src/HOL/Limits.thy
author huffman
Thu, 11 Jun 2009 11:51:12 -0700
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permissions -rw-r--r--
theorem attribute [tendsto_intros]
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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. open S \<and> 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. open S \<and> 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)
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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: open_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. open S \<and> 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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   159
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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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   163
unfolding eventually_at_topological open_dist
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   164
apply safe
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apply fast
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apply (rule_tac x="{x. dist x a < d}" in exI, 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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   169
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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   182
unfolding Bfun_def
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   183
proof (intro exI conjI allI)
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   184
  show "0 < max K 1" by simp
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   185
next
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   186
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
31355
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   187
    using K by (rule eventually_elim1, simp)
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qed
3d18766ddc4b limits of inverse using filters
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   189
3d18766ddc4b limits of inverse using filters
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   190
lemma BfunE:
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  assumes "Bfun f net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   192
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
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   193
using assms unfolding Bfun_def by fast
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   194
3d18766ddc4b limits of inverse using filters
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subsection {* Convergence to Zero *}
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definition
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   199
  Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
31487
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   200
  [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
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2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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lemma ZfunI:
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   203
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
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   204
unfolding Zfun_def by simp
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   205
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lemma ZfunD:
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   207
  "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
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   208
unfolding Zfun_def by simp
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   209
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   210
lemma Zfun_ssubst:
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   211
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
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   212
unfolding Zfun_def by (auto elim!: eventually_rev_mp)
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   213
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   214
lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
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   215
unfolding Zfun_def by simp
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   216
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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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   218
unfolding Zfun_def by simp
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   219
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lemma Zfun_imp_Zfun:
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   221
  assumes f: "Zfun f net"
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   222
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
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   223
  shows "Zfun (\<lambda>x. g x) net"
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   224
proof (cases)
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  assume K: "0 < K"
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  show ?thesis
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   227
  proof (rule ZfunI)
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   228
    fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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    hence "0 < r / K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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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"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   232
      using ZfunD [OF f] by fast
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   233
    with g show "eventually (\<lambda>x. norm (g x) < r) net"
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   234
    proof (rule eventually_elim2)
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   235
      fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   236
      assume *: "norm (g x) \<le> norm (f x) * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   237
      assume "norm (f x) < r / K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   238
      hence "norm (f x) * K < r"
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   239
        by (simp add: pos_less_divide_eq K)
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   240
      thus "norm (g x) < r"
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   241
        by (simp add: order_le_less_trans [OF *])
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    qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   243
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   244
next
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   245
  assume "\<not> 0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   246
  hence K: "K \<le> 0" by (simp only: not_less)
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   247
  show ?thesis
3d18766ddc4b limits of inverse using filters
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   248
  proof (rule ZfunI)
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   249
    fix r :: real
3d18766ddc4b limits of inverse using filters
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   250
    assume "0 < r"
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   251
    from g show "eventually (\<lambda>x. norm (g x) < r) net"
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   252
    proof (rule eventually_elim1)
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   253
      fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   254
      assume "norm (g x) \<le> norm (f x) * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   255
      also have "\<dots> \<le> norm (f x) * 0"
31355
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   256
        using K norm_ge_zero by (rule mult_left_mono)
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   257
      finally show "norm (g x) < r"
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   258
        using `0 < r` by simp
3d18766ddc4b limits of inverse using filters
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   259
    qed
3d18766ddc4b limits of inverse using filters
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   260
  qed
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   261
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   262
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   263
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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   264
by (erule_tac K="1" in Zfun_imp_Zfun, simp)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   265
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   266
lemma Zfun_add:
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   267
  assumes f: "Zfun f net" and g: "Zfun g net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   268
  shows "Zfun (\<lambda>x. f x + g x) net"
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   269
proof (rule ZfunI)
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   270
  fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   271
  hence r: "0 < r / 2" by simp
