src/HOL/Limits.thy
author hoelzl
Tue Mar 26 12:20:58 2013 +0100 (2013-03-26)
changeset 51526 155263089e7b
parent 51524 7cb5ac44ca9e
child 51529 2d2f59e6055a
permissions -rw-r--r--
move SEQ.thy and Lim.thy to Limits.thy
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(*  Title:      Limits.thy
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    Author:     Brian Huffman
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    Author:     Jacques D. Fleuriot, University of Cambridge
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    Author:     Lawrence C Paulson
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    Author:     Jeremy Avigad
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*)
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header {* Limits on Real Vector Spaces *}
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theory Limits
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imports Real_Vector_Spaces
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begin
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(* Unfortunately eventually_within was overwritten by Multivariate_Analysis.
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   Hence it was references as Limits.eventually_within, but now it is Basic_Topology.eventually_within *)
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lemmas eventually_within = eventually_within
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subsection {* Filter going to infinity norm *}
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definition at_infinity :: "'a::real_normed_vector filter" where
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  "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
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lemma eventually_at_infinity:
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  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
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unfolding at_infinity_def
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proof (rule eventually_Abs_filter, rule is_filter.intro)
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  fix P Q :: "'a \<Rightarrow> bool"
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  assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
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  then obtain r s where
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    "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
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  then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
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  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
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qed auto
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lemma at_infinity_eq_at_top_bot:
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  "(at_infinity \<Colon> real filter) = sup at_top at_bot"
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  unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
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proof (intro arg_cong[where f=Abs_filter] ext iffI)
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  fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
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  then guess r ..
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  then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
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  then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
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next
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  fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
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  then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
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  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
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    by (intro exI[of _ "max p (-q)"])
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       (auto simp: abs_real_def)
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qed
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lemma at_top_le_at_infinity:
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  "at_top \<le> (at_infinity :: real filter)"
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  unfolding at_infinity_eq_at_top_bot by simp
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lemma at_bot_le_at_infinity:
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  "at_bot \<le> (at_infinity :: real filter)"
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  unfolding at_infinity_eq_at_top_bot by simp
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subsection {* Boundedness *}
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lemma Bfun_def:
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  "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
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  unfolding Bfun_metric_def norm_conv_dist
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proof safe
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  fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
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  moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
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    by (intro always_eventually) (metis dist_commute dist_triangle)
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  with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
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    by eventually_elim auto
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  with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
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    by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
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qed auto
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lemma BfunI:
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  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
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unfolding Bfun_def
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proof (intro exI conjI allI)
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  show "0 < max K 1" by simp
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next
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  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
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    using K by (rule eventually_elim1, simp)
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qed
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lemma BfunE:
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  assumes "Bfun f F"
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  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
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using assms unfolding Bfun_def by fast
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subsection {* Convergence to Zero *}
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definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
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lemma ZfunI:
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  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
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  unfolding Zfun_def by simp
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lemma ZfunD:
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  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
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  unfolding Zfun_def by simp
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lemma Zfun_ssubst:
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  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
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  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
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lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
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  unfolding Zfun_def by simp
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lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
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  unfolding Zfun_def by simp
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lemma Zfun_imp_Zfun:
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  assumes f: "Zfun f F"
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  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
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  shows "Zfun (\<lambda>x. g x) F"
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proof (cases)
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  assume K: "0 < K"
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  show ?thesis
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  proof (rule ZfunI)
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    fix r::real assume "0 < r"
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    hence "0 < r / K"
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      using K by (rule divide_pos_pos)
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    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
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      using ZfunD [OF f] by fast
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    with g show "eventually (\<lambda>x. norm (g x) < r) F"
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    proof eventually_elim
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      case (elim x)
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      hence "norm (f x) * K < r"
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        by (simp add: pos_less_divide_eq K)
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      thus ?case
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        by (simp add: order_le_less_trans [OF elim(1)])
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    qed
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  qed
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next
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  assume "\<not> 0 < K"
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  hence K: "K \<le> 0" by (simp only: not_less)
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  show ?thesis
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  proof (rule ZfunI)
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    fix r :: real
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    assume "0 < r"
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    from g show "eventually (\<lambda>x. norm (g x) < r) F"
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    proof eventually_elim
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      case (elim x)
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      also have "norm (f x) * K \<le> norm (f x) * 0"
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        using K norm_ge_zero by (rule mult_left_mono)
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      finally show ?case
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        using `0 < r` by simp
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    qed
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  qed
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qed
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lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
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  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
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lemma Zfun_add:
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  assumes f: "Zfun f F" and g: "Zfun g F"
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  shows "Zfun (\<lambda>x. f x + g x) F"
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proof (rule ZfunI)
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  fix r::real assume "0 < r"
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  hence r: "0 < r / 2" by simp
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  have "eventually (\<lambda>x. norm (f x) < r/2) F"
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    using f r by (rule ZfunD)
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  moreover
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  have "eventually (\<lambda>x. norm (g x) < r/2) F"
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    using g r by (rule ZfunD)
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  ultimately
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  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
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  proof eventually_elim
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    case (elim x)
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    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
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      by (rule norm_triangle_ineq)
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    also have "\<dots> < r/2 + r/2"
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      using elim by (rule add_strict_mono)
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    finally show ?case
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      by simp
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  qed
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qed
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lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
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  unfolding Zfun_def by simp
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lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
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  by (simp only: diff_minus Zfun_add Zfun_minus)
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lemma (in bounded_linear) Zfun:
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  assumes g: "Zfun g F"
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  shows "Zfun (\<lambda>x. f (g x)) F"
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proof -
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  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
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    using bounded by fast
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  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
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    by simp
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  with g show ?thesis
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    by (rule Zfun_imp_Zfun)
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qed
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lemma (in bounded_bilinear) Zfun:
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  assumes f: "Zfun f F"
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  assumes g: "Zfun g F"
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  shows "Zfun (\<lambda>x. f x ** g x) F"
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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>x. norm (f x) < r) F"
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    using f r by (rule ZfunD)
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  moreover
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  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
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    using g K' by (rule ZfunD)
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  ultimately
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  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
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  proof eventually_elim
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    case (elim x)
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    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
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      by (rule norm_le)
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    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
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      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
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    also from K have "r * inverse K * K = r"
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      by simp
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    finally show ?case .
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  qed
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qed
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lemma (in bounded_bilinear) Zfun_left:
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  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
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  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
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lemma (in bounded_bilinear) Zfun_right:
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  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
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  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
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lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
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lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
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lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
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lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
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  by (simp only: tendsto_iff Zfun_def dist_norm)
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subsubsection {* Distance and norms *}
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lemma tendsto_norm [tendsto_intros]:
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  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
