author  huffman 
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parent 36656  fec55067ae9b 
child 36822  38a480e0346f 
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 RComplete 
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begin 
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31392  11 
subsection {* Nets *} 
12 

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text {* 

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A net is now defined simply as a filter on a set. 
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The definition also allows nonproper filters. 
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*} 
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locale is_filter = 
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fixes net :: "('a \<Rightarrow> bool) \<Rightarrow> bool" 
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assumes True: "net (\<lambda>x. True)" 
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assumes conj: "net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x) \<Longrightarrow> net (\<lambda>x. P x \<and> Q x)" 
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assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x)" 
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31392  24 
typedef (open) 'a net = 
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"{net :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter net}" 
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proof 
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show "(\<lambda>x. True) \<in> ?net" by (auto intro: is_filter.intro) 
31392  28 
qed 
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lemma is_filter_Rep_net: "is_filter (Rep_net net)" 
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using Rep_net [of net] by simp 
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lemma Abs_net_inverse': 

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assumes "is_filter net" shows "Rep_net (Abs_net net) = net" 
31392  35 
using assms by (simp add: Abs_net_inverse) 
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subsection {* Eventually *} 

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definition 
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eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where 
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[code del]: "eventually P net \<longleftrightarrow> Rep_net net P" 
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lemma eventually_Abs_net: 
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assumes "is_filter net" shows "eventually P (Abs_net net) = net P" 
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unfolding eventually_def using assms by (simp add: Abs_net_inverse) 
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lemma expand_net_eq: 
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shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')" 
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unfolding Rep_net_inject [symmetric] expand_fun_eq eventually_def .. 
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) net" 
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unfolding eventually_def 
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by (rule is_filter.True [OF is_filter_Rep_net]) 
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P net" 
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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 net" 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 net \<Longrightarrow> eventually Q net" 
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unfolding eventually_def 
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by (rule is_filter.mono [OF is_filter_Rep_net]) 
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lemma eventually_conj: 
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assumes P: "eventually (\<lambda>x. P x) net" 
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assumes Q: "eventually (\<lambda>x. Q x) net" 

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shows "eventually (\<lambda>x. P x \<and> Q x) net" 

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using assms unfolding eventually_def 
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by (rule is_filter.conj [OF is_filter_Rep_net]) 
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lemma eventually_mp: 
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net" 
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assumes "eventually (\<lambda>x. P x) net" 

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shows "eventually (\<lambda>x. Q x) net" 

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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 
31392  80 
show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net" 
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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) net" 
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net" 

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shows "eventually (\<lambda>x. Q x) net" 

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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) net \<longleftrightarrow> eventually P net \<and> eventually Q net" 
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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) net" 
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assumes "\<And>i. P i \<Longrightarrow> Q i" 
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shows "eventually (\<lambda>i. Q i) net" 
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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) net" 
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assumes "eventually (\<lambda>i. Q i) net" 

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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) net" 
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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 "net \<le> net'"} means that @{term net} is finer than 
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@{term net'}. *} 
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36630  113 
instantiation net :: (type) complete_lattice 
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begin 
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definition 
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le_net_def [code del]: 
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"net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)" 
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definition 
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less_net_def [code del]: 
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"(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net" 
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definition 
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top_net_def [code del]: 
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"top = Abs_net (\<lambda>P. \<forall>x. P x)" 

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definition 

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bot_net_def [code del]: 
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"bot = Abs_net (\<lambda>P. True)" 
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36630  132 
definition 
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sup_net_def [code del]: 

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"sup net net' = Abs_net (\<lambda>P. eventually P net \<and> eventually P net')" 

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definition 

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inf_net_def [code del]: 

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"inf a b = Abs_net 

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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_net_def [code del]: 

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"Sup A = Abs_net (\<lambda>P. \<forall>net\<in>A. eventually P net)" 

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definition 

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Inf_net_def [code del]: 

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"Inf A = Sup {x::'a net. \<forall>y\<in>A. x \<le> y}" 

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lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" 

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unfolding top_net_def 

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by (rule eventually_Abs_net, rule is_filter.intro, auto) 

