src/HOL/Limits.thy
author huffman
Sat Jun 06 09:11:12 2009 -0700 (2009-06-06)
changeset 31488 5691ccb8d6b5
parent 31487 93938cafc0e6
child 31492 5400beeddb55
permissions -rw-r--r--
generalize tendsto to class topological_space
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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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subsection {* Nets *}
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text {*
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  A net is now defined as a filter base.
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  The definition also allows non-proper filter bases.
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*}
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typedef (open) 'a net =
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  "{net :: 'a set set. (\<exists>A. A \<in> net)
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    \<and> (\<forall>A\<in>net. \<forall>B\<in>net. \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B)}"
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proof
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  show "UNIV \<in> ?net" by auto
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qed
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lemma Rep_net_nonempty: "\<exists>A. A \<in> Rep_net net"
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using Rep_net [of net] by simp
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lemma Rep_net_directed:
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  "A \<in> Rep_net net \<Longrightarrow> B \<in> Rep_net net \<Longrightarrow> \<exists>C\<in>Rep_net net. C \<subseteq> A \<and> C \<subseteq> B"
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using Rep_net [of net] by simp
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lemma Abs_net_inverse':
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  assumes "\<exists>A. A \<in> net"
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  assumes "\<And>A B. A \<in> net \<Longrightarrow> B \<in> net \<Longrightarrow> \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B" 
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  shows "Rep_net (Abs_net net) = net"
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using assms by (simp add: Abs_net_inverse)
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lemma image_nonempty: "\<exists>x. x \<in> A \<Longrightarrow> \<exists>x. x \<in> f ` A"
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by auto
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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> (\<exists>A\<in>Rep_net net. \<forall>x\<in>A. P x)"
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
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unfolding eventually_def using Rep_net_nonempty [of net] by fast
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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 by blast
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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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proof -
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  obtain A where A: "A \<in> Rep_net net" "\<forall>x\<in>A. P x"
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    using P unfolding eventually_def by fast
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  obtain B where B: "B \<in> Rep_net net" "\<forall>x\<in>B. Q x"
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    using Q unfolding eventually_def by fast
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  obtain C where C: "C \<in> Rep_net net" "C \<subseteq> A" "C \<subseteq> B"
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    using Rep_net_directed [OF A(1) B(1)] by fast
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  then have "\<forall>x\<in>C. P x \<and> Q x" "C \<in> Rep_net net"
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    using A(2) B(2) by auto
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  then show ?thesis unfolding eventually_def ..
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qed
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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
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  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 {* Standard Nets *}
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definition
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  sequentially :: "nat net" where
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  [code del]: "sequentially = Abs_net (range (\<lambda>n. {n..}))"
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definition
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  within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
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  [code del]: "net within S = Abs_net ((\<lambda>A. A \<inter> S) ` Rep_net net)"
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definition
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  at :: "'a::topological_space \<Rightarrow> 'a net" where
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  [code del]: "at a = Abs_net ((\<lambda>S. S - {a}) ` {S\<in>topo. a \<in> S})"
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lemma Rep_net_sequentially:
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  "Rep_net sequentially = range (\<lambda>n. {n..})"
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unfolding sequentially_def
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apply (rule Abs_net_inverse')
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apply (rule image_nonempty, simp)
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apply (clarsimp, rename_tac m n)
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apply (rule_tac x="max m n" in exI, auto)
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done
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lemma Rep_net_within:
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  "Rep_net (net within S) = (\<lambda>A. A \<inter> S) ` Rep_net net"
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unfolding within_def
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apply (rule Abs_net_inverse')
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apply (rule image_nonempty, rule Rep_net_nonempty)
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apply (clarsimp, rename_tac A B)
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apply (drule (1) Rep_net_directed)
