src/HOL/Analysis/Elementary_Normed_Spaces.thy
author paulson <lp15@cam.ac.uk>
Thu, 12 Sep 2019 15:32:39 +0100
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permissions -rw-r--r--
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(*  Author:     L C Paulson, University of Cambridge
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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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section \<open>Elementary Normed Vector Spaces\<close>
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theory Elementary_Normed_Spaces
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  imports
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  "HOL-Library.FuncSet"
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  Elementary_Metric_Spaces Linear_Algebra
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  Connected
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begin
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subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas Combining Imports\<close>
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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 (Pi\<^sub>E I F)"
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  by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
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lemma open_sums:
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  fixes T :: "('b::real_normed_vector) set"
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  assumes "open S \<or> open T"
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  shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
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  using assms
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proof
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  assume S: "open S"
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  show ?thesis
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  proof (clarsimp simp: open_dist)
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    fix x y
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    assume "x \<in> S" "y \<in> T"
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    with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
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      by (auto simp: open_dist)
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    then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
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      by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
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    then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
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      using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
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  qed
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next
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  assume T: "open T"
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  show ?thesis
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  proof (clarsimp simp: open_dist)
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    fix x y
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    assume "x \<in> S" "y \<in> T"
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    with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
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      by (auto simp: open_dist)
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    then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
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      by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
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    then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
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      using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
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  qed
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qed
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subsection \<open>Support\<close>
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definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
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  where "support_on s f = {x\<in>s. f x \<noteq> 0}"
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lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
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  by (simp add: support_on_def)
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lemma support_on_simps[simp]:
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  "support_on {} f = {}"
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  "support_on (insert x s) f =
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    (if f x = 0 then support_on s f else insert x (support_on s f))"
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  "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
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  "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
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  "support_on (s - t) f = support_on s f - support_on t f"
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  "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
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  unfolding support_on_def by auto
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lemma support_on_cong:
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  "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
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  by (auto simp: support_on_def)
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lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
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  by (auto simp: support_on_def)
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lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
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  by (auto simp: support_on_def)
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lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)"
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  unfolding support_on_def by auto
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(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
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definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
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  where "supp_sum f S = (\<Sum>x\<in>support_on S f. f x)"
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lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
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  unfolding supp_sum_def by auto
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lemma supp_sum_insert[simp]:
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  "finite (support_on S f) \<Longrightarrow>
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    supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
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  by (simp add: supp_sum_def in_support_on insert_absorb)
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lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
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  by (cases "r = 0")
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     (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
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subsection \<open>Intervals\<close>
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lemma image_affinity_interval:
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  fixes c :: "'a::ordered_real_vector"
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  shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) = 
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           (if {a..b}={} then {}
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            else if 0 \<le> m then {m *\<^sub>R a + c .. m  *\<^sub>R b + c}
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            else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
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         (is "?lhs = ?rhs")
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proof (cases "m=0")
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  case True
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  then show ?thesis
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    by force
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next
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  case False
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  show ?thesis
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  proof
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    show "?lhs \<subseteq> ?rhs"
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      by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
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    show "?rhs \<subseteq> ?lhs"
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    proof (clarsimp, intro conjI impI subsetI)
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      show "\<lbrakk>0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk>
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   127
            \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   128
        apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
70630
2402aa499ffe more rules for ordered real vector spaces
haftmann
parents: 70532
diff changeset
   129
        using False apply (auto simp: pos_le_divideR_eq pos_divideR_le_eq le_diff_eq diff_le_eq)
2402aa499ffe more rules for ordered real vector spaces
haftmann
parents: 70532
diff changeset
   130
        done
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   131
      show "\<lbrakk>\<not> 0 \<le> m; a \<le> b;  x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   132
            \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   133
        apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
70630
2402aa499ffe more rules for ordered real vector spaces
haftmann
parents: 70532
diff changeset
   134
         apply (auto simp add: neg_le_divideR_eq neg_divideR_le_eq not_le le_diff_eq diff_le_eq)
70346
408e15cbd2a6 tuned proofs
haftmann
parents: 70136
diff changeset
   135
        done
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   136
    qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   137
  qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   138
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   139
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   140
subsection \<open>Limit Points\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   141
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   142
lemma islimpt_ball:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   143
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   144
  shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   145
  (is "?lhs \<longleftrightarrow> ?rhs")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   146
proof
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   147
  show ?rhs if ?lhs
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   148
  proof
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   149
    {
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   150
      assume "e \<le> 0"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   151
      then have *: "ball x e = {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   152
        using ball_eq_empty[of x e] by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   153
      have False using \<open>?lhs\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   154
        unfolding * using islimpt_EMPTY[of y] by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   155
    }
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   156
    then show "e > 0" by (metis not_less)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   157
    show "y \<in> cball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   158
      using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   159
        ball_subset_cball[of x e] \<open>?lhs\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   160
      unfolding closed_limpt by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   161
  qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   162
  show ?lhs if ?rhs
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   163
  proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   164
    from that have "e > 0" by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   165
    {
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   166
      fix d :: real
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   167
      assume "d > 0"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   168
      have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   169
      proof (cases "d \<le> dist x y")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   170
        case True
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   171
        then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   172
        proof (cases "x = y")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   173
          case True
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   174
          then have False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   175
            using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   176
          then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   177
            by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   178
        next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   179
          case False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   180
          have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   181
            norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   182
            unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   183
            by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   184
          also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   185
            using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   186
            unfolding scaleR_minus_left scaleR_one
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   187
            by (auto simp: norm_minus_commute)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   188
          also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   189
            unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   190
            unfolding distrib_right using \<open>x\<noteq>y\<close>  by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   191
          also have "\<dots> \<le> e - d/2" using \<open>d \<le> dist x y\<close> and \<open>d>0\<close> and \<open>?rhs\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   192
            by (auto simp: dist_norm)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   193
          finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using \<open>d>0\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   194
            by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   195
          moreover
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   196
