src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author hoelzl
Mon Jun 21 19:33:51 2010 +0200 (2010-06-21)
changeset 37489 44e42d392c6e
parent 37452 8f515d6aded5
child 37490 9de1add14bac
permissions -rw-r--r--
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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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 Glbs
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begin
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(* to be moved elsewhere *)
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lemma euclidean_dist_l2:"dist x (y::'a::euclidean_space) = setL2 (\<lambda>i. dist(x$$i) (y$$i)) {..<DIM('a)}"
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  unfolding dist_norm norm_eq_sqrt_inner setL2_def apply(subst euclidean_inner)
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  apply(auto simp add:power2_eq_square) unfolding euclidean_component.diff ..
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lemma dist_nth_le: "dist (x $$ i) (y $$ i) \<le> dist x (y::'a::euclidean_space)"
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  apply(subst(2) euclidean_dist_l2) apply(cases "i<DIM('a)")
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  apply(rule member_le_setL2) by auto
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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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  unfolding subset_eq Ball_def mem_def by auto
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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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  using openin_clauses by simp
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lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
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  using openin_clauses by simp
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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 show ?rhs by auto
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next
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  assume H: ?rhs
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  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
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  have "openin U ?t" by (simp add: openin_Union)
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  also have "?t = S" using H by auto
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  finally show "openin U S" .
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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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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)
himmelma@33175
   274
  apply atomize
himmelma@33175
   275
  apply (erule_tac x="y" in allE)
himmelma@33175
   276
  apply (erule_tac x="xa" in allE)
himmelma@33175
   277
  by arith
himmelma@33175
   278
himmelma@33175
   279
lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
himmelma@33175
   280
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   281
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
himmelma@33175
   282
hoelzl@33714
   283
lemma openE[elim?]:
hoelzl@33714
   284
  assumes "open S" "x\<in>S" 
hoelzl@33714
   285
  obtains e where "e>0" "ball x e \<subseteq> S"
hoelzl@33714
   286
  using assms unfolding open_contains_ball by auto
hoelzl@33714
   287
himmelma@33175
   288
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   289
  by (metis open_contains_ball subset_eq centre_in_ball)
himmelma@33175
   290
himmelma@33175
   291
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
himmelma@33175
   292
  unfolding mem_ball expand_set_eq
himmelma@33175
   293
  apply (simp add: not_less)
himmelma@33175
   294
  by (metis zero_le_dist order_trans dist_self)
himmelma@33175
   295
himmelma@33175
   296
lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
himmelma@33175
   297
himmelma@33175
   298
subsection{* Basic "localization" results are handy for connectedness. *}
himmelma@33175
   299
himmelma@33175
   300
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
himmelma@33175
   301
  by (auto simp add: openin_subtopology open_openin[symmetric])
himmelma@33175
   302
himmelma@33175
   303
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
himmelma@33175
   304
  by (auto simp add: openin_open)
himmelma@33175
   305
himmelma@33175
   306
lemma open_openin_trans[trans]:
himmelma@33175
   307
 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
himmelma@33175
   308
  by (metis Int_absorb1  openin_open_Int)
himmelma@33175
   309
himmelma@33175
   310
lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
himmelma@33175
   311
  by (auto simp add: openin_open)
himmelma@33175
   312
himmelma@33175
   313
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
himmelma@33175
   314
  by (simp add: closedin_subtopology closed_closedin Int_ac)
himmelma@33175
   315
himmelma@33175
   316
lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
himmelma@33175
   317
  by (metis closedin_closed)
himmelma@33175
   318
himmelma@33175
   319
lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
himmelma@33175
   320
  apply (subgoal_tac "S \<inter> T = T" )
himmelma@33175
   321
  apply auto
himmelma@33175
   322
  apply (frule closedin_closed_Int[of T S])
himmelma@33175
   323
  by simp
himmelma@33175
   324
himmelma@33175
   325
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
himmelma@33175
   326
  by (auto simp add: closedin_closed)
himmelma@33175
   327
himmelma@33175
   328
lemma openin_euclidean_subtopology_iff:
himmelma@33175
   329
  fixes S U :: "'a::metric_space set"
himmelma@33175
   330
  shows "openin (subtopology euclidean U) S
himmelma@33175
   331
  \<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")
himmelma@33175
   332
proof-
himmelma@33175
   333
  {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
himmelma@33175
   334
      by (simp add: open_dist) blast}
himmelma@33175
   335
  moreover
himmelma@33175
   336
  {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"
himmelma@33175
   337
    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)"
himmelma@33175
   338
      by metis
himmelma@33175
   339
    let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
himmelma@33175
   340
    have oT: "open ?T" by auto
himmelma@33175
   341
    { fix x assume "x\<in>S"
himmelma@33175
   342
      hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
himmelma@33175
   343
        apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
himmelma@33175
   344
        by (rule d [THEN conjunct1])
himmelma@33175
   345
      hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }
himmelma@33175
   346
    moreover
himmelma@33175
   347
    { fix y assume "y\<in>?T"
himmelma@33175
   348
      then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
himmelma@33175
   349
      then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
himmelma@33175
   350
      assume "y\<in>U"
himmelma@33175
   351
      hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
himmelma@33175
   352
    ultimately have "S = ?T \<inter> U" by blast
himmelma@33175
   353
    with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
himmelma@33175
   354
  ultimately show ?thesis by blast
himmelma@33175
   355
qed
himmelma@33175
   356
himmelma@33175
   357
text{* These "transitivity" results are handy too. *}
himmelma@33175
   358
himmelma@33175
   359
lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
himmelma@33175
   360
  \<Longrightarrow> openin (subtopology euclidean U) S"
himmelma@33175
   361
  unfolding open_openin openin_open by blast
himmelma@33175
   362
himmelma@33175
   363
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
himmelma@33175
   364
  by (auto simp add: openin_open intro: openin_trans)
himmelma@33175
   365
himmelma@33175
   366
lemma closedin_trans[trans]:
himmelma@33175
   367
 "closedin (subtopology euclidean T) S \<Longrightarrow>
himmelma@33175
   368
           closedin (subtopology euclidean U) T
himmelma@33175
   369
           ==> closedin (subtopology euclidean U) S"
himmelma@33175
   370
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
himmelma@33175
   371
himmelma@33175
   372
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
himmelma@33175
   373
  by (auto simp add: closedin_closed intro: closedin_trans)
himmelma@33175
   374
himmelma@33175
   375
subsection{* Connectedness *}
himmelma@33175
   376
himmelma@33175
   377
definition "connected S \<longleftrightarrow>
himmelma@33175
   378
  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
himmelma@33175
   379
  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
himmelma@33175
   380
himmelma@33175
   381
lemma connected_local:
himmelma@33175
   382
 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
himmelma@33175
   383
                 openin (subtopology euclidean S) e1 \<and>
himmelma@33175
   384
                 openin (subtopology euclidean S) e2 \<and>
himmelma@33175
   385
                 S \<subseteq> e1 \<union> e2 \<and>
himmelma@33175
   386
                 e1 \<inter> e2 = {} \<and>
himmelma@33175
   387
                 ~(e1 = {}) \<and>
himmelma@33175
   388
                 ~(e2 = {}))"
himmelma@33175
   389
unfolding connected_def openin_open by (safe, blast+)
himmelma@33175
   390
huffman@34105
   391
lemma exists_diff:
huffman@34105
   392
  fixes P :: "'a set \<Rightarrow> bool"
huffman@34105
   393
  shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
himmelma@33175
   394
proof-
himmelma@33175
   395
  {assume "?lhs" hence ?rhs by blast }
himmelma@33175
   396
  moreover
himmelma@33175
   397
  {fix S assume H: "P S"
huffman@34105
   398
    have "S = - (- S)" by auto
huffman@34105
   399
    with H have "P (- (- S))" by metis }
himmelma@33175
   400
  ultimately show ?thesis by metis
himmelma@33175
   401
qed
himmelma@33175
   402
himmelma@33175
   403
lemma connected_clopen: "connected S \<longleftrightarrow>
himmelma@33175
   404
        (\<forall>T. openin (subtopology euclidean S) T \<and>
himmelma@33175
   405
            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
himmelma@33175
   406
proof-
huffman@34105
   407
  have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
himmelma@33175
   408
    unfolding connected_def openin_open closedin_closed
himmelma@33175
   409
    apply (subst exists_diff) by blast
huffman@34105
   410
  hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
huffman@34105
   411
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
himmelma@33175
   412
himmelma@33175
   413
  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'))"
himmelma@33175
   414
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
himmelma@33175
   415
    unfolding connected_def openin_open closedin_closed by auto
himmelma@33175
   416
  {fix e2
himmelma@33175
   417
    {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)"
himmelma@33175
   418
        by auto}
himmelma@33175
   419
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
himmelma@33175
   420
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
himmelma@33175
   421
  then show ?thesis unfolding th0 th1 by simp
himmelma@33175
   422
qed
himmelma@33175
   423
himmelma@33175
   424
lemma connected_empty[simp, intro]: "connected {}"
himmelma@33175
   425
  by (simp add: connected_def)
himmelma@33175
   426
himmelma@33175
   427
subsection{* Hausdorff and other separation properties *}
himmelma@33175
   428
huffman@36668
   429
class t0_space = topological_space +
himmelma@33175
   430
  assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
himmelma@33175
   431
huffman@36668
   432
class t1_space = topological_space +
huffman@36668
   433
  assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
huffman@36668
   434
huffman@36668
   435
instance t1_space \<subseteq> t0_space
huffman@36668
   436
proof qed (fast dest: t1_space)
huffman@36668
   437
huffman@36668
   438
lemma separation_t1:
huffman@36668
   439
  fixes x y :: "'a::t1_space"
huffman@36668
   440
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
huffman@36668
   441
  using t1_space[of x y] by blast
huffman@36668
   442
huffman@36668
   443
lemma closed_sing:
huffman@36668
   444
  fixes a :: "'a::t1_space"
huffman@36668
   445
  shows "closed {a}"
huffman@36668
   446
proof -
huffman@36668
   447
  let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
huffman@36668
   448
  have "open ?T" by (simp add: open_Union)
huffman@36668
   449
  also have "?T = - {a}"
huffman@36668
   450
    by (simp add: expand_set_eq separation_t1, auto)
huffman@36668
   451
  finally show "closed {a}" unfolding closed_def .
