author | immler |
Sat, 29 Dec 2018 20:32:09 +0100 | |
changeset 69544 | 5aa5a8d6e5b5 |
child 69611 | 42cc3609fedf |
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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|
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section \<open>Elementary Normed Vector Spaces\<close> |
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|
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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 |
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begin |
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|
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subsection%unimportant \<open>Various Lemmas Combining Imports\<close> |
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|
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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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|
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|
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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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|
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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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72 |
unfolding support_on_def by auto |
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73 |
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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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77 |
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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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79 |
by (auto simp: support_on_def) |
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|
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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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82 |
by (auto simp: support_on_def) |
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83 |
|
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lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)" |
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85 |
unfolding support_on_def by auto |
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86 |
|
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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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90 |
|
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lemma supp_sum_empty[simp]: "supp_sum f {} = 0" |
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92 |
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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98 |
|
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99 |
lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A" |
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100 |
by (cases "r = 0") |
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101 |
(auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong) |
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102 |
|
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103 |
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104 |
subsection \<open>Intervals\<close> |
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105 |
|
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106 |
lemma image_affinity_interval: |
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107 |
fixes c :: "'a::ordered_real_vector" |
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108 |
shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) = |
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109 |
(if {a..b}={} then {} |
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110 |
else if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c} |
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111 |
else {m *\<^sub>R b + c .. m *\<^sub>R a + c})" |
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112 |
(is "?lhs = ?rhs") |
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113 |
proof (cases "m=0") |
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diff
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|
114 |
case True |
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immler
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diff
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|
115 |
then show ?thesis |
5aa5a8d6e5b5
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immler
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diff
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|
116 |
by force |
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|
117 |
next |
5aa5a8d6e5b5
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|
118 |
case False |
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immler
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diff
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|
119 |
show ?thesis |
5aa5a8d6e5b5
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|
120 |
proof |
5aa5a8d6e5b5
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|
121 |
show "?lhs \<subseteq> ?rhs" |
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|
122 |
by (auto simp: scaleR_left_mono scaleR_left_mono_neg) |
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|
123 |
show "?rhs \<subseteq> ?lhs" |
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|
124 |
proof (clarsimp, intro conjI impI subsetI) |
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|
125 |
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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|
126 |
\<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x |
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|
127 |
apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI) |
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immler
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|
128 |
using False apply (auto simp: le_diff_eq pos_le_divideRI) |
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diff
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|
129 |
using diff_le_eq pos_le_divideR_eq by force |
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diff
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|
130 |
show "\<lbrakk>\<not> 0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk> |
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diff
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|
131 |
\<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x |
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diff
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|
132 |
apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI) |
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|
133 |
apply (auto simp: diff_le_eq neg_le_divideR_eq) |
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diff
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|
134 |
using diff_eq_diff_less_eq linordered_field_class.sign_simps(11) minus_diff_eq not_less scaleR_le_cancel_left_neg by fastforce |
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|
135 |
qed |
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diff
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|
136 |
qed |
5aa5a8d6e5b5
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immler
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diff
changeset
|
137 |
qed |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
138 |
|
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immler
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diff
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|
139 |
|
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|
140 |
subsection%unimportant \<open>Various Lemmas on Normed Algebras\<close> |
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|
141 |
|
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immler
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|
142 |
|
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|
143 |
lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)" |
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|
144 |
by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat) |
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immler
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diff
changeset
|
145 |
|
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|
146 |
lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)" |
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|
147 |
by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int) |
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immler
parents:
diff
changeset
|
148 |
|
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|
149 |
lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)" |
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|
150 |
unfolding Nats_def by (rule closed_of_nat_image) |
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immler
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diff
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|
151 |
|
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|
152 |
lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)" |
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diff
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|
153 |
