src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author huffman
Sun Aug 28 20:56:49 2011 -0700 (2011-08-28)
changeset 44571 bd91b77c4cd6
parent 44568 e6f291cb5810
child 44584 08ad27488983
permissions -rw-r--r--
move class perfect_space into RealVector.thy;
use not_open_singleton as perfect_space class axiom;
generalize some lemmas to class perfect_space;
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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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    Author:     Brian Huffman, Portland State University
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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 Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith L2_Norm
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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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  by(auto simp add:power2_eq_square)
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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 as a value *}
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definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
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typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. 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_Collect_eq] .
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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 (simp add: fun_eq_iff)
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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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subsubsection {* 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 mem_Collect_eq
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  by fast+
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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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subsubsection {* 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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subsubsection {* Subspace topology *}
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definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
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lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
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  (is "istopology ?L")
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proof-
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  have "?L {}" by blast
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  {fix A B assume A: "?L A" and B: "?L B"
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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 "?L (A \<inter> B)" by blast}
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  moreover
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  {fix K assume K: "K \<subseteq> Collect ?L"
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    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
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      apply (rule set_eqI)
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      apply (simp add: Ball_def image_iff)
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      by metis
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    from K[unfolded th0 subset_image_iff]
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    obtain Sk where Sk: "Sk \<subseteq> Collect (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)
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    ultimately have "?L (\<Union>K)" by blast}
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  ultimately show ?thesis
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    unfolding subset_eq mem_Collect_eq 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
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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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subsubsection {* The standard Euclidean topology *}
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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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  done
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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_eqI)
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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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text {* Basic "localization" results are handy for connectedness. *}
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lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
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  by (auto simp add: openin_subtopology open_openin[symmetric])
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lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
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  by (auto simp add: openin_open)
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lemma open_openin_trans[trans]:
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 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
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  by (metis Int_absorb1  openin_open_Int)
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lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
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  by (auto simp add: openin_open)
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lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
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  by (simp add: closedin_subtopology closed_closedin Int_ac)
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lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
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  by (metis closedin_closed)
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lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
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  apply (subgoal_tac "S \<inter> T = T" )
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  apply auto
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  apply (frule closedin_closed_Int[of T S])
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  by simp
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lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
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  by (auto simp add: closedin_closed)
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lemma openin_euclidean_subtopology_iff:
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  fixes S U :: "'a::metric_space set"
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  shows "openin (subtopology euclidean U) S
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  \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
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proof
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  assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
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next
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  def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
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  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
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    unfolding T_def
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    apply clarsimp
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    apply (rule_tac x="d - dist x a" in exI)
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    apply (clarsimp simp add: less_diff_eq)
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    apply (erule rev_bexI)
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    apply (rule_tac x=d in exI, clarify)
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   275
    apply (erule le_less_trans [OF dist_triangle])
huffman@44210
   276
    done
huffman@44210
   277
  assume ?rhs hence 2: "S = U \<inter> T"
huffman@44210
   278
    unfolding T_def
huffman@44210
   279
    apply auto
huffman@44210
   280
    apply (drule (1) bspec, erule rev_bexI)
huffman@44210
   281
    apply auto
huffman@44210
   282
    done
huffman@44210
   283
  from 1 2 show ?lhs
huffman@44210
   284
    unfolding openin_open open_dist by fast
huffman@44210
   285
qed
huffman@44210
   286
huffman@44210
   287
text {* These "transitivity" results are handy too *}
huffman@44210
   288
huffman@44210
   289
lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
huffman@44210
   290
  \<Longrightarrow> openin (subtopology euclidean U) S"
huffman@44210
   291
  unfolding open_openin openin_open by blast
huffman@44210
   292
huffman@44210
   293
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
huffman@44210
   294
  by (auto simp add: openin_open intro: openin_trans)
huffman@44210
   295
huffman@44210
   296
lemma closedin_trans[trans]:
huffman@44210
   297
 "closedin (subtopology euclidean T) S \<Longrightarrow>
huffman@44210
   298
           closedin (subtopology euclidean U) T
huffman@44210
   299
           ==> closedin (subtopology euclidean U) S"
huffman@44210
   300
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
huffman@44210
   301
huffman@44210
   302
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
huffman@44210
   303
  by (auto simp add: closedin_closed intro: closedin_trans)
huffman@44210
   304
huffman@44210
   305
huffman@44210
   306
subsection {* Open and closed balls *}
himmelma@33175
   307
himmelma@33175
   308
definition
himmelma@33175
   309
  ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
himmelma@33175
   310
  "ball x e = {y. dist x y < e}"
himmelma@33175
   311
himmelma@33175
   312
definition
himmelma@33175
   313
  cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
himmelma@33175
   314
  "cball x e = {y. dist x y \<le> e}"
himmelma@33175
   315
himmelma@33175
   316
lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
himmelma@33175
   317
lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
himmelma@33175
   318
himmelma@33175
   319
lemma mem_ball_0 [simp]:
himmelma@33175
   320
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   321
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
himmelma@33175
   322
  by (simp add: dist_norm)
himmelma@33175
   323
himmelma@33175
   324
lemma mem_cball_0 [simp]:
himmelma@33175
   325
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   326
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
himmelma@33175
   327
  by (simp add: dist_norm)
himmelma@33175
   328
himmelma@33175
   329
lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
himmelma@33175
   330
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
himmelma@33175
   331
lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
himmelma@33175
   332
lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
himmelma@33175
   333
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
nipkow@39302
   334
  by (simp add: set_eq_iff) arith
himmelma@33175
   335
himmelma@33175
   336
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
nipkow@39302
   337
  by (simp add: set_eq_iff)
himmelma@33175
   338
himmelma@33175
   339
lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
himmelma@33175
   340
  "(a::real) - b < 0 \<longleftrightarrow> a < b"
himmelma@33175
   341
  "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
himmelma@33175
   342
lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
himmelma@33175
   343
  "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
himmelma@33175
   344
himmelma@33175
   345
lemma open_ball[intro, simp]: "open (ball x e)"
huffman@44170
   346
  unfolding open_dist ball_def mem_Collect_eq Ball_def
himmelma@33175
   347
  unfolding dist_commute
himmelma@33175
   348
  apply clarify
himmelma@33175
   349
  apply (rule_tac x="e - dist xa x" in exI)
himmelma@33175
   350
  using dist_triangle_alt[where z=x]
himmelma@33175
   351
  apply (clarsimp simp add: diff_less_iff)
himmelma@33175
   352
  apply atomize
himmelma@33175
   353
  apply (erule_tac x="y" in allE)
himmelma@33175
   354
  apply (erule_tac x="xa" in allE)
himmelma@33175
   355
  by arith
himmelma@33175
   356
himmelma@33175
   357
lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
himmelma@33175
   358
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   359
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
himmelma@33175
   360
hoelzl@33714
   361
lemma openE[elim?]:
hoelzl@33714
   362
  assumes "open S" "x\<in>S" 
hoelzl@33714
   363
  obtains e where "e>0" "ball x e \<subseteq> S"
hoelzl@33714
   364
  using assms unfolding open_contains_ball by auto
hoelzl@33714
   365
himmelma@33175
   366
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   367
  by (metis open_contains_ball subset_eq centre_in_ball)
himmelma@33175
   368
himmelma@33175
   369
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
nipkow@39302
   370
  unfolding mem_ball set_eq_iff
himmelma@33175
   371
  apply (simp add: not_less)
himmelma@33175
   372
  by (metis zero_le_dist order_trans dist_self)
himmelma@33175
   373
himmelma@33175
   374
lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
himmelma@33175
   375
himmelma@33175
   376
himmelma@33175
   377
subsection{* Connectedness *}
himmelma@33175
   378
himmelma@33175
   379
definition "connected S \<longleftrightarrow>
himmelma@33175
   380
  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
himmelma@33175
   381
  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
himmelma@33175
   382
himmelma@33175
   383
lemma connected_local:
himmelma@33175
   384
 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
himmelma@33175
   385
                 openin (subtopology euclidean S) e1 \<and>
himmelma@33175
   386
                 openin (subtopology euclidean S) e2 \<and>
himmelma@33175
   387
                 S \<subseteq> e1 \<union> e2 \<and>
himmelma@33175
   388
                 e1 \<inter> e2 = {} \<and>
himmelma@33175
   389
                 ~(e1 = {}) \<and>
himmelma@33175
   390
                 ~(e2 = {}))"
himmelma@33175
   391
unfolding connected_def openin_open by (safe, blast+)
himmelma@33175
   392
huffman@34105
   393
lemma exists_diff:
huffman@34105
   394
  fixes P :: "'a set \<Rightarrow> bool"
huffman@34105
   395
  shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
himmelma@33175
   396
proof-
himmelma@33175
   397
  {assume "?lhs" hence ?rhs by blast }
himmelma@33175
   398
  moreover
himmelma@33175
   399
  {fix S assume H: "P S"
huffman@34105
   400
    have "S = - (- S)" by auto
huffman@34105
   401
    with H have "P (- (- S))" by metis }
himmelma@33175
   402
  ultimately show ?thesis by metis
himmelma@33175
   403
qed
himmelma@33175
   404
himmelma@33175
   405
lemma connected_clopen: "connected S \<longleftrightarrow>