huffman@36668
   452
qed
huffman@36668
   453
huffman@36668
   454
lemma closed_insert [simp]:
huffman@36668
   455
  fixes a :: "'a::t1_space"
huffman@36668
   456
  assumes "closed S" shows "closed (insert a S)"
huffman@36668
   457
proof -
huffman@36668
   458
  from closed_sing assms
huffman@36668
   459
  have "closed ({a} \<union> S)" by (rule closed_Un)
huffman@36668
   460
  thus "closed (insert a S)" by simp
huffman@36668
   461
qed
huffman@36668
   462
huffman@36668
   463
lemma finite_imp_closed:
huffman@36668
   464
  fixes S :: "'a::t1_space set"
huffman@36668
   465
  shows "finite S \<Longrightarrow> closed S"
huffman@36668
   466
by (induct set: finite, simp_all)
himmelma@33175
   467
himmelma@33175
   468
text {* T2 spaces are also known as Hausdorff spaces. *}
himmelma@33175
   469
huffman@36668
   470
class t2_space = topological_space +
himmelma@33175
   471
  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 = {}"
huffman@36668
   472
huffman@36668
   473
instance t2_space \<subseteq> t1_space
huffman@36668
   474
proof qed (fast dest: hausdorff)
himmelma@33175
   475
himmelma@33175
   476
instance metric_space \<subseteq> t2_space
himmelma@33175
   477
proof
himmelma@33175
   478
  fix x y :: "'a::metric_space"
himmelma@33175
   479
  assume xy: "x \<noteq> y"
himmelma@33175
   480
  let ?U = "ball x (dist x y / 2)"
himmelma@33175
   481
  let ?V = "ball y (dist x y / 2)"
himmelma@33175
   482
  have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
himmelma@33175
   483
               ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
himmelma@33175
   484
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
himmelma@33175
   485
    using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
himmelma@33175
   486
    by (auto simp add: expand_set_eq)
himmelma@33175
   487
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
himmelma@33175
   488
    by blast
himmelma@33175
   489
qed
himmelma@33175
   490
himmelma@33175
   491
lemma separation_t2:
himmelma@33175
   492
  fixes x y :: "'a::t2_space"
himmelma@33175
   493
  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 = {})"
himmelma@33175
   494
  using hausdorff[of x y] by blast
himmelma@33175
   495
himmelma@33175
   496
lemma separation_t0:
himmelma@33175
   497
  fixes x y :: "'a::t0_space"
himmelma@33175
   498
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
himmelma@33175
   499
  using t0_space[of x y] by blast
himmelma@33175
   500
himmelma@33175
   501
subsection{* Limit points *}
himmelma@33175
   502
himmelma@33175
   503
definition
himmelma@33175
   504
  islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
himmelma@33175
   505
    (infixr "islimpt" 60) where
himmelma@33175
   506
  "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
himmelma@33175
   507
himmelma@33175
   508
lemma islimptI:
himmelma@33175
   509
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
himmelma@33175
   510
  shows "x islimpt S"
himmelma@33175
   511
  using assms unfolding islimpt_def by auto
himmelma@33175
   512
himmelma@33175
   513
lemma islimptE:
himmelma@33175
   514
  assumes "x islimpt S" and "x \<in> T" and "open T"
himmelma@33175
   515
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
himmelma@33175
   516
  using assms unfolding islimpt_def by auto
himmelma@33175
   517
himmelma@33175
   518
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
himmelma@33175
   519
himmelma@33175
   520
lemma islimpt_approachable:
himmelma@33175
   521
  fixes x :: "'a::metric_space"
himmelma@33175
   522
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
himmelma@33175
   523
  unfolding islimpt_def
himmelma@33175
   524
  apply auto
himmelma@33175
   525
  apply(erule_tac x="ball x e" in allE)
himmelma@33175
   526
  apply auto
himmelma@33175
   527
  apply(rule_tac x=y in bexI)
himmelma@33175
   528
  apply (auto simp add: dist_commute)
himmelma@33175
   529
  apply (simp add: open_dist, drule (1) bspec)
himmelma@33175
   530
  apply (clarify, drule spec, drule (1) mp, auto)
himmelma@33175
   531
  done
himmelma@33175
   532
himmelma@33175
   533
lemma islimpt_approachable_le:
himmelma@33175
   534
  fixes x :: "'a::metric_space"
himmelma@33175
   535
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
himmelma@33175
   536
  unfolding islimpt_approachable
himmelma@33175
   537
  using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
paulson@33324
   538
  by metis 
himmelma@33175
   539
himmelma@33175
   540
class perfect_space =
himmelma@33175
   541
  (* FIXME: perfect_space should inherit from topological_space *)
himmelma@33175
   542
  assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"
himmelma@33175
   543
himmelma@33175
   544
lemma perfect_choose_dist:
himmelma@33175
   545
  fixes x :: "'a::perfect_space"
himmelma@33175
   546
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
himmelma@33175
   547
using islimpt_UNIV [of x]
himmelma@33175
   548
by (simp add: islimpt_approachable)
himmelma@33175
   549
himmelma@33175
   550
instance real :: perfect_space
himmelma@33175
   551
apply default
himmelma@33175
   552
apply (rule islimpt_approachable [THEN iffD2])
himmelma@33175
   553
apply (clarify, rule_tac x="x + e/2" in bexI)
himmelma@33175
   554
apply (auto simp add: dist_norm)
himmelma@33175
   555
done
himmelma@33175
   556
hoelzl@37489
   557
instance euclidean_space \<subseteq> perfect_space
hoelzl@37489
   558
proof fix x::'a
hoelzl@37489
   559
  { fix e :: real assume "0 < e"
hoelzl@37489
   560
    def a \<equiv> "x $$ 0"
himmelma@33175
   561
    have "a islimpt UNIV" by (rule islimpt_UNIV)
himmelma@33175
   562
    with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
himmelma@33175
   563
      unfolding islimpt_approachable by auto
hoelzl@37489
   564
    def y \<equiv> "\<chi>\<chi> i. if i = 0 then b else x$$i :: 'a"
hoelzl@37489
   565
    from `b \<noteq> a` have "y \<noteq> x" unfolding a_def y_def apply(subst euclidean_eq) apply safe
hoelzl@37489
   566
      apply(erule_tac x=0 in allE) using DIM_positive[where 'a='a] by auto
hoelzl@37489
   567
hoelzl@37489
   568
    have *:"(\<Sum>i<DIM('a). (dist (y $$ i) (x $$ i))\<twosuperior>) = (\<Sum>i\<in>{0}. (dist (y $$ i) (x $$ i))\<twosuperior>)"
hoelzl@37489
   569
      apply(rule setsum_mono_zero_right) unfolding y_def by auto
himmelma@33175
   570
    from `dist b a < e` have "dist y x < e"
hoelzl@37489
   571
      apply(subst euclidean_dist_l2)
hoelzl@37489
   572
      unfolding setL2_def * unfolding y_def a_def using `0 < e` by auto
himmelma@33175
   573
    from `y \<noteq> x` and `dist y x < e`
himmelma@33175
   574
    have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
himmelma@33175
   575
  }
himmelma@33175
   576
  then show "x islimpt UNIV" unfolding islimpt_approachable by blast
himmelma@33175
   577
qed
himmelma@33175
   578
himmelma@33175
   579
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
himmelma@33175
   580
  unfolding closed_def
himmelma@33175
   581
  apply (subst open_subopen)
huffman@34105
   582
  apply (simp add: islimpt_def subset_eq)
huffman@34105
   583
  by (metis ComplE ComplI insertCI insert_absorb mem_def)
himmelma@33175
   584
himmelma@33175
   585
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
himmelma@33175
   586
  unfolding islimpt_def by auto
himmelma@33175
   587
himmelma@33175
   588
lemma finite_set_avoid:
himmelma@33175
   589
  fixes a :: "'a::metric_space"
himmelma@33175
   590
  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
himmelma@33175
   591
proof(induct rule: finite_induct[OF fS])
himmelma@33175
   592
  case 1 thus ?case apply auto by ferrack
himmelma@33175
   593
next
himmelma@33175
   594
  case (2 x F)
himmelma@33175
   595
  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
himmelma@33175
   596
  {assume "x = a" hence ?case using d by auto  }
himmelma@33175
   597
  moreover
himmelma@33175
   598
  {assume xa: "x\<noteq>a"
himmelma@33175
   599
    let ?d = "min d (dist a x)"
himmelma@33175
   600
    have dp: "?d > 0" using xa d(1) using dist_nz by auto
himmelma@33175
   601
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
himmelma@33175
   602
    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
himmelma@33175
   603
  ultimately show ?case by blast
himmelma@33175
   604
qed
himmelma@33175
   605
himmelma@33175
   606
lemma islimpt_finite:
himmelma@33175
   607
  fixes S :: "'a::metric_space set"
himmelma@33175
   608
  assumes fS: "finite S" shows "\<not> a islimpt S"
himmelma@33175
   609
  unfolding islimpt_approachable
himmelma@33175
   610
  using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
himmelma@33175
   611
himmelma@33175
   612
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
himmelma@33175
   613
  apply (rule iffI)
himmelma@33175
   614
  defer
himmelma@33175
   615
  apply (metis Un_upper1 Un_upper2 islimpt_subset)
himmelma@33175
   616
  unfolding islimpt_def
himmelma@33175
   617
  apply (rule ccontr, clarsimp, rename_tac A B)
himmelma@33175
   618
  apply (drule_tac x="A \<inter> B" in spec)
himmelma@33175
   619
  apply (auto simp add: open_Int)
himmelma@33175
   620
  done
himmelma@33175
   621
himmelma@33175
   622
lemma discrete_imp_closed:
himmelma@33175
   623
  fixes S :: "'a::metric_space set"
himmelma@33175
   624
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
himmelma@33175
   625
  shows "closed S"
himmelma@33175
   626
proof-
himmelma@33175
   627
  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
himmelma@33175
   628
    from e have e2: "e/2 > 0" by arith
himmelma@33175
   629
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
himmelma@33175
   630
    let ?m = "min (e/2) (dist x y) "
himmelma@33175
   631
    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
himmelma@33175
   632
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
himmelma@33175
   633
    have th: "dist z y < e" using z y
himmelma@33175
   634
      by (intro dist_triangle_lt [where z=x], simp)
himmelma@33175
   635
    from d[rule_format, OF y(1) z(1) th] y z
himmelma@33175
   636
    have False by (auto simp add: dist_commute)}
himmelma@33175
   637
  then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
himmelma@33175
   638
qed
himmelma@33175
   639
himmelma@33175
   640
subsection{* Interior of a Set *}
himmelma@33175
   641
definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
himmelma@33175
   642
himmelma@33175
   643
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
himmelma@33175
   644
  apply (simp add: expand_set_eq interior_def)
himmelma@33175
   645
  apply (subst (2) open_subopen) by (safe, blast+)
himmelma@33175
   646
himmelma@33175
   647
lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
himmelma@33175
   648
himmelma@33175
   649
lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
himmelma@33175
   650
himmelma@33175
   651
lemma open_interior[simp, intro]: "open(interior S)"
himmelma@33175
   652
  apply (simp add: interior_def)
himmelma@33175
   653
  apply (subst open_subopen) by blast
himmelma@33175
   654
himmelma@33175
   655
lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
himmelma@33175
   656
lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
himmelma@33175
   657
lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
himmelma@33175
   658
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
himmelma@33175
   659
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"
himmelma@33175
   660
  by (metis equalityI interior_maximal interior_subset open_interior)
himmelma@33175
   661
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
himmelma@33175
   662
  apply (simp add: interior_def)
himmelma@33175
   663
  by (metis open_contains_ball centre_in_ball open_ball subset_trans)
himmelma@33175
   664
himmelma@33175
   665
lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
himmelma@33175
   666
  by (metis interior_maximal interior_subset subset_trans)
himmelma@33175
   667
himmelma@33175
   668
lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
himmelma@33175
   669
  apply (rule equalityI, simp)
himmelma@33175
   670
  apply (metis Int_lower1 Int_lower2 subset_interior)
himmelma@33175
   671
  by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
himmelma@33175
   672
himmelma@33175
   673
lemma interior_limit_point [intro]:
himmelma@33175
   674
  fixes x :: "'a::perfect_space"
himmelma@33175
   675
  assumes x: "x \<in> interior S" shows "x islimpt S"
himmelma@33175
   676
proof-
himmelma@33175
   677
  from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
himmelma@33175
   678
    unfolding mem_interior subset_eq Ball_def mem_ball by blast
himmelma@33175
   679
  {
himmelma@33175
   680
    fix d::real assume d: "d>0"
himmelma@33175
   681
    let ?m = "min d e"
himmelma@33175
   682
    have mde2: "0 < ?m" using e(1) d(1) by simp
himmelma@33175
   683
    from perfect_choose_dist [OF mde2, of x]
himmelma@33175
   684
    obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
himmelma@33175
   685
    then have "dist y x < e" "dist y x < d" by simp_all
himmelma@33175
   686
    from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
himmelma@33175
   687
    have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
himmelma@33175
   688
      using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
himmelma@33175
   689
  }
himmelma@33175
   690
  then show ?thesis unfolding islimpt_approachable by blast
himmelma@33175
   691
qed
himmelma@33175
   692
himmelma@33175
   693
lemma interior_closed_Un_empty_interior:
himmelma@33175
   694
  assumes cS: "closed S" and iT: "interior T = {}"
himmelma@33175
   695
  shows "interior(S \<union> T) = interior S"
himmelma@33175
   696
proof
himmelma@33175
   697
  show "interior S \<subseteq> interior (S\<union>T)"
himmelma@33175
   698
    by (rule subset_interior, blast)
himmelma@33175
   699
next
himmelma@33175
   700
  show "interior (S \<union> T) \<subseteq> interior S"
himmelma@33175
   701
  proof
himmelma@33175
   702
    fix x assume "x \<in> interior (S \<union> T)"
himmelma@33175
   703
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
himmelma@33175
   704
      unfolding interior_def by fast
himmelma@33175
   705
    show "x \<in> interior S"
himmelma@33175
   706
    proof (rule ccontr)
himmelma@33175
   707
      assume "x \<notin> interior S"
himmelma@33175
   708
      with `x \<in> R` `open R` obtain y where "y \<in> R - S"
himmelma@33175
   709
        unfolding interior_def expand_set_eq by fast
himmelma@33175
   710
      from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
himmelma@33175
   711
      from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
himmelma@33175
   712
      from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
himmelma@33175
   713
      show "False" unfolding interior_def by fast
himmelma@33175
   714
    qed
himmelma@33175
   715
  qed
himmelma@33175
   716
qed
himmelma@33175
   717
himmelma@33175
   718
himmelma@33175
   719
subsection{* Closure of a Set *}
himmelma@33175
   720
himmelma@33175
   721
definition "closure S = S \<union> {x | x. x islimpt S}"
himmelma@33175
   722
huffman@34105
   723
lemma closure_interior: "closure S = - interior (- S)"
himmelma@33175
   724
proof-
himmelma@33175
   725
  { fix x
huffman@34105
   726
    have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
himmelma@33175
   727
    proof
huffman@34105
   728
      let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"
himmelma@33175
   729
      assume "?lhs"
himmelma@33175
   730
      hence *:"\<not> ?exT x"
himmelma@33175
   731
        unfolding interior_def
himmelma@33175
   732
        by simp
himmelma@33175
   733
      { assume "\<not> ?rhs"
himmelma@33175
   734
        hence False using *
himmelma@33175
   735
          unfolding closure_def islimpt_def
himmelma@33175
   736
          by blast
himmelma@33175
   737
      }
himmelma@33175
   738
      thus "?rhs"
himmelma@33175
   739
        by blast
himmelma@33175
   740
    next
himmelma@33175
   741
      assume "?rhs" thus "?lhs"
himmelma@33175
   742
        unfolding closure_def interior_def islimpt_def
himmelma@33175
   743
        by blast
himmelma@33175
   744
    qed
himmelma@33175
   745
  }
himmelma@33175
   746
  thus ?thesis
himmelma@33175
   747
    by blast
himmelma@33175
   748
qed
himmelma@33175
   749
huffman@34105
   750
lemma interior_closure: "interior S = - (closure (- S))"
himmelma@33175
   751
proof-
himmelma@33175
   752
  { fix x
huffman@34105
   753
    have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"
himmelma@33175
   754
