src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author paulson
Thu, 29 Oct 2009 16:21:43 +0000
changeset 33324 51eb2ffa2189
parent 33270 320a1d67b9ae
child 33714 eb2574ac4173
permissions -rw-r--r--
Tidied up some very ugly proofs
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Library/Topology_Euclidian_Space.thy
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    Author:     Amine Chaieb, University of Cambridge
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    Author:     Robert Himmelmann, TU Muenchen
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*)
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header {* Elementary topology in Euclidean space. *}
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theory Topology_Euclidean_Space
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imports SEQ Euclidean_Space Product_Vector
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begin
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declare fstcart_pastecart[simp] sndcart_pastecart[simp]
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subsection{* General notion of a topology *}
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definition "istopology L \<longleftrightarrow> {} \<in> L \<and> (\<forall>S \<in>L. \<forall>T \<in>L. S \<inter> T \<in> L) \<and> (\<forall>K. K \<subseteq>L \<longrightarrow> \<Union> K \<in> L)"
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typedef (open) 'a topology = "{L::('a set) set. istopology L}"
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  morphisms "openin" "topology"
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  unfolding istopology_def by blast
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lemma istopology_open_in[intro]: "istopology(openin U)"
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  using openin[of U] by blast
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lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
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  using topology_inverse[unfolded mem_def Collect_def] .
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lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
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  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
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lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
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proof-
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  {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
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  moreover
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  {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
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    hence "openin T1 = openin T2" by (metis mem_def set_ext)
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    hence "topology (openin T1) = topology (openin T2)" by simp
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    hence "T1 = T2" unfolding openin_inverse .}
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  ultimately show ?thesis by blast
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qed
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text{* Infer the "universe" from union of all sets in the topology. *}
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definition "topspace T =  \<Union>{S. openin T S}"
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subsection{* Main properties of open sets *}
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lemma openin_clauses:
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  fixes U :: "'a topology"
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  shows "openin U {}"
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  "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
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  "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
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  using openin[of U] unfolding istopology_def Collect_def mem_def
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  by (metis mem_def subset_eq)+
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lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
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  unfolding topspace_def by blast
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lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
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lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
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  by (simp add: openin_clauses)
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lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
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lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
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  using openin_Union[of "{S,T}" U] by auto
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lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
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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")
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proof-
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  {assume ?lhs then have ?rhs by auto }
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  moreover
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  {assume H: ?rhs
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    then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"
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      unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast
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    from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
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    have "\<Union> t`S = S" using t by auto
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    with openin_Union[OF th0] have "openin U S" by simp }
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  ultimately show ?thesis by blast
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qed
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subsection{* Closed sets *}
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definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
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lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
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lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
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lemma closedin_topspace[intro,simp]:
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  "closedin U (topspace U)" by (simp add: closedin_def)
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lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
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  by (auto simp add: Diff_Un closedin_def)
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lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
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lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
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  shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
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lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
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  using closedin_Inter[of "{S,T}" U] by auto
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lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
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lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
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  apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
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  apply (metis openin_subset subset_eq)
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  done
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lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
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  by (simp add: openin_closedin_eq)
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lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
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proof-
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  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
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    by (auto simp add: topspace_def openin_subset)
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  then show ?thesis using oS cT by (auto simp add: closedin_def)
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qed
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lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
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proof-
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  have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
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    by (auto simp add: topspace_def )
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  then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
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qed
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subsection{* Subspace topology. *}
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definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
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lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
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proof-
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  have "{} \<in> ?L" by blast
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  {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
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    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
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    have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
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    then have "A \<inter> B \<in> ?L" by blast}
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  moreover
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  {fix K assume K: "K \<subseteq> ?L"
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    have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
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      apply (rule set_ext)
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      apply (simp add: Ball_def image_iff)
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      by (metis mem_def)
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    from K[unfolded th0 subset_image_iff]
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    obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
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    have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
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    moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
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    ultimately have "\<Union>K \<in> ?L" by blast}
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  ultimately show ?thesis unfolding istopology_def by blast
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qed
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lemma openin_subtopology:
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  "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
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  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
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  by (auto simp add: Collect_def)
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lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
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  by (auto simp add: topspace_def openin_subtopology)
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lemma closedin_subtopology:
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  "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
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  unfolding closedin_def topspace_subtopology
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  apply (simp add: openin_subtopology)
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  apply (rule iffI)
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  apply clarify
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  apply (rule_tac x="topspace U - T" in exI)
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  by auto
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lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
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  unfolding openin_subtopology
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  apply (rule iffI, clarify)
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  apply (frule openin_subset[of U])  apply blast
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  apply (rule exI[where x="topspace U"])
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  by auto
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lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