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apply (clarify, rule_tac x=C in bexI, auto)
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done
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lemma Rep_net_at:
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  "Rep_net (at a) = ((\<lambda>S. S - {a}) ` {S\<in>topo. a \<in> S})"
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unfolding at_def
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apply (rule Abs_net_inverse')
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apply (rule image_nonempty)
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apply (rule_tac x="UNIV" in exI, simp add: topo_UNIV)
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apply (clarsimp, rename_tac S T)
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apply (rule_tac x="S \<inter> T" in exI, auto simp add: topo_Int)
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done
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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 eventually_def Rep_net_sequentially by auto
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lemma eventually_within:
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  "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
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unfolding eventually_def Rep_net_within by auto
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lemma eventually_at_topological:
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  "eventually P (at a) \<longleftrightarrow> (\<exists>S\<in>topo. a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
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unfolding eventually_def Rep_net_at by auto
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lemma eventually_at:
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  fixes a :: "'a::metric_space"
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  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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unfolding eventually_at_topological topo_dist
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apply safe
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apply fast
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apply (rule_tac x="{x. dist x a < d}" in bexI, simp)
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apply clarsimp
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apply (rule_tac x="d - dist x a" in exI, clarsimp)
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apply (simp only: less_diff_eq)
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apply (erule le_less_trans [OF dist_triangle])
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done
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subsection {* Boundedness *}
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definition
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  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)"
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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"
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unfolding Bfun_def
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proof (intro exI conjI allI)
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  show "0 < max K 1" by simp
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next
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  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
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    using K by (rule eventually_elim1, simp)
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qed
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lemma BfunE:
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  assumes "Bfun f net"
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  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
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using assms unfolding Bfun_def by fast
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subsection {* Convergence to Zero *}
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definition
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  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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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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lemma ZfunD:
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  "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
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unfolding Zfun_def by simp
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lemma Zfun_ssubst:
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  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
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unfolding Zfun_def by (auto elim!: eventually_rev_mp)
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lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
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unfolding Zfun_def by simp
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lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
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unfolding Zfun_def by simp
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lemma Zfun_imp_Zfun:
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  assumes f: "Zfun f net"
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  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
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  shows "Zfun (\<lambda>x. g x) net"
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proof (cases)
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  assume K: "0 < K"
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  show ?thesis
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  proof (rule ZfunI)
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    fix r::real assume "0 < r"
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    hence "0 < r / K"
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      using K by (rule divide_pos_pos)
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    then have "eventually (\<lambda>x. norm (f x) < r / K) net"
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      using ZfunD [OF f] by fast
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    with g show "eventually (\<lambda>x. norm (g x) < r) net"
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    proof (rule eventually_elim2)