      unfolding interior_def closure_def islimpt_def
paulson@33324
   755
      by auto
himmelma@33175
   756
  }
himmelma@33175
   757
  thus ?thesis
himmelma@33175
   758
    by blast
himmelma@33175
   759
qed
himmelma@33175
   760
himmelma@33175
   761
lemma closed_closure[simp, intro]: "closed (closure S)"
himmelma@33175
   762
proof-
huffman@34105
   763
  have "closed (- interior (-S))" by blast
himmelma@33175
   764
  thus ?thesis using closure_interior[of S] by simp
himmelma@33175
   765
qed
himmelma@33175
   766
himmelma@33175
   767
lemma closure_hull: "closure S = closed hull S"
himmelma@33175
   768
proof-
himmelma@33175
   769
  have "S \<subseteq> closure S"
himmelma@33175
   770
    unfolding closure_def
himmelma@33175
   771
    by blast
himmelma@33175
   772
  moreover
himmelma@33175
   773
  have "closed (closure S)"
himmelma@33175
   774
    using closed_closure[of S]
himmelma@33175
   775
    by assumption
himmelma@33175
   776
  moreover
himmelma@33175
   777
  { fix t
himmelma@33175
   778
    assume *:"S \<subseteq> t" "closed t"
himmelma@33175
   779
    { fix x
himmelma@33175
   780
      assume "x islimpt S"
himmelma@33175
   781
      hence "x islimpt t" using *(1)
himmelma@33175
   782
        using islimpt_subset[of x, of S, of t]
himmelma@33175
   783
        by blast
himmelma@33175
   784
    }
himmelma@33175
   785
    with * have "closure S \<subseteq> t"
himmelma@33175
   786
      unfolding closure_def
himmelma@33175
   787
      using closed_limpt[of t]
himmelma@33175
   788
      by auto
himmelma@33175
   789
  }
himmelma@33175
   790
  ultimately show ?thesis
himmelma@33175
   791
    using hull_unique[of S, of "closure S", of closed]
himmelma@33175
   792
    unfolding mem_def
himmelma@33175
   793
    by simp
himmelma@33175
   794
qed
himmelma@33175
   795
himmelma@33175
   796
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
himmelma@33175
   797
  unfolding closure_hull
himmelma@33175
   798
  using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]
himmelma@33175
   799
  by (metis mem_def subset_eq)
himmelma@33175
   800
himmelma@33175
   801
lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
himmelma@33175
   802
  using closure_eq[of S]
himmelma@33175
   803
  by simp
himmelma@33175
   804
himmelma@33175
   805
lemma closure_closure[simp]: "closure (closure S) = closure S"
himmelma@33175
   806
  unfolding closure_hull
himmelma@33175
   807
  using hull_hull[of closed S]
himmelma@33175
   808
  by assumption
himmelma@33175
   809
himmelma@33175
   810
lemma closure_subset: "S \<subseteq> closure S"
himmelma@33175
   811
  unfolding closure_hull
himmelma@33175
   812
  using hull_subset[of S closed]
himmelma@33175
   813
  by assumption
himmelma@33175
   814
himmelma@33175
   815
lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
himmelma@33175
   816
  unfolding closure_hull
himmelma@33175
   817
  using hull_mono[of S T closed]
himmelma@33175
   818
  by assumption
himmelma@33175
   819
himmelma@33175
   820
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
himmelma@33175
   821
  using hull_minimal[of S T closed]
himmelma@33175
   822
  unfolding closure_hull mem_def
himmelma@33175
   823
  by simp
himmelma@33175
   824
himmelma@33175
   825
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"
himmelma@33175
   826
  using hull_unique[of S T closed]
himmelma@33175
   827
  unfolding closure_hull mem_def
himmelma@33175
   828
  by simp
himmelma@33175
   829
himmelma@33175
   830
lemma closure_empty[simp]: "closure {} = {}"
himmelma@33175
   831
  using closed_empty closure_closed[of "{}"]
himmelma@33175
   832
  by simp
himmelma@33175
   833
himmelma@33175
   834
lemma closure_univ[simp]: "closure UNIV = UNIV"
himmelma@33175
   835
  using closure_closed[of UNIV]
himmelma@33175
   836
  by simp
himmelma@33175
   837
himmelma@33175
   838
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
himmelma@33175
   839
  using closure_empty closure_subset[of S]
himmelma@33175
   840
  by blast
himmelma@33175
   841
himmelma@33175
   842
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
   843
  using closure_eq[of S] closure_subset[of S]
himmelma@33175
   844
  by simp
himmelma@33175
   845
himmelma@33175
   846
lemma open_inter_closure_eq_empty:
himmelma@33175
   847
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
huffman@34105
   848
  using open_subset_interior[of S "- T"]
huffman@34105
   849
  using interior_subset[of "- T"]
himmelma@33175
   850
  unfolding closure_interior
himmelma@33175
   851
  by auto
himmelma@33175
   852
himmelma@33175
   853
lemma open_inter_closure_subset:
himmelma@33175
   854
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
himmelma@33175
   855
proof
himmelma@33175
   856
  fix x
himmelma@33175
   857
  assume as: "open S" "x \<in> S \<inter> closure T"
himmelma@33175
   858
  { assume *:"x islimpt T"
himmelma@33175
   859
    have "x islimpt (S \<inter> T)"
himmelma@33175
   860
    proof (rule islimptI)
himmelma@33175
   861
      fix A
himmelma@33175
   862
      assume "x \<in> A" "open A"
himmelma@33175
   863
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
himmelma@33175
   864
        by (simp_all add: open_Int)
himmelma@33175
   865
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
himmelma@33175
   866
        by (rule islimptE)
himmelma@33175
   867
      hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
himmelma@33175
   868
        by simp_all
himmelma@33175
   869
      thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
himmelma@33175
   870
    qed
himmelma@33175
   871
  }
himmelma@33175
   872
  then show "x \<in> closure (S \<inter> T)" using as
himmelma@33175
   873
    unfolding closure_def
himmelma@33175
   874
    by blast
himmelma@33175
   875
qed
himmelma@33175
   876
huffman@34105
   877
lemma closure_complement: "closure(- S) = - interior(S)"
himmelma@33175
   878
proof-
huffman@34105
   879
  have "S = - (- S)"
himmelma@33175
   880
    by auto
himmelma@33175
   881
  thus ?thesis
himmelma@33175
   882
    unfolding closure_interior
himmelma@33175
   883
    by auto
himmelma@33175
   884
qed
himmelma@33175
   885
huffman@34105
   886
lemma interior_complement: "interior(- S) = - closure(S)"
himmelma@33175
   887
  unfolding closure_interior
himmelma@33175
   888
  by blast
himmelma@33175
   889
himmelma@33175
   890
subsection{* Frontier (aka boundary) *}
himmelma@33175
   891
himmelma@33175
   892
definition "frontier S = closure S - interior S"
himmelma@33175
   893
himmelma@33175
   894
lemma frontier_closed: "closed(frontier S)"
himmelma@33175
   895
  by (simp add: frontier_def closed_Diff)
himmelma@33175
   896
huffman@34105
   897
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
himmelma@33175
   898
  by (auto simp add: frontier_def interior_closure)
himmelma@33175
   899
himmelma@33175
   900
lemma frontier_straddle:
himmelma@33175
   901
  fixes a :: "'a::metric_space"
himmelma@33175
   902
  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")
himmelma@33175
   903
proof
himmelma@33175
   904
  assume "?lhs"
himmelma@33175
   905
  { fix e::real
himmelma@33175
   906
    assume "e > 0"
himmelma@33175
   907
    let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
himmelma@33175
   908
    { assume "a\<in>S"
himmelma@33175
   909
      have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
himmelma@33175
   910
      moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
himmelma@33175
   911
        unfolding frontier_closures closure_def islimpt_def using `e>0`
himmelma@33175
   912
        by (auto, erule_tac x="ball a e" in allE, auto)
himmelma@33175
   913
      ultimately have ?rhse by auto
himmelma@33175
   914
    }
himmelma@33175
   915
    moreover
himmelma@33175
   916
    { assume "a\<notin>S"
himmelma@33175
   917
      hence ?rhse using `?lhs`
himmelma@33175
   918
        unfolding frontier_closures closure_def islimpt_def
himmelma@33175
   919
        using open_ball[of a e] `e > 0`
paulson@33324
   920
          by simp (metis centre_in_ball mem_ball open_ball) 
himmelma@33175
   921
    }
himmelma@33175
   922
    ultimately have ?rhse by auto
himmelma@33175
   923
  }
himmelma@33175
   924
  thus ?rhs by auto
himmelma@33175
   925
next
himmelma@33175
   926
  assume ?rhs
himmelma@33175
   927
  moreover
himmelma@33175
   928
  { fix T assume "a\<notin>S" and
himmelma@33175
   929
    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"
himmelma@33175
   930
    from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
himmelma@33175
   931
    then obtain e where "e>0" "ball a e \<subseteq> T" by auto
himmelma@33175
   932
    then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
himmelma@33175
   933
    have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
himmelma@33175
   934
      using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
himmelma@33175
   935
  }
himmelma@33175
   936
  hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
himmelma@33175
   937
  moreover
himmelma@33175
   938
  { fix T assume "a \<in> T"  "open T" "a\<in>S"
himmelma@33175
   939
    then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
himmelma@33175
   940
    obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
huffman@34105
   941
    hence "\<exists>y\<in>- S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
himmelma@33175
   942
  }
huffman@34105
   943
  hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
huffman@34105
   944
  ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
himmelma@33175
   945
qed
himmelma@33175
   946
himmelma@33175
   947
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
himmelma@33175
   948
  by (metis frontier_def closure_closed Diff_subset)
himmelma@33175
   949
hoelzl@34964
   950
lemma frontier_empty[simp]: "frontier {} = {}"
huffman@36362
   951
  by (simp add: frontier_def)
himmelma@33175
   952
himmelma@33175
   953
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
   954
proof-
himmelma@33175
   955
  { assume "frontier S \<subseteq> S"
himmelma@33175
   956
    hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
himmelma@33175
   957
    hence "closed S" using closure_subset_eq by auto
himmelma@33175
   958
  }
huffman@36362
   959
  thus ?thesis using frontier_subset_closed[of S] ..
himmelma@33175
   960
qed
himmelma@33175
   961
huffman@34105
   962
lemma frontier_complement: "frontier(- S) = frontier S"
himmelma@33175
   963
  by (auto simp add: frontier_def closure_complement interior_complement)
himmelma@33175
   964
himmelma@33175
   965
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
huffman@34105
   966
  using frontier_complement frontier_subset_eq[of "- S"]
huffman@34105
   967
  unfolding open_closed by auto
himmelma@33175
   968
huffman@36437
   969
subsection {* Nets and the ``eventually true'' quantifier *}
huffman@36437
   970
huffman@36437
   971
text {* Common nets and The "within" modifier for nets. *}
himmelma@33175
   972
himmelma@33175
   973
definition
himmelma@33175
   974
  at_infinity :: "'a::real_normed_vector net" where
huffman@36358
   975
  "at_infinity = Abs_net (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
himmelma@33175
   976
himmelma@33175
   977
definition
himmelma@33175
   978
  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
himmelma@33175
   979
  "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
himmelma@33175
   980
himmelma@33175
   981
text{* Prove That They are all nets. *}
himmelma@33175
   982
huffman@36358
   983
lemma eventually_at_infinity:
huffman@36358
   984
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
himmelma@33175
   985
unfolding at_infinity_def
huffman@36358
   986
proof (rule eventually_Abs_net, rule is_filter.intro)
huffman@36358
   987
  fix P Q :: "'a \<Rightarrow> bool"
huffman@36358
   988
  assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
huffman@36358
   989
  then obtain r s where
huffman@36358
   990
    "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
huffman@36358
   991
  then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
huffman@36358
   992
  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
huffman@36358
   993
qed auto
himmelma@33175
   994
huffman@36437
   995
text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
himmelma@33175
   996
himmelma@33175
   997
definition
himmelma@33175
   998
  trivial_limit :: "'a net \<Rightarrow> bool" where
huffman@36358
   999
  "trivial_limit net \<longleftrightarrow> eventually (\<lambda>x. False) net"
himmelma@33175
  1000
himmelma@33175
  1001
lemma trivial_limit_within:
himmelma@33175
  1002
  shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
himmelma@33175
  1003
proof
himmelma@33175
  1004
  assume "trivial_limit (at a within S)"
himmelma@33175
  1005
  thus "\<not> a islimpt S"
himmelma@33175
  1006
    unfolding trivial_limit_def
huffman@36358
  1007
    unfolding eventually_within eventually_at_topological
himmelma@33175
  1008
    unfolding islimpt_def
himmelma@33175
  1009
    apply (clarsimp simp add: expand_set_eq)
himmelma@33175
  1010
    apply (rename_tac T, rule_tac x=T in exI)
huffman@36358
  1011
    apply (clarsimp, drule_tac x=y in bspec, simp_all)
himmelma@33175
  1012
    done
himmelma@33175
  1013
next
himmelma@33175
  1014
  assume "\<not> a islimpt S"
himmelma@33175
  1015
  thus "trivial_limit (at a within S)"
himmelma@33175
  1016
    unfolding trivial_limit_def
huffman@36358
  1017
    unfolding eventually_within eventually_at_topological
himmelma@33175
  1018
    unfolding islimpt_def
huffman@36358
  1019
    apply clarsimp
huffman@36358
  1020
    apply (rule_tac x=T in exI)
huffman@36358
  1021
    apply auto
himmelma@33175
  1022
    done
himmelma@33175
  1023
qed
himmelma@33175
  1024
himmelma@33175
  1025
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
himmelma@33175
  1026
  using trivial_limit_within [of a UNIV]
himmelma@33175
  1027
  by (simp add: within_UNIV)
himmelma@33175
  1028
himmelma@33175
  1029
lemma trivial_limit_at:
himmelma@33175
  1030
  fixes a :: "'a::perfect_space"
himmelma@33175
  1031
  shows "\<not> trivial_limit (at a)"
himmelma@33175
  1032
  by (simp add: trivial_limit_at_iff)
himmelma@33175
  1033
himmelma@33175
  1034
lemma trivial_limit_at_infinity:
himmelma@33175
  1035
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
himmelma@33175
  1036
  (* FIXME: find a more appropriate type class *)
huffman@36358
  1037
  unfolding trivial_limit_def eventually_at_infinity
huffman@36358
  1038
  apply clarsimp
huffman@36358
  1039
  apply (rule_tac x="scaleR b (sgn 1)" in exI)
himmelma@33175
  1040
  apply (simp add: norm_sgn)
himmelma@33175
  1041
  done
himmelma@33175
  1042
hoelzl@34964
  1043
lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
huffman@36358
  1044
  by (auto simp add: trivial_limit_def eventually_sequentially)
himmelma@33175
  1045
huffman@36437
  1046
text {* Some property holds "sufficiently close" to the limit point. *}
himmelma@33175
  1047
himmelma@33175
  1048
lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
himmelma@33175
  1049
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
himmelma@33175
  1050
unfolding eventually_at dist_nz by auto
himmelma@33175
  1051
himmelma@33175
  1052
lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
himmelma@33175
  1053
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
himmelma@33175
  1054
unfolding eventually_within eventually_at dist_nz by auto
himmelma@33175
  1055
himmelma@33175
  1056
lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
himmelma@33175
  1057
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
himmelma@33175
  1058
unfolding eventually_within
paulson@33324
  1059
by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl) 
himmelma@33175
  1060
himmelma@33175
  1061
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
huffman@36358
  1062
  unfolding trivial_limit_def
huffman@36358
  1063
  by (auto elim: eventually_rev_mp)
himmelma@33175
  1064
himmelma@33175
  1065
lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
huffman@36358
  1066
proof -
huffman@36358
  1067
  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
huffman@36358
  1068
  thus "eventually P net" by simp
huffman@36358
  1069
qed
himmelma@33175
  1070
himmelma@33175
  1071
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
huffman@36358
  1072
  unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
himmelma@33175
  1073
himmelma@33175
  1074
lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
huffman@36358
  1075
  unfolding trivial_limit_def ..