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  shows "subtopology U V = U"
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proof-
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  {fix S
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    {fix T assume T: "openin U T" "S = T \<inter> V"
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      from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
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      have "openin U S" unfolding eq using T by blast}
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    moreover
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    {assume S: "openin U S"
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      hence "\<exists>T. openin U T \<and> S = T \<inter> V"
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        using openin_subset[OF S] UV by auto}
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    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
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  then show ?thesis unfolding topology_eq openin_subtopology by blast
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qed
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lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
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  by (simp add: subtopology_superset)
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lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
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  by (simp add: subtopology_superset)
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subsection{* The universal Euclidean versions are what we use most of the time *}
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definition
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  euclidean :: "'a::topological_space topology" where
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  "euclidean = topology open"
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lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
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  unfolding euclidean_def
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  apply (rule cong[where x=S and y=S])
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  apply (rule topology_inverse[symmetric])
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  apply (auto simp add: istopology_def)
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  by (auto simp add: mem_def subset_eq)
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lemma topspace_euclidean: "topspace euclidean = UNIV"
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  apply (simp add: topspace_def)
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  apply (rule set_ext)
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  by (auto simp add: open_openin[symmetric])
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lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
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  by (simp add: topspace_euclidean topspace_subtopology)
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lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
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  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
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lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
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  by (simp add: open_openin openin_subopen[symmetric])
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subsection{* Open and closed balls. *}
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definition
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  ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
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  "ball x e = {y. dist x y < e}"
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definition
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  cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
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  "cball x e = {y. dist x y \<le> e}"
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lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
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lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
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lemma mem_ball_0 [simp]:
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  fixes x :: "'a::real_normed_vector"
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  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
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  by (simp add: dist_norm)
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lemma mem_cball_0 [simp]:
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  fixes x :: "'a::real_normed_vector"
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  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
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  by (simp add: dist_norm)
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lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
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lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
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lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
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lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
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lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
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  by (simp add: expand_set_eq) arith
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lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
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  by (simp add: expand_set_eq)
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subsection{* Topological properties of open balls *}
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lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
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  "(a::real) - b < 0 \<longleftrightarrow> a < b"
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  "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
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lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
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  "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
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lemma open_ball[intro, simp]: "open (ball x e)"
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  unfolding open_dist ball_def Collect_def Ball_def mem_def
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  unfolding dist_commute
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  apply clarify
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  apply (rule_tac x="e - dist xa x" in exI)
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  using dist_triangle_alt[where z=x]
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  apply (clarsimp simp add: diff_less_iff)
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  apply atomize
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  apply (erule_tac x="y" in allE)
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  apply (erule_tac x="xa" in allE)
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  by arith
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lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
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lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
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  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
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lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
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  by (metis open_contains_ball subset_eq centre_in_ball)
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lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
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  unfolding mem_ball expand_set_eq
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  apply (simp add: not_less)
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  by (metis zero_le_dist order_trans dist_self)
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lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
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subsection{* Basic "localization" results are handy for connectedness. *}
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lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
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  by (auto simp add: openin_subtopology open_openin[symmetric])
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lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
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  by (auto simp add: openin_open)
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lemma open_openin_trans[trans]:
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 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
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  by (metis Int_absorb1  openin_open_Int)
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lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
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  by (auto simp add: openin_open)
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lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
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  by (simp add: closedin_subtopology closed_closedin Int_ac)
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lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
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  by (metis closedin_closed)
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lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
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  apply (subgoal_tac "S \<inter> T = T" )
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  apply auto
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  apply (frule closedin_closed_Int[of T S])
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  by simp
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lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
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  by (auto simp add: closedin_closed)
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lemma openin_euclidean_subtopology_iff:
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  fixes S U :: "'a::metric_space set"
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  shows "openin (subtopology euclidean U) S
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  \<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")
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proof-
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  {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
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      by (simp add: open_dist) blast}
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  moreover
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diff changeset
   326
  {assume SU: "S \<subseteq> U" and H: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   327
    from H obtain d where d: "\<And>x . x\<in> S \<Longrightarrow> d x > 0 \<and> (\<forall>x' \<in> U. dist x' x < d x \<longrightarrow> x' \<in> S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   328
      by metis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   329
    let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   330
    have oT: "open ?T" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   331
    { fix x assume "x\<in>S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   332
      hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   333
        apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   334
        by (rule d [THEN conjunct1])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   335
      hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   336
    moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   337
    { fix y assume "y\<in>?T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   338
      then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   339
      then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   340
      assume "y\<in>U"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   341
      hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   342
    ultimately have "S = ?T \<inter> U" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   343
    with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   344
  ultimately show ?thesis by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   345
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   346
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   347
text{* These "transitivity" results are handy too. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   348
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   349
lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   350
