--- a/src/HOL/Analysis/Abstract_Topology.thy Sat Dec 29 18:40:29 2018 +0000
+++ b/src/HOL/Analysis/Abstract_Topology.thy Sat Dec 29 20:32:09 2018 +0100
@@ -4,9 +4,588 @@
section \<open>Operators involving abstract topology\<close>
theory Abstract_Topology
- imports Topology_Euclidean_Space Path_Connected
+ imports
+ Complex_Main
+ "HOL-Library.Set_Idioms"
+ "HOL-Library.FuncSet"
+ (* Path_Connected *)
begin
+subsection \<open>General notion of a topology as a value\<close>
+
+definition%important "istopology L \<longleftrightarrow>
+ L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union>K))"
+
+typedef%important 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
+ morphisms "openin" "topology"
+ unfolding istopology_def by blast
+
+lemma istopology_openin[intro]: "istopology(openin U)"
+ using openin[of U] by blast
+
+lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
+ using topology_inverse[unfolded mem_Collect_eq] .
+
+lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
+ using topology_inverse[of U] istopology_openin[of "topology U"] by auto
+
+lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
+proof
+ assume "T1 = T2"
+ then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
+next
+ assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
+ then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
+ then have "topology (openin T1) = topology (openin T2)" by simp
+ then show "T1 = T2" unfolding openin_inverse .
+qed
+
+
+text\<open>The "universe": the union of all sets in the topology.\<close>
+definition "topspace T = \<Union>{S. openin T S}"
+
+subsubsection \<open>Main properties of open sets\<close>
+
+proposition openin_clauses:
+ fixes U :: "'a topology"
+ shows
+ "openin U {}"
+ "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
+ "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
+ using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
+
+lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
+ unfolding topspace_def by blast
+
+lemma openin_empty[simp]: "openin U {}"
+ by (rule openin_clauses)
+
+lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
+ by (rule openin_clauses)
+
+lemma openin_Union[intro]: "(\<And>S. S \<in> K \<Longrightarrow> openin U S) \<Longrightarrow> openin U (\<Union>K)"
+ using openin_clauses by blast
+
+lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
+ using openin_Union[of "{S,T}" U] by auto
+
+lemma openin_topspace[intro, simp]: "openin U (topspace U)"
+ by (force simp: openin_Union topspace_def)
+
+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")
+proof
+ assume ?lhs
+ then show ?rhs by auto
+next
+ assume H: ?rhs
+ let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
+ have "openin U ?t" by (force simp: openin_Union)
+ also have "?t = S" using H by auto
+ finally show "openin U S" .
+qed
+
+lemma openin_INT [intro]:
+ assumes "finite I"
+ "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
+ shows "openin T ((\<Inter>i \<in> I. U i) \<inter> topspace T)"
+using assms by (induct, auto simp: inf_sup_aci(2) openin_Int)
+
+lemma openin_INT2 [intro]:
+ assumes "finite I" "I \<noteq> {}"
+ "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
+ shows "openin T (\<Inter>i \<in> I. U i)"
+proof -
+ have "(\<Inter>i \<in> I. U i) \<subseteq> topspace T"
+ using \<open>I \<noteq> {}\<close> openin_subset[OF assms(3)] by auto
+ then show ?thesis
+ using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
+qed
+
+lemma openin_Inter [intro]:
+ assumes "finite \<F>" "\<F> \<noteq> {}" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (\<Inter>\<F>)"
+ by (metis (full_types) assms openin_INT2 image_ident)
+
+lemma openin_Int_Inter:
+ assumes "finite \<F>" "openin T U" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (U \<inter> \<Inter>\<F>)"
+ using openin_Inter [of "insert U \<F>"] assms by auto
+
+
+subsubsection \<open>Closed sets\<close>
+
+definition%important "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
+
+lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
+ by (metis closedin_def)
+
+lemma closedin_empty[simp]: "closedin U {}"
+ by (simp add: closedin_def)
+
+lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
+ by (simp add: closedin_def)
+
+lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
+ by (auto simp: Diff_Un closedin_def)
+
+lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
+ by auto
+
+lemma closedin_Union:
+ assumes "finite S" "\<And>T. T \<in> S \<Longrightarrow> closedin U T"
+ shows "closedin U (\<Union>S)"
+ using assms by induction auto
+
+lemma closedin_Inter[intro]:
+ assumes Ke: "K \<noteq> {}"
+ and Kc: "\<And>S. S \<in>K \<Longrightarrow> closedin U S"
+ shows "closedin U (\<Inter>K)"
+ using Ke Kc unfolding closedin_def Diff_Inter by auto
+
+lemma closedin_INT[intro]:
+ assumes "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)"
+ shows "closedin U (\<Inter>x\<in>A. B x)"
+ apply (rule closedin_Inter)
+ using assms
+ apply auto
+ done
+
+lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
+ using closedin_Inter[of "{S,T}" U] by auto
+
+lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
+ apply (auto simp: closedin_def Diff_Diff_Int inf_absorb2)
+ apply (metis openin_subset subset_eq)
+ done
+
+lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
+ by (simp add: openin_closedin_eq)
+
+lemma openin_diff[intro]:
+ assumes oS: "openin U S"
+ and cT: "closedin U T"
+ shows "openin U (S - T)"
+proof -
+ have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
+ by (auto simp: topspace_def openin_subset)
+ then show ?thesis using oS cT
+ by (auto simp: closedin_def)
+qed
+
+lemma closedin_diff[intro]:
+ assumes oS: "closedin U S"
+ and cT: "openin U T"
+ shows "closedin U (S - T)"
+proof -
+ have "S - T = S \<inter> (topspace U - T)"
+ using closedin_subset[of U S] oS cT by (auto simp: topspace_def)
+ then show ?thesis
+ using oS cT by (auto simp: openin_closedin_eq)
+qed
+
+
+subsection\<open>The discrete topology\<close>
+
+definition discrete_topology where "discrete_topology U \<equiv> topology (\<lambda>S. S \<subseteq> U)"
+
+lemma openin_discrete_topology [simp]: "openin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
+proof -
+ have "istopology (\<lambda>S. S \<subseteq> U)"
+ by (auto simp: istopology_def)
+ then show ?thesis
+ by (simp add: discrete_topology_def topology_inverse')
+qed
+
+lemma topspace_discrete_topology [simp]: "topspace(discrete_topology U) = U"
+ by (meson openin_discrete_topology openin_subset openin_topspace order_refl subset_antisym)
+
+lemma closedin_discrete_topology [simp]: "closedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
+ by (simp add: closedin_def)
+
+lemma discrete_topology_unique:
+ "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> (\<forall>x \<in> U. openin X {x})" (is "?lhs = ?rhs")
+proof
+ assume R: ?rhs
+ then have "openin X S" if "S \<subseteq> U" for S
+ using openin_subopen subsetD that by fastforce
+ moreover have "x \<in> topspace X" if "openin X S" and "x \<in> S" for x S
+ using openin_subset that by blast
+ ultimately
+ show ?lhs
+ using R by (auto simp: topology_eq)
+qed auto
+
+lemma discrete_topology_unique_alt:
+ "discrete_topology U = X \<longleftrightarrow> topspace X \<subseteq> U \<and> (\<forall>x \<in> U. openin X {x})"
+ using openin_subset
+ by (auto simp: discrete_topology_unique)
+
+lemma subtopology_eq_discrete_topology_empty:
+ "X = discrete_topology {} \<longleftrightarrow> topspace X = {}"
+ using discrete_topology_unique [of "{}" X] by auto
+
+lemma subtopology_eq_discrete_topology_sing:
+ "X = discrete_topology {a} \<longleftrightarrow> topspace X = {a}"
+ by (metis discrete_topology_unique openin_topspace singletonD)
+
+
+subsection \<open>Subspace topology\<close>
+
+definition%important "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
+
+lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
+ (is "istopology ?L")
+proof -
+ have "?L {}" by blast
+ {
+ fix A B
+ assume A: "?L A" and B: "?L B"
+ 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
+ have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
+ using Sa Sb by blast+
+ then have "?L (A \<inter> B)" by blast
+ }
+ moreover
+ {
+ fix K
+ assume K: "K \<subseteq> Collect ?L"
+ have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
+ by blast
+ from K[unfolded th0 subset_image_iff]
+ obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
+ by blast
+ have "\<Union>K = (\<Union>Sk) \<inter> V"
+ using Sk by auto
+ moreover have "openin U (\<Union>Sk)"
+ using Sk by (auto simp: subset_eq)
+ ultimately have "?L (\<Union>K)" by blast
+ }
+ ultimately show ?thesis
+ unfolding subset_eq mem_Collect_eq istopology_def by auto
+qed
+
+lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
+ unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
+ by auto
+
+lemma openin_subtopology_Int:
+ "openin X S \<Longrightarrow> openin (subtopology X T) (S \<inter> T)"
+ using openin_subtopology by auto
+
+lemma openin_subtopology_Int2:
+ "openin X T \<Longrightarrow> openin (subtopology X S) (S \<inter> T)"
+ using openin_subtopology by auto
+
+lemma openin_subtopology_diff_closed:
+ "\<lbrakk>S \<subseteq> topspace X; closedin X T\<rbrakk> \<Longrightarrow> openin (subtopology X S) (S - T)"
+ unfolding closedin_def openin_subtopology
+ by (rule_tac x="topspace X - T" in exI) auto
+
+lemma openin_relative_to: "(openin X relative_to S) = openin (subtopology X S)"
+ by (force simp: relative_to_def openin_subtopology)
+
+lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
+ by (auto simp: topspace_def openin_subtopology)
+
+lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
+ unfolding closedin_def topspace_subtopology
+ by (auto simp: openin_subtopology)
+
+lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
+ unfolding openin_subtopology
+ by auto (metis IntD1 in_mono openin_subset)
+
+lemma subtopology_subtopology:
+ "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
+proof -
+ have eq: "\<And>T'. (\<exists>S'. T' = S' \<inter> T \<and> (\<exists>T. openin X T \<and> S' = T \<inter> S)) = (\<exists>Sa. T' = Sa \<inter> (S \<inter> T) \<and> openin X Sa)"
+ by (metis inf_assoc)
+ have "subtopology (subtopology X S) T = topology (\<lambda>Ta. \<exists>Sa. Ta = Sa \<inter> T \<and> openin (subtopology X S) Sa)"
+ by (simp add: subtopology_def)
+ also have "\<dots> = subtopology X (S \<inter> T)"
+ by (simp add: openin_subtopology eq) (simp add: subtopology_def)
+ finally show ?thesis .
+qed
+
+lemma openin_subtopology_alt:
+ "openin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (openin X)"
+ by (simp add: image_iff inf_commute openin_subtopology)
+
+lemma closedin_subtopology_alt:
+ "closedin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (closedin X)"
+ by (simp add: image_iff inf_commute closedin_subtopology)
+
+lemma subtopology_superset:
+ assumes UV: "topspace U \<subseteq> V"
+ shows "subtopology U V = U"
+proof -
+ {
+ fix S
+ {
+ fix T
+ assume T: "openin U T" "S = T \<inter> V"
+ from T openin_subset[OF T(1)] UV have eq: "S = T"
+ by blast
+ have "openin U S"
+ unfolding eq using T by blast
+ }
+ moreover
+ {
+ assume S: "openin U S"
+ then have "\<exists>T. openin U T \<and> S = T \<inter> V"
+ using openin_subset[OF S] UV by auto
+ }
+ ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
+ by blast
+ }
+ then show ?thesis
+ unfolding topology_eq openin_subtopology by blast
+qed
+
+lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
+ by (simp add: subtopology_superset)
+
+lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
+ by (simp add: subtopology_superset)
+
+lemma openin_subtopology_empty:
+ "openin (subtopology U {}) S \<longleftrightarrow> S = {}"
+by (metis Int_empty_right openin_empty openin_subtopology)
+
+lemma closedin_subtopology_empty:
+ "closedin (subtopology U {}) S \<longleftrightarrow> S = {}"
+by (metis Int_empty_right closedin_empty closedin_subtopology)
+
+lemma closedin_subtopology_refl [simp]:
+ "closedin (subtopology U X) X \<longleftrightarrow> X \<subseteq> topspace U"
+by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)
+
+lemma closedin_topspace_empty: "topspace T = {} \<Longrightarrow> (closedin T S \<longleftrightarrow> S = {})"
+ by (simp add: closedin_def)
+
+lemma openin_imp_subset:
+ "openin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
+by (metis Int_iff openin_subtopology subsetI)
+
+lemma closedin_imp_subset:
+ "closedin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
+by (simp add: closedin_def topspace_subtopology)
+
+lemma openin_open_subtopology:
+ "openin X S \<Longrightarrow> openin (subtopology X S) T \<longleftrightarrow> openin X T \<and> T \<subseteq> S"
+ by (metis inf.orderE openin_Int openin_imp_subset openin_subtopology)
+
+lemma closedin_closed_subtopology:
+ "closedin X S \<Longrightarrow> (closedin (subtopology X S) T \<longleftrightarrow> closedin X T \<and> T \<subseteq> S)"
+ by (metis closedin_Int closedin_imp_subset closedin_subtopology inf.orderE)
+
+lemma openin_subtopology_Un:
+ "\<lbrakk>openin (subtopology X T) S; openin (subtopology X U) S\<rbrakk>
+ \<Longrightarrow> openin (subtopology X (T \<union> U)) S"
+by (simp add: openin_subtopology) blast
+
+lemma closedin_subtopology_Un:
+ "\<lbrakk>closedin (subtopology X T) S; closedin (subtopology X U) S\<rbrakk>
+ \<Longrightarrow> closedin (subtopology X (T \<union> U)) S"
+by (simp add: closedin_subtopology) blast
+
+
+subsection \<open>The standard Euclidean topology\<close>
+
+definition%important euclidean :: "'a::topological_space topology"
+ where "euclidean = topology open"
+
+lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
+ unfolding euclidean_def
+ apply (rule cong[where x=S and y=S])
+ apply (rule topology_inverse[symmetric])
+ apply (auto simp: istopology_def)
+ done
+
+declare open_openin [symmetric, simp]
+
+lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
+ by (force simp: topspace_def)
+
+lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
+ by (simp add: topspace_subtopology)
+
+lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
+ by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV)
+
+declare closed_closedin [symmetric, simp]
+
+lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
+ by (metis openin_topspace topspace_euclidean_subtopology)
+
+subsubsection\<open>The most basic facts about the usual topology and metric on R\<close>
+
+abbreviation euclideanreal :: "real topology"
+ where "euclideanreal \<equiv> topology open"
+
+lemma real_openin [simp]: "openin euclideanreal S = open S"
+ by (simp add: euclidean_def open_openin)
+
+lemma topspace_euclideanreal [simp]: "topspace euclideanreal = UNIV"
+ using openin_subset open_UNIV real_openin by blast
+
+lemma topspace_euclideanreal_subtopology [simp]:
+ "topspace (subtopology euclideanreal S) = S"
+ by (simp add: topspace_subtopology)
+
+lemma real_closedin [simp]: "closedin euclideanreal S = closed S"
+ by (simp add: closed_closedin euclidean_def)
+
+subsection \<open>Basic "localization" results are handy for connectedness.\<close>
+
+lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
+ by (auto simp: openin_subtopology)
+
+lemma openin_Int_open:
+ "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
+ \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
+by (metis open_Int Int_assoc openin_open)
+
+lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
+ by (auto simp: openin_open)
+
+lemma open_openin_trans[trans]:
+ "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
+ by (metis Int_absorb1 openin_open_Int)
+
+lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
+ by (auto simp: openin_open)
+
+lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
+ by (simp add: closedin_subtopology Int_ac)
+
+lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
+ by (metis closedin_closed)
+
+lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (auto simp: closedin_closed)
+
+lemma closedin_closed_subset:
+ "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
+ \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
+
+lemma finite_imp_closedin:
+ fixes S :: "'a::t1_space set"
+ shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (simp add: finite_imp_closed closed_subset)
+
+lemma closedin_singleton [simp]:
+ fixes a :: "'a::t1_space"
+ shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
+using closedin_subset by (force intro: closed_subset)
+
+lemma openin_euclidean_subtopology_iff:
+ fixes S U :: "'a::metric_space set"
+ shows "openin (subtopology euclidean U) S \<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")
+proof
+ assume ?lhs
+ then show ?rhs
+ unfolding openin_open open_dist by blast
+next
+ define T where "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
+ have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
+ unfolding T_def
+ apply clarsimp
+ apply (rule_tac x="d - dist x a" in exI)
+ apply (clarsimp simp add: less_diff_eq)
+ by (metis dist_commute dist_triangle_lt)
+ assume ?rhs then have 2: "S = U \<inter> T"
+ unfolding T_def
+ by auto (metis dist_self)
+ from 1 2 show ?lhs
+ unfolding openin_open open_dist by fast
+qed
+
+lemma connected_openin:
+ "connected S \<longleftrightarrow>
+ \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
+ openin (subtopology euclidean S) E2 \<and>
+ S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe)
+ by (simp_all, blast+) (* SLOW *)
+
+lemma connected_openin_eq:
+ "connected S \<longleftrightarrow>
+ \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
+ openin (subtopology euclidean S) E2 \<and>
+ E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
+ E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ apply (simp add: connected_openin, safe, blast)
+ by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)
+
+lemma connected_closedin:
+ "connected S \<longleftrightarrow>
+ (\<nexists>E1 E2.
+ closedin (subtopology euclidean S) E1 \<and>
+ closedin (subtopology euclidean S) E2 \<and>
+ S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ by (auto simp add: connected_closed closedin_closed)
+next
+ assume R: ?rhs
+ then show ?lhs
+ proof (clarsimp simp add: connected_closed closedin_closed)
+ fix A B
+ assume s_sub: "S \<subseteq> A \<union> B" "B \<inter> S \<noteq> {}"
+ and disj: "A \<inter> B \<inter> S = {}"
+ and cl: "closed A" "closed B"
+ have "S \<inter> (A \<union> B) = S"
+ using s_sub(1) by auto
+ have "S - A = B \<inter> S"
+ using Diff_subset_conv Un_Diff_Int disj s_sub(1) by auto
+ then have "S \<inter> A = {}"
+ by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int R cl closedin_closed_Int inf_commute order_refl s_sub(2))
+ then show "A \<inter> S = {}"
+ by blast
+ qed
+qed
+
+lemma connected_closedin_eq:
+ "connected S \<longleftrightarrow>
+ \<not>(\<exists>E1 E2.
+ closedin (subtopology euclidean S) E1 \<and>
+ closedin (subtopology euclidean S) E2 \<and>
+ E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
+ E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ apply (simp add: connected_closedin, safe, blast)
+ by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
+
+text \<open>These "transitivity" results are handy too\<close>
+
+lemma openin_trans[trans]:
+ "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
+ openin (subtopology euclidean U) S"
+ unfolding open_openin openin_open by blast
+
+lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
+ by (auto simp: openin_open intro: openin_trans)
+
+lemma closedin_trans[trans]:
+ "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
+ closedin (subtopology euclidean U) S"
+ by (auto simp: closedin_closed closed_Inter Int_assoc)
+
+lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
+ by (auto simp: closedin_closed intro: closedin_trans)
+
+lemma openin_subtopology_Int_subset:
+ "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
+ by (auto simp: openin_subtopology)
+
+lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
+ using open_subset openin_open_trans openin_subset by fastforce
+
subsection\<open>Derived set (set of limit points)\<close>
@@ -2774,7 +3353,7 @@
obtain \<F> where \<F>: "finite \<F>" "\<F> \<subseteq> \<V>" "S \<subseteq> \<Union>\<F>"
proof -
have 1: "\<forall>U\<in>\<V>. openin X U"
- unfolding \<V>_def using \<U> cont continuous_map by blast
+ unfolding \<V>_def using \<U> cont[unfolded continuous_map] by blast
have 2: "S \<subseteq> \<Union>\<V>"
unfolding \<V>_def using compactin_subset_topspace cpt \<U> by fastforce
show thesis
@@ -2881,4 +3460,97 @@
unfolding embedding_map_def
using homeomorphic_space by blast
+subsection \<open>Continuity\<close>
+
+lemma continuous_on_open:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. openin (subtopology euclidean (f ` S)) T \<longrightarrow>
+ openin (subtopology euclidean S) (S \<inter> f -` T))"
+ unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
+ by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
+
+lemma continuous_on_closed:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. closedin (subtopology euclidean (f ` S)) T \<longrightarrow>
+ closedin (subtopology euclidean S) (S \<inter> f -` T))"
+ unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
+ by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
+
+lemma continuous_on_imp_closedin:
+ assumes "continuous_on S f" "closedin (subtopology euclidean (f ` S)) T"
+ shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
+ using assms continuous_on_closed by blast
+
+subsection%unimportant \<open>Half-global and completely global cases\<close>
+
+lemma continuous_openin_preimage_gen:
+ assumes "continuous_on S f" "open T"
+ shows "openin (subtopology euclidean S) (S \<inter> f -` T)"
+proof -
+ have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
+ by auto
+ have "openin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
+ using openin_open_Int[of T "f ` S", OF assms(2)] unfolding openin_open by auto
+ then show ?thesis
+ using assms(1)[unfolded continuous_on_open, THEN spec[where x="T \<inter> f ` S"]]
+ using * by auto
+qed
+
+lemma continuous_closedin_preimage:
+ assumes "continuous_on S f" and "closed T"
+ shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
+proof -
+ have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
+ by auto
+ have "closedin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
+ using closedin_closed_Int[of T "f ` S", OF assms(2)]
+ by (simp add: Int_commute)
+ then show ?thesis
+ using assms(1)[unfolded continuous_on_closed, THEN spec[where x="T \<inter> f ` S"]]
+ using * by auto
+qed
+
+lemma continuous_openin_preimage_eq:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. open T \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T))"
+apply safe
+apply (simp add: continuous_openin_preimage_gen)
+apply (fastforce simp add: continuous_on_open openin_open)
+done
+
+lemma continuous_closedin_preimage_eq:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. closed T \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` T))"
+apply safe
+apply (simp add: continuous_closedin_preimage)
+apply (fastforce simp add: continuous_on_closed closedin_closed)
+done
+
+lemma continuous_open_preimage:
+ assumes contf: "continuous_on S f" and "open S" "open T"
+ shows "open (S \<inter> f -` T)"
+proof-
+ obtain U where "open U" "(S \<inter> f -` T) = S \<inter> U"
+ using continuous_openin_preimage_gen[OF contf \<open>open T\<close>]
+ unfolding openin_open by auto
+ then show ?thesis
+ using open_Int[of S U, OF \<open>open S\<close>] by auto
+qed
+
+lemma continuous_closed_preimage:
+ assumes contf: "continuous_on S f" and "closed S" "closed T"
+ shows "closed (S \<inter> f -` T)"
+proof-
+ obtain U where "closed U" "(S \<inter> f -` T) = S \<inter> U"
+ using continuous_closedin_preimage[OF contf \<open>closed T\<close>]
+ unfolding closedin_closed by auto
+ then show ?thesis using closed_Int[of S U, OF \<open>closed S\<close>] by auto
+qed
+
+lemma continuous_open_vimage: "open S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> open (f -` S)"
+ by (metis continuous_on_eq_continuous_within open_vimage)
+
+lemma continuous_closed_vimage: "closed S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> closed (f -` S)"
+ by (simp add: closed_vimage continuous_on_eq_continuous_within)
+
end
--- a/src/HOL/Analysis/Connected.thy Sat Dec 29 18:40:29 2018 +0000
+++ b/src/HOL/Analysis/Connected.thy Sat Dec 29 20:32:09 2018 +0100
@@ -5,7 +5,9 @@
section \<open>Connected Components, Homeomorphisms, Baire property, etc\<close>
theory Connected
-imports Topology_Euclidean_Space
+ imports
+ "HOL-Library.Indicator_Function"
+ Topology_Euclidean_Space
begin
subsection%unimportant \<open>More properties of closed balls, spheres, etc\<close>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Elementary_Metric_Spaces.thy Sat Dec 29 20:32:09 2018 +0100
@@ -0,0 +1,1840 @@
+(* Author: L C Paulson, University of Cambridge
+ Author: Amine Chaieb, University of Cambridge
+ Author: Robert Himmelmann, TU Muenchen
+ Author: Brian Huffman, Portland State University
+*)
+
+section \<open>Elementary Metric Spaces\<close>
+
+theory Elementary_Metric_Spaces
+ imports
+ Elementary_Topology
+ Abstract_Topology
+begin
+
+(* FIXME: move elsewhere *)
+
+lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
+ apply auto
+ apply (rule_tac x="d/2" in exI)
+ apply auto
+ done
+
+lemma approachable_lt_le2: \<comment> \<open>like the above, but pushes aside an extra formula\<close>
+ "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
+ apply auto
+ apply (rule_tac x="d/2" in exI, auto)
+ done
+
+lemma triangle_lemma:
+ fixes x y z :: real
+ assumes x: "0 \<le> x"
+ and y: "0 \<le> y"
+ and z: "0 \<le> z"
+ and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
+ shows "x \<le> y + z"
+proof -
+ have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
+ using z y by simp
+ with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
+ by (simp add: power2_eq_square field_simps)
+ from y z have yz: "y + z \<ge> 0"
+ by arith
+ from power2_le_imp_le[OF th yz] show ?thesis .
+qed
+
+subsection \<open>Combination of Elementary and Abstract Topology\<close>
+
+lemma closedin_limpt:
+ "closedin (subtopology euclidean T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
+ apply (simp add: closedin_closed, safe)
+ apply (simp add: closed_limpt islimpt_subset)
+ apply (rule_tac x="closure S" in exI, simp)
+ apply (force simp: closure_def)
+ done
+
+lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
+ by (meson closedin_limpt closed_subset closedin_closed_trans)
+
+lemma connected_closed_set:
+ "closed S
+ \<Longrightarrow> connected S \<longleftrightarrow> (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"
+ unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast
+
+text \<open>If a connnected set is written as the union of two nonempty closed sets, then these sets
+have to intersect.\<close>
+
+lemma connected_as_closed_union:
+ assumes "connected C" "C = A \<union> B" "closed A" "closed B" "A \<noteq> {}" "B \<noteq> {}"
+ shows "A \<inter> B \<noteq> {}"
+by (metis assms closed_Un connected_closed_set)
+
+lemma closedin_subset_trans:
+ "closedin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
+ closedin (subtopology euclidean T) S"
+ by (meson closedin_limpt subset_iff)
+
+lemma openin_subset_trans:
+ "openin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
+ openin (subtopology euclidean T) S"
+ by (auto simp: openin_open)
+
+lemma openin_Times:
+ "openin (subtopology euclidean S) S' \<Longrightarrow> openin (subtopology euclidean T) T' \<Longrightarrow>
+ openin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
+ unfolding openin_open using open_Times by blast
+
+
+subsubsection \<open>Closure\<close>
+
+lemma closure_openin_Int_closure:
+ assumes ope: "openin (subtopology euclidean U) S" and "T \<subseteq> U"
+ shows "closure(S \<inter> closure T) = closure(S \<inter> T)"
+proof
+ obtain V where "open V" and S: "S = U \<inter> V"
+ using ope using openin_open by metis
+ show "closure (S \<inter> closure T) \<subseteq> closure (S \<inter> T)"
+ proof (clarsimp simp: S)
+ fix x
+ assume "x \<in> closure (U \<inter> V \<inter> closure T)"
+ then have "V \<inter> closure T \<subseteq> A \<Longrightarrow> x \<in> closure A" for A
+ by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
+ then have "x \<in> closure (T \<inter> V)"
+ by (metis \<open>open V\<close> closure_closure inf_commute open_Int_closure_subset)
+ then show "x \<in> closure (U \<inter> V \<inter> T)"
+ by (metis \<open>T \<subseteq> U\<close> inf.absorb_iff2 inf_assoc inf_commute)
+ qed
+next
+ show "closure (S \<inter> T) \<subseteq> closure (S \<inter> closure T)"
+ by (meson Int_mono closure_mono closure_subset order_refl)
+qed
+
+corollary infinite_openin:
+ fixes S :: "'a :: t1_space set"
+ shows "\<lbrakk>openin (subtopology euclidean U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
+ by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
+
+subsubsection \<open>Frontier\<close>
+
+lemma connected_Int_frontier:
+ "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
+ apply (simp add: frontier_interiors connected_openin, safe)
+ apply (drule_tac x="s \<inter> interior t" in spec, safe)
+ apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
+ apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
+ done
+
+subsubsection \<open>Compactness\<close>
+
+lemma openin_delete:
+ fixes a :: "'a :: t1_space"
+ shows "openin (subtopology euclidean u) s
+ \<Longrightarrow> openin (subtopology euclidean u) (s - {a})"
+by (metis Int_Diff open_delete openin_open)
+
+
+subsection \<open>Continuity\<close>
+
+lemma interior_image_subset:
+ assumes "inj f" "\<And>x. continuous (at x) f"
+ shows "interior (f ` S) \<subseteq> f ` (interior S)"
+proof
+ fix x assume "x \<in> interior (f ` S)"
+ then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` S" ..