unfolding Ints_def by (rule closed_of_int_image) |
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diff
changeset
|
154 |
|
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immler
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diff
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|
155 |
lemma closed_subset_Ints: |
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diff
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|
156 |
fixes A :: "'a :: real_normed_algebra_1 set" |
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|
157 |
assumes "A \<subseteq> \<int>" |
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|
158 |
shows "closed A" |
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diff
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|
159 |
proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases) |
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immler
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diff
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|
160 |
case (1 x y) |
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diff
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|
161 |
with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto |
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diff
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|
162 |
with \<open>dist y x < 1\<close> show "y = x" |
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immler
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diff
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|
163 |
by (auto elim!: Ints_cases simp: dist_of_int) |
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diff
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|
164 |
qed |
5aa5a8d6e5b5
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immler
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diff
changeset
|
165 |
|
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changeset
|
166 |
subsection \<open>Filters\<close> |
5aa5a8d6e5b5
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|
167 |
|
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|
168 |
definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter" (infixr "indirection" 70) |
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|
169 |
where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}" |
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changeset
|
170 |
|
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immler
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diff
changeset
|
171 |
|
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|
172 |
subsection \<open>Trivial Limits\<close> |
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|
173 |
|
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|
174 |
lemma trivial_limit_at_infinity: |
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diff
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|
175 |
"\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)" |
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immler
parents:
diff
changeset
|
176 |
unfolding trivial_limit_def eventually_at_infinity |
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diff
changeset
|
177 |
apply clarsimp |
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diff
changeset
|
178 |
apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify) |
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immler
parents:
diff
changeset
|
179 |
apply (rule_tac x="scaleR (b / norm x) x" in exI, simp) |
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immler
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diff
changeset
|
180 |
apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def]) |
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immler
parents:
diff
changeset
|
181 |
apply (drule_tac x=UNIV in spec, simp) |
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|
182 |
done |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
183 |
|
5aa5a8d6e5b5
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parents:
diff
changeset
|
184 |
subsection \<open>Limits\<close> |
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diff
changeset
|
185 |
|
5aa5a8d6e5b5
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immler
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|
186 |
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)" |
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|
187 |
by (auto simp: tendsto_iff eventually_at_infinity) |
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immler
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diff
changeset
|
188 |
|
5aa5a8d6e5b5
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immler
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diff
changeset
|
189 |
corollary Lim_at_infinityI [intro?]: |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
190 |
assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e" |
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immler
parents:
diff
changeset
|
191 |
shows "(f \<longlongrightarrow> l) at_infinity" |
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immler
parents:
diff
changeset
|
192 |
apply (simp add: Lim_at_infinity, clarify) |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
193 |
apply (rule ex_forward [OF assms [OF half_gt_zero]], auto) |
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immler
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diff
changeset
|
194 |
done |
5aa5a8d6e5b5
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immler
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diff
changeset
|
195 |
|
5aa5a8d6e5b5
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immler
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diff
changeset
|
196 |
lemma Lim_transform_within_set_eq: |
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immler
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diff
changeset
|
197 |
fixes a l :: "'a::real_normed_vector" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
198 |
shows "eventually (\<lambda>x. x \<in> s \<longleftrightarrow> x \<in> t) (at a) |
5aa5a8d6e5b5
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immler
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diff
changeset
|
199 |
\<Longrightarrow> ((f \<longlongrightarrow> l) (at a within s) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within t))" |
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immler
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diff
changeset
|
200 |
by (force intro: Lim_transform_within_set elim: eventually_mono) |
5aa5a8d6e5b5
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immler
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diff
changeset
|
201 |
|
5aa5a8d6e5b5
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immler
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diff
changeset
|
202 |
lemma Lim_null: |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
203 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
204 |
shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
205 |
by (simp add: Lim dist_norm) |
5aa5a8d6e5b5
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immler
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diff
changeset
|
206 |
|
5aa5a8d6e5b5
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immler
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diff
changeset
|
207 |
lemma Lim_null_comparison: |
5aa5a8d6e5b5
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immler
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diff
changeset
|
208 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
209 |
assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
210 |
shows "(f \<longlongrightarrow> 0) net" |
5aa5a8d6e5b5
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immler
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diff
changeset
|
211 |
using assms(2) |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
212 |
proof (rule metric_tendsto_imp_tendsto) |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
213 |
show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
214 |
using assms(1) by (rule eventually_mono) (simp add: dist_norm) |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
215 |
qed |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
216 |
|
5aa5a8d6e5b5
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immler
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diff
changeset
|
217 |
lemma Lim_transform_bound: |
5aa5a8d6e5b5
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immler
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diff
changeset
|
218 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
5aa5a8d6e5b5
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immler
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changeset
|
219 |
and g :: "'a \<Rightarrow> 'c::real_normed_vector" |
5aa5a8d6e5b5
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immler
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diff
changeset
|
220 |
assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
221 |
and "(g \<longlongrightarrow> 0) net" |
5aa5a8d6e5b5
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immler
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diff
changeset
|
222 |
shows "(f \<longlongrightarrow> 0) net" |
5aa5a8d6e5b5
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immler