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  unfolding norm_conv_dist by (intro tendsto_intros)
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lemma continuous_norm [continuous_intros]:
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  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
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  unfolding continuous_def by (rule tendsto_norm)
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lemma continuous_on_norm [continuous_on_intros]:
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  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
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  unfolding continuous_on_def by (auto intro: tendsto_norm)
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lemma tendsto_norm_zero:
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  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
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  by (drule tendsto_norm, simp)
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lemma tendsto_norm_zero_cancel:
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  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
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  unfolding tendsto_iff dist_norm by simp
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lemma tendsto_norm_zero_iff:
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  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
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  unfolding tendsto_iff dist_norm by simp
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lemma tendsto_rabs [tendsto_intros]:
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  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
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  by (fold real_norm_def, rule tendsto_norm)
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lemma continuous_rabs [continuous_intros]:
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  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
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  unfolding real_norm_def[symmetric] by (rule continuous_norm)
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lemma continuous_on_rabs [continuous_on_intros]:
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  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
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  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
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lemma tendsto_rabs_zero:
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  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
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  by (fold real_norm_def, rule tendsto_norm_zero)
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lemma tendsto_rabs_zero_cancel:
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  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
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  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
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lemma tendsto_rabs_zero_iff:
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  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
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  by (fold real_norm_def, rule tendsto_norm_zero_iff)
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subsubsection {* Addition and subtraction *}
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lemma tendsto_add [tendsto_intros]:
huffman@31349
   296
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   297
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   298
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   299
hoelzl@51478
   300
lemma continuous_add [continuous_intros]:
hoelzl@51478
   301
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   302
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
hoelzl@51478
   303
  unfolding continuous_def by (rule tendsto_add)
hoelzl@51478
   304
hoelzl@51478
   305
lemma continuous_on_add [continuous_on_intros]:
hoelzl@51478
   306
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   307
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
hoelzl@51478
   308
  unfolding continuous_on_def by (auto intro: tendsto_add)
hoelzl@51478
   309
huffman@44194
   310
lemma tendsto_add_zero:
hoelzl@51478
   311
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   312
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   313
  by (drule (1) tendsto_add, simp)
huffman@44194
   314
huffman@31565
   315
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   316
  fixes a :: "'a::real_normed_vector"
huffman@44195
   317
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   318
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   319
hoelzl@51478
   320
lemma continuous_minus [continuous_intros]:
hoelzl@51478
   321
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   322
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
hoelzl@51478
   323
  unfolding continuous_def by (rule tendsto_minus)
hoelzl@51478
   324
hoelzl@51478
   325
lemma continuous_on_minus [continuous_on_intros]:
hoelzl@51478
   326
  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   327
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
hoelzl@51478
   328
  unfolding continuous_on_def by (auto intro: tendsto_minus)
hoelzl@51478
   329
huffman@31349
   330
lemma tendsto_minus_cancel:
huffman@31349
   331
  fixes a :: "'a::real_normed_vector"
huffman@44195
   332
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   333
  by (drule tendsto_minus, simp)
huffman@31349
   334
hoelzl@50330
   335
lemma tendsto_minus_cancel_left:
hoelzl@50330
   336
    "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
hoelzl@50330
   337
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
hoelzl@50330
   338
  by auto
hoelzl@50330
   339
huffman@31565
   340
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   341
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   342
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
huffman@44081
   343
  by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   344
hoelzl@51478
   345
lemma continuous_diff [continuous_intros]:
hoelzl@51478
   346
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   347
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
hoelzl@51478
   348
  unfolding continuous_def by (rule tendsto_diff)
hoelzl@51478
   349
hoelzl@51478
   350
lemma continuous_on_diff [continuous_on_intros]:
hoelzl@51478
   351
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   352
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
hoelzl@51478
   353
  unfolding continuous_on_def by (auto intro: tendsto_diff)
hoelzl@51478
   354
huffman@31588
   355
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   356
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   357
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   358
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   359
proof (cases "finite S")
huffman@31588
   360
  assume "finite S" thus ?thesis using assms
huffman@44194
   361
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
huffman@31588
   362
next
huffman@31588
   363
  assume "\<not> finite S" thus ?thesis
huffman@31588
   364
    by (simp add: tendsto_const)
huffman@31588
   365
qed
huffman@31588
   366
hoelzl@51478
   367
lemma continuous_setsum [continuous_intros]:
hoelzl@51478
   368
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   369
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   370
  unfolding continuous_def by (rule tendsto_setsum)
hoelzl@51478
   371
hoelzl@51478
   372
lemma continuous_on_setsum [continuous_intros]:
hoelzl@51478
   373
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   374
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   375
  unfolding continuous_on_def by (auto intro: tendsto_setsum)
hoelzl@51478
   376
hoelzl@50999
   377
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
hoelzl@50999
   378
huffman@44194
   379
subsubsection {* Linear operators and multiplication *}
huffman@44194
   380
huffman@44282
   381
lemma (in bounded_linear) tendsto:
huffman@44195
   382
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   383
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   384
hoelzl@51478
   385
lemma (in bounded_linear) continuous:
hoelzl@51478
   386
  "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
hoelzl@51478
   387
  using tendsto[of g _ F] by (auto simp: continuous_def)
hoelzl@51478
   388
hoelzl@51478
   389
lemma (in bounded_linear) continuous_on:
hoelzl@51478
   390
  "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
hoelzl@51478
   391
  using tendsto[of g] by (auto simp: continuous_on_def)
hoelzl@51478
   392
huffman@44194
   393
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   394
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   395
  by (drule tendsto, simp only: zero)
huffman@44194
   396
huffman@44282
   397
lemma (in bounded_bilinear) tendsto:
huffman@44195
   398
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   399
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   400
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   401
hoelzl@51478
   402
lemma (in bounded_bilinear) continuous:
hoelzl@51478
   403
  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
hoelzl@51478
   404
  using tendsto[of f _ F g] by (auto simp: continuous_def)
hoelzl@51478
   405
hoelzl@51478
   406
lemma (in bounded_bilinear) continuous_on:
hoelzl@51478
   407
  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
hoelzl@51478
   408
  using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
hoelzl@51478
   409
huffman@44194
   410
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   411
  assumes f: "(f ---> 0) F"
huffman@44195
   412
  assumes g: "(g ---> 0) F"
huffman@44195
   413
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   414
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   415
huffman@44194
   416
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   417
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   418
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   419
huffman@44194
   420
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   421
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   422
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   423
huffman@44282
   424
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   425
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   426
huffman@44282
   427
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   428
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   429
huffman@44282
   430
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   431
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   432
hoelzl@51478
   433
lemmas continuous_of_real [continuous_intros] =
hoelzl@51478
   434
  bounded_linear.continuous [OF bounded_linear_of_real]
hoelzl@51478
   435
hoelzl@51478
   436
lemmas continuous_scaleR [continuous_intros] =
hoelzl@51478
   437
  bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
hoelzl@51478
   438
hoelzl@51478
   439
lemmas continuous_mult [continuous_intros] =
hoelzl@51478
   440
  bounded_bilinear.continuous [OF bounded_bilinear_mult]
hoelzl@51478
   441
hoelzl@51478
   442
lemmas continuous_on_of_real [continuous_on_intros] =
hoelzl@51478
   443
  bounded_linear.continuous_on [OF bounded_linear_of_real]
hoelzl@51478
   444
hoelzl@51478
   445
lemmas continuous_on_scaleR [continuous_on_intros] =
hoelzl@51478
   446
  bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
hoelzl@51478
   447
hoelzl@51478
   448
lemmas continuous_on_mult [continuous_on_intros] =
hoelzl@51478
   449
  bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
hoelzl@51478
   450
huffman@44568
   451
lemmas tendsto_mult_zero =
huffman@44568
   452
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   453
huffman@44568
   454
lemmas tendsto_mult_left_zero =
huffman@44568
   455
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   456
huffman@44568
   457
lemmas tendsto_mult_right_zero =
huffman@44568
   458
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   459
huffman@44194
   460
lemma tendsto_power [tendsto_intros]:
huffman@44194
   461
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   462
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
huffman@44194
   463
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
huffman@44194
   464
hoelzl@51478
   465
lemma continuous_power [continuous_intros]:
hoelzl@51478
   466
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   467
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
hoelzl@51478
   468
  unfolding continuous_def by (rule tendsto_power)
hoelzl@51478
   469
hoelzl@51478
   470
lemma continuous_on_power [continuous_on_intros]:
hoelzl@51478
   471
  fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   472
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
hoelzl@51478
   473
  unfolding continuous_on_def by (auto intro: tendsto_power)
hoelzl@51478
   474
huffman@44194
   475
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   476
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   477
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   478
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   479
proof (cases "finite S")
huffman@44194
   480
  assume "finite S" thus ?thesis using assms
huffman@44194
   481
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
huffman@44194
   482
next
huffman@44194
   483
  assume "\<not> finite S" thus ?thesis
huffman@44194
   484
    by (simp add: tendsto_const)
huffman@44194
   485
qed
huffman@44194
   486
hoelzl@51478
   487
lemma continuous_setprod [continuous_intros]:
hoelzl@51478
   488
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   489
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
hoelzl@51478
   490
  unfolding continuous_def by (rule tendsto_setprod)
hoelzl@51478
   491
hoelzl@51478
   492
lemma continuous_on_setprod [continuous_intros]:
hoelzl@51478
   493
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   494
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
hoelzl@51478
   495
  unfolding continuous_on_def by (auto intro: tendsto_setprod)
hoelzl@51478
   496
huffman@44194
   497
subsubsection {* Inverse and division *}
huffman@31355
   498
huffman@31355
   499
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   500
  assumes f: "Zfun f F"
huffman@44195
   501
  assumes g: "Bfun g F"
huffman@44195
   502
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   503
proof -
huffman@31355
   504
  obtain K where K: "0 \<le> K"
huffman@31355
   505
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   506
    using nonneg_bounded by fast
huffman@31355
   507
  obtain B where B: "0 < B"
huffman@44195
   508
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   509
    using g by (rule BfunE)
huffman@44195
   510
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   511
  using norm_g proof eventually_elim
noschinl@46887
   512
    case (elim x)
huffman@31487
   513
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   514
      by (rule norm_le)
huffman@31487
   515
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   516
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   517
                mult_nonneg_nonneg K elim)
huffman@31487
   518
    also have "\<dots> = norm (f x) * (B * K)"
huffman@31355
   519
      by (rule mult_assoc)
huffman@31487
   520
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   521
  qed
huffman@31487
   522
  with f show ?thesis
huffman@31487
   523
    by (rule Zfun_imp_Zfun)
huffman@31355
   524
qed
huffman@31355
   525
huffman@31355
   526
lemma (in bounded_bilinear) flip:
huffman@31355
   527
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@44081
   528
  apply default
huffman@44081
   529
  apply (rule add_right)
huffman@44081
   530
  apply (rule add_left)
huffman@44081
   531
  apply (rule scaleR_right)
huffman@44081
   532
  apply (rule scaleR_left)
huffman@44081
   533
  apply (subst mult_commute)
huffman@44081
   534
  using bounded by fast
huffman@31355
   535
huffman@31355
   536
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   537
  assumes f: "Bfun f F"
huffman@44195
   538
  assumes g: "Zfun g F"
huffman@44195
   539
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   540
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   541
huffman@31355
   542
lemma Bfun_inverse_lemma:
huffman@31355
   543
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   544
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   545
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   546
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   547
  done
huffman@31355
   548
huffman@31355
   549
lemma Bfun_inverse:
huffman@31355
   550
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   551
  assumes f: "(f ---> a) F"
huffman@31355
   552
  assumes a: "a \<noteq> 0"
huffman@44195
   553
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   554
proof -
huffman@31355
   555
  from a have "0 < norm a" by simp
huffman@31355
   556
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   557
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
   558
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
   559
    using tendstoD [OF f r1] by fast
huffman@44195
   560
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
   561
  proof eventually_elim
noschinl@46887
   562
    case (elim x)
huffman@31487
   563
    hence 1: "norm (f x - a) < r"
huffman@31355
   564
      by (simp add: dist_norm)
huffman@31487
   565
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   566
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   567
      by (rule nonzero_norm_inverse)
huffman@31355
   568
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   569
    proof (rule le_imp_inverse_le)
huffman@31355
   570
      show "0 < norm a - r" using r2 by simp
huffman@31355
   571
    next
huffman@31487
   572
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   573
        by (rule norm_triangle_ineq2)
huffman@31487
   574
      also have "\<dots> = norm (f x - a)"
huffman@31355
   575
        by (rule norm_minus_commute)
huffman@31355
   576
      also have "\<dots> < r" using 1 .