  \<Longrightarrow> openin (subtopology euclidean U) S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   351
  unfolding open_openin openin_open by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   352
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   353
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   354
  by (auto simp add: openin_open intro: openin_trans)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   355
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   356
lemma closedin_trans[trans]:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   357
 "closedin (subtopology euclidean T) S \<Longrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   358
           closedin (subtopology euclidean U) T
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   359
           ==> closedin (subtopology euclidean U) S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   360
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   361
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   362
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   363
  by (auto simp add: closedin_closed intro: closedin_trans)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   364
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   365
subsection{* Connectedness *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   366
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   367
definition "connected S \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   368
  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   369
  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   370
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   371
lemma connected_local:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   372
 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   373
                 openin (subtopology euclidean S) e1 \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   374
                 openin (subtopology euclidean S) e2 \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   375
                 S \<subseteq> e1 \<union> e2 \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   376
                 e1 \<inter> e2 = {} \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   377
                 ~(e1 = {}) \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   378
                 ~(e2 = {}))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   379
unfolding connected_def openin_open by (safe, blast+)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   380
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   381
lemma exists_diff: "(\<exists>S. P(UNIV - S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   382
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   383
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   384
  {assume "?lhs" hence ?rhs by blast }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   385
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   386
  {fix S assume H: "P S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   387
    have "S = UNIV - (UNIV - S)" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   388
    with H have "P (UNIV - (UNIV - S))" by metis }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   389
  ultimately show ?thesis by metis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   390
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   391
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   392
lemma connected_clopen: "connected S \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   393
        (\<forall>T. openin (subtopology euclidean S) T \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   394
            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   395
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   396
  have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (UNIV - e2) \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   397
    unfolding connected_def openin_open closedin_closed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   398
    apply (subst exists_diff) by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   399
  hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   400
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def Compl_eq_Diff_UNIV) by metis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   401
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   402
  have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   403
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   404
    unfolding connected_def openin_open closedin_closed by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   405
  {fix e2
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   406
    {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   407
        by auto}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   408
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   409
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   410
  then show ?thesis unfolding th0 th1 by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   411
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   412
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   413
lemma connected_empty[simp, intro]: "connected {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   414
  by (simp add: connected_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   415
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   416
subsection{* Hausdorff and other separation properties *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   417
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   418
class t0_space =
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   419
  assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   420
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   421
class t1_space =
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   422
  assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<notin> U \<and> x \<notin> V \<and> y \<in> V"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   423
begin
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   424
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   425
subclass t0_space
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   426
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   427
qed (fast dest: t1_space)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   428
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   429
end
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   430
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   431
text {* T2 spaces are also known as Hausdorff spaces. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   432
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   433
class t2_space =
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   434
  assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   435
begin
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   436
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   437
subclass t1_space
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   438
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   439
qed (fast dest: hausdorff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   440
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   441
end
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   442
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   443
instance metric_space \<subseteq> t2_space
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   444
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   445
  fix x y :: "'a::metric_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   446
  assume xy: "x \<noteq> y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   447
  let ?U = "ball x (dist x y / 2)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   448
  let ?V = "ball y (dist x y / 2)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   449
  have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   450
               ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   451
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   452
    using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   453
    by (auto simp add: expand_set_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   454
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   455
    by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   456
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   457
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   458
lemma separation_t2:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   459
  fixes x y :: "'a::t2_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   460
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   461
  using hausdorff[of x y] by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   462
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   463
lemma separation_t1:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   464
  fixes x y :: "'a::t1_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   465
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in>U \<and> y\<notin> U \<and> x\<notin>V \<and> y\<in>V)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   466
  using t1_space[of x y] by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   467
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   468
lemma separation_t0:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   469
  fixes x y :: "'a::t0_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   470
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   471
  using t0_space[of x y] by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   472
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   473
subsection{* Limit points *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   474
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   475
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   476
  islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   477
    (infixr "islimpt" 60) where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   478
  "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   479
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   480
lemma islimptI:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   481
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   482
  shows "x islimpt S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   483
  using assms unfolding islimpt_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   484
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   485
lemma islimptE:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   486
  assumes "x islimpt S" and "x \<in> T" and "open T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   487
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   488
  using assms unfolding islimpt_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   489
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   490
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   491
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   492
lemma islimpt_approachable:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   493
  fixes x :: "'a::metric_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   494
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   495
  unfolding islimpt_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   496
  apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   497
  apply(erule_tac x="ball x e" in allE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   498
  apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   499
  apply(rule_tac x=y in bexI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   500
  apply (auto simp add: dist_commute)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   501
  apply (simp add: open_dist, drule (1) bspec)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   502
  apply (clarify, drule spec, drule (1) mp, auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   503
  done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   504
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   505
lemma islimpt_approachable_le:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   506
  fixes x :: "'a::metric_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   507
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   508
  unfolding islimpt_approachable
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   509
  using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
33324
51eb2ffa2189 Tidied up some very ugly proofs
paulson
parents: 33270
diff changeset
   510
  by metis 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   511
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   512
class perfect_space =
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   513
  (* FIXME: perfect_space should inherit from topological_space *)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   514
  assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   515
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   516
lemma perfect_choose_dist:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   517
  fixes x :: "'a::perfect_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   518
