src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author immler
Tue, 18 Mar 2014 10:12:58 +0100
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permissions -rw-r--r--
removed dependencies on theory Ordered_Euclidean_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
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  Complex_Main
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  "~~/src/HOL/Library/Countable_Set"
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  "~~/src/HOL/Library/FuncSet"
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  Linear_Algebra
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  Norm_Arith
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begin
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lemma dist_0_norm:
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  fixes x :: "'a::real_normed_vector"
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  shows "dist 0 x = norm x"
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unfolding dist_norm by simp
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lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d"
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  using dist_triangle[of y z x] by (simp add: dist_commute)
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(* LEGACY *)
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lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s \<circ> r) ----> l"
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  by (rule LIMSEQ_subseq_LIMSEQ)
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lemma countable_PiE:
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  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"
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  by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
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lemma Lim_within_open:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
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  shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
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  by (fact tendsto_within_open)
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lemma continuous_on_union:
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  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
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  by (fact continuous_on_closed_Un)
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lemma continuous_on_cases:
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  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>
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    \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>
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    continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
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  by (rule continuous_on_If) auto
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subsection {* Topological Basis *}
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context topological_space
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begin
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definition "topological_basis B \<longleftrightarrow>
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  (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
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lemma topological_basis:
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  "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
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  unfolding topological_basis_def
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  apply safe
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     apply fastforce
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    apply fastforce
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   apply (erule_tac x="x" in allE)
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   apply simp
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   apply (rule_tac x="{x}" in exI)
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  apply auto
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  done
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lemma topological_basis_iff:
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  assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
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  shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
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    (is "_ \<longleftrightarrow> ?rhs")
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proof safe
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  fix O' and x::'a
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  assume H: "topological_basis B" "open O'" "x \<in> O'"
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  then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
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  then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
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  then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
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next
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  assume H: ?rhs
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  show "topological_basis B"
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    using assms unfolding topological_basis_def
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  proof safe
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    fix O' :: "'a set"
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    assume "open O'"
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    with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
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      by (force intro: bchoice simp: Bex_def)
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    then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
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      by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
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  qed
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qed
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lemma topological_basisI:
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  assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
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    and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
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  shows "topological_basis B"
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  using assms by (subst topological_basis_iff) auto
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lemma topological_basisE:
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  fixes O'
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  assumes "topological_basis B"
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    and "open O'"
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    and "x \<in> O'"
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  obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
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proof atomize_elim
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  from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
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    by (simp add: topological_basis_def)
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  with topological_basis_iff assms
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  show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
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    using assms by (simp add: Bex_def)
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qed
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lemma topological_basis_open:
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  assumes "topological_basis B"
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    and "X \<in> B"
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  shows "open X"
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  using assms by (simp add: topological_basis_def)
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lemma topological_basis_imp_subbasis:
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  assumes B: "topological_basis B"
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  shows "open = generate_topology B"
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proof (intro ext iffI)
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  fix S :: "'a set"
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  assume "open S"
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  with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
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    unfolding topological_basis_def by blast
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  then show "generate_topology B S"
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    by (auto intro: generate_topology.intros dest: topological_basis_open)
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next
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  fix S :: "'a set"
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  assume "generate_topology B S"
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  then show "open S"
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    by induct (auto dest: topological_basis_open[OF B])
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qed
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lemma basis_dense:
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  fixes B :: "'a set set"
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    and f :: "'a set \<Rightarrow> 'a"
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  assumes "topological_basis B"
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    and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
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  shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
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proof (intro allI impI)
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  fix X :: "'a set"
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  assume "open X" and "X \<noteq> {}"
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  from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]]
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  obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
53255
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   148
  then show "\<exists>B'\<in>B. f B' \<in> X"
addd7b9b2bff tuned proofs;
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   149
    by (auto intro!: choosefrom_basis)
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qed
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   151
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end
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lemma topological_basis_prod:
53255
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  assumes A: "topological_basis A"
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    and B: "topological_basis B"
50882
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  shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
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   158
  unfolding topological_basis_def
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   159
proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
53255
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   160
  fix S :: "('a \<times> 'b) set"
addd7b9b2bff tuned proofs;
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   161
  assume "open S"
50882
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   162
  then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