himmelma@33175
   406
        (\<forall>T. openin (subtopology euclidean S) T \<and>
himmelma@33175
   407
            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
himmelma@33175
   408
proof-
huffman@34105
   409
  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
   410
    unfolding connected_def openin_open closedin_closed
himmelma@33175
   411
    apply (subst exists_diff) by blast
huffman@34105
   412
  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
   413
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
himmelma@33175
   414
himmelma@33175
   415
  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
   416
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
himmelma@33175
   417
    unfolding connected_def openin_open closedin_closed by auto
himmelma@33175
   418
  {fix e2
himmelma@33175
   419
    {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
   420
        by auto}
himmelma@33175
   421
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
himmelma@33175
   422
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
himmelma@33175
   423
  then show ?thesis unfolding th0 th1 by simp
himmelma@33175
   424
qed
himmelma@33175
   425
himmelma@33175
   426
lemma connected_empty[simp, intro]: "connected {}"
himmelma@33175
   427
  by (simp add: connected_def)
himmelma@33175
   428
huffman@44210
   429
himmelma@33175
   430
subsection{* Limit points *}
himmelma@33175
   431
huffman@44207
   432
definition (in topological_space)
huffman@44207
   433
  islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
himmelma@33175
   434
  "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
   435
himmelma@33175
   436
lemma islimptI:
himmelma@33175
   437
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
himmelma@33175
   438
  shows "x islimpt S"
himmelma@33175
   439
  using assms unfolding islimpt_def by auto
himmelma@33175
   440
himmelma@33175
   441
lemma islimptE:
himmelma@33175
   442
  assumes "x islimpt S" and "x \<in> T" and "open T"
himmelma@33175
   443
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
himmelma@33175
   444
  using assms unfolding islimpt_def by auto
himmelma@33175
   445
himmelma@33175
   446
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
himmelma@33175
   447
himmelma@33175
   448
lemma islimpt_approachable:
himmelma@33175
   449
  fixes x :: "'a::metric_space"
himmelma@33175
   450
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
himmelma@33175
   451
  unfolding islimpt_def
himmelma@33175
   452
  apply auto
himmelma@33175
   453
  apply(erule_tac x="ball x e" in allE)
himmelma@33175
   454
  apply auto
himmelma@33175
   455
  apply(rule_tac x=y in bexI)
himmelma@33175
   456
  apply (auto simp add: dist_commute)
himmelma@33175
   457
  apply (simp add: open_dist, drule (1) bspec)
himmelma@33175
   458
  apply (clarify, drule spec, drule (1) mp, auto)
himmelma@33175
   459
  done
himmelma@33175
   460
himmelma@33175
   461
lemma islimpt_approachable_le:
himmelma@33175
   462
  fixes x :: "'a::metric_space"
himmelma@33175
   463
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
himmelma@33175
   464
  unfolding islimpt_approachable
himmelma@33175
   465
  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
   466
  by metis 
himmelma@33175
   467
huffman@44571
   468
lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
huffman@44571
   469
  unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
huffman@44571
   470
huffman@44210
   471
text {* A perfect space has no isolated points. *}
huffman@44210
   472
huffman@44571
   473
lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
huffman@44571
   474
  unfolding islimpt_UNIV_iff by (rule not_open_singleton)
himmelma@33175
   475
himmelma@33175
   476
lemma perfect_choose_dist:
huffman@44072
   477
  fixes x :: "'a::{perfect_space, metric_space}"
himmelma@33175
   478
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
himmelma@33175
   479
using islimpt_UNIV [of x]
himmelma@33175
   480
by (simp add: islimpt_approachable)
himmelma@33175
   481
himmelma@33175
   482
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
himmelma@33175
   483
  unfolding closed_def
himmelma@33175
   484
  apply (subst open_subopen)
huffman@34105
   485
  apply (simp add: islimpt_def subset_eq)
huffman@44170
   486
  by (metis ComplE ComplI)
himmelma@33175
   487
himmelma@33175
   488
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
himmelma@33175
   489
  unfolding islimpt_def by auto
himmelma@33175
   490
himmelma@33175
   491
lemma finite_set_avoid:
himmelma@33175
   492
  fixes a :: "'a::metric_space"
himmelma@33175
   493
  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
himmelma@33175
   494
proof(induct rule: finite_induct[OF fS])
boehmes@41863
   495
  case 1 thus ?case by (auto intro: zero_less_one)
himmelma@33175
   496
next
himmelma@33175
   497
  case (2 x F)
himmelma@33175
   498
  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
   499
  {assume "x = a" hence ?case using d by auto  }
himmelma@33175
   500
  moreover
himmelma@33175
   501
  {assume xa: "x\<noteq>a"
himmelma@33175
   502
    let ?d = "min d (dist a x)"
himmelma@33175
   503
    have dp: "?d > 0" using xa d(1) using dist_nz by auto
himmelma@33175
   504
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
himmelma@33175
   505
    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
himmelma@33175
   506
  ultimately show ?case by blast
himmelma@33175
   507
qed
himmelma@33175
   508
himmelma@33175
   509
lemma islimpt_finite:
himmelma@33175
   510
  fixes S :: "'a::metric_space set"
himmelma@33175
   511
  assumes fS: "finite S" shows "\<not> a islimpt S"
himmelma@33175
   512
  unfolding islimpt_approachable
himmelma@33175
   513
  using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
himmelma@33175
   514
himmelma@33175
   515
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
himmelma@33175
   516
  apply (rule iffI)
himmelma@33175
   517
  defer
himmelma@33175
   518
  apply (metis Un_upper1 Un_upper2 islimpt_subset)
himmelma@33175
   519
  unfolding islimpt_def
himmelma@33175
   520
  apply (rule ccontr, clarsimp, rename_tac A B)
himmelma@33175
   521
  apply (drule_tac x="A \<inter> B" in spec)
himmelma@33175
   522
  apply (auto simp add: open_Int)
himmelma@33175
   523
  done
himmelma@33175
   524
himmelma@33175
   525
lemma discrete_imp_closed:
himmelma@33175
   526
  fixes S :: "'a::metric_space set"
himmelma@33175
   527
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
himmelma@33175
   528
  shows "closed S"
himmelma@33175
   529
proof-
himmelma@33175
   530
  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
himmelma@33175
   531
    from e have e2: "e/2 > 0" by arith
himmelma@33175
   532
    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
   533
    let ?m = "min (e/2) (dist x y) "
himmelma@33175
   534
    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
himmelma@33175
   535
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
himmelma@33175
   536
    have th: "dist z y < e" using z y
himmelma@33175
   537
      by (intro dist_triangle_lt [where z=x], simp)
himmelma@33175
   538
    from d[rule_format, OF y(1) z(1) th] y z
himmelma@33175
   539
    have False by (auto simp add: dist_commute)}
himmelma@33175
   540
  then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
himmelma@33175
   541
qed
himmelma@33175
   542
huffman@44210
   543
huffman@44210
   544
subsection {* Interior of a Set *}
huffman@44210
   545
huffman@44519
   546
definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
huffman@44519
   547
huffman@44519
   548
lemma interiorI [intro?]:
huffman@44519
   549
  assumes "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
   550
  shows "x \<in> interior S"
huffman@44519
   551
  using assms unfolding interior_def by fast
huffman@44519
   552
huffman@44519
   553
lemma interiorE [elim?]:
huffman@44519
   554
  assumes "x \<in> interior S"
huffman@44519
   555
  obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
   556
  using assms unfolding interior_def by fast
huffman@44519
   557
huffman@44519
   558
lemma open_interior [simp, intro]: "open (interior S)"
huffman@44519
   559
  by (simp add: interior_def open_Union)
huffman@44519
   560
huffman@44519
   561
lemma interior_subset: "interior S \<subseteq> S"
huffman@44519
   562
  by (auto simp add: interior_def)
huffman@44519
   563
huffman@44519
   564
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
huffman@44519
   565
  by (auto simp add: interior_def)
huffman@44519
   566
huffman@44519
   567
lemma interior_open: "open S \<Longrightarrow> interior S = S"
huffman@44519
   568
  by (intro equalityI interior_subset interior_maximal subset_refl)
himmelma@33175
   569
himmelma@33175
   570
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
huffman@44519
   571
  by (metis open_interior interior_open)
huffman@44519
   572
huffman@44519
   573
lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
himmelma@33175
   574
  by (metis interior_maximal interior_subset subset_trans)
himmelma@33175
   575
huffman@44519
   576
lemma interior_empty [simp]: "interior {} = {}"
huffman@44519
   577
  using open_empty by (rule interior_open)
huffman@44519
   578
huffman@44522
   579
lemma interior_UNIV [simp]: "interior UNIV = UNIV"
huffman@44522
   580
  using open_UNIV by (rule interior_open)
huffman@44522
   581
huffman@44519
   582
lemma interior_interior [simp]: "interior (interior S) = interior S"
huffman@44519
   583
  using open_interior by (rule interior_open)
huffman@44519
   584
huffman@44522
   585
lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
huffman@44522
   586
  by (auto simp add: interior_def)
huffman@44519
   587
huffman@44519
   588
lemma interior_unique:
huffman@44519
   589
  assumes "T \<subseteq> S" and "open T"
huffman@44519
   590
  assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
huffman@44519
   591
  shows "interior S = T"
huffman@44519
   592
  by (intro equalityI assms interior_subset open_interior interior_maximal)
huffman@44519
   593
huffman@44519
   594
lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
huffman@44522
   595
  by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
huffman@44519
   596
    Int_lower2 interior_maximal interior_subset open_Int open_interior)
huffman@44519
   597
huffman@44519
   598
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
huffman@44519
   599
  using open_contains_ball_eq [where S="interior S"]
huffman@44519
   600
  by (simp add: open_subset_interior)
himmelma@33175
   601
himmelma@33175
   602
lemma interior_limit_point [intro]:
himmelma@33175
   603
  fixes x :: "'a::perfect_space"
himmelma@33175
   604
  assumes x: "x \<in> interior S" shows "x islimpt S"
huffman@44072
   605
  using x islimpt_UNIV [of x]
huffman@44072
   606
  unfolding interior_def islimpt_def
huffman@44072
   607
  apply (clarsimp, rename_tac T T')
huffman@44072
   608
  apply (drule_tac x="T \<inter> T'" in spec)
huffman@44072
   609
  apply (auto simp add: open_Int)
huffman@44072
   610
  done
himmelma@33175
   611
himmelma@33175
   612
lemma interior_closed_Un_empty_interior:
himmelma@33175
   613
  assumes cS: "closed S" and iT: "interior T = {}"
huffman@44519
   614
  shows "interior (S \<union> T) = interior S"
himmelma@33175
   615
proof
huffman@44519
   616
  show "interior S \<subseteq> interior (S \<union> T)"
huffman@44522
   617
    by (rule interior_mono, rule Un_upper1)
himmelma@33175
   618
next
himmelma@33175
   619
  show "interior (S \<union> T) \<subseteq> interior S"
himmelma@33175
   620
  proof
himmelma@33175
   621
    fix x assume "x \<in> interior (S \<union> T)"
huffman@44519
   622
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
himmelma@33175
   623
    show "x \<in> interior S"
himmelma@33175
   624
    proof (rule ccontr)
himmelma@33175
   625
      assume "x \<notin> interior S"
himmelma@33175
   626
      with `x \<in> R` `open R` obtain y where "y \<in> R - S"
huffman@44519
   627
        unfolding interior_def by fast
himmelma@33175
   628
      from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
himmelma@33175
   629
      from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
himmelma@33175
   630
      from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
himmelma@33175
   631
      show "False" unfolding interior_def by fast
himmelma@33175
   632
    qed
himmelma@33175
   633
  qed
himmelma@33175
   634
qed
himmelma@33175
   635
huffman@44365
   636
lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
huffman@44365
   637
proof (rule interior_unique)
huffman@44365
   638
  show "interior A \<times> interior B \<subseteq> A \<times> B"
huffman@44365
   639
    by (intro Sigma_mono interior_subset)
huffman@44365
   640
  show "open (interior A \<times> interior B)"
huffman@44365
   641
    by (intro open_Times open_interior)
huffman@44519
   642
  fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"
huffman@44519
   643
  proof (safe)
huffman@44519
   644
    fix x y assume "(x, y) \<in> T"
huffman@44519
   645
    then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
huffman@44519
   646
      using `open T` unfolding open_prod_def by fast
huffman@44519
   647
    hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
huffman@44519
   648
      using `T \<subseteq> A \<times> B` by auto
huffman@44519
   649
    thus "x \<in> interior A" and "y \<in> interior B"
huffman@44519
   650
      by (auto intro: interiorI)
huffman@44519
   651
  qed
huffman@44365
   652
qed
huffman@44365
   653
himmelma@33175
   654
huffman@44210
   655
subsection {* Closure of a Set *}
himmelma@33175
   656
himmelma@33175
   657
definition "closure S = S \<union> {x | x. x islimpt S}"
himmelma@33175
   658
huffman@44518
   659
lemma interior_closure: "interior S = - (closure (- S))"
huffman@44518
   660
  unfolding interior_def closure_def islimpt_def by auto
huffman@44518
   661
huffman@34105
   662
lemma closure_interior: "closure S = - interior (- S)"
huffman@44518
   663
  unfolding interior_closure by simp
himmelma@33175
   664
himmelma@33175
   665
lemma closed_closure[simp, intro]: "closed (closure S)"
huffman@44518
   666
  unfolding closure_interior by (simp add: closed_Compl)
huffman@44518
   667
huffman@44518
   668
lemma closure_subset: "S \<subseteq> closure S"
huffman@44518
   669
  unfolding closure_def by simp
himmelma@33175
   670
himmelma@33175
   671
lemma closure_hull: "closure S = closed hull S"
huffman@44519
   672
  unfolding hull_def closure_interior interior_def by auto
himmelma@33175
   673
himmelma@33175
   674
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
huffman@44519
   675
  unfolding closure_hull using closed_Inter by (rule hull_eq)
huffman@44519
   676
huffman@44519
   677
lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
huffman@44519
   678
  unfolding closure_eq .
