src/HOL/Limits.thy
author hoelzl
Mon Feb 08 19:53:49 2016 +0100 (2016-02-08)
changeset 62368 106569399cd6
parent 62101 26c0a70f78a3
child 62369 acfc4ad7b76a
permissions -rw-r--r--
add type class for topological monoids
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(*  Title:      HOL/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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section \<open>Limits on Real Vector Spaces\<close>
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theory Limits
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imports Real_Vector_Spaces
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begin
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subsection \<open>Filter going to infinity norm\<close>
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definition at_infinity :: "'a::real_normed_vector filter" where
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  "at_infinity = (INF r. principal {x. r \<le> norm x})"
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lemma eventually_at_infinity: "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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  by (subst eventually_INF_base)
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     (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
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lemma at_infinity_eq_at_top_bot:
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  "(at_infinity :: real filter) = sup at_top at_bot"
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  apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
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                   eventually_at_top_linorder eventually_at_bot_linorder)
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  apply safe
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  apply (rule_tac x="b" in exI, simp)
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  apply (rule_tac x="- b" in exI, simp)
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  apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
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  done
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lemma at_top_le_at_infinity: "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: "at_bot \<le> (at_infinity :: real filter)"
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  unfolding at_infinity_eq_at_top_bot by simp
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lemma filterlim_at_top_imp_at_infinity:
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  fixes f :: "_ \<Rightarrow> real"
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  shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
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  by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
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lemma lim_infinity_imp_sequentially:
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  "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
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by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
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subsubsection \<open>Boundedness\<close>
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definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
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  Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
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abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
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  "Bseq X \<equiv> Bfun X sequentially"
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lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
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lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
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  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
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lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
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  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
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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 \<open>0 < K\<close> 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_mono, 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 blast
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lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
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  unfolding Cauchy_def Bfun_metric_def eventually_sequentially
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  apply (erule_tac x=1 in allE)
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  apply simp
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  apply safe
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  apply (rule_tac x="X M" in exI)
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  apply (rule_tac x=1 in exI)
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  apply (erule_tac x=M in allE)
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  apply simp
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  apply (rule_tac x=M in exI)
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  apply (auto simp: dist_commute)
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  done
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subsubsection \<open>Bounded Sequences\<close>
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lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
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  by (intro BfunI) (auto simp: eventually_sequentially)
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lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
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  by (intro BfunI) (auto simp: eventually_sequentially)
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lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
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  unfolding Bfun_def eventually_sequentially
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proof safe
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  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
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  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
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    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
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       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
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qed auto
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lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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unfolding Bseq_def by auto
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lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
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by (simp add: Bseq_def)
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lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
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by (auto simp add: Bseq_def)
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lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
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proof (elim BseqE, intro bdd_aboveI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
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    by (auto elim!: allE[of _ n])
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qed
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lemma Bseq_bdd_above':
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  "Bseq (X::nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
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proof (elim BseqE, intro bdd_aboveI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "norm (X n) \<le> K"
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    by (auto elim!: allE[of _ n])
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qed
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lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
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proof (elim BseqE, intro bdd_belowI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
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    by (auto elim!: allE[of _ n])
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qed
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lemma Bseq_eventually_mono:
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  assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"
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  shows   "Bseq f"
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proof -
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  from assms(1) obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> norm (g n)"
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    by (auto simp: eventually_at_top_linorder)
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  moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K" by (blast elim!: BseqE)
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  ultimately have "norm (f n) \<le> max K (Max {norm (f n) |n. n < N})" for n
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    apply (cases "n < N")
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    apply (rule max.coboundedI2, rule Max.coboundedI, auto) []
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    apply (rule max.coboundedI1, force intro: order.trans[OF N K])
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    done
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  thus ?thesis by (blast intro: BseqI')
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qed
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lemma lemma_NBseq_def:
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  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
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proof safe
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  fix K :: real
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  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
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  then have "K \<le> real (Suc n)" by auto
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  moreover assume "\<forall>m. norm (X m) \<le> K"
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  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
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    by (blast intro: order_trans)
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  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
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next
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  show "\<And>N. \<forall>n. norm (X n) \<le> real (Suc N) \<Longrightarrow> \<exists>K>0. \<forall>n. norm (X n) \<le> K"
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    using of_nat_0_less_iff by blast
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qed
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text\<open>alternative definition for Bseq\<close>
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lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
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apply (simp add: Bseq_def)
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apply (simp (no_asm) add: lemma_NBseq_def)
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done
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lemma lemma_NBseq_def2:
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     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
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apply (subst lemma_NBseq_def, auto)
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apply (rule_tac x = "Suc N" in exI)
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apply (rule_tac [2] x = N in exI)
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apply (auto simp add: of_nat_Suc)
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 prefer 2 apply (blast intro: order_less_imp_le)
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apply (drule_tac x = n in spec, simp)
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done
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(* yet another definition for Bseq *)
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lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
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by (simp add: Bseq_def lemma_NBseq_def2)
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subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>
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text\<open>alternative formulation for boundedness\<close>
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lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
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apply (unfold Bseq_def, safe)
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apply (rule_tac [2] x = "k + norm x" in exI)
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apply (rule_tac x = K in exI, simp)
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apply (rule exI [where x = 0], auto)
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apply (erule order_less_le_trans, simp)
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apply (drule_tac x=n in spec)
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apply (drule order_trans [OF norm_triangle_ineq2])
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apply simp
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done
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text\<open>alternative formulation for boundedness\<close>
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lemma Bseq_iff3:
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  "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P
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  then obtain K
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    where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
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  from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
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  from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
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    by (auto intro: order_trans norm_triangle_ineq4)
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  then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
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    by simp
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  with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
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next
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  assume ?Q then show ?P by (auto simp add: Bseq_iff2)
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qed
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lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
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apply (simp add: Bseq_def)
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apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
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apply (drule_tac x = n in spec, arith)
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done
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subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>
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lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
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  by (simp add: Bseq_def)
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lemma Bseq_add:
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  assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
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  shows   "Bseq (\<lambda>x. f x + c)"
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proof -
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  from assms obtain K where K: "\<And>x. norm (f x) \<le> K" unfolding Bseq_def by blast
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  {
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    fix x :: nat
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    have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)
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    also have "norm (f x) \<le> K" by (rule K)
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    finally have "norm (f x + c) \<le> K + norm c" by simp
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  }
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  thus ?thesis by (rule BseqI')
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qed
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lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
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  using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto
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lemma Bseq_mult:
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  assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_field)"
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  assumes "Bseq (g :: nat \<Rightarrow> 'a :: real_normed_field)"
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  shows   "Bseq (\<lambda>x. f x * g x)"
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proof -
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  from assms obtain K1 K2 where K: "\<And>x. norm (f x) \<le> K1" "K1 > 0" "\<And>x. norm (g x) \<le> K2" "K2 > 0"
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    unfolding Bseq_def by blast
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  hence "\<And>x. norm (f x * g x) \<le> K1 * K2" by (auto simp: norm_mult intro!: mult_mono)
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  thus ?thesis by (rule BseqI')
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qed
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lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"
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  unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])
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   277
lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
eberlm@61531
   278
proof
eberlm@61531
   279
  assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"
eberlm@61531
   280
  find_theorems "Bfun (\<lambda>_. ?c) _"
eberlm@61531
   281
  from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)
wenzelm@61799
   282
  with \<open>c \<noteq> 0\<close> show "Bseq f" by (simp add: divide_simps)
eberlm@61531
   283
qed (intro Bseq_mult Bfun_const)
eberlm@61531
   284
eberlm@61531
   285
lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
eberlm@61531
   286
  unfolding Bseq_def by auto
eberlm@61531
   287
eberlm@61531
   288
lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
eberlm@61531
   289
  using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
eberlm@61531
   290
eberlm@61531
   291
lemma increasing_Bseq_subseq_iff:
eberlm@61531
   292
  assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"
eberlm@61531
   293
  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
eberlm@61531
   294
proof
eberlm@61531
   295
  assume "Bseq (\<lambda>x. f (g x))"
eberlm@61531
   296
  then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto
eberlm@61531
   297
  {
eberlm@61531
   298
    fix x :: nat
eberlm@61531
   299
    from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
eberlm@61531
   300
      by (auto simp: filterlim_at_top eventually_at_top_linorder)
eberlm@61531
   301
    hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast
eberlm@61531
   302
    also have "norm (f (g y)) \<le> K" by (rule K)
eberlm@61531
   303
    finally have "norm (f x) \<le> K" .
eberlm@61531
   304
  }
eberlm@61531
   305
  thus "Bseq f" by (rule BseqI')
eberlm@61531
   306
qed (insert Bseq_subseq[of f g], simp_all)
eberlm@61531
   307
eberlm@61531
   308
lemma nonneg_incseq_Bseq_subseq_iff:
eberlm@61531
   309
  assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"
eberlm@61531
   310
  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
eberlm@61531
   311
  using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
eberlm@61531
   312
hoelzl@51531
   313
lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
hoelzl@51531
   314
  apply (simp add: subset_eq)
hoelzl@51531
   315
  apply (rule BseqI'[where K="max (norm a) (norm b)"])
hoelzl@51531
   316
  apply (erule_tac x=n in allE)
hoelzl@51531
   317
  apply auto
hoelzl@51531
   318
  done
hoelzl@51531
   319
hoelzl@51531
   320
lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
hoelzl@51531
   321
  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
hoelzl@51531
   322
hoelzl@51531
   323
lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
hoelzl@51531
   324
  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
hoelzl@51531
   325
wenzelm@60758
   326
subsection \<open>Bounded Monotonic Sequences\<close>
hoelzl@51531
   327
wenzelm@60758
   328
subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
hoelzl@51531
   329
hoelzl@51531
   330
(* TODO: delete *)
hoelzl@51531
   331
(* FIXME: one use in NSA/HSEQ.thy *)
wenzelm@61969
   332
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X \<longlonglongrightarrow> L)"
hoelzl@51531
   333
  apply (rule_tac x="X m" in exI)
hoelzl@51531
   334
  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
hoelzl@51531
   335
  unfolding eventually_sequentially
hoelzl@51531
   336
  apply blast
hoelzl@51531
   337
  done
hoelzl@51531
   338
wenzelm@60758
   339
subsection \<open>Convergence to Zero\<close>
huffman@31349
   340
huffman@44081
   341
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   342
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   343
huffman@31349
   344
lemma ZfunI:
huffman@44195
   345
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   346
  unfolding Zfun_def by simp
huffman@31349
   347
huffman@31349
   348
lemma ZfunD:
huffman@44195
   349
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   350
  unfolding Zfun_def by simp
huffman@31349
   351
huffman@31355
   352
lemma Zfun_ssubst:
huffman@44195
   353
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   354
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   355
huffman@44195
   356
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   357
  unfolding Zfun_def by simp
huffman@31349
   358
huffman@44195
   359
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   360
  unfolding Zfun_def by simp
huffman@31349
   361
huffman@31349
   362
lemma Zfun_imp_Zfun:
huffman@44195
   363
  assumes f: "Zfun f F"
huffman@44195
   364
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   365
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   366
proof (cases)
huffman@31349
   367
  assume K: "0 < K"
huffman@31349
   368
  show ?thesis
huffman@31349
   369
  proof (rule ZfunI)
huffman@31349
   370
    fix r::real assume "0 < r"
nipkow@56541
   371
    hence "0 < r / K" using K by simp
huffman@44195
   372
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
lp15@61649
   373
      using ZfunD [OF f] by blast
huffman@44195
   374
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   375
    proof eventually_elim
noschinl@46887
   376
      case (elim x)
huffman@31487
   377
      hence "norm (f x) * K < r"
huffman@31349
   378
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   379
      thus ?case
noschinl@46887
   380
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   381
    qed
huffman@31349
   382
  qed
huffman@31349
   383
next
huffman@31349
   384
  assume "\<not> 0 < K"
huffman@31349
   385
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   386
  show ?thesis
huffman@31355
   387
  proof (rule ZfunI)
huffman@31355
   388
    fix r :: real
huffman@31355
   389
    assume "0 < r"
huffman@44195
   390
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   391
    proof eventually_elim
noschinl@46887
   392
      case (elim x)
noschinl@46887
   393
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   394
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   395
      finally show ?case
wenzelm@60758
   396
        using \<open>0 < r\<close> by simp
huffman@31355
   397
    qed
huffman@31355
   398
  qed
huffman@31349
   399
qed
huffman@31349
   400
huffman@44195
   401
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   402
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   403
huffman@31349
   404
lemma Zfun_add:
huffman@44195
   405
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   406
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   407
proof (rule ZfunI)
huffman@31349
   408
  fix r::real assume "0 < r"
huffman@31349
   409
  hence r: "0 < r / 2" by simp
huffman@44195
   410
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   411
    using f r by (rule ZfunD)
huffman@31349
   412
  moreover
huffman@44195
   413
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   414
    using g r by (rule ZfunD)
huffman@31349
   415
  ultimately
huffman@44195
   416
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   417
  proof eventually_elim
noschinl@46887
   418
    case (elim x)
huffman@31487
   419
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   420
      by (rule norm_triangle_ineq)
huffman@31349
   421
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   422
      using elim by (rule add_strict_mono)
noschinl@46887
   423
    finally show ?case
huffman@31349
   424
      by simp
huffman@31349
   425
  qed
huffman@31349
   426
qed
huffman@31349
   427
huffman@44195
   428
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   429
  unfolding Zfun_def by simp
huffman@31349
   430
huffman@44195
   431
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
haftmann@54230
   432
  using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
huffman@31349
   433
huffman@31349
   434
lemma (in bounded_linear) Zfun:
huffman@44195
   435
  assumes g: "Zfun g F"
huffman@44195
   436
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   437
proof -
huffman@31349
   438
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
lp15@61649
   439
    using bounded by blast
huffman@44195
   440
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   441
    by simp
huffman@31487
   442
  with g show ?thesis
huffman@31349
   443
    by (rule Zfun_imp_Zfun)
huffman@31349
   444
qed
huffman@31349
   445
huffman@31349
   446
lemma (in bounded_bilinear) Zfun:
huffman@44195
   447
  assumes f: "Zfun f F"
huffman@44195
   448
  assumes g: "Zfun g F"
huffman@44195
   449
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   450
proof (rule ZfunI)
huffman@31349
   451
  fix r::real assume r: "0 < r"
huffman@31349
   452
  obtain K where K: "0 < K"
huffman@31349
   453
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
lp15@61649
   454
    using pos_bounded by blast
huffman@31349
   455
  from K have K': "0 < inverse K"
huffman@31349
   456
    by (rule positive_imp_inverse_positive)
huffman@44195
   457
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   458
    using f r by (rule ZfunD)
huffman@31349
   459
  moreover
huffman@44195
   460
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   461
    using g K' by (rule ZfunD)
huffman@31349
   462
  ultimately
huffman@44195
   463
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   464
  proof eventually_elim
noschinl@46887
   465
    case (elim x)
huffman@31487
   466
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   467
      by (rule norm_le)
huffman@31487
   468
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   469
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   470
    also from K have "r * inverse K * K = r"
huffman@31349
   471
      by simp
noschinl@46887
   472
    finally show ?case .