huffman@31487
   577
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   578
    qed
huffman@31487
   579
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   580
  qed
huffman@31355
   581
  thus ?thesis by (rule BfunI)
huffman@31355
   582
qed
huffman@31355
   583
huffman@31565
   584
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   585
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   586
  assumes f: "(f ---> a) F"
huffman@31355
   587
  assumes a: "a \<noteq> 0"
huffman@44195
   588
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
   589
proof -
huffman@31355
   590
  from a have "0 < norm a" by simp
huffman@44195
   591
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   592
    by (rule tendstoD)
huffman@44195
   593
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
   594
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
   595
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
   596
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   597
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
   598
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   599
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
   600
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
   601
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
   602
  ultimately show ?thesis
huffman@44627
   603
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
   604
qed
huffman@31355
   605
hoelzl@51478
   606
lemma continuous_inverse:
hoelzl@51478
   607
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   608
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   609
  shows "continuous F (\<lambda>x. inverse (f x))"
hoelzl@51478
   610
  using assms unfolding continuous_def by (rule tendsto_inverse)
hoelzl@51478
   611
hoelzl@51478
   612
lemma continuous_at_within_inverse[continuous_intros]:
hoelzl@51478
   613
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   614
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   615
  shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
hoelzl@51478
   616
  using assms unfolding continuous_within by (rule tendsto_inverse)
hoelzl@51478
   617
hoelzl@51478
   618
lemma isCont_inverse[continuous_intros, simp]:
hoelzl@51478
   619
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   620
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   621
  shows "isCont (\<lambda>x. inverse (f x)) a"
hoelzl@51478
   622
  using assms unfolding continuous_at by (rule tendsto_inverse)
hoelzl@51478
   623
hoelzl@51478
   624
lemma continuous_on_inverse[continuous_on_intros]:
hoelzl@51478
   625
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   626
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   627
  shows "continuous_on s (\<lambda>x. inverse (f x))"
hoelzl@51478
   628
  using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
hoelzl@51478
   629
huffman@31565
   630
lemma tendsto_divide [tendsto_intros]:
huffman@31355
   631
  fixes a b :: "'a::real_normed_field"
huffman@44195
   632
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
   633
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
   634
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
   635
hoelzl@51478
   636
lemma continuous_divide:
hoelzl@51478
   637
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   638
  assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   639
  shows "continuous F (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   640
  using assms unfolding continuous_def by (rule tendsto_divide)
hoelzl@51478
   641
hoelzl@51478
   642
lemma continuous_at_within_divide[continuous_intros]:
hoelzl@51478
   643
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   644
  assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
hoelzl@51478
   645
  shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   646
  using assms unfolding continuous_within by (rule tendsto_divide)
hoelzl@51478
   647
hoelzl@51478
   648
lemma isCont_divide[continuous_intros, simp]:
hoelzl@51478
   649
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   650
  assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
hoelzl@51478
   651
  shows "isCont (\<lambda>x. (f x) / g x) a"
hoelzl@51478
   652
  using assms unfolding continuous_at by (rule tendsto_divide)
hoelzl@51478
   653
hoelzl@51478
   654
lemma continuous_on_divide[continuous_on_intros]:
hoelzl@51478
   655
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   656
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
hoelzl@51478
   657
  shows "continuous_on s (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   658
  using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
hoelzl@51478
   659
huffman@44194
   660
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
   661
  fixes l :: "'a::real_normed_vector"
huffman@44195
   662
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
   663
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
   664
hoelzl@51478
   665
lemma continuous_sgn:
hoelzl@51478
   666
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   667
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   668
  shows "continuous F (\<lambda>x. sgn (f x))"
hoelzl@51478
   669
  using assms unfolding continuous_def by (rule tendsto_sgn)
hoelzl@51478
   670
hoelzl@51478
   671
lemma continuous_at_within_sgn[continuous_intros]:
hoelzl@51478
   672
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   673
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   674
  shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
hoelzl@51478
   675
  using assms unfolding continuous_within by (rule tendsto_sgn)
hoelzl@51478
   676
hoelzl@51478
   677
lemma isCont_sgn[continuous_intros]:
hoelzl@51478
   678
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   679
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   680
  shows "isCont (\<lambda>x. sgn (f x)) a"
hoelzl@51478
   681
  using assms unfolding continuous_at by (rule tendsto_sgn)
hoelzl@51478
   682
hoelzl@51478
   683
lemma continuous_on_sgn[continuous_on_intros]:
hoelzl@51478
   684
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   685
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   686
  shows "continuous_on s (\<lambda>x. sgn (f x))"
hoelzl@51478
   687
  using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
hoelzl@51478
   688
hoelzl@50325
   689
lemma filterlim_at_infinity:
hoelzl@50325
   690
  fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
hoelzl@50325
   691
  assumes "0 \<le> c"
hoelzl@50325
   692
  shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
hoelzl@50325
   693
  unfolding filterlim_iff eventually_at_infinity
hoelzl@50325
   694
proof safe
hoelzl@50325
   695
  fix P :: "'a \<Rightarrow> bool" and b
hoelzl@50325
   696
  assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
hoelzl@50325
   697
    and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
hoelzl@50325
   698
  have "max b (c + 1) > c" by auto
hoelzl@50325
   699
  with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
hoelzl@50325
   700
    by auto
hoelzl@50325
   701
  then show "eventually (\<lambda>x. P (f x)) F"
hoelzl@50325
   702
  proof eventually_elim
hoelzl@50325
   703
    fix x assume "max b (c + 1) \<le> norm (f x)"
hoelzl@50325
   704
    with P show "P (f x)" by auto
hoelzl@50325
   705
  qed
hoelzl@50325
   706
qed force
hoelzl@50325
   707
hoelzl@50347
   708
subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
hoelzl@50347
   709
hoelzl@50347
   710
text {*
hoelzl@50347
   711
hoelzl@50347
   712
This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
hoelzl@50347
   713
@{term "at_right x"} and also @{term "at_right 0"}.
hoelzl@50347
   714
hoelzl@50347
   715
*}
hoelzl@50347
   716
hoelzl@51471
   717
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
hoelzl@50323
   718
hoelzl@50347
   719
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::real)"
hoelzl@50347
   720
  unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
hoelzl@50347
   721
  by (intro allI ex_cong) (auto simp: dist_real_def field_simps)
hoelzl@50347
   722
hoelzl@50347
   723
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::real)"
hoelzl@50347
   724
  unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
hoelzl@50347
   725
  apply (intro allI ex_cong)
hoelzl@50347
   726
  apply (auto simp: dist_real_def field_simps)
hoelzl@50347
   727
  apply (erule_tac x="-x" in allE)
hoelzl@50347
   728
  apply simp
hoelzl@50347
   729
  done
hoelzl@50347
   730
hoelzl@50347
   731
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::real)"
hoelzl@50347
   732
  unfolding at_def filtermap_nhds_shift[symmetric]
hoelzl@50347
   733
  by (simp add: filter_eq_iff eventually_filtermap eventually_within)
hoelzl@50347
   734
hoelzl@50347
   735
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
hoelzl@50347
   736
  unfolding filtermap_at_shift[symmetric]
hoelzl@50347
   737
  by (simp add: filter_eq_iff eventually_filtermap eventually_within)
hoelzl@50323
   738
hoelzl@50347
   739
lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
hoelzl@50347
   740
  using filtermap_at_right_shift[of "-a" 0] by simp
hoelzl@50347
   741
hoelzl@50347
   742
lemma filterlim_at_right_to_0:
hoelzl@50347
   743
  "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
hoelzl@50347
   744
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
hoelzl@50347
   745
hoelzl@50347
   746
lemma eventually_at_right_to_0:
hoelzl@50347
   747
  "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
hoelzl@50347
   748
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
hoelzl@50347
   749
hoelzl@50347
   750
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::real)"
hoelzl@50347
   751
  unfolding at_def filtermap_nhds_minus[symmetric]
hoelzl@50347
   752
  by (simp add: filter_eq_iff eventually_filtermap eventually_within)
hoelzl@50347
   753
hoelzl@50347
   754
lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
hoelzl@50347
   755
  by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
hoelzl@50323
   756
hoelzl@50347
   757
lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
hoelzl@50347
   758
  by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
hoelzl@50347
   759
hoelzl@50347
   760
lemma filterlim_at_left_to_right:
hoelzl@50347
   761
  "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
hoelzl@50347
   762
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
hoelzl@50347
   763
hoelzl@50347
   764
lemma eventually_at_left_to_right:
hoelzl@50347
   765
  "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
hoelzl@50347
   766
  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
hoelzl@50347
   767
hoelzl@50346
   768
lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
hoelzl@50346
   769
  unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
hoelzl@50346
   770
  by (metis le_minus_iff minus_minus)
hoelzl@50346
   771
hoelzl@50346
   772
lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
hoelzl@50346
   773
  unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
hoelzl@50346
   774
hoelzl@50346
   775
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
hoelzl@50346
   776
  unfolding filterlim_def at_top_mirror filtermap_filtermap ..
