "f x" e, THEN sym] by auto
1.3616 - then obtain d where "d>0" and d:"ball x d \<subseteq> s" unfolding open_contains_ball by auto
1.3617 - { fix y assume "0 < dist y x \<and> dist y x < d"
1.3618 - hence "dist (f y) (f x) < e" using d[unfolded subset_eq Ball_def mem_ball, THEN spec[where x=y]]
1.3619 - using s(3)[THEN bspec[where x=y], unfolded mem_ball] by (auto simp add: dist_commute) }
1.3620 - 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 }
1.3621 - thus ?lhs unfolding continuous_at Lim_at by auto
1.3622 -qed
1.3623 -
1.3624 -lemma continuous_on_open:
1.3625 - "continuous_on s f \<longleftrightarrow>
1.3626 - (\<forall>t. openin (subtopology euclidean (f ` s)) t
1.3627 - --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
1.3628 -proof
1.3629 - assume ?lhs
1.3630 - { fix t assume as:"openin (subtopology euclidean (f ` s)) t"
1.3631 - have "{x \<in> s. f x \<in> t} \<subseteq> s" using as[unfolded openin_euclidean_subtopology_iff] by auto
1.3632 - moreover
1.3633 - { fix x assume as':"x\<in>{x \<in> s. f x \<in> t}"
1.3634 - 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
1.3635 - 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
1.3636 - 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) }
1.3637 - ultimately have "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" unfolding openin_euclidean_subtopology_iff by auto }
1.3638 - thus ?rhs unfolding continuous_on Lim_within using openin by auto
1.3639 -next
1.3640 - assume ?rhs
1.3641 - { fix e::real and x assume "x\<in>s" "e>0"
1.3642 - { 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)"
1.3643 - hence "dist (f x') (f x) < e" using dist_triangle[of "f x'" "f x" "f xa"]
1.3644 - by (auto simp add: dist_commute) }
1.3645 - 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
1.3646 - apply(rule_tac x="e - dist (f xa) (f x)" in exI) using `e>0` by (auto simp add: dist_commute)
1.3647 - 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}"
1.3648 - using `?rhs`[unfolded openin_euclidean_subtopology_iff, THEN spec[where x="ball (f x) e \<inter> f ` s"]] by auto
1.3649 - 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) }
1.3650 - thus ?lhs unfolding continuous_on Lim_within by auto
1.3651 -qed
1.3652 -
1.3653 -(* ------------------------------------------------------------------------- *)
1.3654 -(* Similarly in terms of closed sets. *)
1.3655 -(* ------------------------------------------------------------------------- *)
1.3656 -
1.3657 -lemma continuous_on_closed:
1.3658 - "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")
1.3659 -proof
1.3660 - assume ?lhs
1.3661 - { fix t
1.3662 - have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
1.3663 - have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
1.3664 - assume as:"closedin (subtopology euclidean (f ` s)) t"
1.3665 - hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
1.3666 - 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"]]
1.3667 - unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
1.3668 - thus ?rhs by auto
1.3669 -next
1.3670 - assume ?rhs
1.3671 - { fix t
1.3672 - have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
1.3673 - assume as:"openin (subtopology euclidean (f ` s)) t"
1.3674 - hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
1.3675 - unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
1.3676 - thus ?lhs unfolding continuous_on_open by auto
1.3677 -qed
1.3678 -
1.3679 -text{* Half-global and completely global cases. *}
1.3680 -
1.3681 -lemma continuous_open_in_preimage:
1.3682 - assumes "continuous_on s f" "open t"
1.3683 - shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
1.3684 -proof-
1.3685 - 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
1.3686 - have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
1.3687 - using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
1.3688 - thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
1.3689 -qed
1.3690 -
1.3691 -lemma continuous_closed_in_preimage:
1.3692 - assumes "continuous_on s f" "closed t"
1.3693 - shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
1.3694 -proof-
1.3695 - 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
1.3696 - have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
1.3697 - using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
1.3698 - thus ?thesis
1.3699 - using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
1.3700 -qed
1.3701 -
1.3702 -lemma continuous_open_preimage:
1.3703 - assumes "continuous_on s f" "open s" "open t"
1.3704 - shows "open {x \<in> s. f x \<in> t}"
1.3705 -proof-
1.3706 - obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
1.3707 - using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
1.3708 - thus ?thesis using open_Int[of s T, OF assms(2)] by auto
1.3709 -qed
1.3710 -
1.3711 -lemma continuous_closed_preimage:
1.3712 - assumes "continuous_on s f" "closed s" "closed t"
1.3713 - shows "closed {x \<in> s. f x \<in> t}"
1.3714 -proof-
1.3715 - obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
1.3716 - using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
1.3717 - thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
1.3718 -qed
1.3719 -
1.3720 -lemma continuous_open_preimage_univ:
1.3721 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3722 - shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
1.3723 - using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
1.3724 -
1.3725 -lemma continuous_closed_preimage_univ:
1.3726 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3727 - shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
1.3728 - using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
1.3729 -
1.3730 -lemma continuous_open_vimage:
1.3731 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3732 - shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
1.3733 - unfolding vimage_def by (rule continuous_open_preimage_univ)
1.3734 -
1.3735 -lemma continuous_closed_vimage:
1.3736 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3737 - shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
1.3738 - unfolding vimage_def by (rule continuous_closed_preimage_univ)
1.3739 -
1.3740 -text{* Equality of continuous functions on closure and related results. *}
1.3741 -
1.3742 -lemma continuous_closed_in_preimage_constant:
1.3743 - "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
1.3744 - using continuous_closed_in_preimage[of s f "{a}"] closed_sing by auto
1.3745 -
1.3746 -lemma continuous_closed_preimage_constant:
1.3747 - "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
1.3748 - using continuous_closed_preimage[of s f "{a}"] closed_sing by auto
1.3749 -
1.3750 -lemma continuous_constant_on_closure:
1.3751 - assumes "continuous_on (closure s) f"
1.3752 - "\<forall>x \<in> s. f x = a"
1.3753 - shows "\<forall>x \<in> (closure s). f x = a"
1.3754 - using continuous_closed_preimage_constant[of "closure s" f a]
1.3755 - assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
1.3756 -
1.3757 -lemma image_closure_subset:
1.3758 - assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
1.3759 - shows "f ` (closure s) \<subseteq> t"
1.3760 -proof-
1.3761 - have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
1.3762 - moreover have "closed {x \<in> closure s. f x \<in> t}"
1.3763 - using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
1.3764 - ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
1.3765 - using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
1.3766 - thus ?thesis by auto
1.3767 -qed
1.3768 -
1.3769 -lemma continuous_on_closure_norm_le:
1.3770 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.3771 - assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
1.3772 - shows "norm(f x) \<le> b"
1.3773 -proof-
1.3774 - have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
1.3775 - show ?thesis
1.3776 - using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
1.3777 - unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
1.3778 -qed
1.3779 -
1.3780 -text{* Making a continuous function avoid some value in a neighbourhood. *}
1.3781 -
1.3782 -lemma continuous_within_avoid:
1.3783 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3784 - assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
1.3785 - shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
1.3786 -proof-
1.3787 - 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"
1.3788 - using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
1.3789 - { fix y assume " y\<in>s" "dist x y < d"
1.3790 - hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
1.3791 - apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
1.3792 - thus ?thesis using `d>0` by auto
1.3793 -qed
1.3794 -
1.3795 -lemma continuous_at_avoid:
1.3796 - fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
1.3797 - assumes "continuous (at x) f" "f x \<noteq> a"
1.3798 - shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
1.3799 -using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
1.3800 -
1.3801 -lemma continuous_on_avoid:
1.3802 - assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
1.3803 - shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
1.3804 -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
1.3805 -
1.3806 -lemma continuous_on_open_avoid:
1.3807 - assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
1.3808 - shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
1.3809 -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
1.3810 -
1.3811 -text{* Proving a function is constant by proving open-ness of level set. *}
1.3812 -
1.3813 -lemma continuous_levelset_open_in_cases:
1.3814 - "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
1.3815 - openin (subtopology euclidean s) {x \<in> s. f x = a}
1.3816 - ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
1.3817 -unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
1.3818 -
1.3819 -lemma continuous_levelset_open_in:
1.3820 - "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
1.3821 - openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
1.3822 - (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
1.3823 -using continuous_levelset_open_in_cases[of s f ]
1.3824 -by meson
1.3825 -
1.3826 -lemma continuous_levelset_open:
1.3827 - assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
1.3828 - shows "\<forall>x \<in> s. f x = a"
1.3829 -using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by auto
1.3830 -
1.3831 -text{* Some arithmetical combinations (more to prove). *}
1.3832 -
1.3833 -lemma open_scaling[intro]:
1.3834 - fixes s :: "'a::real_normed_vector set"
1.3835 - assumes "c \<noteq> 0" "open s"
1.3836 - shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
1.3837 -proof-
1.3838 - { fix x assume "x \<in> s"
1.3839 - 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
1.3840 - have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using real_mult_order[OF `e>0`] by auto
1.3841 - moreover
1.3842 - { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
1.3843 - hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
1.3844 - using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
1.3845 - assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
1.3846 - hence "y \<in> op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"] e[THEN spec[where x="(1 / c) *\<^sub>R y"]] assms(1) unfolding dist_norm scaleR_scaleR by auto }
1.3847 - ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto }
1.3848 - thus ?thesis unfolding open_dist by auto
1.3849 -qed
1.3850 -
1.3851 -lemma minus_image_eq_vimage:
1.3852 - fixes A :: "'a::ab_group_add set"
1.3853 - shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
1.3854 - by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
1.3855 -
1.3856 -lemma open_negations:
1.3857 - fixes s :: "'a::real_normed_vector set"
1.3858 - shows "open s ==> open ((\<lambda> x. -x) ` s)"
1.3859 - unfolding scaleR_minus1_left [symmetric]
1.3860 - by (rule open_scaling, auto)
1.3861 -
1.3862 -lemma open_translation:
1.3863 - fixes s :: "'a::real_normed_vector set"
1.3864 - assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
1.3865 -proof-
1.3866 - { 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 }
1.3867 - 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
1.3868 - ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
1.3869 -qed
1.3870 -
1.3871 -lemma open_affinity:
1.3872 - fixes s :: "'a::real_normed_vector set"
1.3873 - assumes "open s" "c \<noteq> 0"
1.3874 - shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
1.3875 -proof-
1.3876 - have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
1.3877 - have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
1.3878 - thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
1.3879 -qed
1.3880 -
1.3881 -lemma interior_translation:
1.3882 - fixes s :: "'a::real_normed_vector set"
1.3883 - shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
1.3884 -proof (rule set_ext, rule)
1.3885 - fix x assume "x \<in> interior (op + a ` s)"
1.3886 - then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
1.3887 - 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
1.3888 - 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
1.3889 -next
1.3890 - fix x assume "x \<in> op + a ` interior s"
1.3891 - 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
1.3892 - { fix z have *:"a + y - z = y + a - z" by auto
1.3893 - assume "z\<in>ball x e"
1.3894 - 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
1.3895 - hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
1.3896 - hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
1.3897 - thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
1.3898 -qed
1.3899 -
1.3900 -subsection {* Preservation of compactness and connectedness under continuous function. *}
1.3901 -
1.3902 -lemma compact_continuous_image:
1.3903 - assumes "continuous_on s f" "compact s"
1.3904 - shows "compact(f ` s)"
1.3905 -proof-
1.3906 - { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
1.3907 - 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
1.3908 - 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
1.3909 - { fix e::real assume "e>0"
1.3910 - 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
1.3911 - 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
1.3912 - { 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 }
1.3913 - hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
1.3914 - 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 }
1.3915 - thus ?thesis unfolding compact_def by auto
1.3916 -qed
1.3917 -
1.3918 -lemma connected_continuous_image:
1.3919 - assumes "continuous_on s f" "connected s"
1.3920 - shows "connected(f ` s)"
1.3921 -proof-
1.3922 - { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
1.3923 - have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
1.3924 - using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
1.3925 - using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
1.3926 - using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
1.3927 - hence False using as(1,2)
1.3928 - using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
1.3929 - thus ?thesis unfolding connected_clopen by auto
1.3930 -qed
1.3931 -
1.3932 -text{* Continuity implies uniform continuity on a compact domain. *}
1.3933 -
1.3934 -lemma compact_uniformly_continuous:
1.3935 - assumes "continuous_on s f" "compact s"
1.3936 - shows "uniformly_continuous_on s f"
1.3937 -proof-
1.3938 - { fix x assume x:"x\<in>s"
1.3939 - 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
1.3940 - 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 }
1.3941 - 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
1.3942 - 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)"
1.3943 - 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
1.3944 -
1.3945 - { fix e::real assume "e>0"
1.3946 -
1.3947 - { 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 }
1.3948 - hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
1.3949 - moreover
1.3950 - { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
1.3951 - 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
1.3952 -
1.3953 - { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
1.3954 - 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
1.3955 - hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
1.3956 - 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`
1.3957 - by (auto simp add: dist_commute)
1.3958 - moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
1.3959 - by (auto simp add: dist_commute)
1.3960 - 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`
1.3961 - by (auto simp add: dist_commute)
1.3962 - ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
1.3963 - by (auto simp add: dist_commute) }
1.3964 - 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 }
1.3965 - thus ?thesis unfolding uniformly_continuous_on_def by auto
1.3966 -qed
1.3967 -
1.3968 -text{* Continuity of inverse function on compact domain. *}
1.3969 -
1.3970 -lemma continuous_on_inverse:
1.3971 - fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
1.3972 - (* TODO: can this be generalized more? *)
1.3973 - assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
1.3974 - shows "continuous_on (f ` s) g"
1.3975 -proof-
1.3976 - have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
1.3977 - { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
1.3978 - then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
1.3979 - have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
1.3980 - unfolding T(2) and Int_left_absorb by auto
1.3981 - moreover have "compact (s \<inter> T)"
1.3982 - using assms(2) unfolding compact_eq_bounded_closed
1.3983 - using bounded_subset[of s "s \<inter> T"] and T(1) by auto
1.3984 - ultimately have "closed (f ` t)" using T(1) unfolding T(2)
1.3985 - using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
1.3986 - moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
1.3987 - ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
1.3988 - unfolding closedin_closed by auto }
1.3989 - thus ?thesis unfolding continuous_on_closed by auto
1.3990 -qed
1.3991 -
1.3992 -subsection{* A uniformly convergent limit of continuous functions is continuous. *}
1.3993 -
1.3994 -lemma norm_triangle_lt:
1.3995 - fixes x y :: "'a::real_normed_vector"
1.3996 - shows "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
1.3997 -by (rule le_less_trans [OF norm_triangle_ineq])
1.3998 -
1.3999 -lemma continuous_uniform_limit:
1.4000 - fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
1.4001 - assumes "\<not> (trivial_limit net)" "eventually (\<lambda>n. continuous_on s (f n)) net"
1.4002 - "\<forall>e>0. eventually (\<lambda>n. \<forall>x \<in> s. norm(f n x - g x) < e) net"
1.4003 - shows "continuous_on s g"
1.4004 -proof-
1.4005 - { fix x and e::real assume "x\<in>s" "e>0"
1.4006 - 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
1.4007 - then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3" "continuous_on s (f n)"
1.4008 - 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
1.4009 - have "e / 3 > 0" using `e>0` by auto
1.4010 - 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"
1.4011 - using n(2)[unfolded continuous_on_def, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
1.4012 - { fix y assume "y\<in>s" "dist y x < d"
1.4013 - hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
1.4014 - 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"]
1.4015 - 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
1.4016 - hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
1.4017 - 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) }
1.4018 - hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto }
1.4019 - thus ?thesis unfolding continuous_on_def by auto
1.4020 -qed
1.4021 -
1.4022 -subsection{* Topological properties of linear functions. *}
1.4023 -
1.4024 -lemma linear_lim_0:
1.4025 - assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
1.4026 -proof-
1.4027 - interpret f: bounded_linear f by fact
1.4028 - have "(f ---> f 0) (at 0)"
1.4029 - using tendsto_ident_at by (rule f.tendsto)
1.4030 - thus ?thesis unfolding f.zero .