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      fix x
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      assume *: "norm (g x) \<le> norm (f x) * K"
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      assume "norm (f x) < r / K"
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      hence "norm (f x) * K < r"
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        by (simp add: pos_less_divide_eq K)
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      thus "norm (g x) < r"
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        by (simp add: order_le_less_trans [OF *])
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    qed
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  qed
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next
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  assume "\<not> 0 < K"
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  hence K: "K \<le> 0" by (simp only: not_less)
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  show ?thesis
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  proof (rule ZfunI)
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    fix r :: real
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    assume "0 < r"
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    from g show "eventually (\<lambda>x. norm (g x) < r) net"
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    proof (rule eventually_elim1)
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      fix x
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      assume "norm (g x) \<le> norm (f x) * K"
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      also have "\<dots> \<le> norm (f x) * 0"
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        using K norm_ge_zero by (rule mult_left_mono)
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      finally show "norm (g x) < r"
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        using `0 < r` by simp
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    qed
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  qed
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qed
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lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
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by (erule_tac K="1" in Zfun_imp_Zfun, simp)
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lemma Zfun_add:
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  assumes f: "Zfun f net" and g: "Zfun g net"
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  shows "Zfun (\<lambda>x. f x + g x) net"
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proof (rule ZfunI)
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  fix r::real assume "0 < r"
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  hence r: "0 < r / 2" by simp
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  have "eventually (\<lambda>x. norm (f x) < r/2) net"
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    using f r by (rule ZfunD)
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  moreover
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  have "eventually (\<lambda>x. norm (g x) < r/2) net"
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    using g r by (rule ZfunD)
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  ultimately
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  show "eventually (\<lambda>x. norm (f x + g x) < r) net"
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  proof (rule eventually_elim2)
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    fix x
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    assume *: "norm (f x) < r/2" "norm (g x) < r/2"
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    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
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      by (rule norm_triangle_ineq)
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    also have "\<dots> < r/2 + r/2"
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      using * by (rule add_strict_mono)
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    finally show "norm (f x + g x) < r"
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      by simp
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  qed
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qed
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lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
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unfolding Zfun_def by simp
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lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
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by (simp only: diff_minus Zfun_add Zfun_minus)
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lemma (in bounded_linear) Zfun:
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  assumes g: "Zfun g net"
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  shows "Zfun (\<lambda>x. f (g x)) net"
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proof -
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  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
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    using bounded by fast
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  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
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    by simp
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  with g show ?thesis
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    by (rule Zfun_imp_Zfun)
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qed
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lemma (in bounded_bilinear) Zfun:
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  assumes f: "Zfun f net"
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  assumes g: "Zfun g net"
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  shows "Zfun (\<lambda>x. f x ** g x) net"
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proof (rule ZfunI)
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  fix r::real assume r: "0 < r"
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  obtain K where K: "0 < K"
huffman@31349
   316
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   317
    using pos_bounded by fast
huffman@31349
   318
  from K have K': "0 < inverse K"
huffman@31349
   319
    by (rule positive_imp_inverse_positive)
huffman@31487
   320
  have "eventually (\<lambda>x. norm (f x) < r) net"
huffman@31487
   321
    using f r by (rule ZfunD)
huffman@31349
   322
  moreover