huffman@44519
   679
huffman@44519
   680
lemma closure_closure [simp]: "closure (closure S) = closure S"
huffman@44518
   681
  unfolding closure_hull by (rule hull_hull)
himmelma@33175
   682
huffman@44522
   683
lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
huffman@44518
   684
  unfolding closure_hull by (rule hull_mono)
himmelma@33175
   685
huffman@44519
   686
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
huffman@44518
   687
  unfolding closure_hull by (rule hull_minimal)
himmelma@33175
   688
huffman@44519
   689
lemma closure_unique:
huffman@44519
   690
  assumes "S \<subseteq> T" and "closed T"
huffman@44519
   691
  assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
huffman@44519
   692
  shows "closure S = T"
huffman@44519
   693
  using assms unfolding closure_hull by (rule hull_unique)
huffman@44519
   694
huffman@44519
   695
lemma closure_empty [simp]: "closure {} = {}"
huffman@44518
   696
  using closed_empty by (rule closure_closed)
himmelma@33175
   697
huffman@44522
   698
lemma closure_UNIV [simp]: "closure UNIV = UNIV"
huffman@44518
   699
  using closed_UNIV by (rule closure_closed)
huffman@44518
   700
huffman@44518
   701
lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
huffman@44518
   702
  unfolding closure_interior by simp
himmelma@33175
   703
himmelma@33175
   704
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
himmelma@33175
   705
  using closure_empty closure_subset[of S]
himmelma@33175
   706
  by blast
himmelma@33175
   707
himmelma@33175
   708
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
   709
  using closure_eq[of S] closure_subset[of S]
himmelma@33175
   710
  by simp
himmelma@33175
   711
himmelma@33175
   712
lemma open_inter_closure_eq_empty:
himmelma@33175
   713
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
huffman@34105
   714
  using open_subset_interior[of S "- T"]
huffman@34105
   715
  using interior_subset[of "- T"]
himmelma@33175
   716
  unfolding closure_interior
himmelma@33175
   717
  by auto
himmelma@33175
   718
himmelma@33175
   719
lemma open_inter_closure_subset:
himmelma@33175
   720
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
himmelma@33175
   721
proof
himmelma@33175
   722
  fix x
himmelma@33175
   723
  assume as: "open S" "x \<in> S \<inter> closure T"
himmelma@33175
   724
  { assume *:"x islimpt T"
himmelma@33175
   725
    have "x islimpt (S \<inter> T)"
himmelma@33175
   726
    proof (rule islimptI)
himmelma@33175
   727
      fix A
himmelma@33175
   728
      assume "x \<in> A" "open A"
himmelma@33175
   729
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
himmelma@33175
   730
        by (simp_all add: open_Int)
himmelma@33175
   731
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
himmelma@33175
   732
        by (rule islimptE)
himmelma@33175
   733
      hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
himmelma@33175
   734
        by simp_all
himmelma@33175
   735
      thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
himmelma@33175
   736
    qed
himmelma@33175
   737
  }
himmelma@33175
   738
  then show "x \<in> closure (S \<inter> T)" using as
himmelma@33175
   739
    unfolding closure_def
himmelma@33175
   740
    by blast
himmelma@33175
   741
qed
himmelma@33175
   742
huffman@44519
   743
lemma closure_complement: "closure (- S) = - interior S"
huffman@44518
   744
  unfolding closure_interior by simp
himmelma@33175
   745
huffman@44519
   746
lemma interior_complement: "interior (- S) = - closure S"
huffman@44518
   747
  unfolding closure_interior by simp
himmelma@33175
   748
huffman@44365
   749
lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
huffman@44519
   750
proof (rule closure_unique)
huffman@44365
   751
  show "A \<times> B \<subseteq> closure A \<times> closure B"
huffman@44365
   752
    by (intro Sigma_mono closure_subset)
huffman@44365
   753
  show "closed (closure A \<times> closure B)"
huffman@44365
   754
    by (intro closed_Times closed_closure)
huffman@44519
   755
  fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
huffman@44365
   756
    apply (simp add: closed_def open_prod_def, clarify)
huffman@44365
   757
    apply (rule ccontr)
huffman@44365
   758
    apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
huffman@44365
   759
    apply (simp add: closure_interior interior_def)
huffman@44365
   760
    apply (drule_tac x=C in spec)
huffman@44365
   761
    apply (drule_tac x=D in spec)
huffman@44365
   762
    apply auto
huffman@44365
   763
    done
huffman@44365
   764
qed
huffman@44365
   765
huffman@44210
   766
huffman@44210
   767
subsection {* Frontier (aka boundary) *}
himmelma@33175
   768
himmelma@33175
   769
definition "frontier S = closure S - interior S"
himmelma@33175
   770
himmelma@33175
   771
lemma frontier_closed: "closed(frontier S)"
himmelma@33175
   772
  by (simp add: frontier_def closed_Diff)
himmelma@33175
   773
huffman@34105
   774
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
himmelma@33175
   775
  by (auto simp add: frontier_def interior_closure)
himmelma@33175
   776
himmelma@33175
   777
lemma frontier_straddle:
himmelma@33175
   778
  fixes a :: "'a::metric_space"
himmelma@33175
   779
  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
   780
proof
himmelma@33175
   781
  assume "?lhs"
himmelma@33175
   782
  { fix e::real
himmelma@33175
   783
    assume "e > 0"
himmelma@33175
   784
    let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
himmelma@33175
   785
    { assume "a\<in>S"
himmelma@33175
   786
      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
   787
      moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
himmelma@33175
   788
        unfolding frontier_closures closure_def islimpt_def using `e>0`
himmelma@33175
   789
        by (auto, erule_tac x="ball a e" in allE, auto)
himmelma@33175
   790
      ultimately have ?rhse by auto
himmelma@33175
   791
    }
himmelma@33175
   792
    moreover
himmelma@33175
   793
    { assume "a\<notin>S"
himmelma@33175
   794
      hence ?rhse using `?lhs`
himmelma@33175
   795
        unfolding frontier_closures closure_def islimpt_def
himmelma@33175
   796
        using open_ball[of a e] `e > 0`
paulson@33324
   797
          by simp (metis centre_in_ball mem_ball open_ball) 
himmelma@33175
   798
    }
himmelma@33175
   799
    ultimately have ?rhse by auto
himmelma@33175
   800
  }
himmelma@33175
   801
  thus ?rhs by auto
himmelma@33175
   802
next
himmelma@33175
   803
  assume ?rhs
himmelma@33175
   804
  moreover
himmelma@33175
   805
  { fix T assume "a\<notin>S" and
himmelma@33175
   806
    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
   807
    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
   808
    then obtain e where "e>0" "ball a e \<subseteq> T" by auto
himmelma@33175
   809
    then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
himmelma@33175
   810
    have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
himmelma@33175
   811
      using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
himmelma@33175
   812
  }
himmelma@33175
   813
  hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
himmelma@33175
   814
  moreover
himmelma@33175
   815
  { fix T assume "a \<in> T"  "open T" "a\<in>S"
himmelma@33175
   816
    then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
himmelma@33175
   817
    obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
huffman@34105
   818
    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
   819
  }
huffman@34105
   820
  hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
huffman@34105
   821
  ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
himmelma@33175
   822
qed
himmelma@33175
   823
himmelma@33175
   824
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
himmelma@33175
   825
  by (metis frontier_def closure_closed Diff_subset)
himmelma@33175
   826
hoelzl@34964
   827
lemma frontier_empty[simp]: "frontier {} = {}"
huffman@36362
   828
  by (simp add: frontier_def)
himmelma@33175
   829
himmelma@33175
   830
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
   831
proof-
himmelma@33175
   832
  { assume "frontier S \<subseteq> S"
himmelma@33175
   833
    hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
himmelma@33175
   834
    hence "closed S" using closure_subset_eq by auto
himmelma@33175
   835
  }
huffman@36362
   836
  thus ?thesis using frontier_subset_closed[of S] ..
himmelma@33175
   837
qed
himmelma@33175
   838
huffman@34105
   839
lemma frontier_complement: "frontier(- S) = frontier S"
himmelma@33175
   840
  by (auto simp add: frontier_def closure_complement interior_complement)
himmelma@33175
   841
himmelma@33175
   842
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
huffman@34105
   843
  using frontier_complement frontier_subset_eq[of "- S"]
huffman@34105
   844
  unfolding open_closed by auto
himmelma@33175
   845
huffman@44210
   846
huffman@44081
   847
subsection {* Filters and the ``eventually true'' quantifier *}
huffman@44081
   848
himmelma@33175
   849
definition
huffman@44081
   850
  at_infinity :: "'a::real_normed_vector filter" where
huffman@44081
   851
  "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
himmelma@33175
   852
himmelma@33175
   853
definition
huffman@44081
   854
  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
huffman@44081
   855
    (infixr "indirection" 70) where
himmelma@33175
   856
  "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
himmelma@33175
   857
huffman@44081
   858
text{* Prove That They are all filters. *}
himmelma@33175
   859
huffman@36358
   860
lemma eventually_at_infinity:
huffman@36358
   861
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
himmelma@33175
   862
unfolding at_infinity_def
huffman@44081
   863
proof (rule eventually_Abs_filter, rule is_filter.intro)
huffman@36358
   864
  fix P Q :: "'a \<Rightarrow> bool"
huffman@36358
   865
  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
   866
  then obtain r s where
huffman@36358
   867
    "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
huffman@36358
   868
  then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
huffman@36358
   869
  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
huffman@36358
   870
qed auto
himmelma@33175
   871
huffman@36437
   872
text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
himmelma@33175
   873
himmelma@33175
   874
lemma trivial_limit_within:
himmelma@33175
   875
  shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
himmelma@33175
   876
proof
himmelma@33175
   877
  assume "trivial_limit (at a within S)"
himmelma@33175
   878
  thus "\<not> a islimpt S"
himmelma@33175
   879
    unfolding trivial_limit_def
huffman@36358
   880
    unfolding eventually_within eventually_at_topological
himmelma@33175
   881
    unfolding islimpt_def
nipkow@39302
   882
    apply (clarsimp simp add: set_eq_iff)
himmelma@33175
   883
    apply (rename_tac T, rule_tac x=T in exI)
huffman@36358
   884
    apply (clarsimp, drule_tac x=y in bspec, simp_all)
himmelma@33175
   885
    done
himmelma@33175
   886
next
himmelma@33175
   887
  assume "\<not> a islimpt S"
himmelma@33175
   888
  thus "trivial_limit (at a within S)"
himmelma@33175
   889
    unfolding trivial_limit_def
huffman@36358
   890
    unfolding eventually_within eventually_at_topological
himmelma@33175
   891
    unfolding islimpt_def
huffman@36358
   892
    apply clarsimp
huffman@36358
   893
    apply (rule_tac x=T in exI)
huffman@36358
   894
    apply auto
himmelma@33175
   895
    done
himmelma@33175
   896
qed
himmelma@33175
   897
himmelma@33175
   898
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
himmelma@33175
   899
  using trivial_limit_within [of a UNIV]
himmelma@33175
   900
  by (simp add: within_UNIV)
himmelma@33175
   901
himmelma@33175
   902
lemma trivial_limit_at:
himmelma@33175
   903
  fixes a :: "'a::perfect_space"
himmelma@33175
   904
  shows "\<not> trivial_limit (at a)"
huffman@44571
   905
  by (rule at_neq_bot)
himmelma@33175
   906
himmelma@33175
   907
lemma trivial_limit_at_infinity:
huffman@44081
   908
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
huffman@36358
   909
  unfolding trivial_limit_def eventually_at_infinity
huffman@36358
   910
  apply clarsimp
huffman@44072
   911
  apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
huffman@44072
   912
   apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
huffman@44072
   913
  apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
huffman@44072
   914
  apply (drule_tac x=UNIV in spec, simp)
himmelma@33175
   915
  done
himmelma@33175
   916
huffman@36437
   917
text {* Some property holds "sufficiently close" to the limit point. *}
himmelma@33175
   918
himmelma@33175
   919
lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
himmelma@33175
   920
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
himmelma@33175
   921
unfolding eventually_at dist_nz by auto
himmelma@33175
   922
himmelma@33175
   923
lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
himmelma@33175
   924
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
himmelma@33175
   925
unfolding eventually_within eventually_at dist_nz by auto
himmelma@33175
   926
himmelma@33175
   927
lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
himmelma@33175
   928
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
himmelma@33175
   929
unfolding eventually_within
paulson@33324
   930
by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl) 
himmelma@33175
   931
himmelma@33175
   932
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
huffman@36358
   933
  unfolding trivial_limit_def
huffman@36358
   934
  by (auto elim: eventually_rev_mp)
himmelma@33175
   935
himmelma@33175
   936
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
huffman@36358
   937
  unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
himmelma@33175
   938
himmelma@33175
   939
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
huffman@44342
   940
  by (simp add: filter_eq_iff)
himmelma@33175
   941
himmelma@33175
   942
text{* Combining theorems for "eventually" *}
himmelma@33175
   943
himmelma@33175
   944
lemma eventually_rev_mono:
himmelma@33175
   945
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
himmelma@33175
   946
using eventually_mono [of P Q] by fast
himmelma@33175
   947
himmelma@33175
   948
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
himmelma@33175
   949
  by (simp add: eventually_False)
himmelma@33175
   950
huffman@44210
   951
huffman@36437
   952
subsection {* Limits *}
himmelma@33175
   953
huffman@44081
   954
text{* Notation Lim to avoid collition with lim defined in analysis *}
huffman@44081
   955
huffman@44081
   956
definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
huffman@44081
   957
  where "Lim A f = (THE l. (f ---> l) A)"
himmelma@33175
   958
himmelma@33175
   959
lemma Lim:
himmelma@33175
   960
 "(f ---> l) net \<longleftrightarrow>
himmelma@33175
   961
        trivial_limit net \<or>
himmelma@33175
   962
        (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
himmelma@33175
   963
  unfolding tendsto_iff trivial_limit_eq by auto
himmelma@33175
   964
himmelma@33175
   965
text{* Show that they yield usual definitions in the various cases. *}
himmelma@33175
   966
himmelma@33175
   967
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
himmelma@33175
   968
           (\<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
   969
  by (auto simp add: tendsto_iff eventually_within_le)
himmelma@33175
   970
himmelma@33175
   971
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
himmelma@33175
   972
        (\<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
   973
  by (auto simp add: tendsto_iff eventually_within)
himmelma@33175
   974
himmelma@33175
   975
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
himmelma@33175
   976
        (\<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
   977
  by (auto simp add: tendsto_iff eventually_at)
himmelma@33175
   978
himmelma@33175
   979
lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
himmelma@33175
   980
  unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
himmelma@33175
   981
himmelma@33175
   982
lemma Lim_at_infinity:
himmelma@33175
   983
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
himmelma@33175
   984
  by (auto simp add: tendsto_iff eventually_at_infinity)
himmelma@33175
   985
himmelma@33175
   986
lemma Lim_sequentially:
himmelma@33175
   987
 "(S ---> l) sequentially \<longleftrightarrow>
himmelma@33175
   988
          (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
huffman@44210
   989
  by (rule LIMSEQ_def) (* FIXME: redundant *)
himmelma@33175
   990
himmelma@33175
   991
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
himmelma@33175
   992
  by (rule topological_tendstoI, auto elim: eventually_rev_mono)
himmelma@33175
   993
himmelma@33175
   994
text{* The expected monotonicity property. *}
himmelma@33175
   995
himmelma@33175
   996
lemma Lim_within_empty: "(f ---> l) (net within {})"
himmelma@33175
   997
  unfolding tendsto_def Limits.eventually_within by simp
himmelma@33175
   998
himmelma@33175
   999
lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
himmelma@33175
  1000
  unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1001
  by (auto elim!: eventually_elim1)
himmelma@33175
  1002
himmelma@33175
  1003
lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
himmelma@33175
  1004
  shows "(f ---> l) (net within (S \<union> T))"
himmelma@33175
  1005
  using assms unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1006
  apply clarify
himmelma@33175
  1007
  apply (drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1008
  apply (drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1009
  apply (auto elim: eventually_elim2)
himmelma@33175
  1010
  done
himmelma@33175
  1011
himmelma@33175
  1012
lemma Lim_Un_univ:
himmelma@33175
  1013
 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
himmelma@33175
  1014
        ==> (f ---> l) net"
himmelma@33175
  1015
  by (metis Lim_Un within_UNIV)
himmelma@33175
  1016
himmelma@33175
  1017
text{* Interrelations between restricted and unrestricted limits. *}
himmelma@33175
  1018
himmelma@33175
  1019
lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
himmelma@33175
  1020
  (* FIXME: rename *)
himmelma@33175
  1021
  unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1022
  apply (clarify, drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1023
  by (auto elim!: eventually_elim1)
himmelma@33175
  1024
huffman@44210
  1025
lemma eventually_within_interior:
huffman@44210
  1026
  assumes "x \<in> interior S"
huffman@44210
  1027
  shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
huffman@44210
  1028
proof-
huffman@44519
  1029
  from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
huffman@44210
  1030
  { assume "?lhs"
huffman@44210
  1031
    then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
huffman@44210
  1032
      unfolding Limits.eventually_within Limits.eventually_at_topological
huffman@44210
  1033
      by auto
huffman@44210
  1034
    with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
huffman@44210
  1035
      by auto
huffman@44210
  1036
    then have "?rhs"
huffman@44210
  1037
      unfolding Limits.eventually_at_topological by auto
huffman@44210
  1038
  } moreover
huffman@44210
  1039
  { assume "?rhs" hence "?lhs"
huffman@44210
  1040
      unfolding Limits.eventually_within
huffman@44210
  1041
      by (auto elim: eventually_elim1)
huffman@44210
  1042
  } ultimately
huffman@44210
  1043
  show "?thesis" ..