+ then have "x \<in> f ` S" by auto
+ then obtain y where y: "y \<in> S" "x = f y" by auto
+ have "open (f -` T)"
+ using assms \<open>open T\<close> by (simp add: continuous_at_imp_continuous_on open_vimage)
+ moreover have "y \<in> vimage f T"
+ using \<open>x = f y\<close> \<open>x \<in> T\<close> by simp
+ moreover have "vimage f T \<subseteq> S"
+ using \<open>T \<subseteq> image f S\<close> \<open>inj f\<close> unfolding inj_on_def subset_eq by auto
+ ultimately have "y \<in> interior S" ..
+ with \<open>x = f y\<close> show "x \<in> f ` interior S" ..
+qed
+
+subsection \<open>Open and closed balls\<close>
+
+definition%important ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
+ where "ball x e = {y. dist x y < e}"
+
+definition%important cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
+ where "cball x e = {y. dist x y \<le> e}"
+
+definition%important sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
+ where "sphere x e = {y. dist x y = e}"
+
+lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
+ by (simp add: ball_def)
+
+lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
+ by (simp add: cball_def)
+
+lemma mem_sphere [simp]: "y \<in> sphere x e \<longleftrightarrow> dist x y = e"
+ by (simp add: sphere_def)
+
+lemma ball_trivial [simp]: "ball x 0 = {}"
+ by (simp add: ball_def)
+
+lemma cball_trivial [simp]: "cball x 0 = {x}"
+ by (simp add: cball_def)
+
+lemma sphere_trivial [simp]: "sphere x 0 = {x}"
+ by (simp add: sphere_def)
+
+lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
+ for x :: "'a::real_normed_vector"
+ by (simp add: dist_norm)
+
+lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
+ for x :: "'a::real_normed_vector"
+ by (simp add: dist_norm)
+
+lemma disjoint_ballI: "dist x y \<ge> r+s \<Longrightarrow> ball x r \<inter> ball y s = {}"
+ using dist_triangle_less_add not_le by fastforce
+
+lemma disjoint_cballI: "dist x y > r + s \<Longrightarrow> cball x r \<inter> cball y s = {}"
+ by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball)
+
+lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
+ for x :: "'a::real_normed_vector"
+ by (simp add: dist_norm)
+
+lemma sphere_empty [simp]: "r < 0 \<Longrightarrow> sphere a r = {}"
+ for a :: "'a::metric_space"
+ by auto
+
+lemma centre_in_ball [simp]: "x \<in> ball x e \<longleftrightarrow> 0 < e"
+ by simp
+
+lemma centre_in_cball [simp]: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
+ by simp
+
+lemma ball_subset_cball [simp, intro]: "ball x e \<subseteq> cball x e"
+ by (simp add: subset_eq)
+
+lemma mem_ball_imp_mem_cball: "x \<in> ball y e \<Longrightarrow> x \<in> cball y e"
+ by (auto simp: mem_ball mem_cball)
+
+lemma sphere_cball [simp,intro]: "sphere z r \<subseteq> cball z r"
+ by force
+
+lemma cball_diff_sphere: "cball a r - sphere a r = ball a r"
+ by auto
+
+lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
+ by (simp add: subset_eq)
+
+lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
+ by (simp add: subset_eq)
+
+lemma mem_ball_leI: "x \<in> ball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> ball y f"
+ by (auto simp: mem_ball mem_cball)
+
+lemma mem_cball_leI: "x \<in> cball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> cball y f"
+ by (auto simp: mem_ball mem_cball)
+
+lemma cball_trans: "y \<in> cball z b \<Longrightarrow> x \<in> cball y a \<Longrightarrow> x \<in> cball z (b + a)"
+ unfolding mem_cball
+proof -
+ have "dist z x \<le> dist z y + dist y x"
+ by (rule dist_triangle)
+ also assume "dist z y \<le> b"
+ also assume "dist y x \<le> a"
+ finally show "dist z x \<le> b + a" by arith
+qed
+
+lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
+ by (simp add: set_eq_iff) arith
+
+lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
+ by (simp add: set_eq_iff)
+
+lemma cball_max_Un: "cball a (max r s) = cball a r \<union> cball a s"
+ by (simp add: set_eq_iff) arith
+
+lemma cball_min_Int: "cball a (min r s) = cball a r \<inter> cball a s"
+ by (simp add: set_eq_iff)
+
+lemma cball_diff_eq_sphere: "cball a r - ball a r = sphere a r"
+ by (auto simp: cball_def ball_def dist_commute)
+
+lemma image_add_ball [simp]:
+ fixes a :: "'a::real_normed_vector"
+ shows "(+) b ` ball a r = ball (a+b) r"
+apply (intro equalityI subsetI)
+apply (force simp: dist_norm)
+apply (rule_tac x="x-b" in image_eqI)
+apply (auto simp: dist_norm algebra_simps)
+done
+
+lemma image_add_cball [simp]:
+ fixes a :: "'a::real_normed_vector"
+ shows "(+) b ` cball a r = cball (a+b) r"
+apply (intro equalityI subsetI)
+apply (force simp: dist_norm)
+apply (rule_tac x="x-b" in image_eqI)
+apply (auto simp: dist_norm algebra_simps)
+done
+
+lemma open_ball [intro, simp]: "open (ball x e)"
+proof -
+ have "open (dist x -` {..<e})"
+ by (intro open_vimage open_lessThan continuous_intros)
+ also have "dist x -` {..<e} = ball x e"
+ by auto
+ finally show ?thesis .
+qed
+
+lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
+ by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute)
+
+lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S) \<Longrightarrow> open S"
+ by (auto simp: open_contains_ball)
+
+lemma openE[elim?]:
+ assumes "open S" "x\<in>S"
+ obtains e where "e>0" "ball x e \<subseteq> S"
+ using assms unfolding open_contains_ball by auto
+
+lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
+ by (metis open_contains_ball subset_eq centre_in_ball)
+
+lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
+ unfolding mem_ball set_eq_iff
+ apply (simp add: not_less)
+ apply (metis zero_le_dist order_trans dist_self)
+ done
+
+lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
+
+lemma closed_cball [iff]: "closed (cball x e)"
+proof -
+ have "closed (dist x -` {..e})"
+ by (intro closed_vimage closed_atMost continuous_intros)
+ also have "dist x -` {..e} = cball x e"
+ by auto
+ finally show ?thesis .
+qed
+
+lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
+proof -
+ {
+ fix x and e::real
+ assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
+ then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
+ }
+ moreover
+ {
+ fix x and e::real
+ assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
+ then have "\<exists>d>0. ball x d \<subseteq> S"
+ unfolding subset_eq
+ apply (rule_tac x="e/2" in exI, auto)
+ done
+ }
+ ultimately show ?thesis
+ unfolding open_contains_ball by auto
+qed
+
+lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
+ by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
+
+lemma eventually_nhds_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z d) (nhds z)"
+ by (rule eventually_nhds_in_open) simp_all
+
+lemma eventually_at_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<in> A) (at z within A)"
+ unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
+
+lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
+ unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
+
+lemma at_within_ball: "e > 0 \<Longrightarrow> dist x y < e \<Longrightarrow> at y within ball x e = at y"
+ by (subst at_within_open) auto
+
+lemma atLeastAtMost_eq_cball:
+ fixes a b::real
+ shows "{a .. b} = cball ((a + b)/2) ((b - a)/2)"
+ by (auto simp: dist_real_def field_simps mem_cball)
+
+lemma greaterThanLessThan_eq_ball:
+ fixes a b::real
+ shows "{a <..< b} = ball ((a + b)/2) ((b - a)/2)"
+ by (auto simp: dist_real_def field_simps mem_ball)
+
+
+subsection \<open>Limit Points\<close>
+
+lemma islimpt_approachable:
+ fixes x :: "'a::metric_space"
+ shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
+ unfolding islimpt_iff_eventually eventually_at by fast
+
+lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
+ for x :: "'a::metric_space"
+ unfolding islimpt_approachable
+ using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
+ THEN arg_cong [where f=Not]]
+ by (simp add: Bex_def conj_commute conj_left_commute)
+
+lemma perfect_choose_dist: "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
+ for x :: "'a::{perfect_space,metric_space}"
+ using islimpt_UNIV [of x] by (simp add: islimpt_approachable)
+
+lemma limpt_of_limpts: "x islimpt {y. y islimpt S} \<Longrightarrow> x islimpt S"
+ for x :: "'a::metric_space"
+ apply (clarsimp simp add: islimpt_approachable)
+ apply (drule_tac x="e/2" in spec)
+ apply (auto simp: simp del: less_divide_eq_numeral1)
+ apply (drule_tac x="dist x' x" in spec)
+ apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
+ apply (erule rev_bexI)
+ apply (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
+ done
+
+lemma closed_limpts: "closed {x::'a::metric_space. x islimpt S}"
+ using closed_limpt limpt_of_limpts by blast
+
+lemma limpt_of_closure: "x islimpt closure S \<longleftrightarrow> x islimpt S"
+ for x :: "'a::metric_space"
+ by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
+
+lemma islimpt_eq_infinite_ball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> ball x e))"
+ apply (simp add: islimpt_eq_acc_point, safe)
+ apply (metis Int_commute open_ball centre_in_ball)
+ by (metis open_contains_ball Int_mono finite_subset inf_commute subset_refl)
+
+lemma islimpt_eq_infinite_cball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> cball x e))"
+ apply (simp add: islimpt_eq_infinite_ball, safe)
+ apply (meson Int_mono ball_subset_cball finite_subset order_refl)
+ by (metis open_ball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq)
+
+
+subsection \<open>?\<close>
+
+lemma finite_ball_include:
+ fixes a :: "'a::metric_space"
+ assumes "finite S"
+ shows "\<exists>e>0. S \<subseteq> ball a e"
+ using assms
+proof induction
+ case (insert x S)
+ then obtain e0 where "e0>0" and e0:"S \<subseteq> ball a e0" by auto
+ define e where "e = max e0 (2 * dist a x)"
+ have "e>0" unfolding e_def using \<open>e0>0\<close> by auto
+ moreover have "insert x S \<subseteq> ball a e"
+ using e0 \<open>e>0\<close> unfolding e_def by auto
+ ultimately show ?case by auto
+qed (auto intro: zero_less_one)
+
+lemma finite_set_avoid:
+ fixes a :: "'a::metric_space"
+ assumes "finite S"
+ shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
+ using assms
+proof induction
+ case (insert x S)
+ then obtain d where "d > 0" and d: "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
+ by blast
+ show ?case
+ proof (cases "x = a")
+ case True
+ with \<open>d > 0 \<close>d show ?thesis by auto
+ next
+ case False
+ let ?d = "min d (dist a x)"
+ from False \<open>d > 0\<close> have dp: "?d > 0"
+ by auto
+ from d have d': "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> ?d \<le> dist a x"
+ by auto
+ with dp False show ?thesis
+ by (metis insert_iff le_less min_less_iff_conj not_less)
+ qed
+qed (auto intro: zero_less_one)
+
+lemma discrete_imp_closed:
+ fixes S :: "'a::metric_space set"
+ assumes e: "0 < e"
+ and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
+ shows "closed S"
+proof -
+ have False if C: "\<And>e. e>0 \<Longrightarrow> \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" for x
+ proof -
+ from e have e2: "e/2 > 0" by arith
+ from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
+ by blast
+ let ?m = "min (e/2) (dist x y) "
+ from e2 y(2) have mp: "?m > 0"
+ by simp
+ from C[OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
+ by blast
+ from z y have "dist z y < e"
+ by (intro dist_triangle_lt [where z=x]) simp
+ from d[rule_format, OF y(1) z(1) this] y z show ?thesis
+ by (auto simp: dist_commute)
+ qed
+ then show ?thesis
+ by (metis islimpt_approachable closed_limpt [where 'a='a])
+qed
+
+
+subsection \<open>Interior\<close>
+
+lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
+ using open_contains_ball_eq [where S="interior S"]
+ by (simp add: open_subset_interior)
+
+
+subsection \<open>Frontier\<close>
+
+lemma frontier_straddle:
+ fixes a :: "'a::metric_space"
+ 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))"
+ unfolding frontier_def closure_interior
+ by (auto simp: mem_interior subset_eq ball_def)
+
+
+subsection \<open>Limits\<close>
+
+proposition Lim: "(f \<longlongrightarrow> l) net \<longleftrightarrow> trivial_limit net \<or> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
+ by (auto simp: tendsto_iff trivial_limit_eq)
+
+text \<open>Show that they yield usual definitions in the various cases.\<close>
+
+proposition Lim_within_le: "(f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow>
+ (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
+ by (auto simp: tendsto_iff eventually_at_le)
+
+proposition Lim_within: "(f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow>
+ (\<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)"
+ by (auto simp: tendsto_iff eventually_at)
+
+corollary Lim_withinI [intro?]:
+ assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e"
+ shows "(f \<longlongrightarrow> l) (at a within S)"
+ apply (simp add: Lim_within, clarify)
+ apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
+ done
+
+proposition Lim_at: "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow>
+ (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
+ by (auto simp: tendsto_iff eventually_at)
+
+lemma Lim_transform_within_set:
+ fixes a :: "'a::metric_space" and l :: "'b::metric_space"
+ shows "\<lbrakk>(f \<longlongrightarrow> l) (at a within S); eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)\<rbrakk>
+ \<Longrightarrow> (f \<longlongrightarrow> l) (at a within T)"
+apply (clarsimp simp: eventually_at Lim_within)
+apply (drule_tac x=e in spec, clarify)
+apply (rename_tac k)
+apply (rule_tac x="min d k" in exI, simp)
+done
+
+text \<open>Another limit point characterization.\<close>
+
+lemma limpt_sequential_inj:
+ fixes x :: "'a::metric_space"
+ shows "x islimpt S \<longleftrightarrow>
+ (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> inj f \<and> (f \<longlongrightarrow> x) sequentially)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then have "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
+ by (force simp: islimpt_approachable)
+ then obtain y where y: "\<And>e. e>0 \<Longrightarrow> y e \<in> S \<and> y e \<noteq> x \<and> dist (y e) x < e"
+ by metis
+ define f where "f \<equiv> rec_nat (y 1) (\<lambda>n fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))"
+ have [simp]: "f 0 = y 1"
+ "f(Suc n) = y (min (inverse(2 ^ (Suc n))) (dist (f n) x))" for n
+ by (simp_all add: f_def)
+ have f: "f n \<in> S \<and> (f n \<noteq> x) \<and> dist (f n) x < inverse(2 ^ n)" for n
+ proof (induction n)
+ case 0 show ?case
+ by (simp add: y)
+ next
+ case (Suc n) then show ?case
+ apply (auto simp: y)
+ by (metis half_gt_zero_iff inverse_positive_iff_positive less_divide_eq_numeral1(1) min_less_iff_conj y zero_less_dist_iff zero_less_numeral zero_less_power)
+ qed
+ show ?rhs
+ proof (rule_tac x=f in exI, intro conjI allI)
+ show "\<And>n. f n \<in> S - {x}"
+ using f by blast
+ have "dist (f n) x < dist (f m) x" if "m < n" for m n
+ using that
+ proof (induction n)
+ case 0 then show ?case by simp
+ next
+ case (Suc n)
+ then consider "m < n" | "m = n" using less_Suc_eq by blast
+ then show ?case
+ proof cases
+ assume "m < n"
+ have "dist (f(Suc n)) x = dist (y (min (inverse(2 ^ (Suc n))) (dist (f n) x))) x"
+ by simp
+ also have "\<dots> < dist (f n) x"
+ by (metis dist_pos_lt f min.strict_order_iff min_less_iff_conj y)
+ also have "\<dots> < dist (f m) x"
+ using Suc.IH \<open>m < n\<close> by blast
+ finally show ?thesis .
+ next
+ assume "m = n" then show ?case
+ by simp (metis dist_pos_lt f half_gt_zero_iff inverse_positive_iff_positive min_less_iff_conj y zero_less_numeral zero_less_power)
+ qed
+ qed
+ then show "inj f"
+ by (metis less_irrefl linorder_injI)
+ show "f \<longlonglongrightarrow> x"
+ apply (rule tendstoI)
+ apply (rule_tac c="nat (ceiling(1/e))" in eventually_sequentiallyI)
+ apply (rule less_trans [OF f [THEN conjunct2, THEN conjunct2]])
+ apply (simp add: field_simps)
+ by (meson le_less_trans mult_less_cancel_left not_le of_nat_less_two_power)
+ qed
+next
+ assume ?rhs
+ then show ?lhs
+ by (fastforce simp add: islimpt_approachable lim_sequentially)
+qed
+
+lemma Lim_dist_ubound:
+ assumes "\<not>(trivial_limit net)"
+ and "(f \<longlongrightarrow> l) net"
+ and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
+ shows "dist a l \<le> e"
+ using assms by (fast intro: tendsto_le tendsto_intros)
+
+
+subsection \<open>Closure and Limit Characterization\<close>
+
+lemma closure_approachable:
+ fixes S :: "'a::metric_space set"
+ shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
+ apply (auto simp: closure_def islimpt_approachable)
+ apply (metis dist_self)
+ done
+
+lemma closure_approachable_le:
+ fixes S :: "'a::metric_space set"
+ shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x \<le> e)"
+ unfolding closure_approachable
+ using dense by force
+
+lemma closure_approachableD:
+ assumes "x \<in> closure S" "e>0"
+ shows "\<exists>y\<in>S. dist x y < e"
+ using assms unfolding closure_approachable by (auto simp: dist_commute)
+
+lemma closed_approachable:
+ fixes S :: "'a::metric_space set"
+ shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
+ by (metis closure_closed closure_approachable)
+
+lemma closure_contains_Inf:
+ fixes S :: "real set"
+ assumes "S \<noteq> {}" "bdd_below S"
+ shows "Inf S \<in> closure S"
+proof -
+ have *: "\<forall>x\<in>S. Inf S \<le> x"
+ using cInf_lower[of _ S] assms by metis
+ {
+ fix e :: real
+ assume "e > 0"
+ then have "Inf S < Inf S + e" by simp
+ with assms obtain x where "x \<in> S" "x < Inf S + e"
+ by (subst (asm) cInf_less_iff) auto
+ with * have "\<exists>x\<in>S. dist x (Inf S) < e"
+ by (intro bexI[of _ x]) (auto simp: dist_real_def)
+ }
+ then show ?thesis unfolding closure_approachable by auto
+qed
+
+lemma not_trivial_limit_within_ball:
+ "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ show ?rhs if ?lhs
+ proof -
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain y where "y \<in> S - {x}" and "dist y x < e"
+ using \<open>?lhs\<close> not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
+ by auto
+ then have "y \<in> S \<inter> ball x e - {x}"
+ unfolding ball_def by (simp add: dist_commute)
+ then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
+ }
+ then show ?thesis by auto
+ qed
+ show ?lhs if ?rhs
+ proof -
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain y where "y \<in> S \<inter> ball x e - {x}"
+ using \<open>?rhs\<close> by blast
+ then have "y \<in> S - {x}" and "dist y x < e"
+ unfolding ball_def by (simp_all add: dist_commute)
+ then have "\<exists>y \<in> S - {x}. dist y x < e"
+ by auto
+ }
+ then show ?thesis
+ using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
+ by auto
+ qed
+qed
+
+subsection \<open>Boundedness\<close>
+
+ (* FIXME: This has to be unified with BSEQ!! *)
+definition%important (in metric_space) bounded :: "'a set \<Rightarrow> bool"
+ where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
+
+lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
+ unfolding bounded_def subset_eq by auto (meson order_trans zero_le_dist)
+
+lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
+ unfolding bounded_def
+ by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
+
+lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
+ unfolding bounded_any_center [where a=0]
+ by (simp add: dist_norm)
+
+lemma bdd_above_norm: "bdd_above (norm ` X) \<longleftrightarrow> bounded X"
+ by (simp add: bounded_iff bdd_above_def)
+
+lemma bounded_norm_comp: "bounded ((\<lambda>x. norm (f x)) ` S) = bounded (f ` S)"
+ by (simp add: bounded_iff)
+
+lemma boundedI:
+ assumes "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
+ shows "bounded S"
+ using assms bounded_iff by blast
+
+lemma bounded_empty [simp]: "bounded {}"
+ by (simp add: bounded_def)
+
+lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"
+ by (metis bounded_def subset_eq)
+
+lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"
+ by (metis bounded_subset interior_subset)
+
+lemma bounded_closure[intro]:
+ assumes "bounded S"
+ shows "bounded (closure S)"
+proof -
+ from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"
+ unfolding bounded_def by auto
+ {
+ fix y
+ assume "y \<in> closure S"
+ then obtain f where f: "\<forall>n. f n \<in> S" "(f \<longlongrightarrow> y) sequentially"
+ unfolding closure_sequential by auto
+ have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
+ then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
+ by (simp add: f(1))
+ have "dist x y \<le> a"
+ apply (rule Lim_dist_ubound [of sequentially f])
+ apply (rule trivial_limit_sequentially)
+ apply (rule f(2))
+ apply fact
+ done
+ }
+ then show ?thesis
+ unfolding bounded_def by auto
+qed
+
+lemma bounded_closure_image: "bounded (f ` closure S) \<Longrightarrow> bounded (f ` S)"
+ by (simp add: bounded_subset closure_subset image_mono)
+
+lemma bounded_cball[simp,intro]: "bounded (cball x e)"
+ apply (simp add: bounded_def)
+ apply (rule_tac x=x in exI)
+ apply (rule_tac x=e in exI, auto)
+ done
+
+lemma bounded_ball[simp,intro]: "bounded (ball x e)"
+ by (metis ball_subset_cball bounded_cball bounded_subset)
+
+lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
+ by (auto simp: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj)
+
+lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"
+ by (induct rule: finite_induct[of F]) auto
+
+lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
+ by (induct set: finite) auto
+
+lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
+proof -
+ have "\<forall>y\<in>{x}. dist x y \<le> 0"
+ by simp
+ then have "bounded {x}"
+ unfolding bounded_def by fast
+ then show ?thesis
+ by (metis insert_is_Un bounded_Un)
+qed
+
+lemma bounded_subset_ballI: "S \<subseteq> ball x r \<Longrightarrow> bounded S"
+ by (meson bounded_ball bounded_subset)
+
+lemma bounded_subset_ballD:
+ assumes "bounded S" shows "\<exists>r. 0 < r \<and> S \<subseteq> ball x r"
+proof -
+ obtain e::real and y where "S \<subseteq> cball y e" "0 \<le> e"
+ using assms by (auto simp: bounded_subset_cball)
+ then show ?thesis
+ apply (rule_tac x="dist x y + e + 1" in exI)
+ apply (simp add: add.commute add_pos_nonneg)
+ apply (erule subset_trans)
+ apply (clarsimp simp add: cball_def)
+ by (metis add_le_cancel_right add_strict_increasing dist_commute dist_triangle_le zero_less_one)
+qed
+
+lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
+ by (induct set: finite) simp_all
+
+lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
+ by (metis Int_lower1 Int_lower2 bounded_subset)
+
+lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"
+ by (metis Diff_subset bounded_subset)
+
+lemma bounded_dist_comp:
+ assumes "bounded (f ` S)" "bounded (g ` S)"
+ shows "bounded ((\<lambda>x. dist (f x) (g x)) ` S)"
+proof -
+ from assms obtain M1 M2 where *: "dist (f x) undefined \<le> M1" "dist undefined (g x) \<le> M2" if "x \<in> S" for x
+ by (auto simp: bounded_any_center[of _ undefined] dist_commute)
+ have "dist (f x) (g x) \<le> M1 + M2" if "x \<in> S" for x
+ using *[OF that]
+ by (rule order_trans[OF dist_triangle add_mono])
+ then show ?thesis
+ by (auto intro!: boundedI)
+qed
+
+
+subsection \<open>Consequences for Real Numbers\<close>
+
+lemma closed_contains_Inf:
+ fixes S :: "real set"
+ shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"
+ by (metis closure_contains_Inf closure_closed)
+
+lemma closed_subset_contains_Inf:
+ fixes A C :: "real set"
+ shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<in> C"
+ by (metis closure_contains_Inf closure_minimal subset_eq)
+
+lemma atLeastAtMost_subset_contains_Inf:
+ fixes A :: "real set" and a b :: real
+ shows "A \<noteq> {} \<Longrightarrow> a \<le> b \<Longrightarrow> A \<subseteq> {a..b} \<Longrightarrow> Inf A \<in> {a..b}"
+ by (rule closed_subset_contains_Inf)
+ (auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a])
+
+subsection \<open>Compactness\<close>
+
+lemma compact_imp_bounded:
+ assumes "compact U"
+ shows "bounded U"
+proof -
+ have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"
+ using assms by auto
+ then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
+ by (metis compactE_image)
+ from \<open>finite D\<close> have "bounded (\<Union>x\<in>D. ball x 1)"
+ by (simp add: bounded_UN)
+ then show "bounded U" using \<open>U \<subseteq> (\<Union>x\<in>D. ball x 1)\<close>
+ by (rule bounded_subset)
+qed
+
+lemma closure_Int_ball_not_empty:
+ assumes "S \<subseteq> closure T" "x \<in> S" "r > 0"
+ shows "T \<inter> ball x r \<noteq> {}"
+ using assms centre_in_ball closure_iff_nhds_not_empty by blast
+
+subsubsection\<open>Totally bounded\<close>
+
+lemma cauchy_def: "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N \<longrightarrow> dist (s m) (s n) < e)"
+ unfolding Cauchy_def by metis
+
+proposition seq_compact_imp_totally_bounded:
+ assumes "seq_compact s"
+ shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)"
+proof -
+ { fix e::real assume "e > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> s \<Longrightarrow> \<not> s \<subseteq> (\<Union>x\<in>k. ball x e)"
+ let ?Q = "\<lambda>x n r. r \<in> s \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < e))"
+ have "\<exists>x. \<forall>n::nat. ?Q x n (x n)"
+ proof (rule dependent_wellorder_choice)
+ fix n x assume "\<And>y. y < n \<Longrightarrow> ?Q x y (x y)"
+ then have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)"
+ using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq)
+ then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)"
+ unfolding subset_eq by auto
+ show "\<exists>r. ?Q x n r"
+ using z by auto
+ qed simp
+ then obtain x where "\<forall>n::nat. x n \<in> s" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"
+ by blast
+ then obtain l r where "l \<in> s" and r:"strict_mono r" and "((x \<circ> r) \<longlongrightarrow> l) sequentially"
+ using assms by (metis seq_compact_def)
+ from this(3) have "Cauchy (x \<circ> r)"
+ using LIMSEQ_imp_Cauchy by auto
+ then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"
+ unfolding cauchy_def using \<open>e > 0\<close> by blast
+ then have False
+ using x[of "r N" "r (N+1)"] r by (auto simp: strict_mono_def) }
+ then show ?thesis
+ by metis
+qed
+
+subsubsection\<open>Heine-Borel theorem\<close>
+
+proposition seq_compact_imp_Heine_Borel:
+ fixes s :: "'a :: metric_space set"
+ assumes "seq_compact s"
+ shows "compact s"
+proof -
+ from seq_compact_imp_totally_bounded[OF \<open>seq_compact s\<close>]
+ obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>f e. ball x e)"
+ unfolding choice_iff' ..