huffman@31349
   473
  qed
huffman@31349
   474
qed
huffman@31349
   475
huffman@31349
   476
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   477
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   478
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   479
huffman@31349
   480
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   481
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   482
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   483
huffman@44282
   484
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   485
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   486
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   487
wenzelm@61973
   488
lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   489
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   490
wenzelm@61973
   491
lemma tendsto_0_le: "\<lbrakk>(f \<longlongrightarrow> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
wenzelm@61973
   492
                     \<Longrightarrow> (g \<longlongrightarrow> 0) F"
lp15@56366
   493
  by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
lp15@56366
   494
wenzelm@60758
   495
subsubsection \<open>Distance and norms\<close>
huffman@36662
   496
hoelzl@51531
   497
lemma tendsto_dist [tendsto_intros]:
hoelzl@51531
   498
  fixes l m :: "'a :: metric_space"
wenzelm@61973
   499
  assumes f: "(f \<longlongrightarrow> l) F" and g: "(g \<longlongrightarrow> m) F"
wenzelm@61973
   500
  shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
hoelzl@51531
   501
proof (rule tendstoI)
hoelzl@51531
   502
  fix e :: real assume "0 < e"
hoelzl@51531
   503
  hence e2: "0 < e/2" by simp
hoelzl@51531
   504
  from tendstoD [OF f e2] tendstoD [OF g e2]
hoelzl@51531
   505
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
hoelzl@51531
   506
  proof (eventually_elim)
hoelzl@51531
   507
    case (elim x)
hoelzl@51531
   508
    then show "dist (dist (f x) (g x)) (dist l m) < e"
hoelzl@51531
   509
      unfolding dist_real_def
hoelzl@51531
   510
      using dist_triangle2 [of "f x" "g x" "l"]
hoelzl@51531
   511
      using dist_triangle2 [of "g x" "l" "m"]
hoelzl@51531
   512
      using dist_triangle3 [of "l" "m" "f x"]
hoelzl@51531
   513
      using dist_triangle [of "f x" "m" "g x"]
hoelzl@51531
   514
      by arith
hoelzl@51531
   515
  qed
hoelzl@51531
   516
qed
hoelzl@51531
   517
hoelzl@51531
   518
lemma continuous_dist[continuous_intros]:
hoelzl@51531
   519
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   520
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   521
  unfolding continuous_def by (rule tendsto_dist)
hoelzl@51531
   522
hoelzl@56371
   523
lemma continuous_on_dist[continuous_intros]:
hoelzl@51531
   524
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   525
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   526
  unfolding continuous_on_def by (auto intro: tendsto_dist)
hoelzl@51531
   527
huffman@31565
   528
lemma tendsto_norm [tendsto_intros]:
wenzelm@61973
   529
  "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
huffman@44081
   530
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   531
hoelzl@51478
   532
lemma continuous_norm [continuous_intros]:
hoelzl@51478
   533
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
hoelzl@51478
   534
  unfolding continuous_def by (rule tendsto_norm)
hoelzl@51478
   535
hoelzl@56371
   536
lemma continuous_on_norm [continuous_intros]:
hoelzl@51478
   537
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
hoelzl@51478
   538
  unfolding continuous_on_def by (auto intro: tendsto_norm)
hoelzl@51478
   539
huffman@36662
   540
lemma tendsto_norm_zero:
wenzelm@61973
   541
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
huffman@44081
   542
  by (drule tendsto_norm, simp)
huffman@36662
   543
huffman@36662
   544
lemma tendsto_norm_zero_cancel:
wenzelm@61973
   545
  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44081
   546
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   547
huffman@36662
   548
lemma tendsto_norm_zero_iff:
wenzelm@61973
   549
  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44081
   550
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   551
huffman@44194
   552
lemma tendsto_rabs [tendsto_intros]:
wenzelm@61973
   553
  "(f \<longlongrightarrow> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
huffman@44194
   554
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   555
hoelzl@51478
   556
lemma continuous_rabs [continuous_intros]:
hoelzl@51478
   557
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   558
  unfolding real_norm_def[symmetric] by (rule continuous_norm)
hoelzl@51478
   559
hoelzl@56371
   560
lemma continuous_on_rabs [continuous_intros]:
hoelzl@51478
   561
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   562
  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
hoelzl@51478
   563
huffman@44194
   564
lemma tendsto_rabs_zero:
wenzelm@61973
   565
  "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
huffman@44194
   566
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   567
huffman@44194
   568
lemma tendsto_rabs_zero_cancel:
wenzelm@61973
   569
  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44194
   570
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   571
huffman@44194
   572
lemma tendsto_rabs_zero_iff:
wenzelm@61973
   573
  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44194
   574
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   575
hoelzl@62368
   576
subsection \<open>Topological Monoid\<close>
hoelzl@62368
   577
hoelzl@62368
   578
class topological_monoid_add = topological_space + monoid_add +
hoelzl@62368
   579
  assumes tendsto_add_Pair: "LIM x (nhds a \<times>\<^sub>F nhds b). fst x + snd x :> nhds (a + b)"
hoelzl@62368
   580
hoelzl@62368
   581
class topological_comm_monoid_add = topological_monoid_add + comm_monoid_add
huffman@44194
   582
huffman@31565
   583
lemma tendsto_add [tendsto_intros]:
hoelzl@62368
   584
  fixes a b :: "'a::topological_monoid_add"
hoelzl@62368
   585
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
hoelzl@62368
   586
  using filterlim_compose[OF tendsto_add_Pair, of "\<lambda>x. (f x, g x)" a b F]
hoelzl@62368
   587
  by (simp add: nhds_prod[symmetric] tendsto_Pair)
huffman@31349
   588
hoelzl@51478
   589
lemma continuous_add [continuous_intros]:
hoelzl@62368
   590
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
hoelzl@51478
   591
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
hoelzl@51478
   592
  unfolding continuous_def by (rule tendsto_add)
hoelzl@51478
   593
hoelzl@56371
   594
lemma continuous_on_add [continuous_intros]:
hoelzl@62368
   595
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
hoelzl@51478
   596
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
hoelzl@51478
   597
  unfolding continuous_on_def by (auto intro: tendsto_add)
hoelzl@51478
   598
huffman@44194
   599
lemma tendsto_add_zero:
hoelzl@62368
   600
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
wenzelm@61973
   601
  shows "\<lbrakk>(f \<longlongrightarrow> 0) F; (g \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
huffman@44194
   602
  by (drule (1) tendsto_add, simp)
huffman@44194
   603
hoelzl@62368
   604
lemma tendsto_setsum [tendsto_intros]:
hoelzl@62368
   605
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@62368
   606
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
hoelzl@62368
   607
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
hoelzl@62368
   608
proof (cases "finite S")
hoelzl@62368
   609
  assume "finite S" thus ?thesis using assms
hoelzl@62368
   610
    by (induct, simp, simp add: tendsto_add)
hoelzl@62368
   611
qed simp
hoelzl@62368
   612
hoelzl@62368
   613
lemma continuous_setsum [continuous_intros]:
hoelzl@62368
   614
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@62368
   615
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@62368
   616
  unfolding continuous_def by (rule tendsto_setsum)
hoelzl@62368
   617
hoelzl@62368
   618
lemma continuous_on_setsum [continuous_intros]:
hoelzl@62368
   619
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@62368
   620
  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@62368
   621
  unfolding continuous_on_def by (auto intro: tendsto_setsum)
hoelzl@62368
   622
hoelzl@62368
   623
subsubsection \<open>Addition and subtraction\<close>
hoelzl@62368
   624
hoelzl@62368
   625
instance real_normed_vector < topological_comm_monoid_add
hoelzl@62368
   626
proof
hoelzl@62368
   627
  fix a b :: 'a show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
hoelzl@62368
   628
    unfolding tendsto_Zfun_iff add_diff_add
hoelzl@62368
   629
    using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
hoelzl@62368
   630
    by (intro Zfun_add)
hoelzl@62368
   631
       (auto simp add: tendsto_Zfun_iff[symmetric] nhds_prod[symmetric] intro!: tendsto_fst)
hoelzl@62368
   632
qed
hoelzl@62368
   633
huffman@31565
   634
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   635
  fixes a :: "'a::real_normed_vector"
wenzelm@61973
   636
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> - a) F"
huffman@44081
   637
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   638
hoelzl@51478
   639
lemma continuous_minus [continuous_intros]:
hoelzl@51478
   640
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   641
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
hoelzl@51478
   642
  unfolding continuous_def by (rule tendsto_minus)
hoelzl@51478
   643
hoelzl@56371
   644
lemma continuous_on_minus [continuous_intros]:
hoelzl@51478
   645
  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   646
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
hoelzl@51478
   647
  unfolding continuous_on_def by (auto intro: tendsto_minus)
hoelzl@51478
   648
huffman@31349
   649
lemma tendsto_minus_cancel:
huffman@31349
   650
  fixes a :: "'a::real_normed_vector"
wenzelm@61973
   651
  shows "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
huffman@44081
   652
  by (drule tendsto_minus, simp)
huffman@31349
   653
hoelzl@50330
   654
lemma tendsto_minus_cancel_left:
wenzelm@61973
   655
    "(f \<longlongrightarrow> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
hoelzl@50330
   656
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
hoelzl@50330
   657
  by auto
hoelzl@50330
   658
huffman@31565
   659
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   660
  fixes a b :: "'a::real_normed_vector"
wenzelm@61973
   661
  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
haftmann@54230
   662
  using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
huffman@31349
   663
hoelzl@51478
   664
lemma continuous_diff [continuous_intros]:
hoelzl@51478
   665
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   666
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
hoelzl@51478
   667
  unfolding continuous_def by (rule tendsto_diff)
hoelzl@51478
   668
hoelzl@56371
   669
lemma continuous_on_diff [continuous_intros]:
lp15@61738
   670
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   671
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
hoelzl@51478
   672
  unfolding continuous_on_def by (auto intro: tendsto_diff)
hoelzl@51478
   673
lp15@61694
   674
lemma continuous_on_op_minus: "continuous_on (s::'a::real_normed_vector set) (op - x)"
lp15@61694
   675
  by (rule continuous_intros | simp)+
lp15@61694
   676
hoelzl@50999
   677
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
hoelzl@50999
   678
wenzelm@60758
   679
subsubsection \<open>Linear operators and multiplication\<close>
huffman@44194
   680
lp15@61806
   681
lemma linear_times:
lp15@61806
   682
  fixes c::"'a::real_algebra" shows "linear (\<lambda>x. c * x)"
lp15@61806
   683
  by (auto simp: linearI distrib_left)
lp15@61806
   684
huffman@44282
   685
lemma (in bounded_linear) tendsto:
wenzelm@61973
   686
  "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
huffman@44081
   687
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   688
hoelzl@51478
   689
lemma (in bounded_linear) continuous:
hoelzl@51478
   690
  "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
hoelzl@51478
   691
  using tendsto[of g _ F] by (auto simp: continuous_def)
hoelzl@51478
   692
hoelzl@51478
   693
lemma (in bounded_linear) continuous_on:
hoelzl@51478
   694
  "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
hoelzl@51478
   695
  using tendsto[of g] by (auto simp: continuous_on_def)
hoelzl@51478
   696
huffman@44194
   697
lemma (in bounded_linear) tendsto_zero:
wenzelm@61973
   698
  "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
huffman@44194
   699
  by (drule tendsto, simp only: zero)
huffman@44194
   700
huffman@44282
   701
lemma (in bounded_bilinear) tendsto:
wenzelm@61973
   702
  "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
huffman@44081
   703
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   704
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   705
hoelzl@51478
   706
lemma (in bounded_bilinear) continuous:
hoelzl@51478
   707
  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
hoelzl@51478
   708
  using tendsto[of f _ F g] by (auto simp: continuous_def)
hoelzl@51478
   709
hoelzl@51478
   710
lemma (in bounded_bilinear) continuous_on:
hoelzl@51478
   711
  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
hoelzl@51478
   712
  using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
hoelzl@51478
   713
huffman@44194
   714
lemma (in bounded_bilinear) tendsto_zero:
wenzelm@61973
   715
  assumes f: "(f \<longlongrightarrow> 0) F"
wenzelm@61973
   716
  assumes g: "(g \<longlongrightarrow> 0) F"
wenzelm@61973
   717
  shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
huffman@44194
   718
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   719
huffman@44194
   720
lemma (in bounded_bilinear) tendsto_left_zero:
wenzelm@61973
   721
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) \<longlongrightarrow> 0) F"
huffman@44194
   722
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   723
huffman@44194
   724
lemma (in bounded_bilinear) tendsto_right_zero:
wenzelm@61973
   725
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) \<longlongrightarrow> 0) F"
huffman@44194
   726
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   727
huffman@44282
   728
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   729
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   730
huffman@44282
   731
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   732
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   733
huffman@44282
   734
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   735
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   736
lp15@61806
   737
lemma tendsto_mult_left:
lp15@61806
   738
  fixes c::"'a::real_normed_algebra"
wenzelm@61973
   739
  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
lp15@61806
   740
by (rule tendsto_mult [OF tendsto_const])
lp15@61806
   741
lp15@61806
   742
lemma tendsto_mult_right:
lp15@61806
   743
  fixes c::"'a::real_normed_algebra"
wenzelm@61973
   744
  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
lp15@61806
   745
by (rule tendsto_mult [OF _ tendsto_const])
lp15@61806
   746
hoelzl@51478
   747
lemmas continuous_of_real [continuous_intros] =
hoelzl@51478
   748
  bounded_linear.continuous [OF bounded_linear_of_real]
hoelzl@51478
   749
hoelzl@51478
   750
lemmas continuous_scaleR [continuous_intros] =
hoelzl@51478
   751
  bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
hoelzl@51478
   752
hoelzl@51478
   753
lemmas continuous_mult [continuous_intros] =
hoelzl@51478
   754
  bounded_bilinear.continuous [OF bounded_bilinear_mult]
hoelzl@51478
   755
hoelzl@56371
   756
lemmas continuous_on_of_real [continuous_intros] =
hoelzl@51478
   757
  bounded_linear.continuous_on [OF bounded_linear_of_real]
hoelzl@51478
   758
hoelzl@56371
   759
lemmas continuous_on_scaleR [continuous_intros] =
hoelzl@51478
   760
  bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
hoelzl@51478
   761
hoelzl@56371
   762
lemmas continuous_on_mult [continuous_intros] =
hoelzl@51478
   763
  bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
hoelzl@51478
   764
huffman@44568
   765
lemmas tendsto_mult_zero =
huffman@44568
   766
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   767
huffman@44568
   768
lemmas tendsto_mult_left_zero =
huffman@44568
   769
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   770
huffman@44568
   771
lemmas tendsto_mult_right_zero =
huffman@44568
   772
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   773
huffman@44194
   774
lemma tendsto_power [tendsto_intros]:
huffman@44194
   775
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
