author  huffman 
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child 44194  0639898074ae 
permissions  rwrr 
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(* Title : Limits.thy 
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Author : Brian Huffman 
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*) 
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header {* Filters and Limits *} 
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theory Limits 
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imports RealVector 
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begin 
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subsection {* Filters *} 
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text {* 

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This definition also allows nonproper filters. 
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*} 
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locale is_filter = 
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fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" 
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assumes True: "F (\<lambda>x. True)" 
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assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)" 
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assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)" 
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typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}" 
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proof 
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show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro) 
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qed 
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lemma is_filter_Rep_filter: "is_filter (Rep_filter A)" 
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using Rep_filter [of A] by simp 
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lemma Abs_filter_inverse': 
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assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" 
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using assms by (simp add: Abs_filter_inverse) 
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subsection {* Eventually *} 

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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" 
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where "eventually P A \<longleftrightarrow> Rep_filter A P" 
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lemma eventually_Abs_filter: 
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assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" 
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unfolding eventually_def using assms by (simp add: Abs_filter_inverse) 
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lemma filter_eq_iff: 
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shows "A = B \<longleftrightarrow> (\<forall>P. eventually P A = eventually P B)" 
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unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. 
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) A" 
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unfolding eventually_def 
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by (rule is_filter.True [OF is_filter_Rep_filter]) 
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P A" 
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proof  
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assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext) 

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thus "eventually P A" by simp 
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qed 
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lemma eventually_mono: 
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"(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P A \<Longrightarrow> eventually Q A" 
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unfolding eventually_def 
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by (rule is_filter.mono [OF is_filter_Rep_filter]) 
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lemma eventually_conj: 
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assumes P: "eventually (\<lambda>x. P x) A" 
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assumes Q: "eventually (\<lambda>x. Q x) A" 
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shows "eventually (\<lambda>x. P x \<and> Q x) A" 
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using assms unfolding eventually_def 
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by (rule is_filter.conj [OF is_filter_Rep_filter]) 
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lemma eventually_mp: 
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A" 
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assumes "eventually (\<lambda>x. P x) A" 
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shows "eventually (\<lambda>x. Q x) A" 
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proof (rule eventually_mono) 
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show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp 
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show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) A" 
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using assms by (rule eventually_conj) 
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qed 
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lemma eventually_rev_mp: 
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assumes "eventually (\<lambda>x. P x) A" 
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A" 
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shows "eventually (\<lambda>x. Q x) A" 
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using assms(2) assms(1) by (rule eventually_mp) 
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lemma eventually_conj_iff: 
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"eventually (\<lambda>x. P x \<and> Q x) A \<longleftrightarrow> eventually P A \<and> eventually Q A" 
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by (auto intro: eventually_conj elim: eventually_rev_mp) 
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lemma eventually_elim1: 
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assumes "eventually (\<lambda>i. P i) A" 
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assumes "\<And>i. P i \<Longrightarrow> Q i" 
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shows "eventually (\<lambda>i. Q i) A" 
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using assms by (auto elim!: eventually_rev_mp) 
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lemma eventually_elim2: 
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assumes "eventually (\<lambda>i. P i) A" 
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assumes "eventually (\<lambda>i. Q i) A" 
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assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i" 
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shows "eventually (\<lambda>i. R i) A" 
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using assms by (auto elim!: eventually_rev_mp) 
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subsection {* Finerthan relation *} 
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text {* @{term "A \<le> B"} means that filter @{term A} is finer than 
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filter @{term B}. *} 
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instantiation filter :: (type) complete_lattice 
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begin 
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definition le_filter_def: 
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"A \<le> B \<longleftrightarrow> (\<forall>P. eventually P B \<longrightarrow> eventually P A)" 
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definition 
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"(A :: 'a filter) < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A" 
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definition 
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"top = Abs_filter (\<lambda>P. \<forall>x. P x)" 
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definition 

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"bot = Abs_filter (\<lambda>P. True)" 
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definition 
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"sup A B = Abs_filter (\<lambda>P. eventually P A \<and> eventually P B)" 
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definition 

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"inf A B = Abs_filter 
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(\<lambda>P. \<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" 
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definition 

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"Sup S = Abs_filter (\<lambda>P. \<forall>A\<in>S. eventually P A)" 
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definition 

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"Inf S = Sup {A::'a filter. \<forall>B\<in>S. A \<le> B}" 
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lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" 