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diff
changeset
|
223 |
using assms(1) tendsto_norm_zero [OF assms(2)] |
5aa5a8d6e5b5
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immler
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diff
changeset
|
224 |
by (rule Lim_null_comparison) |
5aa5a8d6e5b5
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immler
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diff
changeset
|
225 |
|
5aa5a8d6e5b5
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immler
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diff
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|
226 |
lemma lim_null_mult_right_bounded: |
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diff
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|
227 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra" |
5aa5a8d6e5b5
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immler
parents:
diff
changeset
|
228 |
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
|
229 |
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
|
230 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
231 |
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
|
232 |
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
|
233 |
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
|
234 |
apply (rule Lim_null_comparison [OF _ *]) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
235 |
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
|
236 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
237 |
then show ?thesis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
238 |
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
|
239 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
240 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
241 |
lemma lim_null_mult_left_bounded: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
242 |
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
|
243 |
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
|
244 |
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
|
245 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
246 |
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
|
247 |
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
|
248 |
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
|
249 |
apply (rule Lim_null_comparison [OF _ *]) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
250 |
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
|
251 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
252 |
then show ?thesis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
253 |
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
|
254 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
255 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
256 |
lemma lim_null_scaleR_bounded: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
257 |
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
|
258 |
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
|
259 |
proof |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
260 |
fix \<epsilon>::real |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
261 |
assume "0 < \<epsilon>" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
262 |
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
|
263 |
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
|
264 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
265 |
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
|
266 |
by (simp add: mult_left_mono g) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
267 |
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
|
268 |
by (simp add: mult_left_mono) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
269 |
also have "\<dots> < \<epsilon>" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
270 |
by (rule f) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
271 |
finally show ?thesis . |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
272 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
273 |
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
|
274 |
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
|
275 |
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
|
276 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
277 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
278 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
279 |
lemma Lim_norm_ubound: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
280 |
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
|
281 |
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
|
282 |
shows "norm(l) \<le> e" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
283 |
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
|
284 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
285 |
lemma Lim_norm_lbound: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
286 |
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
|
287 |
assumes "\<not> trivial_limit net" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
288 |
and "(f \<longlongrightarrow> l) net" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
289 |
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
|
290 |
shows "e \<le> norm l" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
291 |
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
|
292 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
293 |
text\<open>Limit under bilinear function\<close> |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
294 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
295 |
lemma Lim_bilinear: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
296 |
assumes "(f \<longlongrightarrow> l) net" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
297 |
and "(g \<longlongrightarrow> m) net" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
298 |
and "bounded_bilinear h" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
299 |
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
|
300 |
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
|
301 |
by (rule bounded_bilinear.tendsto) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
302 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
303 |
lemma Lim_at_zero: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
304 |
fixes a :: "'a::real_normed_vector" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
305 |
and l :: "'b::topological_space" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
306 |
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
|
307 |
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
|
308 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
309 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
310 |
subsection%unimportant \<open>Limit Point of Filter\<close> |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
311 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
312 |
lemma netlimit_at_vector: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
313 |
fixes a :: "'a::real_normed_vector" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
314 |
shows "netlimit (at a) = a" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
315 |
proof (cases "\<exists>x. x \<noteq> a") |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
316 |
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
|
317 |
have "\<not> trivial_limit (at a)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
318 |
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
|
319 |
apply clarsimp |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
320 |
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
|
321 |
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
|
322 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
323 |
then show ?thesis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
324 |
by (rule netlimit_within [of a UNIV]) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
325 |
qed simp |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
326 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
327 |
subsection \<open>Boundedness\<close> |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
328 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
329 |
lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>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
|
330 |
apply (simp add: bounded_iff) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
331 |
apply (subgoal_tac "\<And>x (y::real). 0 < 1 + \<bar>y\<bar> \<and> (x \<le> y \<longrightarrow> x \<le> 1 + \<bar>y\<bar>)") |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
332 |
apply metis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
333 |
apply arith |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
334 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
335 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
336 |
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
|
337 |
apply (simp add: bounded_pos) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
338 |
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
|
339 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
340 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
341 |
lemma Bseq_eq_bounded: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
342 |
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
|
343 |
shows "Bseq f \<longleftrightarrow> bounded (range f)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
344 |
unfolding Bseq_def bounded_pos by auto |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