1.4031 -qed
1.4032 -
1.4033 -lemma linear_continuous_at:
1.4034 - assumes "bounded_linear f" shows "continuous (at a) f"
1.4035 - unfolding continuous_at using assms
1.4036 - apply (rule bounded_linear.tendsto)
1.4037 - apply (rule tendsto_ident_at)
1.4038 - done
1.4039 -
1.4040 -lemma linear_continuous_within:
1.4041 - shows "bounded_linear f ==> continuous (at x within s) f"
1.4042 - using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
1.4043 -
1.4044 -lemma linear_continuous_on:
1.4045 - shows "bounded_linear f ==> continuous_on s f"
1.4046 - using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
1.4047 -
1.4048 -text{* Also bilinear functions, in composition form. *}
1.4049 -
1.4050 -lemma bilinear_continuous_at_compose:
1.4051 - shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
1.4052 - ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
1.4053 - unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
1.4054 -
1.4055 -lemma bilinear_continuous_within_compose:
1.4056 - shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
1.4057 - ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
1.4058 - unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
1.4059 -
1.4060 -lemma bilinear_continuous_on_compose:
1.4061 - shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
1.4062 - ==> continuous_on s (\<lambda>x. h (f x) (g x))"
1.4063 - 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
1.4064 - using bilinear_continuous_within_compose[of _ s f g h] by auto
1.4065 -
1.4066 -subsection{* Topological stuff lifted from and dropped to R *}
1.4067 -
1.4068 -
1.4069 -lemma open_real:
1.4070 - fixes s :: "real set" shows
1.4071 - "open s \<longleftrightarrow>
1.4072 - (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
1.4073 - unfolding open_dist dist_norm by simp
1.4074 -
1.4075 -lemma islimpt_approachable_real:
1.4076 - fixes s :: "real set"
1.4077 - shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
1.4078 - unfolding islimpt_approachable dist_norm by simp
1.4079 -
1.4080 -lemma closed_real:
1.4081 - fixes s :: "real set"
1.4082 - shows "closed s \<longleftrightarrow>
1.4083 - (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
1.4084 - --> x \<in> s)"
1.4085 - unfolding closed_limpt islimpt_approachable dist_norm by simp
1.4086 -
1.4087 -lemma continuous_at_real_range:
1.4088 - fixes f :: "'a::real_normed_vector \<Rightarrow> real"
1.4089 - shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
1.4090 - \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
1.4091 - unfolding continuous_at unfolding Lim_at
1.4092 - unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
1.4093 - 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
1.4094 - apply(erule_tac x=e in allE) by auto
1.4095 -
1.4096 -lemma continuous_on_real_range:
1.4097 - fixes f :: "'a::real_normed_vector \<Rightarrow> real"
1.4098 - 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))"
1.4099 - unfolding continuous_on_def dist_norm by simp
1.4100 -
1.4101 -lemma continuous_at_norm: "continuous (at x) norm"
1.4102 - unfolding continuous_at by (intro tendsto_intros)
1.4103 -
1.4104 -lemma continuous_on_norm: "continuous_on s norm"
1.4105 -unfolding continuous_on by (intro ballI tendsto_intros)
1.4106 -
1.4107 -lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
1.4108 -unfolding continuous_at by (intro tendsto_intros)
1.4109 -
1.4110 -lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
1.4111 -unfolding continuous_on by (intro ballI tendsto_intros)
1.4112 -
1.4113 -lemma continuous_at_infnorm: "continuous (at x) infnorm"
1.4114 - unfolding continuous_at Lim_at o_def unfolding dist_norm
1.4115 - apply auto apply (rule_tac x=e in exI) apply auto
1.4116 - using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
1.4117 -
1.4118 -text{* Hence some handy theorems on distance, diameter etc. of/from a set. *}
1.4119 -
1.4120 -lemma compact_attains_sup:
1.4121 - fixes s :: "real set"
1.4122 - assumes "compact s" "s \<noteq> {}"
1.4123 - shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
1.4124 -proof-
1.4125 - from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
1.4126 - { 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"
1.4127 - have "isLub UNIV s (rsup s)" using rsup[OF assms(2)] unfolding setle_def using as(1) by auto
1.4128 - moreover have "isUb UNIV s (rsup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
1.4129 - ultimately have False using isLub_le_isUb[of UNIV s "rsup s" "rsup s - e"] using `e>0` by auto }
1.4130 - thus ?thesis using bounded_has_rsup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rsup s"]]
1.4131 - apply(rule_tac x="rsup s" in bexI) by auto
1.4132 -qed
1.4133 -
1.4134 -lemma compact_attains_inf:
1.4135 - fixes s :: "real set"
1.4136 - assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
1.4137 -proof-
1.4138 - from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
1.4139 - { fix e::real assume as: "\<forall>x\<in>s. x \<ge> rinf s" "rinf s \<notin> s" "0 < e"
1.4140 - "\<forall>x'\<in>s. x' = rinf s \<or> \<not> abs (x' - rinf s) < e"
1.4141 - have "isGlb UNIV s (rinf s)" using rinf[OF assms(2)] unfolding setge_def using as(1) by auto
1.4142 - moreover
1.4143 - { fix x assume "x \<in> s"
1.4144 - hence *:"abs (x - rinf s) = x - rinf s" using as(1)[THEN bspec[where x=x]] by auto
1.4145 - have "rinf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
1.4146 - hence "isLb UNIV s (rinf s + e)" unfolding isLb_def and setge_def by auto
1.4147 - ultimately have False using isGlb_le_isLb[of UNIV s "rinf s" "rinf s + e"] using `e>0` by auto }
1.4148 - thus ?thesis using bounded_has_rinf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rinf s"]]
1.4149 - apply(rule_tac x="rinf s" in bexI) by auto
1.4150 -qed
1.4151 -
1.4152 -lemma continuous_attains_sup:
1.4153 - fixes f :: "'a::metric_space \<Rightarrow> real"
1.4154 - shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
1.4155 - ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
1.4156 - using compact_attains_sup[of "f ` s"]
1.4157 - using compact_continuous_image[of s f] by auto
1.4158 -
1.4159 -lemma continuous_attains_inf:
1.4160 - fixes f :: "'a::metric_space \<Rightarrow> real"
1.4161 - shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
1.4162 - \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
1.4163 - using compact_attains_inf[of "f ` s"]
1.4164 - using compact_continuous_image[of s f] by auto
1.4165 -
1.4166 -lemma distance_attains_sup:
1.4167 - assumes "compact s" "s \<noteq> {}"
1.4168 - shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
1.4169 -proof (rule continuous_attains_sup [OF assms])
1.4170 - { fix x assume "x\<in>s"
1.4171 - have "(dist a ---> dist a x) (at x within s)"
1.4172 - by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
1.4173 - }
1.4174 - thus "continuous_on s (dist a)"
1.4175 - unfolding continuous_on ..
