src/HOL/Limits.thy
author paulson <lp15@cam.ac.uk>
Tue Nov 10 14:18:41 2015 +0000 (2015-11-10)
changeset 61609 77b453bd616f
parent 61552 980dd46a03fb
child 61649 268d88ec9087
permissions -rw-r--r--
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
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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 ---> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) ---> 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_elim1, simp)
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qed
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lemma BfunE:
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  assumes "Bfun f F"
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  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
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using assms unfolding Bfun_def by fast
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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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qed (force simp add: of_nat_Suc)
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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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lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
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proof
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  assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"
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  find_theorems "Bfun (\<lambda>_. ?c) _"
eberlm@61531
   278
  from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)
eberlm@61531
   279
  with `c \<noteq> 0` show "Bseq f" by (simp add: divide_simps)
eberlm@61531
   280
qed (intro Bseq_mult Bfun_const)
eberlm@61531
   281
eberlm@61531
   282
lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
eberlm@61531
   283
  unfolding Bseq_def by auto
eberlm@61531
   284
eberlm@61531
   285
lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
eberlm@61531
   286
  using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
eberlm@61531
   287
eberlm@61531
   288
lemma increasing_Bseq_subseq_iff:
eberlm@61531
   289
  assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"
eberlm@61531
   290
  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
eberlm@61531
   291
proof
eberlm@61531
   292
  assume "Bseq (\<lambda>x. f (g x))"
eberlm@61531
   293
  then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto
eberlm@61531
   294
  {
eberlm@61531
   295
    fix x :: nat
eberlm@61531
   296
    from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
eberlm@61531
   297
      by (auto simp: filterlim_at_top eventually_at_top_linorder)
eberlm@61531
   298
    hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast
eberlm@61531
   299
    also have "norm (f (g y)) \<le> K" by (rule K)
eberlm@61531
   300
    finally have "norm (f x) \<le> K" .
eberlm@61531
   301
  }
eberlm@61531
   302
  thus "Bseq f" by (rule BseqI')
eberlm@61531
   303
qed (insert Bseq_subseq[of f g], simp_all)
eberlm@61531
   304
eberlm@61531
   305
lemma nonneg_incseq_Bseq_subseq_iff:
eberlm@61531
   306
  assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"
eberlm@61531
   307
  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
eberlm@61531
   308
  using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
eberlm@61531
   309
hoelzl@51531
   310
lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
hoelzl@51531
   311
  apply (simp add: subset_eq)
hoelzl@51531
   312
  apply (rule BseqI'[where K="max (norm a) (norm b)"])
hoelzl@51531
   313
  apply (erule_tac x=n in allE)
hoelzl@51531
   314
  apply auto
hoelzl@51531
   315
  done
hoelzl@51531
   316
hoelzl@51531
   317
lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
hoelzl@51531
   318
  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
hoelzl@51531
   319
hoelzl@51531
   320
lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
hoelzl@51531
   321
  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
hoelzl@51531
   322
wenzelm@60758
   323
subsection \<open>Bounded Monotonic Sequences\<close>
hoelzl@51531
   324
wenzelm@60758
   325
subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
hoelzl@51531
   326
hoelzl@51531
   327
(* TODO: delete *)
hoelzl@51531
   328
(* FIXME: one use in NSA/HSEQ.thy *)
hoelzl@51531
   329
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
hoelzl@51531
   330
  apply (rule_tac x="X m" in exI)
hoelzl@51531
   331
  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
hoelzl@51531
   332
  unfolding eventually_sequentially
hoelzl@51531
   333
  apply blast
hoelzl@51531
   334
  done
hoelzl@51531
   335
wenzelm@60758
   336
subsection \<open>Convergence to Zero\<close>
huffman@31349
   337
huffman@44081
   338
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   339
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   340
huffman@31349
   341
lemma ZfunI:
huffman@44195
   342
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   343
  unfolding Zfun_def by simp
huffman@31349
   344
huffman@31349
   345
lemma ZfunD:
huffman@44195
   346
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   347
  unfolding Zfun_def by simp
huffman@31349
   348
huffman@31355
   349
lemma Zfun_ssubst:
huffman@44195
   350
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   351
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   352
huffman@44195
   353
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   354
  unfolding Zfun_def by simp
huffman@31349
   355
huffman@44195
   356
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   357
  unfolding Zfun_def by simp
huffman@31349
   358
huffman@31349
   359
lemma Zfun_imp_Zfun:
huffman@44195
   360
  assumes f: "Zfun f F"
huffman@44195
   361
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   362
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   363
proof (cases)
huffman@31349
   364
  assume K: "0 < K"
huffman@31349
   365
  show ?thesis
huffman@31349
   366
  proof (rule ZfunI)
huffman@31349
   367
    fix r::real assume "0 < r"
nipkow@56541
   368
    hence "0 < r / K" using K by simp
huffman@44195
   369
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
huffman@31487
   370
      using ZfunD [OF f] by fast
huffman@44195
   371
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   372
    proof eventually_elim
noschinl@46887
   373
      case (elim x)
huffman@31487
   374
      hence "norm (f x) * K < r"
huffman@31349
   375
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   376
      thus ?case
noschinl@46887
   377
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   378
    qed
huffman@31349
   379
  qed
huffman@31349
   380
next
huffman@31349
   381
  assume "\<not> 0 < K"
huffman@31349
   382
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   383
  show ?thesis
huffman@31355
   384
  proof (rule ZfunI)
huffman@31355
   385
    fix r :: real
huffman@31355
   386
    assume "0 < r"
huffman@44195