himmelma@33175
  1076
himmelma@33175
  1077
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
himmelma@33175
  1078
  apply (safe elim!: trivial_limit_eventually)
himmelma@33175
  1079
  apply (simp add: eventually_False [symmetric])
himmelma@33175
  1080
  done
himmelma@33175
  1081
himmelma@33175
  1082
text{* Combining theorems for "eventually" *}
himmelma@33175
  1083
himmelma@33175
  1084
lemma eventually_conjI:
himmelma@33175
  1085
  "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
himmelma@33175
  1086
    \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
himmelma@33175
  1087
by (rule eventually_conj)
himmelma@33175
  1088
himmelma@33175
  1089
lemma eventually_rev_mono:
himmelma@33175
  1090
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
himmelma@33175
  1091
using eventually_mono [of P Q] by fast
himmelma@33175
  1092
himmelma@33175
  1093
lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
himmelma@33175
  1094
  by (auto intro!: eventually_conjI elim: eventually_rev_mono)
himmelma@33175
  1095
himmelma@33175
  1096
lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
himmelma@33175
  1097
  by (auto simp add: eventually_False)
himmelma@33175
  1098
himmelma@33175
  1099
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
himmelma@33175
  1100
  by (simp add: eventually_False)
himmelma@33175
  1101
huffman@36437
  1102
subsection {* Limits *}
himmelma@33175
  1103
himmelma@33175
  1104
  text{* Notation Lim to avoid collition with lim defined in analysis *}
himmelma@33175
  1105
definition
himmelma@33175
  1106
  Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b" where
himmelma@33175
  1107
  "Lim net f = (THE l. (f ---> l) net)"
himmelma@33175
  1108
himmelma@33175
  1109
lemma Lim:
himmelma@33175
  1110
 "(f ---> l) net \<longleftrightarrow>
himmelma@33175
  1111
        trivial_limit net \<or>
himmelma@33175
  1112
        (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
himmelma@33175
  1113
  unfolding tendsto_iff trivial_limit_eq by auto
himmelma@33175
  1114
himmelma@33175
  1115
himmelma@33175
  1116
text{* Show that they yield usual definitions in the various cases. *}
himmelma@33175
  1117
himmelma@33175
  1118
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
himmelma@33175
  1119
           (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
himmelma@33175
  1120
  by (auto simp add: tendsto_iff eventually_within_le)
himmelma@33175
  1121
himmelma@33175
  1122
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
himmelma@33175
  1123
        (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
himmelma@33175
  1124
  by (auto simp add: tendsto_iff eventually_within)
himmelma@33175
  1125
himmelma@33175
  1126
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
himmelma@33175
  1127
        (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
himmelma@33175
  1128
  by (auto simp add: tendsto_iff eventually_at)
himmelma@33175
  1129
himmelma@33175
  1130
lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
himmelma@33175
  1131
  unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
himmelma@33175
  1132
himmelma@33175
  1133
lemma Lim_at_infinity:
himmelma@33175
  1134
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
himmelma@33175
  1135
  by (auto simp add: tendsto_iff eventually_at_infinity)
himmelma@33175
  1136
himmelma@33175
  1137
lemma Lim_sequentially:
himmelma@33175
  1138
 "(S ---> l) sequentially \<longleftrightarrow>
himmelma@33175
  1139
          (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
himmelma@33175
  1140
  by (auto simp add: tendsto_iff eventually_sequentially)
himmelma@33175
  1141
himmelma@33175
  1142
lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
himmelma@33175
  1143
  unfolding Lim_sequentially LIMSEQ_def ..
himmelma@33175
  1144
himmelma@33175
  1145
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
himmelma@33175
  1146
  by (rule topological_tendstoI, auto elim: eventually_rev_mono)
himmelma@33175
  1147
himmelma@33175
  1148
text{* The expected monotonicity property. *}
himmelma@33175
  1149
himmelma@33175
  1150
lemma Lim_within_empty: "(f ---> l) (net within {})"
himmelma@33175
  1151
  unfolding tendsto_def Limits.eventually_within by simp
himmelma@33175
  1152
himmelma@33175
  1153
lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
himmelma@33175
  1154
  unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1155
  by (auto elim!: eventually_elim1)
himmelma@33175
  1156
himmelma@33175
  1157
lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
himmelma@33175
  1158
  shows "(f ---> l) (net within (S \<union> T))"
himmelma@33175
  1159
  using assms unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1160
  apply clarify
himmelma@33175
  1161
  apply (drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1162
  apply (drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1163
  apply (auto elim: eventually_elim2)
himmelma@33175
  1164
  done
himmelma@33175
  1165
himmelma@33175
  1166
lemma Lim_Un_univ:
himmelma@33175
  1167
 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
himmelma@33175
  1168
        ==> (f ---> l) net"
himmelma@33175
  1169
  by (metis Lim_Un within_UNIV)
himmelma@33175
  1170
himmelma@33175
  1171
text{* Interrelations between restricted and unrestricted limits. *}
himmelma@33175
  1172
himmelma@33175
  1173
lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
himmelma@33175
  1174
  (* FIXME: rename *)
himmelma@33175
  1175
  unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1176
  apply (clarify, drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1177
  by (auto elim!: eventually_elim1)
himmelma@33175
  1178
himmelma@33175
  1179
lemma Lim_within_open:
himmelma@33175
  1180
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
himmelma@33175
  1181
  assumes"a \<in> S" "open S"
himmelma@33175
  1182
  shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
himmelma@33175
  1183
proof
himmelma@33175
  1184
  assume ?lhs
himmelma@33175
  1185
  { fix A assume "open A" "l \<in> A"
himmelma@33175
  1186
    with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
himmelma@33175
  1187
      by (rule topological_tendstoD)
himmelma@33175
  1188
    hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
himmelma@33175
  1189
      unfolding Limits.eventually_within .
himmelma@33175
  1190
    then obtain T where "open T" "a \<in> T" "\<forall>x\<in>T. x \<noteq> a \<longrightarrow> x \<in> S \<longrightarrow> f x \<in> A"
himmelma@33175
  1191
      unfolding eventually_at_topological by fast
himmelma@33175
  1192
    hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
himmelma@33175
  1193
      using assms by auto
himmelma@33175
  1194
    hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
himmelma@33175
  1195
      by fast
himmelma@33175
  1196
    hence "eventually (\<lambda>x. f x \<in> A) (at a)"
himmelma@33175
  1197
      unfolding eventually_at_topological .
himmelma@33175
  1198
  }
himmelma@33175
  1199
  thus ?rhs by (rule topological_tendstoI)
himmelma@33175
  1200
next
himmelma@33175
  1201
  assume ?rhs
himmelma@33175
  1202
  thus ?lhs by (rule Lim_at_within)
himmelma@33175
  1203
qed
himmelma@33175
  1204
himmelma@33175
  1205
text{* Another limit point characterization. *}
himmelma@33175
  1206
himmelma@33175
  1207
lemma islimpt_sequential:
huffman@36667
  1208
  fixes x :: "'a::metric_space"
himmelma@33175
  1209
  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
himmelma@33175
  1210
    (is "?lhs = ?rhs")
himmelma@33175
  1211
proof
himmelma@33175
  1212
  assume ?lhs
himmelma@33175
  1213
  then obtain f where f:"\<forall>y. y>0 \<longrightarrow> f y \<in> S \<and> f y \<noteq> x \<and> dist (f y) x < y"
himmelma@33175
  1214
    unfolding islimpt_approachable using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto
himmelma@33175
  1215
  { fix n::nat
himmelma@33175
  1216
    have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
himmelma@33175
  1217
  }
himmelma@33175
  1218
  moreover
himmelma@33175
  1219
  { fix e::real assume "e>0"
himmelma@33175
  1220
    hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
himmelma@33175
  1221
    then obtain N::nat where "inverse (real (N + 1)) < e" by auto
himmelma@33175
  1222
    hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
himmelma@33175
  1223
    moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
himmelma@33175
  1224
    ultimately have "\<exists>N::nat. \<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < e" apply(rule_tac x=N in exI) apply auto apply(erule_tac x=n in allE)+ by auto
himmelma@33175
  1225
  }
himmelma@33175
  1226
  hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
himmelma@33175
  1227
    unfolding Lim_sequentially using f by auto
himmelma@33175
  1228
  ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
himmelma@33175
  1229
next
himmelma@33175
  1230
  assume ?rhs
himmelma@33175
  1231
  then obtain f::"nat\<Rightarrow>'a"  where f:"(\<forall>n. f n \<in> S - {x})" "(\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f n) x < e)" unfolding Lim_sequentially by auto
himmelma@33175
  1232
  { fix e::real assume "e>0"
himmelma@33175
  1233
    then obtain N where "dist (f N) x < e" using f(2) by auto
himmelma@33175
  1234
    moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
himmelma@33175
  1235
    ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
himmelma@33175
  1236
  }
himmelma@33175
  1237
  thus ?lhs unfolding islimpt_approachable by auto
himmelma@33175
  1238
qed
himmelma@33175
  1239
himmelma@33175
  1240
text{* Basic arithmetical combining theorems for limits. *}
himmelma@33175
  1241
himmelma@33175
  1242
lemma Lim_linear:
himmelma@33175
  1243
  assumes "(f ---> l) net" "bounded_linear h"
himmelma@33175
  1244
  shows "((\<lambda>x. h (f x)) ---> h l) net"
himmelma@33175
  1245
using `bounded_linear h` `(f ---> l) net`
himmelma@33175
  1246
by (rule bounded_linear.tendsto)
himmelma@33175
  1247
himmelma@33175
  1248
lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
himmelma@33175
  1249
  unfolding tendsto_def Limits.eventually_at_topological by fast
himmelma@33175
  1250
hoelzl@34964
  1251
lemma Lim_const[intro]: "((\<lambda>x. a) ---> a) net" by (rule tendsto_const)
hoelzl@34964
  1252
hoelzl@34964
  1253
lemma Lim_cmul[intro]:
himmelma@33175
  1254
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1255
  shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
himmelma@33175
  1256
  by (intro tendsto_intros)
himmelma@33175
  1257
himmelma@33175
  1258
lemma Lim_neg:
himmelma@33175
  1259
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1260
  shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
himmelma@33175
  1261
  by (rule tendsto_minus)
himmelma@33175
  1262
himmelma@33175
  1263
lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
himmelma@33175
  1264
 "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
himmelma@33175
  1265
  by (rule tendsto_add)
himmelma@33175
  1266
himmelma@33175
  1267
lemma Lim_sub:
himmelma@33175
  1268
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1269
  shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
himmelma@33175
  1270
  by (rule tendsto_diff)
himmelma@33175
  1271
huffman@36437
  1272
lemma Lim_mul:
huffman@36437
  1273
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@36437
  1274
  assumes "(c ---> d) net"  "(f ---> l) net"
huffman@36437
  1275
  shows "((\<lambda>x. c(x) *\<^sub>R f x) ---> (d *\<^sub>R l)) net"
huffman@36437
  1276
  using assms by (rule scaleR.tendsto)
huffman@36437
  1277
huffman@36437
  1278
lemma Lim_inv:
huffman@36437
  1279
  fixes f :: "'a \<Rightarrow> real"
huffman@36437
  1280
  assumes "(f ---> l) (net::'a net)"  "l \<noteq> 0"
huffman@36437
  1281
  shows "((inverse o f) ---> inverse l) net"