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   272
  have "eventually (\<lambda>x. norm (f x) < r/2) net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   273
    using f r by (rule ZfunD)
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   274
  moreover
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   275
  have "eventually (\<lambda>x. norm (g x) < r/2) net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
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diff changeset
   276
    using g r by (rule ZfunD)
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   277
  ultimately
31487
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   278
  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
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diff changeset
   279
  proof (rule eventually_elim2)
31487
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diff changeset
   280
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
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   281
    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
   282
    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
   283
      by (rule norm_triangle_ineq)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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parents:
diff changeset
   284
    also have "\<dots> < r/2 + r/2"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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parents:
diff changeset
   285
      using * by (rule add_strict_mono)
31487
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diff changeset
   286
    finally show "norm (f x + g x) < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   287
      by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   288
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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diff changeset
   289
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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diff changeset
   290
31487
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   291
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
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diff changeset
   292
unfolding Zfun_def by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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diff changeset
   293
31487
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   294
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
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diff changeset
   295
by (simp only: diff_minus Zfun_add Zfun_minus)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   296
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
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   297
lemma (in bounded_linear) Zfun:
31487
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   298
  assumes g: "Zfun g net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   299
  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
   300
proof -
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   301
  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
   302
    using bounded by fast
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   303
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   304
    by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   305
  with g show ?thesis
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   306
    by (rule Zfun_imp_Zfun)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   307
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   308
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   309
lemma (in bounded_bilinear) Zfun:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   310
  assumes f: "Zfun f net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   311
  assumes g: "Zfun g net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   312
  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
   313
proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   314
  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
   315
  obtain K where K: "0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   316
    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
   317
    using pos_bounded by fast
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   318
  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
   319
    by (rule positive_imp_inverse_positive)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   320
  have "eventually (\<lambda>x. norm (f x) < r) net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   321
    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
   322
  moreover
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   323
  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
   324
    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
   325
  ultimately
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   326
  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
   327
  proof (rule eventually_elim2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   328
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   329
    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
   330
    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
   331
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   332
    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
   333
      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
   334
    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
   335
      by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   336
    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
   337
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   338
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   339
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   340
lemma (in bounded_bilinear) Zfun_left:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   341
  "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
   342
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
   343
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   344
lemma (in bounded_bilinear) Zfun_right:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   345
  "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
   346
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
   347
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   348
lemmas Zfun_mult = mult.Zfun
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   349
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
   350
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
   351
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   352
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   353
subsection{* Limits *}
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   354
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   355
definition
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   356
  tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   357
    (infixr "--->" 55)
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   358
where [code del]:
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31488
diff changeset
   359
  "(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
   360
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   361
ML{*
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   362
structure TendstoIntros =
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   363
  NamedThmsFun(val name = "tendsto_intros"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   364
               val description = "introduction rules for tendsto");
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   365
*}
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   366
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   367
setup TendstoIntros.setup
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   368
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   369
lemma topological_tendstoI:
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31488
diff changeset
   370
  "(\<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
   371
    \<Longrightarrow> (f ---> l) net"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   372
  unfolding tendsto_def by auto
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   373
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   374
lemma topological_tendstoD:
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31488
diff changeset
   375
  "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   376
  unfolding tendsto_def by auto
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   377
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   378
lemma tendstoI:
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   379
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   380
  shows "(f ---> l) net"
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   381
apply (rule topological_tendstoI)
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31488
diff changeset
   382
apply (simp add: open_dist)
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   383
apply (drule (1) bspec, clarify)
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   384
apply (drule assms)