+ define K where "K = (\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
+ have "countably_compact s"
+ using \<open>seq_compact s\<close> by (rule seq_compact_imp_countably_compact)
+ then show "compact s"
+ proof (rule countably_compact_imp_compact)
+ show "countable K"
+ unfolding K_def using f
+ by (auto intro: countable_finite countable_subset countable_rat
+ intro!: countable_image countable_SIGMA countable_UN)
+ show "\<forall>b\<in>K. open b" by (auto simp: K_def)
+ next
+ fix T x
+ assume T: "open T" "x \<in> T" and x: "x \<in> s"
+ from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"
+ by auto
+ then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"
+ by auto
+ from Rats_dense_in_real[OF \<open>0 < e / 2\<close>] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"
+ by auto
+ from f[rule_format, of r] \<open>0 < r\<close> \<open>x \<in> s\<close> obtain k where "k \<in> f r" "x \<in> ball k r"
+ by auto
+ from \<open>r \<in> \<rat>\<close> \<open>0 < r\<close> \<open>k \<in> f r\<close> have "ball k r \<in> K"
+ by (auto simp: K_def)
+ then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
+ proof (rule bexI[rotated], safe)
+ fix y
+ assume "y \<in> ball k r"
+ with \<open>r < e / 2\<close> \<open>x \<in> ball k r\<close> have "dist x y < e"
+ by (intro dist_triangle_half_r [of k _ e]) (auto simp: dist_commute)
+ with \<open>ball x e \<subseteq> T\<close> show "y \<in> T"
+ by auto
+ next
+ show "x \<in> ball k r" by fact
+ qed
+ qed
+qed
+
+proposition compact_eq_seq_compact_metric:
+ "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
+ using compact_imp_seq_compact seq_compact_imp_Heine_Borel by blast
+
+proposition compact_def: \<comment> \<open>this is the definition of compactness in HOL Light\<close>
+ "compact (S :: 'a::metric_space set) \<longleftrightarrow>
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
+ unfolding compact_eq_seq_compact_metric seq_compact_def by auto
+
+subsubsection \<open>Complete the chain of compactness variants\<close>
+
+proposition compact_eq_Bolzano_Weierstrass:
+ fixes s :: "'a::metric_space set"
+ shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ using Heine_Borel_imp_Bolzano_Weierstrass[of s] by auto
+next
+ assume ?rhs
+ then show ?lhs
+ unfolding compact_eq_seq_compact_metric by (rule Bolzano_Weierstrass_imp_seq_compact)
+qed
+
+proposition Bolzano_Weierstrass_imp_bounded:
+ "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
+ using compact_imp_bounded unfolding compact_eq_Bolzano_Weierstrass .
+
+
+subsection \<open>Metric spaces with the Heine-Borel property\<close>
+
+text \<open>
+ A metric space (or topological vector space) is said to have the
+ Heine-Borel property if every closed and bounded subset is compact.
+\<close>
+
+class heine_borel = metric_space +
+ assumes bounded_imp_convergent_subsequence:
+ "bounded (range f) \<Longrightarrow> \<exists>l r. strict_mono (r::nat\<Rightarrow>nat) \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+
+proposition bounded_closed_imp_seq_compact:
+ fixes s::"'a::heine_borel set"
+ assumes "bounded s"
+ and "closed s"
+ shows "seq_compact s"
+proof (unfold seq_compact_def, clarify)
+ fix f :: "nat \<Rightarrow> 'a"
+ assume f: "\<forall>n. f n \<in> s"
+ with \<open>bounded s\<close> have "bounded (range f)"
+ by (auto intro: bounded_subset)
+ obtain l r where r: "strict_mono (r :: nat \<Rightarrow> nat)" and l: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using bounded_imp_convergent_subsequence [OF \<open>bounded (range f)\<close>] by auto
+ from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"
+ by simp
+ have "l \<in> s" using \<open>closed s\<close> fr l
+ by (rule closed_sequentially)
+ show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using \<open>l \<in> s\<close> r l by blast
+qed
+
+lemma compact_eq_bounded_closed:
+ fixes s :: "'a::heine_borel set"
+ shows "compact s \<longleftrightarrow> bounded s \<and> closed s"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ using compact_imp_closed compact_imp_bounded
+ by blast
+next
+ assume ?rhs
+ then show ?lhs
+ using bounded_closed_imp_seq_compact[of s]
+ unfolding compact_eq_seq_compact_metric
+ by auto
+qed
+
+lemma compact_Inter:
+ fixes \<F> :: "'a :: heine_borel set set"
+ assumes com: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" and "\<F> \<noteq> {}"
+ shows "compact(\<Inter> \<F>)"
+ using assms
+ by (meson Inf_lower all_not_in_conv bounded_subset closed_Inter compact_eq_bounded_closed)
+
+lemma compact_closure [simp]:
+ fixes S :: "'a::heine_borel set"
+ shows "compact(closure S) \<longleftrightarrow> bounded S"
+by (meson bounded_closure bounded_subset closed_closure closure_subset compact_eq_bounded_closed)
+
+instance%important real :: heine_borel
+proof%unimportant
+ fix f :: "nat \<Rightarrow> real"
+ assume f: "bounded (range f)"
+ obtain r :: "nat \<Rightarrow> nat" where r: "strict_mono r" "monoseq (f \<circ> r)"
+ unfolding comp_def by (metis seq_monosub)
+ then have "Bseq (f \<circ> r)"
+ unfolding Bseq_eq_bounded using f
+ by (metis BseqI' bounded_iff comp_apply rangeI)
+ with r show "\<exists>l r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
+qed
+
+lemma compact_lemma_general:
+ fixes f :: "nat \<Rightarrow> 'a"
+ fixes proj::"'a \<Rightarrow> 'b \<Rightarrow> 'c::heine_borel" (infixl "proj" 60)
+ fixes unproj:: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"
+ assumes finite_basis: "finite basis"
+ assumes bounded_proj: "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. x proj k) ` range f)"
+ assumes proj_unproj: "\<And>e k. k \<in> basis \<Longrightarrow> (unproj e) proj k = e k"
+ assumes unproj_proj: "\<And>x. unproj (\<lambda>k. x proj k) = x"
+ shows "\<forall>d\<subseteq>basis. \<exists>l::'a. \<exists> r::nat\<Rightarrow>nat.
+ strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
+proof safe
+ fix d :: "'b set"
+ assume d: "d \<subseteq> basis"
+ with finite_basis have "finite d"
+ by (blast intro: finite_subset)
+ from this d show "\<exists>l::'a. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and>
+ (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
+ proof (induct d)
+ case empty
+ then show ?case
+ unfolding strict_mono_def by auto
+ next
+ case (insert k d)
+ have k[intro]: "k \<in> basis"
+ using insert by auto
+ have s': "bounded ((\<lambda>x. x proj k) ` range f)"
+ using k
+ by (rule bounded_proj)
+ obtain l1::"'a" and r1 where r1: "strict_mono r1"
+ and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
+ using insert(3) using insert(4) by auto
+ have f': "\<forall>n. f (r1 n) proj k \<in> (\<lambda>x. x proj k) ` range f"
+ by simp
+ have "bounded (range (\<lambda>i. f (r1 i) proj k))"
+ by (metis (lifting) bounded_subset f' image_subsetI s')
+ then obtain l2 r2 where r2:"strict_mono r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) proj k) \<longlongrightarrow> l2) sequentially"
+ using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) proj k"]
+ by (auto simp: o_def)
+ define r where "r = r1 \<circ> r2"
+ have r:"strict_mono r"
+ using r1 and r2 unfolding r_def o_def strict_mono_def by auto
+ moreover
+ define l where "l = unproj (\<lambda>i. if i = k then l2 else l1 proj i)"
+ {
+ fix e::real
+ assume "e > 0"
+ from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
+ by blast
+ from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) proj k) l2 < e) sequentially"
+ by (rule tendstoD)
+ from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e) sequentially"
+ by (rule eventually_subseq)
+ have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) proj i) (l proj i) < e) sequentially"
+ using N1' N2
+ by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def proj_unproj)
+ }
+ ultimately show ?case by auto
+ qed
+qed
+
+lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
+ unfolding bounded_def
+ by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)
+
+lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
+ unfolding bounded_def
+ by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)
+
+instance%important prod :: (heine_borel, heine_borel) heine_borel
+proof%unimportant
+ fix f :: "nat \<Rightarrow> 'a \<times> 'b"
+ assume f: "bounded (range f)"
+ then have "bounded (fst ` range f)"
+ by (rule bounded_fst)
+ then have s1: "bounded (range (fst \<circ> f))"
+ by (simp add: image_comp)
+ obtain l1 r1 where r1: "strict_mono r1" and l1: "(\<lambda>n. fst (f (r1 n))) \<longlonglongrightarrow> l1"
+ using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
+ from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
+ by (auto simp: image_comp intro: bounded_snd bounded_subset)
+ obtain l2 r2 where r2: "strict_mono r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) \<longlongrightarrow> l2) sequentially"
+ using bounded_imp_convergent_subsequence [OF s2]
+ unfolding o_def by fast
+ have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) \<longlongrightarrow> l1) sequentially"
+ using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
+ have l: "((f \<circ> (r1 \<circ> r2)) \<longlongrightarrow> (l1, l2)) sequentially"
+ using tendsto_Pair [OF l1' l2] unfolding o_def by simp
+ have r: "strict_mono (r1 \<circ> r2)"
+ using r1 r2 unfolding strict_mono_def by simp
+ show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using l r by fast
+qed
+
+subsubsection \<open>Completeness\<close>
+
+proposition (in metric_space) completeI:
+ assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f \<longlonglongrightarrow> l"
+ shows "complete s"
+ using assms unfolding complete_def by fast
+
+proposition (in metric_space) completeE:
+ assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"
+ obtains l where "l \<in> s" and "f \<longlonglongrightarrow> l"
+ using assms unfolding complete_def by fast
+
+(* TODO: generalize to uniform spaces *)
+lemma compact_imp_complete:
+ fixes s :: "'a::metric_space set"
+ assumes "compact s"
+ shows "complete s"
+proof -
+ {
+ fix f
+ assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
+ from as(1) obtain l r where lr: "l\<in>s" "strict_mono r" "(f \<circ> r) \<longlonglongrightarrow> l"
+ using assms unfolding compact_def by blast
+
+ note lr' = seq_suble [OF lr(2)]
+ {
+ fix e :: real
+ assume "e > 0"
+ 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 \<open>e > 0\<close>
+ apply (erule_tac x="e/2" in allE, auto)
+ done
+ 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 \<open>e > 0\<close> by auto
+ {
+ fix n :: nat
+ assume n: "n \<ge> max N M"
+ have "dist ((f \<circ> r) n) l < e/2"
+ using n M by auto
+ moreover have "r n \<ge> N"
+ using lr'[of n] n by auto
+ then have "dist (f n) ((f \<circ> r) n) < e / 2"
+ using N and n by auto
+ ultimately have "dist (f n) l < e"
+ using dist_triangle_half_r[of "f (r n)" "f n" e l]
+ by (auto simp: dist_commute)
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
+ }
+ then have "\<exists>l\<in>s. (f \<longlongrightarrow> l) sequentially" using \<open>l\<in>s\<close>
+ unfolding lim_sequentially by auto
+ }
+ then show ?thesis unfolding complete_def by auto
+qed
+
+proposition compact_eq_totally_bounded:
+ "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e))"
+ (is "_ \<longleftrightarrow> ?rhs")
+proof
+ assume assms: "?rhs"
+ then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"
+ by (auto simp: choice_iff')
+
+ show "compact s"
+ proof cases
+ assume "s = {}"
+ then show "compact s" by (simp add: compact_def)
+ next
+ assume "s \<noteq> {}"
+ show ?thesis
+ unfolding compact_def
+ proof safe
+ fix f :: "nat \<Rightarrow> 'a"
+ assume f: "\<forall>n. f n \<in> s"
+
+ define e where "e n = 1 / (2 * Suc n)" for n
+ then have [simp]: "\<And>n. 0 < e n" by auto
+ define B where "B n U = (SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U))" for n U
+ {
+ fix n U
+ assume "infinite {n. f n \<in> U}"
+ then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
+ using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
+ then obtain a where
+ "a \<in> k (e n)"
+ "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..
+ then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
+ by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
+ from someI_ex[OF this]
+ have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
+ unfolding B_def by auto
+ }
+ note B = this
+
+ define F where "F = rec_nat (B 0 UNIV) B"
+ {
+ fix n
+ have "infinite {i. f i \<in> F n}"
+ by (induct n) (auto simp: F_def B)
+ }
+ then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
+ using B by (simp add: F_def)
+ then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
+ using decseq_SucI[of F] by (auto simp: decseq_def)
+
+ obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
+ proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
+ fix k i
+ have "infinite ({n. f n \<in> F k} - {.. i})"
+ using \<open>infinite {n. f n \<in> F k}\<close> by auto
+ from infinite_imp_nonempty[OF this]
+ show "\<exists>x>i. f x \<in> F k"
+ by (simp add: set_eq_iff not_le conj_commute)
+ qed
+
+ define t where "t = rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
+ have "strict_mono t"
+ unfolding strict_mono_Suc_iff by (simp add: t_def sel)
+ moreover have "\<forall>i. (f \<circ> t) i \<in> s"
+ using f by auto
+ moreover
+ {
+ fix n
+ have "(f \<circ> t) n \<in> F n"
+ by (cases n) (simp_all add: t_def sel)
+ }
+ note t = this
+
+ have "Cauchy (f \<circ> t)"
+ proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
+ fix r :: real and N n m
+ assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
+ then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
+ using F_dec t by (auto simp: e_def field_simps of_nat_Suc)
+ with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
+ by (auto simp: subset_eq)
+ with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] \<open>2 * e N < r\<close>
+ show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
+ by (simp add: dist_commute)
+ qed
+
+ ultimately show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ using assms unfolding complete_def by blast
+ qed
+ qed
+qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
+
+lemma cauchy_imp_bounded:
+ assumes "Cauchy s"
+ shows "bounded (range s)"
+proof -
+ 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 by force
+ then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
+ moreover
+ have "bounded (s ` {0..N})"
+ using finite_imp_bounded[of "s ` {1..N}"] by auto
+ then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
+ unfolding bounded_any_center [where a="s N"] by auto
+ ultimately show "?thesis"
+ unfolding bounded_any_center [where a="s N"]
+ apply (rule_tac x="max a 1" in exI, auto)
+ apply (erule_tac x=y in allE)
+ apply (erule_tac x=y in ballE, auto)
+ done
+qed
+
+instance heine_borel < complete_space
+proof
+ fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
+ then have "bounded (range f)"
+ by (rule cauchy_imp_bounded)
+ then have "compact (closure (range f))"
+ unfolding compact_eq_bounded_closed by auto
+ then have "complete (closure (range f))"
+ by (rule compact_imp_complete)
+ moreover have "\<forall>n. f n \<in> closure (range f)"
+ using closure_subset [of "range f"] by auto
+ ultimately have "\<exists>l\<in>closure (range f). (f \<longlongrightarrow> l) sequentially"
+ using \<open>Cauchy f\<close> unfolding complete_def by auto
+ then show "convergent f"
+ unfolding convergent_def by auto
+qed
+
+lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"
+proof (rule completeI)
+ fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
+ then have "convergent f" by (rule Cauchy_convergent)
+ then show "\<exists>l\<in>UNIV. f \<longlonglongrightarrow> l" unfolding convergent_def by simp
+qed
+
+lemma complete_imp_closed:
+ fixes S :: "'a::metric_space set"
+ assumes "complete S"
+ shows "closed S"
+proof (unfold closed_sequential_limits, clarify)
+ fix f x assume "\<forall>n. f n \<in> S" and "f \<longlonglongrightarrow> x"
+ from \<open>f \<longlonglongrightarrow> x\<close> have "Cauchy f"
+ by (rule LIMSEQ_imp_Cauchy)
+ with \<open>complete S\<close> and \<open>\<forall>n. f n \<in> S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
+ by (rule completeE)
+ from \<open>f \<longlonglongrightarrow> x\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "x = l"
+ by (rule LIMSEQ_unique)
+ with \<open>l \<in> S\<close> show "x \<in> S"
+ by simp
+qed
+
+lemma complete_Int_closed:
+ fixes S :: "'a::metric_space set"
+ assumes "complete S" and "closed t"
+ shows "complete (S \<inter> t)"
+proof (rule completeI)
+ fix f assume "\<forall>n. f n \<in> S \<inter> t" and "Cauchy f"
+ then have "\<forall>n. f n \<in> S" and "\<forall>n. f n \<in> t"
+ by simp_all
+ from \<open>complete S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
+ using \<open>\<forall>n. f n \<in> S\<close> and \<open>Cauchy f\<close> by (rule completeE)
+ from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "l \<in> t"
+ by (rule closed_sequentially)
+ with \<open>l \<in> S\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>S \<inter> t. f \<longlonglongrightarrow> l"
+ by fast
+qed
+
+lemma complete_closed_subset:
+ fixes S :: "'a::metric_space set"
+ assumes "closed S" and "S \<subseteq> t" and "complete t"
+ shows "complete S"
+ using assms complete_Int_closed [of t S] by (simp add: Int_absorb1)
+
+lemma complete_eq_closed:
+ fixes S :: "('a::complete_space) set"
+ shows "complete S \<longleftrightarrow> closed S"
+proof
+ assume "closed S" then show "complete S"
+ using subset_UNIV complete_UNIV by (rule complete_closed_subset)
+next
+ assume "complete S" then show "closed S"
+ by (rule complete_imp_closed)
+qed
+
+lemma convergent_eq_Cauchy:
+ fixes S :: "nat \<Rightarrow> 'a::complete_space"
+ shows "(\<exists>l. (S \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy S"
+ unfolding Cauchy_convergent_iff convergent_def ..
+
+lemma convergent_imp_bounded:
+ fixes S :: "nat \<Rightarrow> 'a::metric_space"
+ shows "(S \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range S)"
+ by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
+
+lemma frontier_subset_compact:
+ fixes S :: "'a::heine_borel set"
+ shows "compact S \<Longrightarrow> frontier S \<subseteq> S"
+ using frontier_subset_closed compact_eq_bounded_closed
+ by blast
+
+subsection \<open>Continuity\<close>
+
+text\<open>Derive the epsilon-delta forms, which we often use as "definitions"\<close>
+
+proposition continuous_within_eps_delta:
+ "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)"
+ unfolding continuous_within and Lim_within by fastforce
+
+corollary continuous_at_eps_delta:
+ "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
+ using continuous_within_eps_delta [of x UNIV f] by simp
+
+lemma continuous_at_right_real_increasing:
+ fixes f :: "real \<Rightarrow> real"
+ assumes nondecF: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y"
+ shows "continuous (at_right a) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f (a + d) - f a < e)"
+ apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)
+ apply (intro all_cong ex_cong, safe)
+ apply (erule_tac x="a + d" in allE, simp)
+ apply (simp add: nondecF field_simps)
+ apply (drule nondecF, simp)
+ done
+
+lemma continuous_at_left_real_increasing:
+ assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"
+ shows "(continuous (at_left (a :: real)) f) = (\<forall>e > 0. \<exists>delta > 0. f a - f (a - delta) < e)"
+ apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)
+ apply (intro all_cong ex_cong, safe)
+ apply (erule_tac x="a - d" in allE, simp)
+ apply (simp add: nondecF field_simps)
+ apply (cut_tac x="a - d" and y=x in nondecF, simp_all)
+ done
+
+text\<open>Versions in terms of open balls.\<close>
+
+lemma continuous_within_ball:
+ "continuous (at x within s) f \<longleftrightarrow>
+ (\<forall>e > 0. \<exists>d > 0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ {
+ fix e :: real
+ assume "e > 0"
+ 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 \<open>?lhs\<close>[unfolded continuous_within Lim_within] by auto
+ {
+ fix y
+ assume "y \<in> f ` (ball x d \<inter> s)"
+ then have "y \<in> ball (f x) e"
+ using d(2)
+ apply (auto simp: dist_commute)
+ apply (erule_tac x=xa in ballE, auto)
+ using \<open>e > 0\<close>
+ apply auto
+ done
+ }
+ then have "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e"
+ using \<open>d > 0\<close>
+ unfolding subset_eq ball_def by (auto simp: dist_commute)
+ }
+ then show ?rhs by auto
+next
+ assume ?rhs
+ then show ?lhs
+ unfolding continuous_within Lim_within ball_def subset_eq
+ apply (auto simp: dist_commute)
+ apply (erule_tac x=e in allE, auto)
+ done
+qed
+
+lemma continuous_at_ball:
+ "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
+ apply auto
+ apply (erule_tac x=e in allE, auto)
+ apply (rule_tac x=d in exI, auto)
+ apply (erule_tac x=xa in allE)
+ apply (auto simp: dist_commute)
+ done
+next
+ assume ?rhs
+ then show ?lhs
+ unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
+ apply auto
+ apply (erule_tac x=e in allE, auto)
+ apply (rule_tac x=d in exI, auto)
+ apply (erule_tac x="f xa" in allE)
+ apply (auto simp: dist_commute)
+ done
+qed
+
+text\<open>Define setwise continuity in terms of limits within the set.\<close>
+
+lemma continuous_on_iff:
+ "continuous_on s f \<longleftrightarrow>
+ (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
+ unfolding continuous_on_def Lim_within
+ by (metis dist_pos_lt dist_self)
+
+lemma continuous_within_E:
+ assumes "continuous (at x within s) f" "e>0"
+ obtains d where "d>0" "\<And>x'. \<lbrakk>x'\<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+ using assms apply (simp add: continuous_within_eps_delta)
+ apply (drule spec [of _ e], clarify)
+ apply (rule_tac d="d/2" in that, auto)
+ done
+
+lemma continuous_onI [intro?]:
+ assumes "\<And>x e. \<lbrakk>e > 0; x \<in> s\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
+ shows "continuous_on s f"
+apply (simp add: continuous_on_iff, clarify)
+apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
+done
+
+text\<open>Some simple consequential lemmas.\<close>
+
+lemma continuous_onE:
+ assumes "continuous_on s f" "x\<in>s" "e>0"
+ obtains d where "d>0" "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+ using assms
+ apply (simp add: continuous_on_iff)
+ apply (elim ballE allE)
+ apply (auto intro: that [where d="d/2" for d])
+ done
+
+lemma uniformly_continuous_onE:
+ assumes "uniformly_continuous_on s f" "0 < e"
+ obtains d where "d>0" "\<And>x x'. \<lbrakk>x\<in>s; x'\<in>s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+using assms
+by (auto simp: uniformly_continuous_on_def)
+
+lemma uniformly_continuous_on_sequentially:
+ "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
+ (\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> 0 \<longrightarrow> (\<lambda>n. dist (f(x n)) (f(y n))) \<longlonglongrightarrow> 0)" (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ {
+ fix x y
+ assume x: "\<forall>n. x n \<in> s"
+ and y: "\<forall>n. y n \<in> s"
+ and xy: "((\<lambda>n. dist (x n) (y n)) \<longlongrightarrow> 0) sequentially"
+ {
+ fix e :: real
+ assume "e > 0"
+ 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"
+ using \<open>?lhs\<close>[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
+ obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"
+ using xy[unfolded lim_sequentially dist_norm] and \<open>d>0\<close> by auto
+ {
+ fix n
+ assume "n\<ge>N"
+ then have "dist (f (x n)) (f (y n)) < e"
+ 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
+ by (simp add: dist_commute)
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
+ by auto
+ }
+ then have "((\<lambda>n. dist (f(x n)) (f(y n))) \<longlongrightarrow> 0) sequentially"
+ unfolding lim_sequentially and dist_real_def by auto
+ }
+ then show ?rhs by auto
+next
+ assume ?rhs
+ {
+ assume "\<not> ?lhs"
+ 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
+ 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"
+ 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
+ by (auto simp: dist_commute)
+ define x where "x n = fst (fa (inverse (real n + 1)))" for n
+ define y where "y n = snd (fa (inverse (real n + 1)))" for n
+ 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"
+ unfolding x_def and y_def using fa
+ by auto
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"
+ unfolding real_arch_inverse[of e] by auto
+ {
+ fix n :: nat
+ assume "n \<ge> N"
+ then have "inverse (real n + 1) < inverse (real N)"
+ using of_nat_0_le_iff and \<open>N\<noteq>0\<close> by auto
+ also have "\<dots> < e" using N by auto
+ finally have "inverse (real n + 1) < e" by auto
+ then have "dist (x n) (y n) < e"
+ using xy0[THEN spec[where x=n]] by auto
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto
+ }
+ then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
+ using \<open>?rhs\<close>[THEN spec[where x=x], THEN spec[where x=y]] and xyn
+ unfolding lim_sequentially dist_real_def by auto
+ then have False using fxy and \<open>e>0\<close> by auto
+ }
+ then show ?lhs
+ unfolding uniformly_continuous_on_def by blast
+qed
+
+lemma continuous_closed_imp_Cauchy_continuous:
+ fixes S :: "('a::complete_space) set"
+ shows "\<lbrakk>continuous_on S f; closed S; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f \<circ> \<sigma>)"
+ apply (simp add: complete_eq_closed [symmetric] continuous_on_sequentially)
+ by (meson LIMSEQ_imp_Cauchy complete_def)
+
+text\<open>The usual transformation theorems.\<close>
+
+lemma continuous_transform_within:
+ fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
+ assumes "continuous (at x within s) f"
+ and "0 < d"
+ and "x \<in> s"
+ and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
+ shows "continuous (at x within s) g"
+ using assms
+ unfolding continuous_within
+ by (force intro: Lim_transform_within)
+
+subsubsection%unimportant \<open>Structural rules for uniform continuity\<close>
+
+lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
+ fixes g :: "_::metric_space \<Rightarrow> _"
+ assumes "uniformly_continuous_on s g"
+ shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
+ using assms unfolding uniformly_continuous_on_sequentially
+ unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
+ by (auto intro: tendsto_zero)
+
+
+subsection \<open>With Abstract Topology (TODO: move and remove dependency?)\<close>
+
+lemma openin_contains_ball:
+ "openin (subtopology euclidean t) s \<longleftrightarrow>
+ s \<subseteq> t \<and> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> ball x e \<inter> t \<subseteq> s)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (simp add: openin_open)
+ apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
+ done
+next
+ assume ?rhs
+ then show ?lhs
+ apply (simp add: openin_euclidean_subtopology_iff)
+ by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball)
+qed
+
+lemma openin_contains_cball:
+ "openin (subtopology euclidean t) s \<longleftrightarrow>
+ s \<subseteq> t \<and>
+ (\<forall>x \<in> s. \<exists>e. 0 < e \<and> cball x e \<inter> t \<subseteq> s)"
+apply (simp add: openin_contains_ball)
+apply (rule iffI)
+apply (auto dest!: bspec)
+apply (rule_tac x="e/2" in exI, force+)
+ done
+
+subsection \<open>With abstract Topology\<close>
+
+lemma Times_in_interior_subtopology:
+ fixes U :: "('a::metric_space \<times> 'b::metric_space) set"
+ assumes "(x, y) \<in> U" "openin (subtopology euclidean (S \<times> T)) U"
+ obtains V W where "openin (subtopology euclidean S) V" "x \<in> V"
+ "openin (subtopology euclidean T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
+proof -
+ from assms obtain e where "e > 0" and "U \<subseteq> S \<times> T"
+ and e: "\<And>x' y'. \<lbrakk>x'\<in>S; y'\<in>T; dist (x', y') (x, y) < e\<rbrakk> \<Longrightarrow> (x', y') \<in> U"
+ by (force simp: openin_euclidean_subtopology_iff)
+ with assms have "x \<in> S" "y \<in> T"
+ by auto
+ show ?thesis
+ proof
+ show "openin (subtopology euclidean S) (ball x (e/2) \<inter> S)"
+ by (simp add: Int_commute openin_open_Int)
+ show "x \<in> ball x (e / 2) \<inter> S"
+ by (simp add: \<open>0 < e\<close> \<open>x \<in> S\<close>)
+ show "openin (subtopology euclidean T) (ball y (e/2) \<inter> T)"
+ by (simp add: Int_commute openin_open_Int)
+ show "y \<in> ball y (e / 2) \<inter> T"
+ by (simp add: \<open>0 < e\<close> \<open>y \<in> T\<close>)
+ show "(ball x (e / 2) \<inter> S) \<times> (ball y (e / 2) \<inter> T) \<subseteq> U"
+ by clarify (simp add: e dist_Pair_Pair \<open>0 < e\<close> dist_commute sqrt_sum_squares_half_less)
+ qed
+qed
+
+lemma openin_Times_eq:
+ fixes S :: "'a::metric_space set" and T :: "'b::metric_space set"
+ shows
+ "openin (subtopology euclidean (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
+ S' = {} \<or> T' = {} \<or> openin (subtopology euclidean S) S' \<and> openin (subtopology euclidean T) T'"
+ (is "?lhs = ?rhs")
+proof (cases "S' = {} \<or> T' = {}")
+ case True
+ then show ?thesis by auto
+next
+ case False
+ then obtain x y where "x \<in> S'" "y \<in> T'"
+ by blast
+ show ?thesis
+ proof
+ assume ?lhs
+ have "openin (subtopology euclidean S) S'"
+ apply (subst openin_subopen, clarify)
+ apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
+ using \<open>y \<in> T'\<close>
+ apply auto
+ done
+ moreover have "openin (subtopology euclidean T) T'"
+ apply (subst openin_subopen, clarify)
+ apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
+ using \<open>x \<in> S'\<close>
+ apply auto
+ done
+ ultimately show ?rhs
+ by simp
+ next
+ assume ?rhs
+ with False show ?lhs
+ by (simp add: openin_Times)
+ qed
+qed
+
+lemma closedin_Times:
+ "closedin (subtopology euclidean S) S' \<Longrightarrow> closedin (subtopology euclidean T) T' \<Longrightarrow>
+ closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
+ unfolding closedin_closed using closed_Times by blast
+
+lemma Lim_transform_within_openin:
+ fixes a :: "'a::metric_space"
+ assumes f: "(f \<longlongrightarrow> l) (at a within T)"
+ and "openin (subtopology euclidean T) S" "a \<in> S"
+ and eq: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