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lemma eventually_bot [simp]: "eventually P bot" 
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unfolding bot_net_def 
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by (subst eventually_Abs_net, rule is_filter.intro, auto) 
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36630  157 
lemma eventually_sup: 
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"eventually P (sup net net') \<longleftrightarrow> eventually P net \<and> eventually P net'" 

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unfolding sup_net_def 

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by (rule eventually_Abs_net, rule is_filter.intro) 

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(auto elim!: eventually_rev_mp) 

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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_net_def 

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apply (rule eventually_Abs_net, 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 

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lemma eventually_Sup: 

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"eventually P (Sup A) \<longleftrightarrow> (\<forall>net\<in>A. eventually P net)" 

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unfolding Sup_net_def 

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apply (rule eventually_Abs_net, rule is_filter.intro) 

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apply (auto intro: eventually_conj elim!: eventually_rev_mp) 

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done 

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instance proof 
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fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" 
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by (rule less_net_def) 
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next 
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fix x :: "'a net" show "x \<le> x" 
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unfolding le_net_def by simp 
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next 
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fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z" 
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unfolding le_net_def by simp 
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next 
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fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y" 
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unfolding le_net_def expand_net_eq by fast 
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next 
36630  197 
fix x :: "'a net" show "x \<le> top" 
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unfolding le_net_def eventually_top by (simp add: always_eventually) 

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next 

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fix x :: "'a net" show "bot \<le> x" 
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unfolding le_net_def by simp 
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next 
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fix x y :: "'a net" show "x \<le> sup x y" and "y \<le> sup x y" 

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unfolding le_net_def eventually_sup by simp_all 

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next 

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fix x y z :: "'a net" assume "x \<le> z" and "y \<le> z" thus "sup x y \<le> z" 

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unfolding le_net_def eventually_sup by simp 

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next 

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fix x y :: "'a net" show "inf x y \<le> x" and "inf x y \<le> y" 

210 
unfolding le_net_def eventually_inf by (auto intro: eventually_True) 

211 
next 

212 
fix x y z :: "'a net" assume "x \<le> y" and "x \<le> z" thus "x \<le> inf y z" 

213 
unfolding le_net_def eventually_inf 

214 
by (auto elim!: eventually_mono intro: eventually_conj) 

215 
next 

216 
fix x :: "'a net" and A assume "x \<in> A" thus "x \<le> Sup A" 

217 
unfolding le_net_def eventually_Sup by simp 

218 
next 

219 
fix A and y :: "'a net" assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> y" thus "Sup A \<le> y" 

220 
unfolding le_net_def eventually_Sup by simp 

221 
next 

222 
fix z :: "'a net" and A assume "z \<in> A" thus "Inf A \<le> z" 

223 
unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp 

224 
next 

225 
fix A and x :: "'a net" assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" thus "x \<le> Inf A" 

226 
unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp 

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qed 
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end 
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lemma net_leD: 
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"net \<le> net' \<Longrightarrow> eventually P net' \<Longrightarrow> eventually P net" 
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unfolding le_net_def by simp 
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lemma net_leI: 
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"(\<And>P. eventually P net' \<Longrightarrow> eventually P net) \<Longrightarrow> net \<le> net'" 
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lemma eventually_False: 
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"eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot" 
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unfolding expand_net_eq by (auto elim: eventually_rev_mp) 
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36654  244 
subsection {* Map function for nets *} 
245 

246 
definition 

247 
netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net" 

248 
where [code del]: 

249 
"netmap f net = Abs_net (\<lambda>P. eventually (\<lambda>x. P (f x)) net)" 

250 

251 
lemma eventually_netmap: 

252 
"eventually P (netmap f net) = eventually (\<lambda>x. P (f x)) net" 

253 
unfolding netmap_def 

254 
apply (rule eventually_Abs_net) 

255 
apply (rule is_filter.intro) 

256 
apply (auto elim!: eventually_rev_mp) 

257 
done 

258 

259 
lemma netmap_ident: "netmap (\<lambda>x. x) net = net" 