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unfolding top_filter_def 
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by (rule eventually_Abs_filter, rule is_filter.intro, auto) 
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lemma eventually_bot [simp]: "eventually P bot" 
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unfolding bot_filter_def 
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by (subst eventually_Abs_filter, rule is_filter.intro, auto) 
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lemma eventually_sup: 
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"eventually P (sup A B) \<longleftrightarrow> eventually P A \<and> eventually P B" 
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unfolding sup_filter_def 
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by (rule eventually_Abs_filter, rule is_filter.intro) 
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(auto elim!: eventually_rev_mp) 
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151 
lemma eventually_inf: 

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"eventually P (inf A B) \<longleftrightarrow> 
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(\<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" 
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unfolding inf_filter_def 
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apply (rule eventually_Abs_filter, rule is_filter.intro) 
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apply (fast intro: eventually_True) 
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apply clarify 
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apply (intro exI conjI) 
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apply (erule (1) eventually_conj) 
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apply (erule (1) eventually_conj) 
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apply simp 
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apply auto 
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done 
36630  164 

165 
lemma eventually_Sup: 

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"eventually P (Sup S) \<longleftrightarrow> (\<forall>A\<in>S. eventually P A)" 
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unfolding Sup_filter_def 
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apply (rule eventually_Abs_filter, rule is_filter.intro) 
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apply (auto intro: eventually_conj elim!: eventually_rev_mp) 
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done 
36630  171 

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instance proof 
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fix A B :: "'a filter" show "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A" 
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by (rule less_filter_def) 
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next 
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fix A :: "'a filter" show "A \<le> A" 
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unfolding le_filter_def by simp 
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next 
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fix A B C :: "'a filter" assume "A \<le> B" and "B \<le> C" thus "A \<le> C" 
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unfolding le_filter_def by simp 
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next 
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fix A B :: "'a filter" assume "A \<le> B" and "B \<le> A" thus "A = B" 
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unfolding le_filter_def filter_eq_iff by fast 
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next 
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fix A :: "'a filter" show "A \<le> top" 
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unfolding le_filter_def eventually_top by (simp add: always_eventually) 
36630  187 
next 
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fix A :: "'a filter" show "bot \<le> A" 
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unfolding le_filter_def by simp 
36630  190 
next 
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fix A B :: "'a filter" show "A \<le> sup A B" and "B \<le> sup A B" 
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unfolding le_filter_def eventually_sup by simp_all 
36630  193 
next 
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fix A B C :: "'a filter" assume "A \<le> C" and "B \<le> C" thus "sup A B \<le> C" 
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unfolding le_filter_def eventually_sup by simp 
36630  196 
next 
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fix A B :: "'a filter" show "inf A B \<le> A" and "inf A B \<le> B" 
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unfolding le_filter_def eventually_inf by (auto intro: eventually_True) 
36630  199 
next 
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fix A B C :: "'a filter" assume "A \<le> B" and "A \<le> C" thus "A \<le> inf B C" 
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unfolding le_filter_def eventually_inf 
36630  202 
by (auto elim!: eventually_mono intro: eventually_conj) 
203 
next 

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fix A :: "'a filter" and S assume "A \<in> S" thus "A \<le> Sup S" 
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205 
unfolding le_filter_def eventually_Sup by simp 
36630  206 
next 
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fix S and B :: "'a filter" assume "\<And>A. A \<in> S \<Longrightarrow> A \<le> B" thus "Sup S \<le> B" 
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unfolding le_filter_def eventually_Sup by simp 
36630  209 
next 
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fix C :: "'a filter" and S assume "C \<in> S" thus "Inf S \<le> C" 
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unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp 
36630  212 
next 
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fix S and A :: "'a filter" assume "\<And>B. B \<in> S \<Longrightarrow> A \<le> B" thus "A \<le> Inf S" 
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unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp 
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qed 
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end 
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lemma filter_leD: 
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"A \<le> B \<Longrightarrow> eventually P B \<Longrightarrow> eventually P A" 
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221 
unfolding le_filter_def by simp 
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222 

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lemma filter_leI: 
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"(\<And>P. eventually P B \<Longrightarrow> eventually P A) \<Longrightarrow> A \<le> B" 
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225 
unfolding le_filter_def by simp 
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226 

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lemma eventually_False: 
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"eventually (\<lambda>x. False) A \<longleftrightarrow> A = bot" 
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unfolding filter_eq_iff by (auto elim: eventually_rev_mp) 
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230 

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subsection {* Map function for filters *} 
36654  232 

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definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" 
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where "filtermap f A = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) A)" 
36654  235 

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lemma eventually_filtermap: 
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"eventually P (filtermap f A) = eventually (\<lambda>x. P (f x)) A" 
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unfolding filtermap_def 
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apply (rule eventually_Abs_filter) 
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240 
apply (rule is_filter.intro) 
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241 
apply (auto elim!: eventually_rev_mp) 
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242 
done 
36654  243 