huffman@36437
  1282
  unfolding o_def using assms by (rule tendsto_inverse)
huffman@36437
  1283
huffman@36437
  1284
lemma Lim_vmul:
huffman@36437
  1285
  fixes c :: "'a \<Rightarrow> real" and v :: "'b::real_normed_vector"
huffman@36437
  1286
  shows "(c ---> d) net ==> ((\<lambda>x. c(x) *\<^sub>R v) ---> d *\<^sub>R v) net"
huffman@36437
  1287
  by (intro tendsto_intros)
huffman@36437
  1288
himmelma@33175
  1289
lemma Lim_null:
himmelma@33175
  1290
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1291
  shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
himmelma@33175
  1292
himmelma@33175
  1293
lemma Lim_null_norm:
himmelma@33175
  1294
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1295
  shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
himmelma@33175
  1296
  by (simp add: Lim dist_norm)
himmelma@33175
  1297
himmelma@33175
  1298
lemma Lim_null_comparison:
himmelma@33175
  1299
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1300
  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
himmelma@33175
  1301
  shows "(f ---> 0) net"
himmelma@33175
  1302
proof(simp add: tendsto_iff, rule+)
himmelma@33175
  1303
  fix e::real assume "0<e"
himmelma@33175
  1304
  { fix x
himmelma@33175
  1305
    assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
himmelma@33175
  1306
    hence "dist (f x) 0 < e" by (simp add: dist_norm)
himmelma@33175
  1307
  }
himmelma@33175
  1308
  thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
himmelma@33175
  1309
    using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
himmelma@33175
  1310
    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]
himmelma@33175
  1311
    using assms `e>0` unfolding tendsto_iff by auto
himmelma@33175
  1312
qed
himmelma@33175
  1313
himmelma@33175
  1314
lemma Lim_component:
hoelzl@37489
  1315
  fixes f :: "'a \<Rightarrow> ('a::euclidean_space)"
hoelzl@37489
  1316
  shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $$i) ---> l$$i) net"
himmelma@33175
  1317
  unfolding tendsto_iff
himmelma@33175
  1318
  apply (clarify)
himmelma@33175
  1319
  apply (drule spec, drule (1) mp)
himmelma@33175
  1320
  apply (erule eventually_elim1)
himmelma@33175
  1321
  apply (erule le_less_trans [OF dist_nth_le])
himmelma@33175
  1322
  done
himmelma@33175
  1323
himmelma@33175
  1324
lemma Lim_transform_bound:
himmelma@33175
  1325
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1326
  fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
himmelma@33175
  1327
  assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
himmelma@33175
  1328
  shows "(f ---> 0) net"
himmelma@33175
  1329
proof (rule tendstoI)
himmelma@33175
  1330
  fix e::real assume "e>0"
himmelma@33175
  1331
  { fix x
himmelma@33175
  1332
    assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
himmelma@33175
  1333
    hence "dist (f x) 0 < e" by (simp add: dist_norm)}
himmelma@33175
  1334
  thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
himmelma@33175
  1335
    using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
himmelma@33175
  1336
    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]
himmelma@33175
  1337
    using assms `e>0` unfolding tendsto_iff by blast
himmelma@33175
  1338
qed
himmelma@33175
  1339
himmelma@33175
  1340
text{* Deducing things about the limit from the elements. *}
himmelma@33175
  1341
himmelma@33175
  1342
lemma Lim_in_closed_set:
himmelma@33175
  1343
  assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
himmelma@33175
  1344
  shows "l \<in> S"
himmelma@33175
  1345
proof (rule ccontr)
himmelma@33175
  1346
  assume "l \<notin> S"
himmelma@33175
  1347
  with `closed S` have "open (- S)" "l \<in> - S"
himmelma@33175
  1348
    by (simp_all add: open_Compl)
himmelma@33175
  1349
  with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
himmelma@33175
  1350
    by (rule topological_tendstoD)
himmelma@33175
  1351
  with assms(2) have "eventually (\<lambda>x. False) net"
himmelma@33175
  1352
    by (rule eventually_elim2) simp
himmelma@33175
  1353
  with assms(3) show "False"
himmelma@33175
  1354
    by (simp add: eventually_False)
himmelma@33175
  1355
qed
himmelma@33175
  1356
himmelma@33175
  1357
text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
himmelma@33175
  1358
himmelma@33175
  1359
lemma Lim_dist_ubound:
himmelma@33175
  1360
  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
himmelma@33175
  1361
  shows "dist a l <= e"
himmelma@33175
  1362
proof (rule ccontr)
himmelma@33175
  1363
  assume "\<not> dist a l \<le> e"
himmelma@33175
  1364
  then have "0 < dist a l - e" by simp
himmelma@33175
  1365
  with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
himmelma@33175
  1366
    by (rule tendstoD)
himmelma@33175
  1367
  with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
himmelma@33175
  1368
    by (rule eventually_conjI)
himmelma@33175
  1369
  then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
himmelma@33175
  1370
    using assms(1) eventually_happens by auto
himmelma@33175
  1371
  hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
himmelma@33175
  1372
    by (rule add_le_less_mono)
himmelma@33175
  1373
  hence "dist a (f w) + dist (f w) l < dist a l"
himmelma@33175
  1374
    by simp
himmelma@33175
  1375
  also have "\<dots> \<le> dist a (f w) + dist (f w) l"
himmelma@33175
  1376
    by (rule dist_triangle)
himmelma@33175
  1377
  finally show False by simp
himmelma@33175
  1378
qed
himmelma@33175
  1379
himmelma@33175
  1380
lemma Lim_norm_ubound:
himmelma@33175
  1381
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1382
  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
himmelma@33175
  1383
  shows "norm(l) <= e"
himmelma@33175
  1384
proof (rule ccontr)
himmelma@33175
  1385
  assume "\<not> norm l \<le> e"
himmelma@33175
  1386
  then have "0 < norm l - e" by simp
himmelma@33175
  1387
  with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
himmelma@33175
  1388
    by (rule tendstoD)
himmelma@33175
  1389
  with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
himmelma@33175
  1390
    by (rule eventually_conjI)
himmelma@33175
  1391
  then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
himmelma@33175
  1392
    using assms(1) eventually_happens by auto
himmelma@33175
  1393
  hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
himmelma@33175
  1394
  hence "norm (f w - l) + norm (f w) < norm l" by simp
himmelma@33175
  1395
  hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
himmelma@33175
  1396
  thus False using `\<not> norm l \<le> e` by simp
himmelma@33175
  1397
qed
himmelma@33175
  1398
himmelma@33175
  1399
lemma Lim_norm_lbound:
himmelma@33175
  1400
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1401
  assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
himmelma@33175
  1402
  shows "e \<le> norm l"
himmelma@33175
  1403
proof (rule ccontr)
himmelma@33175
  1404
  assume "\<not> e \<le> norm l"
himmelma@33175
  1405
  then have "0 < e - norm l" by simp
himmelma@33175
  1406
  with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
himmelma@33175
  1407
    by (rule tendstoD)
himmelma@33175
  1408
  with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
himmelma@33175
  1409
    by (rule eventually_conjI)
himmelma@33175
  1410
  then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
himmelma@33175
  1411
    using assms(1) eventually_happens by auto
himmelma@33175
  1412
  hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
himmelma@33175
  1413
  hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
himmelma@33175
  1414
  hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
himmelma@33175
  1415
  thus False by simp
himmelma@33175
  1416
qed
himmelma@33175
  1417
himmelma@33175
  1418
text{* Uniqueness of the limit, when nontrivial. *}
himmelma@33175
  1419
himmelma@33175
  1420
lemma Lim_unique:
himmelma@33175
  1421
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
himmelma@33175
  1422
  assumes "\<not> trivial_limit net"  "(f ---> l) net"  "(f ---> l') net"
himmelma@33175
  1423
  shows "l = l'"
himmelma@33175
  1424
proof (rule ccontr)
himmelma@33175
  1425
  assume "l \<noteq> l'"
himmelma@33175
  1426
  obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
himmelma@33175
  1427
    using hausdorff [OF `l \<noteq> l'`] by fast
himmelma@33175
  1428
  have "eventually (\<lambda>x. f x \<in> U) net"
himmelma@33175
  1429
    using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
himmelma@33175
  1430
  moreover
himmelma@33175
  1431
  have "eventually (\<lambda>x. f x \<in> V) net"
himmelma@33175
  1432
    using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
himmelma@33175
  1433
  ultimately
himmelma@33175
  1434
  have "eventually (\<lambda>x. False) net"
himmelma@33175
  1435
  proof (rule eventually_elim2)
himmelma@33175
  1436
    fix x
himmelma@33175
  1437
    assume "f x \<in> U" "f x \<in> V"
himmelma@33175
  1438
    hence "f x \<in> U \<inter> V" by simp
himmelma@33175
  1439
    with `U \<inter> V = {}` show "False" by simp
himmelma@33175
  1440
  qed
himmelma@33175
  1441
  with `\<not> trivial_limit net` show "False"
himmelma@33175
  1442
    by (simp add: eventually_False)
himmelma@33175
  1443
qed
himmelma@33175
  1444
himmelma@33175
  1445
lemma tendsto_Lim:
himmelma@33175
  1446
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
himmelma@33175
  1447
  shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
himmelma@33175
  1448
  unfolding Lim_def using Lim_unique[of net f] by auto
himmelma@33175
  1449
himmelma@33175
  1450
text{* Limit under bilinear function *}
himmelma@33175
  1451
himmelma@33175
  1452
lemma Lim_bilinear:
himmelma@33175
  1453
  assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
himmelma@33175
  1454
  shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
himmelma@33175
  1455
using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
himmelma@33175
  1456
by (rule bounded_bilinear.tendsto)
himmelma@33175
  1457
himmelma@33175
  1458
text{* These are special for limits out of the same vector space. *}
himmelma@33175
  1459
himmelma@33175
  1460
lemma Lim_within_id: "(id ---> a) (at a within s)"
himmelma@33175
  1461
  unfolding tendsto_def Limits.eventually_within eventually_at_topological
himmelma@33175
  1462
  by auto
himmelma@33175
  1463
huffman@36437
  1464
lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id
huffman@36437
  1465
himmelma@33175
  1466
lemma Lim_at_id: "(id ---> a) (at a)"
himmelma@33175
  1467
apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
himmelma@33175
  1468
himmelma@33175
  1469
lemma Lim_at_zero:
himmelma@33175
  1470
  fixes a :: "'a::real_normed_vector"
himmelma@33175
  1471
  fixes l :: "'b::topological_space"
himmelma@33175
  1472
  shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
himmelma@33175
  1473
proof
himmelma@33175
  1474
  assume "?lhs"
himmelma@33175
  1475
  { fix S assume "open S" "l \<in> S"
himmelma@33175
  1476
    with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
himmelma@33175
  1477
      by (rule topological_tendstoD)
himmelma@33175
  1478
    then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
himmelma@33175
  1479
      unfolding Limits.eventually_at by fast
himmelma@33175
  1480
    { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
himmelma@33175
  1481
      hence "f (a + x) \<in> S" using d
himmelma@33175
  1482
      apply(erule_tac x="x+a" in allE)
haftmann@35820
  1483
      by (auto simp add: add_commute dist_norm dist_commute)
himmelma@33175
  1484
    }
himmelma@33175
  1485
    hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
himmelma@33175
  1486
      using d(1) by auto
himmelma@33175
  1487
    hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
himmelma@33175
  1488
      unfolding Limits.eventually_at .