   387
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   388
    proof eventually_elim
noschinl@46887
   389
      case (elim x)
noschinl@46887
   390
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   391
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   392
      finally show ?case
wenzelm@60758
   393
        using \<open>0 < r\<close> by simp
huffman@31355
   394
    qed
huffman@31355
   395
  qed
huffman@31349
   396
qed
huffman@31349
   397
huffman@44195
   398
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   399
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   400
huffman@31349
   401
lemma Zfun_add:
huffman@44195
   402
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   403
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   404
proof (rule ZfunI)
huffman@31349
   405
  fix r::real assume "0 < r"
huffman@31349
   406
  hence r: "0 < r / 2" by simp
huffman@44195
   407
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   408
    using f r by (rule ZfunD)
huffman@31349
   409
  moreover
huffman@44195
   410
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   411
    using g r by (rule ZfunD)
huffman@31349
   412
  ultimately
huffman@44195
   413
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   414
  proof eventually_elim
noschinl@46887
   415
    case (elim x)
huffman@31487
   416
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   417
      by (rule norm_triangle_ineq)
huffman@31349
   418
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   419
      using elim by (rule add_strict_mono)
noschinl@46887
   420
    finally show ?case
huffman@31349
   421
      by simp
huffman@31349
   422
  qed
huffman@31349
   423
qed
huffman@31349
   424
huffman@44195
   425
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   426
  unfolding Zfun_def by simp
huffman@31349
   427
huffman@44195
   428
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
haftmann@54230
   429
  using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
huffman@31349
   430
huffman@31349
   431
lemma (in bounded_linear) Zfun:
huffman@44195
   432
  assumes g: "Zfun g F"
huffman@44195
   433
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   434
proof -
huffman@31349
   435
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   436
    using bounded by fast
huffman@44195
   437
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   438
    by simp
huffman@31487
   439
  with g show ?thesis
huffman@31349
   440
    by (rule Zfun_imp_Zfun)
huffman@31349
   441
qed
huffman@31349
   442
huffman@31349
   443
lemma (in bounded_bilinear) Zfun:
huffman@44195
   444
  assumes f: "Zfun f F"
huffman@44195
   445
  assumes g: "Zfun g F"
huffman@44195
   446
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   447
proof (rule ZfunI)
huffman@31349
   448
  fix r::real assume r: "0 < r"
huffman@31349
   449
  obtain K where K: "0 < K"
huffman@31349
   450
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   451
    using pos_bounded by fast
huffman@31349
   452
  from K have K': "0 < inverse K"
huffman@31349
   453
    by (rule positive_imp_inverse_positive)
huffman@44195
   454
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   455
    using f r by (rule ZfunD)
huffman@31349
   456
  moreover
huffman@44195
   457
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   458
    using g K' by (rule ZfunD)
huffman@31349
   459
  ultimately
huffman@44195
   460
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   461
  proof eventually_elim
noschinl@46887
   462
    case (elim x)
huffman@31487
   463
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   464
      by (rule norm_le)
huffman@31487
   465
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   466
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   467
    also from K have "r * inverse K * K = r"
huffman@31349
   468
      by simp
noschinl@46887
   469
    finally show ?case .
huffman@31349
   470
  qed
huffman@31349
   471
qed
huffman@31349
   472
huffman@31349
   473
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   474
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   475
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   476
huffman@31349
   477
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   478
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   479
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   480
huffman@44282
   481
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   482
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   483
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   484
huffman@44195
   485
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   486
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   487
lp15@60141
   488
lemma tendsto_0_le: "\<lbrakk>(f ---> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
lp15@56366
   489
                     \<Longrightarrow> (g ---> 0) F"
lp15@56366
   490
  by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
lp15@56366
   491
wenzelm@60758
   492
subsubsection \<open>Distance and norms\<close>
huffman@36662
   493
hoelzl@51531
   494
lemma tendsto_dist [tendsto_intros]:
hoelzl@51531
   495
  fixes l m :: "'a :: metric_space"
hoelzl@51531
   496
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
hoelzl@51531
   497
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
hoelzl@51531
   498
proof (rule tendstoI)
hoelzl@51531
   499
  fix e :: real assume "0 < e"
hoelzl@51531
   500
  hence e2: "0 < e/2" by simp
hoelzl@51531
   501
  from tendstoD [OF f e2] tendstoD [OF g e2]
hoelzl@51531
   502
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
hoelzl@51531
   503
  proof (eventually_elim)
hoelzl@51531
   504
    case (elim x)
hoelzl@51531
   505
    then show "dist (dist (f x) (g x)) (dist l m) < e"
hoelzl@51531
   506
      unfolding dist_real_def
hoelzl@51531
   507
      using dist_triangle2 [of "f x" "g x" "l"]
hoelzl@51531
   508
      using dist_triangle2 [of "g x" "l" "m"]
hoelzl@51531
   509
      using dist_triangle3 [of "l" "m" "f x"]
hoelzl@51531
   510
      using dist_triangle [of "f x" "m" "g x"]
hoelzl@51531
   511
      by arith
hoelzl@51531
   512
  qed
hoelzl@51531
   513
qed
hoelzl@51531
   514
hoelzl@51531
   515
lemma continuous_dist[continuous_intros]:
hoelzl@51531
   516
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   517
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   518
  unfolding continuous_def by (rule tendsto_dist)
hoelzl@51531
   519
hoelzl@56371
   520
lemma continuous_on_dist[continuous_intros]:
hoelzl@51531
   521
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   522
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   523
  unfolding continuous_on_def by (auto intro: tendsto_dist)
hoelzl@51531
   524
huffman@31565
   525
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   526
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   527
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   528
hoelzl@51478
   529