          have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   197
            using \<open>x\<noteq>y\<close>[unfolded dist_nz] \<open>d>0\<close> unfolding scaleR_eq_0_iff
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   198
            by (auto simp: dist_commute)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   199
          moreover
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   200
          have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   201
            unfolding dist_norm
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   202
            apply simp
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   203
            unfolding norm_minus_cancel
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   204
            using \<open>d > 0\<close> \<open>x\<noteq>y\<close>[unfolded dist_nz] dist_commute[of x y]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   205
            unfolding dist_norm
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   206
            apply auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   207
            done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   208
          ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   209
            apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   210
            apply auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   211
            done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   212
        qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   213
      next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   214
        case False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   215
        then have "d > dist x y" by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   216
        show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   217
        proof (cases "x = y")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   218
          case True
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   219
          obtain z where **: "z \<noteq> y" "dist z y < min e d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   220
            using perfect_choose_dist[of "min e d" y]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   221
            using \<open>d > 0\<close> \<open>e>0\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   222
          show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   223
            unfolding \<open>x = y\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   224
            using \<open>z \<noteq> y\<close> **
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   225
            apply (rule_tac x=z in bexI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   226
            apply (auto simp: dist_commute)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   227
            done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   228
        next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   229
          case False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   230
          then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   231
            using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   232
            apply (rule_tac x=x in bexI, auto)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   233
            done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   234
        qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   235
      qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   236
    }
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   237
    then show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   238
      unfolding mem_cball islimpt_approachable mem_ball by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   239
  qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   240
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   241
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   242
lemma closure_ball_lemma:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   243
  fixes x y :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   244
  assumes "x \<noteq> y"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   245
  shows "y islimpt ball x (dist x y)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   246
proof (rule islimptI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   247
  fix T
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   248
  assume "y \<in> T" "open T"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   249
  then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   250
    unfolding open_dist by fast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   251
  (* choose point between x and y, within distance r of y. *)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   252
  define k where "k = min 1 (r / (2 * dist x y))"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   253
  define z where "z = y + scaleR k (x - y)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   254
  have z_def2: "z = x + scaleR (1 - k) (y - x)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   255
    unfolding z_def by (simp add: algebra_simps)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   256
  have "dist z y < r"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   257
    unfolding z_def k_def using \<open>0 < r\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   258
    by (simp add: dist_norm min_def)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   259
  then have "z \<in> T"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   260
    using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   261
  have "dist x z < dist x y"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   262
    unfolding z_def2 dist_norm
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   263
    apply (simp add: norm_minus_commute)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   264
    apply (simp only: dist_norm [symmetric])
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   265
    apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   266
    apply (rule mult_strict_right_mono)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   267
    apply (simp add: k_def \<open>0 < r\<close> \<open>x \<noteq> y\<close>)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   268
    apply (simp add: \<open>x \<noteq> y\<close>)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   269
    done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   270
  then have "z \<in> ball x (dist x y)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   271
    by simp
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   272
  have "z \<noteq> y"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   273
    unfolding z_def k_def using \<open>x \<noteq> y\<close> \<open>0 < r\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   274
    by (simp add: min_def)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   275
  show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   276
    using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   277
    by fast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   278
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   279
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   280
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   281
subsection \<open>Balls and Spheres in Normed Spaces\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   282
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   283
lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   284
  for x :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   285
  by (simp add: dist_norm)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   286
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   287
lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   288
  for x :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   289
  by (simp add: dist_norm)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   290
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   291
lemma closure_ball [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   292
  fixes x :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   293
  shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   294
  apply (rule equalityI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   295
  apply (rule closure_minimal)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   296
  apply (rule ball_subset_cball)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   297
  apply (rule closed_cball)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   298
  apply (rule subsetI, rename_tac y)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   299
  apply (simp add: le_less [where 'a=real])
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   300
  apply (erule disjE)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   301
  apply (rule subsetD [OF closure_subset], simp)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   302
  apply (simp add: closure_def, clarify)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   303
  apply (rule closure_ball_lemma)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   304
  apply simp
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   305
  done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   306
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   307
lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   308
  for x :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   309
  by (simp add: dist_norm)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   310
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   311
(* In a trivial vector space, this fails for e = 0. *)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   312
lemma interior_cball [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   313
  fixes x :: "'a::{real_normed_vector, perfect_space}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   314
  shows "interior (cball x e) = ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   315
proof (cases "e \<ge> 0")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   316
  case False note cs = this
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   317
  from cs have null: "ball x e = {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   318
    using ball_empty[of e x] by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   319
  moreover
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   320
  have "cball x e = {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   321
  proof (rule equals0I)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   322
    fix y
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   323
    assume "y \<in> cball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   324
    then show False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   325
      by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   326
          subset_antisym subset_ball)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   327
  qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   328
  then have "interior (cball x e) = {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   329
    using interior_empty by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   330
  ultimately show ?thesis by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   331
next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   332
  case True note cs = this
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   333
  have "ball x e \<subseteq> cball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   334
    using ball_subset_cball by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   335
  moreover
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   336
  {
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   337
    fix S y
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   338
    assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   339
    then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   340
      unfolding open_dist by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   341
    then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   342
      using perfect_choose_dist [of d] by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   343
    have "xa \<in> S"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   344
      using d[THEN spec[where x = xa]]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   345
      using xa by (auto simp: dist_commute)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   346
    then have xa_cball: "xa \<in> cball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   347
      using as(1) by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   348
    then have "y \<in> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   349
    proof (cases "x = y")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   350
      case True
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   351
      then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   352
      then show "y \<in> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   353
        using \<open>x = y \<close> by simp
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   354
    next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   355
      case False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   356
      have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   357
        unfolding dist_norm
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   358
        using \<open>d>0\<close> norm_ge_zero[of "y - x"] \<open>x \<noteq> y\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   359
      then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   360
        using d as(1)[unfolded subset_eq] by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   361
      have "y - x \<noteq> 0" using \<open>x \<noteq> y\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   362
      hence **:"d / (2 * norm (y - x)) > 0"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   363
        unfolding zero_less_norm_iff[symmetric] using \<open>d>0\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   364
      have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   365
        norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   366
        by (auto simp: dist_norm algebra_simps)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   367
      also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   368
        by (auto simp: algebra_simps)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   369
      also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   370
        using ** by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   371
      also have "\<dots> = (dist y x) + d/2"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   372
        using ** by (auto simp: distrib_right dist_norm)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   373
      finally have "e \<ge> dist x y +d/2"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   374
        using *[unfolded mem_cball] by (auto simp: dist_commute)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   375
      then show "y \<in> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   376
        unfolding mem_ball using \<open>d>0\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   377
    qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   378
  }
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   379
  then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   380
    by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   381
  ultimately show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   382
    using interior_unique[of "ball x e" "cball x e"]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   383
    using open_ball[of x e]
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   384
    by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   385
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   386
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   387
lemma frontier_ball [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   388
  fixes a :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   389
  shows "0 < e \<Longrightarrow> frontier (ball a e) = sphere a e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   390
  by (force simp: frontier_def)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   391
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   392
lemma frontier_cball [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   393
  fixes a :: "'a::{real_normed_vector, perfect_space}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   394
  shows "frontier (cball a e) = sphere a e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   395
  by (force simp: frontier_def)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   396
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   397
corollary compact_sphere [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   398
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   399
  shows "compact (sphere a r)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   400
using compact_frontier [of "cball a r"] by simp
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   401
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   402
corollary bounded_sphere [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   403
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   404
  shows "bounded (sphere a r)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   405
by (simp add: compact_imp_bounded)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   406
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   407
corollary closed_sphere  [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   408
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   409
  shows "closed (sphere a r)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   410
by (simp add: compact_imp_closed)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   411
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   412
lemma image_add_ball [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   413
  fixes a :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   414
  shows "(+) b ` ball a r = ball (a+b) r"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   415
apply (intro equalityI subsetI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   416
apply (force simp: dist_norm)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   417
apply (rule_tac x="x-b" in image_eqI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   418
apply (auto simp: dist_norm algebra_simps)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   419
done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   420
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   421
lemma image_add_cball [simp]:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   422
  fixes a :: "'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   423
  shows "(+) b ` cball a r = cball (a+b) r"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   424
apply (intro equalityI subsetI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   425
apply (force simp: dist_norm)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   426
apply (rule_tac x="x-b" in image_eqI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   427
apply (auto simp: dist_norm algebra_simps)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   428
done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   429
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   430
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 70065
diff changeset
   431
subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas on Normed Algebras\<close>
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   432
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   433
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   434
lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   435
  by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   436
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   437
lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   438
  by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   439
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   440
lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   441
  unfolding Nats_def by (rule closed_of_nat_image)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   442
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   443
lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   444
  unfolding Ints_def by (rule closed_of_int_image)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   445
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   446
lemma closed_subset_Ints:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   447
  fixes A :: "'a :: real_normed_algebra_1 set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   448
  assumes "A \<subseteq> \<int>"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   449
  shows   "closed A"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   450
proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   451
  case (1 x y)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   452
  with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   453
  with \<open>dist y x < 1\<close> show "y = x"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   454
    by (auto elim!: Ints_cases simp: dist_of_int)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   455
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   456
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   457
subsection \<open>Filters\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   458
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   459
definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"  (infixr "indirection" 70)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   460
  where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   461
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   462
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   463
subsection \<open>Trivial Limits\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   464
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   465
lemma trivial_limit_at_infinity:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   466
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   467
  unfolding trivial_limit_def eventually_at_infinity
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   468
  apply clarsimp
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   469
  apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   470
   apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   471
  apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   472
  apply (drule_tac x=UNIV in spec, simp)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   473
  done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   474
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   475
lemma at_within_ball_bot_iff:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   476
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   477
  shows "at x within ball y r = bot \<longleftrightarrow> (r=0 \<or> x \<notin> cball y r)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   478
  unfolding trivial_limit_within 
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   479
  apply (auto simp add:trivial_limit_within islimpt_ball )
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   480
  by (metis le_less_trans less_eq_real_def zero_le_dist)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   481
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   482
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   483
subsection \<open>Limits\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   484
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   485
proposition Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   486
  by (auto simp: tendsto_iff eventually_at_infinity)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   487
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   488
corollary Lim_at_infinityI [intro?]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   489
  assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   490
  shows "(f \<longlongrightarrow> l) at_infinity"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   491
  apply (simp add: Lim_at_infinity, clarify)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   492
  apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   493
  done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   494
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   495
lemma Lim_transform_within_set_eq:
70532
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70380
diff changeset
   496
  fixes a :: "'a::metric_space" and l :: "'b::metric_space"
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70380
diff changeset
   497
  shows "eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)
fcf3b891ccb1 new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents: 70380
diff changeset
   498
         \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within T))"
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   499
  by (force intro: Lim_transform_within_set elim: eventually_mono)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   500
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   501
lemma Lim_null:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   502
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   503
  shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   504
  by (simp add: Lim dist_norm)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   505
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   506
lemma Lim_null_comparison:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   507
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   508
  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   509
  shows "(f \<longlongrightarrow> 0) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   510
  using assms(2)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   511
proof (rule metric_tendsto_imp_tendsto)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   512
  show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   513
    using assms(1) by (rule eventually_mono) (simp add: dist_norm)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   514
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   515
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   516
lemma Lim_transform_bound:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   517
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   518
    and g :: "'a \<Rightarrow> 'c::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   519
  assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   520
    and "(g \<longlongrightarrow> 0) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   521
  shows "(f \<longlongrightarrow> 0) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   522
  using assms(1) tendsto_norm_zero [OF assms(2)]
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   523
  by (rule Lim_null_comparison)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   524
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   525
lemma lim_null_mult_right_bounded:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   526
  fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   527
  assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   528
    shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   529
proof -
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   530
  have *: "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   531
    by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   532
  have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   533
    apply (rule Lim_null_comparison [OF _ *])
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   534
    apply (simp add: eventually_mono [OF g] mult_left_mono)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   535
    done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   536
  then show ?thesis
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   537
    by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   538
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   539
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   540
lemma lim_null_mult_left_bounded:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   541
  fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   542
  assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   543
    shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   544
proof -
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   545
  have *: "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   546
    by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   547
  have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   548
    apply (rule Lim_null_comparison [OF _ *])
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   549
    apply (simp add: eventually_mono [OF g] mult_right_mono)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   550
    done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   551
  then show ?thesis
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   552
    by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   553
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   554
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   555
lemma lim_null_scaleR_bounded:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   556
  assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   557
    shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   558
proof
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   559
  fix \<epsilon>::real
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   560
  assume "0 < \<epsilon>"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   561
  then have B: "0 < \<epsilon> / (abs B + 1)" by simp
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   562
  have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   563
  proof -
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   564
    have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   565
      by (simp add: mult_left_mono g)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   566
    also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   567
      by (simp add: mult_left_mono)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   568
    also have "\<dots> < \<epsilon>"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   569
      by (rule f)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   570
    finally show ?thesis .