himmelma@33175
  1489
  }
himmelma@33175
  1490
  thus "?rhs" by (rule topological_tendstoI)
himmelma@33175
  1491
next
himmelma@33175
  1492
  assume "?rhs"
himmelma@33175
  1493
  { fix S assume "open S" "l \<in> S"
himmelma@33175
  1494
    with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
himmelma@33175
  1495
      by (rule topological_tendstoD)
himmelma@33175
  1496
    then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
himmelma@33175
  1497
      unfolding Limits.eventually_at by fast
himmelma@33175
  1498
    { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
himmelma@33175
  1499
      hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
haftmann@35820
  1500
        by(auto simp add: add_commute dist_norm dist_commute)
himmelma@33175
  1501
    }
himmelma@33175
  1502
    hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
himmelma@33175
  1503
    hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
himmelma@33175
  1504
  }
himmelma@33175
  1505
  thus "?lhs" by (rule topological_tendstoI)
himmelma@33175
  1506
qed
himmelma@33175
  1507
himmelma@33175
  1508
text{* It's also sometimes useful to extract the limit point from the net.  *}
himmelma@33175
  1509
himmelma@33175
  1510
definition
himmelma@33175
  1511
  netlimit :: "'a::t2_space net \<Rightarrow> 'a" where
himmelma@33175
  1512
  "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
himmelma@33175
  1513
himmelma@33175
  1514
lemma netlimit_within:
himmelma@33175
  1515
  assumes "\<not> trivial_limit (at a within S)"
himmelma@33175
  1516
  shows "netlimit (at a within S) = a"
himmelma@33175
  1517
unfolding netlimit_def
himmelma@33175
  1518
apply (rule some_equality)
himmelma@33175
  1519
apply (rule Lim_at_within)
himmelma@33175
  1520
apply (rule Lim_ident_at)
himmelma@33175
  1521
apply (erule Lim_unique [OF assms])
himmelma@33175
  1522
apply (rule Lim_at_within)
himmelma@33175
  1523
apply (rule Lim_ident_at)
himmelma@33175
  1524
done
himmelma@33175
  1525
himmelma@33175
  1526
lemma netlimit_at:
himmelma@33175
  1527
  fixes a :: "'a::perfect_space"
himmelma@33175
  1528
  shows "netlimit (at a) = a"
himmelma@33175
  1529
  apply (subst within_UNIV[symmetric])
himmelma@33175
  1530
  using netlimit_within[of a UNIV]
himmelma@33175
  1531
  by (simp add: trivial_limit_at within_UNIV)
himmelma@33175
  1532
himmelma@33175
  1533
text{* Transformation of limit. *}
himmelma@33175
  1534
himmelma@33175
  1535
lemma Lim_transform:
himmelma@33175
  1536
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1537
  assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
himmelma@33175
  1538
  shows "(g ---> l) net"
himmelma@33175
  1539
proof-
himmelma@33175
  1540
  from assms have "((\<lambda>x. f x - g x - f x) ---> 0 - l) net" using Lim_sub[of "\<lambda>x. f x - g x" 0 net f l] by auto
himmelma@33175
  1541
  thus "?thesis" using Lim_neg [of "\<lambda> x. - g x" "-l" net] by auto
himmelma@33175
  1542
qed
himmelma@33175
  1543
himmelma@33175
  1544
lemma Lim_transform_eventually:
huffman@36667
  1545
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
himmelma@33175
  1546
  apply (rule topological_tendstoI)
himmelma@33175
  1547
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1548
  apply (erule (1) eventually_elim2, simp)
himmelma@33175
  1549
  done
himmelma@33175
  1550
himmelma@33175
  1551
lemma Lim_transform_within:
huffman@36667
  1552
  assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
huffman@36667
  1553
  and "(f ---> l) (at x within S)"
huffman@36667
  1554
  shows "(g ---> l) (at x within S)"
huffman@36667
  1555
proof (rule Lim_transform_eventually)
huffman@36667
  1556
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
huffman@36667
  1557
    unfolding eventually_within
huffman@36667
  1558
    using assms(1,2) by auto
huffman@36667
  1559
  show "(f ---> l) (at x within S)" by fact
huffman@36667
  1560
qed
himmelma@33175
  1561
himmelma@33175
  1562
lemma Lim_transform_at:
huffman@36667
  1563
  assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
huffman@36667
  1564
  and "(f ---> l) (at x)"
huffman@36667
  1565
  shows "(g ---> l) (at x)"
huffman@36667
  1566
proof (rule Lim_transform_eventually)
huffman@36667
  1567
  show "eventually (\<lambda>x. f x = g x) (at x)"
huffman@36667
  1568
    unfolding eventually_at
huffman@36667
  1569
    using assms(1,2) by auto
huffman@36667
  1570
  show "(f ---> l) (at x)" by fact
huffman@36667
  1571
qed
himmelma@33175
  1572
himmelma@33175
  1573
text{* Common case assuming being away from some crucial point like 0. *}
himmelma@33175
  1574
himmelma@33175
  1575
lemma Lim_transform_away_within:
huffman@36669
  1576
  fixes a b :: "'a::t1_space"
huffman@36667
  1577
  assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
himmelma@33175
  1578
  and "(f ---> l) (at a within S)"
himmelma@33175
  1579
  shows "(g ---> l) (at a within S)"
huffman@36669
  1580
proof (rule Lim_transform_eventually)
huffman@36669
  1581
  show "(f ---> l) (at a within S)" by fact
huffman@36669
  1582
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
huffman@36669
  1583
    unfolding Limits.eventually_within eventually_at_topological
huffman@36669
  1584
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
himmelma@33175
  1585
qed
himmelma@33175
  1586
himmelma@33175
  1587
lemma Lim_transform_away_at:
huffman@36669
  1588
  fixes a b :: "'a::t1_space"
himmelma@33175
  1589
  assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
himmelma@33175
  1590
  and fl: "(f ---> l) (at a)"
himmelma@33175
  1591
  shows "(g ---> l) (at a)"
himmelma@33175
  1592
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
himmelma@33175
  1593
  by (auto simp add: within_UNIV)
himmelma@33175
  1594
himmelma@33175
  1595
text{* Alternatively, within an open set. *}
himmelma@33175
  1596
himmelma@33175
  1597
lemma Lim_transform_within_open:
huffman@36667
  1598
  assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
huffman@36667
  1599
  and "(f ---> l) (at a)"
himmelma@33175
  1600
  shows "(g ---> l) (at a)"
huffman@36667
  1601
proof (rule Lim_transform_eventually)
huffman@36667
  1602
  show "eventually (\<lambda>x. f x = g x) (at a)"
huffman@36667
  1603
    unfolding eventually_at_topological
huffman@36667
  1604
    using assms(1,2,3) by auto
huffman@36667
  1605
  show "(f ---> l) (at a)" by fact
himmelma@33175
  1606
qed
himmelma@33175
  1607
himmelma@33175
  1608
text{* A congruence rule allowing us to transform limits assuming not at point. *}
himmelma@33175
  1609
himmelma@33175
  1610
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
himmelma@33175
  1611
huffman@36362
  1612
lemma Lim_cong_within(*[cong add]*):
huffman@36667
  1613
  assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
huffman@36667
  1614
  shows "((\<lambda>x. f x) ---> l) (at a within S) \<longleftrightarrow> ((g ---> l) (at a within S))"
huffman@36667
  1615
  unfolding tendsto_def Limits.eventually_within eventually_at_topological
huffman@36667
  1616
  using assms by simp
huffman@36667
  1617
huffman@36667
  1618
lemma Lim_cong_at(*[cong add]*):
huffman@36667
  1619
  assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
huffman@36667
  1620
  shows "((\<lambda>x. f x) ---> l) (at a) \<longleftrightarrow> ((g ---> l) (at a))"
huffman@36667
  1621
  unfolding tendsto_def eventually_at_topological
huffman@36667
  1622
  using assms by simp
himmelma@33175
  1623
himmelma@33175
  1624
text{* Useful lemmas on closure and set of possible sequential limits.*}
himmelma@33175
  1625
himmelma@33175
  1626
lemma closure_sequential:
huffman@36667
  1627
  fixes l :: "'a::metric_space"
himmelma@33175
  1628
  shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
himmelma@33175
  1629
proof
himmelma@33175
  1630
  assume "?lhs" moreover
himmelma@33175
  1631
  { assume "l \<in> S"
himmelma@33175
  1632
    hence "?rhs" using Lim_const[of l sequentially] by auto
himmelma@33175
  1633
  } moreover
himmelma@33175
  1634
  { assume "l islimpt S"
himmelma@33175
  1635
    hence "?rhs" unfolding islimpt_sequential by auto
himmelma@33175
  1636
  } ultimately
himmelma@33175
  1637
  show "?rhs" unfolding closure_def by auto
himmelma@33175
  1638
next
himmelma@33175
  1639
  assume "?rhs"
himmelma@33175
  1640
  thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
himmelma@33175
  1641
qed
himmelma@33175
  1642
himmelma@33175
  1643
lemma closed_sequential_limits:
himmelma@33175
  1644
  fixes S :: "'a::metric_space set"
himmelma@33175
  1645
  shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
himmelma@33175
  1646
  unfolding closed_limpt
himmelma@33175
  1647
  using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
himmelma@33175
  1648
  by metis
himmelma@33175
  1649
himmelma@33175
  1650
lemma closure_approachable:
himmelma@33175
  1651
  fixes S :: "'a::metric_space set"
himmelma@33175
  1652
  shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
himmelma@33175
  1653
  apply (auto simp add: closure_def islimpt_approachable)
himmelma@33175
  1654
  by (metis dist_self)
himmelma@33175
  1655
himmelma@33175
  1656
lemma closed_approachable:
himmelma@33175
  1657
  fixes S :: "'a::metric_space set"
himmelma@33175
  1658
  shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
himmelma@33175
  1659
  by (metis closure_closed closure_approachable)
himmelma@33175
  1660
himmelma@33175
  1661
text{* Some other lemmas about sequences. *}
himmelma@33175
  1662
huffman@36441
  1663
lemma sequentially_offset:
huffman@36441
  1664
  assumes "eventually (\<lambda>i. P i) sequentially"
huffman@36441
  1665
  shows "eventually (\<lambda>i. P (i + k)) sequentially"
huffman@36441
  1666
  using assms unfolding eventually_sequentially by (metis trans_le_add1)
huffman@36441
  1667
himmelma@33175
  1668
lemma seq_offset:
huffman@36441
  1669
  assumes "(f ---> l) sequentially"
huffman@36441
  1670
  shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
huffman@36441
  1671
  using assms unfolding tendsto_def
huffman@36441
  1672
  by clarify (rule sequentially_offset, simp)
himmelma@33175
  1673
himmelma@33175
  1674
lemma seq_offset_neg:
himmelma@33175
  1675
  "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
himmelma@33175
  1676
  apply (rule topological_tendstoI)
himmelma@33175
  1677
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1678
  apply (simp only: eventually_sequentially)
himmelma@33175
  1679
  apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
himmelma@33175
  1680
  apply metis
himmelma@33175
  1681
  by arith
himmelma@33175
  1682
himmelma@33175
  1683
lemma seq_offset_rev:
himmelma@33175
  1684
  "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
himmelma@33175
  1685
  apply (rule topological_tendstoI)
himmelma@33175
  1686
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1687
  apply (simp only: eventually_sequentially)
himmelma@33175
  1688
  apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
himmelma@33175
  1689
  by metis arith
himmelma@33175
  1690
himmelma@33175
  1691
lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
himmelma@33175
  1692
proof-
himmelma@33175
  1693
  { fix e::real assume "e>0"
himmelma@33175
  1694
    hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
himmelma@33175
  1695
      using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
huffman@36362
  1696
      by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
himmelma@33175
  1697
  }
himmelma@33175
  1698
  thus ?thesis unfolding Lim_sequentially dist_norm by simp
himmelma@33175
  1699
qed
himmelma@33175
  1700
huffman@36437
  1701
subsection {* More properties of closed balls. *}
himmelma@33175
  1702
himmelma@33175
  1703
lemma closed_cball: "closed (cball x e)"
himmelma@33175
  1704
unfolding cball_def closed_def
himmelma@33175
  1705
unfolding Collect_neg_eq [symmetric] not_le
himmelma@33175
  1706
apply (clarsimp simp add: open_dist, rename_tac y)
himmelma@33175
  1707
apply (rule_tac x="dist x y - e" in exI, clarsimp)
himmelma@33175
  1708
apply (rename_tac x')
himmelma@33175
  1709
apply (cut_tac x=x and y=x' and z=y in dist_triangle)
himmelma@33175
  1710
apply simp
himmelma@33175
  1711
done
himmelma@33175
  1712
himmelma@33175
  1713
lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
himmelma@33175
  1714
proof-
himmelma@33175
  1715
  { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
himmelma@33175
  1716
    hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
himmelma@33175
  1717
  } moreover
himmelma@33175
  1718
  { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
himmelma@33175
  1719
    hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
himmelma@33175
  1720
  } ultimately
himmelma@33175
  1721
  show ?thesis unfolding open_contains_ball by auto
himmelma@33175
  1722
qed
himmelma@33175
  1723
himmelma@33175
  1724
lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
himmelma@33175
  1725
  by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
himmelma@33175
  1726
himmelma@33175
  1727
lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
himmelma@33175
  1728
  apply (simp add: interior_def, safe)
himmelma@33175
  1729
  apply (force simp add: open_contains_cball)
himmelma@33175
  1730
  apply (rule_tac x="ball x e" in exI)
huffman@36362
  1731
  apply (simp add: subset_trans [OF ball_subset_cball])
himmelma@33175
  1732
  done
himmelma@33175
  1733
himmelma@33175
  1734
lemma islimpt_ball:
himmelma@33175
  1735
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
himmelma@33175
  1736
  shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
himmelma@33175
  1737
proof
himmelma@33175
  1738
  assume "?lhs"
himmelma@33175
  1739
  { assume "e \<le> 0"
himmelma@33175
  1740
    hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
himmelma@33175
  1741
    have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
himmelma@33175
  1742
  }
himmelma@33175
  1743
  hence "e > 0" by (metis not_less)
himmelma@33175
  1744
  moreover
himmelma@33175
  1745
  have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto
himmelma@33175
  1746
  ultimately show "?rhs" by auto
himmelma@33175
  1747
next
himmelma@33175
  1748
  assume "?rhs" hence "e>0"  by auto
himmelma@33175
  1749
  { fix d::real assume "d>0"
himmelma@33175
  1750
    have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1751
    proof(cases "d \<le> dist x y")
himmelma@33175
  1752
      case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1753
      proof(cases "x=y")
himmelma@33175
  1754
        case True hence False using `d \<le> dist x y` `d>0` by auto
himmelma@33175
  1755
        thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
himmelma@33175
  1756
      next
himmelma@33175
  1757
        case False
himmelma@33175
  1758
himmelma@33175
  1759
        have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
himmelma@33175
  1760
              = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
himmelma@33175
  1761
          unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
himmelma@33175
  1762
        also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
himmelma@33175
  1763
          using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
himmelma@33175
  1764
          unfolding scaleR_minus_left scaleR_one
himmelma@33175
  1765
          by (auto simp add: norm_minus_commute)
himmelma@33175
  1766
        also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
himmelma@33175
  1767
          unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
huffman@36778
  1768
          unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
himmelma@33175
  1769
        also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
himmelma@33175
  1770
        finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
himmelma@33175
  1771
himmelma@33175
  1772
        moreover
himmelma@33175
  1773
himmelma@33175
  1774
        have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
himmelma@33175
  1775
          using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
himmelma@33175
  1776
        moreover
himmelma@33175
  1777
        have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
himmelma@33175
  1778
          using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
himmelma@33175
  1779
          unfolding dist_norm by auto
himmelma@33175
  1780
        ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
himmelma@33175
  1781
      qed
himmelma@33175
  1782
    next
himmelma@33175
  1783
      case False hence "d > dist x y" by auto
himmelma@33175
  1784
      show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1785
      proof(cases "x=y")
himmelma@33175
  1786
        case True
himmelma@33175
  1787
        obtain z where **: "z \<noteq> y" "dist z y < min e d"
himmelma@33175
  1788
          using perfect_choose_dist[of "min e d" y]
himmelma@33175
  1789
          using `d > 0` `e>0` by auto
himmelma@33175
  1790
        show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1791