huffman@31487
   323
  have "eventually (\<lambda>x. norm (g x) < inverse K) net"
huffman@31487
   324
    using g K' by (rule ZfunD)
huffman@31349
   325
  ultimately
huffman@31487
   326
  show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
huffman@31349
   327
  proof (rule eventually_elim2)
huffman@31487
   328
    fix x
huffman@31487
   329
    assume *: "norm (f x) < r" "norm (g x) < inverse K"
huffman@31487
   330
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   331
      by (rule norm_le)
huffman@31487
   332
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
huffman@31349
   333
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
huffman@31349
   334
    also from K have "r * inverse K * K = r"
huffman@31349
   335
      by simp
huffman@31487
   336
    finally show "norm (f x ** g x) < r" .
huffman@31349
   337
  qed
huffman@31349
   338
qed
huffman@31349
   339
huffman@31349
   340
lemma (in bounded_bilinear) Zfun_left:
huffman@31487
   341
  "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
huffman@31349
   342
by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   343
huffman@31349
   344
lemma (in bounded_bilinear) Zfun_right:
huffman@31487
   345
  "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
huffman@31349
   346
by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   347
huffman@31349
   348
lemmas Zfun_mult = mult.Zfun
huffman@31349
   349
lemmas Zfun_mult_right = mult.Zfun_right
huffman@31349
   350
lemmas Zfun_mult_left = mult.Zfun_left
huffman@31349
   351
huffman@31349
   352
huffman@31349
   353
subsection{* Limits *}
huffman@31349
   354
huffman@31349
   355
definition
huffman@31488
   356
  tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
huffman@31488
   357
    (infixr "--->" 55)
huffman@31488
   358
where [code del]:
huffman@31488
   359
  "(f ---> l) net \<longleftrightarrow> (\<forall>S\<in>topo. l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
huffman@31349
   360
huffman@31488
   361
lemma topological_tendstoI:
huffman@31488
   362
  "(\<And>S. S \<in> topo \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
huffman@31487
   363
    \<Longrightarrow> (f ---> l) net"
huffman@31349
   364
  unfolding tendsto_def by auto
huffman@31349
   365
huffman@31488
   366
lemma topological_tendstoD:
huffman@31488
   367
  "(f ---> l) net \<Longrightarrow> S \<in> topo \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
huffman@31488
   368
  unfolding tendsto_def by auto
huffman@31488
   369
huffman@31488
   370
lemma tendstoI:
huffman@31488
   371
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
huffman@31488
   372
  shows "(f ---> l) net"
huffman@31488
   373
apply (rule topological_tendstoI)
huffman@31488
   374
apply (simp add: topo_dist)
huffman@31488
   375
apply (drule (1) bspec, clarify)
huffman@31488
   376
apply (drule assms)
huffman@31488
   377
apply (erule eventually_elim1, simp)
huffman@31488
   378
done
huffman@31488
   379
huffman@31349
   380
lemma tendstoD:
huffman@31487
   381
  "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
huffman@31488
   382
apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
huffman@31488
   383
apply (clarsimp simp add: topo_dist)
huffman@31488
   384
apply (rule_tac x="e - dist x l" in exI, clarsimp)
huffman@31488
   385
apply (simp only: less_diff_eq)
huffman@31488
   386
apply (erule le_less_trans [OF dist_triangle])
huffman@31488
   387
apply simp
huffman@31488
   388
apply simp
huffman@31488
   389
done
huffman@31488
   390
huffman@31488
   391
lemma tendsto_iff:
huffman@31488
   392
  "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
huffman@31488
   393
using tendstoI tendstoD by fast
huffman@31349
   394
huffman@31487
   395
lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
huffman@31488
   396
by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   397
huffman@31487
   398
lemma tendsto_const: "((\<lambda>x. k) ---> k) net"
huffman@31349
   399
by (simp add: tendsto_def)
huffman@31349
   400
huffman@31349
   401
lemma tendsto_norm:
huffman@31349
   402
  fixes a :: "'a::real_normed_vector"
huffman@31487
   403
  shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
huffman@31488
   404
apply (simp add: tendsto_iff dist_norm, safe)
huffman@31349
   405
apply (drule_tac x="e" in spec, safe)
huffman@31349
   406
apply (erule eventually_elim1)
huffman@31349
   407
apply (erule order_le_less_trans [OF norm_triangle_ineq3])
huffman@31349
   408
done
huffman@31349
   409
huffman@31349
   410
lemma add_diff_add:
huffman@31349
   411
  fixes a b c d :: "'a::ab_group_add"
huffman@31349
   412
  shows "(a + c) - (b + d) = (a - b) + (c - d)"
huffman@31349
   413
by simp
huffman@31349
   414
huffman@31349
   415
lemma minus_diff_minus:
huffman@31349
   416
  fixes a b :: "'a::ab_group_add"
huffman@31349
   417
  shows "(- a) - (- b) = - (a - b)"
huffman@31349
   418
by simp
huffman@31349
   419
huffman@31349
   420
lemma tendsto_add:
huffman@31349
   421
  fixes a b :: "'a::real_normed_vector"
huffman@31487
   422
  shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
huffman@31349
   423
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   424
huffman@31349
   425
lemma tendsto_minus:
huffman@31349
   426
  fixes a :: "'a::real_normed_vector"
huffman@31487
   427
  shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
huffman@31349
   428
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   429
huffman@31349
   430
lemma tendsto_minus_cancel:
huffman@31349
   431
  fixes a :: "'a::real_normed_vector"
huffman@31487
   432
  shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
huffman@31349
   433
by (drule tendsto_minus, simp)
huffman@31349
   434
huffman@31349
   435
lemma tendsto_diff:
huffman@31349
   436
  fixes a b :: "'a::real_normed_vector"
huffman@31487
   437
  shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
huffman@31349
   438
by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   439
huffman@31349
   440
lemma (in bounded_linear) tendsto:
huffman@31487
   441
  "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
huffman@31349
   442
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   443
huffman@31349
   444
lemma (in bounded_bilinear) tendsto:
huffman@31487
   445
  "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
huffman@31349
   446
by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@31349
   447
               Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   448
huffman@31355
   449
huffman@31355
   450
subsection {* Continuity of Inverse *}
huffman@31355
   451
huffman@31355
   452
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@31487