1.4176 -qed
1.4177 -
1.4178 -text{* For *minimal* distance, we only need closure, not compactness. *}
1.4179 -
1.4180 -lemma distance_attains_inf:
1.4181 - fixes a :: "'a::heine_borel"
1.4182 - assumes "closed s" "s \<noteq> {}"
1.4183 - shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
1.4184 -proof-
1.4185 - from assms(2) obtain b where "b\<in>s" by auto
1.4186 - let ?B = "cball a (dist b a) \<inter> s"
1.4187 - have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
1.4188 - hence "?B \<noteq> {}" by auto
1.4189 - moreover
1.4190 - { fix x assume "x\<in>?B"
1.4191 - fix e::real assume "e>0"
1.4192 - { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
1.4193 - from as have "\<bar>dist a x' - dist a x\<bar> < e"
1.4194 - unfolding abs_less_iff minus_diff_eq
1.4195 - using dist_triangle2 [of a x' x]
1.4196 - using dist_triangle [of a x x']
1.4197 - by arith
1.4198 - }
1.4199 - hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
1.4200 - using `e>0` by auto
1.4201 - }
1.4202 - hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
1.4203 - unfolding continuous_on Lim_within dist_norm real_norm_def
1.4204 - by fast
1.4205 - moreover have "compact ?B"
1.4206 - using compact_cball[of a "dist b a"]
1.4207 - unfolding compact_eq_bounded_closed
1.4208 - using bounded_Int and closed_Int and assms(1) by auto
1.4209 - 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"
1.4210 - using continuous_attains_inf[of ?B "dist a"] by fastsimp
1.4211 - thus ?thesis by fastsimp
1.4212 -qed
1.4213 -
1.4214 -subsection{* We can now extend limit compositions to consider the scalar multiplier. *}
1.4215 -
1.4216 -lemma Lim_mul:
1.4217 - fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.4218 - assumes "(c ---> d) net" "(f ---> l) net"
1.4219 - shows "((\<lambda>x. c(x) *\<^sub>R f x) ---> (d *\<^sub>R l)) net"
1.4220 - using assms by (rule scaleR.tendsto)
1.4221 -
1.4222 -lemma Lim_vmul:
1.4223 - fixes c :: "'a \<Rightarrow> real" and v :: "'b::real_normed_vector"
1.4224 - shows "(c ---> d) net ==> ((\<lambda>x. c(x) *\<^sub>R v) ---> d *\<^sub>R v) net"
1.4225 - by (intro tendsto_intros)
1.4226 -
1.4227 -lemma continuous_vmul:
1.4228 - fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
1.4229 - shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
1.4230 - unfolding continuous_def using Lim_vmul[of c] by auto
1.4231 -
1.4232 -lemma continuous_mul:
1.4233 - fixes c :: "'a::metric_space \<Rightarrow> real"
1.4234 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.4235 - shows "continuous net c \<Longrightarrow> continuous net f
1.4236 - ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
1.4237 - unfolding continuous_def by (intro tendsto_intros)
1.4238 -
1.4239 -lemma continuous_on_vmul:
1.4240 - fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
1.4241 - shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
1.4242 - unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
1.4243 -
1.4244 -lemma continuous_on_mul:
1.4245 - fixes c :: "'a::metric_space \<Rightarrow> real"
1.4246 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.4247 - shows "continuous_on s c \<Longrightarrow> continuous_on s f
1.4248 - ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
1.4249 - unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
1.4250 -
1.4251 -text{* And so we have continuity of inverse. *}
1.4252 -
1.4253 -lemma Lim_inv:
1.4254 - fixes f :: "'a \<Rightarrow> real"
1.4255 - assumes "(f ---> l) (net::'a net)" "l \<noteq> 0"
1.4256 - shows "((inverse o f) ---> inverse l) net"
1.4257 - unfolding o_def using assms by (rule tendsto_inverse)
1.4258 -
1.4259 -lemma continuous_inv:
1.4260 - fixes f :: "'a::metric_space \<Rightarrow> real"
1.4261 - shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
1.4262 - ==> continuous net (inverse o f)"
1.4263 - unfolding continuous_def using Lim_inv by auto
1.4264 -
1.4265 -lemma continuous_at_within_inv:
1.4266 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
1.4267 - assumes "continuous (at a within s) f" "f a \<noteq> 0"
1.4268 - shows "continuous (at a within s) (inverse o f)"
1.4269 - using assms unfolding continuous_within o_def
1.4270 - by (intro tendsto_intros)
1.4271 -
1.4272 -lemma continuous_at_inv:
1.4273 - fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
1.4274 - shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
1.4275 - ==> continuous (at a) (inverse o f) "
1.4276 - using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
1.4277 -
1.4278 -subsection{* Preservation properties for pasted sets. *}
1.4279 -
1.4280 -lemma bounded_pastecart:
1.4281 - fixes s :: "('a::real_normed_vector ^ _) set" (* FIXME: generalize to metric_space *)
1.4282 - assumes "bounded s" "bounded t"
1.4283 - shows "bounded { pastecart x y | x y . (x \<in> s \<and> y \<in> t)}"
1.4284 -proof-
1.4285 - 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
1.4286 - { fix x y assume "x\<in>s" "y\<in>t"
1.4287 - hence "norm x \<le> a" "norm y \<le> b" using ab by auto
1.4288 - hence "norm (pastecart x y) \<le> a + b" using norm_pastecart[of x y] by auto }
1.4289 - thus ?thesis unfolding bounded_iff by auto
1.4290 -qed
1.4291 -
1.4292 -lemma bounded_Times:
1.4293 - assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
1.4294 -proof-
1.4295 - obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
1.4296 - using assms [unfolded bounded_def] by auto
1.4297 - then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
1.4298 - by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
1.4299 - thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
1.4300 -qed
1.4301 -
1.4302 -lemma closed_pastecart:
1.4303 - fixes s :: "(real ^ 'a::finite) set" (* FIXME: generalize *)
1.4304 - assumes "closed s" "closed t"
1.4305 - shows "closed {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
1.4306 -proof-
1.4307 - { 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"
1.4308 - { 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
1.4309 - moreover
1.4310 - { fix e::real assume "e>0"
1.4311 - 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
1.4312 - { fix n::nat assume "n\<ge>N"
1.4313 - hence "dist (fstcart (x n)) (fstcart l) < e" "dist (sndcart (x n)) (sndcart l) < e"
1.4314 - using N[THEN spec[where x=n]] dist_fstcart[of "x n" l] dist_sndcart[of "x n" l] by auto }
1.4315 - 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 }
1.4316 - ultimately have "fstcart l \<in> s" "sndcart l \<in> t"
1.4317 - using assms(1)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. fstcart (x n)"], THEN spec[where x="fstcart l"]]
1.4318 - using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. sndcart (x n)"], THEN spec[where x="sndcart l"]]
1.4319 - unfolding Lim_sequentially by auto
1.4320 - 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 }
1.4321 - thus ?thesis unfolding closed_sequential_limits by auto
1.4322 -qed
1.4323 -
1.4324 -lemma compact_pastecart:
1.4325 - fixes s t :: "(real ^ _) set"
1.4326 - shows "compact s \<Longrightarrow> compact t ==> compact {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
1.4327 - unfolding compact_eq_bounded_closed using bounded_pastecart[of s t] closed_pastecart[of s t] by auto
1.4328 -
1.4329 -lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
1.4330 -by (induct x) simp
1.4331 -
1.4332 -lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
1.4333 -unfolding compact_def
1.4334 -apply clarify
1.4335 -apply (drule_tac x="fst \<circ> f" in spec)
1.4336 -apply (drule mp, simp add: mem_Times_iff)
1.4337 -apply (clarify, rename_tac l1 r1)
1.4338 -apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
1.4339 -apply (drule mp, simp add: mem_Times_iff)
1.4340 -apply (clarify, rename_tac l2 r2)
1.4341 -apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
1.4342 -apply (rule_tac x="r1 \<circ> r2" in exI)
1.4343 -apply (rule conjI, simp add: subseq_def)
1.4344 -apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
1.4345 -apply (drule (1) tendsto_Pair) back
1.4346 -apply (simp add: o_def)
1.4347 -done
1.4348 -
1.4349 -text{* Hence some useful properties follow quite easily. *}
1.4350 -
1.4351 -lemma compact_scaling:
1.4352 - fixes s :: "'a::real_normed_vector set"
1.4353 - assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
1.4354 -proof-
1.4355 - let ?f = "\<lambda>x. scaleR c x"
1.4356 - have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
1.4357 - show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
1.4358 - using linear_continuous_at[OF *] assms by auto
1.4359 -qed
1.4360 -
1.4361 -lemma compact_negations:
1.4362 - fixes s :: "'a::real_normed_vector set"
1.4363 - assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
1.4364 - using compact_scaling [OF assms, of "- 1"] by auto
1.4365 -
1.4366 -lemma compact_sums:
1.4367 - fixes s t :: "'a::real_normed_vector set"
1.4368 - assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
1.4369 -proof-
1.4370 - have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
1.4371 - apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
1.4372 - have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
1.4373 - unfolding continuous_on by (rule ballI) (intro tendsto_intros)
1.4374 - thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
1.4375 -qed
1.4376 -
1.4377 -lemma compact_differences:
1.4378 - fixes s t :: "'a::real_normed_vector set"
1.4379 - assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
1.4380 -proof-
1.4381 - have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
1.4382 - apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
1.4383 - thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
1.4384 -qed
1.4385 -
1.4386 -lemma compact_translation:
1.4387 - fixes s :: "'a::real_normed_vector set"
1.4388 - assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
1.4389 -proof-
1.4390 - have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
1.4391 - thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
1.4392 -qed
1.4393 -
1.4394 -lemma compact_affinity:
1.4395 - fixes s :: "'a::real_normed_vector set"
1.4396 - assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
1.4397 -proof-
1.4398 - have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
1.4399 - thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
1.4400 -qed
1.4401 -
1.4402 -text{* Hence we get the following. *}
1.4403 -
1.4404 -lemma compact_sup_maxdistance:
1.4405 - fixes s :: "'a::real_normed_vector set"
1.4406 - assumes "compact s" "s \<noteq> {}"
1.4407 - shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
1.4408 -proof-
1.4409 - have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
1.4410 - 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"
1.4411 - using compact_differences[OF assms(1) assms(1)]
1.4412 - using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
1.4413 - from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
1.4414 - thus ?thesis using x(2)[unfolded `x = a - b`] by blast
1.4415 -qed
1.4416 -
1.4417 -text{* We can state this in terms of diameter of a set. *}
1.4418 -
1.4419 -definition "diameter s = (if s = {} then 0::real else rsup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
1.4420 - (* TODO: generalize to class metric_space *)
1.4421 -
1.4422 -lemma diameter_bounded:
1.4423 - assumes "bounded s"
1.4424 - shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
1.4425 - "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
1.4426 -proof-
1.4427 - let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
1.4428 - obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
1.4429 - { fix x y assume "x \<in> s" "y \<in> s"
1.4430 - 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) }
1.4431 - note * = this
1.4432 - { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
1.4433 - have lub:"isLub UNIV ?D (rsup ?D)" using * rsup[of ?D] using `s\<noteq>{}` unfolding setle_def by auto
1.4434 - 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 }
1.4435 - moreover
1.4436 - { fix d::real assume "d>0" "d < diameter s"
1.4437 - hence "s\<noteq>{}" unfolding diameter_def by auto
1.4438 - hence lub:"isLub UNIV ?D (rsup ?D)" using * rsup[of ?D] unfolding setle_def by auto
1.4439 - have "\<exists>d' \<in> ?D. d' > d"
1.4440 - proof(rule ccontr)
1.4441 - assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
1.4442 - hence as:"\<forall>d'\<in>?D. d' \<le> d" apply auto apply(erule_tac x="norm (x - y)" in allE) by auto
1.4443 - hence "isUb UNIV ?D d" unfolding isUb_def unfolding setle_def by auto
1.4444 - thus False using `d < diameter s` `s\<noteq>{}` isLub_le_isUb[OF lub, of d] unfolding diameter_def by auto
1.4445 - qed
1.4446 - hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
1.4447 - ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
1.4448 - "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
1.4449 -qed
1.4450 -
1.4451 -lemma diameter_bounded_bound:
1.4452 - "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
1.4453 - using diameter_bounded by blast
1.4454 -
1.4455 -lemma diameter_compact_attained:
1.4456 - fixes s :: "'a::real_normed_vector set"
1.4457 - assumes "compact s" "s \<noteq> {}"
1.4458 - shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
1.4459 -proof-
1.4460 - have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
1.4461 - 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
1.4462 - 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)"]
1.4463 - unfolding setle_def and diameter_def by auto
1.4464 - thus ?thesis using diameter_bounded(1)[OF b, THEN bspec[where x=x], THEN bspec[where x=y], OF xys] and xys by auto
1.4465 -qed
1.4466 -
1.4467 -text{* Related results with closure as the conclusion. *}
1.4468 -
1.4469 -lemma closed_scaling:
1.4470 - fixes s :: "'a::real_normed_vector set"
1.4471 - assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
1.4472 -proof(cases "s={}")
1.4473 - case True thus ?thesis by auto
1.4474 -next
1.4475 - case False
1.4476 - show ?thesis
1.4477 - proof(cases "c=0")
1.4478 - have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
1.4479 - case True thus ?thesis apply auto unfolding * using closed_sing by auto
1.4480 - next
1.4481 - case False
1.4482 - { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
1.4483 - { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