lemma continuous_norm [continuous_intros]:
hoelzl@51478
   530
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
hoelzl@51478
   531
  unfolding continuous_def by (rule tendsto_norm)
hoelzl@51478
   532
hoelzl@56371
   533
lemma continuous_on_norm [continuous_intros]:
hoelzl@51478
   534
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
hoelzl@51478
   535
  unfolding continuous_on_def by (auto intro: tendsto_norm)
hoelzl@51478
   536
huffman@36662
   537
lemma tendsto_norm_zero:
huffman@44195
   538
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   539
  by (drule tendsto_norm, simp)
huffman@36662
   540
huffman@36662
   541
lemma tendsto_norm_zero_cancel:
huffman@44195
   542
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   543
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   544
huffman@36662
   545
lemma tendsto_norm_zero_iff:
huffman@44195
   546
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   547
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   548
huffman@44194
   549
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   550
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   551
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   552
hoelzl@51478
   553
lemma continuous_rabs [continuous_intros]:
hoelzl@51478
   554
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   555
  unfolding real_norm_def[symmetric] by (rule continuous_norm)
hoelzl@51478
   556
hoelzl@56371
   557
lemma continuous_on_rabs [continuous_intros]:
hoelzl@51478
   558
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   559
  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
hoelzl@51478
   560
huffman@44194
   561
lemma tendsto_rabs_zero:
huffman@44195
   562
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   563
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   564
huffman@44194
   565
lemma tendsto_rabs_zero_cancel:
huffman@44195
   566
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   567
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   568
huffman@44194
   569
lemma tendsto_rabs_zero_iff:
huffman@44195
   570
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   571
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   572
wenzelm@60758
   573
subsubsection \<open>Addition and subtraction\<close>
huffman@44194
   574
huffman@31565
   575
lemma tendsto_add [tendsto_intros]:
huffman@31349
   576
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   577
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   578
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   579
hoelzl@51478
   580
lemma continuous_add [continuous_intros]:
hoelzl@51478
   581
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   582
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
hoelzl@51478
   583
  unfolding continuous_def by (rule tendsto_add)
hoelzl@51478
   584
hoelzl@56371
   585
lemma continuous_on_add [continuous_intros]:
hoelzl@51478
   586
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   587
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
hoelzl@51478
   588
  unfolding continuous_on_def by (auto intro: tendsto_add)
hoelzl@51478
   589
huffman@44194
   590
lemma tendsto_add_zero:
hoelzl@51478
   591
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   592
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   593
  by (drule (1) tendsto_add, simp)
huffman@44194
   594
huffman@31565
   595
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   596
  fixes a :: "'a::real_normed_vector"
huffman@44195
   597
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   598
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   599
hoelzl@51478
   600
lemma continuous_minus [continuous_intros]:
hoelzl@51478
   601
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   602
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
hoelzl@51478
   603
  unfolding continuous_def by (rule tendsto_minus)
hoelzl@51478
   604
hoelzl@56371
   605
lemma continuous_on_minus [continuous_intros]:
hoelzl@51478
   606
  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   607
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
hoelzl@51478
   608
  unfolding continuous_on_def by (auto intro: tendsto_minus)
hoelzl@51478
   609
huffman@31349
   610
lemma tendsto_minus_cancel:
huffman@31349
   611
  fixes a :: "'a::real_normed_vector"
huffman@44195
   612
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   613
  by (drule tendsto_minus, simp)
huffman@31349
   614
hoelzl@50330
   615
lemma tendsto_minus_cancel_left:
hoelzl@50330
   616
    "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
hoelzl@50330
   617
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
hoelzl@50330
   618
  by auto
hoelzl@50330
   619
huffman@31565
   620
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   621
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   622
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
haftmann@54230
   623
  using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
huffman@31349
   624
hoelzl@51478
   625
lemma continuous_diff [continuous_intros]:
hoelzl@51478
   626
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   627
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
hoelzl@51478
   628
  unfolding continuous_def by (rule tendsto_diff)
hoelzl@51478
   629
hoelzl@56371
   630
lemma continuous_on_diff [continuous_intros]:
hoelzl@51478
   631
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   632
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
hoelzl@51478
   633
  unfolding continuous_on_def by (auto intro: tendsto_diff)
hoelzl@51478
   634
huffman@31588
   635
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   636
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   637
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   638
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   639
proof (cases "finite S")
huffman@31588
   640
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   641
    by (induct, simp, simp add: tendsto_add)
hoelzl@58729
   642
qed simp
huffman@31588
   643
hoelzl@51478
   644
lemma continuous_setsum [continuous_intros]:
hoelzl@51478
   645
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   646
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   647
  unfolding continuous_def by (rule tendsto_setsum)
hoelzl@51478
   648
hoelzl@51478
   649
lemma continuous_on_setsum [continuous_intros]:
hoelzl@51478
   650
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   651
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   652
  unfolding continuous_on_def by (auto intro: tendsto_setsum)
hoelzl@51478
   653
hoelzl@50999
   654
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
hoelzl@50999
   655
wenzelm@60758
   656
subsubsection \<open>Linear operators and multiplication\<close>
huffman@44194
   657
huffman@44282
   658
lemma (in bounded_linear) tendsto:
huffman@44195
   659
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   660