          unfolding `x = y`
himmelma@33175
  1792
          using `z \<noteq> y` **
himmelma@33175
  1793
          by (rule_tac x=z in bexI, auto simp add: dist_commute)
himmelma@33175
  1794
      next
himmelma@33175
  1795
        case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1796
          using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
himmelma@33175
  1797
      qed
himmelma@33175
  1798
    qed  }
himmelma@33175
  1799
  thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
himmelma@33175
  1800
qed
himmelma@33175
  1801
himmelma@33175
  1802
lemma closure_ball_lemma:
himmelma@33175
  1803
  fixes x y :: "'a::real_normed_vector"
himmelma@33175
  1804
  assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
himmelma@33175
  1805
proof (rule islimptI)
himmelma@33175
  1806
  fix T assume "y \<in> T" "open T"
himmelma@33175
  1807
  then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
himmelma@33175
  1808
    unfolding open_dist by fast
himmelma@33175
  1809
  (* choose point between x and y, within distance r of y. *)
himmelma@33175
  1810
  def k \<equiv> "min 1 (r / (2 * dist x y))"
himmelma@33175
  1811
  def z \<equiv> "y + scaleR k (x - y)"
himmelma@33175
  1812
  have z_def2: "z = x + scaleR (1 - k) (y - x)"
himmelma@33175
  1813
    unfolding z_def by (simp add: algebra_simps)
himmelma@33175
  1814
  have "dist z y < r"
himmelma@33175
  1815
    unfolding z_def k_def using `0 < r`
himmelma@33175
  1816
    by (simp add: dist_norm min_def)
himmelma@33175
  1817
  hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
himmelma@33175
  1818
  have "dist x z < dist x y"
himmelma@33175
  1819
    unfolding z_def2 dist_norm
himmelma@33175
  1820
    apply (simp add: norm_minus_commute)
himmelma@33175
  1821
    apply (simp only: dist_norm [symmetric])
himmelma@33175
  1822
    apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
himmelma@33175
  1823
    apply (rule mult_strict_right_mono)
himmelma@33175
  1824
    apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
himmelma@33175
  1825
    apply (simp add: zero_less_dist_iff `x \<noteq> y`)
himmelma@33175
  1826
    done
himmelma@33175
  1827
  hence "z \<in> ball x (dist x y)" by simp
himmelma@33175
  1828
  have "z \<noteq> y"
himmelma@33175
  1829
    unfolding z_def k_def using `x \<noteq> y` `0 < r`
himmelma@33175
  1830
    by (simp add: min_def)
himmelma@33175
  1831
  show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
himmelma@33175
  1832
    using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
himmelma@33175
  1833
    by fast
himmelma@33175
  1834
qed
himmelma@33175
  1835
himmelma@33175
  1836
lemma closure_ball:
himmelma@33175
  1837
  fixes x :: "'a::real_normed_vector"
himmelma@33175
  1838
  shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
himmelma@33175
  1839
apply (rule equalityI)
himmelma@33175
  1840
apply (rule closure_minimal)
himmelma@33175
  1841
apply (rule ball_subset_cball)
himmelma@33175
  1842
apply (rule closed_cball)
himmelma@33175
  1843
apply (rule subsetI, rename_tac y)
himmelma@33175
  1844
apply (simp add: le_less [where 'a=real])
himmelma@33175
  1845
apply (erule disjE)
himmelma@33175
  1846
apply (rule subsetD [OF closure_subset], simp)
himmelma@33175
  1847
apply (simp add: closure_def)
himmelma@33175
  1848
apply clarify
himmelma@33175
  1849
apply (rule closure_ball_lemma)
himmelma@33175
  1850
apply (simp add: zero_less_dist_iff)
himmelma@33175
  1851
done
himmelma@33175
  1852
himmelma@33175
  1853
(* In a trivial vector space, this fails for e = 0. *)
himmelma@33175
  1854
lemma interior_cball:
himmelma@33175
  1855
  fixes x :: "'a::{real_normed_vector, perfect_space}"
himmelma@33175
  1856
  shows "interior (cball x e) = ball x e"
himmelma@33175
  1857
proof(cases "e\<ge>0")
himmelma@33175
  1858
  case False note cs = this
himmelma@33175
  1859
  from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
himmelma@33175
  1860
  { fix y assume "y \<in> cball x e"
himmelma@33175
  1861
    hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
himmelma@33175
  1862
  hence "cball x e = {}" by auto
himmelma@33175
  1863
  hence "interior (cball x e) = {}" using interior_empty by auto
himmelma@33175
  1864
  ultimately show ?thesis by blast
himmelma@33175
  1865
next
himmelma@33175
  1866
  case True note cs = this
himmelma@33175
  1867
  have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
himmelma@33175
  1868
  { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
himmelma@33175
  1869
    then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
himmelma@33175
  1870
himmelma@33175
  1871
    then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
himmelma@33175
  1872
      using perfect_choose_dist [of d] by auto
himmelma@33175
  1873
    have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
himmelma@33175
  1874
    hence xa_cball:"xa \<in> cball x e" using as(1) by auto
himmelma@33175
  1875
himmelma@33175
  1876
    hence "y \<in> ball x e" proof(cases "x = y")
himmelma@33175
  1877
      case True
himmelma@33175
  1878
      hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
himmelma@33175
  1879
      thus "y \<in> ball x e" using `x = y ` by simp
himmelma@33175
  1880
    next
himmelma@33175
  1881
      case False
himmelma@33175
  1882
      have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
himmelma@33175
  1883
        using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
himmelma@33175
  1884
      hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
himmelma@33175
  1885
      have "y - x \<noteq> 0" using `x \<noteq> y` by auto
himmelma@33175
  1886
      hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
himmelma@33175
  1887
        using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
himmelma@33175
  1888
himmelma@33175
  1889
      have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
himmelma@33175
  1890
        by (auto simp add: dist_norm algebra_simps)
himmelma@33175
  1891
      also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
himmelma@33175
  1892
        by (auto simp add: algebra_simps)
himmelma@33175
  1893
      also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
himmelma@33175
  1894
        using ** by auto
himmelma@33175
  1895
      also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
himmelma@33175
  1896
      finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
himmelma@33175
  1897
      thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
himmelma@33175
  1898
    qed  }
himmelma@33175
  1899
  hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
himmelma@33175
  1900
  ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
himmelma@33175
  1901
qed
himmelma@33175
  1902
himmelma@33175
  1903
lemma frontier_ball:
himmelma@33175
  1904
  fixes a :: "'a::real_normed_vector"
himmelma@33175
  1905
  shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
huffman@36362
  1906
  apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
himmelma@33175
  1907
  apply (simp add: expand_set_eq)
himmelma@33175
  1908
  by arith
himmelma@33175
  1909
himmelma@33175
  1910
lemma frontier_cball:
himmelma@33175
  1911
  fixes a :: "'a::{real_normed_vector, perfect_space}"
himmelma@33175
  1912
  shows "frontier(cball a e) = {x. dist a x = e}"
huffman@36362
  1913
  apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
himmelma@33175
  1914
  apply (simp add: expand_set_eq)
himmelma@33175
  1915
  by arith
himmelma@33175
  1916
himmelma@33175
  1917
lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
himmelma@33175
  1918
  apply (simp add: expand_set_eq not_le)
himmelma@33175
  1919
  by (metis zero_le_dist dist_self order_less_le_trans)
himmelma@33175
  1920
lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
himmelma@33175
  1921
himmelma@33175
  1922
lemma cball_eq_sing:
himmelma@33175
  1923
  fixes x :: "'a::perfect_space"
himmelma@33175
  1924
  shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
himmelma@33175
  1925
proof (rule linorder_cases)
himmelma@33175
  1926
  assume e: "0 < e"
himmelma@33175
  1927
  obtain a where "a \<noteq> x" "dist a x < e"
himmelma@33175
  1928
    using perfect_choose_dist [OF e] by auto
himmelma@33175
  1929
  hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
himmelma@33175
  1930
  with e show ?thesis by (auto simp add: expand_set_eq)
himmelma@33175
  1931
qed auto
himmelma@33175
  1932
himmelma@33175
  1933
lemma cball_sing:
himmelma@33175
  1934
  fixes x :: "'a::metric_space"
himmelma@33175
  1935
  shows "e = 0 ==> cball x e = {x}"
himmelma@33175
  1936
  by (auto simp add: expand_set_eq)
himmelma@33175
  1937
himmelma@33175
  1938
text{* For points in the interior, localization of limits makes no difference.   *}
himmelma@33175
  1939
himmelma@33175
  1940
lemma eventually_within_interior:
himmelma@33175
  1941
  assumes "x \<in> interior S"
himmelma@33175
  1942
  shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
himmelma@33175
  1943
proof-
himmelma@33175
  1944
  from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
himmelma@33175
  1945
    unfolding interior_def by fast
himmelma@33175
  1946
  { assume "?lhs"
himmelma@33175
  1947
    then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
himmelma@33175
  1948
      unfolding Limits.eventually_within Limits.eventually_at_topological
himmelma@33175
  1949
      by auto
himmelma@33175
  1950
    with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
himmelma@33175
  1951
      by auto
himmelma@33175
  1952
    then have "?rhs"
himmelma@33175
  1953
      unfolding Limits.eventually_at_topological by auto
himmelma@33175
  1954
  } moreover
himmelma@33175
  1955
  { assume "?rhs" hence "?lhs"
himmelma@33175
  1956
      unfolding Limits.eventually_within
himmelma@33175
  1957
      by (auto elim: eventually_elim1)
himmelma@33175
  1958
  } ultimately
himmelma@33175
  1959
  show "?thesis" ..
himmelma@33175
  1960
qed
himmelma@33175
  1961
himmelma@33175
  1962
lemma lim_within_interior:
himmelma@33175
  1963
  "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
himmelma@33175
  1964
  unfolding tendsto_def by (simp add: eventually_within_interior)
himmelma@33175
  1965
himmelma@33175
  1966
lemma netlimit_within_interior:
himmelma@33175
  1967
  fixes x :: "'a::{perfect_space, real_normed_vector}"
himmelma@33175
  1968
    (* FIXME: generalize to perfect_space *)
himmelma@33175
  1969
  assumes "x \<in> interior S"
himmelma@33175
  1970
  shows "netlimit(at x within S) = x" (is "?lhs = ?rhs")
himmelma@33175
  1971
proof-
himmelma@33175
  1972
  from assms obtain e::real where e:"e>0" "ball x e \<subseteq> S" using open_interior[of S] unfolding open_contains_ball using interior_subset[of S] by auto
himmelma@33175
  1973
  hence "\<not> trivial_limit (at x within S)" using islimpt_subset[of x "ball x e" S] unfolding trivial_limit_within islimpt_ball centre_in_cball by auto
himmelma@33175
  1974
  thus ?thesis using netlimit_within by auto
himmelma@33175
  1975
qed
himmelma@33175
  1976
himmelma@33175
  1977
subsection{* Boundedness. *}
himmelma@33175
  1978
himmelma@33175
  1979
  (* FIXME: This has to be unified with BSEQ!! *)
himmelma@33175
  1980
definition
himmelma@33175
  1981
  bounded :: "'a::metric_space set \<Rightarrow> bool" where
himmelma@33175
  1982
  "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
himmelma@33175
  1983
himmelma@33175
  1984
lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
himmelma@33175
  1985
unfolding bounded_def
himmelma@33175
  1986
apply safe
himmelma@33175
  1987
apply (rule_tac x="dist a x + e" in exI, clarify)
himmelma@33175
  1988
apply (drule (1) bspec)
himmelma@33175
  1989
apply (erule order_trans [OF dist_triangle add_left_mono])
himmelma@33175
  1990
apply auto
himmelma@33175
  1991
done
himmelma@33175
  1992
himmelma@33175
  1993
lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
himmelma@33175
  1994
unfolding bounded_any_center [where a=0]
himmelma@33175
  1995
by (simp add: dist_norm)
himmelma@33175
  1996
himmelma@33175
  1997
lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
himmelma@33175
  1998
lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
himmelma@33175
  1999
  by (metis bounded_def subset_eq)
himmelma@33175
  2000
himmelma@33175
  2001
lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
himmelma@33175
  2002
  by (metis bounded_subset interior_subset)
himmelma@33175
  2003
himmelma@33175
  2004
lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
himmelma@33175
  2005
proof-
himmelma@33175
  2006
  from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
himmelma@33175
  2007
  { fix y assume "y \<in> closure S"
himmelma@33175
  2008
    then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
himmelma@33175
  2009
      unfolding closure_sequential by auto
himmelma@33175
  2010
    have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
himmelma@33175
  2011
    hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
himmelma@33175
  2012
      by (rule eventually_mono, simp add: f(1))
himmelma@33175
  2013
    have "dist x y \<le> a"
himmelma@33175
  2014
      apply (rule Lim_dist_ubound [of sequentially f])
himmelma@33175
  2015
      apply (rule trivial_limit_sequentially)
himmelma@33175
  2016
      apply (rule f(2))
himmelma@33175
  2017
      apply fact
himmelma@33175
  2018
      done
himmelma@33175
  2019
  }
himmelma@33175
  2020
  thus ?thesis unfolding bounded_def by auto
himmelma@33175
  2021
qed
himmelma@33175
  2022
himmelma@33175
  2023
lemma bounded_cball[simp,intro]: "bounded (cball x e)"
himmelma@33175
  2024
  apply (simp add: bounded_def)
himmelma@33175
  2025
  apply (rule_tac x=x in exI)
himmelma@33175
  2026
  apply (rule_tac x=e in exI)
himmelma@33175
  2027
  apply auto
himmelma@33175
  2028
  done
himmelma@33175
  2029
himmelma@33175
  2030
lemma bounded_ball[simp,intro]: "bounded(ball x e)"
himmelma@33175
  2031
  by (metis ball_subset_cball bounded_cball bounded_subset)
himmelma@33175
  2032
huffman@36362
  2033
lemma finite_imp_bounded[intro]:
huffman@36362
  2034
  fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
himmelma@33175
  2035
proof-
huffman@36362
  2036
  { fix a and F :: "'a set" assume as:"bounded F"
himmelma@33175
  2037
    then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
himmelma@33175
  2038
    hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
himmelma@33175
  2039
    hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
himmelma@33175
  2040
  }
himmelma@33175
  2041
  thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto
himmelma@33175
  2042
qed
himmelma@33175
  2043
himmelma@33175
  2044
lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
himmelma@33175
  2045
  apply (auto simp add: bounded_def)
himmelma@33175
  2046
  apply (rename_tac x y r s)
himmelma@33175
  2047
  apply (rule_tac x=x in exI)
himmelma@33175
  2048
  apply (rule_tac x="max r (dist x y + s)" in exI)
himmelma@33175
  2049
  apply (rule ballI, rename_tac z, safe)
himmelma@33175
  2050
  apply (drule (1) bspec, simp)
himmelma@33175
  2051
  apply (drule (1) bspec)
himmelma@33175
  2052
  apply (rule min_max.le_supI2)
himmelma@33175
  2053
  apply (erule order_trans [OF dist_triangle add_left_mono])
himmelma@33175
  2054
  done
himmelma@33175
  2055
himmelma@33175
  2056
lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
himmelma@33175
  2057
  by (induct rule: finite_induct[of F], auto)
himmelma@33175
  2058
himmelma@33175
  2059
lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
himmelma@33175
  2060
  apply (simp add: bounded_iff)
himmelma@33175
  2061
  apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
himmelma@33175
  2062
  by metis arith
himmelma@33175
  2063
himmelma@33175
  2064
lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
himmelma@33175
  2065
  by (metis Int_lower1 Int_lower2 bounded_subset)
himmelma@33175
  2066
himmelma@33175
  2067
lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
himmelma@33175
  2068
apply (metis Diff_subset bounded_subset)
himmelma@33175
  2069
done
himmelma@33175
  2070
himmelma@33175
  2071
lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
himmelma@33175
  2072
  by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
himmelma@33175
  2073
himmelma@33175
  2074
lemma not_bounded_UNIV[simp, intro]:
himmelma@33175
  2075
  "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
himmelma@33175
  2076
proof(auto simp add: bounded_pos not_le)
himmelma@33175
  2077
  obtain x :: 'a where "x \<noteq> 0"
himmelma@33175
  2078
    using perfect_choose_dist [OF zero_less_one] by fast
himmelma@33175
  2079
  fix b::real  assume b: "b >0"
himmelma@33175
  2080
  have b1: "b +1 \<ge> 0" using b by simp
himmelma@33175
  2081
  with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
himmelma@33175
  2082
    by (simp add: norm_sgn)
himmelma@33175
  2083
  then show "\<exists>x::'a. b < norm x" ..