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   163
  proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
53255
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parents: 53015
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   164
    fix x y
addd7b9b2bff tuned proofs;
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   165
    assume "(x, y) \<in> S"
50882
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   166
    from open_prod_elim[OF `open S` this]
a382bf90867e move prod instantiation of second_countable_topology to its definition
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   167
    obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
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   168
      by (metis mem_Sigma_iff)
55522
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   169
    moreover
23d2cbac6dce tuned proofs;
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   170
    from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
23d2cbac6dce tuned proofs;
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   171
      by (rule topological_basisE)
23d2cbac6dce tuned proofs;
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   172
    moreover
23d2cbac6dce tuned proofs;
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   173
    from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
23d2cbac6dce tuned proofs;
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parents: 55415
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   174
      by (rule topological_basisE)
50882
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    ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
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   176
      by (intro UN_I[of "(A0, B0)"]) auto
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   177
  qed auto
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qed (metis A B topological_basis_open open_Times)
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   179
53255
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   180
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subsection {* Countable Basis *}
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   183
locale countable_basis =
53640
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  fixes B :: "'a::topological_space set set"
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  assumes is_basis: "topological_basis B"
53282
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   186
    and countable_basis: "countable B"
33175
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parents:
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   187
begin
2083bde13ce1 distinguished session for multivariate analysis
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parents:
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   188
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   189
lemma open_countable_basis_ex:
50087
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   190
  assumes "open X"
50245
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   191
  shows "\<exists>B' \<subseteq> B. X = Union B'"
53255
addd7b9b2bff tuned proofs;
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   192
  using assms countable_basis is_basis
addd7b9b2bff tuned proofs;
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diff changeset
   193
  unfolding topological_basis_def by blast
50245
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   194
dea9363887a6 based countable topological basis on Countable_Set
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   195
lemma open_countable_basisE:
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   196
  assumes "open X"
dea9363887a6 based countable topological basis on Countable_Set
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   197
  obtains B' where "B' \<subseteq> B" "X = Union B'"
53255
addd7b9b2bff tuned proofs;
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diff changeset
   198
  using assms open_countable_basis_ex
addd7b9b2bff tuned proofs;
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   199
  by (atomize_elim) simp
50245
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   200
dea9363887a6 based countable topological basis on Countable_Set
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   201
lemma countable_dense_exists:
53291
f7fa953bd15b tuned proofs;
wenzelm
parents: 53282
diff changeset
   202
  "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
50087
635d73673b5e regularity of measures, therefore:
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diff changeset
   203
proof -
50245
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   204
  let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
dea9363887a6 based countable topological basis on Countable_Set
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diff changeset
   205
  have "countable (?f ` B)" using countable_basis by simp
dea9363887a6 based countable topological basis on Countable_Set
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parents: 50105
diff changeset
   206
  with basis_dense[OF is_basis, of ?f] show ?thesis
dea9363887a6 based countable topological basis on Countable_Set
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parents: 50105
diff changeset
   207
    by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
50087
635d73673b5e regularity of measures, therefore:
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diff changeset
   208
qed
635d73673b5e regularity of measures, therefore:
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diff changeset
   209
635d73673b5e regularity of measures, therefore:
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diff changeset
   210
lemma countable_dense_setE:
50245
dea9363887a6 based countable topological basis on Countable_Set
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diff changeset
   211
  obtains D :: "'a set"
dea9363887a6 based countable topological basis on Countable_Set
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diff changeset
   212
  where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
dea9363887a6 based countable topological basis on Countable_Set
immler
parents: 50105
diff changeset
   213
  using countable_dense_exists by blast
dea9363887a6 based countable topological basis on Countable_Set
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parents: 50105
diff changeset
   214
50087
635d73673b5e regularity of measures, therefore:
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parents: 49962
diff changeset
   215
end
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   216
50883
1421884baf5b introduce first_countable_topology typeclass
hoelzl
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   217
lemma (in first_countable_topology) first_countable_basisE:
1421884baf5b introduce first_countable_topology typeclass
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diff changeset
   218
  obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   219
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   220
  using first_countable_basis[of x]
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   221
  apply atomize_elim
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   222
  apply (elim exE)
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   223
  apply (rule_tac x="range A" in exI)
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   224
  apply auto
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   225
  done
50883
1421884baf5b introduce first_countable_topology typeclass
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parents: 50882
diff changeset
   226
51105
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   227
lemma (in first_countable_topology) first_countable_basis_Int_stableE:
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diff changeset
   228
  obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
a27fcd14c384 fine grained instantiations
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diff changeset
   229
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
a27fcd14c384 fine grained instantiations
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diff changeset
   230
    "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"
a27fcd14c384 fine grained instantiations
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diff changeset
   231
proof atomize_elim
55522
23d2cbac6dce tuned proofs;
wenzelm
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diff changeset
   232
  obtain A' where A':
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   233
    "countable A'"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   234
    "\<And>a. a \<in> A' \<Longrightarrow> x \<in> a"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   235
    "\<And>a. a \<in> A' \<Longrightarrow> open a"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   236
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A'. a \<subseteq> S"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   237
    by (rule first_countable_basisE) blast
51105
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immler
parents: 51103
diff changeset
   238
  def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   239
  then show "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>
51105
a27fcd14c384 fine grained instantiations
immler
parents: 51103
diff changeset
   240
        (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)"
a27fcd14c384 fine grained instantiations
immler
parents: 51103
diff changeset
   241
  proof (safe intro!: exI[where x=A])
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   242
    show "countable A"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   243
      unfolding A_def by (intro countable_image countable_Collect_finite)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   244
    fix a
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   245
    assume "a \<in> A"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   246
    then show "x \<in> a" "open a"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   247
      using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
51105
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diff changeset
   248
  next
52141
eff000cab70f weaker precendence of syntax for big intersection and union on sets
haftmann
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   249
    let ?int = "\<lambda>N. \<Inter>(from_nat_into A' ` N)"
53255
addd7b9b2bff tuned proofs;
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diff changeset
   250
    fix a b
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   251
    assume "a \<in> A" "b \<in> A"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   252
    then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   253
      by (auto simp: A_def)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   254
    then show "a \<inter> b \<in> A"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   255
      by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
51105
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immler
parents: 51103
diff changeset
   256
  next
53255
addd7b9b2bff tuned proofs;
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parents: 53015
diff changeset
   257
    fix S
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   258
    assume "open S" "x \<in> S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   259
    then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   260
    then show "\<exists>a\<in>A. a \<subseteq> S" using a A'
51105
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immler
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diff changeset
   261
      by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])
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diff changeset
   262
  qed
a27fcd14c384 fine grained instantiations
immler
