src/HOL/Limits.thy
author huffman
Tue Jun 02 17:03:22 2009 -0700 (2009-06-02)
changeset 31392 69570155ddf8
parent 31357 df6acdd9dd37
child 31447 97bab1ac463e
permissions -rw-r--r--
replace filters with filter bases
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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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  at :: "'a::metric_space \<Rightarrow> 'a net" where
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  [code del]: "at a = Abs_net ((\<lambda>r. {x. x \<noteq> a \<and> dist x a < r}) ` {r. 0 < r})"
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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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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_at:
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  "Rep_net (at a) = ((\<lambda>r. {x. x \<noteq> a \<and> dist x a < r}) ` {r. 0 < r})"
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unfolding at_def
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apply (rule Abs_net_inverse')
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apply (rule image_nonempty, rule_tac x=1 in exI, simp)
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apply (clarsimp, rename_tac r s)
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apply (rule_tac x="min r s" 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 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_at:
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  "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_def Rep_net_at 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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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 S net = (\<exists>K>0. eventually (\<lambda>i. norm (S i) \<le> K) net)"
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lemma BfunI: assumes K: "eventually (\<lambda>i. norm (X i) \<le> K) net" shows "Bfun X 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>i. norm (X i) \<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 S net"
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  obtains B where "0 < B" and "eventually (\<lambda>i. norm (S i) \<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 S net = (\<forall>r>0. eventually (\<lambda>i. norm (S i) < r) net)"
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lemma ZfunI:
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  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) net) \<Longrightarrow> Zfun S net"
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unfolding Zfun_def by simp
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lemma ZfunD:
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  "\<lbrakk>Zfun S net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) net"
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unfolding Zfun_def by simp
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lemma Zfun_ssubst:
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  "eventually (\<lambda>i. X i = Y i) net \<Longrightarrow> Zfun Y net \<Longrightarrow> Zfun X net"
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unfolding Zfun_def by (auto elim!: eventually_rev_mp)
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lemma Zfun_zero: "Zfun (\<lambda>i. 0) net"
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unfolding Zfun_def by simp
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lemma Zfun_norm_iff: "Zfun (\<lambda>i. norm (S i)) net = Zfun (\<lambda>i. S i) net"
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unfolding Zfun_def by simp
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lemma Zfun_imp_Zfun:
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  assumes X: "Zfun X net"
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  assumes Y: "eventually (\<lambda>i. norm (Y i) \<le> norm (X i) * K) net"
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  shows "Zfun (\<lambda>n. Y n) 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>i. norm (X i) < r / K) net"
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      using ZfunD [OF X] by fast
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    with Y show "eventually (\<lambda>i. norm (Y i) < r) net"
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    proof (rule eventually_elim2)
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      fix i
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      assume *: "norm (Y i) \<le> norm (X i) * K"
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      assume "norm (X i) < r / K"
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      hence "norm (X i) * K < r"
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        by (simp add: pos_less_divide_eq K)
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      thus "norm (Y i) < 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 Y show "eventually (\<lambda>i. norm (Y i) < r) net"
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    proof (rule eventually_elim1)
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      fix i
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      assume "norm (Y i) \<le> norm (X i) * K"
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      also have "\<dots> \<le> norm (X i) * 0"
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        using K norm_ge_zero by (rule mult_left_mono)
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      finally show "norm (Y i) < 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 Y net; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zfun X 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 X: "Zfun X net" and Y: "Zfun Y net"
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  shows "Zfun (\<lambda>n. X n + Y n) 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>i. norm (X i) < r/2) net"
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    using X r by (rule ZfunD)
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  moreover
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  have "eventually (\<lambda>i. norm (Y i) < r/2) net"
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    using Y r by (rule ZfunD)
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  ultimately
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  show "eventually (\<lambda>i. norm (X i + Y i) < r) net"
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  proof (rule eventually_elim2)
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    fix i
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    assume *: "norm (X i) < r/2" "norm (Y i) < r/2"
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    have "norm (X i + Y i) \<le> norm (X i) + norm (Y i)"
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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 (X i + Y i) < 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 X net \<Longrightarrow> Zfun (\<lambda>i. - X i) net"
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unfolding Zfun_def by simp
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lemma Zfun_diff: "\<lbrakk>Zfun X net; Zfun Y net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>i. X i - Y i) 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 X: "Zfun X net"
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  shows "Zfun (\<lambda>n. f (X n)) 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>i. norm (f (X i)) \<le> norm (X i) * K) net"
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    by simp
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  with X 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 X: "Zfun X net"
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  assumes Y: "Zfun Y net"
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  shows "Zfun (\<lambda>n. X n ** Y n) 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"
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    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
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    using pos_bounded by fast
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  from K have K': "0 < inverse K"
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    by (rule positive_imp_inverse_positive)
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  have "eventually (\<lambda>i. norm (X i) < r) net"
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    using X r by (rule ZfunD)
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  moreover
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  have "eventually (\<lambda>i. norm (Y i) < inverse K) net"
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    using Y K' by (rule ZfunD)
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  ultimately
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  show "eventually (\<lambda>i. norm (X i ** Y i) < r) net"
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  proof (rule eventually_elim2)
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    fix i
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    assume *: "norm (X i) < r" "norm (Y i) < inverse K"
huffman@31349
   315
    have "norm (X i ** Y i) \<le> norm (X i) * norm (Y i) * K"
huffman@31349
   316
      by (rule norm_le)
huffman@31349
   317
    also have "norm (X i) * norm (Y i) * K < r * inverse K * K"
huffman@31349
   318
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
huffman@31349
   319
    also from K have "r * inverse K * K = r"
huffman@31349
   320
      by simp
huffman@31349
   321
    finally show "norm (X i ** Y i) < r" .