260 
by (simp add: expand_net_eq eventually_netmap) 

261 

262 
lemma netmap_netmap: "netmap f (netmap g net) = netmap (\<lambda>x. f (g x)) net" 

263 
by (simp add: expand_net_eq eventually_netmap) 

264 

265 
lemma netmap_mono: "net \<le> net' \<Longrightarrow> netmap f net \<le> netmap f net'" 

266 
unfolding le_net_def eventually_netmap by simp 

267 

268 
lemma netmap_bot [simp]: "netmap f bot = bot" 

269 
by (simp add: expand_net_eq eventually_netmap) 

270 

271 

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subsection {* Sequentially *} 
31392  273 

274 
definition 

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sequentially :: "nat net" 
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where [code del]: 
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"sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)" 
31392  278 

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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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281 
unfolding sequentially_def 
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proof (rule eventually_Abs_net, rule is_filter.intro) 
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283 
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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289 

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lemma sequentially_bot [simp]: "sequentially \<noteq> bot" 
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unfolding expand_net_eq eventually_sequentially by auto 
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292 

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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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lemma le_sequentially: 
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"net \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) net)" 
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unfolding le_net_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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subsection {* Standard Nets *} 
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31392  305 
definition 
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within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) 
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where [code del]: 
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"net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)" 
31392  309 

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definition 
36654  311 
nhds :: "'a::topological_space \<Rightarrow> 'a net" 
312 
where [code del]: 

313 
"nhds a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" 

314 

315 
definition 

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at :: "'a::topological_space \<Rightarrow> 'a net" 
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where [code del]: 
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"at a = nhds a within  {a}" 
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31392  320 
lemma eventually_within: 
321 
"eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net" 

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322 
unfolding within_def 
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323 
by (rule eventually_Abs_net, rule is_filter.intro) 
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324 
(auto elim!: eventually_rev_mp) 
31392  325 

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lemma within_UNIV: "net within UNIV = net" 
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327 
unfolding expand_net_eq eventually_within by simp 
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36654  329 
lemma eventually_nhds: 
330 
"eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" 

331 
unfolding nhds_def 

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332 
proof (rule eventually_Abs_net, rule is_filter.intro) 
36654  333 
have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp 
334 
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by  rule 

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335 
next 
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336 
fix P Q 
36654  337 
assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" 
338 
and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" 

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339 
then obtain S T where 
36654  340 
"open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" 
341 
"open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto 

342 
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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by (simp add: open_Int) 
36654  344 
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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345 
qed auto 
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346 

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347 
lemma eventually_nhds_metric: 
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"eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)" 
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349 
unfolding eventually_nhds open_dist 
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350 
apply safe 
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351 
apply fast 
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352 
apply (rule_tac x="{x. dist x a < d}" in exI, simp) 
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353 
apply clarsimp 
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apply (rule_tac x="d  dist x a" in exI, clarsimp) 
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355 
apply (simp only: less_diff_eq) 
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356 
apply (erule le_less_trans [OF dist_triangle]) 
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357 
done 
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358 

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359 
lemma eventually_at_topological: 
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"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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361 
unfolding at_def eventually_within eventually_nhds by simp 
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362 

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363 
lemma eventually_at: 
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fixes a :: "'a::metric_space" 
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365 
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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366 
unfolding at_def eventually_within eventually_nhds_metric by auto 
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367 

31392  368 

31355  369 
subsection {* Boundedness *} 
370 

371 
definition 

31392  372 
Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where 
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[code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)" 
31355  374 

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lemma BfunI: 
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assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net" 
31355  377 
unfolding Bfun_def 
378 
proof (intro exI conjI allI) 

379 
show "0 < max K 1" by simp 

380 
next 

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show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net" 
31355  382 
using K by (rule eventually_elim1, simp) 
383 
qed 

384 

385 
lemma BfunE: 

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386 
assumes "Bfun f net" 
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obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net" 
31355  388 
using assms unfolding Bfun_def by fast 
389 

390 

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391 
subsection {* Convergence to Zero *} 
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392 