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diff changeset
   263
qed
a27fcd14c384 fine grained instantiations
immler
parents: 51103
diff changeset
   264
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
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diff changeset
   265
lemma (in topological_space) first_countableI:
53255
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diff changeset
   266
  assumes "countable A"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   267
    and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   268
    and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   269
  shows "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   270
proof (safe intro!: exI[of _ "from_nat_into A"])
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   271
  fix i
51473
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hoelzl
parents: 51472
diff changeset
   272
  have "A \<noteq> {}" using 2[of UNIV] by auto
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   273
  show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   274
    using range_from_nat_into_subset[OF `A \<noteq> {}`] 1 by auto
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   275
next
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   276
  fix S
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   277
  assume "open S" "x\<in>S" from 2[OF this]
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   278
  show "\<exists>i. from_nat_into A i \<subseteq> S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   279
    using subset_range_from_nat_into[OF `countable A`] by auto
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   280
qed
51350
490f34774a9a eventually nhds represented using sequentially
hoelzl
parents: 51349
diff changeset
   281
50883
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   282
instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   283
proof
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   284
  fix x :: "'a \<times> 'b"
55522
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   285
  obtain A where A:
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   286
      "countable A"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   287
      "\<And>a. a \<in> A \<Longrightarrow> fst x \<in> a"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   288
      "\<And>a. a \<in> A \<Longrightarrow> open a"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   289
      "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   290
    by (rule first_countable_basisE[of "fst x"]) blast
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   291
  obtain B where B:
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   292
      "countable B"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   293
      "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   294
      "\<And>a. a \<in> B \<Longrightarrow> open a"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   295
      "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   296
    by (rule first_countable_basisE[of "snd x"]) blast
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   297
  show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   298
    (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   299
  proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   300
    fix a b
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   301
    assume x: "a \<in> A" "b \<in> B"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53597
diff changeset
   302
    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53597
diff changeset
   303
      unfolding mem_Times_iff
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53597
diff changeset
   304
      by (auto intro: open_Times)
50883
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   305
  next
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   306
    fix S
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   307
    assume "open S" "x \<in> S"
55522
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   308
    then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   309
      by (rule open_prod_elim)
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   310
    moreover
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   311
    from a'b' A(4)[of a'] B(4)[of b']
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   312
    obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   313
      by auto
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   314
    ultimately
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   315
    show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
50883
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   316
      by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   317
  qed (simp add: A B)
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   318
qed
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   319
50881
ae630bab13da renamed countable_basis_space to second_countable_topology
hoelzl
parents: 50526
diff changeset
   320
class second_countable_topology = topological_space +
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   321
  assumes ex_countable_subbasis:
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   322
    "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   323
begin
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   324
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   325
lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   326
proof -
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   327
  from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   328
    by blast
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   329
  let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   330
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   331
  show ?thesis
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   332
  proof (intro exI conjI)
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   333
    show "countable ?B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   334
      by (intro countable_image countable_Collect_finite_subset B)
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   335
    {
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   336
      fix S
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   337
      assume "open S"
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   338
      then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   339
        unfolding B
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   340
      proof induct
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   341
        case UNIV
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   342
        show ?case by (intro exI[of _ "{{}}"]) simp
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   343
      next
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   344
        case (Int a b)
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   345
        then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   346
          and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   347
          by blast
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   348
        show ?case
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   349
          unfolding x y Int_UN_distrib2
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   350
          by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   351
      next
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   352
        case (UN K)
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   353
        then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
55522
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   354
        then obtain k where
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   355
            "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> UNION (k ka) Inter = ka"
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   356
          unfolding bchoice_iff ..
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   357
        then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   358
          by (intro exI[of _ "UNION K k"]) auto
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   359
      next
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   360
        case (Basis S)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   361
        then show ?case
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   362
          by (intro exI[of _ "{{S}}"]) auto
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   363
      qed
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   364
      then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   365
        unfolding subset_image_iff by blast }
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   366
    then show "topological_basis ?B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   367
      unfolding topological_space_class.topological_basis_def
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   368
      by (safe intro!: topological_space_class.open_Inter)
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   369
         (simp_all add: B generate_topology.Basis subset_eq)
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   370
  qed
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   371
qed
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   372
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   373
end
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   374
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   375
sublocale second_countable_topology <
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   376
  countable_basis "SOME B. countable B \<and> topological_basis B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   377
  using someI_ex[OF ex_countable_basis]
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   378
  by unfold_locales safe
50094
84ddcf5364b4 allow arbitrary enumerations of basis in locale for generation of borel sets
immler
parents: 50087
diff changeset
   379
50882
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   380
instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   381
proof
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   382
  obtain A :: "'a set set" where "countable A" "topological_basis A"
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   383
    using ex_countable_basis by auto
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   384
  moreover
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   385
  obtain B :: "'b set set" where "countable B" "topological_basis B"
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   386
    using ex_countable_basis by auto
51343
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   387
  ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   388
    by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis
hoelzl
parents: 51342
diff changeset
   389
      topological_basis_imp_subbasis)
50882
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   390
qed
a382bf90867e move prod instantiation of second_countable_topology to its definition
hoelzl
parents: 50881
diff changeset
   391
50883
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   392
instance second_countable_topology \<subseteq> first_countable_topology
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   393
proof
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   394
  fix x :: 'a
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   395
  def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   396
  then have B: "countable B" "topological_basis B"
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   397
    using countable_basis is_basis
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   398
    by (auto simp: countable_basis is_basis)
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   399
  then show "\<exists>A::nat \<Rightarrow> 'a set.