huffman@31349
   322
  qed
huffman@31349
   323
qed
huffman@31349
   324
huffman@31349
   325
lemma (in bounded_bilinear) Zfun_left:
huffman@31392
   326
  "Zfun X net \<Longrightarrow> Zfun (\<lambda>n. X n ** a) net"
huffman@31349
   327
by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   328
huffman@31349
   329
lemma (in bounded_bilinear) Zfun_right:
huffman@31392
   330
  "Zfun X net \<Longrightarrow> Zfun (\<lambda>n. a ** X n) net"
huffman@31349
   331
by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   332
huffman@31349
   333
lemmas Zfun_mult = mult.Zfun
huffman@31349
   334
lemmas Zfun_mult_right = mult.Zfun_right
huffman@31349
   335
lemmas Zfun_mult_left = mult.Zfun_left
huffman@31349
   336
huffman@31349
   337
huffman@31349
   338
subsection{* Limits *}
huffman@31349
   339
huffman@31349
   340
definition
huffman@31392
   341
  tendsto :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool" where
huffman@31353
   342
  [code del]: "tendsto f l net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
huffman@31349
   343
huffman@31349
   344
lemma tendstoI:
huffman@31349
   345
  "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net)
huffman@31349
   346
    \<Longrightarrow> tendsto f l net"
huffman@31349
   347
  unfolding tendsto_def by auto
huffman@31349
   348
huffman@31349
   349
lemma tendstoD:
huffman@31349
   350
  "tendsto f l net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
huffman@31349
   351
  unfolding tendsto_def by auto
huffman@31349
   352
huffman@31392
   353
lemma tendsto_Zfun_iff: "tendsto (\<lambda>n. X n) L net = Zfun (\<lambda>n. X n - L) net"
huffman@31349
   354
by (simp only: tendsto_def Zfun_def dist_norm)
huffman@31349
   355
huffman@31392
   356
lemma tendsto_const: "tendsto (\<lambda>n. k) k net"
huffman@31349
   357
by (simp add: tendsto_def)
huffman@31349
   358
huffman@31349
   359
lemma tendsto_norm:
huffman@31349
   360
  fixes a :: "'a::real_normed_vector"
huffman@31392
   361
  shows "tendsto X a net \<Longrightarrow> tendsto (\<lambda>n. norm (X n)) (norm a) net"
huffman@31349
   362
apply (simp add: tendsto_def dist_norm, safe)
huffman@31349
   363
apply (drule_tac x="e" in spec, safe)
huffman@31349
   364
apply (erule eventually_elim1)
huffman@31349
   365
apply (erule order_le_less_trans [OF norm_triangle_ineq3])
huffman@31349
   366
done
huffman@31349
   367
huffman@31349
   368
lemma add_diff_add:
huffman@31349
   369
  fixes a b c d :: "'a::ab_group_add"
huffman@31349
   370
  shows "(a + c) - (b + d) = (a - b) + (c - d)"
huffman@31349
   371
by simp
huffman@31349
   372
huffman@31349
   373
lemma minus_diff_minus:
huffman@31349
   374
  fixes a b :: "'a::ab_group_add"
huffman@31349
   375
  shows "(- a) - (- b) = - (a - b)"
huffman@31349
   376
by simp
huffman@31349
   377
huffman@31349
   378
lemma tendsto_add:
huffman@31349
   379
  fixes a b :: "'a::real_normed_vector"
huffman@31392
   380
  shows "\<lbrakk>tendsto X a net; tendsto Y b net\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n + Y n) (a + b) net"
huffman@31349
   381
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   382
huffman@31349
   383
lemma tendsto_minus:
huffman@31349
   384
  fixes a :: "'a::real_normed_vector"
huffman@31392
   385
  shows "tendsto X a net \<Longrightarrow> tendsto (\<lambda>n. - X n) (- a) net"
huffman@31349
   386
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   387
huffman@31349
   388
lemma tendsto_minus_cancel:
huffman@31349
   389
  fixes a :: "'a::real_normed_vector"
huffman@31392
   390
  shows "tendsto (\<lambda>n. - X n) (- a) net \<Longrightarrow> tendsto X a net"
huffman@31349
   391
by (drule tendsto_minus, simp)
huffman@31349
   392
huffman@31349
   393
lemma tendsto_diff:
huffman@31349
   394
  fixes a b :: "'a::real_normed_vector"
huffman@31392
   395
  shows "\<lbrakk>tendsto X a net; tendsto Y b net\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n - Y n) (a - b) net"