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   571
  qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   572
  show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   573
    apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   574
    apply (auto simp: \<open>0 < \<epsilon>\<close> divide_simps * split: if_split_asm)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   575
    done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   576
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   577
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   578
lemma Lim_norm_ubound:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   579
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   580
  assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   581
  shows "norm(l) \<le> e"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   582
  using assms by (fast intro: tendsto_le tendsto_intros)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   583
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   584
lemma Lim_norm_lbound:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   585
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   586
  assumes "\<not> trivial_limit net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   587
    and "(f \<longlongrightarrow> l) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   588
    and "eventually (\<lambda>x. e \<le> norm (f x)) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   589
  shows "e \<le> norm l"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   590
  using assms by (fast intro: tendsto_le tendsto_intros)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   591
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   592
text\<open>Limit under bilinear function\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   593
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   594
lemma Lim_bilinear:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   595
  assumes "(f \<longlongrightarrow> l) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   596
    and "(g \<longlongrightarrow> m) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   597
    and "bounded_bilinear h"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   598
  shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   599
  using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   600
  by (rule bounded_bilinear.tendsto)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   601
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   602
lemma Lim_at_zero:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   603
  fixes a :: "'a::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   604
    and l :: "'b::topological_space"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   605
  shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   606
  using LIM_offset_zero LIM_offset_zero_cancel ..
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   607
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   608
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 70065
diff changeset
   609
subsection\<^marker>\<open>tag unimportant\<close> \<open>Limit Point of Filter\<close>
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   610
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   611
lemma netlimit_at_vector:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   612
  fixes a :: "'a::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   613
  shows "netlimit (at a) = a"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   614
proof (cases "\<exists>x. x \<noteq> a")
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   615
  case True then obtain x where x: "x \<noteq> a" ..
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   616
  have "\<not> trivial_limit (at a)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   617
    unfolding trivial_limit_def eventually_at dist_norm
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   618
    apply clarsimp
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   619
    apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   620
    apply (simp add: norm_sgn sgn_zero_iff x)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   621
    done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   622
  then show ?thesis
70065
cc89a395b5a3 Free_Abelian_Groups finally working; fixed some duplicates; cleaned up some proofs
paulson <lp15@cam.ac.uk>
parents: 69922
diff changeset
   623
    by (rule Lim_ident_at [of a UNIV])
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   624
qed simp
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   625
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   626
subsection \<open>Boundedness\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   627
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   628
lemma continuous_on_closure_norm_le:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   629
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   630
  assumes "continuous_on (closure s) f"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   631
    and "\<forall>y \<in> s. norm(f y) \<le> b"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   632
    and "x \<in> (closure s)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   633
  shows "norm (f x) \<le> b"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   634
proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   635
  have *: "f ` s \<subseteq> cball 0 b"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   636
    using assms(2)[unfolded mem_cball_0[symmetric]] by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   637
  show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   638
    by (meson "*" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   639
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   640
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   641
lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
70380
2b0dca68c3ee More analysis / measure theory material
paulson <lp15@cam.ac.uk>
parents: 70346
diff changeset
   642
  unfolding bounded_iff 
2b0dca68c3ee More analysis / measure theory material
paulson <lp15@cam.ac.uk>
parents: 70346
diff changeset
   643
  by (meson less_imp_le not_le order_trans zero_less_one)
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   644
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   645
lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   646
  apply (simp add: bounded_pos)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   647
  apply (safe; rule_tac x="b+1" in exI; force)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   648
  done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   649
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   650
lemma Bseq_eq_bounded:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   651
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   652
  shows "Bseq f \<longleftrightarrow> bounded (range f)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   653
  unfolding Bseq_def bounded_pos by auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   654
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   655
lemma bounded_linear_image:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   656
  assumes "bounded S"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   657
    and "bounded_linear f"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   658
  shows "bounded (f ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   659
proof -
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   660
  from assms(1) obtain b where "b > 0" and b: "\<forall>x\<in>S. norm x \<le> b"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   661
    unfolding bounded_pos by auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   662
  from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   663
    using bounded_linear.pos_bounded by (auto simp: ac_simps)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   664
  show ?thesis
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   665
    unfolding bounded_pos
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   666
  proof (intro exI, safe)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   667
    show "norm (f x) \<le> B * b" if "x \<in> S" for x
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   668
      by (meson B b less_imp_le mult_left_mono order_trans that)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   669
  qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   670
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   671
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   672
lemma bounded_scaling:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   673
  fixes S :: "'a::real_normed_vector set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   674
  shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   675
  apply (rule bounded_linear_image, assumption)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   676
  apply (rule bounded_linear_scaleR_right)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   677
  done
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   678
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   679
lemma bounded_scaleR_comp:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   680
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   681
  assumes "bounded (f ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   682
  shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   683
  using bounded_scaling[of "f ` S" r] assms
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   684
  by (auto simp: image_image)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   685
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   686
lemma bounded_translation:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   687
  fixes S :: "'a::real_normed_vector set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   688
  assumes "bounded S"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   689
  shows "bounded ((\<lambda>x. a + x) ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   690
proof -
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   691
  from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   692
    unfolding bounded_pos by auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   693
  {
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   694
    fix x
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   695
    assume "x \<in> S"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   696
    then have "norm (a + x) \<le> b + norm a"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   697
      using norm_triangle_ineq[of a x] b by auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   698
  }
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   699
  then show ?thesis
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   700
    unfolding bounded_pos
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   701
    using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   702
    by (auto intro!: exI[of _ "b + norm a"])
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   703
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   704
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   705
lemma bounded_translation_minus:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   706
  fixes S :: "'a::real_normed_vector set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   707
  shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   708
using bounded_translation [of S "-a"] by simp
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   709
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   710
lemma bounded_uminus [simp]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   711
  fixes X :: "'a::real_normed_vector set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   712
  shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   713
by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   714
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   715
lemma uminus_bounded_comp [simp]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   716
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   717