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lemma filtermap_ident: "filtermap (\<lambda>x. x) A = A" 
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by (simp add: filter_eq_iff eventually_filtermap) 
36654  246 

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lemma filtermap_filtermap: 
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"filtermap f (filtermap g A) = filtermap (\<lambda>x. f (g x)) A" 
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by (simp add: filter_eq_iff eventually_filtermap) 
36654  250 

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lemma filtermap_mono: "A \<le> B \<Longrightarrow> filtermap f A \<le> filtermap f B" 
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unfolding le_filter_def eventually_filtermap by simp 
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253 

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lemma filtermap_bot [simp]: "filtermap f bot = bot" 
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by (simp add: filter_eq_iff eventually_filtermap) 
36654  256 

257 

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subsection {* Sequentially *} 
31392  259 

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definition sequentially :: "nat filter" 
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where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)" 
31392  262 

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lemma eventually_sequentially: 
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"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" 
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unfolding sequentially_def 
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proof (rule eventually_Abs_filter, rule is_filter.intro) 
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fix P Q :: "nat \<Rightarrow> bool" 
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assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n" 
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then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto 
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then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp 
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then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" .. 
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qed auto 
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273 

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lemma sequentially_bot [simp]: "sequentially \<noteq> bot" 
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unfolding filter_eq_iff eventually_sequentially by auto 
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276 

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lemma eventually_False_sequentially [simp]: 
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"\<not> eventually (\<lambda>n. False) sequentially" 
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by (simp add: eventually_False) 
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280 

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lemma le_sequentially: 
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"A \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) A)" 
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283 
unfolding le_filter_def eventually_sequentially 
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by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp) 
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285 

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286 

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definition trivial_limit :: "'a filter \<Rightarrow> bool" 
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where "trivial_limit A \<longleftrightarrow> eventually (\<lambda>x. False) A" 
41970  289 

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lemma trivial_limit_sequentially [intro]: "\<not> trivial_limit sequentially" 
41970  291 
by (auto simp add: trivial_limit_def eventually_sequentially) 
292 

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subsection {* Standard filters *} 
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294 

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definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70) 
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where "A within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A)" 
31392  297 

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definition nhds :: "'a::topological_space \<Rightarrow> 'a filter" 
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299 
where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" 
36654  300 

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301 
definition at :: "'a::topological_space \<Rightarrow> 'a filter" 
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302 
where "at a = nhds a within  {a}" 
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303 

31392  304 
lemma eventually_within: 
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"eventually P (A within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A" 
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306 
unfolding within_def 
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307 
by (rule eventually_Abs_filter, rule is_filter.intro) 
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308 
(auto elim!: eventually_rev_mp) 
31392  309 

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lemma within_UNIV: "A within UNIV = A" 
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311 
unfolding filter_eq_iff eventually_within by simp 
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312 

36654  313 
lemma eventually_nhds: 
314 
"eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" 

315 
unfolding nhds_def 

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316 
proof (rule eventually_Abs_filter, rule is_filter.intro) 
36654  317 
have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp 
318 
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by  rule 

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319 
next 
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320 
fix P Q 
36654  321 
assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" 
322 
and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" 

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323 
then obtain S T where 
36654  324 
"open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" 
325 
"open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto 

326 
hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)" 

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327 
by (simp add: open_Int) 
36654  328 
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by  rule 
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329 
qed auto 
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330 

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331 
lemma eventually_nhds_metric: 
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332 
"eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)" 
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333 
unfolding eventually_nhds open_dist 
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334 
apply safe 
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335 
apply fast 
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336 
apply (rule_tac x="{x. dist x a < d}" in exI, simp) 
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337 
apply clarsimp 
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338 
apply (rule_tac x="d  dist x a" in exI, clarsimp) 
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339 
apply (simp only: less_diff_eq) 
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340 
apply (erule le_less_trans [OF dist_triangle]) 
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341 
done 
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342 

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343 
lemma eventually_at_topological: 
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344 
"eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))" 
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345 
unfolding at_def eventually_within eventually_nhds by simp 
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346 

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347 
lemma eventually_at: 
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348 
fixes a :: "'a::metric_space" 
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349 
shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)" 
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350 
unfolding at_def eventually_within eventually_nhds_metric by auto 
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351 

31392  352 

31355  353 
subsection {* Boundedness *} 
354 

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355 
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" 
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356 
where "Bfun f A = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) A)" 
31355  357 

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358 
lemma BfunI: 
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359 
assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) A" shows "Bfun f A" 
31355  360 
unfolding Bfun_def 
361 
proof (intro exI conjI allI) 