wenzelm@61973
   776
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
hoelzl@58729
   777
  by (induct n) (simp_all add: tendsto_mult)
huffman@44194
   778
hoelzl@51478
   779
lemma continuous_power [continuous_intros]:
hoelzl@51478
   780
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   781
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
hoelzl@51478
   782
  unfolding continuous_def by (rule tendsto_power)
hoelzl@51478
   783
hoelzl@56371
   784
lemma continuous_on_power [continuous_intros]:
hoelzl@51478
   785
  fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   786
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
hoelzl@51478
   787
  unfolding continuous_on_def by (auto intro: tendsto_power)
hoelzl@51478
   788
huffman@44194
   789
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   790
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
wenzelm@61973
   791
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F"
wenzelm@61973
   792
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   793
proof (cases "finite S")
huffman@44194
   794
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   795
    by (induct, simp, simp add: tendsto_mult)
hoelzl@58729
   796
qed simp
huffman@44194
   797
hoelzl@51478
   798
lemma continuous_setprod [continuous_intros]:
hoelzl@51478
   799
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   800
  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
   801
  unfolding continuous_def by (rule tendsto_setprod)
hoelzl@51478
   802
hoelzl@51478
   803
lemma continuous_on_setprod [continuous_intros]:
hoelzl@51478
   804
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   805
  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
   806
  unfolding continuous_on_def by (auto intro: tendsto_setprod)
hoelzl@51478
   807
eberlm@61531
   808
lemma tendsto_of_real_iff:
wenzelm@61973
   809
  "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
eberlm@61531
   810
  unfolding tendsto_iff by simp
eberlm@61531
   811
eberlm@61531
   812
lemma tendsto_add_const_iff:
wenzelm@61973
   813
  "((\<lambda>x. c + f x :: 'a :: real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
paulson@62087
   814
  using tendsto_add[OF tendsto_const[of c], of f d]
eberlm@61531
   815
        tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
eberlm@61531
   816
eberlm@61531
   817
wenzelm@60758
   818
subsubsection \<open>Inverse and division\<close>
huffman@31355
   819
huffman@31355
   820
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   821
  assumes f: "Zfun f F"
huffman@44195
   822
  assumes g: "Bfun g F"
huffman@44195
   823
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   824
proof -
huffman@31355
   825
  obtain K where K: "0 \<le> K"
huffman@31355
   826
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
lp15@61649
   827
    using nonneg_bounded by blast
huffman@31355
   828
  obtain B where B: "0 < B"
huffman@44195
   829
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   830
    using g by (rule BfunE)
huffman@44195
   831
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   832
  using norm_g proof eventually_elim
noschinl@46887
   833
    case (elim x)
huffman@31487
   834
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   835
      by (rule norm_le)
huffman@31487
   836
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   837
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   838
                mult_nonneg_nonneg K elim)
huffman@31487
   839
    also have "\<dots> = norm (f x) * (B * K)"
haftmann@57512
   840
      by (rule mult.assoc)
huffman@31487
   841
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   842
  qed
huffman@31487
   843
  with f show ?thesis
huffman@31487
   844
    by (rule Zfun_imp_Zfun)
huffman@31355
   845
qed
huffman@31355
   846
huffman@31355
   847
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   848
  assumes f: "Bfun f F"
huffman@44195
   849
  assumes g: "Zfun g F"
huffman@44195
   850
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   851
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   852
huffman@31355
   853
lemma Bfun_inverse_lemma:
huffman@31355
   854
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   855
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   856
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   857
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   858
  done
huffman@31355
   859
huffman@31355
   860
lemma Bfun_inverse:
huffman@31355
   861
  fixes a :: "'a::real_normed_div_algebra"
wenzelm@61973
   862
  assumes f: "(f \<longlongrightarrow> a) F"
huffman@31355
   863
  assumes a: "a \<noteq> 0"
huffman@44195
   864
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   865
proof -
huffman@31355
   866
  from a have "0 < norm a" by simp
huffman@31355
   867
  hence "\<exists>r>0. r < norm a" by (rule dense)
lp15@61649
   868
  then obtain r where r1: "0 < r" and r2: "r < norm a" by blast
huffman@44195
   869
  have "eventually (\<lambda>x. dist (f x) a < r) F"
lp15@61649
   870
    using tendstoD [OF f r1] by blast
huffman@44195
   871
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
   872
  proof eventually_elim
noschinl@46887
   873
    case (elim x)
huffman@31487
   874
    hence 1: "norm (f x - a) < r"
huffman@31355
   875
      by (simp add: dist_norm)
huffman@31487
   876
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   877
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   878
      by (rule nonzero_norm_inverse)
huffman@31355
   879
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   880
    proof (rule le_imp_inverse_le)
huffman@31355
   881
      show "0 < norm a - r" using r2 by simp
huffman@31355
   882
    next
huffman@31487
   883
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   884
        by (rule norm_triangle_ineq2)
huffman@31487
   885
      also have "\<dots> = norm (f x - a)"
huffman@31355
   886
        by (rule norm_minus_commute)
huffman@31355
   887
      also have "\<dots> < r" using 1 .
huffman@31487
   888
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   889
    qed
huffman@31487
   890
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   891
  qed
huffman@31355
   892
  thus ?thesis by (rule BfunI)
huffman@31355
   893
qed
huffman@31355
   894
huffman@31565
   895
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   896
  fixes a :: "'a::real_normed_div_algebra"
wenzelm@61973
   897
  assumes f: "(f \<longlongrightarrow> a) F"
huffman@31355
   898
  assumes a: "a \<noteq> 0"
wenzelm@61973
   899
  shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
huffman@31355
   900
proof -
huffman@31355
   901
  from a have "0 < norm a" by simp
huffman@44195
   902
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   903
    by (rule tendstoD)
huffman@44195
   904
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
lp15@61810
   905
    unfolding dist_norm by (auto elim!: eventually_mono)
huffman@44627
   906
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
   907
    - (inverse (f x) * (f x - a) * inverse a)) F"
lp15@61810
   908
    by (auto elim!: eventually_mono simp: inverse_diff_inverse)
huffman@44627
   909
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   910
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
   911
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
   912
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
   913
  ultimately show ?thesis
huffman@44627
   914
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
   915
qed
huffman@31355
   916
hoelzl@51478
   917
lemma continuous_inverse:
hoelzl@51478
   918
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   919
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   920
  shows "continuous F (\<lambda>x. inverse (f x))"
hoelzl@51478
   921
  using assms unfolding continuous_def by (rule tendsto_inverse)
hoelzl@51478
   922
hoelzl@51478
   923
lemma continuous_at_within_inverse[continuous_intros]:
hoelzl@51478
   924
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   925
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   926
  shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
hoelzl@51478
   927
  using assms unfolding continuous_within by (rule tendsto_inverse)
hoelzl@51478
   928
hoelzl@51478
   929
lemma isCont_inverse[continuous_intros, simp]:
hoelzl@51478
   930
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   931
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   932
  shows "isCont (\<lambda>x. inverse (f x)) a"
hoelzl@51478
   933
  using assms unfolding continuous_at by (rule tendsto_inverse)
hoelzl@51478
   934
hoelzl@56371
   935
lemma continuous_on_inverse[continuous_intros]:
hoelzl@51478
   936
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   937
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   938
  shows "continuous_on s (\<lambda>x. inverse (f x))"
lp15@61649
   939
  using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)
hoelzl@51478
   940
huffman@31565
   941
lemma tendsto_divide [tendsto_intros]:
huffman@31355
   942
  fixes a b :: "'a::real_normed_field"
wenzelm@61973
   943
  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; b \<noteq> 0\<rbrakk>
wenzelm@61973
   944
    \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
huffman@44282
   945
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
   946
hoelzl@51478
   947
lemma continuous_divide:
hoelzl@51478
   948
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   949
  assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   950
  shows "continuous F (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   951
  using assms unfolding continuous_def by (rule tendsto_divide)
hoelzl@51478
   952
hoelzl@51478
   953
lemma continuous_at_within_divide[continuous_intros]:
hoelzl@51478
   954
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   955
  assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
hoelzl@51478
   956
  shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   957
  using assms unfolding continuous_within by (rule tendsto_divide)
hoelzl@51478
   958
hoelzl@51478
   959
lemma isCont_divide[continuous_intros, simp]:
hoelzl@51478
   960
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   961
  assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
hoelzl@51478
   962
  shows "isCont (\<lambda>x. (f x) / g x) a"
hoelzl@51478
   963
  using assms unfolding continuous_at by (rule tendsto_divide)
hoelzl@51478
   964
hoelzl@56371
   965
lemma continuous_on_divide[continuous_intros]:
hoelzl@51478
   966
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   967
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
hoelzl@51478
   968
  shows "continuous_on s (\<lambda>x. (f x) / (g x))"
lp15@61649
   969
  using assms unfolding continuous_on_def by (blast intro: tendsto_divide)
hoelzl@51478
   970
huffman@44194
   971
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
   972
  fixes l :: "'a::real_normed_vector"
wenzelm@61973
   973
  shows "\<lbrakk>(f \<longlongrightarrow> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
huffman@44194
   974
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
   975
hoelzl@51478
   976
lemma continuous_sgn:
hoelzl@51478
   977
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   978
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   979
  shows "continuous F (\<lambda>x. sgn (f x))"
hoelzl@51478
   980
  using assms unfolding continuous_def by (rule tendsto_sgn)
hoelzl@51478
   981
hoelzl@51478
   982
lemma continuous_at_within_sgn[continuous_intros]:
hoelzl@51478
   983
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   984
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   985
  shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
hoelzl@51478
   986
  using assms unfolding continuous_within by (rule tendsto_sgn)
hoelzl@51478
   987
hoelzl@51478
   988
lemma isCont_sgn[continuous_intros]:
hoelzl@51478
   989
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   990
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   991
  shows "isCont (\<lambda>x. sgn (f x)) a"
hoelzl@51478
   992
  using assms unfolding continuous_at by (rule tendsto_sgn)
hoelzl@51478
   993
hoelzl@56371
   994
lemma continuous_on_sgn[continuous_intros]:
hoelzl@51478
   995
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   996
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   997
  shows "continuous_on s (\<lambda>x. sgn (f x))"
lp15@61649
   998
  using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)
hoelzl@51478
   999
hoelzl@50325
  1000
lemma filterlim_at_infinity:
wenzelm@61076
  1001
  fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"
hoelzl@50325
  1002
  assumes "0 \<le> c"
hoelzl@50325
  1003
  shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
hoelzl@50325
  1004
  unfolding filterlim_iff eventually_at_infinity
hoelzl@50325
  1005
proof safe
hoelzl@50325
  1006
  fix P :: "'a \<Rightarrow> bool" and b
hoelzl@50325
  1007
  assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
hoelzl@50325
  1008
    and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
hoelzl@50325
  1009
  have "max b (c + 1) > c" by auto
hoelzl@50325
  1010
  with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
hoelzl@50325
  1011
    by auto
hoelzl@50325
  1012
  then show "eventually (\<lambda>x. P (f x)) F"
hoelzl@50325
  1013
  proof eventually_elim
hoelzl@50325
  1014
    fix x assume "max b (c + 1) \<le> norm (f x)"
hoelzl@50325
  1015
    with P show "P (f x)" by auto
hoelzl@50325
  1016
  qed
hoelzl@50325
  1017
qed force
hoelzl@50325
  1018
eberlm@61531
  1019
lemma not_tendsto_and_filterlim_at_infinity:
eberlm@61531
  1020
  assumes "F \<noteq> bot"
paulson@62087
  1021
  assumes "(f \<longlongrightarrow> (c :: 'a :: real_normed_vector)) F"
eberlm@61531
  1022
  assumes "filterlim f at_infinity F"
eberlm@61531
  1023
  shows   False
eberlm@61531
  1024
proof -
paulson@62087
  1025
  from tendstoD[OF assms(2), of "1/2"]
eberlm@61531
  1026
    have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp
eberlm@61531
  1027
  moreover from filterlim_at_infinity[of "norm c" f F] assms(3)
eberlm@61531
  1028
    have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
eberlm@61531
  1029
  ultimately have "eventually (\<lambda>x. False) F"
eberlm@61531
  1030
  proof eventually_elim
eberlm@61531
  1031
    fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"
eberlm@61531
  1032
    note B
eberlm@61531
  1033
    also have "norm (f x) = dist (f x) 0" by (simp add: norm_conv_dist)
eberlm@61531
  1034
    also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
eberlm@61531
  1035
    also note A
eberlm@61531
  1036
    finally show False by (simp add: norm_conv_dist)
eberlm@61531
  1037
  qed
eberlm@61531
  1038
  with assms show False by simp
eberlm@61531
  1039
qed
eberlm@61531
  1040
eberlm@61531
  1041
lemma filterlim_at_infinity_imp_not_convergent:
eberlm@61531
  1042
  assumes "filterlim f at_infinity sequentially"
eberlm@61531
  1043
  shows   "\<not>convergent f"
eberlm@61531
  1044
  by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
eberlm@61531
  1045
     (simp_all add: convergent_LIMSEQ_iff)
eberlm@61531
  1046
eberlm@61531
  1047
lemma filterlim_at_infinity_imp_eventually_ne:
eberlm@61531
  1048
  assumes "filterlim f at_infinity F"
eberlm@61531
  1049
  shows   "eventually (\<lambda>z. f z \<noteq> c) F"
eberlm@61531
  1050
proof -
eberlm@61531
  1051
  have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all
eberlm@61531
  1052
  with filterlim_at_infinity[OF order.refl, of f F] assms
eberlm@61531
  1053
    have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast
eberlm@61531
  1054
  thus ?thesis by eventually_elim auto
eberlm@61531
  1055
qed
eberlm@61531
  1056
paulson@62087
  1057
lemma tendsto_of_nat [tendsto_intros]:
eberlm@61531
  1058
  "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"
eberlm@61531
  1059
proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
eberlm@61531
  1060
  fix r :: real assume r: "r > 0"
eberlm@61531
  1061
  def n \<equiv> "nat \<lceil>r\<rceil>"
eberlm@61531
  1062
  from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith
eberlm@61531
  1063
  from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
eberlm@61531
  1064
    by eventually_elim (insert n, simp_all)
eberlm@61531
  1065
qed
eberlm@61531
  1066
eberlm@61531
  1067
wenzelm@60758
  1068
subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
hoelzl@50347
  1069
wenzelm@60758
  1070
text \<open>
hoelzl@50347
  1071
hoelzl@50347
  1072
This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
hoelzl@50347
  1073
@{term "at_right x"} and also @{term "at_right 0"}.