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   519
using islimpt_UNIV [of x]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   520
by (simp add: islimpt_approachable)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   521
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   522
instance real :: perfect_space
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   523
apply default
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   524
apply (rule islimpt_approachable [THEN iffD2])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   525
apply (clarify, rule_tac x="x + e/2" in bexI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   526
apply (auto simp add: dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   527
done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   528
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   529
instance "^" :: (perfect_space, finite) perfect_space
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   530
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   531
  fix x :: "'a ^ 'b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   532
  {
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   533
    fix e :: real assume "0 < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   534
    def a \<equiv> "x $ undefined"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   535
    have "a islimpt UNIV" by (rule islimpt_UNIV)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   536
    with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   537
      unfolding islimpt_approachable by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   538
    def y \<equiv> "Cart_lambda ((Cart_nth x)(undefined := b))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   539
    from `b \<noteq> a` have "y \<noteq> x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   540
      unfolding a_def y_def by (simp add: Cart_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   541
    from `dist b a < e` have "dist y x < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   542
      unfolding dist_vector_def a_def y_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   543
      apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   544
      apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   545
      apply (subst setsum_diff1' [where a=undefined], simp, simp, simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   546
      done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   547
    from `y \<noteq> x` and `dist y x < e`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   548
    have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   549
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   550
  then show "x islimpt UNIV" unfolding islimpt_approachable by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   551
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   552
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   553
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   554
  unfolding closed_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   555
  apply (subst open_subopen)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   556
  apply (simp add: islimpt_def subset_eq Compl_eq_Diff_UNIV)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   557
  by (metis DiffE DiffI UNIV_I insertCI insert_absorb mem_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   558
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   559
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   560
  unfolding islimpt_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   561
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   562
lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x$i}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   563
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   564
  let ?U = "UNIV :: 'n set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   565
  let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   566
  {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   567
    and xi: "x$i < 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   568
    from xi have th0: "-x$i > 0" by arith
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   569
    from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   570
      have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   571
      have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   572
      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   573
        apply (simp only: vector_component)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   574
        by (rule th') auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   575
      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[of "x'-x" i]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   576
        apply (simp add: dist_norm) by norm
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   577
      from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   578
  then show ?thesis unfolding closed_limpt islimpt_approachable
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   579
    unfolding not_le[symmetric] by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   580
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   581
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   582
lemma finite_set_avoid:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   583
  fixes a :: "'a::metric_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   584
  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   585
proof(induct rule: finite_induct[OF fS])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   586
  case 1 thus ?case apply auto by ferrack
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   587
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   588
  case (2 x F)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   589
  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   590
  {assume "x = a" hence ?case using d by auto  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   591
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   592
  {assume xa: "x\<noteq>a"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   593
    let ?d = "min d (dist a x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   594
    have dp: "?d > 0" using xa d(1) using dist_nz by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   595
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   596
    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   597
  ultimately show ?case by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   598
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   599
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   600
lemma islimpt_finite:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   601
  fixes S :: "'a::metric_space set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   602
  assumes fS: "finite S" shows "\<not> a islimpt S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   603
  unfolding islimpt_approachable
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   604
  using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   605
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   606
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   607
  apply (rule iffI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   608
  defer
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   609
  apply (metis Un_upper1 Un_upper2 islimpt_subset)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   610
  unfolding islimpt_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   611
  apply (rule ccontr, clarsimp, rename_tac A B)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   612
  apply (drule_tac x="A \<inter> B" in spec)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   613
  apply (auto simp add: open_Int)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   614
  done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   615
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   616
lemma discrete_imp_closed:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   617
  fixes S :: "'a::metric_space set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   618
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   619
  shows "closed S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   620
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   621
  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   622
    from e have e2: "e/2 > 0" by arith
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   623
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   624
    let ?m = "min (e/2) (dist x y) "
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   625
    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   626
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   627
    have th: "dist z y < e" using z y
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   628
      by (intro dist_triangle_lt [where z=x], simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   629
    from d[rule_format, OF y(1) z(1) th] y z
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   630
    have False by (auto simp add: dist_commute)}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   631
  then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   632
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   633
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   634
subsection{* Interior of a Set *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   635
definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   636
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   637
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   638
  apply (simp add: expand_set_eq interior_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   639
  apply (subst (2) open_subopen) by (safe, blast+)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   640
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   641
lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   642
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   643
lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   644
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   645
lemma open_interior[simp, intro]: "open(interior S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   646
  apply (simp add: interior_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   647
  apply (subst open_subopen) by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   648
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   649
lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   650
lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   651
lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   652
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   653
lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T  \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   654
  by (metis equalityI interior_maximal interior_subset open_interior)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   655
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   656
  apply (simp add: interior_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   657
  by (metis open_contains_ball centre_in_ball open_ball subset_trans)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   658
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   659
lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   660
  by (metis interior_maximal interior_subset subset_trans)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   661
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   662
lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   663
  apply (rule equalityI, simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   664
  apply (metis Int_lower1 Int_lower2 subset_interior)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   665
  by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   666
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   667
lemma interior_limit_point [intro]:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   668
  fixes x :: "'a::perfect_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   669
  assumes x: "x \<in> interior S" shows "x islimpt S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   670
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   671
  from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   672
    unfolding mem_interior subset_eq Ball_def mem_ball by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   673
  {
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   674
    fix d::real assume d: "d>0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   675
    let ?m = "min d e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   676
    have mde2: "0 < ?m" using e(1) d(1) by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   677
    from perfect_choose_dist [OF mde2, of x]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   678
    obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   679
    then have "dist y x < e" "dist y x < d" by simp_all
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   680