362 
show "0 < max K 1" by simp 

363 
next 

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364 
show "eventually (\<lambda>x. norm (f x) \<le> max K 1) A" 
31355  365 
using K by (rule eventually_elim1, simp) 
366 
qed 

367 

368 
lemma BfunE: 

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369 
assumes "Bfun f A" 
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370 
obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) A" 
31355  371 
using assms unfolding Bfun_def by fast 
372 

373 

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374 
subsection {* Convergence to Zero *} 
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375 

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376 
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" 
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377 
where "Zfun f A = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) A)" 
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378 

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379 
lemma ZfunI: 
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380 
"(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A) \<Longrightarrow> Zfun f A" 
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381 
unfolding Zfun_def by simp 
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382 

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383 
lemma ZfunD: 
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384 
"\<lbrakk>Zfun f A; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A" 
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385 
unfolding Zfun_def by simp 
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386 

31355  387 
lemma Zfun_ssubst: 
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388 
"eventually (\<lambda>x. f x = g x) A \<Longrightarrow> Zfun g A \<Longrightarrow> Zfun f A" 
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389 
unfolding Zfun_def by (auto elim!: eventually_rev_mp) 
31355  390 

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391 
lemma Zfun_zero: "Zfun (\<lambda>x. 0) A" 
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392 
unfolding Zfun_def by simp 
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393 

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394 
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) A = Zfun (\<lambda>x. f x) A" 
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395 
unfolding Zfun_def by simp 
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396 

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397 
lemma Zfun_imp_Zfun: 
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398 
assumes f: "Zfun f A" 
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399 
assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) A" 
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400 
shows "Zfun (\<lambda>x. g x) A" 
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401 
proof (cases) 
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402 
assume K: "0 < K" 
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403 
show ?thesis 
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404 
proof (rule ZfunI) 
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405 
fix r::real assume "0 < r" 
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406 
hence "0 < r / K" 
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407 
using K by (rule divide_pos_pos) 
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408 
then have "eventually (\<lambda>x. norm (f x) < r / K) A" 
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409 
using ZfunD [OF f] by fast 
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410 
with g show "eventually (\<lambda>x. norm (g x) < r) A" 
31355  411 
proof (rule eventually_elim2) 
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412 
fix x 
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413 
assume *: "norm (g x) \<le> norm (f x) * K" 
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414 
assume "norm (f x) < r / K" 
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415 
hence "norm (f x) * K < r" 
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416 
by (simp add: pos_less_divide_eq K) 
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417 
thus "norm (g x) < r" 
31355  418 
by (simp add: order_le_less_trans [OF *]) 
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419 
qed 
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420 
qed 
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421 
next 
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422 
assume "\<not> 0 < K" 
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423 
hence K: "K \<le> 0" by (simp only: not_less) 
31355  424 
show ?thesis 
425 
proof (rule ZfunI) 

426 
fix r :: real 

427 
assume "0 < r" 

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428 
from g show "eventually (\<lambda>x. norm (g x) < r) A" 
31355  429 
proof (rule eventually_elim1) 
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430 
fix x 
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431 
assume "norm (g x) \<le> norm (f x) * K" 
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432 
also have "\<dots> \<le> norm (f x) * 0" 
31355  433 
using K norm_ge_zero by (rule mult_left_mono) 
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434 
finally show "norm (g x) < r" 
31355  435 
using `0 < r` by simp 
436 
qed 

437 
qed 

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438 
qed 
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439 

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440 
lemma Zfun_le: "\<lbrakk>Zfun g A; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f A" 
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441 
by (erule_tac K="1" in Zfun_imp_Zfun, simp) 
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442 

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443 
lemma Zfun_add: 
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444 
assumes f: "Zfun f A" and g: "Zfun g A" 
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445 
shows "Zfun (\<lambda>x. f x + g x) A" 
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446 
proof (rule ZfunI) 
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447 
fix r::real assume "0 < r" 
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448 
hence r: "0 < r / 2" by simp 
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449 
have "eventually (\<lambda>x. norm (f x) < r/2) A" 
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diff
changeset

450 
using f r by (rule ZfunD) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

451 
moreover 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

452 
have "eventually (\<lambda>x. norm (g x) < r/2) A" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

453 
using g r by (rule ZfunD) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

454 
ultimately 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

455 
show "eventually (\<lambda>x. norm (f x + g x) < r) A" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

456 
proof (rule eventually_elim2) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

457 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

458 
assume *: "norm (f x) < r/2" "norm (g x) < r/2" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

459 
have "norm (f x + g x) \<le> norm (f x) + norm (g x)" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

460 
by (rule norm_triangle_ineq) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