+ shows "(g \<longlongrightarrow> l) (at a within T)"
+proof -
+ obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball a \<epsilon> \<inter> T \<subseteq> S"
+ using assms by (force simp: openin_contains_ball)
+ then have "a \<in> ball a \<epsilon>"
+ by simp
+ show ?thesis
+ by (rule Lim_transform_within [OF f \<open>0 < \<epsilon>\<close> eq]) (use \<epsilon> in \<open>auto simp: dist_commute subset_iff\<close>)
+qed
+
+lemma closure_Int_ballI:
+ fixes S :: "'a :: metric_space set"
+ assumes "\<And>U. \<lbrakk>openin (subtopology euclidean S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
+ shows "S \<subseteq> closure T"
+proof (clarsimp simp: closure_approachable dist_commute)
+ fix x and e::real
+ assume "x \<in> S" "0 < e"
+ with assms [of "S \<inter> ball x e"] show "\<exists>y\<in>T. dist x y < e"
+ by force
+qed
+
+lemma continuous_on_open_gen:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "f ` S \<subseteq> T"
+ shows "continuous_on S f \<longleftrightarrow>
+ (\<forall>U. openin (subtopology euclidean T) U
+ \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (clarsimp simp: openin_euclidean_subtopology_iff continuous_on_iff)
+ by (metis assms image_subset_iff)
+next
+ have ope: "openin (subtopology euclidean T) (ball y e \<inter> T)" for y e
+ by (simp add: Int_commute openin_open_Int)
+ assume R [rule_format]: ?rhs
+ show ?lhs
+ proof (clarsimp simp add: continuous_on_iff)
+ fix x and e::real
+ assume "x \<in> S" and "0 < e"
+ then have x: "x \<in> S \<inter> (f -` ball (f x) e \<inter> f -` T)"
+ using assms by auto
+ show "\<exists>d>0. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
+ using R [of "ball (f x) e \<inter> T"] x
+ by (fastforce simp add: ope openin_euclidean_subtopology_iff [of S] dist_commute)
+ qed
+qed
+
+lemma continuous_openin_preimage:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ shows
+ "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (subtopology euclidean T) U\<rbrakk>
+ \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U)"
+by (simp add: continuous_on_open_gen)
+
+lemma continuous_on_closed_gen:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "f ` S \<subseteq> T"
+ shows "continuous_on S f \<longleftrightarrow>
+ (\<forall>U. closedin (subtopology euclidean T) U
+ \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` U))"
+ (is "?lhs = ?rhs")
+proof -
+ have *: "U \<subseteq> T \<Longrightarrow> S \<inter> f -` (T - U) = S - (S \<inter> f -` U)" for U
+ using assms by blast
+ show ?thesis
+ proof
+ assume L: ?lhs
+ show ?rhs
+ proof clarify
+ fix U
+ assume "closedin (subtopology euclidean T) U"
+ then show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
+ using L unfolding continuous_on_open_gen [OF assms]
+ by (metis * closedin_def inf_le1 topspace_euclidean_subtopology)
+ qed
+ next
+ assume R [rule_format]: ?rhs
+ show ?lhs
+ unfolding continuous_on_open_gen [OF assms]
+ by (metis * R inf_le1 openin_closedin_eq topspace_euclidean_subtopology)
+ qed
+qed
+
+lemma continuous_closedin_preimage_gen:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (subtopology euclidean T) U"
+ shows "closedin (subtopology euclidean S) (S \<inter> f -` U)"
+using assms continuous_on_closed_gen by blast
+
+lemma continuous_transform_within_openin:
+ fixes a :: "'a::metric_space"
+ assumes "continuous (at a within T) f"
+ and "openin (subtopology euclidean T) S" "a \<in> S"
+ and eq: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
+ shows "continuous (at a within T) g"
+ using assms by (simp add: Lim_transform_within_openin continuous_within)
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Elementary_Normed_Spaces.thy Sat Dec 29 20:32:09 2018 +0100
@@ -0,0 +1,597 @@
+(* Author: L C Paulson, University of Cambridge
+ Author: Amine Chaieb, University of Cambridge
+ Author: Robert Himmelmann, TU Muenchen
+ Author: Brian Huffman, Portland State University
+*)
+
+section \<open>Elementary Normed Vector Spaces\<close>
+
+theory Elementary_Normed_Spaces
+ imports
+ "HOL-Library.FuncSet"
+ Elementary_Metric_Spaces
+begin
+
+subsection%unimportant \<open>Various Lemmas Combining Imports\<close>
+
+lemma countable_PiE:
+ "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
+ by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
+
+
+lemma open_sums:
+ fixes T :: "('b::real_normed_vector) set"
+ assumes "open S \<or> open T"
+ shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+ using assms
+proof
+ assume S: "open S"
+ show ?thesis
+ proof (clarsimp simp: open_dist)
+ fix x y
+ assume "x \<in> S" "y \<in> T"
+ with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
+ by (auto simp: open_dist)
+ then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
+ by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
+ then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
+ using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
+ qed
+next
+ assume T: "open T"
+ show ?thesis
+ proof (clarsimp simp: open_dist)
+ fix x y
+ assume "x \<in> S" "y \<in> T"
+ with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
+ by (auto simp: open_dist)
+ then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
+ by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
+ then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
+ using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
+ qed
+qed
+
+
+subsection \<open>Support\<close>
+
+definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
+ where "support_on s f = {x\<in>s. f x \<noteq> 0}"
+
+lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
+ by (simp add: support_on_def)
+
+lemma support_on_simps[simp]:
+ "support_on {} f = {}"
+ "support_on (insert x s) f =
+ (if f x = 0 then support_on s f else insert x (support_on s f))"
+ "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
+ "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
+ "support_on (s - t) f = support_on s f - support_on t f"
+ "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
+ unfolding support_on_def by auto
+
+lemma support_on_cong:
+ "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
+ by (auto simp: support_on_def)
+
+lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
+ by (auto simp: support_on_def)
+
+lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
+ by (auto simp: support_on_def)
+
+lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)"
+ unfolding support_on_def by auto
+
+(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
+definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+ where "supp_sum f S = (\<Sum>x\<in>support_on S f. f x)"
+
+lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
+ unfolding supp_sum_def by auto
+
+lemma supp_sum_insert[simp]:
+ "finite (support_on S f) \<Longrightarrow>
+ supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
+ by (simp add: supp_sum_def in_support_on insert_absorb)
+
+lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
+ by (cases "r = 0")
+ (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
+
+
+subsection \<open>Intervals\<close>
+
+lemma image_affinity_interval:
+ fixes c :: "'a::ordered_real_vector"
+ shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) =
+ (if {a..b}={} then {}
+ else if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
+ else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
+ (is "?lhs = ?rhs")
+proof (cases "m=0")
+ case True
+ then show ?thesis
+ by force
+next
+ case False
+ show ?thesis
+ proof
+ show "?lhs \<subseteq> ?rhs"
+ by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
+ show "?rhs \<subseteq> ?lhs"
+ proof (clarsimp, intro conjI impI subsetI)
+ show "\<lbrakk>0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk>
+ \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
+ apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
+ using False apply (auto simp: le_diff_eq pos_le_divideRI)
+ using diff_le_eq pos_le_divideR_eq by force
+ show "\<lbrakk>\<not> 0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk>
+ \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
+ apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
+ apply (auto simp: diff_le_eq neg_le_divideR_eq)
+ using diff_eq_diff_less_eq linordered_field_class.sign_simps(11) minus_diff_eq not_less scaleR_le_cancel_left_neg by fastforce
+ qed
+ qed
+qed
+
+
+subsection%unimportant \<open>Various Lemmas on Normed Algebras\<close>
+
+
+lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
+ by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
+
+lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
+ by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
+
+lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
+ unfolding Nats_def by (rule closed_of_nat_image)
+
+lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
+ unfolding Ints_def by (rule closed_of_int_image)
+
+lemma closed_subset_Ints:
+ fixes A :: "'a :: real_normed_algebra_1 set"
+ assumes "A \<subseteq> \<int>"
+ shows "closed A"
+proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
+ case (1 x y)
+ with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto
+ with \<open>dist y x < 1\<close> show "y = x"
+ by (auto elim!: Ints_cases simp: dist_of_int)
+qed
+
+subsection \<open>Filters\<close>
+
+definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter" (infixr "indirection" 70)
+ where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
+
+
+subsection \<open>Trivial Limits\<close>
+
+lemma trivial_limit_at_infinity:
+ "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
+ unfolding trivial_limit_def eventually_at_infinity
+ apply clarsimp
+ apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
+ apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
+ apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
+ apply (drule_tac x=UNIV in spec, simp)
+ done
+
+subsection \<open>Limits\<close>
+
+proposition Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
+ by (auto simp: tendsto_iff eventually_at_infinity)
+
+corollary Lim_at_infinityI [intro?]:
+ assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
+ shows "(f \<longlongrightarrow> l) at_infinity"
+ apply (simp add: Lim_at_infinity, clarify)
+ apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
+ done
+
+lemma Lim_transform_within_set_eq:
+ fixes a l :: "'a::real_normed_vector"
+ shows "eventually (\<lambda>x. x \<in> s \<longleftrightarrow> x \<in> t) (at a)
+ \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within s) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within t))"
+ by (force intro: Lim_transform_within_set elim: eventually_mono)
+
+lemma Lim_null:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
+ by (simp add: Lim dist_norm)
+
+lemma Lim_null_comparison:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
+ shows "(f \<longlongrightarrow> 0) net"
+ using assms(2)
+proof (rule metric_tendsto_imp_tendsto)
+ show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
+ using assms(1) by (rule eventually_mono) (simp add: dist_norm)
+qed
+
+lemma Lim_transform_bound:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ and g :: "'a \<Rightarrow> 'c::real_normed_vector"
+ assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
+ and "(g \<longlongrightarrow> 0) net"
+ shows "(f \<longlongrightarrow> 0) net"
+ using assms(1) tendsto_norm_zero [OF assms(2)]
+ by (rule Lim_null_comparison)
+
+lemma lim_null_mult_right_bounded:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
+ assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
+ shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
+proof -
+ have *: "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
+ by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
+ have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
+ apply (rule Lim_null_comparison [OF _ *])
+ apply (simp add: eventually_mono [OF g] mult_left_mono)
+ done
+ then show ?thesis
+ by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
+qed
+
+lemma lim_null_mult_left_bounded:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
+ assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
+ shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
+proof -
+ have *: "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
+ by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
+ have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
+ apply (rule Lim_null_comparison [OF _ *])
+ apply (simp add: eventually_mono [OF g] mult_right_mono)
+ done
+ then show ?thesis
+ by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
+qed
+
+lemma lim_null_scaleR_bounded:
+ assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
+ shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
+proof
+ fix \<epsilon>::real
+ assume "0 < \<epsilon>"
+ then have B: "0 < \<epsilon> / (abs B + 1)" by simp
+ have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
+ proof -
+ have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
+ by (simp add: mult_left_mono g)
+ also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
+ by (simp add: mult_left_mono)
+ also have "\<dots> < \<epsilon>"
+ by (rule f)
+ finally show ?thesis .
+ qed
+ show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
+ apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
+ apply (auto simp: \<open>0 < \<epsilon>\<close> divide_simps * split: if_split_asm)
+ done
+qed
+
+lemma Lim_norm_ubound:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
+ shows "norm(l) \<le> e"
+ using assms by (fast intro: tendsto_le tendsto_intros)
+
+lemma Lim_norm_lbound:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<not> trivial_limit net"
+ and "(f \<longlongrightarrow> l) net"
+ and "eventually (\<lambda>x. e \<le> norm (f x)) net"
+ shows "e \<le> norm l"
+ using assms by (fast intro: tendsto_le tendsto_intros)
+
+text\<open>Limit under bilinear function\<close>
+
+lemma Lim_bilinear:
+ assumes "(f \<longlongrightarrow> l) net"
+ and "(g \<longlongrightarrow> m) net"
+ and "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
+ using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
+ by (rule bounded_bilinear.tendsto)
+
+lemma Lim_at_zero:
+ fixes a :: "'a::real_normed_vector"
+ and l :: "'b::topological_space"
+ shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
+ using LIM_offset_zero LIM_offset_zero_cancel ..
+
+
+subsection%unimportant \<open>Limit Point of Filter\<close>
+
+lemma netlimit_at_vector:
+ fixes a :: "'a::real_normed_vector"
+ shows "netlimit (at a) = a"
+proof (cases "\<exists>x. x \<noteq> a")
+ case True then obtain x where x: "x \<noteq> a" ..
+ have "\<not> trivial_limit (at a)"
+ unfolding trivial_limit_def eventually_at dist_norm
+ apply clarsimp
+ apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
+ apply (simp add: norm_sgn sgn_zero_iff x)
+ done
+ then show ?thesis
+ by (rule netlimit_within [of a UNIV])
+qed simp
+
+subsection \<open>Boundedness\<close>
+
+lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
+ apply (simp add: bounded_iff)
+ apply (subgoal_tac "\<And>x (y::real). 0 < 1 + \<bar>y\<bar> \<and> (x \<le> y \<longrightarrow> x \<le> 1 + \<bar>y\<bar>)")
+ apply metis
+ apply arith
+ done
+
+lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
+ apply (simp add: bounded_pos)
+ apply (safe; rule_tac x="b+1" in exI; force)
+ done
+
+lemma Bseq_eq_bounded:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "Bseq f \<longleftrightarrow> bounded (range f)"
+ unfolding Bseq_def bounded_pos by auto
+
+lemma bounded_linear_image:
+ assumes "bounded S"
+ and "bounded_linear f"
+ shows "bounded (f ` S)"
+proof -
+ from assms(1) obtain b where "b > 0" and b: "\<forall>x\<in>S. norm x \<le> b"
+ unfolding bounded_pos by auto
+ 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: ac_simps)
+ show ?thesis
+ unfolding bounded_pos
+ proof (intro exI, safe)
+ show "norm (f x) \<le> B * b" if "x \<in> S" for x
+ by (meson B b less_imp_le mult_left_mono order_trans that)
+ qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
+qed
+
+lemma bounded_scaling:
+ fixes S :: "'a::real_normed_vector set"
+ shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
+ apply (rule bounded_linear_image, assumption)
+ apply (rule bounded_linear_scaleR_right)
+ done
+
+lemma bounded_scaleR_comp:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "bounded (f ` S)"
+ shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
+ using bounded_scaling[of "f ` S" r] assms
+ by (auto simp: image_image)
+
+lemma bounded_translation:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "bounded S"
+ shows "bounded ((\<lambda>x. a + x) ` S)"
+proof -
+ from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
+ unfolding bounded_pos by auto
+ {
+ fix x
+ assume "x \<in> S"
+ then have "norm (a + x) \<le> b + norm a"
+ using norm_triangle_ineq[of a x] b by auto
+ }
+ then show ?thesis
+ unfolding bounded_pos
+ using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
+ by (auto intro!: exI[of _ "b + norm a"])
+qed
+
+lemma bounded_translation_minus:
+ fixes S :: "'a::real_normed_vector set"
+ shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
+using bounded_translation [of S "-a"] by simp
+
+lemma bounded_uminus [simp]:
+ fixes X :: "'a::real_normed_vector set"
+ shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
+by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
+
+lemma uminus_bounded_comp [simp]:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ shows "bounded ((\<lambda>x. - f x) ` S) \<longleftrightarrow> bounded (f ` S)"
+ using bounded_uminus[of "f ` S"]
+ by (auto simp: image_image)
+
+lemma bounded_plus_comp:
+ fixes f g::"'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "bounded (f ` S)"
+ assumes "bounded (g ` S)"
+ shows "bounded ((\<lambda>x. f x + g x) ` S)"
+proof -
+ {
+ fix B C
+ assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B" "\<And>x. x\<in>S \<Longrightarrow> norm (g x) \<le> C"
+ then have "\<And>x. x \<in> S \<Longrightarrow> norm (f x + g x) \<le> B + C"
+ by (auto intro!: norm_triangle_le add_mono)
+ } then show ?thesis
+ using assms by (fastforce simp: bounded_iff)
+qed
+
+lemma bounded_plus:
+ fixes S ::"'a::real_normed_vector set"
+ assumes "bounded S" "bounded T"
+ shows "bounded ((\<lambda>(x,y). x + y) ` (S \<times> T))"
+ using bounded_plus_comp [of fst "S \<times> T" snd] assms
+ by (auto simp: split_def split: if_split_asm)
+
+lemma bounded_minus_comp:
+ "bounded (f ` S) \<Longrightarrow> bounded (g ` S) \<Longrightarrow> bounded ((\<lambda>x. f x - g x) ` S)"
+ for f g::"'a \<Rightarrow> 'b::real_normed_vector"
+ using bounded_plus_comp[of "f" S "\<lambda>x. - g x"]
+ by auto
+
+lemma bounded_minus:
+ fixes S ::"'a::real_normed_vector set"
+ assumes "bounded S" "bounded T"
+ shows "bounded ((\<lambda>(x,y). x - y) ` (S \<times> T))"
+ using bounded_minus_comp [of fst "S \<times> T" snd] assms
+ by (auto simp: split_def split: if_split_asm)
+
+lemma not_bounded_UNIV[simp]:
+ "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
+proof (auto simp: bounded_pos not_le)
+ obtain x :: 'a where "x \<noteq> 0"
+ using perfect_choose_dist [OF zero_less_one] by fast
+ fix b :: real
+ assume b: "b >0"
+ have b1: "b +1 \<ge> 0"
+ using b by simp
+ with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
+ by (simp add: norm_sgn)
+ then show "\<exists>x::'a. b < norm x" ..
+qed
+
+corollary cobounded_imp_unbounded:
+ fixes S :: "'a::{real_normed_vector, perfect_space} set"
+ shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
+ using bounded_Un [of S "-S"] by (simp add: sup_compl_top)
+
+
+subsection \<open>Normed spaces with the Heine-Borel property\<close>
+
+lemma not_compact_UNIV[simp]:
+ fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
+ shows "\<not> compact (UNIV::'a set)"
+ by (simp add: compact_eq_bounded_closed)
+
+text\<open>Representing sets as the union of a chain of compact sets.\<close>
+lemma closed_Union_compact_subsets:
+ fixes S :: "'a::{heine_borel,real_normed_vector} set"
+ assumes "closed S"
+ obtains F where "\<And>n. compact(F n)" "\<And>n. F n \<subseteq> S" "\<And>n. F n \<subseteq> F(Suc n)"
+ "(\<Union>n. F n) = S" "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. K \<subseteq> F n"
+proof
+ show "compact (S \<inter> cball 0 (of_nat n))" for n
+ using assms compact_eq_bounded_closed by auto
+next
+ show "(\<Union>n. S \<inter> cball 0 (real n)) = S"
+ by (auto simp: real_arch_simple)
+next
+ fix K :: "'a set"
+ assume "compact K" "K \<subseteq> S"
+ then obtain N where "K \<subseteq> cball 0 N"
+ by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
+ then show "\<exists>N. \<forall>n\<ge>N. K \<subseteq> S \<inter> cball 0 (real n)"
+ by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
+qed auto
+
+
+subsection \<open>Continuity\<close>
+
+subsubsection%unimportant \<open>Structural rules for uniform continuity\<close>
+
+lemma uniformly_continuous_on_dist[continuous_intros]:
+ fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "uniformly_continuous_on s f"
+ and "uniformly_continuous_on s g"
+ shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
+proof -
+ {
+ fix a b c d :: 'b
+ have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
+ using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
+ using dist_triangle3 [of c d a] dist_triangle [of a d b]
+ by arith
+ } note le = this
+ {
+ fix x y
+ assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
+ assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
+ have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
+ by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
+ simp add: le)
+ }
+ then show ?thesis
+ using assms unfolding uniformly_continuous_on_sequentially
+ unfolding dist_real_def by simp
+qed
+
+lemma uniformly_continuous_on_norm[continuous_intros]:
+ fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
+ unfolding norm_conv_dist using assms
+ by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
+
+lemma uniformly_continuous_on_cmul[continuous_intros]:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
+ using bounded_linear_scaleR_right assms
+ by (rule bounded_linear.uniformly_continuous_on)
+
+lemma dist_minus:
+ fixes x y :: "'a::real_normed_vector"
+ shows "dist (- x) (- y) = dist x y"
+ unfolding dist_norm minus_diff_minus norm_minus_cancel ..
+
+lemma uniformly_continuous_on_minus[continuous_intros]:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
+ unfolding uniformly_continuous_on_def dist_minus .
+
+lemma uniformly_continuous_on_add[continuous_intros]:
+ fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ and "uniformly_continuous_on s g"
+ shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
+ using assms
+ unfolding uniformly_continuous_on_sequentially
+ unfolding dist_norm tendsto_norm_zero_iff add_diff_add
+ by (auto intro: tendsto_add_zero)
+
+lemma uniformly_continuous_on_diff[continuous_intros]:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ and "uniformly_continuous_on s g"
+ shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
+ using assms uniformly_continuous_on_add [of s f "- g"]
+ by (simp add: fun_Compl_def uniformly_continuous_on_minus)
+
+
+subsection%unimportant \<open>Topological properties of linear functions\<close>
+
+lemma linear_lim_0:
+ assumes "bounded_linear f"
+ shows "(f \<longlongrightarrow> 0) (at (0))"
+proof -
+ interpret f: bounded_linear f by fact
+ have "(f \<longlongrightarrow> f 0) (at 0)"
+ using tendsto_ident_at by (rule f.tendsto)
+ then show ?thesis unfolding f.zero .
+qed
+
+lemma linear_continuous_at:
+ assumes "bounded_linear f"
+ shows "continuous (at a) f"
+ unfolding continuous_at using assms
+ apply (rule bounded_linear.tendsto)
+ apply (rule tendsto_ident_at)
+ done
+
+lemma linear_continuous_within:
+ "bounded_linear f \<Longrightarrow> continuous (at x within s) f"
+ using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
+
+lemma linear_continuous_on:
+ "bounded_linear f \<Longrightarrow> continuous_on s f"
+ using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Elementary_Topology.thy Sat Dec 29 18:40:29 2018 +0000
+++ b/src/HOL/Analysis/Elementary_Topology.thy Sat Dec 29 20:32:09 2018 +0100
@@ -8,161 +8,15 @@
theory Elementary_Topology
imports
- "HOL-Library.Indicator_Function"
- "HOL-Library.Countable_Set"
- "HOL-Library.FuncSet"
"HOL-Library.Set_Idioms"
- "HOL-Library.Infinite_Set"
Product_Vector
begin
-(* FIXME: move elsewhere *)
-lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
- apply auto
- apply (rule_tac x="d/2" in exI)
- apply auto
- done
-
-lemma approachable_lt_le2: \<comment> \<open>like the above, but pushes aside an extra formula\<close>
- "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
- apply auto
- apply (rule_tac x="d/2" in exI, auto)
- done
-
-lemma triangle_lemma:
- fixes x y z :: real
- assumes x: "0 \<le> x"
- and y: "0 \<le> y"
- and z: "0 \<le> z"
- and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
- shows "x \<le> y + z"
-proof -
- have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
- using z y by simp
- with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
- by (simp add: power2_eq_square field_simps)
- from y z have yz: "y + z \<ge> 0"
- by arith
- from power2_le_imp_le[OF th yz] show ?thesis .
-qed
-
-definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
- where "support_on s f = {x\<in>s. f x \<noteq> 0}"
-
-lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
- by (simp add: support_on_def)
-
-lemma support_on_simps[simp]:
- "support_on {} f = {}"
- "support_on (insert x s) f =
- (if f x = 0 then support_on s f else insert x (support_on s f))"
- "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
- "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
- "support_on (s - t) f = support_on s f - support_on t f"
- "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
- unfolding support_on_def by auto
-
-lemma support_on_cong:
- "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
- by (auto simp: support_on_def)
-
-lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
- by (auto simp: support_on_def)
-
-lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
- by (auto simp: support_on_def)
-
-lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)"
- unfolding support_on_def by auto
-
-(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
-definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
- where "supp_sum f S = (\<Sum>x\<in>support_on S f. f x)"
-
-lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
- unfolding supp_sum_def by auto
-
-lemma supp_sum_insert[simp]:
- "finite (support_on S f) \<Longrightarrow>
- supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
- by (simp add: supp_sum_def in_support_on insert_absorb)
+subsection \<open>TODO: move?\<close>
-lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
- by (cases "r = 0")
- (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
-
-(*END OF SUPPORT, ETC.*)
-
-lemma image_affinity_interval:
- fixes c :: "'a::ordered_real_vector"
- shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) =
- (if {a..b}={} then {}
- else if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
- else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
- (is "?lhs = ?rhs")
-proof (cases "m=0")
- case True
- then show ?thesis
- by force
-next
- case False
- show ?thesis
- proof
- show "?lhs \<subseteq> ?rhs"
- by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
- show "?rhs \<subseteq> ?lhs"
- proof (clarsimp, intro conjI impI subsetI)
- show "\<lbrakk>0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk>
- \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
- apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
- using False apply (auto simp: le_diff_eq pos_le_divideRI)
- using diff_le_eq pos_le_divideR_eq by force
- show "\<lbrakk>\<not> 0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk>
- \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
- apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
- apply (auto simp: diff_le_eq neg_le_divideR_eq)
- using diff_eq_diff_less_eq linordered_field_class.sign_simps(11) minus_diff_eq not_less scaleR_le_cancel_left_neg by fastforce
- qed
- qed
-qed
-
-lemma countable_PiE:
- "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
- by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
-
-lemma open_sums:
- fixes T :: "('b::real_normed_vector) set"
- assumes "open S \<or> open T"
- shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
- using assms
-proof
- assume S: "open S"
- show ?thesis
- proof (clarsimp simp: open_dist)
- fix x y
- assume "x \<in> S" "y \<in> T"
- with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
- by (auto simp: open_dist)
- then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
- by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
- then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
- using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
- qed
-next
- assume T: "open T"
- show ?thesis
- proof (clarsimp simp: open_dist)
- fix x y
- assume "x \<in> S" "y \<in> T"
- with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
- by (auto simp: open_dist)
- then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
- by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
- then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
- using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
- qed
-qed
+lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
+ using openI by auto
subsection \<open>Topological Basis\<close>
@@ -630,824 +484,8 @@
class polish_space = complete_space + second_countable_topology
-subsection \<open>General notion of a topology as a value\<close>
-definition%important "istopology L \<longleftrightarrow>
- L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union>K))"
-
-typedef%important 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
- morphisms "openin" "topology"
- unfolding istopology_def by blast
-
-lemma istopology_openin[intro]: "istopology(openin U)"
- using openin[of U] by blast
-
-lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
- using topology_inverse[unfolded mem_Collect_eq] .