345 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
346 |
lemma bounded_linear_image: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
347 |
assumes "bounded S" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
348 |
and "bounded_linear f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
349 |
shows "bounded (f ` S)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
350 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
351 |
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
|
352 |
unfolding bounded_pos by auto |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
353 |
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
|
354 |
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
|
355 |
show ?thesis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
356 |
unfolding bounded_pos |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
357 |
proof (intro exI, safe) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
358 |
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
|
359 |
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
|
360 |
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
|
361 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
362 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
363 |
lemma bounded_scaling: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
364 |
fixes S :: "'a::real_normed_vector set" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
365 |
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
|
366 |
apply (rule bounded_linear_image, assumption) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
367 |
apply (rule bounded_linear_scaleR_right) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
368 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
369 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
370 |
lemma bounded_scaleR_comp: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
371 |
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
|
372 |
assumes "bounded (f ` S)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
373 |
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
|
374 |
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
|
375 |
by (auto simp: image_image) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
376 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
377 |
lemma bounded_translation: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
378 |
fixes S :: "'a::real_normed_vector set" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
379 |
assumes "bounded S" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
380 |
shows "bounded ((\<lambda>x. a + x) ` S)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
381 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
382 |
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
|
383 |
unfolding bounded_pos by auto |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
384 |
{ |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
385 |
fix x |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
386 |
assume "x \<in> S" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
387 |
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
|
388 |
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
|
389 |
} |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
390 |
then show ?thesis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
391 |
unfolding bounded_pos |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
392 |
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
|
393 |
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
|
394 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
395 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
396 |
lemma bounded_translation_minus: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
397 |
fixes S :: "'a::real_normed_vector set" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
398 |
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
|
399 |
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
|
400 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
401 |
lemma bounded_uminus [simp]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
402 |
fixes X :: "'a::real_normed_vector set" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
403 |
shows "bounded (uminus ` X) \<longleftrightarrow> bounded X" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
404 |
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
|
405 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
406 |
lemma uminus_bounded_comp [simp]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
407 |
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
|
408 |
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
|
409 |
using bounded_uminus[of "f ` S"] |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
410 |
by (auto simp: image_image) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
411 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
412 |
lemma bounded_plus_comp: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
413 |
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
|
414 |
assumes "bounded (f ` S)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
415 |
assumes "bounded (g ` S)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
416 |
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
|
417 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
418 |
{ |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
419 |
fix B C |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
420 |
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
|
421 |
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
|
422 |
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
|
423 |
} then show ?thesis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
424 |
using assms by (fastforce simp: bounded_iff) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
425 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
426 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
427 |
lemma bounded_plus: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
428 |
fixes S ::"'a::real_normed_vector set" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
429 |
assumes "bounded S" "bounded T" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
430 |
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
|
431 |
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
|
432 |
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
|
433 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
434 |
lemma bounded_minus_comp: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
435 |
"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
|
436 |
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
|
437 |
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
|
438 |
by auto |
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 bounded_minus: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
441 |
fixes S ::"'a::real_normed_vector set" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
442 |
assumes "bounded S" "bounded T" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
443 |
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
|
444 |
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
|
445 |
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
|
446 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
447 |
lemma not_bounded_UNIV[simp]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
448 |
"\<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
|
449 |
proof (auto simp: bounded_pos not_le) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
450 |
obtain x :: 'a where "x \<noteq> 0" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
451 |
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
|
452 |
fix b :: real |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
453 |
assume b: "b >0" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
454 |
have b1: "b +1 \<ge> 0" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
455 |
using b by simp |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
456 |
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
|
457 |
by (simp add: norm_sgn) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
458 |
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
|
459 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
460 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
461 |
corollary cobounded_imp_unbounded: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
462 |
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
|
463 |
shows "bounded (- S) \<Longrightarrow> \<not> bounded S" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
464 |
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
|
465 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
466 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
467 |
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
|
468 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
469 |
lemma not_compact_UNIV[simp]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
470 |
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
|
471 |
shows "\<not> compact (UNIV::'a set)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
472 |
by (simp add: compact_eq_bounded_closed) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
473 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
474 |
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
|
475 |