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   400
    (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   401
    by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51472
diff changeset
   402
       (fastforce simp: topological_space_class.topological_basis_def)+
50883
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   403
qed
1421884baf5b introduce first_countable_topology typeclass
hoelzl
parents: 50882
diff changeset
   404
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   405
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   406
subsection {* Polish spaces *}
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   407
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   408
text {* Textbooks define Polish spaces as completely metrizable.
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   409
  We assume the topology to be complete for a given metric. *}
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   410
50881
ae630bab13da renamed countable_basis_space to second_countable_topology
hoelzl
parents: 50526
diff changeset
   411
class polish_space = complete_space + second_countable_topology
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   412
44517
68e8eb0ce8aa minimize imports
huffman
parents: 44516
diff changeset
   413
subsection {* General notion of a topology as a value *}
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   414
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   415
definition "istopology L \<longleftrightarrow>
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   416
  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))"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   417
49834
b27bbb021df1 discontinued obsolete typedef (open) syntax;
wenzelm
parents: 49711
diff changeset
   418
typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   419
  morphisms "openin" "topology"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   420
  unfolding istopology_def by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   421
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   422
lemma istopology_open_in[intro]: "istopology(openin U)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   423
  using openin[of U] by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   424
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   425
lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   426
  using topology_inverse[unfolded mem_Collect_eq] .
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   427
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   428
lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   429
  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   430
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   431
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   432
proof
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   433
  assume "T1 = T2"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   434
  then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   435
next
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   436
  assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   437
  then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   438
  then have "topology (openin T1) = topology (openin T2)" by simp
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   439
  then show "T1 = T2" unfolding openin_inverse .
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   440
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   441
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   442
text{* Infer the "universe" from union of all sets in the topology. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   443
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53597
diff changeset
   444
definition "topspace T = \<Union>{S. openin T S}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   445
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   446
subsubsection {* Main properties of open sets *}
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   447
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   448
lemma openin_clauses:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   449
  fixes U :: "'a topology"
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   450
  shows
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   451
    "openin U {}"
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   452
    "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   453
    "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   454
  using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   455
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   456
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   457
  unfolding topspace_def by blast
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   458
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   459
lemma openin_empty[simp]: "openin U {}"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   460
  by (simp add: openin_clauses)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   461
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   462
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36360
diff changeset
   463
  using openin_clauses by simp
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36360
diff changeset
   464
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36360
diff changeset
   465
lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36360
diff changeset
   466
  using openin_clauses by simp
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   467
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   468
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   469
  using openin_Union[of "{S,T}" U] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   470
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   471
lemma openin_topspace[intro, simp]: "openin U (topspace U)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   472
  by (simp add: openin_Union topspace_def)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   473
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   474
lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   475
  (is "?lhs \<longleftrightarrow> ?rhs")
36584
1535841fc2e9 prove lemma openin_subopen without using choice
huffman
parents: 36442
diff changeset
   476
proof
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   477
  assume ?lhs
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   478
  then show ?rhs by auto
36584
1535841fc2e9 prove lemma openin_subopen without using choice
huffman
parents: 36442
diff changeset
   479
next
1535841fc2e9 prove lemma openin_subopen without using choice
huffman
parents: 36442
diff changeset
   480
  assume H: ?rhs
1535841fc2e9 prove lemma openin_subopen without using choice
huffman
parents: 36442
diff changeset
   481
  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
1535841fc2e9 prove lemma openin_subopen without using choice
huffman
parents: 36442
diff changeset
   482
  have "openin U ?t" by (simp add: openin_Union)
1535841fc2e9 prove lemma openin_subopen without using choice
huffman
parents: 36442
diff changeset
   483
  also have "?t = S" using H by auto
1535841fc2e9 prove lemma openin_subopen without using choice
huffman
parents: 36442
diff changeset
   484
  finally show "openin U S" .