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   661
hoelzl@51478
   662
lemma (in bounded_linear) continuous:
hoelzl@51478
   663
  "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
hoelzl@51478
   664
  using tendsto[of g _ F] by (auto simp: continuous_def)
hoelzl@51478
   665
hoelzl@51478
   666
lemma (in bounded_linear) continuous_on:
hoelzl@51478
   667
  "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
hoelzl@51478
   668
  using tendsto[of g] by (auto simp: continuous_on_def)
hoelzl@51478
   669
huffman@44194
   670
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   671
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   672
  by (drule tendsto, simp only: zero)
huffman@44194
   673
huffman@44282
   674
lemma (in bounded_bilinear) tendsto:
huffman@44195
   675
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   676
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   677
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   678
hoelzl@51478
   679
lemma (in bounded_bilinear) continuous:
hoelzl@51478
   680
  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
hoelzl@51478
   681
  using tendsto[of f _ F g] by (auto simp: continuous_def)
hoelzl@51478
   682
hoelzl@51478
   683
lemma (in bounded_bilinear) continuous_on:
hoelzl@51478
   684
  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
hoelzl@51478
   685
  using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
hoelzl@51478
   686
huffman@44194
   687
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   688
  assumes f: "(f ---> 0) F"
huffman@44195
   689
  assumes g: "(g ---> 0) F"
huffman@44195
   690
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   691
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   692
huffman@44194
   693
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   694
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   695
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   696
huffman@44194
   697
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   698
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   699
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   700
huffman@44282
   701
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   702
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   703
huffman@44282
   704
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   705
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   706
huffman@44282
   707
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   708
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   709
hoelzl@51478
   710
lemmas continuous_of_real [continuous_intros] =
hoelzl@51478
   711
  bounded_linear.continuous [OF bounded_linear_of_real]
hoelzl@51478
   712
hoelzl@51478
   713
lemmas continuous_scaleR [continuous_intros] =
hoelzl@51478
   714
  bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
hoelzl@51478
   715
hoelzl@51478
   716
lemmas continuous_mult [continuous_intros] =
hoelzl@51478
   717
  bounded_bilinear.continuous [OF bounded_bilinear_mult]
hoelzl@51478
   718
hoelzl@56371
   719
lemmas continuous_on_of_real [continuous_intros] =
hoelzl@51478
   720
  bounded_linear.continuous_on [OF bounded_linear_of_real]
hoelzl@51478
   721
hoelzl@56371
   722
lemmas continuous_on_scaleR [continuous_intros] =
hoelzl@51478
   723
  bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
hoelzl@51478
   724
hoelzl@56371
   725
lemmas continuous_on_mult [continuous_intros] =
hoelzl@51478
   726
  bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
hoelzl@51478
   727
huffman@44568
   728
lemmas tendsto_mult_zero =
huffman@44568
   729
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   730
huffman@44568
   731
lemmas tendsto_mult_left_zero =
huffman@44568
   732
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   733
huffman@44568
   734
lemmas tendsto_mult_right_zero =
huffman@44568
   735
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   736
huffman@44194
   737
lemma tendsto_power [tendsto_intros]:
huffman@44194
   738
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   739
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
hoelzl@58729
   740
  by (induct n) (simp_all add: tendsto_mult)
huffman@44194
   741
hoelzl@51478
   742
lemma continuous_power [continuous_intros]:
hoelzl@51478
   743
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   744
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
hoelzl@51478
   745
  unfolding continuous_def by (rule tendsto_power)
hoelzl@51478
   746
hoelzl@56371
   747
lemma continuous_on_power [continuous_intros]:
hoelzl@51478
   748
  fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   749
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
hoelzl@51478
   750
  unfolding continuous_on_def by (auto intro: tendsto_power)
hoelzl@51478
   751
huffman@44194
   752
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   753
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   754
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   755
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   756
proof (cases "finite S")
huffman@44194
   757
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   758
    by (induct, simp, simp add: tendsto_mult)
hoelzl@58729
   759
qed simp
huffman@44194
   760
hoelzl@51478
   761
lemma continuous_setprod [continuous_intros]:
hoelzl@51478
   762
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   763
  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
   764
  unfolding continuous_def by (rule tendsto_setprod)
hoelzl@51478
   765
hoelzl@51478
   766
lemma continuous_on_setprod [continuous_intros]:
hoelzl@51478
   767
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   768
  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
   769
  unfolding continuous_on_def by (auto intro: tendsto_setprod)
hoelzl@51478
   770
eberlm@61531
   771
lemma tendsto_of_real_iff:
eberlm@61531
   772
  "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) ---> of_real c) F \<longleftrightarrow> (f ---> c) F"
eberlm@61531
   773
  unfolding tendsto_iff by simp
eberlm@61531
   774
eberlm@61531
   775
lemma tendsto_add_const_iff:
eberlm@61531
   776
  "((\<lambda>x. c + f x :: 'a :: real_normed_vector) ---> c + d) F \<longleftrightarrow> (f ---> d) F"
eberlm@61531
   777
  using tendsto_add[OF tendsto_const[of c], of f d] 
eberlm@61531
   778
        tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
eberlm@61531
   779
eberlm@61531
   780
wenzelm@60758
   781
subsubsection \<open>Inverse and division\<close>
huffman@31355
   782
huffman@31355
   783
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   784
  assumes f: "Zfun f F"
huffman@44195
   785
  assumes g: "Bfun g F"
huffman@44195
   786
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   787
proof -
huffman@31355
   788
  obtain K where K: "0 \<le> K"
huffman@31355
   789
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   790
    using nonneg_bounded by fast
huffman@31355
   791
  obtain B where B: "0 < B"
huffman@44195