hoelzl@50346
   777
hoelzl@50346
   778
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
hoelzl@50346
   779
  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
hoelzl@50346
   780
hoelzl@50323
   781
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@50323
   782
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@50346
   783
  by (metis leI minus_less_iff order_less_asym)
hoelzl@50323
   784
hoelzl@50323
   785
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@50323
   786
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@50346
   787
  by (metis leI less_minus_iff order_less_asym)
hoelzl@50323
   788
hoelzl@50346
   789
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
hoelzl@50346
   790
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
hoelzl@50346
   791
  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
hoelzl@50346
   792
  by auto
hoelzl@50346
   793
hoelzl@50346
   794
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
hoelzl@50346
   795
  unfolding filterlim_uminus_at_top by simp
hoelzl@50323
   796
hoelzl@50347
   797
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
hoelzl@50347
   798
  unfolding filterlim_at_top_gt[where c=0] eventually_within at_def
hoelzl@50347
   799
proof safe
hoelzl@50347
   800
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50347
   801
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50347
   802
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@50347
   803
  then show "eventually (\<lambda>x. x \<in> - {0} \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
hoelzl@50347
   804
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50347
   805
qed
hoelzl@50347
   806
hoelzl@50347
   807
lemma filterlim_inverse_at_top:
hoelzl@50347
   808
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@50347
   809
  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
hoelzl@50347
   810
     (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)
hoelzl@50347
   811
hoelzl@50347
   812
lemma filterlim_inverse_at_bot_neg:
hoelzl@50347
   813
  "LIM x (at_left (0::real)). inverse x :> at_bot"
hoelzl@50347
   814
  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
hoelzl@50347
   815
hoelzl@50347
   816
lemma filterlim_inverse_at_bot:
hoelzl@50347
   817
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@50347
   818
  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
hoelzl@50347
   819
  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
hoelzl@50347
   820
hoelzl@50325
   821
lemma tendsto_inverse_0:
hoelzl@50325
   822
  fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
hoelzl@50325
   823
  shows "(inverse ---> (0::'a)) at_infinity"
hoelzl@50325
   824
  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
hoelzl@50325
   825
proof safe
hoelzl@50325
   826
  fix r :: real assume "0 < r"
hoelzl@50325
   827
  show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
hoelzl@50325
   828
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
hoelzl@50325
   829
    fix x :: 'a
hoelzl@50325
   830
    from `0 < r` have "0 < inverse (r / 2)" by simp
hoelzl@50325
   831
    also assume *: "inverse (r / 2) \<le> norm x"
hoelzl@50325
   832
    finally show "norm (inverse x) < r"
hoelzl@50325
   833
      using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
hoelzl@50325
   834
  qed
hoelzl@50325
   835
qed
hoelzl@50325
   836
hoelzl@50347
   837
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
hoelzl@50347
   838
proof (rule antisym)
hoelzl@50347
   839
  have "(inverse ---> (0::real)) at_top"
hoelzl@50347
   840
    by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
hoelzl@50347
   841
  then show "filtermap inverse at_top \<le> at_right (0::real)"
hoelzl@50347
   842
    unfolding at_within_eq
hoelzl@50347
   843
    by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)
hoelzl@50347
   844
next
hoelzl@50347
   845
  have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
hoelzl@50347
   846
    using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
hoelzl@50347
   847
  then show "at_right (0::real) \<le> filtermap inverse at_top"
hoelzl@50347
   848
    by (simp add: filtermap_ident filtermap_filtermap)
hoelzl@50347
   849
qed
hoelzl@50347
   850
hoelzl@50347
   851
lemma eventually_at_right_to_top:
hoelzl@50347
   852
  "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
hoelzl@50347
   853
  unfolding at_right_to_top eventually_filtermap ..
hoelzl@50347
   854
hoelzl@50347
   855
lemma filterlim_at_right_to_top:
hoelzl@50347
   856
  "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
hoelzl@50347
   857
  unfolding filterlim_def at_right_to_top filtermap_filtermap ..
hoelzl@50347
   858
hoelzl@50347
   859
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
hoelzl@50347
   860
  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
hoelzl@50347
   861
hoelzl@50347
   862
lemma eventually_at_top_to_right:
hoelzl@50347
   863
  "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
hoelzl@50347
   864
  unfolding at_top_to_right eventually_filtermap ..
hoelzl@50347
   865
hoelzl@50347
   866
lemma filterlim_at_top_to_right:
hoelzl@50347
   867
  "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
hoelzl@50347
   868
  unfolding filterlim_def at_top_to_right filtermap_filtermap ..
hoelzl@50347
   869
hoelzl@50325
   870
lemma filterlim_inverse_at_infinity:
hoelzl@50325
   871
  fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
hoelzl@50325
   872
  shows "filterlim inverse at_infinity (at (0::'a))"
hoelzl@50325
   873
  unfolding filterlim_at_infinity[OF order_refl]
hoelzl@50325
   874
proof safe
hoelzl@50325
   875
  fix r :: real assume "0 < r"
hoelzl@50325
   876
  then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
hoelzl@50325
   877
    unfolding eventually_at norm_inverse
hoelzl@50325
   878
    by (intro exI[of _ "inverse r"])
hoelzl@50325
   879
       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
hoelzl@50325
   880
qed
hoelzl@50325
   881
hoelzl@50325
   882
lemma filterlim_inverse_at_iff:
hoelzl@50325
   883
  fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
hoelzl@50325
   884
  shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
hoelzl@50325
   885
  unfolding filterlim_def filtermap_filtermap[symmetric]
hoelzl@50325
   886
proof
hoelzl@50325
   887
  assume "filtermap g F \<le> at_infinity"
hoelzl@50325
   888
  then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
hoelzl@50325
   889
    by (rule filtermap_mono)
hoelzl@50325
   890
  also have "\<dots> \<le> at 0"
hoelzl@50325
   891
    using tendsto_inverse_0
hoelzl@50325
   892
    by (auto intro!: le_withinI exI[of _ 1]
hoelzl@50325
   893
             simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
hoelzl@50325
   894
  finally show "filtermap inverse (filtermap g F) \<le> at 0" .