   453
  assumes f: "Zfun f net"
huffman@31487
   454
  assumes g: "Bfun g net"
huffman@31487
   455
  shows "Zfun (\<lambda>x. f x ** g x) net"
huffman@31355
   456
proof -
huffman@31355
   457
  obtain K where K: "0 \<le> K"
huffman@31355
   458
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   459
    using nonneg_bounded by fast
huffman@31355
   460
  obtain B where B: "0 < B"
huffman@31487
   461
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
huffman@31487
   462
    using g by (rule BfunE)
huffman@31487
   463
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
huffman@31487
   464
  using norm_g proof (rule eventually_elim1)
huffman@31487
   465
    fix x
huffman@31487
   466
    assume *: "norm (g x) \<le> B"
huffman@31487
   467
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   468
      by (rule norm_le)
huffman@31487
   469
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   470
      by (intro mult_mono' order_refl norm_g norm_ge_zero
huffman@31355
   471
                mult_nonneg_nonneg K *)
huffman@31487
   472
    also have "\<dots> = norm (f x) * (B * K)"
huffman@31355
   473
      by (rule mult_assoc)
huffman@31487
   474
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   475
  qed
huffman@31487
   476
  with f show ?thesis
huffman@31487
   477
    by (rule Zfun_imp_Zfun)
huffman@31355
   478
qed
huffman@31355
   479
huffman@31355
   480
lemma (in bounded_bilinear) flip:
huffman@31355
   481
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@31355
   482
apply default
huffman@31355
   483
apply (rule add_right)
huffman@31355
   484
apply (rule add_left)
huffman@31355
   485
apply (rule scaleR_right)
huffman@31355
   486
apply (rule scaleR_left)
huffman@31355
   487
apply (subst mult_commute)
huffman@31355
   488
using bounded by fast
huffman@31355
   489
huffman@31355
   490
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@31487
   491
  assumes f: "Bfun f net"
huffman@31487
   492
  assumes g: "Zfun g net"
huffman@31487
   493
  shows "Zfun (\<lambda>x. f x ** g x) net"
huffman@31487
   494
using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   495
huffman@31355
   496
lemma inverse_diff_inverse:
huffman@31355
   497
  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
huffman@31355
   498
   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
huffman@31355
   499
by (simp add: algebra_simps)
huffman@31355
   500
huffman@31355
   501
lemma Bfun_inverse_lemma:
huffman@31355
   502
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   503
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@31355
   504
apply (subst nonzero_norm_inverse, clarsimp)
huffman@31355
   505
apply (erule (1) le_imp_inverse_le)
huffman@31355
   506
done
huffman@31355
   507
huffman@31355
   508
lemma Bfun_inverse:
huffman@31355
   509
  fixes a :: "'a::real_normed_div_algebra"
huffman@31487
   510
  assumes f: "(f ---> a) net"
huffman@31355
   511
  assumes a: "a \<noteq> 0"
huffman@31487
   512
  shows "Bfun (\<lambda>x. inverse (f x)) net"
huffman@31355
   513
proof -
huffman@31355
   514
  from a have "0 < norm a" by simp
huffman@31355
   515
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   516
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@31487
   517
  have "eventually (\<lambda>x. dist (f x) a < r) net"
huffman@31487
   518
    using tendstoD [OF f r1] by fast
huffman@31487
   519
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
huffman@31355
   520
  proof (rule eventually_elim1)
huffman@31487
   521
    fix x
huffman@31487
   522
    assume "dist (f x) a < r"
huffman@31487
   523
    hence 1: "norm (f x - a) < r"
huffman@31355
   524
      by (simp add: dist_norm)
huffman@31487
   525
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   526
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   527
      by (rule nonzero_norm_inverse)
huffman@31355
   528
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   529
    proof (rule le_imp_inverse_le)
huffman@31355
   530
      show "0 < norm a - r" using r2 by simp
huffman@31355
   531
    next
huffman@31487
   532
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   533
        by (rule norm_triangle_ineq2)
huffman@31487
   534
      also have "\<dots> = norm (f x - a)"
huffman@31355
   535
        by (rule norm_minus_commute)
huffman@31355
   536
      also have "\<dots> < r" using 1 .
huffman@31487
   537
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   538
    qed
huffman@31487
   539
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   540
  qed
huffman@31355
   541
  thus ?thesis by (rule BfunI)
huffman@31355
   542
qed
huffman@31355
   543
huffman@31355
   544
lemma tendsto_inverse_lemma:
huffman@31355
   545
  fixes a :: "'a::real_normed_div_algebra"
huffman@31487
   546
  shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
huffman@31487
   547
         \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
huffman@31355
   548
apply (subst tendsto_Zfun_iff)
huffman@31355
   549
apply (rule Zfun_ssubst)
huffman@31355
   550
apply (erule eventually_elim1)
huffman@31355
   551
apply (erule (1) inverse_diff_inverse)
huffman@31355
   552
apply (rule Zfun_minus)
huffman@31355
   553
apply (rule Zfun_mult_left)
huffman@31355
   554
apply (rule mult.Bfun_prod_Zfun)
huffman@31355
   555
apply (erule (1) Bfun_inverse)
huffman@31355
   556
apply (simp add: tendsto_Zfun_iff)
huffman@31355
   557
done
huffman@31355
   558
huffman@31355
   559
lemma tendsto_inverse:
huffman@31355
   560
  fixes a :: "'a::real_normed_div_algebra"
huffman@31487
   561
  assumes f: "(f ---> a) net"
huffman@31355
   562
  assumes a: "a \<noteq> 0"
huffman@31487
   563
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
huffman@31355
   564
proof -
huffman@31355
   565
  from a have "0 < norm a" by simp
huffman@31487
   566
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
huffman@31355
   567
    by (rule tendstoD)
huffman@31487
   568
  then have "eventually (\<lambda>x. f x \<noteq> 0) net"
huffman@31355
   569
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@31487
   570
  with f a show ?thesis
huffman@31355
   571
    by (rule tendsto_inverse_lemma)
huffman@31355
   572
qed
huffman@31355
   573
huffman@31355
   574
lemma tendsto_divide:
huffman@31355
   575
  fixes a b :: "'a::real_normed_field"
huffman@31487
   576
  shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
huffman@31487
   577
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
huffman@31355
   578
by (simp add: mult.tendsto tendsto_inverse divide_inverse)
huffman@31355
   579
huffman@31349
   580
end