1.4484 - using as(1)[THEN spec[where x=n]]
1.4485 - using `c\<noteq>0` by (auto simp add: vector_smult_assoc)
1.4486 - }
1.4487 - moreover
1.4488 - { fix e::real assume "e>0"
1.4489 - hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
1.4490 - then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
1.4491 - using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
1.4492 - hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
1.4493 - unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
1.4494 - using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
1.4495 - hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
1.4496 - ultimately have "l \<in> scaleR c ` s"
1.4497 - 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"]]
1.4498 - unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
1.4499 - thus ?thesis unfolding closed_sequential_limits by fast
1.4500 - qed
1.4501 -qed
1.4502 -
1.4503 -lemma closed_negations:
1.4504 - fixes s :: "'a::real_normed_vector set"
1.4505 - assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
1.4506 - using closed_scaling[OF assms, of "- 1"] by simp
1.4507 -
1.4508 -lemma compact_closed_sums:
1.4509 - fixes s :: "'a::real_normed_vector set"
1.4510 - assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
1.4511 -proof-
1.4512 - let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
1.4513 - { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
1.4514 - 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"
1.4515 - using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
1.4516 - obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
1.4517 - using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
1.4518 - have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
1.4519 - using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
1.4520 - hence "l - l' \<in> t"
1.4521 - using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
1.4522 - using f(3) by auto
1.4523 - 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
1.4524 - }
1.4525 - thus ?thesis unfolding closed_sequential_limits by fast
1.4526 -qed
1.4527 -
1.4528 -lemma closed_compact_sums:
1.4529 - fixes s t :: "'a::real_normed_vector set"
1.4530 - assumes "closed s" "compact t"
1.4531 - shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
1.4532 -proof-
1.4533 - 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
1.4534 - apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
1.4535 - thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
1.4536 -qed
1.4537 -
1.4538 -lemma compact_closed_differences:
1.4539 - fixes s t :: "'a::real_normed_vector set"
1.4540 - assumes "compact s" "closed t"
1.4541 - shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
1.4542 -proof-
1.4543 - have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
1.4544 - apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
1.4545 - thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
1.4546 -qed
1.4547 -
1.4548 -lemma closed_compact_differences:
1.4549 - fixes s t :: "'a::real_normed_vector set"
1.4550 - assumes "closed s" "compact t"
1.4551 - shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
1.4552 -proof-
1.4553 - have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
1.4554 - apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
1.4555 - thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
1.4556 -qed
1.4557 -
1.4558 -lemma closed_translation:
1.4559 - fixes a :: "'a::real_normed_vector"
1.4560 - assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
1.4561 -proof-
1.4562 - have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
1.4563 - thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
1.4564 -qed
1.4565 -
1.4566 -lemma translation_UNIV:
1.4567 - fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
1.4568 - apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
1.4569 -
1.4570 -lemma translation_diff:
1.4571 - fixes a :: "'a::ab_group_add"
1.4572 - shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
1.4573 - by auto
1.4574 -
1.4575 -lemma closure_translation:
1.4576 - fixes a :: "'a::real_normed_vector"
1.4577 - shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
1.4578 -proof-
1.4579 - have *:"op + a ` (UNIV - s) = UNIV - op + a ` s"
1.4580 - apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
1.4581 - show ?thesis unfolding closure_interior translation_diff translation_UNIV
1.4582 - using interior_translation[of a "UNIV - s"] unfolding * by auto
1.4583 -qed
1.4584 -
1.4585 -lemma frontier_translation:
1.4586 - fixes a :: "'a::real_normed_vector"
1.4587 - shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
1.4588 - unfolding frontier_def translation_diff interior_translation closure_translation by auto
1.4589 -
1.4590 -subsection{* Separation between points and sets. *}
1.4591 -
1.4592 -lemma separate_point_closed:
1.4593 - fixes s :: "'a::heine_borel set"
1.4594 - shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
1.4595 -proof(cases "s = {}")
1.4596 - case True
1.4597 - thus ?thesis by(auto intro!: exI[where x=1])
1.4598 -next
1.4599 - case False
1.4600 - assume "closed s" "a \<notin> s"
1.4601 - then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using `s \<noteq> {}` distance_attains_inf [of s a] by blast
1.4602 - with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
1.4603 -qed
1.4604 -
1.4605 -lemma separate_compact_closed:
1.4606 - fixes s t :: "'a::{heine_borel, real_normed_vector} set"
1.4607 - (* TODO: does this generalize to heine_borel? *)
1.4608 - assumes "compact s" and "closed t" and "s \<inter> t = {}"
1.4609 - shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
1.4610 -proof-
1.4611 - have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
1.4612 - then obtain d where "d>0" and d:"\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. d \<le> dist 0 x"
1.4613 - using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
1.4614 - { fix x y assume "x\<in>s" "y\<in>t"
1.4615 - hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
1.4616 - hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
1.4617 - by (auto simp add: dist_commute)
1.4618 - hence "d \<le> dist x y" unfolding dist_norm by auto }
1.4619 - thus ?thesis using `d>0` by auto
1.4620 -qed
1.4621 -
1.4622 -lemma separate_closed_compact:
1.4623 - fixes s t :: "'a::{heine_borel, real_normed_vector} set"
1.4624 - assumes "closed s" and "compact t" and "s \<inter> t = {}"
1.4625 - shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
1.4626 -proof-
1.4627 - have *:"t \<inter> s = {}" using assms(3) by auto
1.4628 - show ?thesis using separate_compact_closed[OF assms(2,1) *]
1.4629 - apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
1.4630 - by (auto simp add: dist_commute)
1.4631 -qed
1.4632 -
1.4633 -(* A cute way of denoting open and closed intervals using overloading. *)
1.4634 -
1.4635 -lemma interval: fixes a :: "'a::ord^'n::finite" shows
1.4636 - "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
1.4637 - "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
1.4638 - by (auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
1.4639 -
1.4640 -lemma mem_interval: fixes a :: "'a::ord^'n::finite" shows
1.4641 - "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
1.4642 - "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
1.4643 - using interval[of a b] by(auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
1.4644 -
1.4645 -lemma mem_interval_1: fixes x :: "real^1" shows
1.4646 - "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
1.4647 - "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
1.4648 -by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def forall_1)
1.4649 -
1.4650 -lemma interval_eq_empty: fixes a :: "real^'n::finite" shows
1.4651 - "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
1.4652 - "({a .. b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
1.4653 -proof-
1.4654 - { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
1.4655 - hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval by auto
1.4656 - hence "a$i < b$i" by auto
1.4657 - hence False using as by auto }
1.4658 - moreover
1.4659 - { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
1.4660 - let ?x = "(1/2) *\<^sub>R (a + b)"
1.4661 - { fix i
1.4662 - have "a$i < b$i" using as[THEN spec[where x=i]] by auto
1.4663 - hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
1.4664 - unfolding vector_smult_component and vector_add_component
1.4665 - by (auto simp add: less_divide_eq_number_of1) }
1.4666 - hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
1.4667 - ultimately show ?th1 by blast
1.4668 -
1.4669 - { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
1.4670 - hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval by auto
1.4671 - hence "a$i \<le> b$i" by auto
1.4672 - hence False using as by auto }
1.4673 - moreover
1.4674 - { assume as:"\<forall>i. \<not> (b$i < a$i)"
1.4675 - let ?x = "(1/2) *\<^sub>R (a + b)"
1.4676 - { fix i
1.4677 - have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
1.4678 - hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
1.4679 - unfolding vector_smult_component and vector_add_component
1.4680 - by (auto simp add: less_divide_eq_number_of1) }
1.4681 - hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
1.4682 - ultimately show ?th2 by blast
1.4683 -qed
1.4684 -
1.4685 -lemma interval_ne_empty: fixes a :: "real^'n::finite" shows
1.4686 - "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and
1.4687 - "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
1.4688 - unfolding interval_eq_empty[of a b] by (auto simp add: not_less not_le) (* BH: Why doesn't just "auto" work here? *)
1.4689 -
1.4690 -lemma subset_interval_imp: fixes a :: "real^'n::finite" shows
1.4691 - "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
1.4692 - "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
1.4693 - "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
1.4694 - "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
1.4695 - unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
1.4696 - by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
1.4697 -
1.4698 -lemma interval_sing: fixes a :: "'a::linorder^'n::finite" shows
1.4699 - "{a .. a} = {a} \<and> {a<..<a} = {}"
1.4700 -apply(auto simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
1.4701 -apply (simp add: order_eq_iff)
1.4702 -apply (auto simp add: not_less less_imp_le)
1.4703 -done
1.4704 -
1.4705 -lemma interval_open_subset_closed: fixes a :: "'a::preorder^'n::finite" shows
1.4706 - "{a<..<b} \<subseteq> {a .. b}"
1.4707 -proof(simp add: subset_eq, rule)
1.4708 - fix x
1.4709 - assume x:"x \<in>{a<..<b}"
1.4710 - { fix i
1.4711 - have "a $ i \<le> x $ i"
1.4712 - using x order_less_imp_le[of "a$i" "x$i"]
1.4713 - by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
1.4714 - }
1.4715 - moreover
1.4716 - { fix i
1.4717 - have "x $ i \<le> b $ i"
1.4718 - using x order_less_imp_le[of "x$i" "b$i"]
1.4719 - by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
1.4720 - }
1.4721 - ultimately
1.4722 - show "a \<le> x \<and> x \<le> b"
1.4723 - by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
1.4724 -qed
1.4725 -
1.4726 -lemma subset_interval: fixes a :: "real^'n::finite" shows
1.4727 - "{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
1.4728 - "{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
1.4729 - "{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
1.4730 - "{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)
1.4731 -proof-
1.4732 - show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
1.4733 - 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)
1.4734 - { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i. c$i < d$i"
1.4735 - hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by (auto, drule_tac x=i in spec, simp) (* BH: Why doesn't just "auto" work? *)
1.4736 - fix i
1.4737 - (** TODO combine the following two parts as done in the HOL_light version. **)
1.4738 - { let ?x = "(\<chi> j. (if j=i then ((min (a$j) (d$j))+c$j)/2 else (c$j+d$j)/2))::real^'n"
1.4739 - assume as2: "a$i > c$i"
1.4740 - { fix j
1.4741 - have "c $ j < ?x $ j \<and> ?x $ j < d $ j" unfolding Cart_lambda_beta
1.4742 - apply(cases "j=i") using as(2)[THEN spec[where x=j]]
1.4743 - by (auto simp add: less_divide_eq_number_of1 as2) }
1.4744 - hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
1.4745 - moreover
1.4746 - have "?x\<notin>{a .. b}"
1.4747 - unfolding mem_interval apply auto apply(rule_tac x=i in exI)
1.4748 - using as(2)[THEN spec[where x=i]] and as2
1.4749 - by (auto simp add: less_divide_eq_number_of1)
1.4750 - ultimately have False using as by auto }
1.4751 - hence "a$i \<le> c$i" by(rule ccontr)auto
1.4752 - moreover
1.4753 - { let ?x = "(\<chi> j. (if j=i then ((max (b$j) (c$j))+d$j)/2 else (c$j+d$j)/2))::real^'n"
1.4754 - assume as2: "b$i < d$i"
1.4755 - { fix j
1.4756 - have "d $ j > ?x $ j \<and> ?x $ j > c $ j" unfolding Cart_lambda_beta
1.4757 - apply(cases "j=i") using as(2)[THEN spec[where x=j]]
1.4758 - by (auto simp add: less_divide_eq_number_of1 as2) }
1.4759 - hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
1.4760 - moreover
1.4761 - have "?x\<notin>{a .. b}"
1.4762 - unfolding mem_interval apply auto apply(rule_tac x=i in exI)
1.4763 - using as(2)[THEN spec[where x=i]] and as2
1.4764 - by (auto simp add: less_divide_eq_number_of1)
1.4765 - ultimately have False using as by auto }
1.4766 - hence "b$i \<ge> d$i" by(rule ccontr)auto
1.4767 - ultimately
1.4768 - have "a$i \<le> c$i \<and> d$i \<le> b$i" by auto
1.4769 - } note part1 = this
1.4770 - 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)+
1.4771 - { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i. c$i < d$i"
1.4772 - fix i
1.4773 - from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
1.4774 - hence "a$i \<le> c$i \<and> d$i \<le> b$i" using part1 and as(2) by auto } note * = this
1.4775 - 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)+
1.4776 -qed
1.4777 -
1.4778 -lemma disjoint_interval: fixes a::"real^'n::finite" shows
1.4779 - "{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
1.4780 - "{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
1.4781 - "{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