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   485
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   486
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   487
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   488
subsubsection {* Closed sets *}
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   489
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   490
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   491
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   492
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   493
  by (metis closedin_def)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   494
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   495
lemma closedin_empty[simp]: "closedin U {}"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   496
  by (simp add: closedin_def)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   497
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   498
lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   499
  by (simp add: closedin_def)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   500
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   501
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   502
  by (auto simp add: Diff_Un closedin_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   503
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   504
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   505
  by auto
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   506
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   507
lemma closedin_Inter[intro]:
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   508
  assumes Ke: "K \<noteq> {}"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   509
    and Kc: "\<forall>S \<in>K. closedin U S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   510
  shows "closedin U (\<Inter> K)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   511
  using Ke Kc unfolding closedin_def Diff_Inter by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   512
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   513
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   514
  using closedin_Inter[of "{S,T}" U] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   515
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   516
lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   517
  by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   518
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   519
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   520
  apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   521
  apply (metis openin_subset subset_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   522
  done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   523
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   524
lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   525
  by (simp add: openin_closedin_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   526
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   527
lemma openin_diff[intro]:
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   528
  assumes oS: "openin U S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   529
    and cT: "closedin U T"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   530
  shows "openin U (S - T)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   531
proof -
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   532
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   533
    by (auto simp add: topspace_def openin_subset)
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   534
  then show ?thesis using oS cT
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   535
    by (auto simp add: closedin_def)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   536
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   537
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   538
lemma closedin_diff[intro]:
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   539
  assumes oS: "closedin U S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   540
    and cT: "openin U T"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   541
  shows "closedin U (S - T)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   542
proof -
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   543
  have "S - T = S \<inter> (topspace U - T)"
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   544
    using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   545
  then show ?thesis
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   546
    using oS cT by (auto simp add: openin_closedin_eq)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   547
qed
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   548
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   549
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   550
subsubsection {* Subspace topology *}
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   551
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   552
definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   553
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   554
lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   555
  (is "istopology ?L")
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   556
proof -
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   557
  have "?L {}" by blast
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   558
  {
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   559
    fix A B
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   560
    assume A: "?L A" and B: "?L B"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   561
    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"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   562
      by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   563
    have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   564
      using Sa Sb by blast+
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   565
    then have "?L (A \<inter> B)" by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   566
  }
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   567
  moreover
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   568
  {
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   569
    fix K
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   570
    assume K: "K \<subseteq> Collect ?L"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   571
    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55522
diff changeset
   572
      by blast
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   573
    from K[unfolded th0 subset_image_iff]
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   574
    obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   575
      by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   576
    have "\<Union>K = (\<Union>Sk) \<inter> V"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   577
      using Sk by auto
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   578
    moreover have "openin U (\<Union> Sk)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   579
      using Sk by (auto simp add: subset_eq)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   580
    ultimately have "?L (\<Union>K)" by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   581
  }
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   582
  ultimately show ?thesis
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   583
    unfolding subset_eq mem_Collect_eq istopology_def by blast
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   584
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   585
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   586
lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   587
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   588
  by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   589
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   590
lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   591
  by (auto simp add: topspace_def openin_subtopology)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   592
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   593
lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   594
  unfolding closedin_def topspace_subtopology
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55522
diff changeset
   595
  by (auto simp add: openin_subtopology)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   596
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   597
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   598
  unfolding openin_subtopology
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55522
diff changeset
   599
  by auto (metis IntD1 in_mono openin_subset)
49711
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   600
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   601
lemma subtopology_superset:
e5aaae7eadc9 tuned proofs;
wenzelm
parents: 48125
diff changeset
   602
  assumes UV: "topspace U \<subseteq> V"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   603
  shows "subtopology U V = U"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   604
proof -
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   605
  {
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   606
    fix S
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   607
    {
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   608
      fix T
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   609
      assume T: "openin U T" "S = T \<inter> V"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   610
      from T openin_subset[OF T(1)] UV have eq: "S = T"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   611
        by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   612
      have "openin U S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   613
        unfolding eq using T by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   614
    }
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   615
    moreover
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   616
    {
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   617
      assume S: "openin U S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   618
      then have "\<exists>T. openin U T \<and> S = T \<inter> V"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   619
        using openin_subset[OF S] UV by auto
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   620
    }
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   621
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   622
      by blast
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   623
  }
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   624
  then show ?thesis
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   625
    unfolding topology_eq openin_subtopology by blast
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   626
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   627
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   628
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   629
  by (simp add: subtopology_superset)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   630
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   631
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   632
  by (simp add: subtopology_superset)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   633
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   634
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   635
subsubsection {* The standard Euclidean topology *}
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   636
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   637
definition euclidean :: "'a::topological_space topology"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   638
  where "euclidean = topology open"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   639
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   640
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   641
  unfolding euclidean_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   642
  apply (rule cong[where x=S and y=S])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   643
  apply (rule topology_inverse[symmetric])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   644
  apply (auto simp add: istopology_def)
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   645
  done
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   646
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   647
lemma topspace_euclidean: "topspace euclidean = UNIV"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   648
  apply (simp add: topspace_def)
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   649
  apply (rule set_eqI)
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   650
  apply (auto simp add: open_openin[symmetric])
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   651
  done
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   652
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   653
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   654
  by (simp add: topspace_euclidean topspace_subtopology)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   655
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   656
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   657
  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   658
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   659
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   660
  by (simp add: open_openin openin_subopen[symmetric])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   661
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   662
text {* Basic "localization" results are handy for connectedness. *}
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   663
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   664
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   665
  by (auto simp add: openin_subtopology open_openin[symmetric])
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   666
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   667
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   668
  by (auto simp add: openin_open)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   669
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   670
lemma open_openin_trans[trans]:
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   671
  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   672
  by (metis Int_absorb1  openin_open_Int)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   673
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   674
lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   675
  by (auto simp add: openin_open)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   676
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   677
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   678
  by (simp add: closedin_subtopology closed_closedin Int_ac)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   679
53291
f7fa953bd15b tuned proofs;
wenzelm
parents: 53282
diff changeset
   680
lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   681