himmelma@33175
  2084
qed
himmelma@33175
  2085
himmelma@33175
  2086
lemma bounded_linear_image:
himmelma@33175
  2087
  assumes "bounded S" "bounded_linear f"
himmelma@33175
  2088
  shows "bounded(f ` S)"
himmelma@33175
  2089
proof-
himmelma@33175
  2090
  from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
himmelma@33175
  2091
  from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)
himmelma@33175
  2092
  { fix x assume "x\<in>S"
himmelma@33175
  2093
    hence "norm x \<le> b" using b by auto
himmelma@33175
  2094
    hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
huffman@36778
  2095
      by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
himmelma@33175
  2096
  }
himmelma@33175
  2097
  thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
huffman@36778
  2098
    using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
himmelma@33175
  2099
qed
himmelma@33175
  2100
himmelma@33175
  2101
lemma bounded_scaling:
himmelma@33175
  2102
  fixes S :: "'a::real_normed_vector set"
himmelma@33175
  2103
  shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
himmelma@33175
  2104
  apply (rule bounded_linear_image, assumption)
himmelma@33175
  2105
  apply (rule scaleR.bounded_linear_right)
himmelma@33175
  2106
  done
himmelma@33175
  2107
himmelma@33175
  2108
lemma bounded_translation:
himmelma@33175
  2109
  fixes S :: "'a::real_normed_vector set"
himmelma@33175
  2110
  assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
himmelma@33175
  2111
proof-
himmelma@33175
  2112
  from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
himmelma@33175
  2113
  { fix x assume "x\<in>S"
himmelma@33175
  2114
    hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
himmelma@33175
  2115
  }
himmelma@33175
  2116
  thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
himmelma@33175
  2117
    by (auto intro!: add exI[of _ "b + norm a"])
himmelma@33175
  2118
qed
himmelma@33175
  2119
himmelma@33175
  2120
himmelma@33175
  2121
text{* Some theorems on sups and infs using the notion "bounded". *}
himmelma@33175
  2122
himmelma@33175
  2123
lemma bounded_real:
himmelma@33175
  2124
  fixes S :: "real set"
himmelma@33175
  2125
  shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
himmelma@33175
  2126
  by (simp add: bounded_iff)
himmelma@33175
  2127
paulson@33270
  2128
lemma bounded_has_Sup:
paulson@33270
  2129
  fixes S :: "real set"
paulson@33270
  2130
  assumes "bounded S" "S \<noteq> {}"
paulson@33270
  2131
  shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
paulson@33270
  2132
proof
paulson@33270
  2133
  fix x assume "x\<in>S"
paulson@33270
  2134
  thus "x \<le> Sup S"
paulson@33270
  2135
    by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
paulson@33270
  2136
next
paulson@33270
  2137
  show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
paulson@33270
  2138
    by (metis SupInf.Sup_least)
paulson@33270
  2139
qed
paulson@33270
  2140
paulson@33270
  2141
lemma Sup_insert:
paulson@33270
  2142
  fixes S :: "real set"
paulson@33270
  2143
  shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" 
paulson@33270
  2144
by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) 
paulson@33270
  2145
paulson@33270
  2146
lemma Sup_insert_finite:
paulson@33270
  2147
  fixes S :: "real set"
paulson@33270
  2148
  shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
paulson@33270
  2149
  apply (rule Sup_insert)
paulson@33270
  2150
  apply (rule finite_imp_bounded)
paulson@33270
  2151
  by simp
paulson@33270
  2152
paulson@33270
  2153
lemma bounded_has_Inf:
paulson@33270
  2154
  fixes S :: "real set"
paulson@33270
  2155
  assumes "bounded S"  "S \<noteq> {}"
paulson@33270
  2156
  shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
himmelma@33175
  2157
proof
himmelma@33175
  2158
  fix x assume "x\<in>S"
himmelma@33175
  2159
  from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
paulson@33270
  2160
  thus "x \<ge> Inf S" using `x\<in>S`
paulson@33270
  2161
    by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
himmelma@33175
  2162
next
paulson@33270
  2163
  show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
paulson@33270
  2164
    by (metis SupInf.Inf_greatest)
paulson@33270
  2165
qed
paulson@33270
  2166
paulson@33270
  2167
lemma Inf_insert:
paulson@33270
  2168
  fixes S :: "real set"
paulson@33270
  2169
  shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" 
paulson@33270
  2170
by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal) 
paulson@33270
  2171
lemma Inf_insert_finite:
paulson@33270
  2172
  fixes S :: "real set"
paulson@33270
  2173
  shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
paulson@33270
  2174
  by (rule Inf_insert, rule finite_imp_bounded, simp)
paulson@33270
  2175
himmelma@33175
  2176
himmelma@33175
  2177
(* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
himmelma@33175
  2178
lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
himmelma@33175
  2179
  apply (frule isGlb_isLb)
himmelma@33175
  2180
  apply (frule_tac x = y in isGlb_isLb)
himmelma@33175
  2181
  apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
himmelma@33175
  2182
  done
himmelma@33175
  2183
huffman@36437
  2184
subsection {* Equivalent versions of compactness *}
huffman@36437
  2185
huffman@36437
  2186
subsubsection{* Sequential compactness *}
himmelma@33175
  2187
himmelma@33175
  2188
definition
himmelma@33175
  2189
  compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
himmelma@33175
  2190
  "compact S \<longleftrightarrow>
himmelma@33175
  2191
   (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
himmelma@33175
  2192
       (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
himmelma@33175
  2193
himmelma@33175
  2194
text {*
himmelma@33175
  2195
  A metric space (or topological vector space) is said to have the
himmelma@33175
  2196
  Heine-Borel property if every closed and bounded subset is compact.
himmelma@33175
  2197
*}
himmelma@33175
  2198
himmelma@33175
  2199
class heine_borel =
himmelma@33175
  2200
  assumes bounded_imp_convergent_subsequence:
himmelma@33175
  2201
    "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
himmelma@33175
  2202
      \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
himmelma@33175
  2203
himmelma@33175
  2204
lemma bounded_closed_imp_compact:
himmelma@33175
  2205
  fixes s::"'a::heine_borel set"
himmelma@33175
  2206
  assumes "bounded s" and "closed s" shows "compact s"
himmelma@33175
  2207
proof (unfold compact_def, clarify)
himmelma@33175
  2208
  fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
himmelma@33175
  2209
  obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
himmelma@33175
  2210
    using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
himmelma@33175
  2211
  from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
himmelma@33175
  2212
  have "l \<in> s" using `closed s` fr l
himmelma@33175
  2213
    unfolding closed_sequential_limits by blast
himmelma@33175
  2214
  show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
himmelma@33175
  2215
    using `l \<in> s` r l by blast
himmelma@33175
  2216
qed
himmelma@33175
  2217
himmelma@33175
  2218
lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
himmelma@33175
  2219
proof(induct n)
himmelma@33175
  2220
  show "0 \<le> r 0" by auto
himmelma@33175
  2221
next
himmelma@33175
  2222
  fix n assume "n \<le> r n"
himmelma@33175
  2223
  moreover have "r n < r (Suc n)"
himmelma@33175
  2224
    using assms [unfolded subseq_def] by auto
himmelma@33175
  2225
  ultimately show "Suc n \<le> r (Suc n)" by auto
himmelma@33175
  2226
qed
himmelma@33175
  2227
himmelma@33175
  2228
lemma eventually_subseq:
himmelma@33175
  2229
  assumes r: "subseq r"
himmelma@33175
  2230
  shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
himmelma@33175
  2231
unfolding eventually_sequentially
himmelma@33175
  2232
by (metis subseq_bigger [OF r] le_trans)
himmelma@33175
  2233
himmelma@33175
  2234
lemma lim_subseq:
himmelma@33175
  2235
  "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
himmelma@33175
  2236
unfolding tendsto_def eventually_sequentially o_def
himmelma@33175
  2237
by (metis subseq_bigger le_trans)
himmelma@33175
  2238
himmelma@33175
  2239
lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
himmelma@33175
  2240
  unfolding Ex1_def
himmelma@33175
  2241
  apply (rule_tac x="nat_rec e f" in exI)
himmelma@33175
  2242
  apply (rule conjI)+
himmelma@33175
  2243
apply (rule def_nat_rec_0, simp)
himmelma@33175
  2244
apply (rule allI, rule def_nat_rec_Suc, simp)
himmelma@33175
  2245
apply (rule allI, rule impI, rule ext)
himmelma@33175
  2246
apply (erule conjE)
himmelma@33175
  2247
apply (induct_tac x)
huffman@36362
  2248
apply simp
himmelma@33175
  2249
apply (erule_tac x="n" in allE)
himmelma@33175
  2250
apply (simp)
himmelma@33175
  2251
done
himmelma@33175
  2252
himmelma@33175
  2253
lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
himmelma@33175
  2254
  assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
himmelma@33175
  2255
  shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"
himmelma@33175
  2256
proof-
himmelma@33175
  2257
  have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
himmelma@33175
  2258
  then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
himmelma@33175
  2259
  { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
himmelma@33175
  2260
    { fix n::nat
himmelma@33175
  2261
      obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
himmelma@33175
  2262
      have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
himmelma@33175
  2263
      with n have "s N \<le> t - e" using `e>0` by auto
himmelma@33175
  2264
      hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto  }
himmelma@33175
  2265
    hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
himmelma@33175
  2266
    hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto  }
himmelma@33175
  2267
  thus ?thesis by blast
himmelma@33175
  2268
qed
himmelma@33175
  2269
himmelma@33175
  2270
lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
himmelma@33175
  2271
  assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
himmelma@33175
  2272
  shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
himmelma@33175
  2273
  using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
himmelma@33175
  2274
  unfolding monoseq_def incseq_def
himmelma@33175
  2275
  apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
himmelma@33175
  2276
  unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
himmelma@33175
  2277
hoelzl@37489
  2278
(* TODO: merge this lemma with the ones above *)
hoelzl@37489
  2279
lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
hoelzl@37489
  2280
  assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"
hoelzl@37489
  2281
  shows "\<exists>l. (s ---> l) sequentially"
hoelzl@37489
  2282
proof-
hoelzl@37489
  2283
  obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto
hoelzl@37489
  2284
  { fix m::nat
hoelzl@37489
  2285
    have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"
hoelzl@37489
  2286
      apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
hoelzl@37489
  2287
      apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }
hoelzl@37489 <