   792
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   793
    using g by (rule BfunE)
huffman@44195
   794
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   795
  using norm_g proof eventually_elim
noschinl@46887
   796
    case (elim x)
huffman@31487
   797
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   798
      by (rule norm_le)
huffman@31487
   799
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   800
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   801
                mult_nonneg_nonneg K elim)
huffman@31487
   802
    also have "\<dots> = norm (f x) * (B * K)"
haftmann@57512
   803
      by (rule mult.assoc)
huffman@31487
   804
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   805
  qed
huffman@31487
   806
  with f show ?thesis
huffman@31487
   807
    by (rule Zfun_imp_Zfun)
huffman@31355
   808
qed
huffman@31355
   809
huffman@31355
   810
lemma (in bounded_bilinear) flip:
huffman@31355
   811
  "bounded_bilinear (\<lambda>x y. y ** x)"
wenzelm@61169
   812
  apply standard
huffman@44081
   813
  apply (rule add_right)
huffman@44081
   814
  apply (rule add_left)
huffman@44081
   815
  apply (rule scaleR_right)
huffman@44081
   816
  apply (rule scaleR_left)
haftmann@57512
   817
  apply (subst mult.commute)
wenzelm@61169
   818
  using bounded
wenzelm@61169
   819
  apply fast
wenzelm@61169
   820
  done
huffman@31355
   821
huffman@31355
   822
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   823
  assumes f: "Bfun f F"
huffman@44195
   824
  assumes g: "Zfun g F"
huffman@44195
   825
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   826
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   827
huffman@31355
   828
lemma Bfun_inverse_lemma:
huffman@31355
   829
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   830
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   831
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   832
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   833
  done
huffman@31355
   834
huffman@31355
   835
lemma Bfun_inverse:
huffman@31355
   836
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   837
  assumes f: "(f ---> a) F"
huffman@31355
   838
  assumes a: "a \<noteq> 0"
huffman@44195
   839
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   840
proof -
huffman@31355
   841
  from a have "0 < norm a" by simp
huffman@31355
   842
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   843
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
   844
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
   845
    using tendstoD [OF f r1] by fast
huffman@44195
   846
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
   847
  proof eventually_elim
noschinl@46887
   848
    case (elim x)
huffman@31487
   849
    hence 1: "norm (f x - a) < r"
huffman@31355
   850
      by (simp add: dist_norm)
huffman@31487
   851
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   852
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   853
      by (rule nonzero_norm_inverse)
huffman@31355
   854
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   855
    proof (rule le_imp_inverse_le)
huffman@31355
   856
      show "0 < norm a - r" using r2 by simp
huffman@31355
   857
    next
huffman@31487
   858
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   859
        by (rule norm_triangle_ineq2)
huffman@31487
   860
      also have "\<dots> = norm (f x - a)"
huffman@31355
   861
        by (rule norm_minus_commute)
huffman@31355
   862
      also have "\<dots> < r" using 1 .
huffman@31487
   863
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   864
    qed
huffman@31487
   865
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   866
  qed
huffman@31355
   867
  thus ?thesis by (rule BfunI)
huffman@31355
   868
qed
huffman@31355
   869
huffman@31565
   870
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   871
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   872
  assumes f: "(f ---> a) F"
huffman@31355
   873
  assumes a: "a \<noteq> 0"
huffman@44195
   874
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
   875
proof -
huffman@31355
   876
  from a have "0 < norm a" by simp
huffman@44195
   877
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   878
    by (rule tendstoD)
huffman@44195
   879
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
   880
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
   881
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
   882
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   883
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
   884
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   885
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
   886
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
   887
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
   888
  ultimately show ?thesis
huffman@44627
   889
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
   890
qed
huffman@31355
   891
hoelzl@51478
   892
lemma continuous_inverse:
hoelzl@51478
   893
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   894
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   895
  shows "continuous F (\<lambda>x. inverse (f x))"
hoelzl@51478
   896
  using assms unfolding continuous_def by (rule tendsto_inverse)
hoelzl@51478
   897
hoelzl@51478
   898
lemma continuous_at_within_inverse[continuous_intros]:
hoelzl@51478
   899
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   900
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   901
  shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
hoelzl@51478
   902
  using assms unfolding continuous_within by (rule tendsto_inverse)
hoelzl@51478
   903
hoelzl@51478
   904
lemma isCont_inverse[continuous_intros, simp]:
hoelzl@51478
   905
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   906
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   907
  shows "isCont (\<lambda>x. inverse (f x)) a"
hoelzl@51478
   908
  using assms unfolding continuous_at by (rule tendsto_inverse)
hoelzl@51478
   909
hoelzl@56371
   910
lemma continuous_on_inverse[continuous_intros]:
hoelzl@51478
   911
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   912
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   913
  shows "continuous_on s (\<lambda>x. inverse (f x))"
hoelzl@51478
   914
  using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
hoelzl@51478
   915
huffman@31565
   916
lemma tendsto_divide [tendsto_intros]:
huffman@31355
   917
  fixes a b :: "'a::real_normed_field"
huffman@44195
   918
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
   919
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
   920
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
   921
hoelzl@51478
   922
lemma continuous_divide:
hoelzl@51478
   923
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   924
  assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   925
  shows "continuous F (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   926