hoelzl@50325
   895
next
hoelzl@50325
   896
  assume "filtermap inverse (filtermap g F) \<le> at 0"
hoelzl@50325
   897
  then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
hoelzl@50325
   898
    by (rule filtermap_mono)
hoelzl@50325
   899
  with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
hoelzl@50325
   900
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
hoelzl@50325
   901
qed
hoelzl@50325
   902
hoelzl@50419
   903
lemma tendsto_inverse_0_at_top:
hoelzl@50419
   904
  "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
hoelzl@50419
   905
 by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
hoelzl@50419
   906
hoelzl@50324
   907
text {*
hoelzl@50324
   908
hoelzl@50324
   909
We only show rules for multiplication and addition when the functions are either against a real
hoelzl@50324
   910
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
hoelzl@50324
   911
hoelzl@50324
   912
*}
hoelzl@50324
   913
hoelzl@50324
   914
lemma filterlim_tendsto_pos_mult_at_top: 
hoelzl@50324
   915
  assumes f: "(f ---> c) F" and c: "0 < c"
hoelzl@50324
   916
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
   917
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
   918
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
   919
proof safe
hoelzl@50324
   920
  fix Z :: real assume "0 < Z"
hoelzl@50324
   921
  from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
hoelzl@50324
   922
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
hoelzl@50324
   923
             simp: dist_real_def abs_real_def split: split_if_asm)
hoelzl@50346
   924
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
hoelzl@50324
   925
    unfolding filterlim_at_top by auto
hoelzl@50346
   926
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
   927
  proof eventually_elim
hoelzl@50346
   928
    fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
hoelzl@50346
   929
    with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
hoelzl@50346
   930
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
   931
    with `0 < c` show "Z \<le> f x * g x"
hoelzl@50324
   932
       by simp
hoelzl@50324
   933
  qed
hoelzl@50324
   934
qed
hoelzl@50324
   935
hoelzl@50324
   936
lemma filterlim_at_top_mult_at_top: 
hoelzl@50324
   937
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
   938
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
   939
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
   940
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
   941
proof safe
hoelzl@50324
   942
  fix Z :: real assume "0 < Z"
hoelzl@50346
   943
  from f have "eventually (\<lambda>x. 1 \<le> f x) F"
hoelzl@50324
   944
    unfolding filterlim_at_top by auto
hoelzl@50346
   945
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
   946
    unfolding filterlim_at_top by auto
hoelzl@50346
   947
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
   948
  proof eventually_elim
hoelzl@50346
   949
    fix x assume "1 \<le> f x" "Z \<le> g x"
hoelzl@50346
   950
    with `0 < Z` have "1 * Z \<le> f x * g x"
hoelzl@50346
   951
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
   952
    then show "Z \<le> f x * g x"
hoelzl@50324
   953
       by simp
hoelzl@50324
   954
  qed
hoelzl@50324
   955
qed
hoelzl@50324
   956
hoelzl@50419
   957
lemma filterlim_tendsto_pos_mult_at_bot:
hoelzl@50419
   958
  assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
hoelzl@50419
   959
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@50419
   960
  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
hoelzl@50419
   961
  unfolding filterlim_uminus_at_bot by simp
hoelzl@50419
   962
hoelzl@50324
   963
lemma filterlim_tendsto_add_at_top: 
hoelzl@50324
   964
  assumes f: "(f ---> c) F"
hoelzl@50324
   965
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
   966
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
   967
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
   968
proof safe
hoelzl@50324
   969
  fix Z :: real assume "0 < Z"
hoelzl@50324
   970
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
hoelzl@50324
   971
    by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50346
   972
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
hoelzl@50324
   973
    unfolding filterlim_at_top by auto
hoelzl@50346
   974
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
   975
    by eventually_elim simp
hoelzl@50324
   976
qed
hoelzl@50324
   977
hoelzl@50347
   978
lemma LIM_at_top_divide:
hoelzl@50347
   979
  fixes f g :: "'a \<Rightarrow> real"
hoelzl@50347
   980
  assumes f: "(f ---> a) F" "0 < a"
hoelzl@50347
   981
  assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
hoelzl@50347
   982
  shows "LIM x F. f x / g x :> at_top"
hoelzl@50347
   983
  unfolding divide_inverse
hoelzl@50347
   984
  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
hoelzl@50347
   985
hoelzl@50324
   986
lemma filterlim_at_top_add_at_top: 
hoelzl@50324
   987
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
   988
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
   989
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
   990
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
   991
proof safe
hoelzl@50324
   992
  fix Z :: real assume "0 < Z"
hoelzl@50346
   993
  from f have "eventually (\<lambda>x. 0 \<le> f x) F"
hoelzl@50324
   994
    unfolding filterlim_at_top by auto
hoelzl@50346
   995
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
   996
    unfolding filterlim_at_top by auto
hoelzl@50346
   997
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
   998
    by eventually_elim simp
hoelzl@50324
   999
qed
hoelzl@50324
  1000
hoelzl@50331
  1001
lemma tendsto_divide_0:
hoelzl@50331
  1002
  fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
hoelzl@50331
  1003
  assumes f: "(f ---> c) F"
hoelzl@50331
  1004
  assumes g: "LIM x F. g x :> at_infinity"
hoelzl@50331
  1005
  shows "((\<lambda>x. f x / g x) ---> 0) F"
hoelzl@50331
  1006
  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
hoelzl@50331
  1007
hoelzl@50331
  1008
lemma linear_plus_1_le_power:
hoelzl@50331
  1009
  fixes x :: real
hoelzl@50331
  1010
  assumes x: "0 \<le> x"
hoelzl@50331
  1011
  shows "real n * x + 1 \<le> (x + 1) ^ n"
hoelzl@50331
  1012
proof (induct n)
hoelzl@50331
  1013
  case (Suc n)
hoelzl@50331
  1014
  have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
hoelzl@50331
  1015
    by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
hoelzl@50331
  1016
  also have "\<dots> \<le> (x + 1)^Suc n"
hoelzl@50331
  1017
    using Suc x by (simp add: mult_left_mono)
hoelzl@50331
  1018
  finally show ?case .
hoelzl@50331
  1019
qed simp
hoelzl@50331
  1020
hoelzl@50331
  1021
lemma filterlim_realpow_sequentially_gt1:
hoelzl@50331
  1022
  fixes x :: "'a :: real_normed_div_algebra"
hoelzl@50331
  1023
  assumes x[arith]: "1 < norm x"
hoelzl@50331
  1024
  shows "LIM n sequentially. x ^ n :> at_infinity"
hoelzl@50331
  1025
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
hoelzl@50331
  1026
  fix y :: real assume "0 < y"
hoelzl@50331
  1027
  have "0 < norm x - 1" by simp
hoelzl@50331
  1028
  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
hoelzl@50331
  1029
  also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
hoelzl@50331
  1030
  also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
hoelzl@50331
  1031
  also have "\<dots> = norm x ^ N" by simp
hoelzl@50331
  1032
  finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
hoelzl@50331
  1033
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
hoelzl@50331
  1034
  then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
hoelzl@50331
  1035
    unfolding eventually_sequentially
hoelzl@50331
  1036
    by (auto simp: norm_power)
hoelzl@50331
  1037
qed simp
hoelzl@50331
  1038
hoelzl@51471
  1039
hoelzl@51526
  1040
subsection {* Limits of Sequences *}
hoelzl@51526
  1041
hoelzl@51526
  1042
lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
hoelzl@51526
  1043
  by simp
hoelzl@51526
  1044
hoelzl@51526
  1045
lemma LIMSEQ_iff:
hoelzl@51526
  1046
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1047
  shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
hoelzl@51526
  1048
unfolding LIMSEQ_def dist_norm ..
hoelzl@51526
  1049
hoelzl@51526
  1050
lemma LIMSEQ_I:
hoelzl@51526
  1051
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1052
  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
hoelzl@51526
  1053
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1054
hoelzl@51526
  1055
lemma LIMSEQ_D:
hoelzl@51526
  1056
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1057
  shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
hoelzl@51526
  1058
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1059
hoelzl@51526
  1060
lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
hoelzl@51526
  1061
  unfolding tendsto_def eventually_sequentially
hoelzl@51526
  1062
  by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
hoelzl@51526
  1063
hoelzl@51526
  1064
lemma Bseq_inverse_lemma:
hoelzl@51526
  1065
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51526
  1066
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
hoelzl@51526
  1067
apply (subst nonzero_norm_inverse, clarsimp)
hoelzl@51526
  1068
apply (erule (1) le_imp_inverse_le)
hoelzl@51526
  1069
done
hoelzl@51526
  1070
hoelzl@51526
  1071
lemma Bseq_inverse:
hoelzl@51526
  1072
  fixes a :: "'a::real_normed_div_algebra"
hoelzl@51526
  1073
  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
hoelzl@51526
  1074
  by (rule Bfun_inverse)
hoelzl@51526
  1075
hoelzl@51526
  1076
lemma LIMSEQ_diff_approach_zero:
hoelzl@51526
  1077
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1078
  shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
hoelzl@51526
  1079
  by (drule (1) tendsto_add, simp)
hoelzl@51526
  1080
hoelzl@51526
  1081
lemma LIMSEQ_diff_approach_zero2:
hoelzl@51526
  1082
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1083
  shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
hoelzl@51526
  1084
  by (drule (1) tendsto_diff, simp)
hoelzl@51526
  1085
hoelzl@51526
  1086
text{*An unbounded sequence's inverse tends to 0*}
hoelzl@51526
  1087
hoelzl@51526
  1088
lemma LIMSEQ_inverse_zero:
hoelzl@51526
  1089
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
hoelzl@51526
  1090
  apply (rule filterlim_compose[OF tendsto_inverse_0])
hoelzl@51526
  1091
  apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