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   385
apply (erule eventually_elim1, simp)
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   386
done
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   387
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   388
lemma tendstoD:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   389
  "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   390
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
   391
apply (clarsimp simp add: open_dist)
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   392
apply (rule_tac x="e - dist x l" in exI, clarsimp)
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   393
apply (simp only: less_diff_eq)
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   394
apply (erule le_less_trans [OF dist_triangle])
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   395
apply simp
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   396
apply simp
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   397
done
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   398
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   399
lemma tendsto_iff:
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   400
  "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   401
using tendstoI tendstoD by fast
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   402
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   403
lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   404
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
   405
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   406
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   407
unfolding tendsto_def eventually_at_topological by auto
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   408
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   409
lemma tendsto_ident_at_within [tendsto_intros]:
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   410
  "a \<in> S \<Longrightarrow> ((\<lambda>x. x) ---> a) (at a within S)"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   411
unfolding tendsto_def eventually_within eventually_at_topological by auto
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   412
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   413
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
   414
by (simp add: tendsto_def)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   415
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   416
lemma tendsto_dist [tendsto_intros]:
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   417
  assumes f: "(f ---> l) net" and g: "(g ---> m) net"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   418
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   419
proof (rule tendstoI)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   420
  fix e :: real assume "0 < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   421
  hence e2: "0 < e/2" by simp
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   422
  from tendstoD [OF f e2] tendstoD [OF g e2]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   423
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   424
  proof (rule eventually_elim2)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   425
    fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   426
    then show "dist (dist (f x) (g x)) (dist l m) < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   427
      unfolding dist_real_def
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   428
      using dist_triangle2 [of "f x" "g x" "l"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   429
      using dist_triangle2 [of "g x" "l" "m"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   430
      using dist_triangle3 [of "l" "m" "f x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   431
      using dist_triangle [of "f x" "m" "g x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   432
      by arith
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   433
  qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   434
qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   435
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   436
lemma tendsto_norm [tendsto_intros]:
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   437
  "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   438
apply (simp add: tendsto_iff dist_norm, safe)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   439
apply (drule_tac x="e" in spec, safe)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   440
apply (erule eventually_elim1)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   441
apply (erule order_le_less_trans [OF norm_triangle_ineq3])
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   442
done
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   443
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   444
lemma add_diff_add:
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   445
  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
   446
  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
   447
by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   448
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   449
lemma minus_diff_minus:
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   450
  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
   451
  shows "(- a) - (- b) = - (a - b)"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   452
by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   453
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   454
lemma tendsto_add [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   455
  fixes a b :: "'a::real_normed_vector"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   456
  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
   457
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
   458
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   459
lemma tendsto_minus [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   460
  fixes a :: "'a::real_normed_vector"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   461
  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
   462
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
   463
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   464
lemma tendsto_minus_cancel:
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   465
  fixes a :: "'a::real_normed_vector"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   466
  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
   467
by (drule tendsto_minus, simp)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   468
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   469
lemma tendsto_diff [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   470
  fixes a b :: "'a::real_normed_vector"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   471
  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
   472
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
   473
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   474
lemma (in bounded_linear) tendsto [tendsto_intros]:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   475
  "(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
   476
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
   477
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   478
lemma (in bounded_bilinear) tendsto [tendsto_intros]:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   479
  "\<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
   480
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
   481
               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
   482
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   483
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   484
subsection {* Continuity of Inverse *}
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   485
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   486
lemma (in bounded_bilinear) Zfun_prod_Bfun:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   487
  assumes f: "Zfun f net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   488
  assumes g: "Bfun g net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   489
  shows "Zfun (\<lambda>x. f x ** g x) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   490
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   491
  obtain K where K: "0 \<le> K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   492
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   493
    using nonneg_bounded by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   494
  obtain B where B: "0 < B"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   495
    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
   496
    using g by (rule BfunE)
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   497
  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
   498
  using norm_g proof (rule eventually_elim1)
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   499
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   500
    assume *: "norm (g x) \<le> B"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   501
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   502
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   503