lemma closed_Union_compact_subsets: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
476 |
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
|
477 |
assumes "closed S" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
478 |
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
|
479 |
"(\<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
|
480 |
proof |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
481 |
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
|
482 |
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
|
483 |
next |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
484 |
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
|
485 |
by (auto simp: real_arch_simple) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
486 |
next |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
487 |
fix K :: "'a set" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
488 |
assume "compact K" "K \<subseteq> S" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
489 |
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
|
490 |
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
|
491 |
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
|
492 |
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
|
493 |
qed auto |
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 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
496 |
subsection \<open>Continuity\<close> |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
497 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
498 |
subsubsection%unimportant \<open>Structural rules for uniform continuity\<close> |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
499 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
500 |
lemma uniformly_continuous_on_dist[continuous_intros]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
501 |
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
|
502 |
assumes "uniformly_continuous_on s f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
503 |
and "uniformly_continuous_on s g" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
504 |
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
|
505 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
506 |
{ |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
507 |
fix a b c d :: 'b |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
508 |
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
|
509 |
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
|
510 |
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
|
511 |
by arith |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
512 |
} note le = this |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
513 |
{ |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
514 |
fix x y |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
515 |
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
|
516 |
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
|
517 |
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
|
518 |
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
|
519 |
simp add: le) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
520 |
} |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
521 |
then show ?thesis |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
522 |
using assms unfolding uniformly_continuous_on_sequentially |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
523 |
unfolding dist_real_def by simp |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
524 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
525 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
526 |
lemma uniformly_continuous_on_norm[continuous_intros]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
527 |
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
|
528 |
assumes "uniformly_continuous_on s f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
529 |
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
|
530 |
unfolding norm_conv_dist using assms |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
531 |
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
|
532 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
533 |
lemma uniformly_continuous_on_cmul[continuous_intros]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
534 |
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
|
535 |
assumes "uniformly_continuous_on s f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
536 |
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
|
537 |
using bounded_linear_scaleR_right assms |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
538 |
by (rule bounded_linear.uniformly_continuous_on) |
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 dist_minus: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
541 |
fixes x y :: "'a::real_normed_vector" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
542 |
shows "dist (- x) (- y) = dist x y" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
543 |
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
|
544 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
545 |
lemma uniformly_continuous_on_minus[continuous_intros]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
546 |
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
|
547 |
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
|
548 |
unfolding uniformly_continuous_on_def dist_minus . |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
549 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
550 |
lemma uniformly_continuous_on_add[continuous_intros]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
551 |
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
|
552 |
assumes "uniformly_continuous_on s f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
553 |
and "uniformly_continuous_on s g" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
554 |
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
|
555 |
using assms |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
556 |
unfolding uniformly_continuous_on_sequentially |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
557 |
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
|
558 |
by (auto intro: tendsto_add_zero) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
559 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
560 |
lemma uniformly_continuous_on_diff[continuous_intros]: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
561 |
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
|
562 |
assumes "uniformly_continuous_on s f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
563 |
and "uniformly_continuous_on s g" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
564 |
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
|
565 |
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
|
566 |
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
|
567 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
568 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
569 |
subsection%unimportant \<open>Topological properties of linear functions\<close> |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
570 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
571 |
lemma linear_lim_0: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
572 |
assumes "bounded_linear f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
573 |
shows "(f \<longlongrightarrow> 0) (at (0))" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
574 |
proof - |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
575 |
interpret f: bounded_linear f by fact |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
576 |
have "(f \<longlongrightarrow> f 0) (at 0)" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
577 |
using tendsto_ident_at by (rule f.tendsto) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
578 |
then show ?thesis unfolding f.zero . |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
579 |
qed |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
580 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
581 |
lemma linear_continuous_at: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
582 |
assumes "bounded_linear f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
583 |
shows "continuous (at a) f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
584 |
unfolding continuous_at using assms |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
585 |
apply (rule bounded_linear.tendsto) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
586 |
apply (rule tendsto_ident_at) |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
587 |
done |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
588 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
589 |
lemma linear_continuous_within: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
590 |
"bounded_linear f \<Longrightarrow> continuous (at x within s) f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
591 |
using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
592 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
593 |
lemma linear_continuous_on: |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
594 |
"bounded_linear f \<Longrightarrow> continuous_on s f" |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
595 |
using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto |
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
596 |
|
5aa5a8d6e5b5
split off theorems involving classes below metric_space and real_normed_vector
immler
parents:
diff
changeset
|
597 |
end |