  using assms unfolding continuous_def by (rule tendsto_divide)
hoelzl@51478
   927
hoelzl@51478
   928
lemma continuous_at_within_divide[continuous_intros]:
hoelzl@51478
   929
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   930
  assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
hoelzl@51478
   931
  shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   932
  using assms unfolding continuous_within by (rule tendsto_divide)
hoelzl@51478
   933
hoelzl@51478
   934
lemma isCont_divide[continuous_intros, simp]:
hoelzl@51478
   935
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   936
  assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
hoelzl@51478
   937
  shows "isCont (\<lambda>x. (f x) / g x) a"
hoelzl@51478
   938
  using assms unfolding continuous_at by (rule tendsto_divide)
hoelzl@51478
   939
hoelzl@56371
   940
lemma continuous_on_divide[continuous_intros]:
hoelzl@51478
   941
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   942
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
hoelzl@51478
   943
  shows "continuous_on s (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   944
  using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
hoelzl@51478
   945
huffman@44194
   946
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
   947
  fixes l :: "'a::real_normed_vector"
huffman@44195
   948
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
   949
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
   950
hoelzl@51478
   951
lemma continuous_sgn:
hoelzl@51478
   952
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   953
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   954
  shows "continuous F (\<lambda>x. sgn (f x))"
hoelzl@51478
   955
  using assms unfolding continuous_def by (rule tendsto_sgn)
hoelzl@51478
   956
hoelzl@51478
   957
lemma continuous_at_within_sgn[continuous_intros]:
hoelzl@51478
   958
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   959
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   960
  shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
hoelzl@51478
   961
  using assms unfolding continuous_within by (rule tendsto_sgn)
hoelzl@51478
   962
hoelzl@51478
   963
lemma isCont_sgn[continuous_intros]:
hoelzl@51478
   964
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   965
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   966
  shows "isCont (\<lambda>x. sgn (f x)) a"
hoelzl@51478
   967
  using assms unfolding continuous_at by (rule tendsto_sgn)
hoelzl@51478
   968
hoelzl@56371
   969
lemma continuous_on_sgn[continuous_intros]:
hoelzl@51478
   970
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   971
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   972
  shows "continuous_on s (\<lambda>x. sgn (f x))"
hoelzl@51478
   973
  using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
hoelzl@51478
   974
hoelzl@50325
   975
lemma filterlim_at_infinity:
wenzelm@61076
   976
  fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"
hoelzl@50325
   977
  assumes "0 \<le> c"
hoelzl@50325
   978
  shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
hoelzl@50325
   979
  unfolding filterlim_iff eventually_at_infinity
hoelzl@50325
   980
proof safe
hoelzl@50325
   981
  fix P :: "'a \<Rightarrow> bool" and b
hoelzl@50325
   982
  assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
hoelzl@50325
   983
    and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
hoelzl@50325
   984
  have "max b (c + 1) > c" by auto
hoelzl@50325
   985
  with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
hoelzl@50325
   986
    by auto
hoelzl@50325
   987
  then show "eventually (\<lambda>x. P (f x)) F"
hoelzl@50325
   988
  proof eventually_elim
hoelzl@50325
   989
    fix x assume "max b (c + 1) \<le> norm (f x)"
hoelzl@50325
   990
    with P show "P (f x)" by auto
hoelzl@50325
   991
  qed
hoelzl@50325
   992
qed force
hoelzl@50325
   993
eberlm@61531
   994
lemma not_tendsto_and_filterlim_at_infinity:
eberlm@61531
   995
  assumes "F \<noteq> bot"
eberlm@61531
   996
  assumes "(f ---> (c :: 'a :: real_normed_vector)) F" 
eberlm@61531
   997
  assumes "filterlim f at_infinity F"
eberlm@61531
   998
  shows   False
eberlm@61531
   999
proof -
eberlm@61531
  1000
  from tendstoD[OF assms(2), of "1/2"] 
eberlm@61531
  1001
    have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp
eberlm@61531
  1002
  moreover from filterlim_at_infinity[of "norm c" f F] assms(3)
eberlm@61531
  1003
    have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
eberlm@61531
  1004
  ultimately have "eventually (\<lambda>x. False) F"
eberlm@61531
  1005
  proof eventually_elim
eberlm@61531
  1006
    fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"
eberlm@61531
  1007
    note B
eberlm@61531
  1008
    also have "norm (f x) = dist (f x) 0" by (simp add: norm_conv_dist)
eberlm@61531
  1009
    also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
eberlm@61531
  1010
    also note A
eberlm@61531
  1011
    finally show False by (simp add: norm_conv_dist)
eberlm@61531
  1012
  qed
eberlm@61531
  1013
  with assms show False by simp
eberlm@61531
  1014
qed
eberlm@61531
  1015
eberlm@61531
  1016
lemma filterlim_at_infinity_imp_not_convergent:
eberlm@61531
  1017
  assumes "filterlim f at_infinity sequentially"
eberlm@61531
  1018
  shows   "\<not>convergent f"
eberlm@61531
  1019
  by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
eberlm@61531
  1020
     (simp_all add: convergent_LIMSEQ_iff)
eberlm@61531
  1021
eberlm@61531
  1022
lemma filterlim_at_infinity_imp_eventually_ne:
eberlm@61531
  1023
  assumes "filterlim f at_infinity F"
eberlm@61531
  1024
  shows   "eventually (\<lambda>z. f z \<noteq> c) F"
eberlm@61531
  1025
proof -
eberlm@61531
  1026
  have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all
eberlm@61531
  1027
  with filterlim_at_infinity[OF order.refl, of f F] assms
eberlm@61531
  1028
    have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast
eberlm@61531
  1029
  thus ?thesis by eventually_elim auto
eberlm@61531
  1030
qed
eberlm@61531
  1031
eberlm@61531
  1032
lemma tendsto_of_nat [tendsto_intros]: 
eberlm@61531
  1033
  "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"
eberlm@61531
  1034
proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
eberlm@61531
  1035
  fix r :: real assume r: "r > 0"
eberlm@61531
  1036
  def n \<equiv> "nat \<lceil>r\<rceil>"
eberlm@61531
  1037
  from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith
eberlm@61531
  1038
  from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
eberlm@61531
  1039
    by eventually_elim (insert n, simp_all)
eberlm@61531
  1040
qed
eberlm@61531
  1041
eberlm@61531
  1042
wenzelm@60758
  1043
subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
hoelzl@50347
  1044
wenzelm@60758
  1045
text \<open>
hoelzl@50347
  1046
hoelzl@50347
  1047
This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
hoelzl@50347
  1048
@{term "at_right x"} and also @{term "at_right 0"}.