  by (metis closedin_closed)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   682
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   683
lemma closed_closedin_trans:
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   684
  "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55522
diff changeset
   685
  by (metis closedin_closed inf.absorb2)
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   686
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   687
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   688
  by (auto simp add: closedin_closed)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   689
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   690
lemma openin_euclidean_subtopology_iff:
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   691
  fixes S U :: "'a::metric_space set"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   692
  shows "openin (subtopology euclidean U) S \<longleftrightarrow>
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   693
    S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   694
  (is "?lhs \<longleftrightarrow> ?rhs")
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   695
proof
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   696
  assume ?lhs
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   697
  then show ?rhs
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   698
    unfolding openin_open open_dist by blast
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   699
next
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   700
  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}"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   701
  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   702
    unfolding T_def
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   703
    apply clarsimp
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   704
    apply (rule_tac x="d - dist x a" in exI)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   705
    apply (clarsimp simp add: less_diff_eq)
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55522
diff changeset
   706
    by (metis dist_commute dist_triangle_lt)
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   707
  assume ?rhs then have 2: "S = U \<inter> T"
55775
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55522
diff changeset
   708
    unfolding T_def 
1557a391a858 A bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 55522
diff changeset
   709
    by auto (metis dist_self)
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   710
  from 1 2 show ?lhs
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   711
    unfolding openin_open open_dist by fast
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   712
qed
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   713
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   714
text {* These "transitivity" results are handy too *}
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   715
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   716
lemma openin_trans[trans]:
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   717
  "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   718
    openin (subtopology euclidean U) S"
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   719
  unfolding open_openin openin_open by blast
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   720
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   721
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   722
  by (auto simp add: openin_open intro: openin_trans)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   723
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   724
lemma closedin_trans[trans]:
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   725
  "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   726
    closedin (subtopology euclidean U) S"
44210
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   727
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   728
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   729
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   730
  by (auto simp add: closedin_closed intro: closedin_trans)
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   731
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   732
eba74571833b Topology_Euclidean_Space.thy: organize section headings
huffman
parents: 44207
diff changeset
   733
subsection {* Open and closed balls *}
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   734
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   735
definition ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   736
  where "ball x e = {y. dist x y < e}"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   737
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   738
definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   739
  where "cball x e = {y. dist x y \<le> e}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   740
45776
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   741
lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   742
  by (simp add: ball_def)
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   743
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   744
lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   745
  by (simp add: cball_def)
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   746
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   747
lemma mem_ball_0:
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   748
  fixes x :: "'a::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   749
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   750
  by (simp add: dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   751
45776
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   752
lemma mem_cball_0:
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   753
  fixes x :: "'a::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   754
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   755
  by (simp add: dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   756
45776
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   757
lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   758
  by simp
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   759
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   760
lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   761
  by simp
714100f5fda4 remove mem_(c)ball_0 and centre_in_(c)ball from simpset, as rules mem_(c)ball always match instead
huffman
parents: 45548
diff changeset
   762
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   763
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   764
  by (simp add: subset_eq)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   765
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   766
lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   767
  by (simp add: subset_eq)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   768
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   769
lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   770
  by (simp add: subset_eq)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   771
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   772
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   773
  by (simp add: set_eq_iff) arith
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   774
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   775
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   776
  by (simp add: set_eq_iff)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   777
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   778
lemma diff_less_iff:
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   779
  "(a::real) - b > 0 \<longleftrightarrow> a > b"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   780
  "(a::real) - b < 0 \<longleftrightarrow> a < b"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   781
  "a - b < c \<longleftrightarrow> a < c + b" "a - b > c \<longleftrightarrow> a > c + b"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   782
  by arith+
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   783
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   784
lemma diff_le_iff:
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   785
  "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   786
  "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   787
  "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   788
  "a - b \<ge> c \<longleftrightarrow> a \<ge> c + b"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   789
  by arith+
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   790
54070
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   791
lemma open_ball [intro, simp]: "open (ball x e)"
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   792
proof -
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   793
  have "open (dist x -` {..<e})"
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   794
    by (intro open_vimage open_lessThan continuous_on_intros)
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   795
  also have "dist x -` {..<e} = ball x e"
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   796
    by auto
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   797
  finally show ?thesis .
1a13325269c2 new topological lemmas; tuned proofs
huffman
parents: 53862
diff changeset
   798
qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   799
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   800
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   801
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   802
33714
eb2574ac4173 Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents: 33324
diff changeset
   803
lemma openE[elim?]:
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   804
  assumes "open S" "x\<in>S"
33714
eb2574ac4173 Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents: 33324
diff changeset
   805
  obtains e where "e>0" "ball x e \<subseteq> S"
eb2574ac4173 Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents: 33324
diff changeset
   806
  using assms unfolding open_contains_ball by auto
eb2574ac4173 Added new lemmas to Euclidean Space by Robert Himmelmann
hoelzl
parents: 33324
diff changeset
   807
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   808
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   809
  by (metis open_contains_ball subset_eq centre_in_ball)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   810
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   811
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   812
  unfolding mem_ball set_eq_iff
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   813
  apply (simp add: not_less)
52624
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   814
  apply (metis zero_le_dist order_trans dist_self)
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   815
  done
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   816
53291
f7fa953bd15b tuned proofs;
wenzelm
parents: 53282
diff changeset
   817
lemma ball_empty[intro]: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   818
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   819
lemma euclidean_dist_l2:
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   820
  fixes x y :: "'a :: euclidean_space"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   821
  shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   822
  unfolding dist_norm norm_eq_sqrt_inner setL2_def
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   823
  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   824
56189
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   825
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   826
subsection {* Boxes *}
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   827
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   828
definition (in euclidean_space) eucl_less (infix "<e" 50)
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   829
  where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   830
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   831
definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   832
definition "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
54775
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   833
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   834
lemma box_def: "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
2d3df8633dad prefer box over greaterThanLessThan on euclidean_space
immler
parents: 54489
diff changeset
   835
  and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   836
  and mem_box: "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   837
    "x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   838
  by (auto simp: box_eucl_less eucl_less_def cbox_def)
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   839
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   840
lemma mem_box_real[simp]:
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   841
  "(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b"
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   842
  "(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b"
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   843
  by (auto simp: mem_box)
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   844
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   845
lemma box_real[simp]:
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   846
  fixes a b:: real
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   847
  shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
0268784f60da use cbox to relax class constraints
immler
parents: 56166
diff changeset
   848
  by auto
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   849
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   850
lemma rational_boxes:
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   851
  fixes x :: "'a\<Colon>euclidean_space"
53291
f7fa953bd15b tuned proofs;
wenzelm
parents: 53282
diff changeset
   852
  assumes "e > 0"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   853
  shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   854
proof -
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   855
  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
53291
f7fa953bd15b tuned proofs;
wenzelm
parents: 53282
diff changeset
   856
  then have e: "e' > 0"
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   857
    using assms by (auto intro!: divide_pos_pos simp: DIM_positive)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   858
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   859
  proof
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   860
    fix i
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   861
    from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   862
    show "?th i" by auto
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   863
  qed
55522
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   864
  from choice[OF this] obtain a where
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   865
    a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'" ..