huffman@44210
  1044
qed
huffman@44210
  1045
huffman@44210
  1046
lemma at_within_interior:
huffman@44210
  1047
  "x \<in> interior S \<Longrightarrow> at x within S = at x"
huffman@44210
  1048
  by (simp add: filter_eq_iff eventually_within_interior)
huffman@44210
  1049
huffman@44210
  1050
lemma at_within_open:
huffman@44210
  1051
  "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
huffman@44210
  1052
  by (simp only: at_within_interior interior_open)
huffman@44210
  1053
himmelma@33175
  1054
lemma Lim_within_open:
himmelma@33175
  1055
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
himmelma@33175
  1056
  assumes"a \<in> S" "open S"
huffman@44210
  1057
  shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
huffman@44210
  1058
  using assms by (simp only: at_within_open)
himmelma@33175
  1059
hoelzl@43338
  1060
lemma Lim_within_LIMSEQ:
hoelzl@43338
  1061
  fixes a :: real and L :: "'a::metric_space"
hoelzl@43338
  1062
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
hoelzl@43338
  1063
  shows "(X ---> L) (at a within T)"
hoelzl@43338
  1064
proof (rule ccontr)
hoelzl@43338
  1065
  assume "\<not> (X ---> L) (at a within T)"
hoelzl@43338
  1066
  hence "\<exists>r>0. \<forall>s>0. \<exists>x\<in>T. x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> r \<le> dist (X x) L"
hoelzl@43338
  1067
    unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
hoelzl@43338
  1068
  then obtain r where r: "r > 0" "\<And>s. s > 0 \<Longrightarrow> \<exists>x\<in>T-{a}. \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r" by blast
hoelzl@43338
  1069
hoelzl@43338
  1070
  let ?F = "\<lambda>n::nat. SOME x. x \<in> T \<and> x \<noteq> a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
hoelzl@43338
  1071
  have "\<And>n. \<exists>x. x \<in> T \<and> x \<noteq> a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
hoelzl@43338
  1072
    using r by (simp add: Bex_def)
hoelzl@43338
  1073
  hence F: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"
hoelzl@43338
  1074
    by (rule someI_ex)
hoelzl@43338
  1075
  hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
hoelzl@43338
  1076
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
hoelzl@43338
  1077
    and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
hoelzl@43338
  1078
    by fast+
hoelzl@43338
  1079
hoelzl@43338
  1080
  have "?F ----> a"
hoelzl@43338
  1081
  proof (rule LIMSEQ_I, unfold real_norm_def)
hoelzl@43338
  1082
      fix e::real
hoelzl@43338
  1083
      assume "0 < e"
hoelzl@43338
  1084
        (* choose no such that inverse (real (Suc n)) < e *)
hoelzl@43338
  1085
      then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
hoelzl@43338
  1086
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
hoelzl@43338
  1087
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
hoelzl@43338
  1088
      proof (intro exI allI impI)
hoelzl@43338
  1089
        fix n
hoelzl@43338
  1090
        assume mlen: "m \<le> n"
hoelzl@43338
  1091
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
hoelzl@43338
  1092
          by (rule F2)
hoelzl@43338
  1093
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
hoelzl@43338
  1094
          using mlen by auto
hoelzl@43338
  1095
        also from nodef have
hoelzl@43338
  1096
          "inverse (real (Suc m)) < e" .
hoelzl@43338
  1097
        finally show "\<bar>?F n - a\<bar> < e" .
hoelzl@43338
  1098
      qed
hoelzl@43338
  1099
  qed
hoelzl@43338
  1100
  moreover note `\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
hoelzl@43338
  1101
  ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
hoelzl@43338
  1102
  
hoelzl@43338
  1103
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
hoelzl@43338
  1104
  proof -
hoelzl@43338
  1105
    {
hoelzl@43338
  1106
      fix no::nat
hoelzl@43338
  1107
      obtain n where "n = no + 1" by simp
hoelzl@43338
  1108
      then have nolen: "no \<le> n" by simp
hoelzl@43338
  1109
        (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
hoelzl@43338
  1110
      have "dist (X (?F n)) L \<ge> r"
hoelzl@43338
  1111
        by (rule F3)
hoelzl@43338
  1112
      with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
hoelzl@43338
  1113
    }
hoelzl@43338
  1114
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
hoelzl@43338
  1115
    with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
hoelzl@43338
  1116
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
hoelzl@43338
  1117
  qed
hoelzl@43338
  1118
  ultimately show False by simp
hoelzl@43338
  1119
qed
hoelzl@43338
  1120
hoelzl@43338
  1121
lemma Lim_right_bound:
hoelzl@43338
  1122
  fixes f :: "real \<Rightarrow> real"
hoelzl@43338
  1123
  assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
hoelzl@43338
  1124
  assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
hoelzl@43338
  1125
  shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
hoelzl@43338
  1126
proof cases
hoelzl@43338
  1127
  assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
hoelzl@43338
  1128
next
hoelzl@43338
  1129
  assume [simp]: "{x<..} \<inter> I \<noteq> {}"
hoelzl@43338
  1130
  show ?thesis
hoelzl@43338
  1131
  proof (rule Lim_within_LIMSEQ, safe)
hoelzl@43338
  1132
    fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
hoelzl@43338
  1133
    
hoelzl@43338
  1134
    show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
hoelzl@43338
  1135
    proof (rule LIMSEQ_I, rule ccontr)
hoelzl@43338
  1136
      fix r :: real assume "0 < r"
hoelzl@43338
  1137
      with Inf_close[of "f ` ({x<..} \<inter> I)" r]
hoelzl@43338
  1138
      obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
hoelzl@43338
  1139
      from `x < y` have "0 < y - x" by auto
hoelzl@43338
  1140
      from S(2)[THEN LIMSEQ_D, OF this]
hoelzl@43338
  1141
      obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
hoelzl@43338
  1142
      
hoelzl@43338
  1143
      assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
hoelzl@43338
  1144
      moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
hoelzl@43338
  1145
        using S bnd by (intro Inf_lower[where z=K]) auto
hoelzl@43338
  1146
      ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
hoelzl@43338
  1147
        by (auto simp: not_less field_simps)
hoelzl@43338
  1148
      with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
hoelzl@43338
  1149
      show False by auto
hoelzl@43338
  1150
    qed
hoelzl@43338
  1151
  qed
hoelzl@43338
  1152
qed
hoelzl@43338
  1153
himmelma@33175
  1154
text{* Another limit point characterization. *}
himmelma@33175
  1155
himmelma@33175
  1156
lemma islimpt_sequential:
huffman@36667
  1157
  fixes x :: "'a::metric_space"
himmelma@33175
  1158
  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
himmelma@33175
  1159
    (is "?lhs = ?rhs")
himmelma@33175
  1160
proof
himmelma@33175
  1161
  assume ?lhs
himmelma@33175
  1162
  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
  1163
    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
  1164
  { fix n::nat
himmelma@33175
  1165
    have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
himmelma@33175
  1166
  }
himmelma@33175
  1167
  moreover
himmelma@33175
  1168
  { fix e::real assume "e>0"
himmelma@33175
  1169
    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
  1170
    then obtain N::nat where "inverse (real (N + 1)) < e" by auto
himmelma@33175
  1171
    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
  1172
    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
  1173
    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
  1174
  }
himmelma@33175
  1175
  hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
himmelma@33175
  1176
    unfolding Lim_sequentially using f by auto
himmelma@33175
  1177
  ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
himmelma@33175
  1178
next
himmelma@33175
  1179
  assume ?rhs
himmelma@33175
  1180
  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
  1181
  { fix e::real assume "e>0"
himmelma@33175
  1182
    then obtain N where "dist (f N) x < e" using f(2) by auto
himmelma@33175
  1183
    moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
himmelma@33175
  1184
    ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
himmelma@33175
  1185
  }
himmelma@33175
  1186
  thus ?lhs unfolding islimpt_approachable by auto
himmelma@33175
  1187
qed
himmelma@33175
  1188
huffman@44125
  1189
lemma Lim_inv: (* TODO: delete *)
huffman@44081
  1190
  fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
huffman@44081
  1191
  assumes "(f ---> l) A" and "l \<noteq> 0"
huffman@44081
  1192
  shows "((inverse o f) ---> inverse l) A"
huffman@36437
  1193
  unfolding o_def using assms by (rule tendsto_inverse)
huffman@36437
  1194
himmelma@33175
  1195
lemma Lim_null:
himmelma@33175
  1196
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@44125
  1197
  shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
himmelma@33175
  1198
  by (simp add: Lim dist_norm)
himmelma@33175
  1199
himmelma@33175
  1200
lemma Lim_null_comparison:
himmelma@33175
  1201
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1202
  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
himmelma@33175
  1203
  shows "(f ---> 0) net"
huffman@44252
  1204
proof (rule metric_tendsto_imp_tendsto)
huffman@44252
  1205
  show "(g ---> 0) net" by fact
huffman@44252
  1206
  show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
huffman@44252
  1207
    using assms(1) by (rule eventually_elim1, simp add: dist_norm)
himmelma@33175
  1208
qed
himmelma@33175
  1209
himmelma@33175
  1210
lemma Lim_transform_bound:
himmelma@33175
  1211
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1212
  fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
himmelma@33175
  1213
  assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
himmelma@33175
  1214
  shows "(f ---> 0) net"
huffman@44252
  1215
  using assms(1) tendsto_norm_zero [OF assms(2)]
huffman@44252
  1216
  by (rule Lim_null_comparison)
himmelma@33175
  1217
himmelma@33175
  1218
text{* Deducing things about the limit from the elements. *}
himmelma@33175
  1219
himmelma@33175
  1220
lemma Lim_in_closed_set:
himmelma@33175
  1221
  assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
himmelma@33175
  1222
  shows "l \<in> S"
himmelma@33175
  1223
proof (rule ccontr)
himmelma@33175
  1224
  assume "l \<notin> S"
himmelma@33175
  1225
  with `closed S` have "open (- S)" "l \<in> - S"
himmelma@33175
  1226
    by (simp_all add: open_Compl)
himmelma@33175
  1227
  with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
himmelma@33175
  1228
    by (rule topological_tendstoD)
himmelma@33175
  1229
  with assms(2) have "eventually (\<lambda>x. False) net"
himmelma@33175
  1230
    by (rule eventually_elim2) simp
himmelma@33175
  1231
  with assms(3) show "False"
himmelma@33175
  1232
    by (simp add: eventually_False)
himmelma@33175
  1233
qed
himmelma@33175
  1234
himmelma@33175
  1235
text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
himmelma@33175
  1236
himmelma@33175
  1237
lemma Lim_dist_ubound:
himmelma@33175
  1238
  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
himmelma@33175
  1239
  shows "dist a l <= e"
huffman@44252
  1240
proof-
huffman@44252
  1241
  have "dist a l \<in> {..e}"
huffman@44252
  1242
  proof (rule Lim_in_closed_set)
huffman@44252
  1243
    show "closed {..e}" by simp
huffman@44252
  1244
    show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
huffman@44252
  1245
    show "\<not> trivial_limit net" by fact
huffman@44252
  1246
    show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
huffman@44252
  1247
  qed
huffman@44252
  1248
  thus ?thesis by simp
himmelma@33175
  1249
qed
himmelma@33175
  1250
himmelma@33175
  1251
lemma Lim_norm_ubound:
himmelma@33175
  1252
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1253
  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
himmelma@33175
  1254
  shows "norm(l) <= e"
huffman@44252
  1255
proof-
huffman@44252
  1256
  have "norm l \<in> {..e}"
huffman@44252
  1257
  proof (rule Lim_in_closed_set)
huffman@44252
  1258
    show "closed {..e}" by simp
huffman@44252
  1259
    show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
huffman@44252
  1260
    show "\<not> trivial_limit net" by fact
huffman@44252
  1261
    show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
huffman@44252
  1262
  qed
huffman@44252
  1263
  thus ?thesis by simp
himmelma@33175
  1264
qed
himmelma@33175
  1265
himmelma@33175
  1266
lemma Lim_norm_lbound:
himmelma@33175
  1267
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1268
  assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
himmelma@33175
  1269
  shows "e \<le> norm l"
huffman@44252
  1270
proof-
huffman@44252
  1271
  have "norm l \<in> {e..}"
huffman@44252
  1272
  proof (rule Lim_in_closed_set)
huffman@44252
  1273
    show "closed {e..}" by simp
huffman@44252
  1274
    show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
huffman@44252
  1275
    show "\<not> trivial_limit net" by fact
huffman@44252
  1276
    show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
huffman@44252
  1277
  qed
huffman@44252
  1278
  thus ?thesis by simp
himmelma@33175
  1279
qed
himmelma@33175
  1280
himmelma@33175
  1281
text{* Uniqueness of the limit, when nontrivial. *}
himmelma@33175
  1282
himmelma@33175
  1283
lemma tendsto_Lim:
himmelma@33175
  1284
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
himmelma@33175
  1285
  shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
hoelzl@41970
  1286
  unfolding Lim_def using tendsto_unique[of net f] by auto
himmelma@33175
  1287
himmelma@33175
  1288
text{* Limit under bilinear function *}
himmelma@33175
  1289
himmelma@33175
  1290
lemma Lim_bilinear:
himmelma@33175
  1291
  assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
himmelma@33175
  1292
  shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
himmelma@33175
  1293
using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
himmelma@33175
  1294
by (rule bounded_bilinear.tendsto)
himmelma@33175
  1295
himmelma@33175
  1296
text{* These are special for limits out of the same vector space. *}
himmelma@33175
  1297
himmelma@33175
  1298
lemma Lim_within_id: "(id ---> a) (at a within s)"
himmelma@33175
  1299
  unfolding tendsto_def Limits.eventually_within eventually_at_topological
himmelma@33175
  1300
  by auto
himmelma@33175
  1301
himmelma@33175
  1302
lemma Lim_at_id: "(id ---> a) (at a)"
himmelma@33175
  1303
apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
himmelma@33175
  1304
himmelma@33175
  1305
lemma Lim_at_zero:
himmelma@33175
  1306
  fixes a :: "'a::real_normed_vector"
himmelma@33175
  1307
  fixes l :: "'b::topological_space"
himmelma@33175
  1308
  shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
huffman@44252
  1309
  using LIM_offset_zero LIM_offset_zero_cancel ..