huffman@31349
   396
by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   397
huffman@31349
   398
lemma (in bounded_linear) tendsto:
huffman@31392
   399
  "tendsto X a net \<Longrightarrow> tendsto (\<lambda>n. f (X n)) (f a) net"
huffman@31349
   400
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   401
huffman@31349
   402
lemma (in bounded_bilinear) tendsto:
huffman@31392
   403
  "\<lbrakk>tendsto X a net; tendsto Y b net\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n ** Y n) (a ** b) net"
huffman@31349
   404
by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@31349
   405
               Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   406
huffman@31355
   407
huffman@31355
   408
subsection {* Continuity of Inverse *}
huffman@31355
   409
huffman@31355
   410
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@31392
   411
  assumes X: "Zfun X net"
huffman@31392
   412
  assumes Y: "Bfun Y net"
huffman@31392
   413
  shows "Zfun (\<lambda>n. X n ** Y n) net"
huffman@31355
   414
proof -
huffman@31355
   415
  obtain K where K: "0 \<le> K"
huffman@31355
   416
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   417
    using nonneg_bounded by fast
huffman@31355
   418
  obtain B where B: "0 < B"
huffman@31392
   419
    and norm_Y: "eventually (\<lambda>i. norm (Y i) \<le> B) net"
huffman@31355
   420
    using Y by (rule BfunE)
huffman@31392
   421
  have "eventually (\<lambda>i. norm (X i ** Y i) \<le> norm (X i) * (B * K)) net"
huffman@31355
   422
  using norm_Y proof (rule eventually_elim1)
huffman@31355
   423
    fix i
huffman@31355
   424
    assume *: "norm (Y i) \<le> B"
huffman@31355
   425
    have "norm (X i ** Y i) \<le> norm (X i) * norm (Y i) * K"
huffman@31355
   426
      by (rule norm_le)
huffman@31355
   427
    also have "\<dots> \<le> norm (X i) * B * K"
huffman@31355
   428
      by (intro mult_mono' order_refl norm_Y norm_ge_zero
huffman@31355
   429
                mult_nonneg_nonneg K *)
huffman@31355
   430
    also have "\<dots> = norm (X i) * (B * K)"
huffman@31355
   431
      by (rule mult_assoc)
huffman@31355
   432
    finally show "norm (X i ** Y i) \<le> norm (X i) * (B * K)" .
huffman@31355
   433
  qed
huffman@31355
   434
  with X show ?thesis
huffman@31355
   435
  by (rule Zfun_imp_Zfun)
huffman@31355
   436
qed
huffman@31355
   437
huffman@31355
   438
lemma (in bounded_bilinear) flip:
huffman@31355
   439
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@31355
   440
apply default
huffman@31355
   441
apply (rule add_right)
huffman@31355
   442
apply (rule add_left)
huffman@31355
   443
apply (rule scaleR_right)
huffman@31355
   444
apply (rule scaleR_left)
huffman@31355
   445
apply (subst mult_commute)
huffman@31355
   446
using bounded by fast
huffman@31355
   447
huffman@31355
   448
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@31392
   449
  assumes X: "Bfun X net"
huffman@31392
   450
  assumes Y: "Zfun Y net"
huffman@31392
   451
  shows "Zfun (\<lambda>n. X n ** Y n) net"
huffman@31355
   452
using flip Y X by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   453
huffman@31355
   454
lemma inverse_diff_inverse:
huffman@31355
   455
  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
huffman@31355
   456
   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
huffman@31355
   457
by (simp add: algebra_simps)
huffman@31355
   458
huffman@31355
   459
lemma Bfun_inverse_lemma:
huffman@31355
   460
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   461
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@31355
   462
apply (subst nonzero_norm_inverse, clarsimp)
huffman@31355
   463
apply (erule (1) le_imp_inverse_le)
huffman@31355
   464
done
huffman@31355
   465
huffman@31355
   466
lemma Bfun_inverse:
huffman@31355
   467
  fixes a :: "'a::real_normed_div_algebra"
huffman@31392
   468
  assumes X: "tendsto X a net"