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   866
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   867
  proof
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   868
    fix i
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   869
    from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   870
    show "?th i" by auto
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   871
  qed
55522
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   872
  from choice[OF this] obtain b where
23d2cbac6dce tuned proofs;
wenzelm
parents: 55415
diff changeset
   873
    b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'" ..
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   874
  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   875
  show ?thesis
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   876
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   877
    fix y :: 'a
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   878
    assume *: "y \<in> box ?a ?b"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52625
diff changeset
   879
    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   880
      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   881
    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   882
    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   883
      fix i :: "'a"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   884
      assume i: "i \<in> Basis"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   885
      have "a i < y\<bullet>i \<and> y\<bullet>i < b i"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   886
        using * i by (auto simp: box_def)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   887
      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   888
        using a by auto
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   889
      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   890
        using b by auto
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   891
      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   892
        by auto
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   893
      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   894
        unfolding e'_def by (auto simp: dist_real_def)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52625
diff changeset
   895
      then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   896
        by (rule power_strict_mono) auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52625
diff changeset
   897
      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   898
        by (simp add: power_divide)
635d73673b5e regularity of measures, therefore:
immler
parents: 49962
diff changeset
   899
    qed auto
53255
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   900
    also have "\<dots> = e"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   901
      using `0 < e` by (simp add: real_eq_of_nat)
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   902
    finally show "y \<in> ball x e"
addd7b9b2bff tuned proofs;
wenzelm
parents: 53015
diff changeset
   903
      by (auto simp: ball_def)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   904
  qed (insert a b, auto simp: box_def)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   905
qed
51103
5dd7b89a16de generalized
immler
parents: 51102
diff changeset
   906
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   907
lemma open_UNION_box:
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   908
  fixes M :: "'a\<Colon>euclidean_space set"
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   909
  assumes "open M"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   910
  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   911
  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52625
diff changeset
   912
  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50324
diff changeset
   913
  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
52624
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   914
proof -
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   915
  {
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   916
    fix x assume "x \<in> M"
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   917
    obtain e where e: "e > 0" "ball x e \<subseteq> M"
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   918
      using openE[OF `open M` `x \<in> M`] by auto
53282
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   919
    moreover obtain a b where ab:
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   920
      "x \<in> box a b"
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   921
      "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   922
      "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"
9d6e263fa921 tuned proofs;
wenzelm
parents: 53255
diff changeset
   923
      "box a b \<subseteq> ball x e"
52624
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   924
      using rational_boxes[OF e(1)] by metis
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   925
    ultimately have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   926
       by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   927
          (auto simp: euclidean_representation I_def a'_def b'_def)
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   928
  }
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   929
  then show ?thesis by (auto simp: I_def)
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   930
qed
8a7b59a81088 tuned proofs;
wenzelm
parents: 52141
diff changeset
   931
56189
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   932
lemma box_eq_empty:
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   933
  fixes a :: "'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   934
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   935
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   936
proof -
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   937
  {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   938
    fix i x
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   939
    assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>box a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   940
    then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   941
      unfolding mem_box by (auto simp: box_def)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   942
    then have "a\<bullet>i < b\<bullet>i" by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   943
    then have False using as by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   944
  }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   945
  moreover
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   946
  {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   947
    assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   948
    let ?x = "(1/2) *\<^sub>R (a + b)"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   949
    {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   950
      fix i :: 'a
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   951
      assume i: "i \<in> Basis"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   952
      have "a\<bullet>i < b\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   953
        using as[THEN bspec[where x=i]] i by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   954
      then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   955
        by (auto simp: inner_add_left)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   956
    }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   957
    then have "box a b \<noteq> {}"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   958
      using mem_box(1)[of "?x" a b] by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   959
  }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   960
  ultimately show ?th1 by blast
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   961
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   962
  {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   963
    fix i x
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   964
    assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>cbox a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   965
    then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   966
      unfolding mem_box by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   967
    then have "a\<bullet>i \<le> b\<bullet>i" by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   968
    then have False using as by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   969
  }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   970
  moreover
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   971
  {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   972
    assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   973
    let ?x = "(1/2) *\<^sub>R (a + b)"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   974
    {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   975
      fix i :: 'a
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   976
      assume i:"i \<in> Basis"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   977
      have "a\<bullet>i \<le> b\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   978
        using as[THEN bspec[where x=i]] i by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   979
      then have "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   980
        by (auto simp: inner_add_left)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   981
    }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   982
    then have "cbox a b \<noteq> {}"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   983
      using mem_box(2)[of "?x" a b] by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   984
  }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   985
  ultimately show ?th2 by blast
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   986
qed
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   987
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   988
lemma box_ne_empty:
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   989
  fixes a :: "'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   990
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   991
  and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   992
  unfolding box_eq_empty[of a b] by fastforce+
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   993
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   994
lemma
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   995
  fixes a :: "'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   996
  shows cbox_sing: "cbox a a = {a}"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   997
    and box_sing: "box a a = {}"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   998
  unfolding set_eq_iff mem_box eq_iff [symmetric]