hoelzl@51526
  1092
  apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
hoelzl@51526
  1093
  done
hoelzl@51526
  1094
hoelzl@51526
  1095
text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
hoelzl@51526
  1096
hoelzl@51526
  1097
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
hoelzl@51526
  1098
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
hoelzl@51526
  1099
            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
hoelzl@51526
  1100
hoelzl@51526
  1101
text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
hoelzl@51526
  1102
infinity is now easily proved*}
hoelzl@51526
  1103
hoelzl@51526
  1104
lemma LIMSEQ_inverse_real_of_nat_add:
hoelzl@51526
  1105
     "(%n. r + inverse(real(Suc n))) ----> r"
hoelzl@51526
  1106
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
hoelzl@51526
  1107
hoelzl@51526
  1108
lemma LIMSEQ_inverse_real_of_nat_add_minus:
hoelzl@51526
  1109
     "(%n. r + -inverse(real(Suc n))) ----> r"
hoelzl@51526
  1110
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
hoelzl@51526
  1111
  by auto
hoelzl@51526
  1112
hoelzl@51526
  1113
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
hoelzl@51526
  1114
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
hoelzl@51526
  1115
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
hoelzl@51526
  1116
  by auto
hoelzl@51526
  1117
hoelzl@51526
  1118
subsection {* Convergence on sequences *}
hoelzl@51526
  1119
hoelzl@51526
  1120
lemma convergent_add:
hoelzl@51526
  1121
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1122
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1123
  assumes "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1124
  shows "convergent (\<lambda>n. X n + Y n)"
hoelzl@51526
  1125
  using assms unfolding convergent_def by (fast intro: tendsto_add)
hoelzl@51526
  1126
hoelzl@51526
  1127
lemma convergent_setsum:
hoelzl@51526
  1128
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1129
  assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
hoelzl@51526
  1130
  shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
hoelzl@51526
  1131
proof (cases "finite A")
hoelzl@51526
  1132
  case True from this and assms show ?thesis
hoelzl@51526
  1133
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
hoelzl@51526
  1134
qed (simp add: convergent_const)
hoelzl@51526
  1135
hoelzl@51526
  1136
lemma (in bounded_linear) convergent:
hoelzl@51526
  1137
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1138
  shows "convergent (\<lambda>n. f (X n))"
hoelzl@51526
  1139
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1140
hoelzl@51526
  1141
lemma (in bounded_bilinear) convergent:
hoelzl@51526
  1142
  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1143
  shows "convergent (\<lambda>n. X n ** Y n)"
hoelzl@51526
  1144
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1145
hoelzl@51526
  1146
lemma convergent_minus_iff:
hoelzl@51526
  1147
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1148
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
hoelzl@51526
  1149
apply (simp add: convergent_def)
hoelzl@51526
  1150
apply (auto dest: tendsto_minus)
hoelzl@51526
  1151
apply (drule tendsto_minus, auto)
hoelzl@51526
  1152
done
hoelzl@51526
  1153
hoelzl@51526
  1154
subsection {* Bounded Monotonic Sequences *}
hoelzl@51526
  1155
hoelzl@51526
  1156
subsubsection {* Bounded Sequences *}
hoelzl@51526
  1157
hoelzl@51526
  1158
lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
hoelzl@51526
  1159
  by (intro BfunI) (auto simp: eventually_sequentially)
hoelzl@51526
  1160
hoelzl@51526
  1161
lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
hoelzl@51526
  1162
  by (intro BfunI) (auto simp: eventually_sequentially)
hoelzl@51526
  1163
hoelzl@51526
  1164
lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
hoelzl@51526
  1165
  unfolding Bfun_def eventually_sequentially
hoelzl@51526
  1166
proof safe
hoelzl@51526
  1167
  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
hoelzl@51526
  1168
  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
hoelzl@51526
  1169
    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] min_max.less_supI2)
hoelzl@51526
  1170
       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
hoelzl@51526
  1171
qed auto
hoelzl@51526
  1172
hoelzl@51526
  1173
lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
hoelzl@51526
  1174
unfolding Bseq_def by auto
hoelzl@51526
  1175
hoelzl@51526
  1176
lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
hoelzl@51526
  1177
by (simp add: Bseq_def)
hoelzl@51526
  1178
hoelzl@51526
  1179
lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
hoelzl@51526
  1180
by (auto simp add: Bseq_def)
hoelzl@51526
  1181
hoelzl@51526
  1182
lemma lemma_NBseq_def:
hoelzl@51526
  1183
  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
hoelzl@51526
  1184
proof safe
hoelzl@51526
  1185
  fix K :: real
hoelzl@51526
  1186
  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
hoelzl@51526
  1187
  then have "K \<le> real (Suc n)" by auto
hoelzl@51526
  1188
  moreover assume "\<forall>m. norm (X m) \<le> K"
hoelzl@51526
  1189
  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
hoelzl@51526
  1190
    by (blast intro: order_trans)
hoelzl@51526
  1191
  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
hoelzl@51526
  1192
qed (force simp add: real_of_nat_Suc)
hoelzl@51526
  1193
hoelzl@51526
  1194
text{* alternative definition for Bseq *}
hoelzl@51526
  1195
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
hoelzl@51526
  1196
apply (simp add: Bseq_def)
hoelzl@51526
  1197
apply (simp (no_asm) add: lemma_NBseq_def)
hoelzl@51526
  1198
done
hoelzl@51526
  1199
hoelzl@51526
  1200
lemma lemma_NBseq_def2:
hoelzl@51526
  1201
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
hoelzl@51526
  1202
apply (subst lemma_NBseq_def, auto)
hoelzl@51526
  1203
apply (rule_tac x = "Suc N" in exI)
hoelzl@51526
  1204
apply (rule_tac [2] x = N in exI)
hoelzl@51526
  1205
apply (auto simp add: real_of_nat_Suc)
hoelzl@51526
  1206
 prefer 2 apply (blast intro: order_less_imp_le)
hoelzl@51526
  1207
apply (drule_tac x = n in spec, simp)
hoelzl@51526
  1208
done
hoelzl@51526
  1209
hoelzl@51526
  1210
(* yet another definition for Bseq *)
hoelzl@51526
  1211
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
hoelzl@51526
  1212
by (simp add: Bseq_def lemma_NBseq_def2)
hoelzl@51526
  1213
hoelzl@51526
  1214
subsubsection{*A Few More Equivalence Theorems for Boundedness*}
hoelzl@51526
  1215
hoelzl@51526
  1216
text{*alternative formulation for boundedness*}
hoelzl@51526
  1217
lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
hoelzl@51526
  1218
apply (unfold Bseq_def, safe)
hoelzl@51526
  1219
apply (rule_tac [2] x = "k + norm x" in exI)
hoelzl@51526
  1220
apply (rule_tac x = K in exI, simp)
hoelzl@51526
  1221
apply (rule exI [where x = 0], auto)
hoelzl@51526
  1222
apply (erule order_less_le_trans, simp)
hoelzl@51526
  1223
apply (drule_tac x=n in spec, fold diff_minus)
hoelzl@51526
  1224
apply (drule order_trans [OF norm_triangle_ineq2])
hoelzl@51526
  1225
apply simp
hoelzl@51526
  1226
done
hoelzl@51526
  1227
hoelzl@51526
  1228
text{*alternative formulation for boundedness*}
hoelzl@51526
  1229
lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
hoelzl@51526
  1230
apply safe
hoelzl@51526
  1231
apply (simp add: Bseq_def, safe)
hoelzl@51526
  1232
apply (rule_tac x = "K + norm (X N)" in exI)
hoelzl@51526
  1233
apply auto
hoelzl@51526
  1234
apply (erule order_less_le_trans, simp)
hoelzl@51526
  1235
apply (rule_tac x = N in exI, safe)
hoelzl@51526
  1236
apply (drule_tac x = n in spec)
hoelzl@51526
  1237
apply (rule order_trans [OF norm_triangle_ineq], simp)
hoelzl@51526
  1238
apply (auto simp add: Bseq_iff2)
hoelzl@51526
  1239
done
hoelzl@51526
  1240
hoelzl@51526
  1241
lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
hoelzl@51526
  1242
apply (simp add: Bseq_def)
hoelzl@51526
  1243
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
hoelzl@51526
  1244
apply (drule_tac x = n in spec, arith)
hoelzl@51526
  1245
done
hoelzl@51526
  1246
hoelzl@51526
  1247
subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
hoelzl@51526
  1248
hoelzl@51526
  1249
lemma Bseq_isUb:
hoelzl@51526
  1250
  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
hoelzl@51526
  1251
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
hoelzl@51526
  1252
hoelzl@51526
  1253
text{* Use completeness of reals (supremum property)
hoelzl@51526
  1254
   to show that any bounded sequence has a least upper bound*}
hoelzl@51526
  1255
hoelzl@51526
  1256
lemma Bseq_isLub:
hoelzl@51526
  1257
  "!!(X::nat=>real). Bseq X ==>
hoelzl@51526
  1258
   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
hoelzl@51526
  1259
by (blast intro: reals_complete Bseq_isUb)
hoelzl@51526
  1260
hoelzl@51526
  1261
subsubsection{*A Bounded and Monotonic Sequence Converges*}
hoelzl@51526
  1262
hoelzl@51526
  1263
(* TODO: delete *)
hoelzl@51526
  1264
(* FIXME: one use in NSA/HSEQ.thy *)
hoelzl@51526
  1265
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
hoelzl@51526
  1266
  apply (rule_tac x="X m" in exI)
hoelzl@51526
  1267
  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
hoelzl@51526
  1268
  unfolding eventually_sequentially
hoelzl@51526
  1269
  apply blast
hoelzl@51526
  1270
  done
hoelzl@51526
  1271
hoelzl@51526
  1272
text {* A monotone sequence converges to its least upper bound. *}
hoelzl@51526
  1273
hoelzl@51526
  1274
lemma isLub_mono_imp_LIMSEQ:
hoelzl@51526
  1275
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1276
  assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
hoelzl@51526
  1277
  assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
hoelzl@51526
  1278
  shows "X ----> u"
hoelzl@51526
  1279
proof (rule LIMSEQ_I)
hoelzl@51526
  1280
  have 1: "\<forall>n. X n \<le> u"
hoelzl@51526
  1281
    using isLubD2 [OF u] by auto
hoelzl@51526
  1282
  have "\<forall>y. (\<forall>n. X n \<le> y) \<longrightarrow> u \<le> y"
hoelzl@51526
  1283
    using isLub_le_isUb [OF u] by (auto simp add: isUb_def setle_def)
hoelzl@51526
  1284
  hence 2: "\<forall>y<u. \<exists>n. y < X n"
hoelzl@51526
  1285
    by (metis not_le)
hoelzl@51526
  1286
  fix r :: real assume "0 < r"
hoelzl@51526
  1287
  hence "u - r < u" by simp
hoelzl@51526
  1288
  hence "\<exists>m. u - r < X m" using 2 by simp
hoelzl@51526
  1289
  then obtain m where "u - r < X m" ..