huffman@31355
   469
  assumes a: "a \<noteq> 0"
huffman@31392
   470
  shows "Bfun (\<lambda>n. inverse (X n)) net"
huffman@31355
   471
proof -
huffman@31355
   472
  from a have "0 < norm a" by simp
huffman@31355
   473
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   474
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@31392
   475
  have "eventually (\<lambda>i. dist (X i) a < r) net"
huffman@31355
   476
    using tendstoD [OF X r1] by fast
huffman@31392
   477
  hence "eventually (\<lambda>i. norm (inverse (X i)) \<le> inverse (norm a - r)) net"
huffman@31355
   478
  proof (rule eventually_elim1)
huffman@31355
   479
    fix i
huffman@31355
   480
    assume "dist (X i) a < r"
huffman@31355
   481
    hence 1: "norm (X i - a) < r"
huffman@31355
   482
      by (simp add: dist_norm)
huffman@31355
   483
    hence 2: "X i \<noteq> 0" using r2 by auto
huffman@31355
   484
    hence "norm (inverse (X i)) = inverse (norm (X i))"
huffman@31355
   485
      by (rule nonzero_norm_inverse)
huffman@31355
   486
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   487
    proof (rule le_imp_inverse_le)
huffman@31355
   488
      show "0 < norm a - r" using r2 by simp
huffman@31355
   489
    next
huffman@31355
   490
      have "norm a - norm (X i) \<le> norm (a - X i)"
huffman@31355
   491
        by (rule norm_triangle_ineq2)
huffman@31355
   492
      also have "\<dots> = norm (X i - a)"
huffman@31355
   493
        by (rule norm_minus_commute)
huffman@31355
   494
      also have "\<dots> < r" using 1 .
huffman@31355
   495
      finally show "norm a - r \<le> norm (X i)" by simp
huffman@31355
   496
    qed
huffman@31355
   497
    finally show "norm (inverse (X i)) \<le> inverse (norm a - r)" .
huffman@31355
   498
  qed
huffman@31355
   499
  thus ?thesis by (rule BfunI)
huffman@31355
   500
qed
huffman@31355
   501
huffman@31355
   502
lemma tendsto_inverse_lemma:
huffman@31355
   503
  fixes a :: "'a::real_normed_div_algebra"
huffman@31392
   504
  shows "\<lbrakk>tendsto X a net; a \<noteq> 0; eventually (\<lambda>i. X i \<noteq> 0) net\<rbrakk>
huffman@31392
   505
         \<Longrightarrow> tendsto (\<lambda>i. inverse (X i)) (inverse a) net"
huffman@31355
   506
apply (subst tendsto_Zfun_iff)
huffman@31355
   507
apply (rule Zfun_ssubst)
huffman@31355
   508
apply (erule eventually_elim1)
huffman@31355
   509
apply (erule (1) inverse_diff_inverse)
huffman@31355
   510
apply (rule Zfun_minus)
huffman@31355
   511
apply (rule Zfun_mult_left)
huffman@31355
   512
apply (rule mult.Bfun_prod_Zfun)
huffman@31355
   513
apply (erule (1) Bfun_inverse)
huffman@31355
   514
apply (simp add: tendsto_Zfun_iff)
huffman@31355
   515
done
huffman@31355
   516
huffman@31355
   517
lemma tendsto_inverse:
huffman@31355
   518
  fixes a :: "'a::real_normed_div_algebra"
huffman@31392
   519
  assumes X: "tendsto X a net"
huffman@31355
   520
  assumes a: "a \<noteq> 0"
huffman@31392
   521
  shows "tendsto (\<lambda>i. inverse (X i)) (inverse a) net"
huffman@31355
   522
proof -
huffman@31355
   523
  from a have "0 < norm a" by simp
huffman@31392
   524
  with X have "eventually (\<lambda>i. dist (X i) a < norm a) net"
huffman@31355
   525
    by (rule tendstoD)
huffman@31392
   526
  then have "eventually (\<lambda>i. X i \<noteq> 0) net"
huffman@31355
   527
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@31355
   528
  with X a show ?thesis
huffman@31355
   529
    by (rule tendsto_inverse_lemma)
huffman@31355
   530
qed
huffman@31355
   531
huffman@31355
   532
lemma tendsto_divide:
huffman@31355
   533
  fixes a b :: "'a::real_normed_field"
huffman@31392
   534
  shows "\<lbrakk>tendsto X a net; tendsto Y b net; b \<noteq> 0\<rbrakk>
huffman@31392
   535
    \<Longrightarrow> tendsto (\<lambda>n. X n / Y n) (a / b) net"
huffman@31355
   536
by (simp add: mult.tendsto tendsto_inverse divide_inverse)
huffman@31355
   537
huffman@31349
   538
end