1.4782 - "{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)
1.4783 -proof-
1.4784 - let ?z = "(\<chi> i. ((max (a$i) (c$i)) + (min (b$i) (d$i))) / 2)::real^'n"
1.4785 - show ?th1 ?th2 ?th3 ?th4
1.4786 - unfolding expand_set_eq and Int_iff and empty_iff and mem_interval and all_conj_distrib[THEN sym] and eq_False
1.4787 - apply (auto elim!: allE[where x="?z"])
1.4788 - apply ((rule_tac x=x in exI, force) | (rule_tac x=i in exI, force))+
1.4789 - done
1.4790 -qed
1.4791 -
1.4792 -lemma inter_interval: fixes a :: "'a::linorder^'n::finite" shows
1.4793 - "{a .. b} \<inter> {c .. d} = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
1.4794 - unfolding expand_set_eq and Int_iff and mem_interval
1.4795 - by (auto simp add: less_divide_eq_number_of1 intro!: bexI)
1.4796 -
1.4797 -(* Moved interval_open_subset_closed a bit upwards *)
1.4798 -
1.4799 -lemma open_interval_lemma: fixes x :: "real" shows
1.4800 - "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
1.4801 - by(rule_tac x="min (x - a) (b - x)" in exI, auto)
1.4802 -
1.4803 -lemma open_interval: fixes a :: "real^'n::finite" shows "open {a<..<b}"
1.4804 -proof-
1.4805 - { fix x assume x:"x\<in>{a<..<b}"
1.4806 - { fix i
1.4807 - have "\<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i"
1.4808 - using x[unfolded mem_interval, THEN spec[where x=i]]
1.4809 - using open_interval_lemma[of "a$i" "x$i" "b$i"] by auto }
1.4810 -
1.4811 - hence "\<forall>i. \<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i" by auto
1.4812 - 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)"
1.4813 - 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
1.4814 -
1.4815 - let ?d = "Min (range d)"
1.4816 - have **:"finite (range d)" "range d \<noteq> {}" by auto
1.4817 - have "?d>0" unfolding Min_gr_iff[OF **] using d by auto
1.4818 - moreover
1.4819 - { fix x' assume as:"dist x' x < ?d"
1.4820 - { fix i
1.4821 - have "\<bar>x'$i - x $ i\<bar> < d i"
1.4822 - using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
1.4823 - unfolding vector_minus_component and Min_gr_iff[OF **] by auto
1.4824 - hence "a $ i < x' $ i" "x' $ i < b $ i" using d[THEN spec[where x=i]] by auto }
1.4825 - hence "a < x' \<and> x' < b" unfolding vector_less_def by auto }
1.4826 - 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)
1.4827 - }
1.4828 - thus ?thesis unfolding open_dist using open_interval_lemma by auto
1.4829 -qed
1.4830 -
1.4831 -lemma closed_interval: fixes a :: "real^'n::finite" shows "closed {a .. b}"
1.4832 -proof-
1.4833 - { 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"*)
1.4834 - { assume xa:"a$i > x$i"
1.4835 - 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
1.4836 - hence False unfolding mem_interval and dist_norm
1.4837 - using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xa by(auto elim!: allE[where x=i])
1.4838 - } hence "a$i \<le> x$i" by(rule ccontr)auto
1.4839 - moreover
1.4840 - { assume xb:"b$i < x$i"
1.4841 - 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
1.4842 - hence False unfolding mem_interval and dist_norm
1.4843 - using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xb by(auto elim!: allE[where x=i])
1.4844 - } hence "x$i \<le> b$i" by(rule ccontr)auto
1.4845 - ultimately
1.4846 - have "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" by auto }
1.4847 - thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
1.4848 -qed
1.4849 -
1.4850 -lemma interior_closed_interval: fixes a :: "real^'n::finite" shows
1.4851 - "interior {a .. b} = {a<..<b}" (is "?L = ?R")
1.4852 -proof(rule subset_antisym)
1.4853 - show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
1.4854 -next
1.4855 - { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
1.4856 - then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
1.4857 - 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
1.4858 - { fix i
1.4859 - have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
1.4860 - "dist (x + (e / 2) *\<^sub>R basis i) x < e"
1.4861 - unfolding dist_norm apply auto
1.4862 - unfolding norm_minus_cancel using norm_basis[of i] and `e>0` by auto
1.4863 - hence "a $ i \<le> (x - (e / 2) *\<^sub>R basis i) $ i"
1.4864 - "(x + (e / 2) *\<^sub>R basis i) $ i \<le> b $ i"
1.4865 - using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
1.4866 - and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
1.4867 - unfolding mem_interval by (auto elim!: allE[where x=i])
1.4868 - hence "a $ i < x $ i" and "x $ i < b $ i"
1.4869 - unfolding vector_minus_component and vector_add_component
1.4870 - unfolding vector_smult_component and basis_component using `e>0` by auto }
1.4871 - hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
1.4872 - thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
1.4873 -qed
1.4874 -
1.4875 -lemma bounded_closed_interval: fixes a :: "real^'n::finite" shows
1.4876 - "bounded {a .. b}"
1.4877 -proof-
1.4878 - let ?b = "\<Sum>i\<in>UNIV. \<bar>a$i\<bar> + \<bar>b$i\<bar>"
1.4879 - { fix x::"real^'n" assume x:"\<forall>i. a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
1.4880 - { fix i
1.4881 - have "\<bar>x$i\<bar> \<le> \<bar>a$i\<bar> + \<bar>b$i\<bar>" using x[THEN spec[where x=i]] by auto }
1.4882 - hence "(\<Sum>i\<in>UNIV. \<bar>x $ i\<bar>) \<le> ?b" by(rule setsum_mono)
1.4883 - hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
1.4884 - thus ?thesis unfolding interval and bounded_iff by auto
1.4885 -qed
1.4886 -
1.4887 -lemma bounded_interval: fixes a :: "real^'n::finite" shows
1.4888 - "bounded {a .. b} \<and> bounded {a<..<b}"
1.4889 - using bounded_closed_interval[of a b]
1.4890 - using interval_open_subset_closed[of a b]
1.4891 - using bounded_subset[of "{a..b}" "{a<..<b}"]
1.4892 - by simp
1.4893 -
1.4894 -lemma not_interval_univ: fixes a :: "real^'n::finite" shows
1.4895 - "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
1.4896 - using bounded_interval[of a b]
1.4897 - by auto
1.4898 -
1.4899 -lemma compact_interval: fixes a :: "real^'n::finite" shows
1.4900 - "compact {a .. b}"
1.4901 - using bounded_closed_imp_compact using bounded_interval[of a b] using closed_interval[of a b] by auto
1.4902 -
1.4903 -lemma open_interval_midpoint: fixes a :: "real^'n::finite"
1.4904 - assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
1.4905 -proof-
1.4906 - { fix i
1.4907 - have "a $ i < ((1 / 2) *\<^sub>R (a + b)) $ i \<and> ((1 / 2) *\<^sub>R (a + b)) $ i < b $ i"
1.4908 - using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
1.4909 - unfolding vector_smult_component and vector_add_component
1.4910 - by(auto simp add: less_divide_eq_number_of1) }
1.4911 - thus ?thesis unfolding mem_interval by auto
1.4912 -qed
1.4913 -
1.4914 -lemma open_closed_interval_convex: fixes x :: "real^'n::finite"
1.4915 - assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
1.4916 - shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
1.4917 -proof-
1.4918 - { fix i
1.4919 - have "a $ i = e * a$i + (1 - e) * a$i" unfolding left_diff_distrib by simp
1.4920 - also have "\<dots> < e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
1.4921 - using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
1.4922 - using x unfolding mem_interval apply simp
1.4923 - using y unfolding mem_interval apply simp
1.4924 - done
1.4925 - finally have "a $ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i" by auto
1.4926 - moreover {
1.4927 - have "b $ i = e * b$i + (1 - e) * b$i" unfolding left_diff_distrib by simp
1.4928 - also have "\<dots> > e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
1.4929 - using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
1.4930 - using x unfolding mem_interval apply simp
1.4931 - using y unfolding mem_interval apply simp
1.4932 - done
1.4933 - finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i < b $ i" by auto
1.4934 - } ultimately have "a $ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i < b $ i" by auto }
1.4935 - thus ?thesis unfolding mem_interval by auto
1.4936 -qed
1.4937 -
1.4938 -lemma closure_open_interval: fixes a :: "real^'n::finite"
1.4939 - assumes "{a<..<b} \<noteq> {}"
1.4940 - shows "closure {a<..<b} = {a .. b}"
1.4941 -proof-
1.4942 - have ab:"a < b" using assms[unfolded interval_ne_empty] unfolding vector_less_def by auto
1.4943 - let ?c = "(1 / 2) *\<^sub>R (a + b)"
1.4944 - { fix x assume as:"x \<in> {a .. b}"
1.4945 - def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
1.4946 - { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
1.4947 - have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
1.4948 - have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
1.4949 - x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
1.4950 - by (auto simp add: algebra_simps)
1.4951 - 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
1.4952 - hence False using fn unfolding f_def using xc by(auto simp add: vector_mul_lcancel vector_ssub_ldistrib) }
1.4953 - moreover
1.4954 - { assume "\<not> (f ---> x) sequentially"
1.4955 - { fix e::real assume "e>0"
1.4956 - 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
1.4957 - then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1.4958 - 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)
1.4959 - hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
1.4960 - hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
1.4961 - unfolding Lim_sequentially by(auto simp add: dist_norm)
1.4962 - hence "(f ---> x) sequentially" unfolding f_def
1.4963 - using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
1.4964 - using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
1.4965 - ultimately have "x \<in> closure {a<..<b}"
1.4966 - using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
1.4967 - thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
1.4968 -qed
1.4969 -
1.4970 -lemma bounded_subset_open_interval_symmetric: fixes s::"(real^'n::finite) set"
1.4971 - assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
1.4972 -proof-
1.4973 - obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
1.4974 - def a \<equiv> "(\<chi> i. b+1)::real^'n"
1.4975 - { fix x assume "x\<in>s"
1.4976 - fix i
1.4977 - 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]
1.4978 - unfolding vector_uminus_component and a_def and Cart_lambda_beta by auto
1.4979 - }
1.4980 - thus ?thesis by(auto intro: exI[where x=a] simp add: vector_less_def)
1.4981 -qed
1.4982 -
1.4983 -lemma bounded_subset_open_interval:
1.4984 - fixes s :: "(real ^ 'n::finite) set"
1.4985 - shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
1.4986 - by (auto dest!: bounded_subset_open_interval_symmetric)
1.4987 -
1.4988 -lemma bounded_subset_closed_interval_symmetric:
1.4989 - fixes s :: "(real ^ 'n::finite) set"
1.4990 - assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
1.4991 -proof-
1.4992 - obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
1.4993 - thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
1.4994 -qed
1.4995 -
1.4996 -lemma bounded_subset_closed_interval:
1.4997 - fixes s :: "(real ^ 'n::finite) set"
1.4998 - shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
1.4999 - using bounded_subset_closed_interval_symmetric[of s] by auto
1.5000 -
1.5001 -lemma frontier_closed_interval:
1.5002 - fixes a b :: "real ^ _"
1.5003 - shows "frontier {a .. b} = {a .. b} - {a<..<b}"
1.5004 - unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
1.5005 -
1.5006 -lemma frontier_open_interval:
1.5007 - fixes a b :: "real ^ _"
1.5008 - shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
1.5009 -proof(cases "{a<..<b} = {}")
1.5010 - case True thus ?thesis using frontier_empty by auto
1.5011 -next
1.5012 - case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
1.5013 -qed
1.5014 -
1.5015 -lemma inter_interval_mixed_eq_empty: fixes a :: "real^'n::finite"
1.5016 - assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
1.5017 - unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
1.5018 -
1.5019 -
1.5020 -(* Some special cases for intervals in R^1. *)
1.5021 -
1.5022 -lemma all_1: "(\<forall>x::1. P x) \<longleftrightarrow> P 1"
1.5023 - by (metis num1_eq_iff)
1.5024 -
1.5025 -lemma ex_1: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
1.5026 - by auto (metis num1_eq_iff)
1.5027 -
1.5028 -lemma interval_cases_1: fixes x :: "real^1" shows
1.5029 - "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
1.5030 - by(simp add: Cart_eq vector_less_def vector_less_eq_def all_1, auto)
1.5031 -
1.5032 -lemma in_interval_1: fixes x :: "real^1" shows
1.5033 - "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
1.5034 - (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
1.5035 -by(simp add: Cart_eq vector_less_def vector_less_eq_def all_1 dest_vec1_def)
1.5036 -
1.5037 -lemma interval_eq_empty_1: fixes a :: "real^1" shows
1.5038 - "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
1.5039 - "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
1.5040 - unfolding interval_eq_empty and ex_1 and dest_vec1_def by auto
1.5041 -
1.5042 -lemma subset_interval_1: fixes a :: "real^1" shows
1.5043 - "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
1.5044 - dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
1.5045 - "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
1.5046 - dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
1.5047 - "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
1.5048 - dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
1.5049 - "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
1.5050 - dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
1.5051 - unfolding subset_interval[of a b c d] unfolding all_1 and dest_vec1_def by auto
1.5052 -
1.5053 -lemma eq_interval_1: fixes a :: "real^1" shows
1.5054 - "{a .. b} = {c .. d} \<longleftrightarrow>
1.5055 - dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
1.5056 - dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
1.5057 -using set_eq_subset[of "{a .. b}" "{c .. d}"]
1.5058 -using subset_interval_1(1)[of a b c d]
1.5059 -using subset_interval_1(1)[of c d a b]
1.5060 -by auto (* FIXME: slow *)
1.5061 -
1.5062 -lemma disjoint_interval_1: fixes a :: "real^1" shows
1.5063 - "{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"
1.5064 - "{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"