hoelzl@50347
  1074
wenzelm@60758
  1075
\<close>
hoelzl@50347
  1076
hoelzl@51471
  1077
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
hoelzl@50323
  1078
hoelzl@51641
  1079
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
hoelzl@60721
  1080
  by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
hoelzl@60721
  1081
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@50347
  1082
hoelzl@51641
  1083
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
hoelzl@60721
  1084
  by (rule filtermap_fun_inverse[where g=uminus])
hoelzl@60721
  1085
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@51641
  1086
hoelzl@51641
  1087
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
hoelzl@51641
  1088
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50347
  1089
hoelzl@50347
  1090
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
hoelzl@51641
  1091
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50323
  1092
hoelzl@50347
  1093
lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
hoelzl@50347
  1094
  using filtermap_at_right_shift[of "-a" 0] by simp
hoelzl@50347
  1095
hoelzl@50347
  1096
lemma filterlim_at_right_to_0:
hoelzl@50347
  1097
  "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
hoelzl@50347
  1098
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
hoelzl@50347
  1099
hoelzl@50347
  1100
lemma eventually_at_right_to_0:
hoelzl@50347
  1101
  "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
hoelzl@50347
  1102
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
hoelzl@50347
  1103
hoelzl@51641
  1104
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
hoelzl@51641
  1105
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
  1106
hoelzl@50347
  1107
lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
hoelzl@51641
  1108
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50323
  1109
hoelzl@50347
  1110
lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
hoelzl@51641
  1111
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
  1112
hoelzl@50347
  1113
lemma filterlim_at_left_to_right:
hoelzl@50347
  1114
  "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
hoelzl@50347
  1115
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
hoelzl@50347
  1116
hoelzl@50347
  1117
lemma eventually_at_left_to_right:
hoelzl@50347
  1118
  "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
hoelzl@50347
  1119
  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
hoelzl@50347
  1120
hoelzl@60721
  1121
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@60721
  1122
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@60721
  1123
  by (metis leI minus_less_iff order_less_asym)
hoelzl@60721
  1124
hoelzl@60721
  1125
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@60721
  1126
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@60721
  1127
  by (metis leI less_minus_iff order_less_asym)
hoelzl@60721
  1128
hoelzl@50346
  1129
lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
hoelzl@60721
  1130
  by (rule filtermap_fun_inverse[symmetric, of uminus])
hoelzl@60721
  1131
     (auto intro: filterlim_uminus_at_bot_at_top filterlim_uminus_at_top_at_bot)
hoelzl@50346
  1132
hoelzl@50346
  1133
lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
hoelzl@50346
  1134
  unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
hoelzl@50346
  1135
hoelzl@50346
  1136
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
hoelzl@50346
  1137
  unfolding filterlim_def at_top_mirror filtermap_filtermap ..
hoelzl@50346
  1138
hoelzl@50346
  1139
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
hoelzl@50346
  1140
  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
hoelzl@50346
  1141
hoelzl@50346
  1142
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
hoelzl@50346
  1143
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
hoelzl@50346
  1144
  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
hoelzl@50346
  1145
  by auto
hoelzl@50346
  1146
hoelzl@50346
  1147
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
hoelzl@50346
  1148
  unfolding filterlim_uminus_at_top by simp
hoelzl@50323
  1149
hoelzl@50347
  1150
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
hoelzl@51641
  1151
  unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
hoelzl@50347
  1152
proof safe
hoelzl@50347
  1153
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50347
  1154
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50347
  1155
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@51641
  1156
  then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
lp15@61810
  1157
    by (auto elim!: eventually_mono simp: inverse_eq_divide field_simps)
hoelzl@50347
  1158
qed
hoelzl@50347
  1159
hoelzl@50325
  1160
lemma tendsto_inverse_0:
wenzelm@61076
  1161
  fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
wenzelm@61973
  1162
  shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
hoelzl@50325
  1163
  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
hoelzl@50325
  1164
proof safe
hoelzl@50325
  1165
  fix r :: real assume "0 < r"
hoelzl@50325
  1166
  show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
hoelzl@50325
  1167
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
hoelzl@50325
  1168
    fix x :: 'a
wenzelm@60758
  1169
    from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
hoelzl@50325
  1170
    also assume *: "inverse (r / 2) \<le> norm x"
hoelzl@50325
  1171
    finally show "norm (inverse x) < r"
wenzelm@60758
  1172
      using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
hoelzl@50325
  1173
  qed
hoelzl@50325
  1174
qed
hoelzl@50325
  1175
eberlm@61552
  1176
lemma tendsto_add_filterlim_at_infinity:
wenzelm@61973
  1177
  assumes "(f \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
eberlm@61552
  1178
  assumes "filterlim g at_infinity F"
eberlm@61552
  1179
  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
eberlm@61552
  1180
proof (subst filterlim_at_infinity[OF order_refl], safe)
eberlm@61552
  1181
  fix r :: real assume r: "r > 0"
wenzelm@61973
  1182
  from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F" by (rule tendsto_norm)
eberlm@61552
  1183
  hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
paulson@62087
  1184
  moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all
eberlm@61552
  1185
  with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
eberlm@61552
  1186
    unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all
eberlm@61552
  1187
  ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
eberlm@61552
  1188
  proof eventually_elim
eberlm@61552
  1189
    fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
eberlm@61552
  1190
    from A B have "r \<le> norm (g x) - norm (f x)" by simp
eberlm@61552
  1191
    also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)
eberlm@61552
  1192
    finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)
eberlm@61552
  1193
  qed
eberlm@61552
  1194
qed
eberlm@61552
  1195
eberlm@61552
  1196
lemma tendsto_add_filterlim_at_infinity':
eberlm@61552
  1197
  assumes "filterlim f at_infinity F"
wenzelm@61973
  1198
  assumes "(g \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
eberlm@61552
  1199
  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
eberlm@61552
  1200
  by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
eberlm@61552
  1201
hoelzl@60721
  1202
lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
hoelzl@60721
  1203
  unfolding filterlim_at
hoelzl@60721
  1204
  by (auto simp: eventually_at_top_dense)
hoelzl@60721
  1205
     (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
hoelzl@60721
  1206
hoelzl@60721
  1207
lemma filterlim_inverse_at_top:
wenzelm@61973
  1208
  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@60721
  1209
  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
lp15@61810
  1210
     (simp add: filterlim_def eventually_filtermap eventually_mono at_within_def le_principal)
hoelzl@60721
  1211
hoelzl@60721
  1212
lemma filterlim_inverse_at_bot_neg:
hoelzl@60721
  1213
  "LIM x (at_left (0::real)). inverse x :> at_bot"
hoelzl@60721
  1214
  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
hoelzl@60721
  1215
hoelzl@60721
  1216
lemma filterlim_inverse_at_bot:
wenzelm@61973
  1217
  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@60721
  1218
  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
hoelzl@60721
  1219
  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
hoelzl@60721
  1220
hoelzl@50347
  1221
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
hoelzl@60721
  1222
  by (intro filtermap_fun_inverse[symmetric, where g=inverse])
hoelzl@60721
  1223
     (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)
hoelzl@50347
  1224
hoelzl@50347
  1225
lemma eventually_at_right_to_top:
hoelzl@50347
  1226
  "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
hoelzl@50347
  1227
  unfolding at_right_to_top eventually_filtermap ..
hoelzl@50347
  1228
hoelzl@50347
  1229
lemma filterlim_at_right_to_top:
hoelzl@50347
  1230
  "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
hoelzl@50347
  1231
  unfolding filterlim_def at_right_to_top filtermap_filtermap ..
hoelzl@50347
  1232
hoelzl@50347
  1233
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
hoelzl@50347
  1234
  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
hoelzl@50347
  1235
hoelzl@50347
  1236
lemma eventually_at_top_to_right:
hoelzl@50347
  1237
  "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
hoelzl@50347
  1238
  unfolding at_top_to_right eventually_filtermap ..
hoelzl@50347
  1239
hoelzl@50347
  1240
lemma filterlim_at_top_to_right:
hoelzl@50347
  1241
  "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
hoelzl@50347
  1242
  unfolding filterlim_def at_top_to_right filtermap_filtermap ..
hoelzl@50347
  1243
hoelzl@50325
  1244
lemma filterlim_inverse_at_infinity:
wenzelm@61076
  1245
  fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1246
  shows "filterlim inverse at_infinity (at (0::'a))"
hoelzl@50325
  1247
  unfolding filterlim_at_infinity[OF order_refl]
hoelzl@50325
  1248
proof safe
hoelzl@50325
  1249
  fix r :: real assume "0 < r"
hoelzl@50325
  1250
  then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
hoelzl@50325
  1251
    unfolding eventually_at norm_inverse
hoelzl@50325
  1252
    by (intro exI[of _ "inverse r"])
hoelzl@50325
  1253
       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
hoelzl@50325
  1254
qed
hoelzl@50325
  1255
hoelzl@50325
  1256
lemma filterlim_inverse_at_iff:
wenzelm@61076
  1257
  fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1258
  shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
hoelzl@50325
  1259
  unfolding filterlim_def filtermap_filtermap[symmetric]
hoelzl@50325
  1260
proof
hoelzl@50325
  1261
  assume "filtermap g F \<le> at_infinity"
hoelzl@50325
  1262
  then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
hoelzl@50325
  1263
    by (rule filtermap_mono)
hoelzl@50325
  1264
  also have "\<dots> \<le> at 0"
hoelzl@51641
  1265
    using tendsto_inverse_0[where 'a='b]
hoelzl@51641
  1266
    by (auto intro!: exI[of _ 1]
hoelzl@51641
  1267
             simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
hoelzl@50325
  1268
  finally show "filtermap inverse (filtermap g F) \<le> at 0" .