-
-lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
- using topology_inverse[of U] istopology_openin[of "topology U"] by auto
-
-lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
-proof
- assume "T1 = T2"
- then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
-next
- assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
- then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
- then have "topology (openin T1) = topology (openin T2)" by simp
- then show "T1 = T2" unfolding openin_inverse .
-qed
-
-
-text\<open>The "universe": the union of all sets in the topology.\<close>
-definition "topspace T = \<Union>{S. openin T S}"
-
-subsubsection \<open>Main properties of open sets\<close>
-
-proposition openin_clauses:
- fixes U :: "'a topology"
- shows
- "openin U {}"
- "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
- "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
- using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
-
-lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
- unfolding topspace_def by blast
-
-lemma openin_empty[simp]: "openin U {}"
- by (rule openin_clauses)
-
-lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
- by (rule openin_clauses)
-
-lemma openin_Union[intro]: "(\<And>S. S \<in> K \<Longrightarrow> openin U S) \<Longrightarrow> openin U (\<Union>K)"
- using openin_clauses by blast
-
-lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
- using openin_Union[of "{S,T}" U] by auto
-
-lemma openin_topspace[intro, simp]: "openin U (topspace U)"
- by (force simp: openin_Union topspace_def)
-
-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")
-proof
- assume ?lhs
- then show ?rhs by auto
-next
- assume H: ?rhs
- let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
- have "openin U ?t" by (force simp: openin_Union)
- also have "?t = S" using H by auto
- finally show "openin U S" .
-qed
-
-lemma openin_INT [intro]:
- assumes "finite I"
- "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
- shows "openin T ((\<Inter>i \<in> I. U i) \<inter> topspace T)"
-using assms by (induct, auto simp: inf_sup_aci(2) openin_Int)
-
-lemma openin_INT2 [intro]:
- assumes "finite I" "I \<noteq> {}"
- "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
- shows "openin T (\<Inter>i \<in> I. U i)"
-proof -
- have "(\<Inter>i \<in> I. U i) \<subseteq> topspace T"
- using \<open>I \<noteq> {}\<close> openin_subset[OF assms(3)] by auto
- then show ?thesis
- using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
-qed
-
-lemma openin_Inter [intro]:
- assumes "finite \<F>" "\<F> \<noteq> {}" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (\<Inter>\<F>)"
- by (metis (full_types) assms openin_INT2 image_ident)
-
-lemma openin_Int_Inter:
- assumes "finite \<F>" "openin T U" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (U \<inter> \<Inter>\<F>)"
- using openin_Inter [of "insert U \<F>"] assms by auto
-
-
-subsubsection \<open>Closed sets\<close>
-
-definition%important "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
-
-lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
- by (metis closedin_def)
-
-lemma closedin_empty[simp]: "closedin U {}"
- by (simp add: closedin_def)
-
-lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
- by (simp add: closedin_def)
-
-lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
- by (auto simp: Diff_Un closedin_def)
-
-lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
- by auto
-
-lemma closedin_Union:
- assumes "finite S" "\<And>T. T \<in> S \<Longrightarrow> closedin U T"
- shows "closedin U (\<Union>S)"
- using assms by induction auto
-
-lemma closedin_Inter[intro]:
- assumes Ke: "K \<noteq> {}"
- and Kc: "\<And>S. S \<in>K \<Longrightarrow> closedin U S"
- shows "closedin U (\<Inter>K)"
- using Ke Kc unfolding closedin_def Diff_Inter by auto
-
-lemma closedin_INT[intro]:
- assumes "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)"
- shows "closedin U (\<Inter>x\<in>A. B x)"
- apply (rule closedin_Inter)
- using assms
- apply auto
- done
-
-lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
- using closedin_Inter[of "{S,T}" U] by auto
-
-lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
- apply (auto simp: closedin_def Diff_Diff_Int inf_absorb2)
- apply (metis openin_subset subset_eq)
- done
-
-lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
- by (simp add: openin_closedin_eq)
-
-lemma openin_diff[intro]:
- assumes oS: "openin U S"
- and cT: "closedin U T"
- shows "openin U (S - T)"
-proof -
- have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
- by (auto simp: topspace_def openin_subset)
- then show ?thesis using oS cT
- by (auto simp: closedin_def)
-qed
-
-lemma closedin_diff[intro]:
- assumes oS: "closedin U S"
- and cT: "openin U T"
- shows "closedin U (S - T)"
-proof -
- have "S - T = S \<inter> (topspace U - T)"
- using closedin_subset[of U S] oS cT by (auto simp: topspace_def)
- then show ?thesis
- using oS cT by (auto simp: openin_closedin_eq)
-qed
-
-
-subsection\<open>The discrete topology\<close>
-
-definition discrete_topology where "discrete_topology U \<equiv> topology (\<lambda>S. S \<subseteq> U)"
-
-lemma openin_discrete_topology [simp]: "openin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
-proof -
- have "istopology (\<lambda>S. S \<subseteq> U)"
- by (auto simp: istopology_def)
- then show ?thesis
- by (simp add: discrete_topology_def topology_inverse')
-qed
-
-lemma topspace_discrete_topology [simp]: "topspace(discrete_topology U) = U"
- by (meson openin_discrete_topology openin_subset openin_topspace order_refl subset_antisym)
-
-lemma closedin_discrete_topology [simp]: "closedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
- by (simp add: closedin_def)
-
-lemma discrete_topology_unique:
- "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> (\<forall>x \<in> U. openin X {x})" (is "?lhs = ?rhs")
-proof
- assume R: ?rhs
- then have "openin X S" if "S \<subseteq> U" for S
- using openin_subopen subsetD that by fastforce
- moreover have "x \<in> topspace X" if "openin X S" and "x \<in> S" for x S
- using openin_subset that by blast
- ultimately
- show ?lhs
- using R by (auto simp: topology_eq)
-qed auto
-
-lemma discrete_topology_unique_alt:
- "discrete_topology U = X \<longleftrightarrow> topspace X \<subseteq> U \<and> (\<forall>x \<in> U. openin X {x})"
- using openin_subset
- by (auto simp: discrete_topology_unique)
-
-lemma subtopology_eq_discrete_topology_empty:
- "X = discrete_topology {} \<longleftrightarrow> topspace X = {}"
- using discrete_topology_unique [of "{}" X] by auto
-
-lemma subtopology_eq_discrete_topology_sing:
- "X = discrete_topology {a} \<longleftrightarrow> topspace X = {a}"
- by (metis discrete_topology_unique openin_topspace singletonD)
-
-
-subsection \<open>Subspace topology\<close>
-
-definition%important "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
-
-lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
- (is "istopology ?L")
-proof -
- have "?L {}" by blast
- {
- fix A B
- assume A: "?L A" and B: "?L B"
- 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
- have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
- using Sa Sb by blast+
- then have "?L (A \<inter> B)" by blast
- }
- moreover
- {
- fix K
- assume K: "K \<subseteq> Collect ?L"
- have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
- by blast
- from K[unfolded th0 subset_image_iff]
- obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
- by blast
- have "\<Union>K = (\<Union>Sk) \<inter> V"
- using Sk by auto
- moreover have "openin U (\<Union>Sk)"
- using Sk by (auto simp: subset_eq)
- ultimately have "?L (\<Union>K)" by blast
- }
- ultimately show ?thesis
- unfolding subset_eq mem_Collect_eq istopology_def by auto
-qed
-
-lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
- unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
- by auto
-
-lemma openin_subtopology_Int:
- "openin X S \<Longrightarrow> openin (subtopology X T) (S \<inter> T)"
- using openin_subtopology by auto
-
-lemma openin_subtopology_Int2:
- "openin X T \<Longrightarrow> openin (subtopology X S) (S \<inter> T)"
- using openin_subtopology by auto
-
-lemma openin_subtopology_diff_closed:
- "\<lbrakk>S \<subseteq> topspace X; closedin X T\<rbrakk> \<Longrightarrow> openin (subtopology X S) (S - T)"
- unfolding closedin_def openin_subtopology
- by (rule_tac x="topspace X - T" in exI) auto
-
-lemma openin_relative_to: "(openin X relative_to S) = openin (subtopology X S)"
- by (force simp: relative_to_def openin_subtopology)
-
-lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
- by (auto simp: topspace_def openin_subtopology)
-
-lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
- unfolding closedin_def topspace_subtopology
- by (auto simp: openin_subtopology)
-
-lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
- unfolding openin_subtopology
- by auto (metis IntD1 in_mono openin_subset)
-
-lemma subtopology_subtopology:
- "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
-proof -
- have eq: "\<And>T'. (\<exists>S'. T' = S' \<inter> T \<and> (\<exists>T. openin X T \<and> S' = T \<inter> S)) = (\<exists>Sa. T' = Sa \<inter> (S \<inter> T) \<and> openin X Sa)"
- by (metis inf_assoc)
- have "subtopology (subtopology X S) T = topology (\<lambda>Ta. \<exists>Sa. Ta = Sa \<inter> T \<and> openin (subtopology X S) Sa)"
- by (simp add: subtopology_def)
- also have "\<dots> = subtopology X (S \<inter> T)"
- by (simp add: openin_subtopology eq) (simp add: subtopology_def)
- finally show ?thesis .
-qed
-
-lemma openin_subtopology_alt:
- "openin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (openin X)"
- by (simp add: image_iff inf_commute openin_subtopology)
-
-lemma closedin_subtopology_alt:
- "closedin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (closedin X)"
- by (simp add: image_iff inf_commute closedin_subtopology)
-
-lemma subtopology_superset:
- assumes UV: "topspace U \<subseteq> V"
- shows "subtopology U V = U"
-proof -
- {
- fix S
- {
- fix T
- assume T: "openin U T" "S = T \<inter> V"
- from T openin_subset[OF T(1)] UV have eq: "S = T"
- by blast
- have "openin U S"
- unfolding eq using T by blast
- }
- moreover
- {
- assume S: "openin U S"
- then have "\<exists>T. openin U T \<and> S = T \<inter> V"
- using openin_subset[OF S] UV by auto
- }
- ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
- by blast
- }
- then show ?thesis
- unfolding topology_eq openin_subtopology by blast
-qed
-
-lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
- by (simp add: subtopology_superset)
-
-lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
- by (simp add: subtopology_superset)
-
-lemma openin_subtopology_empty:
- "openin (subtopology U {}) S \<longleftrightarrow> S = {}"
-by (metis Int_empty_right openin_empty openin_subtopology)
-
-lemma closedin_subtopology_empty:
- "closedin (subtopology U {}) S \<longleftrightarrow> S = {}"
-by (metis Int_empty_right closedin_empty closedin_subtopology)
-
-lemma closedin_subtopology_refl [simp]:
- "closedin (subtopology U X) X \<longleftrightarrow> X \<subseteq> topspace U"
-by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)
-
-lemma closedin_topspace_empty: "topspace T = {} \<Longrightarrow> (closedin T S \<longleftrightarrow> S = {})"
- by (simp add: closedin_def)
-
-lemma openin_imp_subset:
- "openin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
-by (metis Int_iff openin_subtopology subsetI)
-
-lemma closedin_imp_subset:
- "closedin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
-by (simp add: closedin_def topspace_subtopology)
-
-lemma openin_open_subtopology:
- "openin X S \<Longrightarrow> openin (subtopology X S) T \<longleftrightarrow> openin X T \<and> T \<subseteq> S"
- by (metis inf.orderE openin_Int openin_imp_subset openin_subtopology)
-
-lemma closedin_closed_subtopology:
- "closedin X S \<Longrightarrow> (closedin (subtopology X S) T \<longleftrightarrow> closedin X T \<and> T \<subseteq> S)"
- by (metis closedin_Int closedin_imp_subset closedin_subtopology inf.orderE)
-
-lemma openin_subtopology_Un:
- "\<lbrakk>openin (subtopology X T) S; openin (subtopology X U) S\<rbrakk>
- \<Longrightarrow> openin (subtopology X (T \<union> U)) S"
-by (simp add: openin_subtopology) blast
-
-lemma closedin_subtopology_Un:
- "\<lbrakk>closedin (subtopology X T) S; closedin (subtopology X U) S\<rbrakk>
- \<Longrightarrow> closedin (subtopology X (T \<union> U)) S"
-by (simp add: closedin_subtopology) blast
-
-
-subsection \<open>The standard Euclidean topology\<close>
-
-definition%important euclidean :: "'a::topological_space topology"
- where "euclidean = topology open"
-
-lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
- unfolding euclidean_def
- apply (rule cong[where x=S and y=S])
- apply (rule topology_inverse[symmetric])
- apply (auto simp: istopology_def)
- done
-
-declare open_openin [symmetric, simp]
-
-lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
- by (force simp: topspace_def)
-
-lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
- by (simp add: topspace_subtopology)
-
-lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
- by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV)
-
-declare closed_closedin [symmetric, simp]
-
-lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
- using openI by auto
-
-lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
- by (metis openin_topspace topspace_euclidean_subtopology)
-
-subsubsection\<open>The most basic facts about the usual topology and metric on R\<close>
-
-abbreviation euclideanreal :: "real topology"
- where "euclideanreal \<equiv> topology open"
-
-lemma real_openin [simp]: "openin euclideanreal S = open S"
- by (simp add: euclidean_def open_openin)
-
-lemma topspace_euclideanreal [simp]: "topspace euclideanreal = UNIV"
- using openin_subset open_UNIV real_openin by blast
-
-lemma topspace_euclideanreal_subtopology [simp]:
- "topspace (subtopology euclideanreal S) = S"
- by (simp add: topspace_subtopology)
-
-lemma real_closedin [simp]: "closedin euclideanreal S = closed S"
- by (simp add: closed_closedin euclidean_def)
-
-subsection \<open>Basic "localization" results are handy for connectedness.\<close>
-
-lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
- by (auto simp: openin_subtopology)
-
-lemma openin_Int_open:
- "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
- \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
-by (metis open_Int Int_assoc openin_open)
-
-lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
- by (auto simp: openin_open)
-
-lemma open_openin_trans[trans]:
- "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
- by (metis Int_absorb1 openin_open_Int)
-
-lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
- by (auto simp: openin_open)
-
-lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
- by (simp add: closedin_subtopology Int_ac)
-
-lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
- by (metis closedin_closed)
-
-lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
- by (auto simp: closedin_closed)
-
-lemma closedin_closed_subset:
- "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
- \<Longrightarrow> closedin (subtopology euclidean T) S"
- by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
-
-lemma finite_imp_closedin:
- fixes S :: "'a::t1_space set"
- shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
- by (simp add: finite_imp_closed closed_subset)
-
-lemma closedin_singleton [simp]:
- fixes a :: "'a::t1_space"
- shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
-using closedin_subset by (force intro: closed_subset)
-
-lemma openin_euclidean_subtopology_iff:
- fixes S U :: "'a::metric_space set"
- shows "openin (subtopology euclidean U) S \<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")
-proof
- assume ?lhs
- then show ?rhs
- unfolding openin_open open_dist by blast
-next
- define T where "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
- have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
- unfolding T_def
- apply clarsimp
- apply (rule_tac x="d - dist x a" in exI)
- apply (clarsimp simp add: less_diff_eq)
- by (metis dist_commute dist_triangle_lt)
- assume ?rhs then have 2: "S = U \<inter> T"
- unfolding T_def
- by auto (metis dist_self)
- from 1 2 show ?lhs
- unfolding openin_open open_dist by fast
-qed
-
-lemma connected_openin:
- "connected S \<longleftrightarrow>
- \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
- openin (subtopology euclidean S) E2 \<and>
- S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
- apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe)
- apply (simp_all, blast+) (* SLOW *)
- done
-
-lemma connected_openin_eq:
- "connected S \<longleftrightarrow>
- \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
- openin (subtopology euclidean S) E2 \<and>
- E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
- E1 \<noteq> {} \<and> E2 \<noteq> {})"
- apply (simp add: connected_openin, safe, blast)
- by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)
-
-lemma connected_closedin:
- "connected S \<longleftrightarrow>
- (\<nexists>E1 E2.
- closedin (subtopology euclidean S) E1 \<and>
- closedin (subtopology euclidean S) E2 \<and>
- S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- by (auto simp add: connected_closed closedin_closed)
-next
- assume R: ?rhs
- then show ?lhs
- proof (clarsimp simp add: connected_closed closedin_closed)
- fix A B
- assume s_sub: "S \<subseteq> A \<union> B" "B \<inter> S \<noteq> {}"
- and disj: "A \<inter> B \<inter> S = {}"
- and cl: "closed A" "closed B"
- have "S \<inter> (A \<union> B) = S"
- using s_sub(1) by auto
- have "S - A = B \<inter> S"
- using Diff_subset_conv Un_Diff_Int disj s_sub(1) by auto
- then have "S \<inter> A = {}"
- by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int R cl closedin_closed_Int inf_commute order_refl s_sub(2))
- then show "A \<inter> S = {}"
- by blast
- qed
-qed
-
-lemma connected_closedin_eq:
- "connected S \<longleftrightarrow>
- \<not>(\<exists>E1 E2.
- closedin (subtopology euclidean S) E1 \<and>
- closedin (subtopology euclidean S) E2 \<and>
- E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
- E1 \<noteq> {} \<and> E2 \<noteq> {})"
- apply (simp add: connected_closedin, safe, blast)
- by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
-
-text \<open>These "transitivity" results are handy too\<close>
-
-lemma openin_trans[trans]:
- "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
- openin (subtopology euclidean U) S"
- unfolding open_openin openin_open by blast
-
-lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
- by (auto simp: openin_open intro: openin_trans)
-
-lemma closedin_trans[trans]:
- "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
- closedin (subtopology euclidean U) S"
- by (auto simp: closedin_closed closed_Inter Int_assoc)
-
-lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
- by (auto simp: closedin_closed intro: closedin_trans)
-
-lemma openin_subtopology_Int_subset:
- "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
- by (auto simp: openin_subtopology)
-
-lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
- using open_subset openin_open_trans openin_subset by fastforce
-
-
-subsection \<open>Open and closed balls\<close>
-
-definition%important ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "ball x e = {y. dist x y < e}"
-
-definition%important cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "cball x e = {y. dist x y \<le> e}"
-
-definition%important sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "sphere x e = {y. dist x y = e}"
-
-lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
- by (simp add: ball_def)
-
-lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
- by (simp add: cball_def)
-
-lemma mem_sphere [simp]: "y \<in> sphere x e \<longleftrightarrow> dist x y = e"
- by (simp add: sphere_def)
-
-lemma ball_trivial [simp]: "ball x 0 = {}"
- by (simp add: ball_def)
-
-lemma cball_trivial [simp]: "cball x 0 = {x}"
- by (simp add: cball_def)
-
-lemma sphere_trivial [simp]: "sphere x 0 = {x}"
- by (simp add: sphere_def)
-
-lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
- for x :: "'a::real_normed_vector"
- by (simp add: dist_norm)
-
-lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
- for x :: "'a::real_normed_vector"
- by (simp add: dist_norm)
-
-lemma disjoint_ballI: "dist x y \<ge> r+s \<Longrightarrow> ball x r \<inter> ball y s = {}"
- using dist_triangle_less_add not_le by fastforce
-
-lemma disjoint_cballI: "dist x y > r + s \<Longrightarrow> cball x r \<inter> cball y s = {}"
- by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball)
-
-lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
- for x :: "'a::real_normed_vector"
- by (simp add: dist_norm)
-
-lemma sphere_empty [simp]: "r < 0 \<Longrightarrow> sphere a r = {}"
- for a :: "'a::metric_space"
- by auto
-
-lemma centre_in_ball [simp]: "x \<in> ball x e \<longleftrightarrow> 0 < e"
- by simp
-
-lemma centre_in_cball [simp]: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
- by simp
-
-lemma ball_subset_cball [simp, intro]: "ball x e \<subseteq> cball x e"
- by (simp add: subset_eq)
-
-lemma mem_ball_imp_mem_cball: "x \<in> ball y e \<Longrightarrow> x \<in> cball y e"
- by (auto simp: mem_ball mem_cball)
-
-lemma sphere_cball [simp,intro]: "sphere z r \<subseteq> cball z r"
- by force
-
-lemma cball_diff_sphere: "cball a r - sphere a r = ball a r"
- by auto
-
-lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
- by (simp add: subset_eq)
-
-lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
- by (simp add: subset_eq)
-
-lemma mem_ball_leI: "x \<in> ball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> ball y f"
- by (auto simp: mem_ball mem_cball)
-
-lemma mem_cball_leI: "x \<in> cball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> cball y f"
- by (auto simp: mem_ball mem_cball)
-
-lemma cball_trans: "y \<in> cball z b \<Longrightarrow> x \<in> cball y a \<Longrightarrow> x \<in> cball z (b + a)"
- unfolding mem_cball
-proof -
- have "dist z x \<le> dist z y + dist y x"
- by (rule dist_triangle)
- also assume "dist z y \<le> b"
- also assume "dist y x \<le> a"
- finally show "dist z x \<le> b + a" by arith
-qed
-
-lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
- by (simp add: set_eq_iff) arith
-
-lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
- by (simp add: set_eq_iff)
-
-lemma cball_max_Un: "cball a (max r s) = cball a r \<union> cball a s"
- by (simp add: set_eq_iff) arith
-
-lemma cball_min_Int: "cball a (min r s) = cball a r \<inter> cball a s"
- by (simp add: set_eq_iff)
-
-lemma cball_diff_eq_sphere: "cball a r - ball a r = sphere a r"
- by (auto simp: cball_def ball_def dist_commute)
-
-lemma image_add_ball [simp]:
- fixes a :: "'a::real_normed_vector"
- shows "(+) b ` ball a r = ball (a+b) r"
-apply (intro equalityI subsetI)
-apply (force simp: dist_norm)
-apply (rule_tac x="x-b" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma image_add_cball [simp]:
- fixes a :: "'a::real_normed_vector"
- shows "(+) b ` cball a r = cball (a+b) r"
-apply (intro equalityI subsetI)
-apply (force simp: dist_norm)
-apply (rule_tac x="x-b" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma open_ball [intro, simp]: "open (ball x e)"
-proof -
- have "open (dist x -` {..<e})"
- by (intro open_vimage open_lessThan continuous_intros)
- also have "dist x -` {..<e} = ball x e"
- by auto
- finally show ?thesis .
-qed
-
-lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
- by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute)
-
-lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S) \<Longrightarrow> open S"
- by (auto simp: open_contains_ball)
-
-lemma openE[elim?]:
- assumes "open S" "x\<in>S"
- obtains e where "e>0" "ball x e \<subseteq> S"
- using assms unfolding open_contains_ball by auto
-
-lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
- by (metis open_contains_ball subset_eq centre_in_ball)
-
-lemma openin_contains_ball:
- "openin (subtopology euclidean t) s \<longleftrightarrow>
- s \<subseteq> t \<and> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> ball x e \<inter> t \<subseteq> s)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (simp add: openin_open)
- apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
- done
-next
- assume ?rhs
- then show ?lhs
- apply (simp add: openin_euclidean_subtopology_iff)
- by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball)
-qed
-
-lemma openin_contains_cball:
- "openin (subtopology euclidean t) s \<longleftrightarrow>
- s \<subseteq> t \<and>
- (\<forall>x \<in> s. \<exists>e. 0 < e \<and> cball x e \<inter> t \<subseteq> s)"
-apply (simp add: openin_contains_ball)
-apply (rule iffI)
-apply (auto dest!: bspec)
-apply (rule_tac x="e/2" in exI, force+)
-done
-
-lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
- unfolding mem_ball set_eq_iff
- apply (simp add: not_less)
- apply (metis zero_le_dist order_trans dist_self)
- done
-
-lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
-
-lemma closed_cball [iff]: "closed (cball x e)"
-proof -
- have "closed (dist x -` {..e})"
- by (intro closed_vimage closed_atMost continuous_intros)
- also have "dist x -` {..e} = cball x e"
- by auto
- finally show ?thesis .