1.5065 - "{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"
1.5066 - "{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"
1.5067 - unfolding disjoint_interval and dest_vec1_def ex_1 by auto
1.5068 -
1.5069 -lemma open_closed_interval_1: fixes a :: "real^1" shows
1.5070 - "{a<..<b} = {a .. b} - {a, b}"
1.5071 - 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
1.5072 -
1.5073 -lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
1.5074 - 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
1.5075 -
1.5076 -(* Some stuff for half-infinite intervals too; FIXME: notation? *)
1.5077 -
1.5078 -lemma closed_interval_left: fixes b::"real^'n::finite"
1.5079 - shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
1.5080 -proof-
1.5081 - { fix i
1.5082 - 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"
1.5083 - { assume "x$i > b$i"
1.5084 - 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
1.5085 - hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto }
1.5086 - hence "x$i \<le> b$i" by(rule ccontr)auto }
1.5087 - thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
1.5088 -qed
1.5089 -
1.5090 -lemma closed_interval_right: fixes a::"real^'n::finite"
1.5091 - shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
1.5092 -proof-
1.5093 - { fix i
1.5094 - 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"
1.5095 - { assume "a$i > x$i"
1.5096 - 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
1.5097 - hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto }
1.5098 - hence "a$i \<le> x$i" by(rule ccontr)auto }
1.5099 - thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
1.5100 -qed
1.5101 -
1.5102 -subsection{* Intervals in general, including infinite and mixtures of open and closed. *}
1.5103 -
1.5104 -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)"
1.5105 -
1.5106 -lemma is_interval_interval: "is_interval {a .. b::real^'n::finite}" (is ?th1) "is_interval {a<..<b}" (is ?th2) proof -
1.5107 - have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
1.5108 - show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
1.5109 - by(meson real_le_trans le_less_trans less_le_trans *)+ qed
1.5110 -
1.5111 -lemma is_interval_empty:
1.5112 - "is_interval {}"
1.5113 - unfolding is_interval_def
1.5114 - by simp
1.5115 -
1.5116 -lemma is_interval_univ:
1.5117 - "is_interval UNIV"
1.5118 - unfolding is_interval_def
1.5119 - by simp
1.5120 -
1.5121 -subsection{* Closure of halfspaces and hyperplanes. *}
1.5122 -
1.5123 -lemma Lim_inner:
1.5124 - assumes "(f ---> l) net" shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
1.5125 - by (intro tendsto_intros assms)
1.5126 -
1.5127 -lemma continuous_at_inner: "continuous (at x) (inner a)"
1.5128 - unfolding continuous_at by (intro tendsto_intros)
1.5129 -
1.5130 -lemma continuous_on_inner:
1.5131 - fixes s :: "'a::real_inner set"
1.5132 - shows "continuous_on s (inner a)"
1.5133 - unfolding continuous_on by (rule ballI) (intro tendsto_intros)
1.5134 -
1.5135 -lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
1.5136 -proof-
1.5137 - have "\<forall>x. continuous (at x) (inner a)"
1.5138 - unfolding continuous_at by (rule allI) (intro tendsto_intros)
1.5139 - hence "closed (inner a -` {..b})"
1.5140 - using closed_real_atMost by (rule continuous_closed_vimage)
1.5141 - moreover have "{x. inner a x \<le> b} = inner a -` {..b}" by auto
1.5142 - ultimately show ?thesis by simp
1.5143 -qed
1.5144 -
1.5145 -lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
1.5146 - using closed_halfspace_le[of "-a" "-b"] unfolding inner_minus_left by auto
1.5147 -
1.5148 -lemma closed_hyperplane: "closed {x. inner a x = b}"
1.5149 -proof-
1.5150 - have "{x. inner a x = b} = {x. inner a x \<ge> b} \<inter> {x. inner a x \<le> b}" by auto
1.5151 - thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
1.5152 -qed
1.5153 -
1.5154 -lemma closed_halfspace_component_le:
1.5155 - shows "closed {x::real^'n::finite. x$i \<le> a}"
1.5156 - using closed_halfspace_le[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
1.5157 -
1.5158 -lemma closed_halfspace_component_ge:
1.5159 - shows "closed {x::real^'n::finite. x$i \<ge> a}"
1.5160 - using closed_halfspace_ge[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
1.5161 -
1.5162 -text{* Openness of halfspaces. *}
1.5163 -
1.5164 -lemma open_halfspace_lt: "open {x. inner a x < b}"
1.5165 -proof-
1.5166 - have "UNIV - {x. b \<le> inner a x} = {x. inner a x < b}" by auto
1.5167 - thus ?thesis using closed_halfspace_ge[unfolded closed_def Compl_eq_Diff_UNIV, of b a] by auto
1.5168 -qed
1.5169 -
1.5170 -lemma open_halfspace_gt: "open {x. inner a x > b}"
1.5171 -proof-
1.5172 - have "UNIV - {x. b \<ge> inner a x} = {x. inner a x > b}" by auto
1.5173 - thus ?thesis using closed_halfspace_le[unfolded closed_def Compl_eq_Diff_UNIV, of a b] by auto
1.5174 -qed
1.5175 -
1.5176 -lemma open_halfspace_component_lt:
1.5177 - shows "open {x::real^'n::finite. x$i < a}"
1.5178 - using open_halfspace_lt[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
1.5179 -
1.5180 -lemma open_halfspace_component_gt:
1.5181 - shows "open {x::real^'n::finite. x$i > a}"
1.5182 - using open_halfspace_gt[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
1.5183 -
1.5184 -text{* This gives a simple derivation of limit component bounds. *}
1.5185 -
1.5186 -lemma Lim_component_le: fixes f :: "'a \<Rightarrow> real^'n::finite"
1.5187 - assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$i \<le> b) net"
1.5188 - shows "l$i \<le> b"
1.5189 -proof-
1.5190 - { fix x have "x \<in> {x::real^'n. inner (basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this
1.5191 - show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
1.5192 - using closed_halfspace_le[of "(basis i)::real^'n" b] and assms(1,2,3) by auto
1.5193 -qed
1.5194 -
1.5195 -lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> real^'n::finite"
1.5196 - assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net"
1.5197 - shows "b \<le> l$i"
1.5198 -proof-
1.5199 - { fix x have "x \<in> {x::real^'n. inner (basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this
1.5200 - show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
1.5201 - using closed_halfspace_ge[of b "(basis i)::real^'n"] and assms(1,2,3) by auto
1.5202 -qed
1.5203 -
1.5204 -lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> real^'n::finite"
1.5205 - assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
1.5206 - shows "l$i = b"
1.5207 - 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
1.5208 -
1.5209 -lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
1.5210 - "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
1.5211 - using Lim_component_le[of f l net 1 b] unfolding dest_vec1_def by auto
1.5212 -
1.5213 -lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows
1.5214 - "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
1.5215 - using Lim_component_ge[of f l net b 1] unfolding dest_vec1_def by auto
1.5216 -
1.5217 -text{* Limits relative to a union. *}
1.5218 -
1.5219 -lemma eventually_within_Un:
1.5220 - "eventually P (net within (s \<union> t)) \<longleftrightarrow>
1.5221 - eventually P (net within s) \<and> eventually P (net within t)"
1.5222 - unfolding Limits.eventually_within
1.5223 - by (auto elim!: eventually_rev_mp)
1.5224 -
1.5225 -lemma Lim_within_union:
1.5226 - "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
1.5227 - (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
1.5228 - unfolding tendsto_def
1.5229 - by (auto simp add: eventually_within_Un)
1.5230 -
1.5231 -lemma continuous_on_union:
1.5232 - assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
1.5233 - shows "continuous_on (s \<union> t) f"
1.5234 - using assms unfolding continuous_on unfolding Lim_within_union
1.5235 - unfolding Lim unfolding trivial_limit_within unfolding closed_limpt by auto
1.5236 -
1.5237 -lemma continuous_on_cases:
1.5238 - assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
1.5239 - "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
1.5240 - shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
1.5241 -proof-
1.5242 - let ?h = "(\<lambda>x. if P x then f x else g x)"
1.5243 - have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
1.5244 - hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
1.5245 - moreover
1.5246 - have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
1.5247 - hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
1.5248 - ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
1.5249 -qed
1.5250 -
1.5251 -
1.5252 -text{* Some more convenient intermediate-value theorem formulations. *}
1.5253 -
1.5254 -lemma connected_ivt_hyperplane:
1.5255 - assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
1.5256 - shows "\<exists>z \<in> s. inner a z = b"
1.5257 -proof(rule ccontr)
1.5258 - assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
1.5259 - let ?A = "{x. inner a x < b}"
1.5260 - let ?B = "{x. inner a x > b}"
1.5261 - have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
1.5262 - moreover have "?A \<inter> ?B = {}" by auto
1.5263 - moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
1.5264 - 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
1.5265 -qed
1.5266 -
1.5267 -lemma connected_ivt_component: fixes x::"real^'n::finite" shows
1.5268 - "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)"
1.5269 - using connected_ivt_hyperplane[of s x y "(basis k)::real^'n" a] by (auto simp add: inner_basis)
1.5270 -
1.5271 -text{* Also more convenient formulations of monotone convergence. *}
1.5272 -
1.5273 -lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real^1"
1.5274 - assumes "bounded {s n| n::nat. True}" "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
1.5275 - shows "\<exists>l. (s ---> l) sequentially"
1.5276 -proof-
1.5277 - 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
1.5278 - { fix m::nat
1.5279 - have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
1.5280 - apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq) }
1.5281 - hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
1.5282 - 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
1.5283 - thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)
1.5284 - unfolding dist_norm unfolding abs_dest_vec1 and dest_vec1_sub by auto
1.5285 -qed
1.5286 -
1.5287 -subsection{* Basic homeomorphism definitions. *}
1.5288 -
1.5289 -definition "homeomorphism s t f g \<equiv>
1.5290 - (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
1.5291 - (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
1.5292 -
1.5293 -definition
1.5294 - homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
1.5295 - (infixr "homeomorphic" 60) where
1.5296 - homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
1.5297 -
1.5298 -lemma homeomorphic_refl: "s homeomorphic s"
1.5299 - unfolding homeomorphic_def
1.5300 - unfolding homeomorphism_def
1.5301 - using continuous_on_id
1.5302 - apply(rule_tac x = "(\<lambda>x. x)" in exI)
1.5303 - apply(rule_tac x = "(\<lambda>x. x)" in exI)
1.5304 - by blast
1.5305 -
1.5306 -lemma homeomorphic_sym:
1.5307 - "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
1.5308 -unfolding homeomorphic_def
1.5309 -unfolding homeomorphism_def
1.5310 -by blast (* FIXME: slow *)
1.5311 -
1.5312 -lemma homeomorphic_trans:
1.5313 - assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
1.5314 -proof-
1.5315 - obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x" "f1 ` s = t" "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1 ` t = s" "continuous_on t g1"
1.5316 - using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
1.5317 - obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x" "f2 ` t = u" "continuous_on t f2" "\<forall>y\<in>u. f2 (g2 y) = y" "g2 ` u = t" "continuous_on u g2"
1.5318 - using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
1.5319 -
1.5320 - { fix x assume "x\<in>s" hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x" using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2) by auto }
1.5321 - moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
1.5322 - moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
1.5323 - moreover { fix y assume "y\<in>u" hence "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y" using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5) by auto }
1.5324 - moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
1.5325 - moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
1.5326 - ultimately show ?thesis unfolding homeomorphic_def homeomorphism_def apply(rule_tac x="f2 \<circ> f1" in exI) apply(rule_tac x="g1 \<circ> g2" in exI) by auto
1.5327 -qed
1.5328 -
1.5329 -lemma homeomorphic_minimal:
1.5330 - "s homeomorphic t \<longleftrightarrow>
1.5331 - (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
1.5332 - (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
1.5333 - continuous_on s f \<and> continuous_on t g)"
1.5334 -unfolding homeomorphic_def homeomorphism_def
1.5335 -apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
1.5336 -apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
1.5337 -unfolding image_iff
1.5338 -apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
1.5339 -apply auto apply(rule_tac x="g x" in bexI) apply auto
1.5340 -apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
1.5341 -apply auto apply(rule_tac x="f x" in bexI) by auto
1.5342 -
1.5343 -subsection{* Relatively weak hypotheses if a set is compact. *}
1.5344 -
1.5345 -definition "inv_on f s = (\<lambda>x. SOME y. y\<in>s \<and> f y = x)"
1.5346 -
1.5347 -lemma assumes "inj_on f s" "x\<in>s"
1.5348 - shows "inv_on f s (f x) = x"
1.5349 - using assms unfolding inj_on_def inv_on_def by auto
1.5350 -
1.5351 -lemma homeomorphism_compact:
1.5352 - fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
1.5353 - (* class constraint due to continuous_on_inverse *)
1.5354 - assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
1.5355 - shows "\<exists>g. homeomorphism s t f g"
1.5356 -proof-
1.5357 - def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
1.5358 - have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
1.5359 - { fix y assume "y\<in>t"
1.5360 - then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
1.5361 - hence "g (f x) = x" using g by auto