hoelzl@50325
  1269
next
hoelzl@50325
  1270
  assume "filtermap inverse (filtermap g F) \<le> at 0"
hoelzl@50325
  1271
  then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
hoelzl@50325
  1272
    by (rule filtermap_mono)
hoelzl@50325
  1273
  with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
hoelzl@50325
  1274
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
hoelzl@50325
  1275
qed
hoelzl@50325
  1276
eberlm@61531
  1277
lemma tendsto_mult_filterlim_at_infinity:
wenzelm@61973
  1278
  assumes "F \<noteq> bot" "(f \<longlongrightarrow> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
eberlm@61531
  1279
  assumes "filterlim g at_infinity F"
eberlm@61531
  1280
  shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"
eberlm@61531
  1281
proof -
wenzelm@61973
  1282
  have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
eberlm@61531
  1283
    by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
eberlm@61531
  1284
  hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
eberlm@61531
  1285
    unfolding filterlim_at using assms
eberlm@61531
  1286
    by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
eberlm@61531
  1287
  thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
eberlm@61531
  1288
qed
eberlm@61531
  1289
wenzelm@61973
  1290
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
hoelzl@51641
  1291
 by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
hoelzl@50419
  1292
eberlm@61531
  1293
lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
eberlm@61531
  1294
  by (rule filterlim_subseq) (auto simp: subseq_def)
eberlm@61531
  1295
eberlm@61531
  1296
lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"
eberlm@61531
  1297
  by (rule filterlim_subseq) (auto simp: subseq_def)
lp15@59613
  1298
lp15@59613
  1299
lemma at_to_infinity:
wenzelm@61076
  1300
  fixes x :: "'a :: {real_normed_field,field}"
lp15@59613
  1301
  shows "(at (0::'a)) = filtermap inverse at_infinity"
lp15@59613
  1302
proof (rule antisym)
wenzelm@61973
  1303
  have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
lp15@59613
  1304
    by (fact tendsto_inverse_0)
lp15@59613
  1305
  then show "filtermap inverse at_infinity \<le> at (0::'a)"
lp15@59613
  1306
    apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
lp15@59613
  1307
    apply (rule_tac x="1" in exI, auto)
lp15@59613
  1308
    done
lp15@59613
  1309
next
lp15@59613
  1310
  have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
lp15@59613
  1311
    using filterlim_inverse_at_infinity unfolding filterlim_def
lp15@59613
  1312
    by (rule filtermap_mono)
lp15@59613
  1313
  then show "at (0::'a) \<le> filtermap inverse at_infinity"
lp15@59613
  1314
    by (simp add: filtermap_ident filtermap_filtermap)
lp15@59613
  1315
qed
lp15@59613
  1316
lp15@59613
  1317
lemma lim_at_infinity_0:
haftmann@59867
  1318
  fixes l :: "'a :: {real_normed_field,field}"
wenzelm@61973
  1319
  shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f o inverse) \<longlongrightarrow> l) (at (0::'a))"
lp15@59613
  1320
by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
lp15@59613
  1321
lp15@59613
  1322
lemma lim_zero_infinity:
haftmann@59867
  1323
  fixes l :: "'a :: {real_normed_field,field}"
wenzelm@61973
  1324
  shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
lp15@59613
  1325
by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
lp15@59613
  1326
lp15@59613
  1327
wenzelm@60758
  1328
text \<open>
hoelzl@50324
  1329
hoelzl@50324
  1330
We only show rules for multiplication and addition when the functions are either against a real
hoelzl@50324
  1331
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
hoelzl@50324
  1332
wenzelm@60758
  1333
\<close>
hoelzl@50324
  1334
lp15@60141
  1335
lemma filterlim_tendsto_pos_mult_at_top:
wenzelm@61973
  1336
  assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c"
hoelzl@50324
  1337
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1338
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1339
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1340
proof safe
hoelzl@50324
  1341
  fix Z :: real assume "0 < Z"
wenzelm@60758
  1342
  from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
lp15@61810
  1343
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
hoelzl@50324
  1344
             simp: dist_real_def abs_real_def split: split_if_asm)
hoelzl@50346
  1345
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
hoelzl@50324
  1346
    unfolding filterlim_at_top by auto
hoelzl@50346
  1347
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1348
  proof eventually_elim
hoelzl@50346
  1349
    fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
wenzelm@60758
  1350
    with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
hoelzl@50346
  1351
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
wenzelm@60758
  1352
    with \<open>0 < c\<close> show "Z \<le> f x * g x"
hoelzl@50324
  1353
       by simp
hoelzl@50324
  1354
  qed
hoelzl@50324
  1355
qed
hoelzl@50324
  1356
lp15@60141
  1357
lemma filterlim_at_top_mult_at_top:
hoelzl@50324
  1358
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1359
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1360
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1361
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1362
proof safe
hoelzl@50324
  1363
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1364
  from f have "eventually (\<lambda>x. 1 \<le> f x) F"
hoelzl@50324
  1365
    unfolding filterlim_at_top by auto
hoelzl@50346
  1366
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1367
    unfolding filterlim_at_top by auto
hoelzl@50346
  1368
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1369
  proof eventually_elim
hoelzl@50346
  1370
    fix x assume "1 \<le> f x" "Z \<le> g x"
wenzelm@60758
  1371
    with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
hoelzl@50346
  1372
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
  1373
    then show "Z \<le> f x * g x"
hoelzl@50324
  1374
       by simp
hoelzl@50324
  1375
  qed
hoelzl@50324
  1376
qed
hoelzl@50324
  1377
hoelzl@50419
  1378
lemma filterlim_tendsto_pos_mult_at_bot:
wenzelm@61973
  1379
  assumes "(f \<longlongrightarrow> c) F" "0 < (c::real)" "filterlim g at_bot F"
hoelzl@50419
  1380
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@50419
  1381
  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
hoelzl@50419
  1382
  unfolding filterlim_uminus_at_bot by simp
hoelzl@50419
  1383
hoelzl@60182
  1384
lemma filterlim_tendsto_neg_mult_at_bot:
wenzelm@61973
  1385
  assumes c: "(f \<longlongrightarrow> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
hoelzl@60182
  1386
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@60182
  1387
  using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
hoelzl@60182
  1388
  unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
hoelzl@60182
  1389
hoelzl@56330
  1390
lemma filterlim_pow_at_top:
hoelzl@56330
  1391
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1392
  assumes "0 < n" and f: "LIM x F. f x :> at_top"
hoelzl@56330
  1393
  shows "LIM x F. (f x)^n :: real :> at_top"
wenzelm@60758
  1394
using \<open>0 < n\<close> proof (induct n)
hoelzl@56330
  1395
  case (Suc n) with f show ?case
hoelzl@56330
  1396
    by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
hoelzl@56330
  1397
qed simp
hoelzl@56330
  1398
hoelzl@56330
  1399
lemma filterlim_pow_at_bot_even:
hoelzl@56330
  1400
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1401
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
hoelzl@56330
  1402
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
hoelzl@56330
  1403
hoelzl@56330
  1404
lemma filterlim_pow_at_bot_odd:
hoelzl@56330
  1405
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1406
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
hoelzl@56330
  1407
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
hoelzl@56330
  1408
lp15@60141
  1409
lemma filterlim_tendsto_add_at_top:
wenzelm@61973
  1410
  assumes f: "(f \<longlongrightarrow> c) F"
hoelzl@50324
  1411
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1412
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1413
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1414
proof safe
hoelzl@50324
  1415
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1416
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
lp15@61810
  1417
    by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
hoelzl@50346
  1418
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
hoelzl@50324
  1419
    unfolding filterlim_at_top by auto
hoelzl@50346
  1420
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1421
    by eventually_elim simp
hoelzl@50324
  1422
qed
hoelzl@50324
  1423
hoelzl@50347
  1424
lemma LIM_at_top_divide:
hoelzl@50347
  1425
  fixes f g :: "'a \<Rightarrow> real"
wenzelm@61973
  1426
  assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
wenzelm@61973
  1427
  assumes g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
hoelzl@50347
  1428
  shows "LIM x F. f x / g x :> at_top"
hoelzl@50347
  1429
  unfolding divide_inverse
hoelzl@50347
  1430
  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
hoelzl@50347
  1431
lp15@60141
  1432
lemma filterlim_at_top_add_at_top:
hoelzl@50324
  1433
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1434
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1435
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1436
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1437
proof safe
hoelzl@50324
  1438
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1439
  from f have "eventually (\<lambda>x. 0 \<le> f x) F"
hoelzl@50324
  1440
    unfolding filterlim_at_top by auto
hoelzl@50346
  1441
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1442
    unfolding filterlim_at_top by auto
hoelzl@50346
  1443
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1444
    by eventually_elim simp
hoelzl@50324
  1445
qed
hoelzl@50324
  1446
hoelzl@50331
  1447
lemma tendsto_divide_0:
wenzelm@61076
  1448
  fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
wenzelm@61973
  1449
  assumes f: "(f \<longlongrightarrow> c) F"
hoelzl@50331
  1450
  assumes g: "LIM x F. g x :> at_infinity"
wenzelm@61973
  1451
  shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
hoelzl@50331
  1452
  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
hoelzl@50331
  1453
hoelzl@50331
  1454
lemma linear_plus_1_le_power:
hoelzl@50331
  1455
  fixes x :: real
hoelzl@50331
  1456
  assumes x: "0 \<le> x"
hoelzl@50331
  1457
  shows "real n * x + 1 \<le> (x + 1) ^ n"
hoelzl@50331
  1458
proof (induct n)
hoelzl@50331
  1459
  case (Suc n)
hoelzl@50331
  1460
  have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
lp15@61609
  1461
    by (simp add: field_simps of_nat_Suc x)
hoelzl@50331
  1462
  also have "\<dots> \<le> (x + 1)^Suc n"
hoelzl@50331
  1463
    using Suc x by (simp add: mult_left_mono)
hoelzl@50331
  1464
  finally show ?case .
hoelzl@50331
  1465
qed simp
hoelzl@50331
  1466
hoelzl@50331
  1467
lemma filterlim_realpow_sequentially_gt1:
hoelzl@50331
  1468
  fixes x :: "'a :: real_normed_div_algebra"
hoelzl@50331
  1469
  assumes x[arith]: "1 < norm x"
hoelzl@50331
  1470
  shows "LIM n sequentially. x ^ n :> at_infinity"
hoelzl@50331
  1471
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
hoelzl@50331
  1472
  fix y :: real assume "0 < y"
hoelzl@50331
  1473
  have "0 < norm x - 1" by simp
hoelzl@50331
  1474
  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
hoelzl@50331
  1475
  also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
hoelzl@50331
  1476
  also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
hoelzl@50331
  1477
  also have "\<dots> = norm x ^ N" by simp
hoelzl@50331
  1478
  finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
hoelzl@50331
  1479
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
hoelzl@50331
  1480
  then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
hoelzl@50331
  1481
    unfolding eventually_sequentially
hoelzl@50331
  1482
    by (auto simp: norm_power)
hoelzl@50331
  1483
qed simp
hoelzl@50331
  1484
hoelzl@51471
  1485
wenzelm@60758
  1486
subsection \<open>Limits of Sequences\<close>
hoelzl@51526
  1487
hoelzl@62368
  1488
lemma [trans]: "X = Y \<Longrightarrow> Y \<longlonglongrightarrow> z \<Longrightarrow> X \<longlonglongrightarrow> z"
hoelzl@51526
  1489
  by simp
hoelzl@51526
  1490
hoelzl@51526
  1491
lemma LIMSEQ_iff:
hoelzl@51526
  1492
  fixes L :: "'a::real_normed_vector"
wenzelm@61969
  1493
  shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
lp15@60017
  1494
unfolding lim_sequentially dist_norm ..
hoelzl@51526
  1495
hoelzl@51526
  1496
lemma LIMSEQ_I:
hoelzl@51526
  1497
  fixes L :: "'a::real_normed_vector"
wenzelm@61969
  1498
  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
hoelzl@51526
  1499
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1500
hoelzl@51526
  1501
lemma LIMSEQ_D:
hoelzl@51526
  1502
  fixes L :: "'a::real_normed_vector"
wenzelm@61969
  1503
  shows "\<lbrakk>X \<longlonglongrightarrow> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
hoelzl@51526
  1504
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1505
wenzelm@61969
  1506
lemma LIMSEQ_linear: "\<lbrakk> X \<longlonglongrightarrow> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
hoelzl@51526
  1507
  unfolding tendsto_def eventually_sequentially
haftmann@57512
  1508
  by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
hoelzl@51526
  1509
hoelzl@51526
  1510
lemma Bseq_inverse_lemma:
hoelzl@51526
  1511
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51526
  1512
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
hoelzl@51526
  1513
apply (subst nonzero_norm_inverse, clarsimp)
hoelzl@51526
  1514
apply (erule (1) le_imp_inverse_le)
hoelzl@51526
  1515
done
hoelzl@51526
  1516
hoelzl@51526
  1517
lemma Bseq_inverse:
hoelzl@51526
  1518
  fixes a :: "'a::real_normed_div_algebra"
wenzelm@61969
  1519
  shows "\<lbrakk>X \<longlonglongrightarrow> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
hoelzl@51526
  1520
  by (rule Bfun_inverse)
hoelzl@51526
  1521
wenzelm@60758
  1522
text\<open>Transformation of limit.\<close>
lp15@60141
  1523
lp15@60141
  1524
lemma eventually_at2:
lp15@60141
  1525
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
paulson@62087
  1526
  unfolding eventually_at by auto
lp15@60141
  1527
lp15@60141
  1528
lemma Lim_transform:
lp15@60141
  1529
  fixes a b :: "'a::real_normed_vector"
wenzelm@61973
  1530
  shows "\<lbrakk>(g \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> a) F"
lp15@60141
  1531
  using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
lp15@60141
  1532
lp15@60141
  1533
lemma Lim_transform2:
lp15@60141
  1534
  fixes a b :: "'a::real_normed_vector"
wenzelm@61973
  1535
  shows "\<lbrakk>(f \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> a) F"
lp15@60141
  1536
  by (erule Lim_transform) (simp add: tendsto_minus_cancel)
lp15@60141
  1537
lp15@60141
  1538
lemma Lim_transform_eventually:
wenzelm@61973
  1539
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> (g \<longlongrightarrow> l) net"
lp15@60141
  1540
  apply (rule topological_tendstoI)
lp15@60141
  1541
  apply (drule (2) topological_tendstoD)
lp15@60141
  1542
  apply (erule (1) eventually_elim2, simp)
lp15@60141
  1543
  done
lp15@60141
  1544
lp15@60141
  1545
lemma Lim_transform_within:
paulson@62087
  1546
  assumes "(f \<longlongrightarrow> l) (at x within S)"
paulson@62087
  1547
    and "0 < d"
paulson@62087
  1548
    and "\<And>x'. \<lbrakk>x'\<in>S; 0 < dist x' x; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
wenzelm@61973
  1549