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
   999
  by (auto intro!: euclidean_eqI[where 'a='a])
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1000
     (metis all_not_in_conv nonempty_Basis)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1001
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1002
lemma subset_box_imp:
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1003
  fixes a :: "'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1004
  shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1005
    and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> box a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1006
    and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> cbox a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1007
     and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> box a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1008
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1009
   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1010
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1011
lemma box_subset_cbox:
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1012
  fixes a :: "'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1013
  shows "box a b \<subseteq> cbox a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1014
  unfolding subset_eq [unfolded Ball_def] mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1015
  by (fast intro: less_imp_le)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1016
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1017
lemma subset_box:
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1018
  fixes a :: "'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1019
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1020
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1021
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1022
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1023
proof -
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1024
  show ?th1
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1025
    unfolding subset_eq and Ball_def and mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1026
    by (auto intro: order_trans)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1027
  show ?th2
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1028
    unfolding subset_eq and Ball_def and mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1029
    by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1030
  {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1031
    assume as: "box c d \<subseteq> cbox a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1032
    then have "box c d \<noteq> {}"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1033
      unfolding box_eq_empty by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1034
    fix i :: 'a
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1035
    assume i: "i \<in> Basis"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1036
    (** TODO combine the following two parts as done in the HOL_light version. **)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1037
    {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1038
      let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1039
      assume as2: "a\<bullet>i > c\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1040
      {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1041
        fix j :: 'a
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1042
        assume j: "j \<in> Basis"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1043
        then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1044
          apply (cases "j = i")
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1045
          using as(2)[THEN bspec[where x=j]] i
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1046
          apply (auto simp add: as2)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1047
          done
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1048
      }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1049
      then have "?x\<in>box c d"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1050
        using i unfolding mem_box by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1051
      moreover
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1052
      have "?x \<notin> cbox a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1053
        unfolding mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1054
        apply auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1055
        apply (rule_tac x=i in bexI)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1056
        using as(2)[THEN bspec[where x=i]] and as2 i
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1057
        apply auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1058
        done
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1059
      ultimately have False using as by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1060
    }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1061
    then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1062
    moreover
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1063
    {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1064
      let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1065
      assume as2: "b\<bullet>i < d\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1066
      {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1067
        fix j :: 'a
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1068
        assume "j\<in>Basis"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1069
        then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1070
          apply (cases "j = i")
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1071
          using as(2)[THEN bspec[where x=j]]
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1072
          apply (auto simp add: as2)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1073
          done
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1074
      }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1075
      then have "?x\<in>box c d"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1076
        unfolding mem_box by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1077
      moreover
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1078
      have "?x\<notin>cbox a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1079
        unfolding mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1080
        apply auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1081
        apply (rule_tac x=i in bexI)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1082
        using as(2)[THEN bspec[where x=i]] and as2 using i
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1083
        apply auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1084
        done
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1085
      ultimately have False using as by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1086
    }
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1087
    then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1088
    ultimately
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1089
    have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1090
  } note part1 = this
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1091
  show ?th3
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1092
    unfolding subset_eq and Ball_def and mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1093
    apply (rule, rule, rule, rule)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1094
    apply (rule part1)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1095
    unfolding subset_eq and Ball_def and mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1096
    prefer 4
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1097
    apply auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1098
    apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1099
    done
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1100
  {
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1101
    assume as: "box c d \<subseteq> box a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1102
    fix i :: 'a
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1103
    assume i:"i\<in>Basis"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1104
    from as(1) have "box c d \<subseteq> cbox a b"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1105
      using box_subset_cbox[of a b] by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1106
    then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1107
      using part1 and as(2) using i by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1108
  } note * = this
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1109
  show ?th4
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1110
    unfolding subset_eq and Ball_def and mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1111
    apply (rule, rule, rule, rule)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1112
    apply (rule *)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1113
    unfolding subset_eq and Ball_def and mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1114
    prefer 4
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1115
    apply auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1116
    apply (erule_tac x=xa in allE, simp)+
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1117
    done
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1118
qed
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1119
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1120
lemma inter_interval:
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1121
  fixes a :: "'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1122
  shows "cbox a b \<inter> cbox c d =
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1123
    cbox (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1124
  unfolding set_eq_iff and Int_iff and mem_box
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1125
  by auto
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1126
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1127
lemma disjoint_interval:
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1128
  fixes a::"'a::euclidean_space"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1129
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1130
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1131
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1132
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1133
proof -
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset
  1134
  let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
c4daa97ac57a removed dependencies on theory Ordered_Euclidean_Space
immler
parents: 56188
diff changeset <