1.5362 - hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
1.5363 - hence g':"\<forall>x\<in>t. f (g x) = x" by auto
1.5364 - moreover
1.5365 - { fix x
1.5366 - 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"])
1.5367 - moreover
1.5368 - { assume "x\<in>g ` t"
1.5369 - then obtain y where y:"y\<in>t" "g y = x" by auto
1.5370 - then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
1.5371 - 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 }
1.5372 - ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" by auto }
1.5373 - hence "g ` t = s" by auto
1.5374 - ultimately
1.5375 - show ?thesis unfolding homeomorphism_def homeomorphic_def
1.5376 - 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
1.5377 -qed
1.5378 -
1.5379 -lemma homeomorphic_compact:
1.5380 - fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
1.5381 - (* class constraint due to continuous_on_inverse *)
1.5382 - shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
1.5383 - \<Longrightarrow> s homeomorphic t"
1.5384 - unfolding homeomorphic_def by(metis homeomorphism_compact)
1.5385 -
1.5386 -text{* Preservation of topological properties. *}
1.5387 -
1.5388 -lemma homeomorphic_compactness:
1.5389 - "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
1.5390 -unfolding homeomorphic_def homeomorphism_def
1.5391 -by (metis compact_continuous_image)
1.5392 -
1.5393 -text{* Results on translation, scaling etc. *}
1.5394 -
1.5395 -lemma homeomorphic_scaling:
1.5396 - fixes s :: "'a::real_normed_vector set"
1.5397 - assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
1.5398 - unfolding homeomorphic_minimal
1.5399 - apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
1.5400 - apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
1.5401 - using assms apply auto
1.5402 - using continuous_on_cmul[OF continuous_on_id] by auto
1.5403 -
1.5404 -lemma homeomorphic_translation:
1.5405 - fixes s :: "'a::real_normed_vector set"
1.5406 - shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
1.5407 - unfolding homeomorphic_minimal
1.5408 - apply(rule_tac x="\<lambda>x. a + x" in exI)
1.5409 - apply(rule_tac x="\<lambda>x. -a + x" in exI)
1.5410 - using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
1.5411 -
1.5412 -lemma homeomorphic_affinity:
1.5413 - fixes s :: "'a::real_normed_vector set"
1.5414 - assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
1.5415 -proof-
1.5416 - have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
1.5417 - show ?thesis
1.5418 - using homeomorphic_trans
1.5419 - using homeomorphic_scaling[OF assms, of s]
1.5420 - using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
1.5421 -qed
1.5422 -
1.5423 -lemma homeomorphic_balls:
1.5424 - fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
1.5425 - assumes "0 < d" "0 < e"
1.5426 - shows "(ball a d) homeomorphic (ball b e)" (is ?th)
1.5427 - "(cball a d) homeomorphic (cball b e)" (is ?cth)
1.5428 -proof-
1.5429 - have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
1.5430 - show ?th unfolding homeomorphic_minimal
1.5431 - apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
1.5432 - apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
1.5433 - using assms apply (auto simp add: dist_commute)
1.5434 - unfolding dist_norm
1.5435 - apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
1.5436 - unfolding continuous_on
1.5437 - by (intro ballI tendsto_intros, simp, assumption)+
1.5438 -next
1.5439 - have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
1.5440 - show ?cth unfolding homeomorphic_minimal
1.5441 - apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
1.5442 - apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
1.5443 - using assms apply (auto simp add: dist_commute)
1.5444 - unfolding dist_norm
1.5445 - apply (auto simp add: pos_divide_le_eq)
1.5446 - unfolding continuous_on
1.5447 - by (intro ballI tendsto_intros, simp, assumption)+
1.5448 -qed
1.5449 -
1.5450 -text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
1.5451 -
1.5452 -lemma cauchy_isometric:
1.5453 - fixes x :: "nat \<Rightarrow> real ^ 'n::finite"
1.5454 - assumes e:"0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and xs:"\<forall>n::nat. x n \<in> s" and cf:"Cauchy(f o x)"
1.5455 - shows "Cauchy x"
1.5456 -proof-
1.5457 - interpret f: bounded_linear f by fact
1.5458 - { fix d::real assume "d>0"
1.5459 - then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
1.5460 - 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
1.5461 - { fix n assume "n\<ge>N"
1.5462 - hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
1.5463 - moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
1.5464 - using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
1.5465 - using normf[THEN bspec[where x="x n - x N"]] by auto
1.5466 - ultimately have "norm (x n - x N) < d" using `e>0`
1.5467 - using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
1.5468 - hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
1.5469 - thus ?thesis unfolding cauchy and dist_norm by auto
1.5470 -qed
1.5471 -
1.5472 -lemma complete_isometric_image:
1.5473 - fixes f :: "real ^ _ \<Rightarrow> real ^ _"
1.5474 - assumes "0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and cs:"complete s"
1.5475 - shows "complete(f ` s)"
1.5476 -proof-
1.5477 - { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
1.5478 - then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)" unfolding image_iff and Bex_def
1.5479 - using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
1.5480 - hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
1.5481 - hence "f \<circ> x = g" unfolding expand_fun_eq by auto
1.5482 - then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
1.5483 - using cs[unfolded complete_def, THEN spec[where x="x"]]
1.5484 - using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
1.5485 - hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
1.5486 - using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
1.5487 - unfolding `f \<circ> x = g` by auto }
1.5488 - thus ?thesis unfolding complete_def by auto
1.5489 -qed
1.5490 -
1.5491 -lemma dist_0_norm:
1.5492 - fixes x :: "'a::real_normed_vector"
1.5493 - shows "dist 0 x = norm x"
1.5494 -unfolding dist_norm by simp
1.5495 -
1.5496 -lemma injective_imp_isometric: fixes f::"real^'m::finite \<Rightarrow> real^'n::finite"
1.5497 - assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
1.5498 - shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
1.5499 -proof(cases "s \<subseteq> {0::real^'m}")
1.5500 - case True
1.5501 - { fix x assume "x \<in> s"
1.5502 - hence "x = 0" using True by auto
1.5503 - hence "norm x \<le> norm (f x)" by auto }
1.5504 - thus ?thesis by(auto intro!: exI[where x=1])
1.5505 -next
1.5506 - interpret f: bounded_linear f by fact
1.5507 - case False
1.5508 - then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
1.5509 - from False have "s \<noteq> {}" by auto
1.5510 - let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
1.5511 - let ?S' = "{x::real^'m. x\<in>s \<and> norm x = norm a}"
1.5512 - let ?S'' = "{x::real^'m. norm x = norm a}"
1.5513 -
1.5514 - have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by (auto simp add: norm_minus_cancel)
1.5515 - hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
1.5516 - moreover have "?S' = s \<inter> ?S''" by auto
1.5517 - ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
1.5518 - moreover have *:"f ` ?S' = ?S" by auto
1.5519 - ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
1.5520 - hence "closed ?S" using compact_imp_closed by auto
1.5521 - moreover have "?S \<noteq> {}" using a by auto
1.5522 - 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
1.5523 - 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
1.5524 -
1.5525 - let ?e = "norm (f b) / norm b"
1.5526 - have "norm b > 0" using ba and a and norm_ge_zero by auto
1.5527 - 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
1.5528 - ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
1.5529 - moreover
1.5530 - { fix x assume "x\<in>s"
1.5531 - hence "norm (f b) / norm b * norm x \<le> norm (f x)"
1.5532 - proof(cases "x=0")
1.5533 - case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
1.5534 - next
1.5535 - case False
1.5536 - hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
1.5537 - have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def smult_conv_scaleR] by auto
1.5538 - hence "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}" using `x\<in>s` and `x\<noteq>0` by auto
1.5539 - thus "norm (f b) / norm b * norm x \<le> norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
1.5540 - unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
1.5541 - by (auto simp add: real_mult_commute pos_le_divide_eq pos_divide_le_eq)
1.5542 - qed }
1.5543 - ultimately
1.5544 - show ?thesis by auto
1.5545 -qed
1.5546 -
1.5547 -lemma closed_injective_image_subspace:
1.5548 - fixes f :: "real ^ _ \<Rightarrow> real ^ _"
1.5549 - assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
1.5550 - shows "closed(f ` s)"
1.5551 -proof-
1.5552 - 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
1.5553 - show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
1.5554 - unfolding complete_eq_closed[THEN sym] by auto
1.5555 -qed
1.5556 -
1.5557 -subsection{* Some properties of a canonical subspace. *}
1.5558 -
1.5559 -lemma subspace_substandard:
1.5560 - "subspace {x::real^'n. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
1.5561 - unfolding subspace_def by(auto simp add: vector_add_component vector_smult_component elim!: ballE)
1.5562 -
1.5563 -lemma closed_substandard:
1.5564 - "closed {x::real^'n::finite. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
1.5565 -proof-
1.5566 - let ?D = "{i. P i}"
1.5567 - let ?Bs = "{{x::real^'n. inner (basis i) x = 0}| i. i \<in> ?D}"
1.5568 - { fix x
1.5569 - { assume "x\<in>?A"
1.5570 - hence x:"\<forall>i\<in>?D. x $ i = 0" by auto
1.5571 - hence "x\<in> \<Inter> ?Bs" by(auto simp add: inner_basis x) }
1.5572 - moreover
1.5573 - { assume x:"x\<in>\<Inter>?Bs"
1.5574 - { fix i assume i:"i \<in> ?D"
1.5575 - then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (basis i) x = 0}" by auto
1.5576 - hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto }
1.5577 - hence "x\<in>?A" by auto }
1.5578 - ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
1.5579 - hence "?A = \<Inter> ?Bs" by auto
1.5580 - thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
1.5581 -qed
1.5582 -
1.5583 -lemma dim_substandard:
1.5584 - shows "dim {x::real^'n::finite. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
1.5585 -proof-
1.5586 - let ?D = "UNIV::'n set"
1.5587 - let ?B = "(basis::'n\<Rightarrow>real^'n) ` d"
1.5588 -
1.5589 - let ?bas = "basis::'n \<Rightarrow> real^'n"
1.5590 -
1.5591 - have "?B \<subseteq> ?A" by auto
1.5592 -
1.5593 - moreover
1.5594 - { fix x::"real^'n" assume "x\<in>?A"
1.5595 - with finite[of d]
1.5596 - have "x\<in> span ?B"
1.5597 - proof(induct d arbitrary: x)
1.5598 - case empty hence "x=0" unfolding Cart_eq by auto
1.5599 - thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
1.5600 - next
1.5601 - case (insert k F)
1.5602 - hence *:"\<forall>i. i \<notin> insert k F \<longrightarrow> x $ i = 0" by auto
1.5603 - have **:"F \<subseteq> insert k F" by auto
1.5604 - def y \<equiv> "x - x$k *\<^sub>R basis k"
1.5605 - have y:"x = y + (x$k) *\<^sub>R basis k" unfolding y_def by auto
1.5606 - { fix i assume i':"i \<notin> F"
1.5607 - hence "y $ i = 0" unfolding y_def unfolding vector_minus_component
1.5608 - and vector_smult_component and basis_component
1.5609 - using *[THEN spec[where x=i]] by auto }
1.5610 - hence "y \<in> span (basis ` (insert k F))" using insert(3)
1.5611 - using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
1.5612 - using image_mono[OF **, of basis] by auto
1.5613 - moreover
1.5614 - have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
1.5615 - hence "x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
1.5616 - using span_mul [where 'a=real, unfolded smult_conv_scaleR] by auto
1.5617 - ultimately
1.5618 - have "y + x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
1.5619 - using span_add by auto
1.5620 - thus ?case using y by auto
1.5621 - qed
1.5622 - }
1.5623 - hence "?A \<subseteq> span ?B" by auto
1.5624 -
1.5625 - moreover
1.5626 - { fix x assume "x \<in> ?B"
1.5627 - hence "x\<in>{(basis i)::real^'n |i. i \<in> ?D}" using assms by auto }
1.5628 - hence "independent ?B" using independent_mono[OF independent_stdbasis, of ?B] and assms by auto
1.5629 -
1.5630 - moreover
1.5631 - have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
1.5632 - hence *:"inj_on (basis::'n\<Rightarrow>real^'n) d" using subset_inj_on[OF basis_inj, of "d"] by auto
1.5633 - have "?B hassize (card d)" unfolding hassize_def and card_image[OF *] by auto
1.5634 -
1.5635 - ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
1.5636 -qed
1.5637 -
1.5638 -text{* Hence closure and completeness of all subspaces. *}
1.5639 -
1.5640 -lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
1.5641 -apply (induct n)
1.5642 -apply (rule_tac x="{}" in exI, simp)
1.5643 -apply clarsimp
1.5644 -apply (subgoal_tac "\<exists>x. x \<notin> A")
1.5645 -apply (erule exE)
1.5646 -apply (rule_tac x="insert x A" in exI, simp)
1.5647 -apply (subgoal_tac "A \<noteq> UNIV", auto)
1.5648 -done
1.5649 -
1.5650 -lemma closed_subspace: fixes s::"(real^'n::finite) set"
1.5651 - assumes "subspace s" shows "closed s"
1.5652 -proof-
1.5653 - have "dim s \<le> card (UNIV :: 'n set)" using dim_subset_univ by auto
1.5654 - then obtain d::"'n set" where t: "card d = dim s"
1.5655 - using closed_subspace_lemma by auto
1.5656 - let ?t = "{x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"
1.5657 - obtain f where f:"bounded_linear f" "f ` ?t = s" "inj_on f ?t"
1.5658 - using subspace_isomorphism[unfolded linear_conv_bounded_linear, OF subspace_substandard[of "\<lambda>i. i \<notin> d"] assms]
1.5659 - using dim_substandard[of d] and t by auto
1.5660 - interpret f: bounded_linear f by fact
1.5661 - have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
1.5662 - by(erule_tac x=0 in ballE) auto
1.5663 - moreover have "closed ?t" using closed_substandard .
1.5664 - moreover have "subspace ?t" using subspace_substandard .