  shows "(g \<longlongrightarrow> l) (at x within S)"
lp15@60141
  1550
proof (rule Lim_transform_eventually)
lp15@60141
  1551
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
paulson@62087
  1552
    using assms by (auto simp: eventually_at)
wenzelm@61973
  1553
  show "(f \<longlongrightarrow> l) (at x within S)" by fact
lp15@60141
  1554
qed
lp15@60141
  1555
wenzelm@60758
  1556
text\<open>Common case assuming being away from some crucial point like 0.\<close>
hoelzl@51526
  1557
lp15@60141
  1558
lemma Lim_transform_away_within:
lp15@60141
  1559
  fixes a b :: "'a::t1_space"
lp15@60141
  1560
  assumes "a \<noteq> b"
lp15@60141
  1561
    and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
wenzelm@61973
  1562
    and "(f \<longlongrightarrow> l) (at a within S)"
wenzelm@61973
  1563
  shows "(g \<longlongrightarrow> l) (at a within S)"
lp15@60141
  1564
proof (rule Lim_transform_eventually)
wenzelm@61973
  1565
  show "(f \<longlongrightarrow> l) (at a within S)" by fact
lp15@60141
  1566
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
lp15@60141
  1567
    unfolding eventually_at_topological
lp15@60141
  1568
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
lp15@60141
  1569
qed
lp15@60141
  1570
lp15@60141
  1571
lemma Lim_transform_away_at:
lp15@60141
  1572
  fixes a b :: "'a::t1_space"
lp15@60141
  1573
  assumes ab: "a\<noteq>b"
lp15@60141
  1574
    and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
wenzelm@61973
  1575
    and fl: "(f \<longlongrightarrow> l) (at a)"
wenzelm@61973
  1576
  shows "(g \<longlongrightarrow> l) (at a)"
lp15@60141
  1577
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
lp15@60141
  1578
wenzelm@60758
  1579
text\<open>Alternatively, within an open set.\<close>
hoelzl@51526
  1580
lp15@60141
  1581
lemma Lim_transform_within_open:
paulson@62087
  1582
  assumes "(f \<longlongrightarrow> l) (at a within T)"
paulson@62087
  1583
    and "open s" and "a \<in> s"
paulson@62087
  1584
    and "\<And>x. \<lbrakk>x\<in>s; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
paulson@62087
  1585
  shows "(g \<longlongrightarrow> l) (at a within T)"
lp15@60141
  1586
proof (rule Lim_transform_eventually)
paulson@62087
  1587
  show "eventually (\<lambda>x. f x = g x) (at a within T)"
lp15@60141
  1588
    unfolding eventually_at_topological
paulson@62087
  1589
    using assms by auto
paulson@62087
  1590
  show "(f \<longlongrightarrow> l) (at a within T)" by fact
lp15@60141
  1591
qed
lp15@60141
  1592
wenzelm@60758
  1593
text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
lp15@60141
  1594
lp15@60141
  1595
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
lp15@60141
  1596
lp15@60141
  1597
lemma Lim_cong_within(*[cong add]*):
lp15@60141
  1598
  assumes "a = b"
lp15@60141
  1599
    and "x = y"
lp15@60141
  1600
    and "S = T"
lp15@60141
  1601
    and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
wenzelm@61973
  1602
  shows "(f \<longlongrightarrow> x) (at a within S) \<longleftrightarrow> (g \<longlongrightarrow> y) (at b within T)"
lp15@60141
  1603
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1604
  using assms by simp
lp15@60141
  1605
lp15@60141
  1606
lemma Lim_cong_at(*[cong add]*):
lp15@60141
  1607
  assumes "a = b" "x = y"
lp15@60141
  1608
    and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
wenzelm@61973
  1609
  shows "((\<lambda>x. f x) \<longlongrightarrow> x) (at a) \<longleftrightarrow> ((g \<longlongrightarrow> y) (at a))"
lp15@60141
  1610
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1611
  using assms by simp
wenzelm@60758
  1612
text\<open>An unbounded sequence's inverse tends to 0\<close>
hoelzl@51526
  1613
hoelzl@51526
  1614
lemma LIMSEQ_inverse_zero:
wenzelm@61969
  1615
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
hoelzl@51526
  1616
  apply (rule filterlim_compose[OF tendsto_inverse_0])
hoelzl@51526
  1617
  apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
hoelzl@51526
  1618
  apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
hoelzl@51526
  1619
  done
hoelzl@51526
  1620
wenzelm@60758
  1621
text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
hoelzl@51526
  1622
wenzelm@61969
  1623
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) \<longlonglongrightarrow> 0"
hoelzl@51526
  1624
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
hoelzl@51526
  1625
            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
hoelzl@51526
  1626
wenzelm@60758
  1627
text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
wenzelm@60758
  1628
infinity is now easily proved\<close>
hoelzl@51526
  1629
hoelzl@51526
  1630
lemma LIMSEQ_inverse_real_of_nat_add:
wenzelm@61969
  1631
     "(%n. r + inverse(real(Suc n))) \<longlonglongrightarrow> r"
hoelzl@51526
  1632
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
hoelzl@51526
  1633
hoelzl@51526
  1634
lemma LIMSEQ_inverse_real_of_nat_add_minus:
wenzelm@61969
  1635
     "(%n. r + -inverse(real(Suc n))) \<longlonglongrightarrow> r"
hoelzl@51526
  1636
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
hoelzl@51526
  1637
  by auto
hoelzl@51526
  1638
hoelzl@51526
  1639
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
wenzelm@61969
  1640
     "(%n. r*( 1 + -inverse(real(Suc n)))) \<longlonglongrightarrow> r"
hoelzl@51526
  1641
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
hoelzl@51526
  1642
  by auto
hoelzl@51526
  1643
wenzelm@61973
  1644
lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
eberlm@61524
  1645
  using lim_1_over_n by (simp add: inverse_eq_divide)
eberlm@61524
  1646
wenzelm@61969
  1647
lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
eberlm@61524
  1648
proof (rule Lim_transform_eventually)
eberlm@61524
  1649
  show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
eberlm@61524
  1650
    using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
wenzelm@61969
  1651
  have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
eberlm@61524
  1652
    by (intro tendsto_add tendsto_const lim_inverse_n)
wenzelm@61969
  1653
  thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1" by simp
eberlm@61524
  1654
qed
eberlm@61524
  1655
wenzelm@61969
  1656
lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
eberlm@61524
  1657
proof (rule Lim_transform_eventually)
paulson@62087
  1658
  show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) =
eberlm@61524
  1659
                        of_nat n / of_nat (Suc n)) sequentially"
paulson@62087
  1660
    using eventually_gt_at_top[of "0::nat"]
eberlm@61524
  1661
    by eventually_elim (simp add: field_simps del: of_nat_Suc)
wenzelm@61969
  1662
  have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
eberlm@61524
  1663
    by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
wenzelm@61969
  1664
  thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1" by simp
eberlm@61524
  1665
qed
eberlm@61524
  1666
wenzelm@60758
  1667
subsection \<open>Convergence on sequences\<close>
hoelzl@51526
  1668
eberlm@61531
  1669
lemma convergent_cong:
eberlm@61531
  1670
  assumes "eventually (\<lambda>x. f x = g x) sequentially"
eberlm@61531
  1671
  shows   "convergent f \<longleftrightarrow> convergent g"
eberlm@61531
  1672
  unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)
eberlm@61531
  1673
eberlm@61531
  1674
lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
eberlm@61531
  1675
  by (auto simp: convergent_def LIMSEQ_Suc_iff)
eberlm@61531
  1676
eberlm@61531
  1677
lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
eberlm@61531
  1678
proof (induction m arbitrary: f)
eberlm@61531
  1679
  case (Suc m)
eberlm@61531
  1680
  have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp
eberlm@61531
  1681
  also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)
eberlm@61531
  1682
  also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)
eberlm@61531
  1683
  finally show ?case .
eberlm@61531
  1684
qed simp_all
eberlm@61531
  1685
hoelzl@51526
  1686
lemma convergent_add:
hoelzl@51526
  1687
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1688
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1689
  assumes "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1690
  shows "convergent (\<lambda>n. X n + Y n)"
lp15@61649
  1691
  using assms unfolding convergent_def by (blast intro: tendsto_add)
hoelzl@51526
  1692
hoelzl@51526
  1693
lemma convergent_setsum:
hoelzl@51526
  1694
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1695
  assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
hoelzl@51526
  1696
  shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
hoelzl@51526
  1697
proof (cases "finite A")
hoelzl@51526
  1698
  case True from this and assms show ?thesis
hoelzl@51526
  1699
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
hoelzl@51526
  1700
qed (simp add: convergent_const)
hoelzl@51526
  1701
hoelzl@51526
  1702
lemma (in bounded_linear) convergent:
hoelzl@51526
  1703
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1704
  shows "convergent (\<lambda>n. f (X n))"
lp15@61649
  1705
  using assms unfolding convergent_def by (blast intro: tendsto)
hoelzl@51526
  1706
hoelzl@51526
  1707
lemma (in bounded_bilinear) convergent:
hoelzl@51526
  1708
  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1709
  shows "convergent (\<lambda>n. X n ** Y n)"
lp15@61649
  1710
  using assms unfolding convergent_def by (blast intro: tendsto)
hoelzl@51526
  1711
hoelzl@51526
  1712
lemma convergent_minus_iff:
hoelzl@51526
  1713
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1714
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
hoelzl@51526
  1715
apply (simp add: convergent_def)
hoelzl@51526
  1716
apply (auto dest: tendsto_minus)
hoelzl@51526
  1717
apply (drule tendsto_minus, auto)
hoelzl@51526
  1718
done
hoelzl@51526
  1719
eberlm@61531
  1720
lemma convergent_diff:
eberlm@61531
  1721
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
eberlm@61531
  1722
  assumes "convergent (\<lambda>n. X n)"
eberlm@61531
  1723
  assumes "convergent (\<lambda>n. Y n)"
eberlm@61531
  1724
  shows "convergent (\<lambda>n. X n - Y n)"
lp15@61649
  1725
  using assms unfolding convergent_def by (blast intro: tendsto_diff)
eberlm@61531
  1726
eberlm@61531
  1727
lemma convergent_norm:
eberlm@61531
  1728
  assumes "convergent f"
eberlm@61531
  1729
  shows   "convergent (\<lambda>n. norm (f n))"
eberlm@61531
  1730
proof -
wenzelm@61969
  1731
  from assms have "f \<longlonglongrightarrow> lim f" by (simp add: convergent_LIMSEQ_iff)
wenzelm@61969
  1732
  hence "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)" by (rule tendsto_norm)
eberlm@61531
  1733
  thus ?thesis by (auto simp: convergent_def)
eberlm@61531
  1734
qed
eberlm@61531
  1735
paulson@62087
  1736
lemma convergent_of_real:
eberlm@61531
  1737
  "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"
eberlm@61531
  1738
  unfolding convergent_def by (blast intro!: tendsto_of_real)
eberlm@61531
  1739
paulson@62087
  1740
lemma convergent_add_const_iff:
eberlm@61531
  1741
  "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1742
proof
eberlm@61531
  1743
  assume "convergent (\<lambda>n. c + f n)"
eberlm@61531
  1744
  from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp
eberlm@61531
  1745
next
eberlm@61531
  1746
  assume "convergent f"
eberlm@61531
  1747
  from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp
eberlm@61531
  1748
qed
eberlm@61531
  1749
paulson@62087
  1750
lemma convergent_add_const_right_iff:
eberlm@61531
  1751
  "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1752
  using convergent_add_const_iff[of c f] by (simp add: add_ac)
eberlm@61531
  1753
paulson@62087
  1754
lemma convergent_diff_const_right_iff:
eberlm@61531
  1755
  "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1756
  using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)
eberlm@61531
  1757
eberlm@61531
  1758
lemma convergent_mult:
eberlm@61531
  1759
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
eberlm@61531
  1760
  assumes "convergent (\<lambda>n. X n)"
eberlm@61531
  1761
  assumes "convergent (\<lambda>n. Y n)"
eberlm@61531
  1762
  shows "convergent (\<lambda>n. X n * Y n)"
lp15@61649
  1763
  using assms unfolding convergent_def by (blast intro: tendsto_mult)
eberlm@61531
  1764
eberlm@61531
  1765
lemma convergent_mult_const_iff:
eberlm@61531
  1766
  assumes "c \<noteq> 0"
eberlm@61531
  1767
  shows   "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"
eberlm@61531
  1768
proof
eberlm@61531
  1769
  assume "convergent (\<lambda>n. c * f n)"
paulson@62087
  1770
  from assms convergent_mult[OF this convergent_const[of "inverse c"]]
eberlm@61531
  1771
    show "convergent f" by (simp add: field_simps)
eberlm@61531
  1772
next
eberlm@61531
  1773
  assume "convergent f"
eberlm@61531
  1774
  from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp
eberlm@61531
  1775
qed
eberlm@61531
  1776
eberlm@61531
  1777
lemma convergent_mult_const_right_iff:
eberlm@61531
  1778
  assumes "c \<noteq> 0"
eberlm@61531
  1779
  shows   "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"
eberlm@61531
  1780
  using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
eberlm@61531
  1781
eberlm@61531
  1782
lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
eberlm@61531
  1783
  by (simp add: Cauchy_Bseq convergent_Cauchy)
eberlm@61531
  1784
hoelzl@51526
  1785
wenzelm@60758
  1786
text \<open>A monotone sequence converges to its least upper bound.\<close>
hoelzl@51526
  1787
hoelzl@54263
  1788
lemma LIMSEQ_incseq_SUP:
hoelzl@54263
  1789
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1790
  assumes u: "bdd_above (range X)"
hoelzl@54263
  1791
  assumes X: "incseq X"
wenzelm@61969
  1792
  shows "X \<longlonglongrightarrow> (SUP i. X i)"
hoelzl@54263
  1793
  by (rule order_tendstoI)
hoelzl@54263
  1794
     (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
hoelzl@51526
  1795
hoelzl@54263
  1796
lemma LIMSEQ_decseq_INF:
hoelzl@54263
  1797
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1798
  assumes u: "bdd_below (range X)"
hoelzl@54263
  1799
  assumes X: "decseq X"
wenzelm@61969
  1800
  shows "X \<longlonglongrightarrow> (INF i. X i)"
hoelzl@54263
  1801
  by (rule order_tendstoI)
hoelzl@54263
  1802
     (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
hoelzl@51526
  1803
wenzelm@60758
  1804
text\<open>Main monotonicity theorem\<close>
hoelzl@51526
  1805
hoelzl@51526
  1806
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1807
  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
hoelzl@54263
  1808
hoelzl@54263
  1809
lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1810
  by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
hoelzl@51526
  1811
eberlm@61531
  1812
lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
eberlm@61531
  1813
  using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
eberlm@61531
  1814
eberlm@61531
  1815
lemma Bseq_monoseq_convergent'_inc:
eberlm@61531
  1816
  "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"
eberlm@61531
  1817
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
eberlm@61531
  1818
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
eberlm@61531
  1819
eberlm@61531
  1820
lemma Bseq_monoseq_convergent'_dec:
eberlm@61531
  1821
  "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"
eberlm@61531
  1822
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
eberlm@61531
  1823
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
eberlm@61531
  1824
hoelzl@51526
  1825
lemma Cauchy_iff:
hoelzl@51526
  1826
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1827
  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
  1828
  unfolding Cauchy_def dist_norm ..