-qed
-
-lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
-proof -
- {
- fix x and e::real
- assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
- then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
- }
- moreover
- {
- fix x and e::real
- assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
- then have "\<exists>d>0. ball x d \<subseteq> S"
- unfolding subset_eq
- apply (rule_tac x="e/2" in exI, auto)
- done
- }
- ultimately show ?thesis
- unfolding open_contains_ball by auto
-qed
-
-lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
- by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
-
-lemma eventually_nhds_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z d) (nhds z)"
- by (rule eventually_nhds_in_open) simp_all
-
-lemma eventually_at_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<in> A) (at z within A)"
- unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
-
-lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
- unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
-
-lemma at_within_ball: "e > 0 \<Longrightarrow> dist x y < e \<Longrightarrow> at y within ball x e = at y"
- by (subst at_within_open) auto
-
-lemma atLeastAtMost_eq_cball:
- fixes a b::real
- shows "{a .. b} = cball ((a + b)/2) ((b - a)/2)"
- by (auto simp: dist_real_def field_simps mem_cball)
-
-lemma greaterThanLessThan_eq_ball:
- fixes a b::real
- shows "{a <..< b} = ball ((a + b)/2) ((b - a)/2)"
- by (auto simp: dist_real_def field_simps mem_ball)
-
-
-subsection \<open>Limit points\<close>
+subsection \<open>Limit Points\<close>
definition%important (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60)
where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
@@ -1468,18 +506,6 @@
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
unfolding islimpt_def by fast
-lemma islimpt_approachable:
- fixes x :: "'a::metric_space"
- shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
- unfolding islimpt_iff_eventually eventually_at by fast
-
-lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
- for x :: "'a::metric_space"
- unfolding islimpt_approachable
- using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
- THEN arg_cong [where f=Not]]
- by (simp add: Bex_def conj_commute conj_left_commute)
-
lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
@@ -1492,10 +518,6 @@
for x :: "'a::perfect_space"
unfolding islimpt_UNIV_iff by (rule not_open_singleton)
-lemma perfect_choose_dist: "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
- for x :: "'a::{perfect_space,metric_space}"
- using islimpt_UNIV [of x] by (simp add: islimpt_approachable)
-
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
unfolding closed_def
apply (subst open_subopen)
@@ -1506,95 +528,160 @@
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
by (auto simp: islimpt_def)
-lemma finite_ball_include:
- fixes a :: "'a::metric_space"
- assumes "finite S"
- shows "\<exists>e>0. S \<subseteq> ball a e"
- using assms
-proof induction
- case (insert x S)
- then obtain e0 where "e0>0" and e0:"S \<subseteq> ball a e0" by auto
- define e where "e = max e0 (2 * dist a x)"
- have "e>0" unfolding e_def using \<open>e0>0\<close> by auto
- moreover have "insert x S \<subseteq> ball a e"
- using e0 \<open>e>0\<close> unfolding e_def by auto
- ultimately show ?case by auto
-qed (auto intro: zero_less_one)
-
-lemma finite_set_avoid:
- fixes a :: "'a::metric_space"
- assumes "finite S"
- shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
- using assms
-proof induction
- case (insert x S)
- then obtain d where "d > 0" and d: "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
- by blast
- show ?case
- proof (cases "x = a")
- case True
- with \<open>d > 0 \<close>d show ?thesis by auto
- next
- case False
- let ?d = "min d (dist a x)"
- from False \<open>d > 0\<close> have dp: "?d > 0"
- by auto
- from d have d': "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> ?d \<le> dist a x"
- by auto
- with dp False show ?thesis
- by (metis insert_iff le_less min_less_iff_conj not_less)
- qed
-qed (auto intro: zero_less_one)
-
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
by (simp add: islimpt_iff_eventually eventually_conj_iff)
-lemma discrete_imp_closed:
- fixes S :: "'a::metric_space set"
- assumes e: "0 < e"
- and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
- shows "closed S"
-proof -
- have False if C: "\<And>e. e>0 \<Longrightarrow> \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" for x
- proof -
- from e have e2: "e/2 > 0" by arith
- from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
- by blast
- let ?m = "min (e/2) (dist x y) "
- from e2 y(2) have mp: "?m > 0"
- by simp
- from C[OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
- by blast
- from z y have "dist z y < e"
- by (intro dist_triangle_lt [where z=x]) simp
- from d[rule_format, OF y(1) z(1) this] y z show ?thesis
- by (auto simp: dist_commute)
+
+lemma islimpt_insert:
+ fixes x :: "'a::t1_space"
+ shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
+proof
+ assume *: "x islimpt (insert a s)"
+ show "x islimpt s"
+ proof (rule islimptI)
+ fix t
+ assume t: "x \<in> t" "open t"
+ show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
+ proof (cases "x = a")
+ case True
+ obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
+ using * t by (rule islimptE)
+ with \<open>x = a\<close> show ?thesis by auto
+ next
+ case False
+ with t have t': "x \<in> t - {a}" "open (t - {a})"
+ by (simp_all add: open_Diff)
+ obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
+ using * t' by (rule islimptE)
+ then show ?thesis by auto
+ qed
qed
- then show ?thesis
- by (metis islimpt_approachable closed_limpt [where 'a='a])
+next
+ assume "x islimpt s"
+ then show "x islimpt (insert a s)"
+ by (rule islimpt_subset) auto
+qed
+
+lemma islimpt_finite:
+ fixes x :: "'a::t1_space"
+ shows "finite s \<Longrightarrow> \<not> x islimpt s"
+ by (induct set: finite) (simp_all add: islimpt_insert)
+
+lemma islimpt_Un_finite:
+ fixes x :: "'a::t1_space"
+ shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
+ by (simp add: islimpt_Un islimpt_finite)
+
+lemma islimpt_eq_acc_point:
+ fixes l :: "'a :: t1_space"
+ shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
+proof (safe intro!: islimptI)
+ fix U
+ assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
+ then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
+ by (auto intro: finite_imp_closed)
+ then show False
+ by (rule islimptE) auto
+next
+ fix T
+ assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
+ then have "infinite (T \<inter> S - {l})"
+ by auto
+ then have "\<exists>x. x \<in> (T \<inter> S - {l})"
+ unfolding ex_in_conv by (intro notI) simp
+ then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
+ by auto
qed
-lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
- by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
-
-lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
- by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
-
-lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
- unfolding Nats_def by (rule closed_of_nat_image)
+lemma acc_point_range_imp_convergent_subsequence:
+ fixes l :: "'a :: first_countable_topology"
+ assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
+ shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+proof -
+ from countable_basis_at_decseq[of l]
+ obtain A where A:
+ "\<And>i. open (A i)"
+ "\<And>i. l \<in> A i"
+ "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by blast
+ define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
+ {
+ fix n i
+ have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
+ using l A by auto
+ then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
+ unfolding ex_in_conv by (intro notI) simp
+ then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
+ by auto
+ then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
+ by (auto simp: not_le)
+ then have "i < s n i" "f (s n i) \<in> A (Suc n)"
+ unfolding s_def by (auto intro: someI2_ex)
+ }
+ note s = this
+ define r where "r = rec_nat (s 0 0) s"
+ have "strict_mono r"
+ by (auto simp: r_def s strict_mono_Suc_iff)
+ moreover
+ have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
+ proof (rule topological_tendstoI)
+ fix S
+ assume "open S" "l \<in> S"
+ with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by auto
+ moreover
+ {
+ fix i
+ assume "Suc 0 \<le> i"
+ then have "f (r i) \<in> A i"
+ by (cases i) (simp_all add: r_def s)
+ }
+ then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
+ by (auto simp: eventually_sequentially)
+ ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
+ by eventually_elim auto
+ qed
+ ultimately show "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ by (auto simp: convergent_def comp_def)
+qed
-lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
- unfolding Ints_def by (rule closed_of_int_image)
+lemma islimpt_range_imp_convergent_subsequence:
+ fixes l :: "'a :: {t1_space, first_countable_topology}"
+ assumes l: "l islimpt (range f)"
+ shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ using l unfolding islimpt_eq_acc_point
+ by (rule acc_point_range_imp_convergent_subsequence)
-lemma closed_subset_Ints:
- fixes A :: "'a :: real_normed_algebra_1 set"
- assumes "A \<subseteq> \<int>"
- shows "closed A"
-proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
- case (1 x y)
- with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto
- with \<open>dist y x < 1\<close> show "y = x"
- by (auto elim!: Ints_cases simp: dist_of_int)
+lemma sequence_unique_limpt:
+ fixes f :: "nat \<Rightarrow> 'a::t2_space"
+ assumes "(f \<longlongrightarrow> l) sequentially"
+ and "l' islimpt (range f)"
+ shows "l' = l"
+proof (rule ccontr)
+ assume "l' \<noteq> l"
+ obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
+ using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
+ have "eventually (\<lambda>n. f n \<in> t) sequentially"
+ using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
+ then obtain N where "\<forall>n\<ge>N. f n \<in> t"
+ unfolding eventually_sequentially by auto
+
+ have "UNIV = {..<N} \<union> {N..}"
+ by auto
+ then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
+ using assms(2) by simp
+ then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
+ by (simp add: image_Un)
+ then have "l' islimpt (f ` {N..})"
+ by (simp add: islimpt_Un_finite)
+ then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
+ using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
+ then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
+ by auto
+ with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
+ by simp
+ with \<open>s \<inter> t = {}\<close> show False
+ by simp
qed
@@ -1659,10 +746,6 @@
by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
Int_lower2 interior_maximal interior_subset open_Int open_interior)
-lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
- using open_contains_ball_eq [where S="interior S"]
- by (simp add: open_subset_interior)
-
lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
using interior_subset[of s] by (subst eventually_nhds) blast
@@ -1874,159 +957,12 @@
done
qed
-lemma closure_openin_Int_closure:
- assumes ope: "openin (subtopology euclidean U) S" and "T \<subseteq> U"
- shows "closure(S \<inter> closure T) = closure(S \<inter> T)"
-proof
- obtain V where "open V" and S: "S = U \<inter> V"
- using ope using openin_open by metis
- show "closure (S \<inter> closure T) \<subseteq> closure (S \<inter> T)"
- proof (clarsimp simp: S)
- fix x
- assume "x \<in> closure (U \<inter> V \<inter> closure T)"
- then have "V \<inter> closure T \<subseteq> A \<Longrightarrow> x \<in> closure A" for A
- by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
- then have "x \<in> closure (T \<inter> V)"
- by (metis \<open>open V\<close> closure_closure inf_commute open_Int_closure_subset)
- then show "x \<in> closure (U \<inter> V \<inter> T)"
- by (metis \<open>T \<subseteq> U\<close> inf.absorb_iff2 inf_assoc inf_commute)
- qed
-next
- show "closure (S \<inter> T) \<subseteq> closure (S \<inter> closure T)"
- by (meson Int_mono closure_mono closure_subset order_refl)
-qed
-
lemma islimpt_in_closure: "(x islimpt S) = (x\<in>closure(S-{x}))"
unfolding closure_def using islimpt_punctured by blast
lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
-lemma limpt_of_limpts: "x islimpt {y. y islimpt S} \<Longrightarrow> x islimpt S"
- for x :: "'a::metric_space"
- apply (clarsimp simp add: islimpt_approachable)
- apply (drule_tac x="e/2" in spec)
- apply (auto simp: simp del: less_divide_eq_numeral1)
- apply (drule_tac x="dist x' x" in spec)
- apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
- apply (erule rev_bexI)
- apply (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
- done
-
-lemma closed_limpts: "closed {x::'a::metric_space. x islimpt S}"
- using closed_limpt limpt_of_limpts by blast
-
-lemma limpt_of_closure: "x islimpt closure S \<longleftrightarrow> x islimpt S"
- for x :: "'a::metric_space"
- by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
-
-lemma closedin_limpt:
- "closedin (subtopology euclidean T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
- apply (simp add: closedin_closed, safe)
- apply (simp add: closed_limpt islimpt_subset)
- apply (rule_tac x="closure S" in exI, simp)
- apply (force simp: closure_def)
- done
-
-lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
- by (meson closedin_limpt closed_subset closedin_closed_trans)
-
-lemma connected_closed_set:
- "closed S
- \<Longrightarrow> connected S \<longleftrightarrow> (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"
- unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast
-
-text \<open>If a connnected set is written as the union of two nonempty closed sets, then these sets
-have to intersect.\<close>
-
-lemma connected_as_closed_union:
- assumes "connected C" "C = A \<union> B" "closed A" "closed B" "A \<noteq> {}" "B \<noteq> {}"
- shows "A \<inter> B \<noteq> {}"
-by (metis assms closed_Un connected_closed_set)
-
-lemma closedin_subset_trans:
- "closedin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
- closedin (subtopology euclidean T) S"
- by (meson closedin_limpt subset_iff)
-
-lemma openin_subset_trans:
- "openin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
- openin (subtopology euclidean T) S"
- by (auto simp: openin_open)
-
-lemma openin_Times:
- "openin (subtopology euclidean S) S' \<Longrightarrow> openin (subtopology euclidean T) T' \<Longrightarrow>
- openin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
- unfolding openin_open using open_Times by blast
-
-lemma Times_in_interior_subtopology:
- fixes U :: "('a::metric_space \<times> 'b::metric_space) set"
- assumes "(x, y) \<in> U" "openin (subtopology euclidean (S \<times> T)) U"
- obtains V W where "openin (subtopology euclidean S) V" "x \<in> V"
- "openin (subtopology euclidean T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
-proof -
- from assms obtain e where "e > 0" and "U \<subseteq> S \<times> T"
- and e: "\<And>x' y'. \<lbrakk>x'\<in>S; y'\<in>T; dist (x', y') (x, y) < e\<rbrakk> \<Longrightarrow> (x', y') \<in> U"
- by (force simp: openin_euclidean_subtopology_iff)
- with assms have "x \<in> S" "y \<in> T"
- by auto
- show ?thesis
- proof
- show "openin (subtopology euclidean S) (ball x (e/2) \<inter> S)"
- by (simp add: Int_commute openin_open_Int)
- show "x \<in> ball x (e / 2) \<inter> S"
- by (simp add: \<open>0 < e\<close> \<open>x \<in> S\<close>)
- show "openin (subtopology euclidean T) (ball y (e/2) \<inter> T)"
- by (simp add: Int_commute openin_open_Int)
- show "y \<in> ball y (e / 2) \<inter> T"
- by (simp add: \<open>0 < e\<close> \<open>y \<in> T\<close>)
- show "(ball x (e / 2) \<inter> S) \<times> (ball y (e / 2) \<inter> T) \<subseteq> U"
- by clarify (simp add: e dist_Pair_Pair \<open>0 < e\<close> dist_commute sqrt_sum_squares_half_less)
- qed
-qed
-
-lemma openin_Times_eq:
- fixes S :: "'a::metric_space set" and T :: "'b::metric_space set"
- shows
- "openin (subtopology euclidean (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
- S' = {} \<or> T' = {} \<or> openin (subtopology euclidean S) S' \<and> openin (subtopology euclidean T) T'"
- (is "?lhs = ?rhs")
-proof (cases "S' = {} \<or> T' = {}")
- case True
- then show ?thesis by auto
-next
- case False
- then obtain x y where "x \<in> S'" "y \<in> T'"
- by blast
- show ?thesis
- proof
- assume ?lhs
- have "openin (subtopology euclidean S) S'"
- apply (subst openin_subopen, clarify)
- apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
- using \<open>y \<in> T'\<close>
- apply auto
- done
- moreover have "openin (subtopology euclidean T) T'"
- apply (subst openin_subopen, clarify)
- apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
- using \<open>x \<in> S'\<close>
- apply auto
- done
- ultimately show ?rhs
- by simp
- next
- assume ?rhs
- with False show ?lhs
- by (simp add: openin_Times)
- qed
-qed
-
-lemma closedin_Times:
- "closedin (subtopology euclidean S) S' \<Longrightarrow> closedin (subtopology euclidean T) T' \<Longrightarrow>
- closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
- unfolding closedin_closed using closed_Times by blast
-
lemma bdd_below_closure:
fixes A :: "real set"
assumes "bdd_below A"
@@ -2075,12 +1011,6 @@
by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
qed
-lemma frontier_straddle:
- fixes a :: "'a::metric_space"
- 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))"
- unfolding frontier_def closure_interior
- by (auto simp: mem_interior subset_eq ball_def)
-
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
by (metis frontier_def closure_closed Diff_subset)
@@ -2118,14 +1048,6 @@
lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
-lemma connected_Int_frontier:
- "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
- apply (simp add: frontier_interiors connected_openin, safe)
- apply (drule_tac x="s \<inter> interior t" in spec, safe)
- apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
- apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
- done
-
lemma closure_Un_frontier: "closure S = S \<union> frontier S"
proof -
have "S \<union> interior S = S"
@@ -2137,9 +1059,6 @@
subsection%unimportant \<open>Filters and the ``eventually true'' quantifier\<close>
-definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter" (infixr "indirection" 70)
- where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
-
text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
@@ -2167,16 +1086,6 @@
for a :: "'a::perfect_space"
by (rule at_neq_bot)
-lemma trivial_limit_at_infinity:
- "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
- unfolding trivial_limit_def eventually_at_infinity
- apply clarsimp
- apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
- apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
- apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
- apply (drule_tac x=UNIV in spec, simp)
- done
-
lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
using islimpt_in_closure by (metis trivial_limit_within)
@@ -2203,82 +1112,9 @@
subsection \<open>Limits\<close>
-proposition Lim: "(f \<longlongrightarrow> l) net \<longleftrightarrow> trivial_limit net \<or> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
- by (auto simp: tendsto_iff trivial_limit_eq)
-
-text \<open>Show that they yield usual definitions in the various cases.\<close>
-
-proposition Lim_within_le: "(f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow>
- (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
- by (auto simp: tendsto_iff eventually_at_le)
-
-proposition Lim_within: "(f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow>
- (\<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)"
- by (auto simp: tendsto_iff eventually_at)
-
-corollary Lim_withinI [intro?]:
- assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e"
- shows "(f \<longlongrightarrow> l) (at a within S)"
- apply (simp add: Lim_within, clarify)
- apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
- done
-
-proposition Lim_at: "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow>
- (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
- by (auto simp: tendsto_iff eventually_at)
-
-proposition Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
- by (auto simp: tendsto_iff eventually_at_infinity)
-
-corollary Lim_at_infinityI [intro?]:
- assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
- shows "(f \<longlongrightarrow> l) at_infinity"
- apply (simp add: Lim_at_infinity, clarify)
- apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
- done
-
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f \<longlongrightarrow> l) net"
by (rule topological_tendstoI) (auto elim: eventually_mono)
-lemma Lim_transform_within_set:
- fixes a :: "'a::metric_space" and l :: "'b::metric_space"
- shows "\<lbrakk>(f \<longlongrightarrow> l) (at a within S); eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)\<rbrakk>
- \<Longrightarrow> (f \<longlongrightarrow> l) (at a within T)"
-apply (clarsimp simp: eventually_at Lim_within)
-apply (drule_tac x=e in spec, clarify)
-apply (rename_tac k)
-apply (rule_tac x="min d k" in exI, simp)
-done
-
-lemma Lim_transform_within_set_eq:
- fixes a l :: "'a::real_normed_vector"
- shows "eventually (\<lambda>x. x \<in> s \<longleftrightarrow> x \<in> t) (at a)
- \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within s) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within t))"
- by (force intro: Lim_transform_within_set elim: eventually_mono)
-
-lemma Lim_transform_within_openin:
- fixes a :: "'a::metric_space"
- assumes f: "(f \<longlongrightarrow> l) (at a within T)"
- and "openin (subtopology euclidean T) S" "a \<in> S"
- and eq: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
- shows "(g \<longlongrightarrow> l) (at a within T)"
-proof -
- obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball a \<epsilon> \<inter> T \<subseteq> S"
- using assms by (force simp: openin_contains_ball)
- then have "a \<in> ball a \<epsilon>"
- by simp
- show ?thesis
- by (rule Lim_transform_within [OF f \<open>0 < \<epsilon>\<close> eq]) (use \<epsilon> in \<open>auto simp: dist_commute subset_iff\<close>)
-qed
-
-lemma continuous_transform_within_openin:
- fixes a :: "'a::metric_space"
- assumes "continuous (at a within T) f"
- and "openin (subtopology euclidean T) S" "a \<in> S"
- and eq: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
- shows "continuous (at a within T) g"
- using assms by (simp add: Lim_transform_within_openin continuous_within)
-
text \<open>The expected monotonicity property.\<close>
lemma Lim_Un:
@@ -2364,73 +1200,6 @@
qed
qed
-text \<open>Another limit point characterization.\<close>
-
-lemma limpt_sequential_inj:
- fixes x :: "'a::metric_space"
- shows "x islimpt S \<longleftrightarrow>
- (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> inj f \<and> (f \<longlongrightarrow> x) sequentially)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then have "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
- by (force simp: islimpt_approachable)
- then obtain y where y: "\<And>e. e>0 \<Longrightarrow> y e \<in> S \<and> y e \<noteq> x \<and> dist (y e) x < e"
- by metis
- define f where "f \<equiv> rec_nat (y 1) (\<lambda>n fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))"
- have [simp]: "f 0 = y 1"
- "f(Suc n) = y (min (inverse(2 ^ (Suc n))) (dist (f n) x))" for n
- by (simp_all add: f_def)
- have f: "f n \<in> S \<and> (f n \<noteq> x) \<and> dist (f n) x < inverse(2 ^ n)" for n
- proof (induction n)
- case 0 show ?case
- by (simp add: y)
- next
- case (Suc n) then show ?case
- apply (auto simp: y)
- by (metis half_gt_zero_iff inverse_positive_iff_positive less_divide_eq_numeral1(1) min_less_iff_conj y zero_less_dist_iff zero_less_numeral zero_less_power)
- qed
- show ?rhs
- proof (rule_tac x=f in exI, intro conjI allI)
- show "\<And>n. f n \<in> S - {x}"
- using f by blast
- have "dist (f n) x < dist (f m) x" if "m < n" for m n
- using that
- proof (induction n)
- case 0 then show ?case by simp
- next
- case (Suc n)
- then consider "m < n" | "m = n" using less_Suc_eq by blast
- then show ?case
- proof cases
- assume "m < n"
- have "dist (f(Suc n)) x = dist (y (min (inverse(2 ^ (Suc n))) (dist (f n) x))) x"
- by simp
- also have "\<dots> < dist (f n) x"
- by (metis dist_pos_lt f min.strict_order_iff min_less_iff_conj y)
- also have "\<dots> < dist (f m) x"
- using Suc.IH \<open>m < n\<close> by blast
- finally show ?thesis .
- next
- assume "m = n" then show ?case
- by simp (metis dist_pos_lt f half_gt_zero_iff inverse_positive_iff_positive min_less_iff_conj y zero_less_numeral zero_less_power)
- qed
- qed
- then show "inj f"
- by (metis less_irrefl linorder_injI)
- show "f \<longlonglongrightarrow> x"
- apply (rule tendstoI)
- apply (rule_tac c="nat (ceiling(1/e))" in eventually_sequentiallyI)
- apply (rule less_trans [OF f [THEN conjunct2, THEN conjunct2]])
- apply (simp add: field_simps)
- by (meson le_less_trans mult_less_cancel_left not_le of_nat_less_two_power)
- qed
-next
- assume ?rhs
- then show ?lhs
- by (fastforce simp add: islimpt_approachable lim_sequentially)
-qed
-
(*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
lemma islimpt_sequential:
fixes x :: "'a::first_countable_topology"
@@ -2477,83 +1246,6 @@
qed
qed
-lemma Lim_null:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
- by (simp add: Lim dist_norm)
-
-lemma Lim_null_comparison:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
- shows "(f \<longlongrightarrow> 0) net"
- using assms(2)
-proof (rule metric_tendsto_imp_tendsto)
- show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
- using assms(1) by (rule eventually_mono) (simp add: dist_norm)
-qed
-
-lemma Lim_transform_bound:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- and g :: "'a \<Rightarrow> 'c::real_normed_vector"
- assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
- and "(g \<longlongrightarrow> 0) net"
- shows "(f \<longlongrightarrow> 0) net"
- using assms(1) tendsto_norm_zero [OF assms(2)]
- by (rule Lim_null_comparison)
-
-lemma lim_null_mult_right_bounded:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
- assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
- shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
-proof -
- have *: "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
- by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
- have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
- apply (rule Lim_null_comparison [OF _ *])
- apply (simp add: eventually_mono [OF g] mult_left_mono)
- done
- then show ?thesis
- by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
-qed
-
-lemma lim_null_mult_left_bounded:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
- assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
- shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
-proof -
- have *: "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
- by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
- have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
- apply (rule Lim_null_comparison [OF _ *])
- apply (simp add: eventually_mono [OF g] mult_right_mono)
- done
- then show ?thesis
- by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
-qed
-
-lemma lim_null_scaleR_bounded:
- assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
- shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
-proof
- fix \<epsilon>::real
- assume "0 < \<epsilon>"
- then have B: "0 < \<epsilon> / (abs B + 1)" by simp
- have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
- proof -
- have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
- by (simp add: mult_left_mono g)
- also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
- by (simp add: mult_left_mono)
- also have "\<dots> < \<epsilon>"
- by (rule f)
- finally show ?thesis .
- qed
- show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
- apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
- apply (auto simp: \<open>0 < \<epsilon>\<close> divide_simps * split: if_split_asm)
- done
-qed
-
text\<open>Deducing things about the limit from the elements.\<close>
lemma Lim_in_closed_set:
@@ -2573,40 +1265,7 @@
by (simp add: eventually_False)
qed
-text\<open>Need to prove closed(cball(x,e)) before deducing this as a corollary.\<close>
-
-lemma Lim_dist_ubound:
- assumes "\<not>(trivial_limit net)"
- and "(f \<longlongrightarrow> l) net"
- and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
- shows "dist a l \<le> e"
- using assms by (fast intro: tendsto_le tendsto_intros)
-
-lemma Lim_norm_ubound:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
- shows "norm(l) \<le> e"
- using assms by (fast intro: tendsto_le tendsto_intros)
-
-lemma Lim_norm_lbound:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "\<not> trivial_limit net"
- and "(f \<longlongrightarrow> l) net"
- and "eventually (\<lambda>x. e \<le> norm (f x)) net"
- shows "e \<le> norm l"
- using assms by (fast intro: tendsto_le tendsto_intros)
-
-text\<open>Limit under bilinear function\<close>
-
-lemma Lim_bilinear:
- assumes "(f \<longlongrightarrow> l) net"
- and "(g \<longlongrightarrow> m) net"
- and "bounded_bilinear h"
- shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
- using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
- by (rule bounded_bilinear.tendsto)
-
-text\<open>These are special for limits out of the same vector space.\<close>
+text\<open>These are special for limits out of the same topological space.\<close>
lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
unfolding id_def by (rule tendsto_ident_at)
@@ -2614,12 +1273,6 @@
lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
unfolding id_def by (rule tendsto_ident_at)
-lemma Lim_at_zero:
- fixes a :: "'a::real_normed_vector"
- and l :: "'b::topological_space"
- shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
- using LIM_offset_zero LIM_offset_zero_cancel ..
-
text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
@@ -2643,22 +1296,6 @@
shows "netlimit (at x within S) = x"
using assms by (metis at_within_interior netlimit_at)
-lemma netlimit_at_vector:
- fixes a :: "'a::real_normed_vector"
- shows "netlimit (at a) = a"
-proof (cases "\<exists>x. x \<noteq> a")
- case True then obtain x where x: "x \<noteq> a" ..