461 
also have "\<dots> < r/2 + r/2" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

462 
using * by (rule add_strict_mono) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

463 
finally show "norm (f x + g x) < r" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

464 
by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

465 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

466 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

467 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

468 
lemma Zfun_minus: "Zfun f A \<Longrightarrow> Zfun (\<lambda>x.  f x) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

469 
unfolding Zfun_def by simp 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

470 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

471 
lemma Zfun_diff: "\<lbrakk>Zfun f A; Zfun g A\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x  g x) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

472 
by (simp only: diff_minus Zfun_add Zfun_minus) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

473 

2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

474 
lemma (in bounded_linear) Zfun: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

475 
assumes g: "Zfun g A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

476 
shows "Zfun (\<lambda>x. f (g x)) A" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

477 
proof  
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

478 
obtain K where "\<And>x. norm (f x) \<le> norm x * K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

479 
using bounded by fast 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

480 
then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) A" 
31355  481 
by simp 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

482 
with g show ?thesis 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

483 
by (rule Zfun_imp_Zfun) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

484 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

485 

2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

486 
lemma (in bounded_bilinear) Zfun: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

487 
assumes f: "Zfun f A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

488 
assumes g: "Zfun g A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

489 
shows "Zfun (\<lambda>x. f x ** g x) A" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

490 
proof (rule ZfunI) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

491 
fix r::real assume r: "0 < r" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

492 
obtain K where K: "0 < K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

493 
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

494 
using pos_bounded by fast 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

495 
from K have K': "0 < inverse K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

496 
by (rule positive_imp_inverse_positive) 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

497 
have "eventually (\<lambda>x. norm (f x) < r) A" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

498 
using f r by (rule ZfunD) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

499 
moreover 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

500 
have "eventually (\<lambda>x. norm (g x) < inverse K) A" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

501 
using g K' by (rule ZfunD) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

502 
ultimately 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

503 
show "eventually (\<lambda>x. norm (f x ** g x) < r) A" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

504 
proof (rule eventually_elim2) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

505 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

506 
assume *: "norm (f x) < r" "norm (g x) < inverse K" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

507 
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

508 
by (rule norm_le) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

509 
also have "norm (f x) * norm (g x) * K < r * inverse K * K" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

510 
by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

511 
also from K have "r * inverse K * K = r" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

512 
by simp 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

513 
finally show "norm (f x ** g x) < r" . 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

514 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

515 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

516 

2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

517 
lemma (in bounded_bilinear) Zfun_left: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

518 
"Zfun f A \<Longrightarrow> Zfun (\<lambda>x. f x ** a) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

519 
by (rule bounded_linear_left [THEN bounded_linear.Zfun]) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

520 

2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

521 
lemma (in bounded_bilinear) Zfun_right: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

522 
"Zfun f A \<Longrightarrow> Zfun (\<lambda>x. a ** f x) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

523 
by (rule bounded_linear_right [THEN bounded_linear.Zfun]) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

524 

2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

525 
lemmas Zfun_mult = mult.Zfun 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

526 
lemmas Zfun_mult_right = mult.Zfun_right 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

527 
lemmas Zfun_mult_left = mult.Zfun_left 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

528 

2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

529 

31902  530 
subsection {* Limits *} 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

531 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

532 
definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool" 
37767  533 
(infixr ">" 55) where 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

534 
"(f > l) A \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) A)" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

535 

31902  536 
ML {* 
537 
structure Tendsto_Intros = Named_Thms 

538 
( 

539 
val name = "tendsto_intros" 

540 
val description = "introduction rules for tendsto" 

541 
) 

31565  542 
*} 
543 

31902  544 
setup Tendsto_Intros.setup 
31565  545 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

546 
lemma tendsto_mono: "A \<le> A' \<Longrightarrow> (f > l) A' \<Longrightarrow> (f > l) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

547 
unfolding tendsto_def le_filter_def by fast 
36656
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset

548 

31488  549 
lemma topological_tendstoI: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

550 
"(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

551 
\<Longrightarrow> (f > l) A" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

552 
unfolding tendsto_def by auto 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

553 

31488  554 
lemma topological_tendstoD: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

555 
"(f > l) A \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A" 
31488  556 
unfolding tendsto_def by auto 
557 

558 
lemma tendstoI: 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

559 
assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

560 
shows "(f > l) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

561 
apply (rule topological_tendstoI) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

562 
apply (simp add: open_dist) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

563 
apply (drule (1) bspec, clarify) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

564 
apply (drule assms) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

565 
apply (erule eventually_elim1, simp) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