    also have "\<dots> \<le> norm (f x) * B * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   504
      by (intro mult_mono' order_refl norm_g norm_ge_zero
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   505
                mult_nonneg_nonneg K *)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   506
    also have "\<dots> = norm (f x) * (B * K)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   507
      by (rule mult_assoc)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   508
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   509
  qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   510
  with f show ?thesis
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   511
    by (rule Zfun_imp_Zfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   512
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   513
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   514
lemma (in bounded_bilinear) flip:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   515
  "bounded_bilinear (\<lambda>x y. y ** x)"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   516
apply default
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   517
apply (rule add_right)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   518
apply (rule add_left)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   519
apply (rule scaleR_right)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   520
apply (rule scaleR_left)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   521
apply (subst mult_commute)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   522
using bounded by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   523
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   524
lemma (in bounded_bilinear) Bfun_prod_Zfun:
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   525
  assumes f: "Bfun f net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   526
  assumes g: "Zfun g net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   527
  shows "Zfun (\<lambda>x. f x ** g x) net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   528
using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   529
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   530
lemma inverse_diff_inverse:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   531
  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   532
   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   533
by (simp add: algebra_simps)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   534
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   535
lemma Bfun_inverse_lemma:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   536
  fixes x :: "'a::real_normed_div_algebra"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   537
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   538
apply (subst nonzero_norm_inverse, clarsimp)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   539
apply (erule (1) le_imp_inverse_le)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   540
done
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   541
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   542
lemma Bfun_inverse:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   543
  fixes a :: "'a::real_normed_div_algebra"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   544
  assumes f: "(f ---> a) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   545
  assumes a: "a \<noteq> 0"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   546
  shows "Bfun (\<lambda>x. inverse (f x)) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   547
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   548
  from a have "0 < norm a" by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   549
  hence "\<exists>r>0. r < norm a" by (rule dense)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   550
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   551
  have "eventually (\<lambda>x. dist (f x) a < r) net"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   552
    using tendstoD [OF f r1] by fast
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   553
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   554
  proof (rule eventually_elim1)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   555
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   556
    assume "dist (f x) a < r"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   557
    hence 1: "norm (f x - a) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   558
      by (simp add: dist_norm)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   559
    hence 2: "f x \<noteq> 0" using r2 by auto
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   560
    hence "norm (inverse (f x)) = inverse (norm (f x))"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   561
      by (rule nonzero_norm_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   562
    also have "\<dots> \<le> inverse (norm a - r)"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   563
    proof (rule le_imp_inverse_le)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   564
      show "0 < norm a - r" using r2 by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   565
    next
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   566
      have "norm a - norm (f x) \<le> norm (a - f x)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   567
        by (rule norm_triangle_ineq2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   568
      also have "\<dots> = norm (f x - a)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   569
        by (rule norm_minus_commute)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   570
      also have "\<dots> < r" using 1 .
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   571
      finally show "norm a - r \<le> norm (f x)" by simp
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   572
    qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   573
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   574
  qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   575
  thus ?thesis by (rule BfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   576
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   577
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   578
lemma tendsto_inverse_lemma:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   579
  fixes a :: "'a::real_normed_div_algebra"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   580
  shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   581
         \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   582
apply (subst tendsto_Zfun_iff)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   583
apply (rule Zfun_ssubst)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   584
apply (erule eventually_elim1)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   585
apply (erule (1) inverse_diff_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   586
apply (rule Zfun_minus)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   587
apply (rule Zfun_mult_left)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   588
apply (rule mult.Bfun_prod_Zfun)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   589
apply (erule (1) Bfun_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   590
apply (simp add: tendsto_Zfun_iff)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   591
done
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   592
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   593
lemma tendsto_inverse [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   594
  fixes a :: "'a::real_normed_div_algebra"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   595
  assumes f: "(f ---> a) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   596
  assumes a: "a \<noteq> 0"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   597
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   598
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   599
  from a have "0 < norm a" by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   600
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   601
    by (rule tendstoD)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   602
  then have "eventually (\<lambda>x. f x \<noteq> 0) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   603
    unfolding dist_norm by (auto elim!: eventually_elim1)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   604
  with f a show ?thesis
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   605
    by (rule tendsto_inverse_lemma)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   606
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   607
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   608
lemma tendsto_divide [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   609
  fixes a b :: "'a::real_normed_field"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   610
  shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   611
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   612
by (simp add: mult.tendsto tendsto_inverse divide_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   613
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   614
end