- have "\<not> trivial_limit (at a)"
- unfolding trivial_limit_def eventually_at dist_norm
- apply clarsimp
- apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
- apply (simp add: norm_sgn sgn_zero_iff x)
- done
- then show ?thesis
- by (rule netlimit_within [of a UNIV])
-qed simp
-
-
text\<open>Useful lemmas on closure and set of possible sequential limits.\<close>
lemma closure_sequential:
@@ -2689,111 +1326,6 @@
shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
by (metis closure_sequential closure_subset_eq subset_iff)
-lemma closure_approachable:
- fixes S :: "'a::metric_space set"
- shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
- apply (auto simp: closure_def islimpt_approachable)
- apply (metis dist_self)
- done
-
-lemma closure_approachable_le:
- fixes S :: "'a::metric_space set"
- shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x \<le> e)"
- unfolding closure_approachable
- using dense by force
-
-lemma closure_approachableD:
- assumes "x \<in> closure S" "e>0"
- shows "\<exists>y\<in>S. dist x y < e"
- using assms unfolding closure_approachable by (auto simp: dist_commute)
-
-lemma closed_approachable:
- fixes S :: "'a::metric_space set"
- shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
- by (metis closure_closed closure_approachable)
-
-lemma closure_contains_Inf:
- fixes S :: "real set"
- assumes "S \<noteq> {}" "bdd_below S"
- shows "Inf S \<in> closure S"
-proof -
- have *: "\<forall>x\<in>S. Inf S \<le> x"
- using cInf_lower[of _ S] assms by metis
- {
- fix e :: real
- assume "e > 0"
- then have "Inf S < Inf S + e" by simp
- with assms obtain x where "x \<in> S" "x < Inf S + e"
- by (subst (asm) cInf_less_iff) auto
- with * have "\<exists>x\<in>S. dist x (Inf S) < e"
- by (intro bexI[of _ x]) (auto simp: dist_real_def)
- }
- then show ?thesis unfolding closure_approachable by auto
-qed
-
-lemma closure_Int_ballI:
- fixes S :: "'a :: metric_space set"
- assumes "\<And>U. \<lbrakk>openin (subtopology euclidean S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
- shows "S \<subseteq> closure T"
-proof (clarsimp simp: closure_approachable dist_commute)
- fix x and e::real
- assume "x \<in> S" "0 < e"
- with assms [of "S \<inter> ball x e"] show "\<exists>y\<in>T. dist x y < e"
- by force
-qed
-
-lemma closed_contains_Inf:
- fixes S :: "real set"
- shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"
- by (metis closure_contains_Inf closure_closed)
-
-lemma closed_subset_contains_Inf:
- fixes A C :: "real set"
- shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<in> C"
- by (metis closure_contains_Inf closure_minimal subset_eq)
-
-lemma atLeastAtMost_subset_contains_Inf:
- fixes A :: "real set" and a b :: real
- shows "A \<noteq> {} \<Longrightarrow> a \<le> b \<Longrightarrow> A \<subseteq> {a..b} \<Longrightarrow> Inf A \<in> {a..b}"
- by (rule closed_subset_contains_Inf)
- (auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a])
-
-lemma not_trivial_limit_within_ball:
- "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof
- show ?rhs if ?lhs
- proof -
- {
- fix e :: real
- assume "e > 0"
- then obtain y where "y \<in> S - {x}" and "dist y x < e"
- using \<open>?lhs\<close> not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
- by auto
- then have "y \<in> S \<inter> ball x e - {x}"
- unfolding ball_def by (simp add: dist_commute)
- then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
- }
- then show ?thesis by auto
- qed
- show ?lhs if ?rhs
- proof -
- {
- fix e :: real
- assume "e > 0"
- then obtain y where "y \<in> S \<inter> ball x e - {x}"
- using \<open>?rhs\<close> by blast
- then have "y \<in> S - {x}" and "dist y x < e"
- unfolding ball_def by (simp_all add: dist_commute)
- then have "\<exists>y \<in> S - {x}. dist y x < e"
- by auto
- }
- then show ?thesis
- using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
- by auto
- qed
-qed
-
lemma tendsto_If_within_closures:
assumes f: "x \<in> s \<union> (closure s \<inter> closure t) \<Longrightarrow>
(f \<longlongrightarrow> l x) (at x within s \<union> (closure s \<inter> closure t))"
@@ -2817,275 +1349,6 @@
qed
-subsection \<open>Boundedness\<close>
-
- (* FIXME: This has to be unified with BSEQ!! *)
-definition%important (in metric_space) bounded :: "'a set \<Rightarrow> bool"
- where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
-
-lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
- unfolding bounded_def subset_eq by auto (meson order_trans zero_le_dist)
-
-lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
- unfolding bounded_def
- by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
-
-lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
- unfolding bounded_any_center [where a=0]
- by (simp add: dist_norm)
-
-lemma bdd_above_norm: "bdd_above (norm ` X) \<longleftrightarrow> bounded X"
- by (simp add: bounded_iff bdd_above_def)
-
-lemma bounded_norm_comp: "bounded ((\<lambda>x. norm (f x)) ` S) = bounded (f ` S)"
- by (simp add: bounded_iff)
-
-lemma boundedI:
- assumes "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
- shows "bounded S"
- using assms bounded_iff by blast
-
-lemma bounded_empty [simp]: "bounded {}"
- by (simp add: bounded_def)
-
-lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"
- by (metis bounded_def subset_eq)
-
-lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"
- by (metis bounded_subset interior_subset)
-
-lemma bounded_closure[intro]:
- assumes "bounded S"
- shows "bounded (closure S)"
-proof -
- from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"
- unfolding bounded_def by auto
- {
- fix y
- assume "y \<in> closure S"
- then obtain f where f: "\<forall>n. f n \<in> S" "(f \<longlongrightarrow> y) sequentially"
- unfolding closure_sequential by auto
- have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
- then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
- by (simp add: f(1))
- have "dist x y \<le> a"
- apply (rule Lim_dist_ubound [of sequentially f])
- apply (rule trivial_limit_sequentially)
- apply (rule f(2))
- apply fact
- done
- }
- then show ?thesis
- unfolding bounded_def by auto
-qed
-
-lemma bounded_closure_image: "bounded (f ` closure S) \<Longrightarrow> bounded (f ` S)"
- by (simp add: bounded_subset closure_subset image_mono)
-
-lemma bounded_cball[simp,intro]: "bounded (cball x e)"
- apply (simp add: bounded_def)
- apply (rule_tac x=x in exI)
- apply (rule_tac x=e in exI, auto)
- done
-
-lemma bounded_ball[simp,intro]: "bounded (ball x e)"
- by (metis ball_subset_cball bounded_cball bounded_subset)
-
-lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
- by (auto simp: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj)
-
-lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"
- by (induct rule: finite_induct[of F]) auto
-
-lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
- by (induct set: finite) auto
-
-lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
-proof -
- have "\<forall>y\<in>{x}. dist x y \<le> 0"
- by simp
- then have "bounded {x}"
- unfolding bounded_def by fast
- then show ?thesis
- by (metis insert_is_Un bounded_Un)
-qed
-
-lemma bounded_subset_ballI: "S \<subseteq> ball x r \<Longrightarrow> bounded S"
- by (meson bounded_ball bounded_subset)
-
-lemma bounded_subset_ballD:
- assumes "bounded S" shows "\<exists>r. 0 < r \<and> S \<subseteq> ball x r"
-proof -
- obtain e::real and y where "S \<subseteq> cball y e" "0 \<le> e"
- using assms by (auto simp: bounded_subset_cball)
- then show ?thesis
- apply (rule_tac x="dist x y + e + 1" in exI)
- apply (simp add: add.commute add_pos_nonneg)
- apply (erule subset_trans)
- apply (clarsimp simp add: cball_def)
- by (metis add_le_cancel_right add_strict_increasing dist_commute dist_triangle_le zero_less_one)
-qed
-
-lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
- by (induct set: finite) simp_all
-
-lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
- apply (simp add: bounded_iff)
- apply (subgoal_tac "\<And>x (y::real). 0 < 1 + \<bar>y\<bar> \<and> (x \<le> y \<longrightarrow> x \<le> 1 + \<bar>y\<bar>)")
- apply metis
- apply arith
- done
-
-lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
- apply (simp add: bounded_pos)
- apply (safe; rule_tac x="b+1" in exI; force)
- done
-
-lemma Bseq_eq_bounded:
- fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
- shows "Bseq f \<longleftrightarrow> bounded (range f)"
- unfolding Bseq_def bounded_pos by auto
-
-lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
- by (metis Int_lower1 Int_lower2 bounded_subset)
-
-lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"
- by (metis Diff_subset bounded_subset)
-
-lemma not_bounded_UNIV[simp]:
- "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
-proof (auto simp: bounded_pos not_le)
- obtain x :: 'a where "x \<noteq> 0"
- using perfect_choose_dist [OF zero_less_one] by fast
- fix b :: real
- assume b: "b >0"
- have b1: "b +1 \<ge> 0"
- using b by simp
- with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
- by (simp add: norm_sgn)
- then show "\<exists>x::'a. b < norm x" ..
-qed
-
-corollary cobounded_imp_unbounded:
- fixes S :: "'a::{real_normed_vector, perfect_space} set"
- shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
- using bounded_Un [of S "-S"] by (simp add: sup_compl_top)
-
-lemma bounded_dist_comp:
- assumes "bounded (f ` S)" "bounded (g ` S)"
- shows "bounded ((\<lambda>x. dist (f x) (g x)) ` S)"
-proof -
- from assms obtain M1 M2 where *: "dist (f x) undefined \<le> M1" "dist undefined (g x) \<le> M2" if "x \<in> S" for x
- by (auto simp: bounded_any_center[of _ undefined] dist_commute)
- have "dist (f x) (g x) \<le> M1 + M2" if "x \<in> S" for x
- using *[OF that]
- by (rule order_trans[OF dist_triangle add_mono])
- then show ?thesis
- by (auto intro!: boundedI)
-qed
-
-lemma bounded_linear_image:
- assumes "bounded S"
- and "bounded_linear f"
- shows "bounded (f ` S)"
-proof -
- from assms(1) obtain b where "b > 0" and b: "\<forall>x\<in>S. norm x \<le> b"
- unfolding bounded_pos by auto
- 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: ac_simps)
- show ?thesis
- unfolding bounded_pos
- proof (intro exI, safe)
- show "norm (f x) \<le> B * b" if "x \<in> S" for x
- by (meson B b less_imp_le mult_left_mono order_trans that)
- qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
-qed
-
-lemma bounded_scaling:
- fixes S :: "'a::real_normed_vector set"
- shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
- apply (rule bounded_linear_image, assumption)
- apply (rule bounded_linear_scaleR_right)
- done
-
-lemma bounded_scaleR_comp:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "bounded (f ` S)"
- shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
- using bounded_scaling[of "f ` S" r] assms
- by (auto simp: image_image)
-
-lemma bounded_translation:
- fixes S :: "'a::real_normed_vector set"
- assumes "bounded S"
- shows "bounded ((\<lambda>x. a + x) ` S)"
-proof -
- from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
- unfolding bounded_pos by auto
- {
- fix x
- assume "x \<in> S"
- then have "norm (a + x) \<le> b + norm a"
- using norm_triangle_ineq[of a x] b by auto
- }
- then show ?thesis
- unfolding bounded_pos
- using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
- by (auto intro!: exI[of _ "b + norm a"])
-qed
-
-lemma bounded_translation_minus:
- fixes S :: "'a::real_normed_vector set"
- shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
-using bounded_translation [of S "-a"] by simp
-
-lemma bounded_uminus [simp]:
- fixes X :: "'a::real_normed_vector set"
- shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
-by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
-
-lemma uminus_bounded_comp [simp]:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "bounded ((\<lambda>x. - f x) ` S) \<longleftrightarrow> bounded (f ` S)"
- using bounded_uminus[of "f ` S"]
- by (auto simp: image_image)
-
-lemma bounded_plus_comp:
- fixes f g::"'a \<Rightarrow> 'b::real_normed_vector"
- assumes "bounded (f ` S)"
- assumes "bounded (g ` S)"
- shows "bounded ((\<lambda>x. f x + g x) ` S)"
-proof -
- {
- fix B C
- assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B" "\<And>x. x\<in>S \<Longrightarrow> norm (g x) \<le> C"
- then have "\<And>x. x \<in> S \<Longrightarrow> norm (f x + g x) \<le> B + C"
- by (auto intro!: norm_triangle_le add_mono)
- } then show ?thesis
- using assms by (fastforce simp: bounded_iff)
-qed
-
-lemma bounded_plus:
- fixes S ::"'a::real_normed_vector set"
- assumes "bounded S" "bounded T"
- shows "bounded ((\<lambda>(x,y). x + y) ` (S \<times> T))"
- using bounded_plus_comp [of fst "S \<times> T" snd] assms
- by (auto simp: split_def split: if_split_asm)
-
-lemma bounded_minus_comp:
- "bounded (f ` S) \<Longrightarrow> bounded (g ` S) \<Longrightarrow> bounded ((\<lambda>x. f x - g x) ` S)"
- for f g::"'a \<Rightarrow> 'b::real_normed_vector"
- using bounded_plus_comp[of "f" S "\<lambda>x. - g x"]
- by auto
-
-lemma bounded_minus:
- fixes S ::"'a::real_normed_vector set"
- assumes "bounded S" "bounded T"
- shows "bounded ((\<lambda>(x,y). x - y) ` (S \<times> T))"
- using bounded_minus_comp [of fst "S \<times> T" snd] assms
- by (auto simp: split_def split: if_split_asm)
-
-
subsection \<open>Compactness\<close>
subsubsection \<open>Bolzano-Weierstrass property\<close>
@@ -3139,58 +1402,6 @@
using g(2) using finite_subset by auto
qed
-lemma acc_point_range_imp_convergent_subsequence:
- fixes l :: "'a :: first_countable_topology"
- assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
- shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
-proof -
- from countable_basis_at_decseq[of l]
- obtain A where A:
- "\<And>i. open (A i)"
- "\<And>i. l \<in> A i"
- "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
- by blast
- define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
- {
- fix n i
- have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
- using l A by auto
- then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
- unfolding ex_in_conv by (intro notI) simp
- then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
- by auto
- then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
- by (auto simp: not_le)
- then have "i < s n i" "f (s n i) \<in> A (Suc n)"
- unfolding s_def by (auto intro: someI2_ex)
- }
- note s = this
- define r where "r = rec_nat (s 0 0) s"
- have "strict_mono r"
- by (auto simp: r_def s strict_mono_Suc_iff)
- moreover
- have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
- proof (rule topological_tendstoI)
- fix S
- assume "open S" "l \<in> S"
- with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
- by auto
- moreover
- {
- fix i
- assume "Suc 0 \<le> i"
- then have "f (r i) \<in> A i"
- by (cases i) (simp_all add: r_def s)
- }
- then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
- by (auto simp: eventually_sequentially)
- ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
- by eventually_elim auto
- qed
- ultimately show "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- by (auto simp: convergent_def comp_def)
-qed
-
lemma sequence_infinite_lemma:
fixes f :: "nat \<Rightarrow> 'a::t1_space"
assumes "\<forall>n. f n \<noteq> l"
@@ -3212,130 +1423,6 @@
by auto
qed
-lemma closure_insert:
- fixes x :: "'a::t1_space"
- shows "closure (insert x s) = insert x (closure s)"
- apply (rule closure_unique)
- apply (rule insert_mono [OF closure_subset])
- apply (rule closed_insert [OF closed_closure])
- apply (simp add: closure_minimal)
- done
-
-lemma islimpt_insert:
- fixes x :: "'a::t1_space"
- shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
-proof
- assume *: "x islimpt (insert a s)"
- show "x islimpt s"
- proof (rule islimptI)
- fix t
- assume t: "x \<in> t" "open t"
- show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
- proof (cases "x = a")
- case True
- obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
- using * t by (rule islimptE)
- with \<open>x = a\<close> show ?thesis by auto
- next
- case False
- with t have t': "x \<in> t - {a}" "open (t - {a})"
- by (simp_all add: open_Diff)
- obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
- using * t' by (rule islimptE)
- then show ?thesis by auto
- qed
- qed
-next
- assume "x islimpt s"
- then show "x islimpt (insert a s)"
- by (rule islimpt_subset) auto
-qed
-
-lemma islimpt_finite:
- fixes x :: "'a::t1_space"
- shows "finite s \<Longrightarrow> \<not> x islimpt s"
- by (induct set: finite) (simp_all add: islimpt_insert)
-
-lemma islimpt_Un_finite:
- fixes x :: "'a::t1_space"
- shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
- by (simp add: islimpt_Un islimpt_finite)
-
-lemma islimpt_eq_acc_point:
- fixes l :: "'a :: t1_space"
- shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
-proof (safe intro!: islimptI)
- fix U
- assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
- then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
- by (auto intro: finite_imp_closed)
- then show False
- by (rule islimptE) auto
-next
- fix T
- assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
- then have "infinite (T \<inter> S - {l})"
- by auto
- then have "\<exists>x. x \<in> (T \<inter> S - {l})"
- unfolding ex_in_conv by (intro notI) simp
- then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
- by auto
-qed
-
-corollary infinite_openin:
- fixes S :: "'a :: t1_space set"
- shows "\<lbrakk>openin (subtopology euclidean U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
- by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
-
-lemma islimpt_range_imp_convergent_subsequence:
- fixes l :: "'a :: {t1_space, first_countable_topology}"
- assumes l: "l islimpt (range f)"
- shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- using l unfolding islimpt_eq_acc_point
- by (rule acc_point_range_imp_convergent_subsequence)
-
-lemma islimpt_eq_infinite_ball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> ball x e))"
- apply (simp add: islimpt_eq_acc_point, safe)
- apply (metis Int_commute open_ball centre_in_ball)
- by (metis open_contains_ball Int_mono finite_subset inf_commute subset_refl)
-
-lemma islimpt_eq_infinite_cball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> cball x e))"
- apply (simp add: islimpt_eq_infinite_ball, safe)
- apply (meson Int_mono ball_subset_cball finite_subset order_refl)
- by (metis open_ball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq)
-
-lemma sequence_unique_limpt:
- fixes f :: "nat \<Rightarrow> 'a::t2_space"
- assumes "(f \<longlongrightarrow> l) sequentially"
- and "l' islimpt (range f)"
- shows "l' = l"
-proof (rule ccontr)
- assume "l' \<noteq> l"
- obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
- using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
- have "eventually (\<lambda>n. f n \<in> t) sequentially"
- using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
- then obtain N where "\<forall>n\<ge>N. f n \<in> t"
- unfolding eventually_sequentially by auto
-
- have "UNIV = {..<N} \<union> {N..}"
- by auto
- then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
- using assms(2) by simp
- then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
- by (simp add: image_Un)
- then have "l' islimpt (f ` {N..})"
- by (simp add: islimpt_Un_finite)
- then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
- using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
- then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
- by auto
- with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
- by simp
- with \<open>s \<inter> t = {}\<close> show False
- by simp
-qed
-
lemma Bolzano_Weierstrass_imp_closed:
fixes s :: "'a::{first_countable_topology,t2_space} set"
assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
@@ -3362,19 +1449,15 @@
unfolding closed_sequential_limits by fast
qed
-lemma compact_imp_bounded:
- assumes "compact U"
- shows "bounded U"
-proof -
- have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"
- using assms by auto
- then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
- by (metis compactE_image)
- from \<open>finite D\<close> have "bounded (\<Union>x\<in>D. ball x 1)"
- by (simp add: bounded_UN)
- then show "bounded U" using \<open>U \<subseteq> (\<Union>x\<in>D. ball x 1)\<close>
- by (rule bounded_subset)
-qed
+lemma closure_insert:
+ fixes x :: "'a::t1_space"
+ shows "closure (insert x s) = insert x (closure s)"
+ apply (rule closure_unique)
+ apply (rule insert_mono [OF closure_subset])
+ apply (rule closed_insert [OF closed_closure])
+ apply (simp add: closure_minimal)
+ done
+
text\<open>In particular, some common special cases.\<close>
@@ -3435,11 +1518,6 @@
shows "open s \<Longrightarrow> open (s - {x})"
by (simp add: open_Diff)
-lemma openin_delete:
- fixes a :: "'a :: t1_space"
- shows "openin (subtopology euclidean u) s
- \<Longrightarrow> openin (subtopology euclidean u) (s - {a})"
-by (metis Int_Diff open_delete openin_open)
text\<open>Compactness expressed with filters\<close>
@@ -3464,11 +1542,6 @@
by (simp add: closure_subset open_Compl)
qed
-corollary closure_Int_ball_not_empty:
- assumes "S \<subseteq> closure T" "x \<in> S" "r > 0"
- shows "T \<inter> ball x r \<noteq> {}"
- using assms centre_in_ball closure_iff_nhds_not_empty by blast
-
lemma compact_filter:
"compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
@@ -3892,702 +1965,8 @@
by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
-subsubsection\<open>Totally bounded\<close>
-
-lemma cauchy_def: "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N \<longrightarrow> dist (s m) (s n) < e)"
- unfolding Cauchy_def by metis
-
-proposition seq_compact_imp_totally_bounded:
- assumes "seq_compact s"
- shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)"
-proof -
- { fix e::real assume "e > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> s \<Longrightarrow> \<not> s \<subseteq> (\<Union>x\<in>k. ball x e)"
- let ?Q = "\<lambda>x n r. r \<in> s \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < e))"
- have "\<exists>x. \<forall>n::nat. ?Q x n (x n)"
- proof (rule dependent_wellorder_choice)
- fix n x assume "\<And>y. y < n \<Longrightarrow> ?Q x y (x y)"
- then have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)"
- using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq)
- then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)"
- unfolding subset_eq by auto
- show "\<exists>r. ?Q x n r"
- using z by auto
- qed simp
- then obtain x where "\<forall>n::nat. x n \<in> s" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"
- by blast
- then obtain l r where "l \<in> s" and r:"strict_mono r" and "((x \<circ> r) \<longlongrightarrow> l) sequentially"
- using assms by (metis seq_compact_def)
- from this(3) have "Cauchy (x \<circ> r)"
- using LIMSEQ_imp_Cauchy by auto
- then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"
- unfolding cauchy_def using \<open>e > 0\<close> by blast
- then have False
- using x[of "r N" "r (N+1)"] r by (auto simp: strict_mono_def) }
- then show ?thesis
- by metis
-qed
-
-subsubsection\<open>Heine-Borel theorem\<close>
-
-proposition seq_compact_imp_Heine_Borel:
- fixes s :: "'a :: metric_space set"
- assumes "seq_compact s"
- shows "compact s"
-proof -
- from seq_compact_imp_totally_bounded[OF \<open>seq_compact s\<close>]
- obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>f e. ball x e)"
- unfolding choice_iff' ..
- define K where "K = (\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
- have "countably_compact s"
- using \<open>seq_compact s\<close> by (rule seq_compact_imp_countably_compact)
- then show "compact s"
- proof (rule countably_compact_imp_compact)
- show "countable K"
- unfolding K_def using f
- by (auto intro: countable_finite countable_subset countable_rat
- intro!: countable_image countable_SIGMA countable_UN)
- show "\<forall>b\<in>K. open b" by (auto simp: K_def)
- next
- fix T x
- assume T: "open T" "x \<in> T" and x: "x \<in> s"
- from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"
- by auto
- then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"
- by auto
- from Rats_dense_in_real[OF \<open>0 < e / 2\<close>] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"
- by auto
- from f[rule_format, of r] \<open>0 < r\<close> \<open>x \<in> s\<close> obtain k where "k \<in> f r" "x \<in> ball k r"
- by auto
- from \<open>r \<in> \<rat>\<close> \<open>0 < r\<close> \<open>k \<in> f r\<close> have "ball k r \<in> K"
- by (auto simp: K_def)
- then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
- proof (rule bexI[rotated], safe)
- fix y
- assume "y \<in> ball k r"
- with \<open>r < e / 2\<close> \<open>x \<in> ball k r\<close> have "dist x y < e"
- by (intro dist_triangle_half_r [of k _ e]) (auto simp: dist_commute)
- with \<open>ball x e \<subseteq> T\<close> show "y \<in> T"
- by auto
- next
- show "x \<in> ball k r" by fact
- qed
- qed
-qed
-
-proposition compact_eq_seq_compact_metric:
- "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
- using compact_imp_seq_compact seq_compact_imp_Heine_Borel by blast
-
-proposition compact_def: \<comment> \<open>this is the definition of compactness in HOL Light\<close>
- "compact (S :: 'a::metric_space set) \<longleftrightarrow>
- (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
- unfolding compact_eq_seq_compact_metric seq_compact_def by auto
-
-subsubsection \<open>Complete the chain of compactness variants\<close>
-
-proposition compact_eq_Bolzano_Weierstrass:
- fixes s :: "'a::metric_space set"
- shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- using Heine_Borel_imp_Bolzano_Weierstrass[of s] by auto
-next
- assume ?rhs
- then show ?lhs
- unfolding compact_eq_seq_compact_metric by (rule Bolzano_Weierstrass_imp_seq_compact)
-qed
-
-proposition Bolzano_Weierstrass_imp_bounded:
- "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
- using compact_imp_bounded unfolding compact_eq_Bolzano_Weierstrass .
-
-
-subsection \<open>Metric spaces with the Heine-Borel property\<close>
-
-text \<open>
- A metric space (or topological vector space) is said to have the
- Heine-Borel property if every closed and bounded subset is compact.
-\<close>
-
-class heine_borel = metric_space +
- assumes bounded_imp_convergent_subsequence:
- "bounded (range f) \<Longrightarrow> \<exists>l r. strict_mono (r::nat\<Rightarrow>nat) \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
-
-proposition bounded_closed_imp_seq_compact:
- fixes s::"'a::heine_borel set"
- assumes "bounded s"
- and "closed s"
- shows "seq_compact s"
-proof (unfold seq_compact_def, clarify)
- fix f :: "nat \<Rightarrow> 'a"
- assume f: "\<forall>n. f n \<in> s"
- with \<open>bounded s\<close> have "bounded (range f)"
- by (auto intro: bounded_subset)
- obtain l r where r: "strict_mono (r :: nat \<Rightarrow> nat)" and l: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using bounded_imp_convergent_subsequence [OF \<open>bounded (range f)\<close>] by auto
- from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"
- by simp
- have "l \<in> s" using \<open>closed s\<close> fr l
- by (rule closed_sequentially)
- show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using \<open>l \<in> s\<close> r l by blast
-qed
-
-lemma compact_eq_bounded_closed:
- fixes s :: "'a::heine_borel set"
- shows "compact s \<longleftrightarrow> bounded s \<and> closed s"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- using compact_imp_closed compact_imp_bounded
- by blast
-next
- assume ?rhs
- then show ?lhs
- using bounded_closed_imp_seq_compact[of s]
- unfolding compact_eq_seq_compact_metric
- by auto
-qed
-
-lemma compact_Inter:
- fixes \<F> :: "'a :: heine_borel set set"
- assumes com: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" and "\<F> \<noteq> {}"
- shows "compact(\<Inter> \<F>)"
- using assms
- by (meson Inf_lower all_not_in_conv bounded_subset closed_Inter compact_eq_bounded_closed)
-
-lemma compact_closure [simp]:
- fixes S :: "'a::heine_borel set"
- shows "compact(closure S) \<longleftrightarrow> bounded S"
-by (meson bounded_closure bounded_subset closed_closure closure_subset compact_eq_bounded_closed)
-
-lemma not_compact_UNIV[simp]:
- fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
- shows "\<not> compact (UNIV::'a set)"
- by (simp add: compact_eq_bounded_closed)
-
-text\<open>Representing sets as the union of a chain of compact sets.\<close>
-lemma closed_Union_compact_subsets:
- fixes S :: "'a::{heine_borel,real_normed_vector} set"
- assumes "closed S"
- obtains F where "\<And>n. compact(F n)" "\<And>n. F n \<subseteq> S" "\<And>n. F n \<subseteq> F(Suc n)"
- "(\<Union>n. F n) = S" "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. K \<subseteq> F n"
-proof
- show "compact (S \<inter> cball 0 (of_nat n))" for n
- using assms compact_eq_bounded_closed by auto
-next
- show "(\<Union>n. S \<inter> cball 0 (real n)) = S"
- by (auto simp: real_arch_simple)
-next
- fix K :: "'a set"
- assume "compact K" "K \<subseteq> S"
- then obtain N where "K \<subseteq> cball 0 N"
- by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
- then show "\<exists>N. \<forall>n\<ge>N. K \<subseteq> S \<inter> cball 0 (real n)"
- by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
-qed auto
-
-instance%important real :: heine_borel
-proof%unimportant
- fix f :: "nat \<Rightarrow> real"
- assume f: "bounded (range f)"
- obtain r :: "nat \<Rightarrow> nat" where r: "strict_mono r" "monoseq (f \<circ> r)"
- unfolding comp_def by (metis seq_monosub)
- then have "Bseq (f \<circ> r)"
- unfolding Bseq_eq_bounded using f by (force intro: bounded_subset)
- with r show "\<exists>l r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
-qed
-
-lemma compact_lemma_general:
- fixes f :: "nat \<Rightarrow> 'a"
- fixes proj::"'a \<Rightarrow> 'b \<Rightarrow> 'c::heine_borel" (infixl "proj" 60)
- fixes unproj:: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"
- assumes finite_basis: "finite basis"
- assumes bounded_proj: "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. x proj k) ` range f)"
- assumes proj_unproj: "\<And>e k. k \<in> basis \<Longrightarrow> (unproj e) proj k = e k"
- assumes unproj_proj: "\<And>x. unproj (\<lambda>k. x proj k) = x"
- shows "\<forall>d\<subseteq>basis. \<exists>l::'a. \<exists> r::nat\<Rightarrow>nat.
- strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
-proof safe
- fix d :: "'b set"
- assume d: "d \<subseteq> basis"
- with finite_basis have "finite d"
- by (blast intro: finite_subset)
- from this d show "\<exists>l::'a. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and>
- (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
- proof (induct d)
- case empty
- then show ?case
- unfolding strict_mono_def by auto
- next
- case (insert k d)
- have k[intro]: "k \<in> basis"
- using insert by auto
- have s': "bounded ((\<lambda>x. x proj k) ` range f)"
- using k
- by (rule bounded_proj)
- obtain l1::"'a" and r1 where r1: "strict_mono r1"
- and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
- using insert(3) using insert(4) by auto
- have f': "\<forall>n. f (r1 n) proj k \<in> (\<lambda>x. x proj k) ` range f"
- by simp
- have "bounded (range (\<lambda>i. f (r1 i) proj k))"
- by (metis (lifting) bounded_subset f' image_subsetI s')
- then obtain l2 r2 where r2:"strict_mono r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) proj k) \<longlongrightarrow> l2) sequentially"
- using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) proj k"]
- by (auto simp: o_def)
- define r where "r = r1 \<circ> r2"
- have r:"strict_mono r"
- using r1 and r2 unfolding r_def o_def strict_mono_def by auto
- moreover
- define l where "l = unproj (\<lambda>i. if i = k then l2 else l1 proj i)"
- {
- fix e::real
- assume "e > 0"
- from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
- by blast
- from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) proj k) l2 < e) sequentially"
- by (rule tendstoD)
- from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e) sequentially"
- by (rule eventually_subseq)
- have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) proj i) (l proj i) < e) sequentially"
- using N1' N2
- by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def proj_unproj)
- }
- ultimately show ?case by auto
- qed
-qed
-
-lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
- unfolding bounded_def
- by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)
-
-lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
- unfolding bounded_def
- by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)
-
-instance%important prod :: (heine_borel, heine_borel) heine_borel
-proof%unimportant
- fix f :: "nat \<Rightarrow> 'a \<times> 'b"
- assume f: "bounded (range f)"
- then have "bounded (fst ` range f)"
- by (rule bounded_fst)
- then have s1: "bounded (range (fst \<circ> f))"
- by (simp add: image_comp)
- obtain l1 r1 where r1: "strict_mono r1" and l1: "(\<lambda>n. fst (f (r1 n))) \<longlonglongrightarrow> l1"
- using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
- from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
- by (auto simp: image_comp intro: bounded_snd bounded_subset)
- obtain l2 r2 where r2: "strict_mono r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) \<longlongrightarrow> l2) sequentially"
- using bounded_imp_convergent_subsequence [OF s2]
- unfolding o_def by fast
- have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) \<longlongrightarrow> l1) sequentially"
- using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
- have l: "((f \<circ> (r1 \<circ> r2)) \<longlongrightarrow> (l1, l2)) sequentially"
- using tendsto_Pair [OF l1' l2] unfolding o_def by simp
- have r: "strict_mono (r1 \<circ> r2)"
- using r1 r2 unfolding strict_mono_def by simp
- show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using l r by fast
-qed
-
-subsubsection \<open>Completeness\<close>
-
-proposition (in metric_space) completeI:
- assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f \<longlonglongrightarrow> l"
- shows "complete s"
- using assms unfolding complete_def by fast
-
-proposition (in metric_space) completeE:
- assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"
- obtains l where "l \<in> s" and "f \<longlonglongrightarrow> l"
- using assms unfolding complete_def by fast
-
-(* TODO: generalize to uniform spaces *)
-lemma compact_imp_complete:
- fixes s :: "'a::metric_space set"
- assumes "compact s"
- shows "complete s"
-proof -
- {
- fix f
- assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
- from as(1) obtain l r where lr: "l\<in>s" "strict_mono r" "(f \<circ> r) \<longlonglongrightarrow> l"
- using assms unfolding compact_def by blast
-
- note lr' = seq_suble [OF lr(2)]
- {
- fix e :: real
- assume "e > 0"
- 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 \<open>e > 0\<close>
- apply (erule_tac x="e/2" in allE, auto)
- done
- 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 \<open>e > 0\<close> by auto
- {
- fix n :: nat
- assume n: "n \<ge> max N M"
- have "dist ((f \<circ> r) n) l < e/2"
- using n M by auto
- moreover have "r n \<ge> N"
- using lr'[of n] n by auto
- then have "dist (f n) ((f \<circ> r) n) < e / 2"
- using N and n by auto
- ultimately have "dist (f n) l < e"
- using dist_triangle_half_r[of "f (r n)" "f n" e l]
- by (auto simp: dist_commute)
- }
- then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
- }
- then have "\<exists>l\<in>s. (f \<longlongrightarrow> l) sequentially" using \<open>l\<in>s\<close>
- unfolding lim_sequentially by auto
- }
- then show ?thesis unfolding complete_def by auto
-qed
-
-proposition compact_eq_totally_bounded:
- "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e))"
- (is "_ \<longleftrightarrow> ?rhs")
-proof
- assume assms: "?rhs"
- then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"
- by (auto simp: choice_iff')
-
- show "compact s"
- proof cases
- assume "s = {}"
- then show "compact s" by (simp add: compact_def)
- next
- assume "s \<noteq> {}"
- show ?thesis
- unfolding compact_def
- proof safe
- fix f :: "nat \<Rightarrow> 'a"
- assume f: "\<forall>n. f n \<in> s"
-
- define e where "e n = 1 / (2 * Suc n)" for n
- then have [simp]: "\<And>n. 0 < e n" by auto
- define B where "B n U = (SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U))" for n U
- {
- fix n U
- assume "infinite {n. f n \<in> U}"
- then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
- using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
- then obtain a where
- "a \<in> k (e n)"
- "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..
- then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
- by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
- from someI_ex[OF this]
- have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
- unfolding B_def by auto
- }
- note B = this
-
- define F where "F = rec_nat (B 0 UNIV) B"
- {
- fix n
- have "infinite {i. f i \<in> F n}"
- by (induct n) (auto simp: F_def B)
- }
- then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
- using B by (simp add: F_def)
- then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
- using decseq_SucI[of F] by (auto simp: decseq_def)
-
- obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
- proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
- fix k i
- have "infinite ({n. f n \<in> F k} - {.. i})"
- using \<open>infinite {n. f n \<in> F k}\<close> by auto
- from infinite_imp_nonempty[OF this]
- show "\<exists>x>i. f x \<in> F k"
- by (simp add: set_eq_iff not_le conj_commute)
- qed
-
- define t where "t = rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
- have "strict_mono t"
- unfolding strict_mono_Suc_iff by (simp add: t_def sel)
- moreover have "\<forall>i. (f \<circ> t) i \<in> s"
- using f by auto
- moreover
- {
- fix n
- have "(f \<circ> t) n \<in> F n"
- by (cases n) (simp_all add: t_def sel)
- }
- note t = this
-
- have "Cauchy (f \<circ> t)"
- proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
- fix r :: real and N n m
- assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
- then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
- using F_dec t by (auto simp: e_def field_simps of_nat_Suc)
- with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
- by (auto simp: subset_eq)
- with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] \<open>2 * e N < r\<close>
- show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
- by (simp add: dist_commute)
- qed
-
- ultimately show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- using assms unfolding complete_def by blast
- qed
- qed
-qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
-
-lemma cauchy_imp_bounded:
- assumes "Cauchy s"
- shows "bounded (range s)"
-proof -
- 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 by force
- then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
- moreover
- have "bounded (s ` {0..N})"
- using finite_imp_bounded[of "s ` {1..N}"] by auto
- then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
- unfolding bounded_any_center [where a="s N"] by auto
- ultimately show "?thesis"
- unfolding bounded_any_center [where a="s N"]
- apply (rule_tac x="max a 1" in exI, auto)
- apply (erule_tac x=y in allE)
- apply (erule_tac x=y in ballE, auto)
- done
-qed
-
-instance heine_borel < complete_space
-proof
- fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
- then have "bounded (range f)"
- by (rule cauchy_imp_bounded)
- then have "compact (closure (range f))"
- unfolding compact_eq_bounded_closed by auto
- then have "complete (closure (range f))"
- by (rule compact_imp_complete)
- moreover have "\<forall>n. f n \<in> closure (range f)"
- using closure_subset [of "range f"] by auto
- ultimately have "\<exists>l\<in>closure (range f). (f \<longlongrightarrow> l) sequentially"
- using \<open>Cauchy f\<close> unfolding complete_def by auto
- then show "convergent f"
- unfolding convergent_def by auto
-qed
-
-lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"
-proof (rule completeI)
- fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
- then have "convergent f" by (rule Cauchy_convergent)
- then show "\<exists>l\<in>UNIV. f \<longlonglongrightarrow> l" unfolding convergent_def by simp
-qed
-
-lemma complete_imp_closed:
- fixes S :: "'a::metric_space set"
- assumes "complete S"
- shows "closed S"
-proof (unfold closed_sequential_limits, clarify)
- fix f x assume "\<forall>n. f n \<in> S" and "f \<longlonglongrightarrow> x"
- from \<open>f \<longlonglongrightarrow> x\<close> have "Cauchy f"
- by (rule LIMSEQ_imp_Cauchy)
- with \<open>complete S\<close> and \<open>\<forall>n. f n \<in> S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
- by (rule completeE)
- from \<open>f \<longlonglongrightarrow> x\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "x = l"
- by (rule LIMSEQ_unique)
- with \<open>l \<in> S\<close> show "x \<in> S"
- by simp
-qed
-
-lemma complete_Int_closed:
- fixes S :: "'a::metric_space set"
- assumes "complete S" and "closed t"
- shows "complete (S \<inter> t)"
-proof (rule completeI)
- fix f assume "\<forall>n. f n \<in> S \<inter> t" and "Cauchy f"
- then have "\<forall>n. f n \<in> S" and "\<forall>n. f n \<in> t"
- by simp_all
- from \<open>complete S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
- using \<open>\<forall>n. f n \<in> S\<close> and \<open>Cauchy f\<close> by (rule completeE)
- from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "l \<in> t"
- by (rule closed_sequentially)
- with \<open>l \<in> S\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>S \<inter> t. f \<longlonglongrightarrow> l"
- by fast
-qed
-
-lemma complete_closed_subset:
- fixes S :: "'a::metric_space set"
- assumes "closed S" and "S \<subseteq> t" and "complete t"
- shows "complete S"
- using assms complete_Int_closed [of t S] by (simp add: Int_absorb1)
-
-lemma complete_eq_closed:
- fixes S :: "('a::complete_space) set"
- shows "complete S \<longleftrightarrow> closed S"
-proof
- assume "closed S" then show "complete S"
- using subset_UNIV complete_UNIV by (rule complete_closed_subset)
-next
- assume "complete S" then show "closed S"
- by (rule complete_imp_closed)
-qed
-
-lemma convergent_eq_Cauchy:
- fixes S :: "nat \<Rightarrow> 'a::complete_space"
- shows "(\<exists>l. (S \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy S"
- unfolding Cauchy_convergent_iff convergent_def ..
-
-lemma convergent_imp_bounded:
- fixes S :: "nat \<Rightarrow> 'a::metric_space"
- shows "(S \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range S)"
- by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
-
-lemma frontier_subset_compact:
- fixes S :: "'a::heine_borel set"
- shows "compact S \<Longrightarrow> frontier S \<subseteq> S"
- using frontier_subset_closed compact_eq_bounded_closed
- by blast
-
subsection \<open>Continuity\<close>
-text\<open>Derive the epsilon-delta forms, which we often use as "definitions"\<close>
-
-proposition continuous_within_eps_delta:
- "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)"
- unfolding continuous_within and Lim_within by fastforce
-
-corollary continuous_at_eps_delta:
- "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
- using continuous_within_eps_delta [of x UNIV f] by simp
-
-lemma continuous_at_right_real_increasing:
- fixes f :: "real \<Rightarrow> real"
- assumes nondecF: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y"
- shows "continuous (at_right a) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f (a + d) - f a < e)"
- apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)
- apply (intro all_cong ex_cong, safe)
- apply (erule_tac x="a + d" in allE, simp)
- apply (simp add: nondecF field_simps)
- apply (drule nondecF, simp)
- done
-
-lemma continuous_at_left_real_increasing:
- assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"
- shows "(continuous (at_left (a :: real)) f) = (\<forall>e > 0. \<exists>delta > 0. f a - f (a - delta) < e)"
- apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)
- apply (intro all_cong ex_cong, safe)
- apply (erule_tac x="a - d" in allE, simp)
- apply (simp add: nondecF field_simps)
- apply (cut_tac x="a - d" and y=x in nondecF, simp_all)
- done
-
-text\<open>Versions in terms of open balls.\<close>
-
-lemma continuous_within_ball:
- "continuous (at x within s) f \<longleftrightarrow>
- (\<forall>e > 0. \<exists>d > 0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- {
- fix e :: real
- assume "e > 0"
- 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 \<open>?lhs\<close>[unfolded continuous_within Lim_within] by auto
- {
- fix y
- assume "y \<in> f ` (ball x d \<inter> s)"
- then have "y \<in> ball (f x) e"
- using d(2)
- apply (auto simp: dist_commute)
- apply (erule_tac x=xa in ballE, auto)
- using \<open>e > 0\<close>
- apply auto
- done
- }
- then have "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e"
- using \<open>d > 0\<close>
- unfolding subset_eq ball_def by (auto simp: dist_commute)
- }
- then show ?rhs by auto
-next
- assume ?rhs
- then show ?lhs
- unfolding continuous_within Lim_within ball_def subset_eq
- apply (auto simp: dist_commute)
- apply (erule_tac x=e in allE, auto)
- done
-qed
-
-lemma continuous_at_ball:
- "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
- apply auto
- apply (erule_tac x=e in allE, auto)
- apply (rule_tac x=d in exI, auto)
- apply (erule_tac x=xa in allE)
- apply (auto simp: dist_commute)
- done
-next
- assume ?rhs
- then show ?lhs
- unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
- apply auto
- apply (erule_tac x=e in allE, auto)
- apply (rule_tac x=d in exI, auto)
- apply (erule_tac x="f xa" in allE)
- apply (auto simp: dist_commute)
- done
-qed
-
-text\<open>Define setwise continuity in terms of limits within the set.\<close>
-
-lemma continuous_on_iff:
- "continuous_on s f \<longleftrightarrow>
- (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
- unfolding continuous_on_def Lim_within
- by (metis dist_pos_lt dist_self)
-
-lemma continuous_within_E:
- assumes "continuous (at x within s) f" "e>0"
- obtains d where "d>0" "\<And>x'. \<lbrakk>x'\<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
- using assms apply (simp add: continuous_within_eps_delta)
- apply (drule spec [of _ e], clarify)
- apply (rule_tac d="d/2" in that, auto)
- done
-
-lemma continuous_onI [intro?]:
- assumes "\<And>x e. \<lbrakk>e > 0; x \<in> s\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
- shows "continuous_on s f"
-apply (simp add: continuous_on_iff, clarify)
-apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
-done
-
-text\<open>Some simple consequential lemmas.\<close>
-
-lemma continuous_onE:
- assumes "continuous_on s f" "x\<in>s" "e>0"
- obtains d where "d>0" "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
- using assms
- apply (simp add: continuous_on_iff)
- apply (elim ballE allE)
- apply (auto intro: that [where d="d/2" for d])
- done
-
-lemma uniformly_continuous_onE:
- assumes "uniformly_continuous_on s f" "0 < e"
- obtains d where "d>0" "\<And>x x'. \<lbrakk>x\<in>s; x'\<in>s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
-using assms
-by (auto simp: uniformly_continuous_on_def)
-
lemma continuous_at_imp_continuous_within:
"continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
@@ -4669,7 +2048,7 @@
using continuous_within_sequentiallyI[of _ s f] by auto
lemma continuous_on_sequentially:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
+ fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
shows "continuous_on s f \<longleftrightarrow>
(\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
--> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
@@ -4688,181 +2067,6 @@
by auto
qed
-lemma uniformly_continuous_on_sequentially:
- "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
- (\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> 0 \<longrightarrow> (\<lambda>n. dist (f(x n)) (f(y n))) \<longlonglongrightarrow> 0)" (is "?lhs = ?rhs")
-proof
- assume ?lhs
- {
- fix x y
- assume x: "\<forall>n. x n \<in> s"
- and y: "\<forall>n. y n \<in> s"
- and xy: "((\<lambda>n. dist (x n) (y n)) \<longlongrightarrow> 0) sequentially"
- {
- fix e :: real
- assume "e > 0"
- 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"
- using \<open>?lhs\<close>[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
- obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"
- using xy[unfolded lim_sequentially dist_norm] and \<open>d>0\<close> by auto
- {
- fix n
- assume "n\<ge>N"
- then have "dist (f (x n)) (f (y n)) < e"
- 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
- by (simp add: dist_commute)
- }
- then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
- by auto
- }
- then have "((\<lambda>n. dist (f(x n)) (f(y n))) \<longlongrightarrow> 0) sequentially"
- unfolding lim_sequentially and dist_real_def by auto
- }
- then show ?rhs by auto
-next
- assume ?rhs
- {
- assume "\<not> ?lhs"
- 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
- 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"
- 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
- by (auto simp: dist_commute)
- define x where "x n = fst (fa (inverse (real n + 1)))" for n
- define y where "y n = snd (fa (inverse (real n + 1)))" for n
- 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"
- unfolding x_def and y_def using fa
- by auto
- {
- fix e :: real
- assume "e > 0"
- then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"
- unfolding real_arch_inverse[of e] by auto
- {
- fix n :: nat
- assume "n \<ge> N"
- then have "inverse (real n + 1) < inverse (real N)"
- using of_nat_0_le_iff and \<open>N\<noteq>0\<close> by auto
- also have "\<dots> < e" using N by auto
- finally have "inverse (real n + 1) < e" by auto
- then have "dist (x n) (y n) < e"
- using xy0[THEN spec[where x=n]] by auto
- }
- then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto
- }
- then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
- using \<open>?rhs\<close>[THEN spec[where x=x], THEN spec[where x=y]] and xyn
- unfolding lim_sequentially dist_real_def by auto
- then have False using fxy and \<open>e>0\<close> by auto
- }
- then show ?lhs
- unfolding uniformly_continuous_on_def by blast
-qed
-
-lemma continuous_closed_imp_Cauchy_continuous:
- fixes S :: "('a::complete_space) set"
- shows "\<lbrakk>continuous_on S f; closed S; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f \<circ> \<sigma>)"
- apply (simp add: complete_eq_closed [symmetric] continuous_on_sequentially)
- by (meson LIMSEQ_imp_Cauchy complete_def)
-
-text\<open>The usual transformation theorems.\<close>
-
-lemma continuous_transform_within:
- fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
- assumes "continuous (at x within s) f"
- and "0 < d"
- and "x \<in> s"
- and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
- shows "continuous (at x within s) g"
- using assms
- unfolding continuous_within
- by (force intro: Lim_transform_within)
-
-
-subsubsection%unimportant \<open>Structural rules for uniform continuity\<close>
-
-lemma uniformly_continuous_on_dist[continuous_intros]:
- fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "uniformly_continuous_on s f"
- and "uniformly_continuous_on s g"
- shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
-proof -
- {
- fix a b c d :: 'b
- have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
- using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
- using dist_triangle3 [of c d a] dist_triangle [of a d b]
- by arith
- } note le = this
- {
- fix x y
- assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
- assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
- have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
- by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
- simp add: le)
- }
- then show ?thesis
- using assms unfolding uniformly_continuous_on_sequentially
- unfolding dist_real_def by simp
-qed
-
-lemma uniformly_continuous_on_norm[continuous_intros]:
- fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
- assumes "uniformly_continuous_on s f"
- shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
- unfolding norm_conv_dist using assms
- by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
-
-lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
- fixes g :: "_::metric_space \<Rightarrow> _"
- assumes "uniformly_continuous_on s g"
- shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
- using assms unfolding uniformly_continuous_on_sequentially
- unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
- by (auto intro: tendsto_zero)
-
-lemma uniformly_continuous_on_cmul[continuous_intros]:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "uniformly_continuous_on s f"
- shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
- using bounded_linear_scaleR_right assms
- by (rule bounded_linear.uniformly_continuous_on)
-
-lemma dist_minus:
- fixes x y :: "'a::real_normed_vector"
- shows "dist (- x) (- y) = dist x y"
- unfolding dist_norm minus_diff_minus norm_minus_cancel ..
-
-lemma uniformly_continuous_on_minus[continuous_intros]:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
- unfolding uniformly_continuous_on_def dist_minus .
-
-lemma uniformly_continuous_on_add[continuous_intros]:
- fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "uniformly_continuous_on s f"
- and "uniformly_continuous_on s g"
- shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
- using assms
- unfolding uniformly_continuous_on_sequentially
- unfolding dist_norm tendsto_norm_zero_iff add_diff_add
- by (auto intro: tendsto_add_zero)
-
-lemma uniformly_continuous_on_diff[continuous_intros]:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "uniformly_continuous_on s f"
- and "uniformly_continuous_on s g"
- shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
- using assms uniformly_continuous_on_add [of s f "- g"]
- by (simp add: fun_Compl_def uniformly_continuous_on_minus)
-
text \<open>Continuity in terms of open preimages.\<close>
lemma continuous_at_open:
@@ -4886,213 +2090,4 @@
using T_def by (auto elim!: eventually_mono)
qed
-lemma continuous_on_open:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. openin (subtopology euclidean (f ` S)) T \<longrightarrow>
- openin (subtopology euclidean S) (S \<inter> f -` T))"
- unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
- by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
-
-lemma continuous_on_open_gen:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "f ` S \<subseteq> T"
- shows "continuous_on S f \<longleftrightarrow>
- (\<forall>U. openin (subtopology euclidean T) U
- \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (clarsimp simp: openin_euclidean_subtopology_iff continuous_on_iff)
- by (metis assms image_subset_iff)
-next
- have ope: "openin (subtopology euclidean T) (ball y e \<inter> T)" for y e
- by (simp add: Int_commute openin_open_Int)
- assume R [rule_format]: ?rhs
- show ?lhs
- proof (clarsimp simp add: continuous_on_iff)
- fix x and e::real
- assume "x \<in> S" and "0 < e"
- then have x: "x \<in> S \<inter> (f -` ball (f x) e \<inter> f -` T)"
- using assms by auto
- show "\<exists>d>0. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
- using R [of "ball (f x) e \<inter> T"] x
- by (fastforce simp add: ope openin_euclidean_subtopology_iff [of S] dist_commute)
- qed
-qed
-
-lemma continuous_openin_preimage:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- shows
- "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (subtopology euclidean T) U\<rbrakk>
- \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U)"
-by (simp add: continuous_on_open_gen)
-
-text \<open>Similarly in terms of closed sets.\<close>
-
-lemma continuous_on_closed:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. closedin (subtopology euclidean (f ` S)) T \<longrightarrow>
- closedin (subtopology euclidean S) (S \<inter> f -` T))"
- unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
- by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
-
-lemma continuous_on_closed_gen:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "f ` S \<subseteq> T"
- shows "continuous_on S f \<longleftrightarrow>
- (\<forall>U. closedin (subtopology euclidean T) U
- \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` U))"
- (is "?lhs = ?rhs")
-proof -
- have *: "U \<subseteq> T \<Longrightarrow> S \<inter> f -` (T - U) = S - (S \<inter> f -` U)" for U
- using assms by blast
- show ?thesis
- proof
- assume L: ?lhs
- show ?rhs
- proof clarify
- fix U
- assume "closedin (subtopology euclidean T) U"
- then show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
- using L unfolding continuous_on_open_gen [OF assms]
- by (metis * closedin_def inf_le1 topspace_euclidean_subtopology)
- qed
- next
- assume R [rule_format]: ?rhs
- show ?lhs
- unfolding continuous_on_open_gen [OF assms]
- by (metis * R inf_le1 openin_closedin_eq topspace_euclidean_subtopology)
- qed
-qed
-
-lemma continuous_closedin_preimage_gen:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (subtopology euclidean T) U"
- shows "closedin (subtopology euclidean S) (S \<inter> f -` U)"
-using assms continuous_on_closed_gen by blast
-
-lemma continuous_on_imp_closedin:
- assumes "continuous_on S f" "closedin (subtopology euclidean (f ` S)) T"
- shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
-using assms continuous_on_closed by blast
-
-subsection%unimportant \<open>Half-global and completely global cases\<close>
-
-lemma continuous_openin_preimage_gen:
- assumes "continuous_on S f" "open T"
- shows "openin (subtopology euclidean S) (S \<inter> f -` T)"
-proof -
- have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
- by auto
- have "openin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
- using openin_open_Int[of T "f ` S", OF assms(2)] unfolding openin_open by auto
- then show ?thesis
- using assms(1)[unfolded continuous_on_open, THEN spec[where x="T \<inter> f ` S"]]
- using * by auto
-qed
-
-lemma continuous_closedin_preimage:
- assumes "continuous_on S f" and "closed T"
- shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
-proof -
- have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
- by auto
- have "closedin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
- using closedin_closed_Int[of T "f ` S", OF assms(2)]
- by (simp add: Int_commute)
- then show ?thesis
- using assms(1)[unfolded continuous_on_closed, THEN spec[where x="T \<inter> f ` S"]]
- using * by auto
-qed
-
-lemma continuous_openin_preimage_eq:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. open T \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T))"
-apply safe
-apply (simp add: continuous_openin_preimage_gen)
-apply (fastforce simp add: continuous_on_open openin_open)
-done
-
-lemma continuous_closedin_preimage_eq:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. closed T \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` T))"
-apply safe
-apply (simp add: continuous_closedin_preimage)
-apply (fastforce simp add: continuous_on_closed closedin_closed)
-done
-
-lemma continuous_open_preimage:
- assumes contf: "continuous_on S f" and "open S" "open T"
- shows "open (S \<inter> f -` T)"
-proof-
- obtain U where "open U" "(S \<inter> f -` T) = S \<inter> U"
- using continuous_openin_preimage_gen[OF contf \<open>open T\<close>]
- unfolding openin_open by auto
- then show ?thesis
- using open_Int[of S U, OF \<open>open S\<close>] by auto
-qed
-
-lemma continuous_closed_preimage:
- assumes contf: "continuous_on S f" and "closed S" "closed T"
- shows "closed (S \<inter> f -` T)"
-proof-
- obtain U where "closed U" "(S \<inter> f -` T) = S \<inter> U"
- using continuous_closedin_preimage[OF contf \<open>closed T\<close>]
- unfolding closedin_closed by auto
- then show ?thesis using closed_Int[of S U, OF \<open>closed S\<close>] by auto
-qed
-
-lemma continuous_open_vimage: "open S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> open (f -` S)"
- by (metis continuous_on_eq_continuous_within open_vimage)
-
-lemma continuous_closed_vimage: "closed S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> closed (f -` S)"
- by (simp add: closed_vimage continuous_on_eq_continuous_within)
-
-lemma interior_image_subset:
- assumes "inj f" "\<And>x. continuous (at x) f"
- shows "interior (f ` S) \<subseteq> f ` (interior S)"
-proof
- fix x assume "x \<in> interior (f ` S)"
- then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` S" ..
- then have "x \<in> f ` S" by auto
- then obtain y where y: "y \<in> S" "x = f y" by auto
- have "open (f -` T)"
- using assms \<open>open T\<close> by (simp add: continuous_at_imp_continuous_on open_vimage)
- moreover have "y \<in> vimage f T"
- using \<open>x = f y\<close> \<open>x \<in> T\<close> by simp
- moreover have "vimage f T \<subseteq> S"
- using \<open>T \<subseteq> image f S\<close> \<open>inj f\<close> unfolding inj_on_def subset_eq by auto
- ultimately have "y \<in> interior S" ..
- with \<open>x = f y\<close> show "x \<in> f ` interior S" ..
-qed
-
-subsection%unimportant \<open>Topological properties of linear functions\<close>
-
-lemma linear_lim_0:
- assumes "bounded_linear f"
- shows "(f \<longlongrightarrow> 0) (at (0))"
-proof -
- interpret f: bounded_linear f by fact
- have "(f \<longlongrightarrow> f 0) (at 0)"
- using tendsto_ident_at by (rule f.tendsto)
- then show ?thesis unfolding f.zero .
-qed
-
-lemma linear_continuous_at:
- assumes "bounded_linear f"
- shows "continuous (at a) f"
- unfolding continuous_at using assms
- apply (rule bounded_linear.tendsto)
- apply (rule tendsto_ident_at)
- done
-
-lemma linear_continuous_within:
- "bounded_linear f \<Longrightarrow> continuous (at x within s) f"
- using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
-
-lemma linear_continuous_on:
- "bounded_linear f \<Longrightarrow> continuous_on s f"
- using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
-
end
\ No newline at end of file
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Sat Dec 29 18:40:29 2018 +0000
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Sat Dec 29 20:32:09 2018 +0100
@@ -8,7 +8,7 @@
theory Topology_Euclidean_Space
imports
- Elementary_Topology
+ Elementary_Normed_Spaces
Linear_Algebra
Norm_Arith
begin