    from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   681
    have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   682
      using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   683
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   684
  then show ?thesis unfolding islimpt_approachable by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   685
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   686
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   687
lemma interior_closed_Un_empty_interior:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   688
  assumes cS: "closed S" and iT: "interior T = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   689
  shows "interior(S \<union> T) = interior S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   690
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   691
  show "interior S \<subseteq> interior (S\<union>T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   692
    by (rule subset_interior, blast)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   693
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   694
  show "interior (S \<union> T) \<subseteq> interior S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   695
  proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   696
    fix x assume "x \<in> interior (S \<union> T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   697
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   698
      unfolding interior_def by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   699
    show "x \<in> interior S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   700
    proof (rule ccontr)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   701
      assume "x \<notin> interior S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   702
      with `x \<in> R` `open R` obtain y where "y \<in> R - S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   703
        unfolding interior_def expand_set_eq by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   704
      from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   705
      from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   706
      from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   707
      show "False" unfolding interior_def by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   708
    qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   709
  qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   710
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   711
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   712
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   713
subsection{* Closure of a Set *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   714
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   715
definition "closure S = S \<union> {x | x. x islimpt S}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   716
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   717
lemma closure_interior: "closure S = UNIV - interior (UNIV - S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   718
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   719
  { fix x
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   720
    have "x\<in>UNIV - interior (UNIV - S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   721
    proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   722
      let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> UNIV - S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   723
      assume "?lhs"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   724
      hence *:"\<not> ?exT x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   725
        unfolding interior_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   726
        by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   727
      { assume "\<not> ?rhs"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   728
        hence False using *
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   729
          unfolding closure_def islimpt_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   730
          by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   731
      }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   732
      thus "?rhs"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   733
        by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   734
    next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   735
      assume "?rhs" thus "?lhs"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   736
        unfolding closure_def interior_def islimpt_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   737
        by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   738
    qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   739
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   740
  thus ?thesis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   741
    by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   742
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   743
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   744
lemma interior_closure: "interior S = UNIV - (closure (UNIV - S))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   745
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   746
  { fix x
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   747
    have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   748
      unfolding interior_def closure_def islimpt_def
33324
51eb2ffa2189 Tidied up some very ugly proofs
paulson
parents: 33270
diff changeset
   749
      by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   750
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   751
  thus ?thesis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   752
    by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   753
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   754
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   755
lemma closed_closure[simp, intro]: "closed (closure S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   756
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   757
  have "closed (UNIV - interior (UNIV -S))" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   758
  thus ?thesis using closure_interior[of S] by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   759
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   760
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   761
lemma closure_hull: "closure S = closed hull S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   762
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   763
  have "S \<subseteq> closure S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   764
    unfolding closure_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   765
    by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   766
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   767
  have "closed (closure S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   768
    using closed_closure[of S]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   769
    by assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   770
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   771
  { fix t
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   772
    assume *:"S \<subseteq> t" "closed t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   773
    { fix x
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   774
      assume "x islimpt S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   775
      hence "x islimpt t" using *(1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   776
        using islimpt_subset[of x, of S, of t]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   777
        by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   778
    }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   779
    with * have "closure S \<subseteq> t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   780
      unfolding closure_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   781
      using closed_limpt[of t]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   782
      by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   783
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   784
  ultimately show ?thesis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   785
    using hull_unique[of S, of "closure S", of closed]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   786
    unfolding mem_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   787
    by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   788
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   789
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   790
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   791
  unfolding closure_hull
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   792
  using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   793
  by (metis mem_def subset_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   794
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   795
lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   796
  using closure_eq[of S]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   797
  by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   798
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   799
lemma closure_closure[simp]: "closure (closure S) = closure S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   800
  unfolding closure_hull
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   801
  using hull_hull[of closed S]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   802
  by assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   803
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   804
lemma closure_subset: "S \<subseteq> closure S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   805
  unfolding closure_hull
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   806
  using hull_subset[of S closed]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   807
  by assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   808
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   809
lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   810
  unfolding closure_hull
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   811
  using hull_mono[of S T closed]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   812
  by assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   813
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   814
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   815
  using hull_minimal[of S T closed]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   816
  unfolding closure_hull mem_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   817
  by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   818
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   819
lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   820
  using hull_unique[of S T closed]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   821
  unfolding closure_hull mem_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   822
  by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   823
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   824
lemma closure_empty[simp]: "closure {} = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   825
  using closed_empty closure_closed[of "{}"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   826
  by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   827
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   828
lemma closure_univ[simp]: "closure UNIV = UNIV"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   829
  using closure_closed[of UNIV]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   830
  by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   831
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   832
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   833
  using closure_empty closure_subset[of S]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   834
  by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   835
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   836
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   837
  using closure_eq[of S] closure_subset[of S]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   838
  by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   839
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   840
lemma open_inter_closure_eq_empty:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   841
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   842
  using open_subset_interior[of S "UNIV - T"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   843
  using interior_subset[of "UNIV - T"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   844
  unfolding closure_interior
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   845
  by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   846
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   847
lemma open_inter_closure_subset:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   848
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   849
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   850
  fix x