hoelzl@51526
  1290
  with X have "\<forall>n\<ge>m. u - r < X n"
hoelzl@51526
  1291
    by (fast intro: less_le_trans)
hoelzl@51526
  1292
  hence "\<exists>m. \<forall>n\<ge>m. u - r < X n" ..
hoelzl@51526
  1293
  thus "\<exists>m. \<forall>n\<ge>m. norm (X n - u) < r"
hoelzl@51526
  1294
    using 1 by (simp add: diff_less_eq add_commute)
hoelzl@51526
  1295
qed
hoelzl@51526
  1296
hoelzl@51526
  1297
text{*A standard proof of the theorem for monotone increasing sequence*}
hoelzl@51526
  1298
hoelzl@51526
  1299
lemma Bseq_mono_convergent:
hoelzl@51526
  1300
   "Bseq X \<Longrightarrow> \<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> convergent (X::nat=>real)"
hoelzl@51526
  1301
  by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)
hoelzl@51526
  1302
hoelzl@51526
  1303
lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
hoelzl@51526
  1304
  by (simp add: Bseq_def)
hoelzl@51526
  1305
hoelzl@51526
  1306
text{*Main monotonicity theorem*}
hoelzl@51526
  1307
hoelzl@51526
  1308
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@51526
  1309
  by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
hoelzl@51526
  1310
            Bseq_mono_convergent)
hoelzl@51526
  1311
hoelzl@51526
  1312
lemma Cauchy_iff:
hoelzl@51526
  1313
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1314
  shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
hoelzl@51526
  1315
  unfolding Cauchy_def dist_norm ..
hoelzl@51526
  1316
hoelzl@51526
  1317
lemma CauchyI:
hoelzl@51526
  1318
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1319
  shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
hoelzl@51526
  1320
by (simp add: Cauchy_iff)
hoelzl@51526
  1321
hoelzl@51526
  1322
lemma CauchyD:
hoelzl@51526
  1323
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1324
  shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
hoelzl@51526
  1325
by (simp add: Cauchy_iff)
hoelzl@51526
  1326
hoelzl@51526
  1327
lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
hoelzl@51526
  1328
  apply (simp add: subset_eq)
hoelzl@51526
  1329
  apply (rule BseqI'[where K="max (norm a) (norm b)"])
hoelzl@51526
  1330
  apply (erule_tac x=n in allE)
hoelzl@51526
  1331
  apply auto
hoelzl@51526
  1332
  done
hoelzl@51526
  1333
hoelzl@51526
  1334
lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
hoelzl@51526
  1335
  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
hoelzl@51526
  1336
hoelzl@51526
  1337
lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
hoelzl@51526
  1338
  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
hoelzl@51526
  1339
hoelzl@51526
  1340
lemma incseq_convergent:
hoelzl@51526
  1341
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1342
  assumes "incseq X" and "\<forall>i. X i \<le> B"
hoelzl@51526
  1343
  obtains L where "X ----> L" "\<forall>i. X i \<le> L"
hoelzl@51526
  1344
proof atomize_elim
hoelzl@51526
  1345
  from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
hoelzl@51526
  1346
  obtain L where "X ----> L"
hoelzl@51526
  1347
    by (auto simp: convergent_def monoseq_def incseq_def)
hoelzl@51526
  1348
  with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
hoelzl@51526
  1349
    by (auto intro!: exI[of _ L] incseq_le)
hoelzl@51526
  1350
qed
hoelzl@51526
  1351
hoelzl@51526
  1352
lemma decseq_convergent:
hoelzl@51526
  1353
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1354
  assumes "decseq X" and "\<forall>i. B \<le> X i"
hoelzl@51526
  1355
  obtains L where "X ----> L" "\<forall>i. L \<le> X i"
hoelzl@51526
  1356
proof atomize_elim
hoelzl@51526
  1357
  from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
hoelzl@51526
  1358
  obtain L where "X ----> L"
hoelzl@51526
  1359
    by (auto simp: convergent_def monoseq_def decseq_def)
hoelzl@51526
  1360
  with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
hoelzl@51526
  1361
    by (auto intro!: exI[of _ L] decseq_le)
hoelzl@51526
  1362
qed
hoelzl@51526
  1363
hoelzl@51526
  1364
subsubsection {* Cauchy Sequences are Bounded *}
hoelzl@51526
  1365
hoelzl@51526
  1366
text{*A Cauchy sequence is bounded -- this is the standard
hoelzl@51526
  1367
  proof mechanization rather than the nonstandard proof*}
hoelzl@51526
  1368
hoelzl@51526
  1369
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
hoelzl@51526
  1370
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
hoelzl@51526
  1371
apply (clarify, drule spec, drule (1) mp)
hoelzl@51526
  1372
apply (simp only: norm_minus_commute)
hoelzl@51526
  1373
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
hoelzl@51526
  1374
apply simp
hoelzl@51526
  1375
done
hoelzl@51526
  1376
hoelzl@51526
  1377
class banach = real_normed_vector + complete_space
hoelzl@51526
  1378
hoelzl@51526
  1379
instance real :: banach by default
hoelzl@51526
  1380
hoelzl@51526
  1381
subsection {* Power Sequences *}
hoelzl@51526
  1382
hoelzl@51526
  1383
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
hoelzl@51526
  1384
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
hoelzl@51526
  1385
  also fact that bounded and monotonic sequence converges.*}
hoelzl@51526
  1386
hoelzl@51526
  1387
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
hoelzl@51526
  1388
apply (simp add: Bseq_def)
hoelzl@51526
  1389
apply (rule_tac x = 1 in exI)
hoelzl@51526
  1390
apply (simp add: power_abs)
hoelzl@51526
  1391
apply (auto dest: power_mono)
hoelzl@51526
  1392
done
hoelzl@51526
  1393
hoelzl@51526
  1394
lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
hoelzl@51526
  1395
apply (clarify intro!: mono_SucI2)
hoelzl@51526
  1396
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
hoelzl@51526
  1397
done
hoelzl@51526
  1398
hoelzl@51526
  1399
lemma convergent_realpow:
hoelzl@51526
  1400
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
hoelzl@51526
  1401
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
hoelzl@51526
  1402
hoelzl@51526
  1403
lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
hoelzl@51526
  1404
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1405
hoelzl@51526
  1406
lemma LIMSEQ_realpow_zero:
hoelzl@51526
  1407
  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1408
proof cases
hoelzl@51526
  1409
  assume "0 \<le> x" and "x \<noteq> 0"
hoelzl@51526
  1410
  hence x0: "0 < x" by simp
hoelzl@51526
  1411
  assume x1: "x < 1"
hoelzl@51526
  1412
  from x0 x1 have "1 < inverse x"
hoelzl@51526
  1413
    by (rule one_less_inverse)
hoelzl@51526
  1414
  hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
hoelzl@51526
  1415
    by (rule LIMSEQ_inverse_realpow_zero)
hoelzl@51526
  1416
  thus ?thesis by (simp add: power_inverse)
hoelzl@51526
  1417
qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)
hoelzl@51526
  1418
hoelzl@51526
  1419
lemma LIMSEQ_power_zero:
hoelzl@51526
  1420
  fixes x :: "'a::{real_normed_algebra_1}"
hoelzl@51526
  1421
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1422
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
hoelzl@51526
  1423
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
hoelzl@51526
  1424
apply (simp add: power_abs norm_power_ineq)
hoelzl@51526
  1425
done
hoelzl@51526
  1426
hoelzl@51526
  1427
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
hoelzl@51526
  1428
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1429
hoelzl@51526
  1430
text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
hoelzl@51526
  1431
hoelzl@51526
  1432
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
hoelzl@51526
  1433
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
hoelzl@51526
  1434
hoelzl@51526
  1435
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
hoelzl@51526
  1436
  by (rule LIMSEQ_power_zero) simp
hoelzl@51526
  1437
hoelzl@51526
  1438
hoelzl@51526
  1439
subsection {* Limits of Functions *}
hoelzl@51526
  1440
hoelzl@51526
  1441
lemma LIM_eq:
hoelzl@51526
  1442
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1443
  shows "f -- a --> L =
hoelzl@51526
  1444
     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
hoelzl@51526
  1445
by (simp add: LIM_def dist_norm)
hoelzl@51526
  1446
hoelzl@51526
  1447
lemma LIM_I:
hoelzl@51526
  1448
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1449
  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
hoelzl@51526
  1450
      ==> f -- a --> L"
hoelzl@51526
  1451
by (simp add: LIM_eq)
hoelzl@51526
  1452
hoelzl@51526
  1453
lemma LIM_D:
hoelzl@51526
  1454
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1455
  shows "[| f -- a --> L; 0<r |]
hoelzl@51526
  1456
      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
hoelzl@51526
  1457
by (simp add: LIM_eq)
hoelzl@51526
  1458
hoelzl@51526
  1459
lemma LIM_offset:
hoelzl@51526
  1460
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1461
  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
hoelzl@51526
  1462
apply (rule topological_tendstoI)
hoelzl@51526
  1463
apply (drule (2) topological_tendstoD)
hoelzl@51526
  1464
apply (simp only: eventually_at dist_norm)
hoelzl@51526
  1465
apply (clarify, rule_tac x=d in exI, safe)
hoelzl@51526
  1466
apply (drule_tac x="x + k" in spec)
hoelzl@51526
  1467
apply (simp add: algebra_simps)
hoelzl@51526
  1468
done
hoelzl@51526
  1469
hoelzl@51526
  1470
lemma LIM_offset_zero:
hoelzl@51526
  1471