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definition 
31392  394 
Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where 
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[code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)" 
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396 

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lemma ZfunI: 
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"(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net" 
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unfolding Zfun_def by simp 
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400 

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

402 
"\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

403 
unfolding Zfun_def by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

404 

31355  405 
lemma Zfun_ssubst: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

406 
"eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net" 
31355  407 
unfolding Zfun_def by (auto elim!: eventually_rev_mp) 
408 

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

409 
lemma Zfun_zero: "Zfun (\<lambda>x. 0) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

410 
unfolding Zfun_def by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

411 

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

412 
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

413 
unfolding Zfun_def by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

414 

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

415 
lemma Zfun_imp_Zfun: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

416 
assumes f: "Zfun f net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

417 
assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

418 
shows "Zfun (\<lambda>x. g x) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

419 
proof (cases) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

420 
assume K: "0 < K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

421 
show ?thesis 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

422 
proof (rule ZfunI) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

423 
fix r::real assume "0 < r" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

424 
hence "0 < r / K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

425 
using K by (rule divide_pos_pos) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

426 
then have "eventually (\<lambda>x. norm (f x) < r / K) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

427 
using ZfunD [OF f] by fast 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

428 
with g show "eventually (\<lambda>x. norm (g x) < r) net" 
31355  429 
proof (rule eventually_elim2) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

430 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

431 
assume *: "norm (g x) \<le> norm (f x) * K" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

432 
assume "norm (f x) < r / K" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

433 
hence "norm (f x) * K < r" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

434 
by (simp add: pos_less_divide_eq K) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

435 
thus "norm (g x) < r" 
31355  436 
by (simp add: order_le_less_trans [OF *]) 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

437 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

438 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

439 
next 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

440 
assume "\<not> 0 < K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

441 
hence K: "K \<le> 0" by (simp only: not_less) 
31355  442 
show ?thesis 
443 
proof (rule ZfunI) 

444 
fix r :: real 

445 
assume "0 < r" 

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

446 
from g show "eventually (\<lambda>x. norm (g x) < r) net" 
31355  447 
proof (rule eventually_elim1) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

448 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

449 
assume "norm (g x) \<le> norm (f x) * K" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

450 
also have "\<dots> \<le> norm (f x) * 0" 
31355  451 
using K norm_ge_zero by (rule mult_left_mono) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

452 
finally show "norm (g x) < r" 
31355  453 
using `0 < r` by simp 
454 
qed 

455 
qed 

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

456 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

457 

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

458 
lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

459 
by (erule_tac K="1" in Zfun_imp_Zfun, simp) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

460 

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

461 
lemma Zfun_add: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

462 
assumes f: "Zfun f net" and g: "Zfun g net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

463 
shows "Zfun (\<lambda>x. f x + g x) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

464 
proof (rule ZfunI) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

465 
fix r::real assume "0 < r" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

466 
hence r: "0 < r / 2" by simp 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

467 
have "eventually (\<lambda>x. norm (f x) < r/2) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

468 
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

469 
moreover 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

470 
have "eventually (\<lambda>x. norm (g x) < r/2) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

471 
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

472 
ultimately 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

473 
show "eventually (\<lambda>x. norm (f x + g x) < r) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

474 
proof (rule eventually_elim2) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

475 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

476 
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

477 
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

478 
by (rule norm_triangle_ineq) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

479 
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

480 
using * by (rule add_strict_mono) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

481 
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

482 
by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

483 
qed 
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 

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

486 
lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x.  f x) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

487 
unfolding Zfun_def by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

488 

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

489 
lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x  g x) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

490 
by (simp only: diff_minus Zfun_add Zfun_minus) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

491 

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

492 
lemma (in bounded_linear) Zfun: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

493 
assumes g: "Zfun g net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

494 
shows "Zfun (\<lambda>x. f (g x)) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

495 
proof  
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

496 
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

497 
using bounded by fast 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

498 
then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net" 
31355  499 
by simp 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

500 
with g show ?thesis 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

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

502 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

503 

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

504 
lemma (in bounded_bilinear) Zfun: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