  shows "bounded ((\<lambda>x. - f x) ` S) \<longleftrightarrow> bounded (f ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   718
  using bounded_uminus[of "f ` S"]
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   719
  by (auto simp: image_image)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   720
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   721
lemma bounded_plus_comp:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   722
  fixes f g::"'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   723
  assumes "bounded (f ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   724
  assumes "bounded (g ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   725
  shows "bounded ((\<lambda>x. f x + g x) ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   726
proof -
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   727
  {
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   728
    fix B C
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   729
    assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B" "\<And>x. x\<in>S \<Longrightarrow> norm (g x) \<le> C"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   730
    then have "\<And>x. x \<in> S \<Longrightarrow> norm (f x + g x) \<le> B + C"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   731
      by (auto intro!: norm_triangle_le add_mono)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   732
  } then show ?thesis
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   733
    using assms by (fastforce simp: bounded_iff)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   734
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   735
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   736
lemma bounded_plus:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   737
  fixes S ::"'a::real_normed_vector set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   738
  assumes "bounded S" "bounded T"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   739
  shows "bounded ((\<lambda>(x,y). x + y) ` (S \<times> T))"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   740
  using bounded_plus_comp [of fst "S \<times> T" snd] assms
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   741
  by (auto simp: split_def split: if_split_asm)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   742
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   743
lemma bounded_minus_comp:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   744
  "bounded (f ` S) \<Longrightarrow> bounded (g ` S) \<Longrightarrow> bounded ((\<lambda>x. f x - g x) ` S)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   745
  for f g::"'a \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   746
  using bounded_plus_comp[of "f" S "\<lambda>x. - g x"]
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   747
  by auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   748
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   749
lemma bounded_minus:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   750
  fixes S ::"'a::real_normed_vector set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   751
  assumes "bounded S" "bounded T"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   752
  shows "bounded ((\<lambda>(x,y). x - y) ` (S \<times> T))"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   753
  using bounded_minus_comp [of fst "S \<times> T" snd] assms
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   754
  by (auto simp: split_def split: if_split_asm)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   755
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   756
lemma not_bounded_UNIV[simp]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   757
  "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   758
proof (auto simp: bounded_pos not_le)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   759
  obtain x :: 'a where "x \<noteq> 0"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   760
    using perfect_choose_dist [OF zero_less_one] by fast
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   761
  fix b :: real
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   762
  assume b: "b >0"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   763
  have b1: "b +1 \<ge> 0"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   764
    using b by simp
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   765
  with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   766
    by (simp add: norm_sgn)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   767
  then show "\<exists>x::'a. b < norm x" ..
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   768
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   769
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   770
corollary cobounded_imp_unbounded:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   771
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   772
    shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   773
  using bounded_Un [of S "-S"]  by (simp add: sup_compl_top)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   774
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 70065
diff changeset
   775
subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations among convergence and absolute convergence for power series\<close>
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   776
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   777
lemma summable_imp_bounded:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   778
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   779
  shows "summable f \<Longrightarrow> bounded (range f)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   780
by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   781
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   782
lemma summable_imp_sums_bounded:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   783
   "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   784
by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   785
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   786
lemma power_series_conv_imp_absconv_weak:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   787
  fixes a:: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" and w :: 'a
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   788
  assumes sum: "summable (\<lambda>n. a n * z ^ n)" and no: "norm w < norm z"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   789
    shows "summable (\<lambda>n. of_real(norm(a n)) * w ^ n)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   790
proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   791
  obtain M where M: "\<And>x. norm (a x * z ^ x) \<le> M"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   792
    using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   793
  then have *: "summable (\<lambda>n. norm (a n) * norm w ^ n)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   794
    by (rule_tac M=M in Abel_lemma) (auto simp: norm_mult norm_power intro: no)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   795
  show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   796
    apply (rule series_comparison_complex [of "(\<lambda>n. of_real(norm(a n) * norm w ^ n))"])
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   797
    apply (simp only: summable_complex_of_real *)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   798
    apply (auto simp: norm_mult norm_power)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   799
    done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   800
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   801
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   802
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   803
subsection \<open>Normed spaces with the Heine-Borel property\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   804
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   805
lemma not_compact_UNIV[simp]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   806
  fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   807
  shows "\<not> compact (UNIV::'a set)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   808
    by (simp add: compact_eq_bounded_closed)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   809
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
   810
lemma not_compact_space_euclideanreal [simp]: "\<not> compact_space euclideanreal"
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
   811
  by (simp add: compact_space_def)
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
   812
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   813
text\<open>Representing sets as the union of a chain of compact sets.\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   814
lemma closed_Union_compact_subsets:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   815
  fixes S :: "'a::{heine_borel,real_normed_vector} set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   816
  assumes "closed S"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   817
  obtains F where "\<And>n. compact(F n)" "\<And>n. F n \<subseteq> S" "\<And>n. F n \<subseteq> F(Suc n)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   818
                  "(\<Union>n. F n) = S" "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. K \<subseteq> F n"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   819
proof
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   820
  show "compact (S \<inter> cball 0 (of_nat n))" for n
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   821
    using assms compact_eq_bounded_closed by auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   822
next
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   823
  show "(\<Union>n. S \<inter> cball 0 (real n)) = S"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   824
    by (auto simp: real_arch_simple)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   825
next
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   826
  fix K :: "'a set"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   827
  assume "compact K" "K \<subseteq> S"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   828
  then obtain N where "K \<subseteq> cball 0 N"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   829
    by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   830
  then show "\<exists>N. \<forall>n\<ge>N. K \<subseteq> S \<inter> cball 0 (real n)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   831
    by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   832
qed auto
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
   833
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   834
subsection \<open>Intersecting chains of compact sets and the Baire property\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   835
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   836
proposition bounded_closed_chain:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   837
  fixes \<F> :: "'a::heine_borel set set"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   838
  assumes "B \<in> \<F>" "bounded B" and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" and "{} \<notin> \<F>"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   839
      and chain: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   840
    shows "\<Inter>\<F> \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   841
proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   842
  have "B \<inter> \<Inter>\<F> \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   843
  proof (rule compact_imp_fip)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   844
    show "compact B" "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   845
      by (simp_all add: assms compact_eq_bounded_closed)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   846
    show "\<lbrakk>finite \<G>; \<G> \<subseteq> \<F>\<rbrakk> \<Longrightarrow> B \<inter> \<Inter>\<G> \<noteq> {}" for \<G>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   847
    proof (induction \<G> rule: finite_induct)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   848
      case empty
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   849