himmelma@33175
  1310
huffman@44081
  1311
text{* It's also sometimes useful to extract the limit point from the filter. *}
himmelma@33175
  1312
himmelma@33175
  1313
definition
huffman@44081
  1314
  netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
himmelma@33175
  1315
  "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
himmelma@33175
  1316
himmelma@33175
  1317
lemma netlimit_within:
himmelma@33175
  1318
  assumes "\<not> trivial_limit (at a within S)"
himmelma@33175
  1319
  shows "netlimit (at a within S) = a"
himmelma@33175
  1320
unfolding netlimit_def
himmelma@33175
  1321
apply (rule some_equality)
himmelma@33175
  1322
apply (rule Lim_at_within)
huffman@44568
  1323
apply (rule tendsto_ident_at)
hoelzl@41970
  1324
apply (erule tendsto_unique [OF assms])
himmelma@33175
  1325
apply (rule Lim_at_within)
huffman@44568
  1326
apply (rule tendsto_ident_at)
himmelma@33175
  1327
done
himmelma@33175
  1328
himmelma@33175
  1329
lemma netlimit_at:
huffman@44072
  1330
  fixes a :: "'a::{perfect_space,t2_space}"
himmelma@33175
  1331
  shows "netlimit (at a) = a"
himmelma@33175
  1332
  apply (subst within_UNIV[symmetric])
himmelma@33175
  1333
  using netlimit_within[of a UNIV]
himmelma@33175
  1334
  by (simp add: trivial_limit_at within_UNIV)
himmelma@33175
  1335
huffman@44210
  1336
lemma lim_within_interior:
huffman@44210
  1337
  "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
huffman@44210
  1338
  by (simp add: at_within_interior)
huffman@44210
  1339
huffman@44210
  1340
lemma netlimit_within_interior:
huffman@44210
  1341
  fixes x :: "'a::{t2_space,perfect_space}"
huffman@44210
  1342
  assumes "x \<in> interior S"
huffman@44210
  1343
  shows "netlimit (at x within S) = x"
huffman@44210
  1344
using assms by (simp add: at_within_interior netlimit_at)
huffman@44210
  1345
himmelma@33175
  1346
text{* Transformation of limit. *}
himmelma@33175
  1347
himmelma@33175
  1348
lemma Lim_transform:
himmelma@33175
  1349
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1350
  assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
himmelma@33175
  1351
  shows "(g ---> l) net"
huffman@44252
  1352
  using tendsto_diff [OF assms(2) assms(1)] by simp
himmelma@33175
  1353
himmelma@33175
  1354
lemma Lim_transform_eventually:
huffman@36667
  1355
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
himmelma@33175
  1356
  apply (rule topological_tendstoI)
himmelma@33175
  1357
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1358
  apply (erule (1) eventually_elim2, simp)
himmelma@33175
  1359
  done
himmelma@33175
  1360
himmelma@33175
  1361
lemma Lim_transform_within:
huffman@36667
  1362
  assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
huffman@36667
  1363
  and "(f ---> l) (at x within S)"
huffman@36667
  1364
  shows "(g ---> l) (at x within S)"
huffman@36667
  1365
proof (rule Lim_transform_eventually)
huffman@36667
  1366
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
huffman@36667
  1367
    unfolding eventually_within
huffman@36667
  1368
    using assms(1,2) by auto
huffman@36667
  1369
  show "(f ---> l) (at x within S)" by fact
huffman@36667
  1370
qed
himmelma@33175
  1371
himmelma@33175
  1372
lemma Lim_transform_at:
huffman@36667
  1373
  assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
huffman@36667
  1374
  and "(f ---> l) (at x)"
huffman@36667
  1375
  shows "(g ---> l) (at x)"
huffman@36667
  1376
proof (rule Lim_transform_eventually)
huffman@36667
  1377
  show "eventually (\<lambda>x. f x = g x) (at x)"
huffman@36667
  1378
    unfolding eventually_at
huffman@36667
  1379
    using assms(1,2) by auto
huffman@36667
  1380
  show "(f ---> l) (at x)" by fact
huffman@36667
  1381
qed
himmelma@33175
  1382
himmelma@33175
  1383
text{* Common case assuming being away from some crucial point like 0. *}
himmelma@33175
  1384
himmelma@33175
  1385
lemma Lim_transform_away_within:
huffman@36669
  1386
  fixes a b :: "'a::t1_space"
huffman@36667
  1387
  assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
himmelma@33175
  1388
  and "(f ---> l) (at a within S)"
himmelma@33175
  1389
  shows "(g ---> l) (at a within S)"
huffman@36669
  1390
proof (rule Lim_transform_eventually)
huffman@36669
  1391
  show "(f ---> l) (at a within S)" by fact
huffman@36669
  1392
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
huffman@36669
  1393
    unfolding Limits.eventually_within eventually_at_topological
huffman@36669
  1394
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
himmelma@33175
  1395
qed
himmelma@33175
  1396
himmelma@33175
  1397
lemma Lim_transform_away_at:
huffman@36669
  1398
  fixes a b :: "'a::t1_space"
himmelma@33175
  1399
  assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
himmelma@33175
  1400
  and fl: "(f ---> l) (at a)"
himmelma@33175
  1401
  shows "(g ---> l) (at a)"
himmelma@33175
  1402
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
himmelma@33175
  1403
  by (auto simp add: within_UNIV)
himmelma@33175
  1404
himmelma@33175
  1405
text{* Alternatively, within an open set. *}
himmelma@33175
  1406
himmelma@33175
  1407
lemma Lim_transform_within_open:
huffman@36667
  1408
  assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
huffman@36667
  1409
  and "(f ---> l) (at a)"
himmelma@33175
  1410
  shows "(g ---> l) (at a)"
huffman@36667
  1411
proof (rule Lim_transform_eventually)
huffman@36667
  1412
  show "eventually (\<lambda>x. f x = g x) (at a)"
huffman@36667
  1413
    unfolding eventually_at_topological
huffman@36667
  1414
    using assms(1,2,3) by auto
huffman@36667
  1415
  show "(f ---> l) (at a)" by fact
himmelma@33175
  1416
qed
himmelma@33175
  1417
himmelma@33175
  1418
text{* A congruence rule allowing us to transform limits assuming not at point. *}
himmelma@33175
  1419
himmelma@33175
  1420
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
himmelma@33175
  1421
huffman@36362
  1422
lemma Lim_cong_within(*[cong add]*):
hoelzl@43338
  1423
  assumes "a = b" "x = y" "S = T"
hoelzl@43338
  1424
  assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
hoelzl@43338
  1425
  shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
huffman@36667
  1426
  unfolding tendsto_def Limits.eventually_within eventually_at_topological
huffman@36667
  1427
  using assms by simp
huffman@36667
  1428
huffman@36667
  1429
lemma Lim_cong_at(*[cong add]*):
hoelzl@43338
  1430
  assumes "a = b" "x = y"
huffman@36667
  1431
  assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
hoelzl@43338
  1432
  shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
huffman@36667
  1433
  unfolding tendsto_def eventually_at_topological
huffman@36667
  1434
  using assms by simp
himmelma@33175
  1435
himmelma@33175
  1436
text{* Useful lemmas on closure and set of possible sequential limits.*}
himmelma@33175
  1437
himmelma@33175
  1438
lemma closure_sequential:
huffman@36667
  1439
  fixes l :: "'a::metric_space"
himmelma@33175
  1440
  shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
himmelma@33175
  1441
proof
himmelma@33175
  1442
  assume "?lhs" moreover
himmelma@33175
  1443
  { assume "l \<in> S"
huffman@44125
  1444
    hence "?rhs" using tendsto_const[of l sequentially] by auto
himmelma@33175
  1445
  } moreover
himmelma@33175
  1446
  { assume "l islimpt S"
himmelma@33175
  1447
    hence "?rhs" unfolding islimpt_sequential by auto
himmelma@33175
  1448
  } ultimately
himmelma@33175
  1449
  show "?rhs" unfolding closure_def by auto
himmelma@33175
  1450
next
himmelma@33175
  1451
  assume "?rhs"
himmelma@33175
  1452
  thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
himmelma@33175
  1453
qed
himmelma@33175
  1454
himmelma@33175
  1455
lemma closed_sequential_limits:
himmelma@33175
  1456
  fixes S :: "'a::metric_space set"
himmelma@33175
  1457
  shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
himmelma@33175
  1458
  unfolding closed_limpt
himmelma@33175
  1459
  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
  1460
  by metis
himmelma@33175
  1461
himmelma@33175
  1462
lemma closure_approachable:
himmelma@33175
  1463
  fixes S :: "'a::metric_space set"
himmelma@33175
  1464
  shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
himmelma@33175
  1465
  apply (auto simp add: closure_def islimpt_approachable)
himmelma@33175
  1466
  by (metis dist_self)
himmelma@33175
  1467
himmelma@33175
  1468
lemma closed_approachable:
himmelma@33175
  1469
  fixes S :: "'a::metric_space set"
himmelma@33175
  1470
  shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
himmelma@33175
  1471
  by (metis closure_closed closure_approachable)
himmelma@33175
  1472
himmelma@33175
  1473
text{* Some other lemmas about sequences. *}
himmelma@33175
  1474
huffman@36441
  1475
lemma sequentially_offset:
huffman@36441
  1476
  assumes "eventually (\<lambda>i. P i) sequentially"
huffman@36441
  1477
  shows "eventually (\<lambda>i. P (i + k)) sequentially"
huffman@36441
  1478
  using assms unfolding eventually_sequentially by (metis trans_le_add1)
huffman@36441
  1479
himmelma@33175
  1480
lemma seq_offset:
huffman@36441
  1481
  assumes "(f ---> l) sequentially"
huffman@36441
  1482
  shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
huffman@36441
  1483
  using assms unfolding tendsto_def
huffman@36441
  1484
  by clarify (rule sequentially_offset, simp)
himmelma@33175
  1485
himmelma@33175
  1486
lemma seq_offset_neg:
himmelma@33175
  1487
  "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
himmelma@33175
  1488
  apply (rule topological_tendstoI)
himmelma@33175
  1489
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1490
  apply (simp only: eventually_sequentially)
himmelma@33175
  1491
  apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
himmelma@33175
  1492
  apply metis
himmelma@33175
  1493
  by arith
himmelma@33175
  1494
himmelma@33175
  1495
lemma seq_offset_rev:
himmelma@33175
  1496
  "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
himmelma@33175
  1497
  apply (rule topological_tendstoI)
himmelma@33175
  1498
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1499
  apply (simp only: eventually_sequentially)
himmelma@33175
  1500
  apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
himmelma@33175
  1501
  by metis arith
himmelma@33175
  1502
himmelma@33175
  1503
lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
himmelma@33175
  1504
proof-
himmelma@33175
  1505
  { fix e::real assume "e>0"
himmelma@33175
  1506
    hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
himmelma@33175
  1507
      using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
huffman@36362
  1508
      by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
himmelma@33175
  1509
  }
himmelma@33175
  1510
  thus ?thesis unfolding Lim_sequentially dist_norm by simp
himmelma@33175
  1511
qed
himmelma@33175
  1512
huffman@44210
  1513
subsection {* More properties of closed balls *}
himmelma@33175
  1514
himmelma@33175
  1515
lemma closed_cball: "closed (cball x e)"
himmelma@33175
  1516
unfolding cball_def closed_def
himmelma@33175
  1517
unfolding Collect_neg_eq [symmetric] not_le
himmelma@33175
  1518
apply (clarsimp simp add: open_dist, rename_tac y)
himmelma@33175
  1519
apply (rule_tac x="dist x y - e" in exI, clarsimp)
himmelma@33175
  1520
apply (rename_tac x')
himmelma@33175
  1521
apply (cut_tac x=x and y=x' and z=y in dist_triangle)
himmelma@33175
  1522
apply simp
himmelma@33175
  1523
done
himmelma@33175
  1524
himmelma@33175
  1525
lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
himmelma@33175
  1526
proof-
himmelma@33175
  1527
  { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
himmelma@33175
  1528
    hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
himmelma@33175
  1529
  } moreover
himmelma@33175
  1530
  { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
himmelma@33175
  1531
    hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
himmelma@33175
  1532
  } ultimately
himmelma@33175
  1533
  show ?thesis unfolding open_contains_ball by auto
himmelma@33175
  1534
qed
himmelma@33175
  1535
himmelma@33175
  1536
lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
huffman@44170
  1537
  by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
himmelma@33175
  1538
himmelma@33175
  1539
lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
himmelma@33175
  1540
  apply (simp add: interior_def, safe)