505 
assumes f: "Zfun f net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

506 
assumes g: "Zfun g net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

507 
shows "Zfun (\<lambda>x. f x ** g x) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

508 
proof (rule ZfunI) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

509 
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

510 
obtain K where K: "0 < K" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

511 
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

512 
using pos_bounded by fast 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

513 
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

514 
by (rule positive_imp_inverse_positive) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

515 
have "eventually (\<lambda>x. norm (f x) < r) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

516 
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

517 
moreover 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

518 
have "eventually (\<lambda>x. norm (g x) < inverse K) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

519 
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

520 
ultimately 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

521 
show "eventually (\<lambda>x. norm (f x ** g x) < r) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

522 
proof (rule eventually_elim2) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

523 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

524 
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

525 
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

526 
by (rule norm_le) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

527 
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

528 
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

529 
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

530 
by simp 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

531 
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

532 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

533 
qed 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

534 

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

535 
lemma (in bounded_bilinear) Zfun_left: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

536 
"Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

537 
by (rule bounded_linear_left [THEN bounded_linear.Zfun]) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

538 

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

539 
lemma (in bounded_bilinear) Zfun_right: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

540 
"Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

541 
by (rule bounded_linear_right [THEN bounded_linear.Zfun]) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

542 

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

543 
lemmas Zfun_mult = mult.Zfun 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

544 
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

545 
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

546 

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

547 

31902  548 
subsection {* Limits *} 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

549 

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

550 
definition 
31488  551 
tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool" 
552 
(infixr ">" 55) 

553 
where [code del]: 

31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31488
diff
changeset

554 
"(f > l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

555 

31902  556 
ML {* 
557 
structure Tendsto_Intros = Named_Thms 

558 
( 

559 
val name = "tendsto_intros" 

560 
val description = "introduction rules for tendsto" 

561 
) 

31565  562 
*} 
563 

31902  564 
setup Tendsto_Intros.setup 
31565  565 

36656
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset

566 
lemma tendsto_mono: "net \<le> net' \<Longrightarrow> (f > l) net' \<Longrightarrow> (f > l) net" 
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset

567 
unfolding tendsto_def le_net_def by fast 
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset

568 

31488  569 
lemma topological_tendstoI: 
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31488
diff
changeset

570 
"(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

571 
\<Longrightarrow> (f > l) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

572 
unfolding tendsto_def by auto 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

573 

31488  574 
lemma topological_tendstoD: 
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31488
diff
changeset

575 
"(f > l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net" 
31488  576 
unfolding tendsto_def by auto 
577 

578 
lemma tendstoI: 

579 
assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net" 

580 
shows "(f > l) net" 

581 
apply (rule topological_tendstoI) 

31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31488
diff
changeset

582 
apply (simp add: open_dist) 
31488  583 
apply (drule (1) bspec, clarify) 
584 
apply (drule assms) 

585 
apply (erule eventually_elim1, simp) 

586 
done 

587 

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

588 
lemma tendstoD: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

589 
"(f > l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net" 
31488  590 
apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD) 
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31488
diff
changeset

591 
apply (clarsimp simp add: open_dist) 
31488  592 
apply (rule_tac x="e  dist x l" in exI, clarsimp) 
593 
apply (simp only: less_diff_eq) 

594 
apply (erule le_less_trans [OF dist_triangle]) 

595 
apply simp 

596 
apply simp 

597 
done 

598 

599 
lemma tendsto_iff: 

600 
"(f > l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)" 

601 
using tendstoI tendstoD by fast 

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

602 

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

603 
lemma tendsto_Zfun_iff: "(f > a) net = Zfun (\<lambda>x. f x  a) net" 
31488  604 
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

605 

31565  606 
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) > a) (at a)" 
607 
unfolding tendsto_def eventually_at_topological by auto 

608 

609 
lemma tendsto_ident_at_within [tendsto_intros]: 

36655  610 
"((\<lambda>x. x) > a) (at a within S)" 
31565  611 
unfolding tendsto_def eventually_within eventually_at_topological by auto 
612 

613 
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) > k) net" 