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   851
  assume as: "open S" "x \<in> S \<inter> closure T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   852
  { assume *:"x islimpt T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   853
    have "x islimpt (S \<inter> T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   854
    proof (rule islimptI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   855
      fix A
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   856
      assume "x \<in> A" "open A"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   857
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   858
        by (simp_all add: open_Int)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   859
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   860
        by (rule islimptE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   861
      hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   862
        by simp_all
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   863
      thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   864
    qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   865
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   866
  then show "x \<in> closure (S \<inter> T)" using as
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   867
    unfolding closure_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   868
    by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   869
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   870
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   871
lemma closure_complement: "closure(UNIV - S) = UNIV - interior(S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   872
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   873
  have "S = UNIV - (UNIV - S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   874
    by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   875
  thus ?thesis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   876
    unfolding closure_interior
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   877
    by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   878
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   879
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   880
lemma interior_complement: "interior(UNIV - S) = UNIV - closure(S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   881
  unfolding closure_interior
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   882
  by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   883
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   884
subsection{* Frontier (aka boundary) *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   885
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   886
definition "frontier S = closure S - interior S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   887
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   888
lemma frontier_closed: "closed(frontier S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   889
  by (simp add: frontier_def closed_Diff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   890
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   891
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(UNIV - S))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   892
  by (auto simp add: frontier_def interior_closure)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   893
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   894
lemma frontier_straddle:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   895
  fixes a :: "'a::metric_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   896
  shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))" (is "?lhs \<longleftrightarrow> ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   897
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   898
  assume "?lhs"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   899
  { fix e::real
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   900
    assume "e > 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   901
    let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   902
    { assume "a\<in>S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   903
      have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   904
      moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   905
        unfolding frontier_closures closure_def islimpt_def using `e>0`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   906
        by (auto, erule_tac x="ball a e" in allE, auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   907
      ultimately have ?rhse by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   908
    }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   909
    moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   910
    { assume "a\<notin>S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   911
      hence ?rhse using `?lhs`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   912
        unfolding frontier_closures closure_def islimpt_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   913
        using open_ball[of a e] `e > 0`
33324
51eb2ffa2189 Tidied up some very ugly proofs
paulson
parents: 33270
diff changeset
   914
          by simp (metis centre_in_ball mem_ball open_ball) 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   915
    }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   916
    ultimately have ?rhse by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   917
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   918
  thus ?rhs by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   919
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   920
  assume ?rhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   921
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   922
  { fix T assume "a\<notin>S" and
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   923
    as:"\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)" "a \<notin> S" "a \<in> T" "open T"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   924
    from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   925
    then obtain e where "e>0" "ball a e \<subseteq> T" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   926
    then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   927
    have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   928
      using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   929
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   930
  hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   931
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   932
  { fix T assume "a \<in> T"  "open T" "a\<in>S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   933
    then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   934
    obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   935
    hence "\<exists>y\<in>UNIV - S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   936
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   937
  hence "a islimpt (UNIV - S) \<or> a\<notin>S" unfolding islimpt_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   938
  ultimately show ?lhs unfolding frontier_closures using closure_def[of "UNIV - S"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   939
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   940
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   941
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   942
  by (metis frontier_def closure_closed Diff_subset)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   943
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   944
lemma frontier_empty: "frontier {} = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   945
  by (simp add: frontier_def closure_empty)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   946
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   947
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   948
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   949
  { assume "frontier S \<subseteq> S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   950
    hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   951
    hence "closed S" using closure_subset_eq by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   952
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   953
  thus ?thesis using frontier_subset_closed[of S] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   954
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   955
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   956
lemma frontier_complement: "frontier(UNIV - S) = frontier S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   957
  by (auto simp add: frontier_def closure_complement interior_complement)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   958
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   959
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   960
  using frontier_complement frontier_subset_eq[of "UNIV - S"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   961
  unfolding open_closed Compl_eq_Diff_UNIV by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   962
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   963
subsection{* Common nets and The "within" modifier for nets. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   964
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   965
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   966
  at_infinity :: "'a::real_normed_vector net" where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   967
  "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   968
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   969
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   970
  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   971
  "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   972
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   973
text{* Prove That They are all nets. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   974
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   975
lemma Rep_net_at_infinity:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   976
  "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   977
unfolding at_infinity_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   978
apply (rule Abs_net_inverse')
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   979
apply (rule image_nonempty, simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   980
apply (clarsimp, rename_tac r s)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   981
apply (rule_tac x="max r s" in exI, auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   982
done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   983
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   984
lemma within_UNIV: "net within UNIV = net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   985
  by (simp add: Rep_net_inject [symmetric] Rep_net_within)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   986
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   987
subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   988
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   989
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   990
  trivial_limit :: "'a net \<Rightarrow> bool" where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   991
  "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   992
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   993
lemma trivial_limit_within:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   994
  shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   995
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   996
  assume "trivial_limit (at a within S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   997
  thus "\<not> a islimpt S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   998
    unfolding trivial_limit_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   999
    unfolding Rep_net_within Rep_net_at
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1000
    unfolding islimpt_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1001
    apply (clarsimp simp add: expand_set_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1002
    apply (rename_tac T, rule_tac x=T in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1003
    apply (clarsimp, drule_tac x=y in spec, simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1004
    done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1005
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1006
  assume "\<not> a islimpt S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1007
  thus "trivial_limit (at a within S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1008
    unfolding trivial_limit_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1009
    unfolding Rep_net_within Rep_net_at
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1010
    unfolding islimpt_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1011
    apply (clarsimp simp add: image_image)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1012
    apply (rule_tac x=T in image_eqI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1013