566 
done 
31488  567 

31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

568 
lemma tendstoD: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

569 
"(f > l) A \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

570 
apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

571 
apply (clarsimp simp add: open_dist) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

572 
apply (rule_tac x="e  dist x l" in exI, clarsimp) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

573 
apply (simp only: less_diff_eq) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

574 
apply (erule le_less_trans [OF dist_triangle]) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

575 
apply simp 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

576 
apply simp 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

577 
done 
31488  578 

579 
lemma tendsto_iff: 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

580 
"(f > l) A \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) A)" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

581 
using tendstoI tendstoD by fast 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

582 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

583 
lemma tendsto_Zfun_iff: "(f > a) A = Zfun (\<lambda>x. f x  a) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

584 
by (simp only: tendsto_iff Zfun_def dist_norm) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

585 

31565  586 
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) > a) (at a)" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

587 
unfolding tendsto_def eventually_at_topological by auto 
31565  588 

589 
lemma tendsto_ident_at_within [tendsto_intros]: 

36655  590 
"((\<lambda>x. x) > a) (at a within S)" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

591 
unfolding tendsto_def eventually_within eventually_at_topological by auto 
31565  592 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

593 
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) > k) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

594 
by (simp add: tendsto_def) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

595 

36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

596 
lemma tendsto_const_iff: 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

597 
fixes k l :: "'a::metric_space" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

598 
assumes "A \<noteq> bot" shows "((\<lambda>n. k) > l) A \<longleftrightarrow> k = l" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

599 
apply (safe intro!: tendsto_const) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

600 
apply (rule ccontr) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

601 
apply (drule_tac e="dist k l" in tendstoD) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

602 
apply (simp add: zero_less_dist_iff) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

603 
apply (simp add: eventually_False assms) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

604 
done 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

605 

31565  606 
lemma tendsto_dist [tendsto_intros]: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

607 
assumes f: "(f > l) A" and g: "(g > m) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

608 
shows "((\<lambda>x. dist (f x) (g x)) > dist l m) A" 
31565  609 
proof (rule tendstoI) 
610 
fix e :: real assume "0 < e" 

611 
hence e2: "0 < e/2" by simp 

612 
from tendstoD [OF f e2] tendstoD [OF g e2] 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

613 
show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) A" 
31565  614 
proof (rule eventually_elim2) 
615 
fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2" 

616 
then show "dist (dist (f x) (g x)) (dist l m) < e" 

617 
unfolding dist_real_def 

618 
using dist_triangle2 [of "f x" "g x" "l"] 

619 
using dist_triangle2 [of "g x" "l" "m"] 

620 
using dist_triangle3 [of "l" "m" "f x"] 

621 
using dist_triangle [of "f x" "m" "g x"] 

622 
by arith 

623 
qed 

624 
qed 

625 

36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

626 
lemma norm_conv_dist: "norm x = dist x 0" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

627 
unfolding dist_norm by simp 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

628 

31565  629 
lemma tendsto_norm [tendsto_intros]: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

630 
"(f > a) A \<Longrightarrow> ((\<lambda>x. norm (f x)) > norm a) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

631 
unfolding norm_conv_dist by (intro tendsto_intros) 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

632 

621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

633 
lemma tendsto_norm_zero: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

634 
"(f > 0) A \<Longrightarrow> ((\<lambda>x. norm (f x)) > 0) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

635 
by (drule tendsto_norm, simp) 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

636 

621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

637 
lemma tendsto_norm_zero_cancel: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

638 
"((\<lambda>x. norm (f x)) > 0) A \<Longrightarrow> (f > 0) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

639 
unfolding tendsto_iff dist_norm by simp 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

640 

621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

641 
lemma tendsto_norm_zero_iff: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

642 
"((\<lambda>x. norm (f x)) > 0) A \<longleftrightarrow> (f > 0) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

643 
unfolding tendsto_iff dist_norm by simp 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

644 

31565  645 
lemma tendsto_add [tendsto_intros]: 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

646 
fixes a b :: "'a::real_normed_vector" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

647 
shows "\<lbrakk>(f > a) A; (g > b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) > a + b) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

648 
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

649 

31565  650 
lemma tendsto_minus [tendsto_intros]: 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

651 
fixes a :: "'a::real_normed_vector" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

652 
shows "(f > a) A \<Longrightarrow> ((\<lambda>x.  f x) >  a) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

653 
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

654 

2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

655 
lemma tendsto_minus_cancel: 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

656 
fixes a :: "'a::real_normed_vector" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

657 
shows "((\<lambda>x.  f x) >  a) A \<Longrightarrow> (f > a) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

658 
by (drule tendsto_minus, simp) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

659 

31565  660 
lemma tendsto_diff [tendsto_intros]: 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