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1472
  shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
hoelzl@51526
  1473
by (drule_tac k="a" in LIM_offset, simp add: add_commute)
hoelzl@51526
  1474
hoelzl@51526
  1475
lemma LIM_offset_zero_cancel:
hoelzl@51526
  1476
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1477
  shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
hoelzl@51526
  1478
by (drule_tac k="- a" in LIM_offset, simp)
hoelzl@51526
  1479
hoelzl@51526
  1480
lemma LIM_zero:
hoelzl@51526
  1481
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1482
  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
hoelzl@51526
  1483
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1484
hoelzl@51526
  1485
lemma LIM_zero_cancel:
hoelzl@51526
  1486
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1487
  shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
hoelzl@51526
  1488
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1489
hoelzl@51526
  1490
lemma LIM_zero_iff:
hoelzl@51526
  1491
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1492
  shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
hoelzl@51526
  1493
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1494
hoelzl@51526
  1495
lemma LIM_imp_LIM:
hoelzl@51526
  1496
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1497
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  1498
  assumes f: "f -- a --> l"
hoelzl@51526
  1499
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
hoelzl@51526
  1500
  shows "g -- a --> m"
hoelzl@51526
  1501
  by (rule metric_LIM_imp_LIM [OF f],
hoelzl@51526
  1502
    simp add: dist_norm le)
hoelzl@51526
  1503
hoelzl@51526
  1504
lemma LIM_equal2:
hoelzl@51526
  1505
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1506
  assumes 1: "0 < R"
hoelzl@51526
  1507
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
hoelzl@51526
  1508
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
hoelzl@51526
  1509
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
hoelzl@51526
  1510
hoelzl@51526
  1511
lemma LIM_compose2:
hoelzl@51526
  1512
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1513
  assumes f: "f -- a --> b"
hoelzl@51526
  1514
  assumes g: "g -- b --> c"
hoelzl@51526
  1515
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
hoelzl@51526
  1516
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51526
  1517
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
hoelzl@51526
  1518
hoelzl@51526
  1519
lemma real_LIM_sandwich_zero:
hoelzl@51526
  1520
  fixes f g :: "'a::topological_space \<Rightarrow> real"
hoelzl@51526
  1521
  assumes f: "f -- a --> 0"
hoelzl@51526
  1522
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
hoelzl@51526
  1523
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
hoelzl@51526
  1524
  shows "g -- a --> 0"
hoelzl@51526
  1525
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
hoelzl@51526
  1526
  fix x assume x: "x \<noteq> a"
hoelzl@51526
  1527
  have "norm (g x - 0) = g x" by (simp add: 1 x)
hoelzl@51526
  1528
  also have "g x \<le> f x" by (rule 2 [OF x])
hoelzl@51526
  1529
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
hoelzl@51526
  1530
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
hoelzl@51526
  1531
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
hoelzl@51526
  1532
qed
hoelzl@51526
  1533
hoelzl@51526
  1534
hoelzl@51526
  1535
subsection {* Continuity *}
hoelzl@51526
  1536
hoelzl@51526
  1537
lemma LIM_isCont_iff:
hoelzl@51526
  1538
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1539
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
hoelzl@51526
  1540
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
hoelzl@51526
  1541
hoelzl@51526
  1542
lemma isCont_iff:
hoelzl@51526
  1543
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1544
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
hoelzl@51526
  1545
by (simp add: isCont_def LIM_isCont_iff)
hoelzl@51526
  1546
hoelzl@51526
  1547
lemma isCont_LIM_compose2:
hoelzl@51526
  1548
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1549
  assumes f [unfolded isCont_def]: "isCont f a"
hoelzl@51526
  1550
  assumes g: "g -- f a --> l"
hoelzl@51526
  1551
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
hoelzl@51526
  1552
  shows "(\<lambda>x. g (f x)) -- a --> l"
hoelzl@51526
  1553
by (rule LIM_compose2 [OF f g inj])
hoelzl@51526
  1554
hoelzl@51526
  1555
hoelzl@51526
  1556
lemma isCont_norm [simp]:
hoelzl@51526
  1557
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1558
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
hoelzl@51526
  1559
  by (fact continuous_norm)
hoelzl@51526
  1560
hoelzl@51526
  1561
lemma isCont_rabs [simp]:
hoelzl@51526
  1562
  fixes f :: "'a::t2_space \<Rightarrow> real"
hoelzl@51526
  1563
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
hoelzl@51526
  1564
  by (fact continuous_rabs)
hoelzl@51526
  1565
hoelzl@51526
  1566
lemma isCont_add [simp]:
hoelzl@51526
  1567
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1568
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
hoelzl@51526
  1569
  by (fact continuous_add)
hoelzl@51526
  1570
hoelzl@51526
  1571
lemma isCont_minus [simp]:
hoelzl@51526
  1572
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1573
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
hoelzl@51526
  1574
  by (fact continuous_minus)
hoelzl@51526
  1575
hoelzl@51526
  1576
lemma isCont_diff [simp]:
hoelzl@51526
  1577
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1578
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
hoelzl@51526
  1579
  by (fact continuous_diff)
hoelzl@51526
  1580
hoelzl@51526
  1581
lemma isCont_mult [simp]:
hoelzl@51526
  1582
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
hoelzl@51526
  1583
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
hoelzl@51526
  1584
  by (fact continuous_mult)
hoelzl@51526
  1585
hoelzl@51526
  1586
lemma (in bounded_linear) isCont:
hoelzl@51526
  1587
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
hoelzl@51526
  1588
  by (fact continuous)
hoelzl@51526
  1589
hoelzl@51526
  1590
lemma (in bounded_bilinear) isCont:
hoelzl@51526
  1591
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
hoelzl@51526
  1592
  by (fact continuous)
hoelzl@51526
  1593
hoelzl@51526
  1594
lemmas isCont_scaleR [simp] = 
hoelzl@51526
  1595
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
hoelzl@51526
  1596
hoelzl@51526
  1597
lemmas isCont_of_real [simp] =
hoelzl@51526
  1598
  bounded_linear.isCont [OF bounded_linear_of_real]
hoelzl@51526
  1599
hoelzl@51526
  1600
lemma isCont_power [simp]:
hoelzl@51526
  1601
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51526
  1602
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
hoelzl@51526
  1603
  by (fact continuous_power)
hoelzl@51526
  1604
hoelzl@51526
  1605
lemma isCont_setsum [simp]:
hoelzl@51526
  1606
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  1607
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
hoelzl@51526
  1608
  by (auto intro: continuous_setsum)
hoelzl@51526
  1609
hoelzl@51526
  1610
lemmas isCont_intros =
hoelzl@51526
  1611
  isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
hoelzl@51526
  1612
  isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
hoelzl@51526
  1613
  isCont_of_real isCont_power isCont_sgn isCont_setsum
hoelzl@51526
  1614
hoelzl@51526
  1615
subsection {* Uniform Continuity *}
hoelzl@51526
  1616
hoelzl@51526
  1617
lemma (in bounded_linear) isUCont: "isUCont f"
hoelzl@51526
  1618
unfolding isUCont_def dist_norm
hoelzl@51526
  1619
proof (intro allI impI)
hoelzl@51526
  1620
  fix r::real assume r: "0 < r"
hoelzl@51526
  1621
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
hoelzl@51526
  1622
    using pos_bounded by fast
hoelzl@51526
  1623
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
hoelzl@51526
  1624
  proof (rule exI, safe)
hoelzl@51526
  1625
    from r K show "0 < r / K" by (rule divide_pos_pos)
hoelzl@51526
  1626
  next
hoelzl@51526
  1627
    fix x y :: 'a
hoelzl@51526
  1628
    assume xy: "norm (x - y) < r / K"
hoelzl@51526
  1629
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
hoelzl@51526
  1630
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
hoelzl@51526
  1631
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
hoelzl@51526
  1632
    finally show "norm (f x - f y) < r" .
hoelzl@51526
  1633
  qed
hoelzl@51526
  1634
qed
hoelzl@51526
  1635
hoelzl@51526
  1636
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51526
  1637
by (rule isUCont [THEN isUCont_Cauchy])
hoelzl@51526
  1638
hoelzl@51526
  1639
hoelzl@51526
  1640
lemma LIM_less_bound: 
hoelzl@51526
  1641
  fixes f :: "real \<Rightarrow> real"
hoelzl@51526
  1642
  assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
hoelzl@51526
  1643
  shows "0 \<le> f x"
hoelzl@51526
  1644
proof (rule tendsto_le_const)
hoelzl@51526
  1645
  show "(f ---> f x) (at_left x)"
hoelzl@51526
  1646
    using `isCont f x` by (simp add: filterlim_at_split isCont_def)
hoelzl@51526
  1647
  show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
hoelzl@51526
  1648
    using ev by (auto simp: eventually_within_less dist_real_def intro!: exI[of _ "x - b"])
hoelzl@51526
  1649
qed simp
hoelzl@51471
  1650
huffman@31349
  1651
end
hoelzl@50324
  1652