    apply (auto simp add: expand_set_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1014
    done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1015
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1016
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1017
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1018
  using trivial_limit_within [of a UNIV]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1019
  by (simp add: within_UNIV)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1020
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1021
lemma trivial_limit_at:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1022
  fixes a :: "'a::perfect_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1023
  shows "\<not> trivial_limit (at a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1024
  by (simp add: trivial_limit_at_iff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1025
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1026
lemma trivial_limit_at_infinity:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1027
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1028
  (* FIXME: find a more appropriate type class *)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1029
  unfolding trivial_limit_def Rep_net_at_infinity
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1030
  apply (clarsimp simp add: expand_set_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1031
  apply (drule_tac x="scaleR r (sgn 1)" in spec)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1032
  apply (simp add: norm_sgn)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1033
  done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1034
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1035
lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1036
  by (auto simp add: trivial_limit_def Rep_net_sequentially)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1037
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1038
subsection{* Some property holds "sufficiently close" to the limit point. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1039
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1040
lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1041
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1042
unfolding eventually_at dist_nz by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1043
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1044
lemma eventually_at_infinity:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1045
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1046
unfolding eventually_def Rep_net_at_infinity by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1047
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1048
lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1049
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1050
unfolding eventually_within eventually_at dist_nz by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1051
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1052
lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1053
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1054
unfolding eventually_within
33324
51eb2ffa2189 Tidied up some very ugly proofs
paulson
parents: 33270
diff changeset
  1055
by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl) 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1056
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1057
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1058
  unfolding eventually_def trivial_limit_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1059
  using Rep_net_nonempty [of net] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1060
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1061
lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1062
  unfolding eventually_def trivial_limit_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1063
  using Rep_net_nonempty [of net] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1064
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1065
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1066
  unfolding trivial_limit_def eventually_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1067
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1068
lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1069
  unfolding trivial_limit_def eventually_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1070
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1071
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1072
  apply (safe elim!: trivial_limit_eventually)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1073
  apply (simp add: eventually_False [symmetric])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1074
  done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1075
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1076
text{* Combining theorems for "eventually" *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1077
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1078
lemma eventually_conjI:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1079
  "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1080
    \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1081
by (rule eventually_conj)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1082
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1083
lemma eventually_rev_mono:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1084
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1085
using eventually_mono [of P Q] by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1086
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1087
lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1088
  by (auto intro!: eventually_conjI elim: eventually_rev_mono)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1089
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1090
lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1091
  by (auto simp add: eventually_False)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1092
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1093
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1094
  by (simp add: eventually_False)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1095
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1096
subsection{* Limits, defined as vacuously true when the limit is trivial. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1097
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1098
  text{* Notation Lim to avoid collition with lim defined in analysis *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1099
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1100
  Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b" where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1101
  "Lim net f = (THE l. (f ---> l) net)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1102
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1103
lemma Lim:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1104
 "(f ---> l) net \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1105
        trivial_limit net \<or>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1106
        (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1107
  unfolding tendsto_iff trivial_limit_eq by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1108
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1109
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1110
text{* Show that they yield usual definitions in the various cases. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1111
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1112
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1113
           (\<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)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1114
  by (auto simp add: tendsto_iff eventually_within_le)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1115
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1116
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1117
        (\<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)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1118
  by (auto simp add: tendsto_iff eventually_within)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1119
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1120
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1121
        (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1122
  by (auto simp add: tendsto_iff eventually_at)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1123
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1124
lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1125
  unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1126
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1127
lemma Lim_at_infinity:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1128
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1129
  by (auto simp add: tendsto_iff eventually_at_infinity)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1130
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1131
lemma Lim_sequentially:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1132
 "(S ---> l) sequentially \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1133
          (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1134
  by (auto simp add: tendsto_iff eventually_sequentially)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1135
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1136
lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1137
  unfolding Lim_sequentially LIMSEQ_def ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1138
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1139
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1140
  by (rule topological_tendstoI, auto elim: eventually_rev_mono)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1141
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1142
text{* The expected monotonicity property. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1143
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1144
lemma Lim_within_empty: "(f ---> l) (net within {})"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1145
  unfolding tendsto_def Limits.eventually_within by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1146
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1147
lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1148
  unfolding tendsto_def Limits.eventually_within
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1149
  by (auto elim!: eventually_elim1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1150
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1151
lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1152
  shows "(f ---> l) (net within (S \<union> T))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1153
  using assms unfolding tendsto_def Limits.eventually_within
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1154
  apply clarify
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1155
  apply (drule spec, drule (1) mp, drule (1) mp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1156
  apply (drule spec, drule (1) mp, drule (1) mp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1157
  apply (auto elim: eventually_elim2)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1158
  done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1159
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1160
lemma Lim_Un_univ:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1161
 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1162
        ==> (f ---> l) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1163
  by (metis Lim_Un within_UNIV)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1164
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1165
text{* Interrelations between restricted and unrestricted limits. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1166
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1167
lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1168
  (* FIXME: rename *)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1169
  unfolding tendsto_def Limits.eventually_within
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1170
  apply (clarify, drule spec, drule (1) mp, drule (1) mp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1171
  by (auto elim!: eventually_elim1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1172
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1173
lemma Lim_within_open:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1174
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1175
  assumes"a \<in> S" "open S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1176
  shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1177
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1178
  assume ?lhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1179
  { fix A assume "open A" "l \<in> A"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1180
    with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1181
      by (rule topological_tendstoD)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1182
    hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1183
      unfolding Limits.eventually_within .