661 
fixes a b :: "'a::real_normed_vector" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

662 
shows "\<lbrakk>(f > a) A; (g > b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x  g x) > a  b) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

663 
by (simp add: diff_minus tendsto_add tendsto_minus) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

664 

31588  665 
lemma tendsto_setsum [tendsto_intros]: 
666 
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector" 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

667 
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i > a i) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

668 
shows "((\<lambda>x. \<Sum>i\<in>S. f i x) > (\<Sum>i\<in>S. a i)) A" 
31588  669 
proof (cases "finite S") 
670 
assume "finite S" thus ?thesis using assms 

671 
proof (induct set: finite) 

672 
case empty show ?case 

673 
by (simp add: tendsto_const) 

674 
next 

675 
case (insert i F) thus ?case 

676 
by (simp add: tendsto_add) 

677 
qed 

678 
next 

679 
assume "\<not> finite S" thus ?thesis 

680 
by (simp add: tendsto_const) 

681 
qed 

682 

31565  683 
lemma (in bounded_linear) tendsto [tendsto_intros]: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

684 
"(g > a) A \<Longrightarrow> ((\<lambda>x. f (g x)) > f a) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

685 
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

686 

31565  687 
lemma (in bounded_bilinear) tendsto [tendsto_intros]: 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

688 
"\<lbrakk>(f > a) A; (g > b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) > a ** b) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

689 
by (simp only: tendsto_Zfun_iff prod_diff_prod 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

690 
Zfun_add Zfun Zfun_left Zfun_right) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

691 

31355  692 

693 
subsection {* Continuity of Inverse *} 

694 

695 
lemma (in bounded_bilinear) Zfun_prod_Bfun: 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

696 
assumes f: "Zfun f A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

697 
assumes g: "Bfun g A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

698 
shows "Zfun (\<lambda>x. f x ** g x) A" 
31355  699 
proof  
700 
obtain K where K: "0 \<le> K" 

701 
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K" 

702 
using nonneg_bounded by fast 

703 
obtain B where B: "0 < B" 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

704 
and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) A" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

705 
using g by (rule BfunE) 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

706 
have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) A" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

707 
using norm_g proof (rule eventually_elim1) 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

708 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

709 
assume *: "norm (g x) \<le> B" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

710 
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" 
31355  711 
by (rule norm_le) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

712 
also have "\<dots> \<le> norm (f x) * B * K" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

713 
by (intro mult_mono' order_refl norm_g norm_ge_zero 
31355  714 
mult_nonneg_nonneg K *) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

715 
also have "\<dots> = norm (f x) * (B * K)" 
31355  716 
by (rule mult_assoc) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

717 
finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" . 
31355  718 
qed 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

719 
with f show ?thesis 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

720 
by (rule Zfun_imp_Zfun) 
31355  721 
qed 
722 

723 
lemma (in bounded_bilinear) flip: 

724 
"bounded_bilinear (\<lambda>x y. y ** x)" 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

725 
apply default 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

726 
apply (rule add_right) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

727 
apply (rule add_left) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

728 
apply (rule scaleR_right) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

729 
apply (rule scaleR_left) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

730 
apply (subst mult_commute) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

731 
using bounded by fast 
31355  732 

733 
lemma (in bounded_bilinear) Bfun_prod_Zfun: 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

734 
assumes f: "Bfun f A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

735 
assumes g: "Zfun g A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

736 
shows "Zfun (\<lambda>x. f x ** g x) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

737 
using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun) 
31355  738 

739 
lemma Bfun_inverse_lemma: 

740 
fixes x :: "'a::real_normed_div_algebra" 

741 
shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r" 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

742 
apply (subst nonzero_norm_inverse, clarsimp) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

743 
apply (erule (1) le_imp_inverse_le) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

744 
done 
31355  745 

746 
lemma Bfun_inverse: 

747 
fixes a :: "'a::real_normed_div_algebra" 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

748 
assumes f: "(f > a) A" 
31355  749 
assumes a: "a \<noteq> 0" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

750 
shows "Bfun (\<lambda>x. inverse (f x)) A" 
31355  751 
proof  
752 
from a have "0 < norm a" by simp 

753 
hence "\<exists>r>0. r < norm a" by (rule dense) 

754 
then obtain r where r1: "0 < r" and r2: "r < norm a" by fast 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

755 
have "eventually (\<lambda>x. dist (f x) a < r) A" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

756 
using tendstoD [OF f r1] by fast 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

757 
hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a  r)) A" 
31355  758 
proof (rule eventually_elim1) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

759 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

760 
assume "dist (f x) a < r" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