      with assms show ?case by force
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   850
    next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   851
      case (insert U \<G>)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   852
      then have "U \<in> \<F>" and ne: "B \<inter> \<Inter>\<G> \<noteq> {}" by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   853
      then consider "B \<subseteq> U" | "U \<subseteq> B"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   854
          using \<open>B \<in> \<F>\<close> chain by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   855
        then show ?case
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   856
        proof cases
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   857
          case 1
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   858
          then show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   859
            using Int_left_commute ne by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   860
        next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   861
          case 2
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   862
          have "U \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   863
            using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   864
          moreover
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   865
          have False if "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. x \<notin> Y"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   866
          proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   867
            have "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. Y \<subseteq> U"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   868
              by (metis chain contra_subsetD insert.prems insert_subset that)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   869
            then obtain Y where "Y \<in> \<G>" "Y \<subseteq> U"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   870
              by (metis all_not_in_conv \<open>U \<noteq> {}\<close>)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   871
            moreover obtain x where "x \<in> \<Inter>\<G>"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   872
              by (metis Int_emptyI ne)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   873
            ultimately show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   874
              by (metis Inf_lower subset_eq that)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   875
          qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   876
          with 2 show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   877
            by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   878
        qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   879
      qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   880
  qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   881
  then show ?thesis by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   882
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   883
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   884
corollary compact_chain:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   885
  fixes \<F> :: "'a::heine_borel set set"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   886
  assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" "{} \<notin> \<F>"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   887
          "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   888
    shows "\<Inter> \<F> \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   889
proof (cases "\<F> = {}")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   890
  case True
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   891
  then show ?thesis by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   892
next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   893
  case False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   894
  show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   895
    by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   896
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   897
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   898
lemma compact_nest:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   899
  fixes F :: "'a::linorder \<Rightarrow> 'b::heine_borel set"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   900
  assumes F: "\<And>n. compact(F n)" "\<And>n. F n \<noteq> {}" and mono: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
69745
aec42cee2521 more canonical and less specialized syntax
nipkow
parents: 69712
diff changeset
   901
  shows "\<Inter>(range F) \<noteq> {}"
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   902
proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   903
  have *: "\<And>S T. S \<in> range F \<and> T \<in> range F \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   904
    by (metis mono image_iff le_cases)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   905
  show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   906
    apply (rule compact_chain [OF _ _ *])
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   907
    using F apply (blast intro: dest: *)+
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   908
    done
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   909
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   910
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   911
text\<open>The Baire property of dense sets\<close>
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   912
theorem Baire:
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   913
  fixes S::"'a::{real_normed_vector,heine_borel} set"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   914
  assumes "closed S" "countable \<G>"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   915
      and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (top_of_set S) T \<and> S \<subseteq> closure T"
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   916
 shows "S \<subseteq> closure(\<Inter>\<G>)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   917
proof (cases "\<G> = {}")
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   918
  case True
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   919
  then show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   920
    using closure_subset by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   921
next
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   922
  let ?g = "from_nat_into \<G>"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   923
  case False
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   924
  then have gin: "?g n \<in> \<G>" for n
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   925
    by (simp add: from_nat_into)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   926
  show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   927
  proof (clarsimp simp: closure_approachable)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   928
    fix x and e::real
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   929
    assume "x \<in> S" "0 < e"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   930
    obtain TF where opeF: "\<And>n. openin (top_of_set S) (TF n)"
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   931
               and ne: "\<And>n. TF n \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   932
               and subg: "\<And>n. S \<inter> closure(TF n) \<subseteq> ?g n"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   933
               and subball: "\<And>n. closure(TF n) \<subseteq> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   934
               and decr: "\<And>n. TF(Suc n) \<subseteq> TF n"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   935
    proof -
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   936
      have *: "\<exists>Y. (openin (top_of_set S) Y \<and> Y \<noteq> {} \<and>
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   937
                   S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   938
        if opeU: "openin (top_of_set S) U" and "U \<noteq> {}" and cloU: "closure U \<subseteq> ball x e" for U n
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   939
      proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   940
        obtain T where T: "open T" "U = T \<inter> S"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   941
          using \<open>openin (top_of_set S) U\<close> by (auto simp: openin_subtopology)
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   942
        with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   943
          using gin ope by fastforce
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   944
        then have "T \<inter> ?g n \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   945
          using \<open>open T\<close> open_Int_closure_eq_empty by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   946
        then obtain y where "y \<in> U" "y \<in> ?g n"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   947
          using T ope [of "?g n", OF gin] by (blast dest:  openin_imp_subset)
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   948
        moreover have "openin (top_of_set S) (U \<inter> ?g n)"
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   949
          using gin ope opeU by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   950
        ultimately obtain d where U: "U \<inter> ?g n \<subseteq> S" and "d > 0" and d: "ball y d \<inter> S \<subseteq> U \<inter> ?g n"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   951
          by (force simp: openin_contains_ball)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   952
        show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   953
        proof (intro exI conjI)
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   954
          show "openin (top_of_set S) (S \<inter> ball y (d/2))"
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   955
            by (simp add: openin_open_Int)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   956
          show "S \<inter> ball y (d/2) \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   957
            using \<open>0 < d\<close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   958
          have "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> closure (ball y (d/2))"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   959
            using closure_mono by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   960
          also have "... \<subseteq> ?g n"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   961
            using \<open>d > 0\<close> d by force
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   962
          finally show "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> ?g n" .
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   963
          have "closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> ball y d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   964
          proof -
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   965
            have "closure (ball y (d/2)) \<subseteq> ball y d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   966
              using \<open>d > 0\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   967
            then have "closure (S \<inter> ball y (d/2)) \<subseteq> ball y d"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   968
              by (meson closure_mono inf.cobounded2 subset_trans)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   969
            then show ?thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   970
              by (simp add: \<open>closed S\<close> closure_minimal)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   971
          qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   972
          also have "...  \<subseteq> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   973
            using cloU closure_subset d by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   974
          finally show "closure (S \<inter> ball y (d/2)) \<subseteq> ball x e" .