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1184
    then obtain T where "open T" "a \<in> T" "\<forall>x\<in>T. x \<noteq> a \<longrightarrow> x \<in> S \<longrightarrow> f x \<in> A"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1185
      unfolding eventually_at_topological by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1186
    hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1187
      using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1188
    hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1189
      by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1190
    hence "eventually (\<lambda>x. f x \<in> A) (at a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1191
      unfolding eventually_at_topological .
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1192
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1193
  thus ?rhs by (rule topological_tendstoI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1194
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1195
  assume ?rhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1196
  thus ?lhs by (rule Lim_at_within)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1197
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1198
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1199
text{* Another limit point characterization. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1200
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1201
lemma islimpt_sequential:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1202
  fixes x :: "'a::metric_space" (* FIXME: generalize to topological_space *)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1203
  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1204
    (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1205
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1206
  assume ?lhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1207
  then obtain f where f:"\<forall>y. y>0 \<longrightarrow> f y \<in> S \<and> f y \<noteq> x \<and> dist (f y) x < y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1208
    unfolding islimpt_approachable using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1209
  { fix n::nat
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1210
    have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1211
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1212
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1213
  { fix e::real assume "e>0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1214
    hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1215
    then obtain N::nat where "inverse (real (N + 1)) < e" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1216
    hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1217
    moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1218
    ultimately have "\<exists>N::nat. \<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < e" apply(rule_tac x=N in exI) apply auto apply(erule_tac x=n in allE)+ by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1219
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1220
  hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1221
    unfolding Lim_sequentially using f by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1222
  ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1223
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1224
  assume ?rhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1225
  then obtain f::"nat\<Rightarrow>'a"  where f:"(\<forall>n. f n \<in> S - {x})" "(\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f n) x < e)" unfolding Lim_sequentially by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1226
  { fix e::real assume "e>0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1227
    then obtain N where "dist (f N) x < e" using f(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1228
    moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1229
    ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1230
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1231
  thus ?lhs unfolding islimpt_approachable by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1232
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1233
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1234
text{* Basic arithmetical combining theorems for limits. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1235
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1236
lemma Lim_linear:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1237
  assumes "(f ---> l) net" "bounded_linear h"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1238
  shows "((\<lambda>x. h (f x)) ---> h l) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1239
using `bounded_linear h` `(f ---> l) net`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1240
by (rule bounded_linear.tendsto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1241
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1242
lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1243
  unfolding tendsto_def Limits.eventually_at_topological by fast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1244
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1245
lemma Lim_const: "((\<lambda>x. a) ---> a) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1246
  by (rule tendsto_const)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1247
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1248
lemma Lim_cmul:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1249
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1250
  shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1251
  by (intro tendsto_intros)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1252
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1253
lemma Lim_neg:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1254
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1255
  shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1256
  by (rule tendsto_minus)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1257
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1258
lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1259
 "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1260
  by (rule tendsto_add)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1261
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1262
lemma Lim_sub:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1263
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1264
  shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1265
  by (rule tendsto_diff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1266
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1267
lemma Lim_null:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1268
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1269
  shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1270
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1271
lemma Lim_null_norm:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1272
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1273
  shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1274
  by (simp add: Lim dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1275
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1276
lemma Lim_null_comparison:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1277
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1278
  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1279
  shows "(f ---> 0) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1280
proof(simp add: tendsto_iff, rule+)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1281
  fix e::real assume "0<e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1282
  { fix x
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1283
    assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1284
    hence "dist (f x) 0 < e" by (simp add: dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1285
  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1286
  thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1287
    using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1288
    using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (g x) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1289
    using assms `e>0` unfolding tendsto_iff by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1290
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1291
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1292
lemma Lim_component:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1293
  fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1294
  shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1295
  unfolding tendsto_iff
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1296
  apply (clarify)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1297
  apply (drule spec, drule (1) mp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1298
  apply (erule eventually_elim1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1299
  apply (erule le_less_trans [OF dist_nth_le])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1300
  done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1301
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1302
lemma Lim_transform_bound:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1303
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1304
  fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1305
  assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1306
  shows "(f ---> 0) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1307
proof (rule tendstoI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1308
  fix e::real assume "e>0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1309
  { fix x
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1310
    assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1311
    hence "dist (f x) 0 < e" by (simp add: dist_norm)}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1312
  thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1313
    using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1314
    using eventually_mono[of "\<lambda>x. norm (f x) \<le> norm (g x) \<and> dist (g x) 0 < e" "\<lambda>x. dist (f x) 0 < e" net]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1315
    using assms `e>0` unfolding tendsto_iff by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1316
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1317
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1318
text{* Deducing things about the limit from the elements. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1319
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1320
lemma Lim_in_closed_set:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1321
  assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1322
  shows "l \<in> S"