himmelma@33175
  1541
  apply (force simp add: open_contains_cball)
himmelma@33175
  1542
  apply (rule_tac x="ball x e" in exI)
huffman@36362
  1543
  apply (simp add: subset_trans [OF ball_subset_cball])
himmelma@33175
  1544
  done
himmelma@33175
  1545
himmelma@33175
  1546
lemma islimpt_ball:
himmelma@33175
  1547
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
himmelma@33175
  1548
  shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
himmelma@33175
  1549
proof
himmelma@33175
  1550
  assume "?lhs"
himmelma@33175
  1551
  { assume "e \<le> 0"
himmelma@33175
  1552
    hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
himmelma@33175
  1553
    have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
himmelma@33175
  1554
  }
himmelma@33175
  1555
  hence "e > 0" by (metis not_less)
himmelma@33175
  1556
  moreover
himmelma@33175
  1557
  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
  1558
  ultimately show "?rhs" by auto
himmelma@33175
  1559
next
himmelma@33175
  1560
  assume "?rhs" hence "e>0"  by auto
himmelma@33175
  1561
  { fix d::real assume "d>0"
himmelma@33175
  1562
    have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1563
    proof(cases "d \<le> dist x y")
himmelma@33175
  1564
      case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1565
      proof(cases "x=y")
himmelma@33175
  1566
        case True hence False using `d \<le> dist x y` `d>0` by auto
himmelma@33175
  1567
        thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
himmelma@33175
  1568
      next
himmelma@33175
  1569
        case False
himmelma@33175
  1570
himmelma@33175
  1571
        have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
himmelma@33175
  1572
              = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
himmelma@33175
  1573
          unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
himmelma@33175
  1574
        also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
himmelma@33175
  1575
          using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
himmelma@33175
  1576
          unfolding scaleR_minus_left scaleR_one
himmelma@33175
  1577
          by (auto simp add: norm_minus_commute)
himmelma@33175
  1578
        also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
himmelma@33175
  1579
          unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
huffman@36778
  1580
          unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
himmelma@33175
  1581
        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
  1582
        finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
himmelma@33175
  1583
himmelma@33175
  1584
        moreover
himmelma@33175
  1585
himmelma@33175
  1586
        have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
himmelma@33175
  1587
          using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
himmelma@33175
  1588
        moreover
himmelma@33175
  1589
        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
  1590
          using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
himmelma@33175
  1591
          unfolding dist_norm by auto
himmelma@33175
  1592
        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
  1593
      qed
himmelma@33175
  1594
    next
himmelma@33175
  1595
      case False hence "d > dist x y" by auto
himmelma@33175
  1596
      show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1597
      proof(cases "x=y")
himmelma@33175
  1598
        case True
himmelma@33175
  1599
        obtain z where **: "z \<noteq> y" "dist z y < min e d"
himmelma@33175
  1600
          using perfect_choose_dist[of "min e d" y]
himmelma@33175
  1601
          using `d > 0` `e>0` by auto
himmelma@33175
  1602
        show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1603
          unfolding `x = y`
himmelma@33175
  1604
          using `z \<noteq> y` **
himmelma@33175
  1605
          by (rule_tac x=z in bexI, auto simp add: dist_commute)
himmelma@33175
  1606
      next
himmelma@33175
  1607
        case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  1608
          using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
himmelma@33175
  1609
      qed
himmelma@33175
  1610
    qed  }
himmelma@33175
  1611
  thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
himmelma@33175
  1612
qed
himmelma@33175
  1613
himmelma@33175
  1614
lemma closure_ball_lemma:
himmelma@33175
  1615
  fixes x y :: "'a::real_normed_vector"
himmelma@33175
  1616
  assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
himmelma@33175
  1617
proof (rule islimptI)
himmelma@33175
  1618
  fix T assume "y \<in> T" "open T"
himmelma@33175
  1619
  then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
himmelma@33175
  1620
    unfolding open_dist by fast
himmelma@33175
  1621
  (* choose point between x and y, within distance r of y. *)
himmelma@33175
  1622
  def k \<equiv> "min 1 (r / (2 * dist x y))"
himmelma@33175
  1623
  def z \<equiv> "y + scaleR k (x - y)"
himmelma@33175
  1624
  have z_def2: "z = x + scaleR (1 - k) (y - x)"
himmelma@33175
  1625
    unfolding z_def by (simp add: algebra_simps)
himmelma@33175
  1626
  have "dist z y < r"
himmelma@33175
  1627
    unfolding z_def k_def using `0 < r`
himmelma@33175
  1628
    by (simp add: dist_norm min_def)
himmelma@33175
  1629
  hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
himmelma@33175
  1630
  have "dist x z < dist x y"
himmelma@33175
  1631
    unfolding z_def2 dist_norm
himmelma@33175
  1632
    apply (simp add: norm_minus_commute)
himmelma@33175
  1633
    apply (simp only: dist_norm [symmetric])
himmelma@33175
  1634
    apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
himmelma@33175
  1635
    apply (rule mult_strict_right_mono)
himmelma@33175
  1636
    apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
himmelma@33175
  1637
    apply (simp add: zero_less_dist_iff `x \<noteq> y`)
himmelma@33175
  1638
    done
himmelma@33175
  1639
  hence "z \<in> ball x (dist x y)" by simp
himmelma@33175
  1640
  have "z \<noteq> y"
himmelma@33175
  1641
    unfolding z_def k_def using `x \<noteq> y` `0 < r`
himmelma@33175
  1642
    by (simp add: min_def)
himmelma@33175
  1643
  show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
himmelma@33175
  1644
    using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
himmelma@33175
  1645
    by fast
himmelma@33175
  1646
qed
himmelma@33175
  1647
himmelma@33175
  1648
lemma closure_ball:
himmelma@33175
  1649
  fixes x :: "'a::real_normed_vector"
himmelma@33175
  1650
  shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
himmelma@33175
  1651
apply (rule equalityI)
himmelma@33175
  1652
apply (rule closure_minimal)
himmelma@33175
  1653
apply (rule ball_subset_cball)
himmelma@33175
  1654
apply (rule closed_cball)
himmelma@33175
  1655
apply (rule subsetI, rename_tac y)
himmelma@33175
  1656
apply (simp add: le_less [where 'a=real])
himmelma@33175
  1657
apply (erule disjE)
himmelma@33175
  1658
apply (rule subsetD [OF closure_subset], simp)
himmelma@33175
  1659
apply (simp add: closure_def)
himmelma@33175
  1660
apply clarify
himmelma@33175
  1661
apply (rule closure_ball_lemma)
himmelma@33175
  1662
apply (simp add: zero_less_dist_iff)
himmelma@33175
  1663
done
himmelma@33175
  1664
himmelma@33175
  1665
(* In a trivial vector space, this fails for e = 0. *)
himmelma@33175
  1666
lemma interior_cball:
himmelma@33175
  1667
  fixes x :: "'a::{real_normed_vector, perfect_space}"
himmelma@33175
  1668
  shows "interior (cball x e) = ball x e"
himmelma@33175
  1669
proof(cases "e\<ge>0")
himmelma@33175
  1670
  case False note cs = this
himmelma@33175
  1671
  from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
himmelma@33175
  1672
  { fix y assume "y \<in> cball x e"
himmelma@33175
  1673
    hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
himmelma@33175
  1674
  hence "cball x e = {}" by auto
himmelma@33175
  1675
  hence "interior (cball x e) = {}" using interior_empty by auto
himmelma@33175
  1676
  ultimately show ?thesis by blast
himmelma@33175
  1677
next
himmelma@33175
  1678
  case True note cs = this
himmelma@33175
  1679
  have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
himmelma@33175
  1680
  { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
himmelma@33175
  1681
    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
  1682
himmelma@33175
  1683
    then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
himmelma@33175
  1684
      using perfect_choose_dist [of d] by auto
himmelma@33175
  1685
    have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
himmelma@33175
  1686
    hence xa_cball:"xa \<in> cball x e" using as(1) by auto
himmelma@33175
  1687
himmelma@33175
  1688
    hence "y \<in> ball x e" proof(cases "x = y")
himmelma@33175
  1689
      case True
himmelma@33175
  1690
      hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
himmelma@33175
  1691
      thus "y \<in> ball x e" using `x = y ` by simp
himmelma@33175
  1692
    next
himmelma@33175
  1693
      case False
himmelma@33175
  1694
      have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
himmelma@33175
  1695
        using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
himmelma@33175
  1696
      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
  1697
      have "y - x \<noteq> 0" using `x \<noteq> y` by auto
himmelma@33175
  1698
      hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
himmelma@33175
  1699
        using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
himmelma@33175
  1700
himmelma@33175
  1701
      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
  1702
        by (auto simp add: dist_norm algebra_simps)
himmelma@33175
  1703
      also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
himmelma@33175
  1704
        by (auto simp add: algebra_simps)
himmelma@33175
  1705
      also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
himmelma@33175
  1706
        using ** by auto
himmelma@33175
  1707
      also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
himmelma@33175
  1708
      finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
himmelma@33175
  1709
      thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
himmelma@33175
  1710
    qed  }
himmelma@33175
  1711
  hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
himmelma@33175
  1712
  ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
himmelma@33175
  1713
qed
himmelma@33175
  1714
himmelma@33175
  1715
lemma frontier_ball:
himmelma@33175
  1716
  fixes a :: "'a::real_normed_vector"
himmelma@33175
  1717
  shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
huffman@36362
  1718
  apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
nipkow@39302
  1719
  apply (simp add: set_eq_iff)
himmelma@33175
  1720
  by arith
himmelma@33175
  1721
himmelma@33175
  1722
lemma frontier_cball:
himmelma@33175
  1723
  fixes a :: "'a::{real_normed_vector, perfect_space}"
himmelma@33175
  1724
  shows "frontier(cball a e) = {x. dist a x = e}"
huffman@36362
  1725
  apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
nipkow@39302
  1726
  apply (simp add: set_eq_iff)
himmelma@33175
  1727
  by arith
himmelma@33175
  1728
himmelma@33175
  1729
lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
nipkow@39302
  1730
  apply (simp add: set_eq_iff not_le)
himmelma@33175
  1731
  by (metis zero_le_dist dist_self order_less_le_trans)
himmelma@33175
  1732
lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
himmelma@33175
  1733
himmelma@33175
  1734
lemma cball_eq_sing:
huffman@44072
  1735
  fixes x :: "'a::{metric_space,perfect_space}"
himmelma@33175
  1736
  shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
himmelma@33175
  1737
proof (rule linorder_cases)
himmelma@33175
  1738
  assume e: "0 < e"
himmelma@33175
  1739
  obtain a where "a \<noteq> x" "dist a x < e"
himmelma@33175
  1740
    using perfect_choose_dist [OF e] by auto
himmelma@33175
  1741
  hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
nipkow@39302
  1742
  with e show ?thesis by (auto simp add: set_eq_iff)
himmelma@33175
  1743
qed auto
himmelma@33175
  1744
himmelma@33175
  1745
lemma cball_sing:
himmelma@33175
  1746
  fixes x :: "'a::metric_space"
himmelma@33175
  1747
  shows "e = 0 ==> cball x e = {x}"
nipkow@39302
  1748
  by (auto simp add: set_eq_iff)