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

614 
by (simp add: tendsto_def) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

615 

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

616 
lemma tendsto_const_iff: 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

617 
fixes k l :: "'a::metric_space" 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

618 
assumes "net \<noteq> bot" shows "((\<lambda>n. k) > l) net \<longleftrightarrow> k = l" 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

619 
apply (safe intro!: tendsto_const) 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

620 
apply (rule ccontr) 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

621 
apply (drule_tac e="dist k l" in tendstoD) 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

622 
apply (simp add: zero_less_dist_iff) 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

623 
apply (simp add: eventually_False assms) 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

624 
done 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

625 

31565  626 
lemma tendsto_dist [tendsto_intros]: 
627 
assumes f: "(f > l) net" and g: "(g > m) net" 

628 
shows "((\<lambda>x. dist (f x) (g x)) > dist l m) net" 

629 
proof (rule tendstoI) 

630 
fix e :: real assume "0 < e" 

631 
hence e2: "0 < e/2" by simp 

632 
from tendstoD [OF f e2] tendstoD [OF g e2] 

633 
show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net" 

634 
proof (rule eventually_elim2) 

635 
fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2" 

636 
then show "dist (dist (f x) (g x)) (dist l m) < e" 

637 
unfolding dist_real_def 

638 
using dist_triangle2 [of "f x" "g x" "l"] 

639 
using dist_triangle2 [of "g x" "l" "m"] 

640 
using dist_triangle3 [of "l" "m" "f x"] 

641 
using dist_triangle [of "f x" "m" "g x"] 

642 
by arith 

643 
qed 

644 
qed 

645 

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

646 
lemma norm_conv_dist: "norm x = dist x 0" 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

647 
unfolding dist_norm by simp 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

648 

31565  649 
lemma tendsto_norm [tendsto_intros]: 
650 
"(f > a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) > norm a) net" 

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

651 
unfolding norm_conv_dist by (intro tendsto_intros) 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

652 

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

653 
lemma tendsto_norm_zero: 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

654 
"(f > 0) net \<Longrightarrow> ((\<lambda>x. norm (f x)) > 0) net" 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

655 
by (drule tendsto_norm, simp) 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

656 

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

657 
lemma tendsto_norm_zero_cancel: 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

658 
"((\<lambda>x. norm (f x)) > 0) net \<Longrightarrow> (f > 0) net" 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

659 
unfolding tendsto_iff dist_norm by simp 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

660 

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

661 
lemma tendsto_norm_zero_iff: 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

662 
"((\<lambda>x. norm (f x)) > 0) net \<longleftrightarrow> (f > 0) net" 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset

663 
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

664 

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

665 
lemma add_diff_add: 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

666 
fixes a b c d :: "'a::ab_group_add" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

667 
shows "(a + c)  (b + d) = (a  b) + (c  d)" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

668 
by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

669 

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

670 
lemma minus_diff_minus: 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

671 
fixes a b :: "'a::ab_group_add" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

672 
shows "( a)  ( b) =  (a  b)" 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

673 
by simp 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

674 

31565  675 
lemma tendsto_add [tendsto_intros]: 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

676 
fixes a b :: "'a::real_normed_vector" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

677 
shows "\<lbrakk>(f > a) net; (g > b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) > a + b) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

678 
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

679 

31565  680 
lemma tendsto_minus [tendsto_intros]: 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

681 
fixes a :: "'a::real_normed_vector" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

682 
shows "(f > a) net \<Longrightarrow> ((\<lambda>x.  f x) >  a) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

683 
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

684 

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

685 
lemma tendsto_minus_cancel: 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

686 
fixes a :: "'a::real_normed_vector" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

687 
shows "((\<lambda>x.  f x) >  a) net \<Longrightarrow> (f > a) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

688 
by (drule tendsto_minus, simp) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

689 

31565  690 
lemma tendsto_diff [tendsto_intros]: 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

691 
fixes a b :: "'a::real_normed_vector" 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

692 
shows "\<lbrakk>(f > a) net; (g > b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x  g x) > a  b) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