761 
hence 1: "norm (f x  a) < r" 
31355  762 
by (simp add: dist_norm) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

763 
hence 2: "f x \<noteq> 0" using r2 by auto 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

764 
hence "norm (inverse (f x)) = inverse (norm (f x))" 
31355  765 
by (rule nonzero_norm_inverse) 
766 
also have "\<dots> \<le> inverse (norm a  r)" 

767 
proof (rule le_imp_inverse_le) 

768 
show "0 < norm a  r" using r2 by simp 

769 
next 

31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

770 
have "norm a  norm (f x) \<le> norm (a  f x)" 
31355  771 
by (rule norm_triangle_ineq2) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

772 
also have "\<dots> = norm (f x  a)" 
31355  773 
by (rule norm_minus_commute) 
774 
also have "\<dots> < r" using 1 . 

31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

775 
finally show "norm a  r \<le> norm (f x)" by simp 
31355  776 
qed 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

777 
finally show "norm (inverse (f x)) \<le> inverse (norm a  r)" . 
31355  778 
qed 
779 
thus ?thesis by (rule BfunI) 

780 
qed 

781 

782 
lemma tendsto_inverse_lemma: 

783 
fixes a :: "'a::real_normed_div_algebra" 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

784 
shows "\<lbrakk>(f > a) A; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) A\<rbrakk> 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

785 
\<Longrightarrow> ((\<lambda>x. inverse (f x)) > inverse a) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

786 
apply (subst tendsto_Zfun_iff) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
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diff
changeset

787 
apply (rule Zfun_ssubst) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
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diff
changeset

788 
apply (erule eventually_elim1) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
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diff
changeset

789 
apply (erule (1) inverse_diff_inverse) 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

790 
apply (rule Zfun_minus) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

791 
apply (rule Zfun_mult_left) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
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diff
changeset

792 
apply (rule mult.Bfun_prod_Zfun) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

793 
apply (erule (1) Bfun_inverse) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

794 
apply (simp add: tendsto_Zfun_iff) 
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

795 
done 
31355  796 

31565  797 
lemma tendsto_inverse [tendsto_intros]: 
31355  798 
fixes a :: "'a::real_normed_div_algebra" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

799 
assumes f: "(f > a) A" 
31355  800 
assumes a: "a \<noteq> 0" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

801 
shows "((\<lambda>x. inverse (f x)) > inverse a) A" 
31355  802 
proof  
803 
from a have "0 < norm a" by simp 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

804 
with f have "eventually (\<lambda>x. dist (f x) a < norm a) A" 
31355  805 
by (rule tendstoD) 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

806 
then have "eventually (\<lambda>x. f x \<noteq> 0) A" 
31355  807 
unfolding dist_norm by (auto elim!: eventually_elim1) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

808 
with f a show ?thesis 
31355  809 
by (rule tendsto_inverse_lemma) 
810 
qed 

811 

31565  812 
lemma tendsto_divide [tendsto_intros]: 
31355  813 
fixes a b :: "'a::real_normed_field" 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

814 
shows "\<lbrakk>(f > a) A; (g > b) A; b \<noteq> 0\<rbrakk> 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

815 
\<Longrightarrow> ((\<lambda>x. f x / g x) > a / b) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

816 
by (simp add: mult.tendsto tendsto_inverse divide_inverse) 
31355  817 

41970  818 
lemma tendsto_unique: 
819 
fixes f :: "'a \<Rightarrow> 'b::t2_space" 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

820 
assumes "\<not> trivial_limit A" "(f > l) A" "(f > l') A" 
41970  821 
shows "l = l'" 
822 
proof (rule ccontr) 

823 
assume "l \<noteq> l'" 

824 
obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}" 

825 
using hausdorff [OF `l \<noteq> l'`] by fast 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

826 
have "eventually (\<lambda>x. f x \<in> U) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

827 
using `(f > l) A` `open U` `l \<in> U` by (rule topological_tendstoD) 
41970  828 
moreover 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

829 
have "eventually (\<lambda>x. f x \<in> V) A" 
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

830 
using `(f > l') A` `open V` `l' \<in> V` by (rule topological_tendstoD) 
41970  831 
ultimately 
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

832 
have "eventually (\<lambda>x. False) A" 
41970  833 
proof (rule eventually_elim2) 
834 
fix x 

835 
assume "f x \<in> U" "f x \<in> V" 

836 
hence "f x \<in> U \<inter> V" by simp 

837 
with `U \<inter> V = {}` show "False" by simp 

838 
qed 

44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset

839 
with `\<not> trivial_limit A` show "False" 
41970  840 
by (simp add: trivial_limit_def) 
841 
qed 

842 

31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

843 
end 