himmelma@33175
  1749
huffman@44210
  1750
huffman@44210
  1751
subsection {* Boundedness *}
himmelma@33175
  1752
himmelma@33175
  1753
  (* FIXME: This has to be unified with BSEQ!! *)
huffman@44207
  1754
definition (in metric_space)
huffman@44207
  1755
  bounded :: "'a set \<Rightarrow> bool" where
himmelma@33175
  1756
  "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
himmelma@33175
  1757
himmelma@33175
  1758
lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
himmelma@33175
  1759
unfolding bounded_def
himmelma@33175
  1760
apply safe
himmelma@33175
  1761
apply (rule_tac x="dist a x + e" in exI, clarify)
himmelma@33175
  1762
apply (drule (1) bspec)
himmelma@33175
  1763
apply (erule order_trans [OF dist_triangle add_left_mono])
himmelma@33175
  1764
apply auto
himmelma@33175
  1765
done
himmelma@33175
  1766
himmelma@33175
  1767
lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
himmelma@33175
  1768
unfolding bounded_any_center [where a=0]
himmelma@33175
  1769
by (simp add: dist_norm)
himmelma@33175
  1770
himmelma@33175
  1771
lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
himmelma@33175
  1772
lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
himmelma@33175
  1773
  by (metis bounded_def subset_eq)
himmelma@33175
  1774
himmelma@33175
  1775
lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
himmelma@33175
  1776
  by (metis bounded_subset interior_subset)
himmelma@33175
  1777
himmelma@33175
  1778
lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
himmelma@33175
  1779
proof-
himmelma@33175
  1780
  from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
himmelma@33175
  1781
  { fix y assume "y \<in> closure S"
himmelma@33175
  1782
    then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
himmelma@33175
  1783
      unfolding closure_sequential by auto
himmelma@33175
  1784
    have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
himmelma@33175
  1785
    hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
himmelma@33175
  1786
      by (rule eventually_mono, simp add: f(1))
himmelma@33175
  1787
    have "dist x y \<le> a"
himmelma@33175
  1788
      apply (rule Lim_dist_ubound [of sequentially f])
himmelma@33175
  1789
      apply (rule trivial_limit_sequentially)
himmelma@33175
  1790
      apply (rule f(2))
himmelma@33175
  1791
      apply fact
himmelma@33175
  1792
      done
himmelma@33175
  1793
  }
himmelma@33175
  1794
  thus ?thesis unfolding bounded_def by auto
himmelma@33175
  1795
qed
himmelma@33175
  1796
himmelma@33175
  1797
lemma bounded_cball[simp,intro]: "bounded (cball x e)"
himmelma@33175
  1798
  apply (simp add: bounded_def)
himmelma@33175
  1799
  apply (rule_tac x=x in exI)
himmelma@33175
  1800
  apply (rule_tac x=e in exI)
himmelma@33175
  1801
  apply auto
himmelma@33175
  1802
  done
himmelma@33175
  1803
himmelma@33175
  1804
lemma bounded_ball[simp,intro]: "bounded(ball x e)"
himmelma@33175
  1805
  by (metis ball_subset_cball bounded_cball bounded_subset)
himmelma@33175
  1806
huffman@36362
  1807
lemma finite_imp_bounded[intro]:
huffman@36362
  1808
  fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
himmelma@33175
  1809
proof-
huffman@36362
  1810
  { fix a and F :: "'a set" assume as:"bounded F"
himmelma@33175
  1811
    then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
himmelma@33175
  1812
    hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
himmelma@33175
  1813
    hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
himmelma@33175
  1814
  }
himmelma@33175
  1815
  thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto
himmelma@33175
  1816
qed
himmelma@33175
  1817
himmelma@33175
  1818
lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
himmelma@33175
  1819
  apply (auto simp add: bounded_def)
himmelma@33175
  1820
  apply (rename_tac x y r s)
himmelma@33175
  1821
  apply (rule_tac x=x in exI)
himmelma@33175
  1822
  apply (rule_tac x="max r (dist x y + s)" in exI)
himmelma@33175
  1823
  apply (rule ballI, rename_tac z, safe)
himmelma@33175
  1824
  apply (drule (1) bspec, simp)
himmelma@33175
  1825
  apply (drule (1) bspec)
himmelma@33175
  1826
  apply (rule min_max.le_supI2)
himmelma@33175
  1827
  apply (erule order_trans [OF dist_triangle add_left_mono])
himmelma@33175
  1828
  done
himmelma@33175
  1829
himmelma@33175
  1830
lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
himmelma@33175
  1831
  by (induct rule: finite_induct[of F], auto)
himmelma@33175
  1832
himmelma@33175
  1833
lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
himmelma@33175
  1834
  apply (simp add: bounded_iff)
himmelma@33175
  1835
  apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
himmelma@33175
  1836
  by metis arith
himmelma@33175
  1837
himmelma@33175
  1838
lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
himmelma@33175
  1839
  by (metis Int_lower1 Int_lower2 bounded_subset)
himmelma@33175
  1840
himmelma@33175
  1841
lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
himmelma@33175
  1842
apply (metis Diff_subset bounded_subset)
himmelma@33175
  1843
done
himmelma@33175
  1844
himmelma@33175
  1845
lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
himmelma@33175
  1846
  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
  1847
himmelma@33175
  1848
lemma not_bounded_UNIV[simp, intro]:
himmelma@33175
  1849
  "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
himmelma@33175
  1850
proof(auto simp add: bounded_pos not_le)
himmelma@33175
  1851
  obtain x :: 'a where "x \<noteq> 0"
himmelma@33175
  1852
    using perfect_choose_dist [OF zero_less_one] by fast
himmelma@33175
  1853
  fix b::real  assume b: "b >0"
himmelma@33175
  1854
  have b1: "b +1 \<ge> 0" using b by simp
himmelma@33175
  1855
  with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
himmelma@33175
  1856
    by (simp add: norm_sgn)
himmelma@33175
  1857
  then show "\<exists>x::'a. b < norm x" ..
himmelma@33175
  1858
qed
himmelma@33175
  1859
himmelma@33175
  1860
lemma bounded_linear_image:
himmelma@33175
  1861
  assumes "bounded S" "bounded_linear f"
himmelma@33175
  1862
  shows "bounded(f ` S)"
himmelma@33175
  1863
proof-
himmelma@33175
  1864
  from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
himmelma@33175
  1865
  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
  1866
  { fix x assume "x\<in>S"
himmelma@33175
  1867
    hence "norm x \<le> b" using b by auto
himmelma@33175
  1868
    hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
huffman@36778
  1869
      by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
himmelma@33175
  1870
  }
himmelma@33175
  1871
  thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
huffman@36778
  1872
    using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
himmelma@33175
  1873
qed
himmelma@33175
  1874
himmelma@33175
  1875
lemma bounded_scaling:
himmelma@33175
  1876
  fixes S :: "'a::real_normed_vector set"
himmelma@33175
  1877
  shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
himmelma@33175
  1878
  apply (rule bounded_linear_image, assumption)
huffman@44282
  1879
  apply (rule bounded_linear_scaleR_right)
himmelma@33175
  1880
  done
himmelma@33175
  1881
himmelma@33175
  1882
lemma bounded_translation:
himmelma@33175
  1883
  fixes S :: "'a::real_normed_vector set"
himmelma@33175
  1884
  assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
himmelma@33175
  1885
proof-
himmelma@33175
  1886
  from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
himmelma@33175
  1887
  { fix x assume "x\<in>S"
himmelma@33175
  1888
    hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
himmelma@33175
  1889
  }
himmelma@33175
  1890
  thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
himmelma@33175
  1891
    by (auto intro!: add exI[of _ "b + norm a"])
himmelma@33175
  1892
qed
himmelma@33175
  1893
himmelma@33175
  1894
himmelma@33175
  1895
text{* Some theorems on sups and infs using the notion "bounded". *}
himmelma@33175
  1896
himmelma@33175
  1897
lemma bounded_real:
himmelma@33175
  1898
  fixes S :: "real set"
himmelma@33175
  1899
  shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
himmelma@33175
  1900
  by (simp add: bounded_iff)
himmelma@33175
  1901
paulson@33270
  1902
lemma bounded_has_Sup:
paulson@33270
  1903
  fixes S :: "real set"
paulson@33270
  1904
  assumes "bounded S" "S \<noteq> {}"
paulson@33270
  1905
  shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
paulson@33270
  1906
proof
paulson@33270
  1907
  fix x assume "x\<in>S"
paulson@33270
  1908
  thus "x \<le> Sup S"
paulson@33270
  1909
    by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
paulson@33270
  1910
next
paulson@33270
  1911
  show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
paulson@33270
  1912
    by (metis SupInf.Sup_least)
paulson@33270
  1913
qed
paulson@33270
  1914
paulson@33270
  1915
lemma Sup_insert:
paulson@33270
  1916
  fixes S :: "real set"
paulson@33270
  1917
  shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" 
paulson@33270
  1918
by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) 
paulson@33270
  1919
paulson@33270
  1920
lemma Sup_insert_finite:
paulson@33270
  1921
  fixes S :: "real set"
paulson@33270
  1922
  shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
paulson@33270
  1923
  apply (rule Sup_insert)
paulson@33270
  1924
  apply (rule finite_imp_bounded)
paulson@33270
  1925
  by simp
paulson@33270
  1926
paulson@33270
  1927
lemma bounded_has_Inf:
paulson@33270
  1928
  fixes S :: "real set"
paulson@33270
  1929
  assumes "bounded S"  "S \<noteq> {}"
paulson@33270
  1930
  shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
himmelma@33175
  1931
proof
himmelma@33175
  1932
  fix x assume "x\<in>S"
himmelma@33175
  1933
  from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
paulson@33270
  1934
  thus "x \<ge> Inf S" using `x\<in>S`
paulson@33270
  1935
    by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
himmelma@33175
  1936
next
paulson@33270
  1937
  show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
paulson@33270
  1938
    by (metis SupInf.Inf_greatest)
paulson@33270
  1939
qed
paulson@33270
  1940
paulson@33270
  1941
lemma Inf_insert:
paulson@33270
  1942
  fixes S :: "real set"
paulson@33270
  1943
  shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" 
paulson@33270
  1944
by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal) 
paulson@33270
  1945
lemma Inf_insert_finite:
paulson@33270
  1946
  fixes S :: "real set"
paulson@33270
  1947
  shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
paulson@33270
  1948
  by (rule Inf_insert, rule finite_imp_bounded, simp)
paulson@33270
  1949
himmelma@33175
  1950
(* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
himmelma@33175
  1951
lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
himmelma@33175
  1952
  apply (frule isGlb_isLb)
himmelma@33175
  1953
  apply (frule_tac x = y in isGlb_isLb)
himmelma@33175
  1954
  apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
himmelma@33175
  1955
  done
himmelma@33175
  1956
huffman@44210
  1957
huffman@36437
  1958
subsection {* Equivalent versions of compactness *}
huffman@36437
  1959
huffman@36437
  1960
subsubsection{* Sequential compactness *}
himmelma@33175
  1961
himmelma@33175
  1962
definition
himmelma@33175
  1963
  compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
himmelma@33175
  1964
  "compact S \<longleftrightarrow>
himmelma@33175
  1965
   (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
himmelma@33175
  1966
       (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
himmelma@33175
  1967
huffman@44075
  1968
lemma compactI:
huffman@44075
  1969
  assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
huffman@44075
  1970
  shows "compact S"
huffman@44075
  1971
  unfolding compact_def using assms by fast
huffman@44075
  1972
huffman@44075
  1973
lemma compactE:
huffman@44075
  1974
  assumes "compact S" "\<forall>n. f n \<in> S"
huffman@44075
  1975
  obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
huffman@44075
  1976
  using assms unfolding compact_def by fast
huffman@44075
  1977
himmelma@33175
  1978
text {*
himmelma@33175
  1979
  A metric space (or topological vector space) is said to have the
himmelma@33175
  1980
  Heine-Borel property if every closed and bounded subset is compact.