693 
by (simp add: diff_minus tendsto_add tendsto_minus) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

694 

31588  695 
lemma tendsto_setsum [tendsto_intros]: 
696 
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector" 

697 
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i > a i) net" 

698 
shows "((\<lambda>x. \<Sum>i\<in>S. f i x) > (\<Sum>i\<in>S. a i)) net" 

699 
proof (cases "finite S") 

700 
assume "finite S" thus ?thesis using assms 

701 
proof (induct set: finite) 

702 
case empty show ?case 

703 
by (simp add: tendsto_const) 

704 
next 

705 
case (insert i F) thus ?case 

706 
by (simp add: tendsto_add) 

707 
qed 

708 
next 

709 
assume "\<not> finite S" thus ?thesis 

710 
by (simp add: tendsto_const) 

711 
qed 

712 

31565  713 
lemma (in bounded_linear) tendsto [tendsto_intros]: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

714 
"(g > a) net \<Longrightarrow> ((\<lambda>x. f (g x)) > f a) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

715 
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

716 

31565  717 
lemma (in bounded_bilinear) tendsto [tendsto_intros]: 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

718 
"\<lbrakk>(f > a) net; (g > b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) > a ** b) net" 
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

719 
by (simp only: tendsto_Zfun_iff prod_diff_prod 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

720 
Zfun_add Zfun Zfun_left Zfun_right) 
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset

721 

31355  722 

723 
subsection {* Continuity of Inverse *} 

724 

725 
lemma (in bounded_bilinear) Zfun_prod_Bfun: 

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

726 
assumes f: "Zfun f net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

727 
assumes g: "Bfun g net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

728 
shows "Zfun (\<lambda>x. f x ** g x) net" 
31355  729 
proof  
730 
obtain K where K: "0 \<le> K" 

731 
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K" 

732 
using nonneg_bounded by fast 

733 
obtain B where B: "0 < B" 

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

734 
and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

735 
using g by (rule BfunE) 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

736 
have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

737 
using norm_g proof (rule eventually_elim1) 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

738 
fix x 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

739 
assume *: "norm (g x) \<le> B" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

740 
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" 
31355  741 
by (rule norm_le) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

742 
also have "\<dots> \<le> norm (f x) * B * K" 
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

743 
by (intro mult_mono' order_refl norm_g norm_ge_zero 
31355  744 
mult_nonneg_nonneg K *) 
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset

745 
also have "\<dots> = norm (f x) * (B * K)" 
31355  746 
by (rule mult_assoc) 
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747 
finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" . 
31355  748 
qed 
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749 
with f show ?thesis 
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750 
by (rule Zfun_imp_Zfun) 
31355  751 
qed 
752 

753 
lemma (in bounded_bilinear) flip: 

754 
"bounded_bilinear (\<lambda>x y. y ** x)" 

755 
apply default 

756 
apply (rule add_right) 

757 
apply (rule add_left) 

758 
apply (rule scaleR_right) 

759 
apply (rule scaleR_left) 

760 
apply (subst mult_commute) 

761 
using bounded by fast 

762 

763 
lemma (in bounded_bilinear) Bfun_prod_Zfun: 

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764 
assumes f: "Bfun f net" 
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765 
assumes g: "Zfun g net" 
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766 
shows "Zfun (\<lambda>x. f x ** g x) net" 
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767 
using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun) 
31355  768 

769 
lemma inverse_diff_inverse: 

770 
"\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk> 

771 
\<Longrightarrow> inverse a  inverse b =  (inverse a * (a  b) * inverse b)" 

772 
by (simp add: algebra_simps) 

773 

774 
lemma Bfun_inverse_lemma: 

775 
fixes x :: "'a::real_normed_div_algebra" 

776 
shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r" 

777 
apply (subst nonzero_norm_inverse, clarsimp) 

778 
apply (erule (1) le_imp_inverse_le) 

779 
done 

780 

781 
lemma Bfun_inverse: 

782 
fixes a :: "'a::real_normed_div_algebra" 