1.5665 - ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
1.5666 - unfolding f(2) using f(1) by auto
1.5667 -qed
1.5668 -
1.5669 -lemma complete_subspace:
1.5670 - fixes s :: "(real ^ _) set" shows "subspace s ==> complete s"
1.5671 - using complete_eq_closed closed_subspace
1.5672 - by auto
1.5673 -
1.5674 -lemma dim_closure:
1.5675 - fixes s :: "(real ^ _) set"
1.5676 - shows "dim(closure s) = dim s" (is "?dc = ?d")
1.5677 -proof-
1.5678 - have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
1.5679 - using closed_subspace[OF subspace_span, of s]
1.5680 - using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
1.5681 - thus ?thesis using dim_subset[OF closure_subset, of s] by auto
1.5682 -qed
1.5683 -
1.5684 -text{* Affine transformations of intervals. *}
1.5685 -
1.5686 -lemma affinity_inverses:
1.5687 - assumes m0: "m \<noteq> (0::'a::field)"
1.5688 - shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
1.5689 - "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
1.5690 - using m0
1.5691 -apply (auto simp add: expand_fun_eq vector_add_ldistrib vector_smult_assoc)
1.5692 -by (simp add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
1.5693 -
1.5694 -lemma real_affinity_le:
1.5695 - "0 < (m::'a::ordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
1.5696 - by (simp add: field_simps inverse_eq_divide)
1.5697 -
1.5698 -lemma real_le_affinity:
1.5699 - "0 < (m::'a::ordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
1.5700 - by (simp add: field_simps inverse_eq_divide)
1.5701 -
1.5702 -lemma real_affinity_lt:
1.5703 - "0 < (m::'a::ordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
1.5704 - by (simp add: field_simps inverse_eq_divide)
1.5705 -
1.5706 -lemma real_lt_affinity:
1.5707 - "0 < (m::'a::ordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
1.5708 - by (simp add: field_simps inverse_eq_divide)
1.5709 -
1.5710 -lemma real_affinity_eq:
1.5711 - "(m::'a::ordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
1.5712 - by (simp add: field_simps inverse_eq_divide)
1.5713 -
1.5714 -lemma real_eq_affinity:
1.5715 - "(m::'a::ordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
1.5716 - by (simp add: field_simps inverse_eq_divide)
1.5717 -
1.5718 -lemma vector_affinity_eq:
1.5719 - assumes m0: "(m::'a::field) \<noteq> 0"
1.5720 - shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
1.5721 -proof
1.5722 - assume h: "m *s x + c = y"
1.5723 - hence "m *s x = y - c" by (simp add: ring_simps)
1.5724 - hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
1.5725 - then show "x = inverse m *s y + - (inverse m *s c)"
1.5726 - using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
1.5727 -next
1.5728 - assume h: "x = inverse m *s y + - (inverse m *s c)"
1.5729 - show "m *s x + c = y" unfolding h diff_minus[symmetric]
1.5730 - using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
1.5731 -qed
1.5732 -
1.5733 -lemma vector_eq_affinity:
1.5734 - "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
1.5735 - using vector_affinity_eq[where m=m and x=x and y=y and c=c]
1.5736 - by metis
1.5737 -
1.5738 -lemma image_affinity_interval: fixes m::real
1.5739 - fixes a b c :: "real^'n::finite"
1.5740 - shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
1.5741 - (if {a .. b} = {} then {}
1.5742 - else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
1.5743 - else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
1.5744 -proof(cases "m=0")
1.5745 - { fix x assume "x \<le> c" "c \<le> x"
1.5746 - hence "x=c" unfolding vector_less_eq_def and Cart_eq by (auto intro: order_antisym) }
1.5747 - moreover case True
1.5748 - moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: vector_less_eq_def)
1.5749 - ultimately show ?thesis by auto
1.5750 -next
1.5751 - case False
1.5752 - { fix y assume "a \<le> y" "y \<le> b" "m > 0"
1.5753 - hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
1.5754 - unfolding vector_less_eq_def by(auto simp add: vector_smult_component vector_add_component)
1.5755 - } moreover
1.5756 - { fix y assume "a \<le> y" "y \<le> b" "m < 0"
1.5757 - hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
1.5758 - unfolding vector_less_eq_def by(auto simp add: vector_smult_component vector_add_component mult_left_mono_neg elim!:ballE)
1.5759 - } moreover
1.5760 - { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
1.5761 - hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
1.5762 - unfolding image_iff Bex_def mem_interval vector_less_eq_def
1.5763 - apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
1.5764 - intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
1.5765 - by(auto simp add: pos_le_divide_eq pos_divide_le_eq real_mult_commute diff_le_iff)
1.5766 - } moreover
1.5767 - { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
1.5768 - hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
1.5769 - unfolding image_iff Bex_def mem_interval vector_less_eq_def
1.5770 - apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
1.5771 - intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
1.5772 - by(auto simp add: neg_le_divide_eq neg_divide_le_eq real_mult_commute diff_le_iff)
1.5773 - }
1.5774 - ultimately show ?thesis using False by auto
1.5775 -qed
1.5776 -
1.5777 -lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n::finite)) ` {a..b} =
1.5778 - (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
1.5779 - using image_affinity_interval[of m 0 a b] by auto
1.5780 -
1.5781 -subsection{* Banach fixed point theorem (not really topological...) *}
1.5782 -
1.5783 -lemma banach_fix:
1.5784 - assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
1.5785 - lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
1.5786 - shows "\<exists>! x\<in>s. (f x = x)"
1.5787 -proof-
1.5788 - have "1 - c > 0" using c by auto
1.5789 -
1.5790 - from s(2) obtain z0 where "z0 \<in> s" by auto
1.5791 - def z \<equiv> "\<lambda>n. (f ^^ n) z0"
1.5792 - { fix n::nat
1.5793 - have "z n \<in> s" unfolding z_def
1.5794 - proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
1.5795 - next case Suc thus ?case using f by auto qed }
1.5796 - note z_in_s = this
1.5797 -
1.5798 - def d \<equiv> "dist (z 0) (z 1)"
1.5799 -
1.5800 - have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
1.5801 - { fix n::nat
1.5802 - have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
1.5803 - proof(induct n)
1.5804 - case 0 thus ?case unfolding d_def by auto
1.5805 - next
1.5806 - case (Suc m)
1.5807 - hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
1.5808 - using `0 \<le> c` using mult_mono1_class.mult_mono1[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
1.5809 - thus ?case using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
1.5810 - unfolding fzn and mult_le_cancel_left by auto
1.5811 - qed
1.5812 - } note cf_z = this
1.5813 -
1.5814 - { fix n m::nat
1.5815 - have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
1.5816 - proof(induct n)
1.5817 - case 0 show ?case by auto
1.5818 - next
1.5819 - case (Suc k)
1.5820 - have "(1 - c) * dist (z m) (z (m + Suc k)) \<le> (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
1.5821 - using dist_triangle and c by(auto simp add: dist_triangle)
1.5822 - also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
1.5823 - using cf_z[of "m + k"] and c by auto
1.5824 - also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
1.5825 - using Suc by (auto simp add: ring_simps)
1.5826 - also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
1.5827 - unfolding power_add by (auto simp add: ring_simps)
1.5828 - also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
1.5829 - using c by (auto simp add: ring_simps)
1.5830 - finally show ?case by auto
1.5831 - qed
1.5832 - } note cf_z2 = this
1.5833 - { fix e::real assume "e>0"
1.5834 - hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
1.5835 - proof(cases "d = 0")
1.5836 - case True
1.5837 - hence "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def by (auto simp add: pos_prod_le[OF `1 - c > 0`])
1.5838 - thus ?thesis using `e>0` by auto
1.5839 - next
1.5840 - case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
1.5841 - by (metis False d_def real_less_def)
1.5842 - hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
1.5843 - using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
1.5844 - then obtain N where N:"c ^ N < e * (1 - c) / d" using real_arch_pow_inv[of "e * (1 - c) / d" c] and c by auto
1.5845 - { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
1.5846 - have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
1.5847 - have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
1.5848 - hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
1.5849 - using real_mult_order[OF `d>0`, of "1 - c ^ (m - n)"]
1.5850 - using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
1.5851 - using `0 < 1 - c` by auto
1.5852 -
1.5853 - have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
1.5854 - using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
1.5855 - by (auto simp add: real_mult_commute dist_commute)
1.5856 - also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
1.5857 - using mult_right_mono[OF * order_less_imp_le[OF **]]
1.5858 - unfolding real_mult_assoc by auto
1.5859 - also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
1.5860 - using mult_strict_right_mono[OF N **] unfolding real_mult_assoc by auto
1.5861 - also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
1.5862 - also have "\<dots> \<le> e" using c and `1 - c ^ (m - n) > 0` and `e>0` using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
1.5863 - finally have "dist (z m) (z n) < e" by auto
1.5864 - } note * = this
1.5865 - { fix m n::nat assume as:"N\<le>m" "N\<le>n"
1.5866 - hence "dist (z n) (z m) < e"
1.5867 - proof(cases "n = m")
1.5868 - case True thus ?thesis using `e>0` by auto
1.5869 - next
1.5870 - case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
1.5871 - qed }
1.5872 - thus ?thesis by auto
1.5873 - qed
1.5874 - }
1.5875 - hence "Cauchy z" unfolding cauchy_def by auto
1.5876 - then obtain x where "x\<in>s" and x:"(z ---> x) sequentially" using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
1.5877 -
1.5878 - def e \<equiv> "dist (f x) x"
1.5879 - have "e = 0" proof(rule ccontr)
1.5880 - assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
1.5881 - by (metis dist_eq_0_iff dist_nz e_def)
1.5882 - then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
1.5883 - using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
1.5884 - hence N':"dist (z N) x < e / 2" by auto
1.5885 -
1.5886 - have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
1.5887 - using zero_le_dist[of "z N" x] and c
1.5888 - by (metis dist_eq_0_iff dist_nz order_less_asym real_less_def)
1.5889 - have "dist (f (z N)) (f x) \<le> c * dist (z N) x" using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
1.5890 - using z_in_s[of N] `x\<in>s` using c by auto
1.5891 - also have "\<dots> < e / 2" using N' and c using * by auto
1.5892 - finally show False unfolding fzn
1.5893 - using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
1.5894 - unfolding e_def by auto
1.5895 - qed
1.5896 - hence "f x = x" unfolding e_def by auto
1.5897 - moreover
1.5898 - { fix y assume "f y = y" "y\<in>s"
1.5899 - hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
1.5900 - using `x\<in>s` and `f x = x` by auto
1.5901 - hence "dist x y = 0" unfolding mult_le_cancel_right1
1.5902 - using c and zero_le_dist[of x y] by auto
1.5903 - hence "y = x" by auto
1.5904 - }
1.5905 - ultimately show ?thesis unfolding Bex1_def using `x\<in>s` by blast+
1.5906 -qed
1.5907 -
1.5908 -subsection{* Edelstein fixed point theorem. *}
1.5909 -
1.5910 -lemma edelstein_fix:
1.5911 - fixes s :: "'a::real_normed_vector set"
1.5912 - assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
1.5913 - and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
1.5914 - shows "\<exists>! x\<in>s. g x = x"
1.5915 -proof(cases "\<exists>x\<in>s. g x \<noteq> x")
1.5916 - obtain x where "x\<in>s" using s(2) by auto
1.5917 - case False hence g:"\<forall>x\<in>s. g x = x" by auto
1.5918 - { fix y assume "y\<in>s"
1.5919 - hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
1.5920 - unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
1.5921 - unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
1.5922 - thus ?thesis unfolding Bex1_def using `x\<in>s` and g by blast+
1.5923 -next
1.5924 - case True
1.5925 - then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
1.5926 - { fix x y assume "x \<in> s" "y \<in> s"
1.5927 - hence "dist (g x) (g y) \<le> dist x y"
1.5928 - using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
1.5929 - def y \<equiv> "g x"
1.5930 - have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
1.5931 - def f \<equiv> "\<lambda>n. g ^^ n"
1.5932 - have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
1.5933 - have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
1.5934 - { fix n::nat and z assume "z\<in>s"
1.5935 - have "f n z \<in> s" unfolding f_def
1.5936 - proof(induct n)
1.5937 - case 0 thus ?case using `z\<in>s` by simp
1.5938 - next
1.5939 - case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
1.5940 - qed } note fs = this
1.5941 - { fix m n ::nat assume "m\<le>n"
1.5942 - fix w z assume "w\<in>s" "z\<in>s"
1.5943 - have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
1.5944 - proof(induct n)
1.5945 - case 0 thus ?case by auto
1.5946 - next
1.5947 - case (Suc n)
1.5948 - thus ?case proof(cases "m\<le>n")
1.5949 - case True thus ?thesis using Suc(1)
1.5950 - using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
1.5951 - next
1.5952 - case False hence mn:"m = Suc n" using Suc(2) by simp
1.5953 - show ?thesis unfolding mn by auto
1.5954 - qed
1.5955 - qed } note distf = this
1.5956 -
1.5957 - def h \<equiv> "\<lambda>n. (f n x, f n y)"
1.5958 - let ?s2 = "s \<times> s"
1.5959 - obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
1.5960 - using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
1.5961 - using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
1.5962 - def a \<equiv> "fst l" def b \<equiv> "snd l"
1.5963 - have lab:"l = (a, b)" unfolding a_def b_def by simp
1.5964 - have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
1.5965 -
1.5966 - have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
1.5967 - and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
1.5968 - using lr
1.5969 - unfolding o_def a_def b_def by (simp_all add: tendsto_intros)
1.5970 -
1.5971 - { fix n::nat
1.5972 - have *:"\<And>fx fy (x::'a) y. dist fx fy \<le> dist x y \<Longrightarrow> \<not> (dist (fx - fy) (a - b) < dist a b - dist x y)" unfolding dist_norm by norm
1.5973 - { fix x y :: 'a
1.5974 - have "dist (-x) (-y) = dist x y" unfolding dist_norm
1.5975 - using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
1.5976 -
1.5977 - { assume as:"dist a b > dist (f n x) (f n y)"
1.5978 - then obtain Na Nb where "\<forall>m\<ge>Na. dist (f (r m) x) a < (dist a b - dist (f n x) (f n y)) / 2"
1.5979 - and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
1.5980 - using lima limb unfolding h_def Lim_sequentially by (fastsimp simp del: less_divide_eq_number_of1)
1.5981 - hence "dist (f (r (Na + Nb + n)) x - f (r (Na + Nb + n)) y) (a - b) < dist a b - dist (f n x) (f n y)"
1.5982 - apply(erule_tac x="Na+Nb+n" in allE)
1.5983 - apply(erule_tac x="Na+Nb+n" in allE) apply simp
1.5984 - using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
1.5985 - "-b" "- f (r (Na + Nb + n)) y"]
1.5986 - unfolding ** unfolding group_simps(12) by (auto simp add: dist_commute)
1.5987 - moreover
1.5988 - have "dist (f (r (Na + Nb + n)) x - f (r (Na + Nb + n)) y) (a - b) \<ge> dist a b - dist (f n x) (f n y)"
1.5989 - using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
1.5990 - using subseq_bigger[OF r, of "Na+Nb+n"]
1.5991 - using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
1.5992 - ultimately have False by simp
1.5993 - }
1.5994 - hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
1.5995 - note ab_fn = this
1.5996 -
1.5997 - have [simp]:"a = b" proof(rule ccontr)
1.5998 - def e \<equiv> "dist a b - dist (g a) (g b)"
1.5999 - assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastsimp
1.6000 - hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
1.6001 - using lima limb unfolding Lim_sequentially
1.6002 - apply (auto elim!: allE[where x="e/2"]) apply(rule_tac x="r (max N Na)" in exI) unfolding h_def by fastsimp
1.6003 - then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
1.6004 - have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
1.6005 - using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
1.6006 - moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
1.6007 - using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
1.6008 - ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
1.6009 - thus False unfolding e_def using ab_fn[of "Suc n"] by norm
1.6010 - qed
1.6011 -
1.6012 - have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
1.6013 - { fix x y assume "x\<in>s" "y\<in>s" moreover
1.6014 - fix e::real assume "e>0" ultimately
1.6015 - have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
1.6016 - hence "continuous_on s g" unfolding continuous_on_def by auto
1.6017 -
1.6018 - hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
1.6019 - apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
1.6020 - using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
1.6021 - hence "g a = a" using Lim_unique[OF trivial_limit_sequentially limb, of "g a"]
1.6022 - unfolding `a=b` and o_assoc by auto
1.6023 - moreover
1.6024 - { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
1.6025 - hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
1.6026 - using `g a = a` and `a\<in>s` by auto }
1.6027 - ultimately show "\<exists>!x\<in>s. g x = x" unfolding Bex1_def using `a\<in>s` by blast
1.6028 -qed
1.6029 -
1.6030 -end