himmelma@33175
  1981
*}
himmelma@33175
  1982
huffman@44207
  1983
class heine_borel = metric_space +
himmelma@33175
  1984
  assumes bounded_imp_convergent_subsequence:
himmelma@33175
  1985
    "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
himmelma@33175
  1986
      \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
himmelma@33175
  1987
himmelma@33175
  1988
lemma bounded_closed_imp_compact:
himmelma@33175
  1989
  fixes s::"'a::heine_borel set"
himmelma@33175
  1990
  assumes "bounded s" and "closed s" shows "compact s"
himmelma@33175
  1991
proof (unfold compact_def, clarify)
himmelma@33175
  1992
  fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
himmelma@33175
  1993
  obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
himmelma@33175
  1994
    using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
himmelma@33175
  1995
  from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
himmelma@33175
  1996
  have "l \<in> s" using `closed s` fr l
himmelma@33175
  1997
    unfolding closed_sequential_limits by blast
himmelma@33175
  1998
  show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
himmelma@33175
  1999
    using `l \<in> s` r l by blast
himmelma@33175
  2000
qed
himmelma@33175
  2001
himmelma@33175
  2002
lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
himmelma@33175
  2003
proof(induct n)
himmelma@33175
  2004
  show "0 \<le> r 0" by auto
himmelma@33175
  2005
next
himmelma@33175
  2006
  fix n assume "n \<le> r n"
himmelma@33175
  2007
  moreover have "r n < r (Suc n)"
himmelma@33175
  2008
    using assms [unfolded subseq_def] by auto
himmelma@33175
  2009
  ultimately show "Suc n \<le> r (Suc n)" by auto
himmelma@33175
  2010
qed
himmelma@33175
  2011
himmelma@33175
  2012
lemma eventually_subseq:
himmelma@33175
  2013
  assumes r: "subseq r"
himmelma@33175
  2014
  shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
himmelma@33175
  2015
unfolding eventually_sequentially
himmelma@33175
  2016
by (metis subseq_bigger [OF r] le_trans)
himmelma@33175
  2017
himmelma@33175
  2018
lemma lim_subseq:
himmelma@33175
  2019
  "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
himmelma@33175
  2020
unfolding tendsto_def eventually_sequentially o_def
himmelma@33175
  2021
by (metis subseq_bigger le_trans)
himmelma@33175
  2022
himmelma@33175
  2023
lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
himmelma@33175
  2024
  unfolding Ex1_def
himmelma@33175
  2025
  apply (rule_tac x="nat_rec e f" in exI)
himmelma@33175
  2026
  apply (rule conjI)+
himmelma@33175
  2027
apply (rule def_nat_rec_0, simp)
himmelma@33175
  2028
apply (rule allI, rule def_nat_rec_Suc, simp)
himmelma@33175
  2029
apply (rule allI, rule impI, rule ext)
himmelma@33175
  2030
apply (erule conjE)
himmelma@33175
  2031
apply (induct_tac x)
huffman@36362
  2032
apply simp
himmelma@33175
  2033
apply (erule_tac x="n" in allE)
himmelma@33175
  2034
apply (simp)
himmelma@33175
  2035
done
himmelma@33175
  2036
himmelma@33175
  2037
lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
himmelma@33175
  2038
  assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
himmelma@33175
  2039
  shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"
himmelma@33175
  2040
proof-
himmelma@33175
  2041
  have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
himmelma@33175
  2042
  then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
himmelma@33175
  2043
  { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
himmelma@33175
  2044
    { fix n::nat
himmelma@33175
  2045
      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
  2046
      have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
himmelma@33175
  2047
      with n have "s N \<le> t - e" using `e>0` by auto
himmelma@33175
  2048
      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
  2049
    hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
himmelma@33175
  2050
    hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto  }
himmelma@33175
  2051
  thus ?thesis by blast
himmelma@33175
  2052
qed
himmelma@33175
  2053
himmelma@33175
  2054
lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
himmelma@33175
  2055
  assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
himmelma@33175
  2056
  shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
himmelma@33175
  2057
  using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
himmelma@33175
  2058
  unfolding monoseq_def incseq_def
himmelma@33175
  2059
  apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
himmelma@33175
  2060
  unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
himmelma@33175
  2061
hoelzl@37489
  2062
(* TODO: merge this lemma with the ones above *)
hoelzl@37489
  2063
lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
hoelzl@37489
  2064
  assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"
hoelzl@37489
  2065
  shows "\<exists>l. (s ---> l) sequentially"
hoelzl@37489
  2066
proof-
hoelzl@37489
  2067
  obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto
hoelzl@37489
  2068
  { fix m::nat
hoelzl@37489
  2069
    have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"
hoelzl@37489
  2070
      apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
hoelzl@37489
  2071
      apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }
hoelzl@37489
  2072
  hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
hoelzl@37489
  2073
  then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar> (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a]
hoelzl@37489
  2074
    unfolding monoseq_def by auto
hoelzl@37489
  2075
  thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
hoelzl@37489
  2076
    unfolding dist_norm  by auto
hoelzl@37489
  2077
qed
hoelzl@37489
  2078
himmelma@33175
  2079
lemma compact_real_lemma:
himmelma@33175
  2080
  assumes "\<forall>n::nat. abs(s n) \<le> b"
himmelma@33175
  2081
  shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
himmelma@33175
  2082
proof-
himmelma@33175
  2083
  obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
himmelma@33175
  2084
    using seq_monosub[of s] by auto
himmelma@33175
  2085
  thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
himmelma@33175
  2086
    unfolding tendsto_iff dist_norm eventually_sequentially by auto
himmelma@33175
  2087
qed
himmelma@33175
  2088
himmelma@33175
  2089
instance real :: heine_borel
himmelma@33175
  2090
proof
himmelma@33175
  2091
  fix s :: "real set" and f :: "nat \<Rightarrow> real"
himmelma@33175
  2092
  assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
himmelma@33175
  2093
  then obtain b where b: "\<forall>n. abs (f n) \<le> b"
himmelma@33175
  2094
    unfolding bounded_iff by auto
himmelma@33175
  2095
  obtain l :: real and r :: "nat \<Rightarrow> nat" where
himmelma@33175
  2096
    r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
himmelma@33175
  2097
    using compact_real_lemma [OF b] by auto
himmelma@33175
  2098
  thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
himmelma@33175
  2099
    by auto
himmelma@33175
  2100
qed
himmelma@33175
  2101
huffman@44138
  2102
lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
huffman@44138
  2103
  apply (erule bounded_linear_image)
huffman@44138
  2104
  apply (rule bounded_linear_euclidean_component)
huffman@44138
  2105
  done
himmelma@33175
  2106
himmelma@33175
  2107
lemma compact_lemma:
hoelzl@37489
  2108
  fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
himmelma@33175
  2109
  assumes "bounded s" and "\<forall>n. f n \<in> s"
hoelzl@37489
  2110
  shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
hoelzl@37489
  2111
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
himmelma@33175
  2112
proof
hoelzl@37489
  2113
  fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
hoelzl@37489
  2114
  have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
hoelzl@37489
  2115
  hence "\<exists>l::'a. \<exists>r. subseq r \<and>
hoelzl@37489
  2116
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
himmelma@33175
  2117
  proof(induct d) case empty thus ?case unfolding subseq_def by auto
hoelzl@37489
  2118
  next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
hoelzl@37489
  2119
    have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
hoelzl@37489
  2120
    obtain l1::"'a" and r1 where r1:"subseq r1" and
hoelzl@37489
  2121
      lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
hoelzl@37489
  2122
      using insert(3) using insert(4) by auto
hoelzl@37489
  2123
    have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
hoelzl@37489
  2124
    obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
himmelma@33175
  2125
      using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
himmelma@33175
  2126
    def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
himmelma@33175
  2127
      using r1 and r2 unfolding r_def o_def subseq_def by auto
himmelma@33175
  2128
    moreover
hoelzl@37489
  2129
    def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
himmelma@33175
  2130
    { fix e::real assume "e>0"
hoelzl@37489
  2131
      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
hoelzl@37489
  2132
      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
hoelzl@37489
  2133
      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
himmelma@33175
  2134
        by (rule eventually_subseq)
hoelzl@37489
  2135
      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
hoelzl@37489
  2136
        using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
hoelzl@37489
  2137
        using insert.prems by auto
himmelma@33175
  2138
    }
himmelma@33175
  2139
    ultimately show ?case by auto
himmelma@33175
  2140
  qed
hoelzl@37489
  2141
  thus "\<exists>l::'a. \<exists>r. subseq r \<and>
hoelzl@37489
  2142
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
hoelzl@37489
  2143
    apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
hoelzl@37489
  2144
    apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe 
hoelzl@37489
  2145
    apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
hoelzl@37489
  2146
    apply(erule_tac x=i in ballE) 
hoelzl@37489
  2147
  proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
hoelzl@37489
  2148
    assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
hoelzl@37489
  2149
    hence *:"i\<ge>DIM('a)" by auto
hoelzl@37489
  2150
    thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
hoelzl@37489
  2151
  qed
hoelzl@37489
  2152
qed
hoelzl@37489
  2153
hoelzl@37489
  2154
instance euclidean_space \<subseteq> heine_borel
himmelma@33175
  2155
proof
hoelzl@37489
  2156
  fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
himmelma@33175
  2157
  assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
hoelzl@37489
  2158
  then obtain l::'a and r where r: "subseq r"
hoelzl@37489
  2159
    and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
himmelma@33175
  2160
    using compact_lemma [OF s f] by blast
hoelzl@37489
  2161
  let ?d = "{..<DIM('a)}"
himmelma@33175
  2162
  { fix e::real assume "e>0"
himmelma@33175
  2163
    hence "0 < e / (real_of_nat (card ?d))"
hoelzl@37489
  2164
      using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
  2165
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
himmelma@33175
  2166
      by simp
himmelma@33175
  2167
    moreover
hoelzl@37489
  2168
    { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
  2169
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
hoelzl@37489
  2170
        apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
himmelma@33175
  2171
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
hoelzl@37489
  2172
        apply(rule setsum_strict_mono) using n by auto
hoelzl@37489
  2173
      finally have "dist (f (r n)) l < e" unfolding setsum_constant
hoelzl@37489
  2174
        using DIM_positive[where 'a='a] by auto
himmelma@33175
  2175
    }
himmelma@33175
  2176
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
himmelma@33175
  2177
      by (rule eventually_elim1)
himmelma@33175
  2178
  }
himmelma@33175
  2179
  hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
himmelma@33175
  2180
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
himmelma@33175
  2181
qed
himmelma@33175
  2182
himmelma@33175
  2183
lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
himmelma@33175
  2184
unfolding bounded_def
himmelma@33175
  2185
apply clarify
himmelma@33175
  2186
apply (rule_tac x="a" in exI)
himmelma@33175
  2187
apply (rule_tac x="e" in exI)
himmelma@33175
  2188
apply clarsimp
himmelma@33175
  2189
apply (drule (1) bspec)
himmelma@33175
  2190
apply (simp add: dist_Pair_Pair)
himmelma@33175
  2191
apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
himmelma@33175
  2192
done
himmelma@33175
  2193
himmelma@33175
  2194
lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
himmelma@33175
  2195
unfolding bounded_def
himmelma@33175
  2196
apply clarify
himmelma@33175
  2197
apply (rule_tac x="b" in exI)
himmelma@33175
  2198
apply (rule_tac x="e" in exI)
himmelma@33175
  2199
apply clarsimp
himmelma@33175
  2200
apply (drule (1) bspec)
himmelma@33175
  2201
apply (simp add: dist_Pair_Pair)
himmelma@33175
  2202
apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
himmelma@33175
  2203
done
himmelma@33175
  2204
haftmann@37678
  2205
instance prod :: (heine_borel, heine_borel) heine_borel
himmelma@33175
  2206
proof
himmelma@33175
  2207
  fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
himmelma@33175
  2208
  assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
himmelma@33175
  2209
  from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
himmelma@33175
  2210
  from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
himmelma@33175
  2211
  obtain l1 r1 where r1: "subseq r1"
himmelma@33175
  2212
    and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
himmelma@33175
  2213
    using bounded_imp_convergent_subsequence [OF s1 f1]
himmelma@33175
  2214
    unfolding o_def by fast
himmelma@33175
  2215
  from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
himmelma@33175
  2216
  from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
himmelma@33175
  2217
  obtain l2 r2 where r2: "subseq r2"
himmelma@33175
  2218
    and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
himmelma@33175
  2219
    using bounded_imp_convergent_subsequence [OF s2 f2]
himmelma@33175
  2220
    unfolding o_def by fast
himmelma@33175
  2221
  have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
himmelma@33175
  2222
    using lim_subseq [OF r2 l1] unfolding o_def .
himmelma@33175
  2223
  have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
himmelma@33175
  2224
    using tendsto_Pair [OF l1' l2] unfolding o_def by simp
himmelma@33175
  2225
  have r: "subseq (r1 \<circ> r2)"
himmelma@33175
  2226
    using r1 r2 unfolding subseq_def by simp
himmelma@33175
  2227
  show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
himmelma@33175
  2228
    using l r by fast
himmelma@33175
  2229
qed
himmelma@33175
  2230
huffman@36437
  2231
subsubsection{* Completeness *}
himmelma@33175
  2232
himmelma@33175
  2233
lemma cauchy_def:
himmelma@33175
  2234
  "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
himmelma@33175
  2235
unfolding Cauchy_def by blast
himmelma@33175
  2236
himmelma@33175
  2237
definition
himmelma@33175
  2238
  complete :: "'a::metric_space set \<Rightarrow> bool" where
himmelma@33175
  2239
  "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
himmelma@33175
  2240
                      --> (\<exists>l \<in> s. (f ---> l) sequentially))"
himmelma@33175
  2241
himmelma@33175
  2242
lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
himmelma@33175
  2243
proof-
himmelma@33175
  2244
  { assume ?rhs
himmelma@33175
  2245
    { fix e::real
himmelma@33175
  2246
      assume "e>0"
himmelma@33175
  2247
      with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
himmelma@33175
  2248
        by (erule_tac x="e/2" in allE) auto
himmelma@33175
  2249
      { fix n m
himmelma@33175
  2250
        assume nm:"N \<le> m \<and> N \<le> n"
himmelma@33175
  2251
        hence "dist (s m) (s n) < e" using N
himmelma@33175
  2252
          using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
himmelma@33175
  2253
          by blast
himmelma@33175
  2254
      }
himmelma@33175
  2255
      hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
himmelma@33175
  2256
        by blast