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   975
          show "S \<inter> ball y (d/2) \<subseteq> U"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   976
            using ball_divide_subset_numeral d by blast
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   977
        qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   978
      qed
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   979
      let ?\<Phi> = "\<lambda>n X. openin (top_of_set S) X \<and> X \<noteq> {} \<and>
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   980
                      S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   981
      have "closure (S \<inter> ball x (e / 2)) \<subseteq> closure(ball x (e/2))"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   982
        by (simp add: closure_mono)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   983
      also have "...  \<subseteq> ball x e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   984
        using \<open>e > 0\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   985
      finally have "closure (S \<inter> ball x (e / 2)) \<subseteq> ball x e" .
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   986
      moreover have"openin (top_of_set S) (S \<inter> ball x (e / 2))" "S \<inter> ball x (e / 2) \<noteq> {}"
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   987
        using \<open>0 < e\<close> \<open>x \<in> S\<close> by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   988
      ultimately obtain Y where Y: "?\<Phi> 0 Y \<and> Y \<subseteq> S \<inter> ball x (e / 2)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   989
            using * [of "S \<inter> ball x (e/2)" 0] by metis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   990
      show thesis
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   991
      proof (rule exE [OF dependent_nat_choice [of ?\<Phi> "\<lambda>n X Y. Y \<subseteq> X"]])
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   992
        show "\<exists>x. ?\<Phi> 0 x"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   993
          using Y by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   994
        show "\<exists>Y. ?\<Phi> (Suc n) Y \<and> Y \<subseteq> X" if "?\<Phi> n X" for X n
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   995
          using that by (blast intro: *)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   996
      qed (use that in metis)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   997
    qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   998
    have "(\<Inter>n. S \<inter> closure (TF n)) \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
   999
    proof (rule compact_nest)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1000
      show "\<And>n. compact (S \<inter> closure (TF n))"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1001
        by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF \<open>closed S\<close>])
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1002
      show "\<And>n. S \<inter> closure (TF n) \<noteq> {}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1003
        by (metis Int_absorb1 opeF \<open>closed S\<close> closure_eq_empty closure_minimal ne openin_imp_subset)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1004
      show "\<And>m n. m \<le> n \<Longrightarrow> S \<inter> closure (TF n) \<subseteq> S \<inter> closure (TF m)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1005
        by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1006
    qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1007
    moreover have "(\<Inter>n. S \<inter> closure (TF n)) \<subseteq> {y \<in> \<Inter>\<G>. dist y x < e}"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1008
    proof (clarsimp, intro conjI)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1009
      fix y
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1010
      assume "y \<in> S" and y: "\<forall>n. y \<in> closure (TF n)"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1011
      then show "\<forall>T\<in>\<G>. y \<in> T"
69712
dc85b5b3a532 renamings and new material
paulson <lp15@cam.ac.uk>
parents: 69661
diff changeset
  1012
        by (metis Int_iff from_nat_into_surj [OF \<open>countable \<G>\<close>] subsetD subg)
69611
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1013
      show "dist y x < e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1014
        by (metis y dist_commute mem_ball subball subsetCE)
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1015
    qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1016
    ultimately show "\<exists>y \<in> \<Inter>\<G>. dist y x < e"
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1017
      by auto
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1018
  qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1019
qed
42cc3609fedf moved some material from Connected.thy to more appropriate places
immler
parents: 69544
diff changeset
  1020
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1021
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1022
subsection \<open>Continuity\<close>
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1023
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 70065
diff changeset
  1024
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Structural rules for uniform continuity\<close>
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1025
69613
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1026
lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1027
  fixes g :: "_::metric_space \<Rightarrow> _"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1028
  assumes "uniformly_continuous_on s g"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1029
  shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1030
  using assms unfolding uniformly_continuous_on_sequentially
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1031
  unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1032
  by (auto intro: tendsto_zero)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1033
69544
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1034
lemma uniformly_continuous_on_dist[continuous_intros]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1035
  fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1036
  assumes "uniformly_continuous_on s f"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1037
    and "uniformly_continuous_on s g"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1038
  shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1039
proof -
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1040
  {
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1041
    fix a b c d :: 'b
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1042
    have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1043
      using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1044
      using dist_triangle3 [of c d a] dist_triangle [of a d b]
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1045
      by arith
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1046
  } note le = this
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1047
  {
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1048
    fix x y
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1049
    assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1050
    assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1051
    have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1052
      by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1053
        simp add: le)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1054
  }
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1055
  then show ?thesis
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1056
    using assms unfolding uniformly_continuous_on_sequentially
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1057
    unfolding dist_real_def by simp
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1058
qed
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1059
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1060
lemma uniformly_continuous_on_norm[continuous_intros]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1061
  fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1062
  assumes "uniformly_continuous_on s f"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1063
  shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1064
  unfolding norm_conv_dist using assms
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1065
  by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1066
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1067
lemma uniformly_continuous_on_cmul[continuous_intros]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1068
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1069
  assumes "uniformly_continuous_on s f"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1070
  shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1071
  using bounded_linear_scaleR_right assms
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1072
  by (rule bounded_linear.uniformly_continuous_on)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1073
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1074
lemma dist_minus:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1075
  fixes x y :: "'a::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1076
  shows "dist (- x) (- y) = dist x y"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1077
  unfolding dist_norm minus_diff_minus norm_minus_cancel ..
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1078
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1079
lemma uniformly_continuous_on_minus[continuous_intros]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1080
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1081
  shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1082
  unfolding uniformly_continuous_on_def dist_minus .