hoelzl@51526
  1829
hoelzl@51526
  1830
lemma CauchyI:
hoelzl@51526
  1831
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1832
  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
  1833
by (simp add: Cauchy_iff)
hoelzl@51526
  1834
hoelzl@51526
  1835
lemma CauchyD:
hoelzl@51526
  1836
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1837
  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
  1838
by (simp add: Cauchy_iff)
hoelzl@51526
  1839
hoelzl@51526
  1840
lemma incseq_convergent:
hoelzl@51526
  1841
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1842
  assumes "incseq X" and "\<forall>i. X i \<le> B"
wenzelm@61969
  1843
  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
hoelzl@51526
  1844
proof atomize_elim
wenzelm@60758
  1845
  from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
wenzelm@61969
  1846
  obtain L where "X \<longlonglongrightarrow> L"
hoelzl@51526
  1847
    by (auto simp: convergent_def monoseq_def incseq_def)
wenzelm@61969
  1848
  with \<open>incseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. X i \<le> L)"
hoelzl@51526
  1849
    by (auto intro!: exI[of _ L] incseq_le)
hoelzl@51526
  1850
qed
hoelzl@51526
  1851
hoelzl@51526
  1852
lemma decseq_convergent:
hoelzl@51526
  1853
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1854
  assumes "decseq X" and "\<forall>i. B \<le> X i"
wenzelm@61969
  1855
  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
hoelzl@51526
  1856
proof atomize_elim
wenzelm@60758
  1857
  from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
wenzelm@61969
  1858
  obtain L where "X \<longlonglongrightarrow> L"
hoelzl@51526
  1859
    by (auto simp: convergent_def monoseq_def decseq_def)
wenzelm@61969
  1860
  with \<open>decseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. L \<le> X i)"
hoelzl@51526
  1861
    by (auto intro!: exI[of _ L] decseq_le)
hoelzl@51526
  1862
qed
hoelzl@51526
  1863
wenzelm@60758
  1864
subsubsection \<open>Cauchy Sequences are Bounded\<close>
hoelzl@51526
  1865
wenzelm@60758
  1866
text\<open>A Cauchy sequence is bounded -- this is the standard
wenzelm@60758
  1867
  proof mechanization rather than the nonstandard proof\<close>
hoelzl@51526
  1868
hoelzl@51526
  1869
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
hoelzl@51526
  1870
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
hoelzl@51526
  1871
apply (clarify, drule spec, drule (1) mp)
hoelzl@51526
  1872
apply (simp only: norm_minus_commute)
hoelzl@51526
  1873
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
hoelzl@51526
  1874
apply simp
hoelzl@51526
  1875
done
hoelzl@51526
  1876
wenzelm@60758
  1877
subsection \<open>Power Sequences\<close>
hoelzl@51526
  1878
wenzelm@60758
  1879
text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
hoelzl@51526
  1880
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
wenzelm@60758
  1881
  also fact that bounded and monotonic sequence converges.\<close>
hoelzl@51526
  1882
hoelzl@51526
  1883
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
hoelzl@51526
  1884
apply (simp add: Bseq_def)
hoelzl@51526
  1885
apply (rule_tac x = 1 in exI)
hoelzl@51526
  1886
apply (simp add: power_abs)
hoelzl@51526
  1887
apply (auto dest: power_mono)
hoelzl@51526
  1888
done
hoelzl@51526
  1889
hoelzl@51526
  1890
lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
hoelzl@51526
  1891
apply (clarify intro!: mono_SucI2)
hoelzl@51526
  1892
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
hoelzl@51526
  1893
done
hoelzl@51526
  1894
hoelzl@51526
  1895
lemma convergent_realpow:
hoelzl@51526
  1896
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
hoelzl@51526
  1897
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
hoelzl@51526
  1898
wenzelm@61969
  1899
lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
hoelzl@51526
  1900
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1901
hoelzl@51526
  1902
lemma LIMSEQ_realpow_zero:
wenzelm@61969
  1903
  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
hoelzl@51526
  1904
proof cases
hoelzl@51526
  1905
  assume "0 \<le> x" and "x \<noteq> 0"
hoelzl@51526
  1906
  hence x0: "0 < x" by simp
hoelzl@51526
  1907
  assume x1: "x < 1"
hoelzl@51526
  1908
  from x0 x1 have "1 < inverse x"
hoelzl@51526
  1909
    by (rule one_less_inverse)
wenzelm@61969
  1910
  hence "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
hoelzl@51526
  1911
    by (rule LIMSEQ_inverse_realpow_zero)
hoelzl@51526
  1912
  thus ?thesis by (simp add: power_inverse)
hoelzl@58729
  1913
qed (rule LIMSEQ_imp_Suc, simp)
hoelzl@51526
  1914
hoelzl@51526
  1915
lemma LIMSEQ_power_zero:
hoelzl@51526
  1916
  fixes x :: "'a::{real_normed_algebra_1}"
wenzelm@61969
  1917
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
hoelzl@51526
  1918
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
hoelzl@51526
  1919
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
hoelzl@51526
  1920
apply (simp add: power_abs norm_power_ineq)
hoelzl@51526
  1921
done
hoelzl@51526
  1922
wenzelm@61969
  1923
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
hoelzl@51526
  1924
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1925
wenzelm@60758
  1926
text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
hoelzl@51526
  1927
wenzelm@61969
  1928
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
hoelzl@51526
  1929
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
hoelzl@51526
  1930
wenzelm@61969
  1931
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) \<longlonglongrightarrow> 0"
hoelzl@51526
  1932
  by (rule LIMSEQ_power_zero) simp
hoelzl@51526
  1933
hoelzl@51526
  1934
wenzelm@60758
  1935
subsection \<open>Limits of Functions\<close>
hoelzl@51526
  1936
hoelzl@51526
  1937
lemma LIM_eq:
hoelzl@51526
  1938
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
wenzelm@61976
  1939
  shows "f \<midarrow>a\<rightarrow> L =
hoelzl@51526
  1940
     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
hoelzl@51526
  1941
by (simp add: LIM_def dist_norm)
hoelzl@51526
  1942
hoelzl@51526
  1943
lemma LIM_I:
hoelzl@51526
  1944
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1945
  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
wenzelm@61976
  1946
      ==> f \<midarrow>a\<rightarrow> L"
hoelzl@51526
  1947
by (simp add: LIM_eq)
hoelzl@51526
  1948
hoelzl@51526
  1949
lemma LIM_D:
hoelzl@51526
  1950
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
wenzelm@61976
  1951
  shows "[| f \<midarrow>a\<rightarrow> L; 0<r |]
hoelzl@51526
  1952
      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
hoelzl@51526
  1953
by (simp add: LIM_eq)
hoelzl@51526
  1954
hoelzl@51526
  1955
lemma LIM_offset:
hoelzl@51526
  1956
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  1957
  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
hoelzl@51641
  1958
  unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
hoelzl@51526
  1959
hoelzl@51526
  1960
lemma LIM_offset_zero:
hoelzl@51526
  1961
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  1962
  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
haftmann@57512
  1963
by (drule_tac k="a" in LIM_offset, simp add: add.commute)
hoelzl@51526
  1964
hoelzl@51526
  1965
lemma LIM_offset_zero_cancel:
hoelzl@51526
  1966
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  1967
  shows "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
hoelzl@51526
  1968
by (drule_tac k="- a" in LIM_offset, simp)
hoelzl@51526
  1969
hoelzl@51642
  1970
lemma LIM_offset_zero_iff:
hoelzl@51642
  1971
  fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
wenzelm@61976
  1972
  shows  "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
hoelzl@51642
  1973
  using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
hoelzl@51642
  1974
hoelzl@51526
  1975
lemma LIM_zero:
hoelzl@51526
  1976
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@61973
  1977
  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
hoelzl@51526
  1978
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1979
hoelzl@51526
  1980
lemma LIM_zero_cancel:
hoelzl@51526
  1981
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@61973
  1982
  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
hoelzl@51526
  1983
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1984
hoelzl@51526
  1985
lemma LIM_zero_iff:
hoelzl@51526
  1986
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@61973
  1987
  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
hoelzl@51526
  1988
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1989
hoelzl@51526
  1990
lemma LIM_imp_LIM:
hoelzl@51526
  1991
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1992
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
wenzelm@61976
  1993
  assumes f: "f \<midarrow>a\<rightarrow> l"
hoelzl@51526
  1994
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
wenzelm@61976
  1995
  shows "g \<midarrow>a\<rightarrow> m"
hoelzl@51526
  1996
  by (rule metric_LIM_imp_LIM [OF f],
hoelzl@51526
  1997
    simp add: dist_norm le)
hoelzl@51526
  1998
hoelzl@51526
  1999
lemma LIM_equal2:
hoelzl@51526
  2000
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  2001
  assumes 1: "0 < R"
hoelzl@51526
  2002
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
wenzelm@61976
  2003
  shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l"
hoelzl@51526
  2004
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
hoelzl@51526
  2005
hoelzl@51526
  2006
lemma LIM_compose2:
hoelzl@51526
  2007
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  2008
  assumes f: "f \<midarrow>a\<rightarrow> b"
wenzelm@61976
  2009
  assumes g: "g \<midarrow>b\<rightarrow> c"
hoelzl@51526
  2010
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
wenzelm@61976
  2011
  shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
hoelzl@51526
  2012
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
hoelzl@51526
  2013
hoelzl@51526
  2014
lemma real_LIM_sandwich_zero:
hoelzl@51526
  2015
  fixes f g :: "'a::topological_space \<Rightarrow> real"
wenzelm@61976
  2016
  assumes f: "f \<midarrow>a\<rightarrow> 0"
hoelzl@51526
  2017
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
hoelzl@51526
  2018
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
wenzelm@61976
  2019
  shows "g \<midarrow>a\<rightarrow> 0"
hoelzl@51526
  2020
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
hoelzl@51526
  2021
  fix x assume x: "x \<noteq> a"
hoelzl@51526
  2022
  have "norm (g x - 0) = g x" by (simp add: 1 x)
hoelzl@51526
  2023
  also have "g x \<le> f x" by (rule 2 [OF x])
hoelzl@51526
  2024
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
hoelzl@51526
  2025
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
hoelzl@51526
  2026
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
hoelzl@51526
  2027
qed
hoelzl@51526
  2028
hoelzl@51526
  2029
wenzelm@60758
  2030
subsection \<open>Continuity\<close>
hoelzl@51526
  2031
hoelzl@51526
  2032
lemma LIM_isCont_iff:
hoelzl@51526
  2033
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
wenzelm@61976
  2034
  shows "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
hoelzl@51526
  2035
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
hoelzl@51526
  2036
hoelzl@51526
  2037
lemma isCont_iff:
hoelzl@51526
  2038
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
wenzelm@61976
  2039
  shows "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
hoelzl@51526
  2040
by (simp add: isCont_def LIM_isCont_iff)
hoelzl@51526
  2041
hoelzl@51526
  2042
lemma isCont_LIM_compose2:
hoelzl@51526
  2043
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  2044
  assumes f [unfolded isCont_def]: "isCont f a"
wenzelm@61976
  2045
  assumes g: "g \<midarrow>f a\<rightarrow> l"
hoelzl@51526
  2046
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
wenzelm@61976
  2047
  shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
hoelzl@51526
  2048
by (rule LIM_compose2 [OF f g inj])
hoelzl@51526
  2049
hoelzl@51526
  2050
hoelzl@51526
  2051
lemma isCont_norm [simp]:
hoelzl@51526
  2052
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2053
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
hoelzl@51526
  2054
  by (fact continuous_norm)
hoelzl@51526
  2055
hoelzl@51526
  2056
lemma isCont_rabs [simp]:
hoelzl@51526
  2057
  fixes f :: "'a::t2_space \<Rightarrow> real"
hoelzl@51526
  2058
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
hoelzl@51526
  2059
  by (fact continuous_rabs)
hoelzl@51526
  2060
hoelzl@51526
  2061
lemma isCont_add [simp]:
hoelzl@62368
  2062
  fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
hoelzl@51526
  2063
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
hoelzl@51526
  2064
  by (fact continuous_add)
hoelzl@51526
  2065
hoelzl@51526
  2066
lemma isCont_minus [simp]:
hoelzl@51526
  2067
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2068
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
hoelzl@51526
  2069
  by (fact continuous_minus)
hoelzl@51526
  2070
hoelzl@51526
  2071
lemma isCont_diff [simp]:
hoelzl@51526
  2072
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2073
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
hoelzl@51526
  2074
  by (fact continuous_diff)
hoelzl@51526
  2075
hoelzl@51526
  2076
lemma isCont_mult [simp]:
hoelzl@51526
  2077
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
hoelzl@51526
  2078
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
hoelzl@51526
  2079
  by (fact continuous_mult)
hoelzl@51526
  2080
hoelzl@51526
  2081
lemma (in bounded_linear) isCont:
hoelzl@51526
  2082
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
hoelzl@51526
  2083
  by (fact continuous)
hoelzl@51526
  2084
hoelzl@51526
  2085
lemma (in bounded_bilinear) isCont:
hoelzl@51526
  2086
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
hoelzl@51526
  2087
  by (fact continuous)
hoelzl@51526
  2088
lp15@60141
  2089
lemmas isCont_scaleR [simp] =
hoelzl@51526
  2090
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
hoelzl@51526
  2091
hoelzl@51526
  2092
lemmas isCont_of_real [simp] =
hoelzl@51526
  2093
  bounded_linear.isCont [OF bounded_linear_of_real]
hoelzl@51526
  2094
hoelzl@51526
  2095
lemma isCont_power [simp]:
hoelzl@51526
  2096
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51526
  2097
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
hoelzl@51526
  2098
  by (fact continuous_power)
hoelzl@51526
  2099
hoelzl@51526
  2100
lemma isCont_setsum [simp]:
hoelzl@62368
  2101
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@51526
  2102
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
hoelzl@51526
  2103
  by (auto intro: continuous_setsum)
hoelzl@51526
  2104
wenzelm@60758
  2105
subsection \<open>Uniform Continuity\<close>
hoelzl@51526
  2106
hoelzl@51531
  2107
definition
hoelzl@51531
  2108
  isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
hoelzl@51531
  2109
  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
hoelzl@51531
  2110
hoelzl@51531
  2111
lemma isUCont_isCont: "isUCont f ==> isCont f x"
hoelzl@51531
  2112
by (simp add: isUCont_def isCont_def LIM_def, force)
hoelzl@51531
  2113
hoelzl@51531
  2114
lemma isUCont_Cauchy:
hoelzl@51531
  2115
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51531
  2116
unfolding isUCont_def
hoelzl@51531
  2117
apply (rule metric_CauchyI)
hoelzl@51531
  2118
apply (drule_tac x=e in spec, safe)
hoelzl@51531
  2119
apply (drule_tac e=s in metric_CauchyD, safe)
hoelzl@51531
  2120
apply (rule_tac x=M in exI, simp)
hoelzl@51531
  2121
done
hoelzl@51531
  2122
hoelzl@51526
  2123
lemma (in bounded_linear) isUCont: "isUCont f"
hoelzl@51526
  2124
unfolding isUCont_def dist_norm
hoelzl@51526
  2125
proof (intro allI impI)
hoelzl@51526
  2126
  fix r::real assume r: "0 < r"
hoelzl@51526
  2127
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
lp15@61649
  2128
    using pos_bounded by blast
hoelzl@51526
  2129
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
hoelzl@51526
  2130
  proof (rule exI, safe)
nipkow@56541
  2131
    from r K show "0 < r / K" by simp
hoelzl@51526
  2132
  next
hoelzl@51526
  2133
    fix x y :: 'a
hoelzl@51526
  2134
    assume xy: "norm (x - y) < r / K"
hoelzl@51526
  2135
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
hoelzl@51526
  2136
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
hoelzl@51526
  2137
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
hoelzl@51526
  2138
    finally show "norm (f x - f y) < r" .