hoelzl@50347
  1049
wenzelm@60758
  1050
\<close>
hoelzl@50347
  1051
hoelzl@51471
  1052
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
hoelzl@50323
  1053
hoelzl@51641
  1054
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
hoelzl@60721
  1055
  by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
hoelzl@60721
  1056
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@50347
  1057
hoelzl@51641
  1058
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
hoelzl@60721
  1059
  by (rule filtermap_fun_inverse[where g=uminus])
hoelzl@60721
  1060
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@51641
  1061
hoelzl@51641
  1062
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
hoelzl@51641
  1063
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50347
  1064
hoelzl@50347
  1065
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
hoelzl@51641
  1066
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50323
  1067
hoelzl@50347
  1068
lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
hoelzl@50347
  1069
  using filtermap_at_right_shift[of "-a" 0] by simp
hoelzl@50347
  1070
hoelzl@50347
  1071
lemma filterlim_at_right_to_0:
hoelzl@50347
  1072
  "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
hoelzl@50347
  1073
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
hoelzl@50347
  1074
hoelzl@50347
  1075
lemma eventually_at_right_to_0:
hoelzl@50347
  1076
  "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
hoelzl@50347
  1077
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
hoelzl@50347
  1078
hoelzl@51641
  1079
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
hoelzl@51641
  1080
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
  1081
hoelzl@50347
  1082
lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
hoelzl@51641
  1083
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50323
  1084
hoelzl@50347
  1085
lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
hoelzl@51641
  1086
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
  1087
hoelzl@50347
  1088
lemma filterlim_at_left_to_right:
hoelzl@50347
  1089
  "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
hoelzl@50347
  1090
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
hoelzl@50347
  1091
hoelzl@50347
  1092
lemma eventually_at_left_to_right:
hoelzl@50347
  1093
  "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
hoelzl@50347
  1094
  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
hoelzl@50347
  1095
hoelzl@60721
  1096
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@60721
  1097
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@60721
  1098
  by (metis leI minus_less_iff order_less_asym)
hoelzl@60721
  1099
hoelzl@60721
  1100
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@60721
  1101
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@60721
  1102
  by (metis leI less_minus_iff order_less_asym)
hoelzl@60721
  1103
hoelzl@50346
  1104
lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
hoelzl@60721
  1105
  by (rule filtermap_fun_inverse[symmetric, of uminus])
hoelzl@60721
  1106
     (auto intro: filterlim_uminus_at_bot_at_top filterlim_uminus_at_top_at_bot)
hoelzl@50346
  1107
hoelzl@50346
  1108
lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
hoelzl@50346
  1109
  unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
hoelzl@50346
  1110
hoelzl@50346
  1111
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
hoelzl@50346
  1112
  unfolding filterlim_def at_top_mirror filtermap_filtermap ..
hoelzl@50346
  1113
hoelzl@50346
  1114
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
hoelzl@50346
  1115
  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
hoelzl@50346
  1116
hoelzl@50346
  1117
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
hoelzl@50346
  1118
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
hoelzl@50346
  1119
  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
hoelzl@50346
  1120
  by auto
hoelzl@50346
  1121
hoelzl@50346
  1122
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
hoelzl@50346
  1123
  unfolding filterlim_uminus_at_top by simp
hoelzl@50323
  1124
hoelzl@50347
  1125
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
hoelzl@51641
  1126
  unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
hoelzl@50347
  1127
proof safe
hoelzl@50347
  1128
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50347
  1129
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50347
  1130
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@51641
  1131
  then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
hoelzl@50347
  1132
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50347
  1133
qed
hoelzl@50347
  1134
hoelzl@50325
  1135
lemma tendsto_inverse_0:
wenzelm@61076
  1136
  fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@50325
  1137
  shows "(inverse ---> (0::'a)) at_infinity"
hoelzl@50325
  1138
  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
hoelzl@50325
  1139
proof safe
hoelzl@50325
  1140
  fix r :: real assume "0 < r"
hoelzl@50325
  1141
  show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
hoelzl@50325
  1142
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
hoelzl@50325
  1143
    fix x :: 'a
wenzelm@60758
  1144
    from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
hoelzl@50325
  1145
    also assume *: "inverse (r / 2) \<le> norm x"
hoelzl@50325
  1146
    finally show "norm (inverse x) < r"
wenzelm@60758
  1147
      using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
hoelzl@50325
  1148
  qed
hoelzl@50325
  1149
qed
hoelzl@50325
  1150
eberlm@61552
  1151
lemma tendsto_add_filterlim_at_infinity:
eberlm@61552
  1152
  assumes "(f ---> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
eberlm@61552
  1153
  assumes "filterlim g at_infinity F"
eberlm@61552
  1154
  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
eberlm@61552
  1155
proof (subst filterlim_at_infinity[OF order_refl], safe)
eberlm@61552
  1156
  fix r :: real assume r: "r > 0"
eberlm@61552
  1157
  from assms(1) have "((\<lambda>x. norm (f x)) ---> norm c) F" by (rule tendsto_norm)
eberlm@61552
  1158
  hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
eberlm@61552
  1159
  moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all 
eberlm@61552
  1160
  with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
eberlm@61552
  1161
    unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all
eberlm@61552
  1162
  ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
eberlm@61552
  1163
  proof eventually_elim
eberlm@61552
  1164
    fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
eberlm@61552
  1165
    from A B have "r \<le> norm (g x) - norm (f x)" by simp
eberlm@61552
  1166
    also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)
eberlm@61552
  1167
    finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)
eberlm@61552
  1168
  qed
eberlm@61552
  1169
qed
eberlm@61552
  1170
eberlm@61552
  1171
lemma tendsto_add_filterlim_at_infinity':
eberlm@61552
  1172
  assumes "filterlim f at_infinity F"
eberlm@61552
  1173
  assumes "(g ---> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
eberlm@61552
  1174
  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
eberlm@61552
  1175
  by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
eberlm@61552
  1176
hoelzl@60721
  1177
lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
hoelzl@60721
  1178
  unfolding filterlim_at
hoelzl@60721
  1179
  by (auto simp: eventually_at_top_dense)
hoelzl@60721
  1180
     (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
hoelzl@60721
  1181
hoelzl@60721
  1182
lemma filterlim_inverse_at_top:
hoelzl@60721
  1183
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@60721
  1184
  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
hoelzl@60721
  1185
     (simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)
hoelzl@60721
  1186
hoelzl@60721
  1187
lemma filterlim_inverse_at_bot_neg:
hoelzl@60721
  1188
  "LIM x (at_left (0::real)). inverse x :> at_bot"
hoelzl@60721
  1189
  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
hoelzl@60721
  1190
hoelzl@60721
  1191
lemma filterlim_inverse_at_bot:
hoelzl@60721
  1192
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@60721
  1193
  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
hoelzl@60721
  1194
  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
hoelzl@60721
  1195
hoelzl@50347
  1196
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
hoelzl@60721
  1197
  by (intro filtermap_fun_inverse[symmetric, where g=inverse])
hoelzl@60721
  1198
     (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)
hoelzl@50347
  1199
hoelzl@50347
  1200
lemma eventually_at_right_to_top:
hoelzl@50347
  1201
  "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
hoelzl@50347
  1202
  unfolding at_right_to_top eventually_filtermap ..