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783 
assumes f: "(f > a) net" 
31355  784 
assumes a: "a \<noteq> 0" 
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785 
shows "Bfun (\<lambda>x. inverse (f x)) net" 
31355  786 
proof  
787 
from a have "0 < norm a" by simp 

788 
hence "\<exists>r>0. r < norm a" by (rule dense) 

789 
then obtain r where r1: "0 < r" and r2: "r < norm a" by fast 

31487
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790 
have "eventually (\<lambda>x. dist (f x) a < r) net" 
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changeset

791 
using tendstoD [OF f r1] by fast 
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792 
hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a  r)) net" 
31355  793 
proof (rule eventually_elim1) 
31487
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794 
fix x 
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changeset

795 
assume "dist (f x) a < r" 
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796 
hence 1: "norm (f x  a) < r" 
31355  797 
by (simp add: dist_norm) 
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798 
hence 2: "f x \<noteq> 0" using r2 by auto 
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799 
hence "norm (inverse (f x)) = inverse (norm (f x))" 
31355  800 
by (rule nonzero_norm_inverse) 
801 
also have "\<dots> \<le> inverse (norm a  r)" 

802 
proof (rule le_imp_inverse_le) 

803 
show "0 < norm a  r" using r2 by simp 

804 
next 

31487
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805 
have "norm a  norm (f x) \<le> norm (a  f x)" 
31355  806 
by (rule norm_triangle_ineq2) 
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807 
also have "\<dots> = norm (f x  a)" 
31355  808 
by (rule norm_minus_commute) 
809 
also have "\<dots> < r" using 1 . 

31487
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changeset

810 
finally show "norm a  r \<le> norm (f x)" by simp 
31355  811 
qed 
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812 
finally show "norm (inverse (f x)) \<le> inverse (norm a  r)" . 
31355  813 
qed 
814 
thus ?thesis by (rule BfunI) 

815 
qed 

816 

817 
lemma tendsto_inverse_lemma: 

818 
fixes a :: "'a::real_normed_div_algebra" 

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changeset

819 
shows "\<lbrakk>(f > a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk> 
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changeset

820 
\<Longrightarrow> ((\<lambda>x. inverse (f x)) > inverse a) net" 
31355  821 
apply (subst tendsto_Zfun_iff) 
822 
apply (rule Zfun_ssubst) 

823 
apply (erule eventually_elim1) 

824 
apply (erule (1) inverse_diff_inverse) 

825 
apply (rule Zfun_minus) 

826 
apply (rule Zfun_mult_left) 

827 
apply (rule mult.Bfun_prod_Zfun) 

828 
apply (erule (1) Bfun_inverse) 

829 
apply (simp add: tendsto_Zfun_iff) 

830 
done 

831 

31565  832 
lemma tendsto_inverse [tendsto_intros]: 
31355  833 
fixes a :: "'a::real_normed_div_algebra" 
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changeset

834 
assumes f: "(f > a) net" 
31355  835 
assumes a: "a \<noteq> 0" 
31487
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changeset

836 
shows "((\<lambda>x. inverse (f x)) > inverse a) net" 
31355  837 
proof  
838 
from a have "0 < norm a" by simp 

31487
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changeset

839 
with f have "eventually (\<lambda>x. dist (f x) a < norm a) net" 
31355  840 
by (rule tendstoD) 
31487
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changeset

841 
then have "eventually (\<lambda>x. f x \<noteq> 0) net" 
31355  842 
unfolding dist_norm by (auto elim!: eventually_elim1) 
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changeset

843 
with f a show ?thesis 
31355  844 
by (rule tendsto_inverse_lemma) 
845 
qed 

846 

31565  847 
lemma tendsto_divide [tendsto_intros]: 
31355  848 
fixes a b :: "'a::real_normed_field" 
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changeset

849 
shows "\<lbrakk>(f > a) net; (g > b) net; b \<noteq> 0\<rbrakk> 
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850 
\<Longrightarrow> ((\<lambda>x. f x / g x) > a / b) net" 
31355  851 
by (simp add: mult.tendsto tendsto_inverse divide_inverse) 
852 

31349
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new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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853 
end 