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1083
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1084
lemma uniformly_continuous_on_add[continuous_intros]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1085
  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1086
  assumes "uniformly_continuous_on s f"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1087
    and "uniformly_continuous_on s g"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1088
  shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1089
  using assms
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1090
  unfolding uniformly_continuous_on_sequentially
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1091
  unfolding dist_norm tendsto_norm_zero_iff add_diff_add
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1092
  by (auto intro: tendsto_add_zero)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1093
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1094
lemma uniformly_continuous_on_diff[continuous_intros]:
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1095
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1096
  assumes "uniformly_continuous_on s f"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1097
    and "uniformly_continuous_on s g"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1098
  shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1099
  using assms uniformly_continuous_on_add [of s f "- g"]
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1100
    by (simp add: fun_Compl_def uniformly_continuous_on_minus)
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1101
5aa5a8d6e5b5 split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff changeset
  1102
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 70065
diff changeset
  1103
subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic Preserves Topological Properties\<close>
69613
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1104
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1105
lemma open_scaling[intro]:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1106
  fixes s :: "'a::real_normed_vector set"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1107
  assumes "c \<noteq> 0"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1108
    and "open s"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1109
  shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1110
proof -
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1111
  {
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1112
    fix x
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1113
    assume "x \<in> s"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1114
    then obtain e where "e>0"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1115
      and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1116
      by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1117
    have "e * \<bar>c\<bar> > 0"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1118
      using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1119
    moreover
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1120
    {
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1121
      fix y
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1122
      assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1123
      then have "norm ((1 / c) *\<^sub>R y - x) < e"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1124
        unfolding dist_norm
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1125
        using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1126
          assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1127
      then have "y \<in> (*\<^sub>R) c ` s"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1128
        using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "(*\<^sub>R) c"]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1129
        using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1130
        using assms(1)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1131
        unfolding dist_norm scaleR_scaleR
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1132
        by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1133
    }
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1134
    ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> (*\<^sub>R) c ` s"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1135
      apply (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1136
      done
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1137
  }
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1138
  then show ?thesis unfolding open_dist by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1139
qed
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1140
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1141
lemma minus_image_eq_vimage:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1142
  fixes A :: "'a::ab_group_add set"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1143
  shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1144
  by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1145
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1146
lemma open_negations:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1147
  fixes S :: "'a::real_normed_vector set"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1148
  shows "open S \<Longrightarrow> open ((\<lambda>x. - x) ` S)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1149
  using open_scaling [of "- 1" S] by simp
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1150
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1151
lemma open_translation:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1152
  fixes S :: "'a::real_normed_vector set"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1153
  assumes "open S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1154
  shows "open((\<lambda>x. a + x) ` S)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1155
proof -
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1156
  {
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1157
    fix x
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1158
    have "continuous (at x) (\<lambda>x. x - a)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1159
      by (intro continuous_diff continuous_ident continuous_const)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1160
  }
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1161
  moreover have "{x. x - a \<in> S} = (+) a ` S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1162
    by force
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1163
  ultimately show ?thesis
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1164
    by (metis assms continuous_open_vimage vimage_def)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1165
qed
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1166
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1167
lemma open_neg_translation:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1168
  fixes s :: "'a::real_normed_vector set"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1169
  assumes "open s"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1170
  shows "open((\<lambda>x. a - x) ` s)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1171
  using open_translation[OF open_negations[OF assms], of a]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1172
  by (auto simp: image_image)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1173
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1174
lemma open_affinity:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1175
  fixes S :: "'a::real_normed_vector set"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1176
  assumes "open S"  "c \<noteq> 0"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1177
  shows "open ((\<lambda>x. a + c *\<^sub>R x) ` S)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1178
proof -
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1179
  have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1180
    unfolding o_def ..
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1181
  have "(+) a ` (*\<^sub>R) c ` S = ((+) a \<circ> (*\<^sub>R) c) ` S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1182
    by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1183
  then show ?thesis
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1184
    using assms open_translation[of "(*\<^sub>R) c ` S" a]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1185
    unfolding *
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1186
    by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1187
qed
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1188
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1189
lemma interior_translation:
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69617
diff changeset
  1190
  "interior ((+) a ` S) = (+) a ` (interior S)" for S :: "'a::real_normed_vector set"
69613
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1191
proof (rule set_eqI, rule)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1192
  fix x
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1193
  assume "x \<in> interior ((+) a ` S)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1194
  then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) a ` S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1195
    unfolding mem_interior by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1196
  then have "ball (x - a) e \<subseteq> S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1197
    unfolding subset_eq Ball_def mem_ball dist_norm
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1198
    by (auto simp: diff_diff_eq)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1199
  then show "x \<in> (+) a ` interior S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1200
    unfolding image_iff
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1201
    apply (rule_tac x="x - a" in bexI)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1202
    unfolding mem_interior
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1203
    using \<open>e > 0\<close>
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1204
    apply auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1205
    done
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1206
next
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1207
  fix x
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1208
  assume "x \<in> (+) a ` interior S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1209
  then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1210
    unfolding image_iff Bex_def mem_interior by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1211
  {
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1212
    fix z
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1213
    have *: "a + y - z = y + a - z" by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1214
    assume "z \<in> ball x e"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1215
    then have "z - a \<in> S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1216
      using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1217
      unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1218
      by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1219
    then have "z \<in> (+) a ` S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1220
      unfolding image_iff by (auto intro!: bexI[where x="z - a"])
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1221
  }
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1222
  then have "ball x e \<subseteq> (+) a ` S"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1223
    unfolding subset_eq by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1224
  then show "x \<in> interior ((+) a ` S)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1225
    unfolding mem_interior using \<open>e > 0\<close> by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1226
qed
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1227
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69617
diff changeset
  1228
lemma interior_translation_subtract:
a03a63b81f44 tuned proofs
haftmann
parents: 69617
diff changeset
  1229
  "interior ((\<lambda>x. x - a) ` S) = (\<lambda>x. x - a) ` interior S" for S :: "'a::real_normed_vector set"
a03a63b81f44 tuned proofs
haftmann
parents: 69617
diff changeset
  1230
  using interior_translation [of "- a"] by (simp cong: image_cong_simp)
a03a63b81f44 tuned proofs
haftmann
parents: 69617
diff changeset
  1231
a03a63b81f44 tuned proofs
haftmann
parents: 69617
diff changeset
  1232
69613
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1233
lemma compact_scaling:
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1234
  fixes s :: "'a::real_normed_vector set"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1235
  assumes "compact s"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1236
  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1237
proof -
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1238
  let ?f = "\<lambda>x. scaleR c x"
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1239
  have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1240
  show ?thesis
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1241
    using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1242
    using linear_continuous_at[OF *] assms
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1243
    by auto
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1244
qed
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset
  1245
1331e57b54c6 moved material from Connected.thy to more appropriate places
immler
parents: 69611
diff changeset