hoelzl@51526
  2139
  qed
hoelzl@51526
  2140
qed
hoelzl@51526
  2141
hoelzl@51526
  2142
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51526
  2143
by (rule isUCont [THEN isUCont_Cauchy])
hoelzl@51526
  2144
lp15@60141
  2145
lemma LIM_less_bound:
hoelzl@51526
  2146
  fixes f :: "real \<Rightarrow> real"
hoelzl@51526
  2147
  assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
hoelzl@51526
  2148
  shows "0 \<le> f x"
hoelzl@51526
  2149
proof (rule tendsto_le_const)
wenzelm@61973
  2150
  show "(f \<longlongrightarrow> f x) (at_left x)"
wenzelm@60758
  2151
    using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
hoelzl@51526
  2152
  show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
hoelzl@51641
  2153
    using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
hoelzl@51526
  2154
qed simp
hoelzl@51471
  2155
hoelzl@51529
  2156
wenzelm@60758
  2157
subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
hoelzl@51529
  2158
hoelzl@51529
  2159
lemma nested_sequence_unique:
wenzelm@61969
  2160
  assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0"
wenzelm@61969
  2161
  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f \<longlonglongrightarrow> l) \<and> ((\<forall>n. l \<le> g n) \<and> g \<longlonglongrightarrow> l)"
hoelzl@51529
  2162
proof -
hoelzl@51529
  2163
  have "incseq f" unfolding incseq_Suc_iff by fact
hoelzl@51529
  2164
  have "decseq g" unfolding decseq_Suc_iff by fact
hoelzl@51529
  2165
hoelzl@51529
  2166
  { fix n
wenzelm@60758
  2167
    from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
wenzelm@60758
  2168
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
wenzelm@61969
  2169
  then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
wenzelm@60758
  2170
    using incseq_convergent[OF \<open>incseq f\<close>] by auto
hoelzl@51529
  2171
  moreover
hoelzl@51529
  2172
  { fix n
wenzelm@60758
  2173
    from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
wenzelm@60758
  2174
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
wenzelm@61969
  2175
  then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
wenzelm@60758
  2176
    using decseq_convergent[OF \<open>decseq g\<close>] by auto
wenzelm@61969
  2177
  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
hoelzl@51529
  2178
  ultimately show ?thesis by auto
hoelzl@51529
  2179
qed
hoelzl@51529
  2180
hoelzl@51529
  2181
lemma Bolzano[consumes 1, case_names trans local]:
hoelzl@51529
  2182
  fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
hoelzl@51529
  2183
  assumes [arith]: "a \<le> b"
hoelzl@51529
  2184
  assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
hoelzl@51529
  2185
  assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
hoelzl@51529
  2186
  shows "P a b"
hoelzl@51529
  2187
proof -
blanchet@55415
  2188
  def bisect \<equiv> "rec_nat (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
hoelzl@51529
  2189
  def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
hoelzl@51529
  2190
  have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
hoelzl@51529
  2191
    and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
hoelzl@51529
  2192
    by (simp_all add: l_def u_def bisect_def split: prod.split)
hoelzl@51529
  2193
hoelzl@51529
  2194
  { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
hoelzl@51529
  2195
wenzelm@61969
  2196
  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
hoelzl@51529
  2197
  proof (safe intro!: nested_sequence_unique)
hoelzl@51529
  2198
    fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
hoelzl@51529
  2199
  next
hoelzl@51529
  2200
    { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
wenzelm@61969
  2201
    then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0" by (simp add: LIMSEQ_divide_realpow_zero)
hoelzl@51529
  2202
  qed fact
wenzelm@61969
  2203
  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x" by auto
hoelzl@51529
  2204
  obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
wenzelm@60758
  2205
    using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
hoelzl@51529
  2206
hoelzl@51529
  2207
  show "P a b"
hoelzl@51529
  2208
  proof (rule ccontr)
lp15@60141
  2209
    assume "\<not> P a b"
hoelzl@51529
  2210
    { fix n have "\<not> P (l n) (u n)"
hoelzl@51529
  2211
      proof (induct n)
hoelzl@51529
  2212
        case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
wenzelm@60758
  2213
      qed (simp add: \<open>\<not> P a b\<close>) }
hoelzl@51529
  2214
    moreover
hoelzl@51529
  2215
    { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
wenzelm@61969
  2216
        using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  2217
      moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
wenzelm@61969
  2218
        using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  2219
      ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
hoelzl@51529
  2220
      proof eventually_elim
hoelzl@51529
  2221
        fix n assume "x - d / 2 < l n" "u n < x + d / 2"
hoelzl@51529
  2222
        from add_strict_mono[OF this] have "u n - l n < d" by simp
hoelzl@51529
  2223
        with x show "P (l n) (u n)" by (rule d)
hoelzl@51529
  2224
      qed }
hoelzl@51529
  2225
    ultimately show False by simp
hoelzl@51529
  2226
  qed
hoelzl@51529
  2227
qed
hoelzl@51529
  2228
hoelzl@51529
  2229
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
hoelzl@51529
  2230
proof (cases "a \<le> b", rule compactI)
hoelzl@51529
  2231
  fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
hoelzl@51529
  2232
  def T == "{a .. b}"
hoelzl@51529
  2233
  from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
hoelzl@51529
  2234
  proof (induct rule: Bolzano)
hoelzl@51529
  2235
    case (trans a b c)
hoelzl@51529
  2236
    then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
hoelzl@51529
  2237
    from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
hoelzl@51529
  2238
      by (auto simp: *)
hoelzl@51529
  2239
    with trans show ?case
hoelzl@51529
  2240
      unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
hoelzl@51529
  2241
  next
hoelzl@51529
  2242
    case (local x)
hoelzl@51529
  2243
    then have "x \<in> \<Union>C" using C by auto
hoelzl@51529
  2244
    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
hoelzl@51529
  2245
    then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
hoelzl@62101
  2246
      by (auto simp: open_dist dist_real_def subset_eq Ball_def abs_less_iff)
wenzelm@60758
  2247
    with \<open>c \<in> C\<close> show ?case
hoelzl@51529
  2248
      by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
hoelzl@51529
  2249
  qed
hoelzl@51529
  2250
qed simp
hoelzl@51529
  2251
hoelzl@51529
  2252
hoelzl@57447
  2253
lemma continuous_image_closed_interval:
hoelzl@57447
  2254
  fixes a b and f :: "real \<Rightarrow> real"
hoelzl@57447
  2255
  defines "S \<equiv> {a..b}"
hoelzl@57447
  2256
  assumes "a \<le> b" and f: "continuous_on S f"
hoelzl@57447
  2257
  shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
hoelzl@57447
  2258
proof -
hoelzl@57447
  2259
  have S: "compact S" "S \<noteq> {}"
wenzelm@60758
  2260
    using \<open>a \<le> b\<close> by (auto simp: S_def)
hoelzl@57447
  2261
  obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
hoelzl@57447
  2262
    using continuous_attains_sup[OF S f] by auto
hoelzl@57447
  2263
  moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
hoelzl@57447
  2264
    using continuous_attains_inf[OF S f] by auto
hoelzl@57447
  2265
  moreover have "connected (f`S)"
hoelzl@57447
  2266
    using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
hoelzl@57447
  2267
  ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
hoelzl@57447
  2268
    by (auto simp: connected_iff_interval)
hoelzl@57447
  2269
  then show ?thesis
hoelzl@57447
  2270
    by auto
hoelzl@57447
  2271
qed
hoelzl@57447
  2272
lp15@60974
  2273
lemma open_Collect_positive:
lp15@60974
  2274
 fixes f :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  2275
 assumes f: "continuous_on s f"
lp15@60974
  2276
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
lp15@60974
  2277
 using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
lp15@60974
  2278
 by (auto simp: Int_def field_simps)
lp15@60974
  2279
lp15@60974
  2280
lemma open_Collect_less_Int:
lp15@60974
  2281
 fixes f g :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  2282
 assumes f: "continuous_on s f" and g: "continuous_on s g"
lp15@60974
  2283
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
lp15@60974
  2284
 using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
lp15@60974
  2285
lp15@60974
  2286
wenzelm@60758
  2287
subsection \<open>Boundedness of continuous functions\<close>
hoelzl@51529
  2288
wenzelm@60758
  2289
text\<open>By bisection, function continuous on closed interval is bounded above\<close>
hoelzl@51529
  2290
hoelzl@51529
  2291
lemma isCont_eq_Ub:
hoelzl@51529
  2292
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2293
  shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2294
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
hoelzl@51529
  2295
  using continuous_attains_sup[of "{a .. b}" f]
hoelzl@51529
  2296
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2297
hoelzl@51529
  2298
lemma isCont_eq_Lb:
hoelzl@51529
  2299
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2300
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2301
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
hoelzl@51529
  2302
  using continuous_attains_inf[of "{a .. b}" f]
hoelzl@51529
  2303
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2304
hoelzl@51529
  2305
lemma isCont_bounded:
hoelzl@51529
  2306
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2307
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
hoelzl@51529
  2308
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2309
hoelzl@51529
  2310
lemma isCont_has_Ub:
hoelzl@51529
  2311
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2312
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2313
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
hoelzl@51529
  2314
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2315
hoelzl@51529
  2316
(*HOL style here: object-level formulations*)
hoelzl@51529
  2317
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
hoelzl@51529
  2318
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2319
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2320
  by (blast intro: IVT)
hoelzl@51529
  2321
hoelzl@51529
  2322
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
hoelzl@51529
  2323
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2324
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2325
  by (blast intro: IVT2)
hoelzl@51529
  2326
hoelzl@51529
  2327
lemma isCont_Lb_Ub:
hoelzl@51529
  2328
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  2329
  assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
lp15@60141
  2330
  shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
hoelzl@51529
  2331
               (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
hoelzl@51529
  2332
proof -
hoelzl@51529
  2333
  obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
hoelzl@51529
  2334
    using isCont_eq_Ub[OF assms] by auto
hoelzl@51529
  2335
  obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
hoelzl@51529
  2336
    using isCont_eq_Lb[OF assms] by auto
hoelzl@51529
  2337
  show ?thesis
hoelzl@51529
  2338
    using IVT[of f L _ M] IVT2[of f L _ M] M L assms
hoelzl@51529
  2339
    apply (rule_tac x="f L" in exI)
hoelzl@51529
  2340
    apply (rule_tac x="f M" in exI)
hoelzl@51529
  2341
    apply (cases "L \<le> M")
hoelzl@51529
  2342
    apply (simp, metis order_trans)
hoelzl@51529
  2343
    apply (simp, metis order_trans)
hoelzl@51529
  2344
    done
hoelzl@51529
  2345
qed
hoelzl@51529
  2346
hoelzl@51529
  2347
wenzelm@60758
  2348
text\<open>Continuity of inverse function\<close>
hoelzl@51529
  2349
hoelzl@51529
  2350
lemma isCont_inverse_function:
hoelzl@51529
  2351
  fixes f g :: "real \<Rightarrow> real"
hoelzl@51529
  2352
  assumes d: "0 < d"
hoelzl@51529
  2353
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
hoelzl@51529
  2354
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
hoelzl@51529
  2355
  shows "isCont g (f x)"
hoelzl@51529
  2356
proof -
hoelzl@51529
  2357
  let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
hoelzl@51529
  2358
hoelzl@51529
  2359
  have f: "continuous_on ?D f"
hoelzl@51529
  2360
    using cont by (intro continuous_at_imp_continuous_on ballI) auto
hoelzl@51529
  2361
  then have g: "continuous_on (f`?D) g"
hoelzl@51529
  2362
    using inj by (intro continuous_on_inv) auto
hoelzl@51529
  2363
hoelzl@51529
  2364
  from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
hoelzl@51529
  2365
    by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
hoelzl@51529
  2366
  with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
hoelzl@51529
  2367
    by (rule continuous_on_subset)
hoelzl@51529
  2368
  moreover
hoelzl@51529
  2369
  have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
hoelzl@51529
  2370
    using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
hoelzl@51529
  2371
  then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
hoelzl@51529
  2372
    by auto
hoelzl@51529
  2373
  ultimately
hoelzl@51529
  2374
  show ?thesis
hoelzl@51529
  2375
    by (simp add: continuous_on_eq_continuous_at)
hoelzl@51529
  2376
qed