hoelzl@50347
  1203
hoelzl@50347
  1204
lemma filterlim_at_right_to_top:
hoelzl@50347
  1205
  "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
hoelzl@50347
  1206
  unfolding filterlim_def at_right_to_top filtermap_filtermap ..
hoelzl@50347
  1207
hoelzl@50347
  1208
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
hoelzl@50347
  1209
  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
hoelzl@50347
  1210
hoelzl@50347
  1211
lemma eventually_at_top_to_right:
hoelzl@50347
  1212
  "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
hoelzl@50347
  1213
  unfolding at_top_to_right eventually_filtermap ..
hoelzl@50347
  1214
hoelzl@50347
  1215
lemma filterlim_at_top_to_right:
hoelzl@50347
  1216
  "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
hoelzl@50347
  1217
  unfolding filterlim_def at_top_to_right filtermap_filtermap ..
hoelzl@50347
  1218
hoelzl@50325
  1219
lemma filterlim_inverse_at_infinity:
wenzelm@61076
  1220
  fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1221
  shows "filterlim inverse at_infinity (at (0::'a))"
hoelzl@50325
  1222
  unfolding filterlim_at_infinity[OF order_refl]
hoelzl@50325
  1223
proof safe
hoelzl@50325
  1224
  fix r :: real assume "0 < r"
hoelzl@50325
  1225
  then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
hoelzl@50325
  1226
    unfolding eventually_at norm_inverse
hoelzl@50325
  1227
    by (intro exI[of _ "inverse r"])
hoelzl@50325
  1228
       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
hoelzl@50325
  1229
qed
hoelzl@50325
  1230
hoelzl@50325
  1231
lemma filterlim_inverse_at_iff:
wenzelm@61076
  1232
  fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1233
  shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
hoelzl@50325
  1234
  unfolding filterlim_def filtermap_filtermap[symmetric]
hoelzl@50325
  1235
proof
hoelzl@50325
  1236
  assume "filtermap g F \<le> at_infinity"
hoelzl@50325
  1237
  then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
hoelzl@50325
  1238
    by (rule filtermap_mono)
hoelzl@50325
  1239
  also have "\<dots> \<le> at 0"
hoelzl@51641
  1240
    using tendsto_inverse_0[where 'a='b]
hoelzl@51641
  1241
    by (auto intro!: exI[of _ 1]
hoelzl@51641
  1242
             simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
hoelzl@50325
  1243
  finally show "filtermap inverse (filtermap g F) \<le> at 0" .
hoelzl@50325
  1244
next
hoelzl@50325
  1245
  assume "filtermap inverse (filtermap g F) \<le> at 0"
hoelzl@50325
  1246
  then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
hoelzl@50325
  1247
    by (rule filtermap_mono)
hoelzl@50325
  1248
  with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
hoelzl@50325
  1249
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
hoelzl@50325
  1250
qed
hoelzl@50325
  1251
eberlm@61531
  1252
lemma tendsto_mult_filterlim_at_infinity:
eberlm@61531
  1253
  assumes "F \<noteq> bot" "(f ---> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
eberlm@61531
  1254
  assumes "filterlim g at_infinity F"
eberlm@61531
  1255
  shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"
eberlm@61531
  1256
proof -
eberlm@61531
  1257
  have "((\<lambda>x. inverse (f x) * inverse (g x)) ---> inverse c * 0) F"
eberlm@61531
  1258
    by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
eberlm@61531
  1259
  hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
eberlm@61531
  1260
    unfolding filterlim_at using assms
eberlm@61531
  1261
    by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
eberlm@61531
  1262
  thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
eberlm@61531
  1263
qed
eberlm@61531
  1264
hoelzl@51641
  1265
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
hoelzl@51641
  1266
 by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
hoelzl@50419
  1267
eberlm@61531
  1268
lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
eberlm@61531
  1269
  by (rule filterlim_subseq) (auto simp: subseq_def)
eberlm@61531
  1270
eberlm@61531
  1271
lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"
eberlm@61531
  1272
  by (rule filterlim_subseq) (auto simp: subseq_def)
lp15@59613
  1273
lp15@59613
  1274
lemma at_to_infinity:
wenzelm@61076
  1275
  fixes x :: "'a :: {real_normed_field,field}"
lp15@59613
  1276
  shows "(at (0::'a)) = filtermap inverse at_infinity"
lp15@59613
  1277
proof (rule antisym)
lp15@59613
  1278
  have "(inverse ---> (0::'a)) at_infinity"
lp15@59613
  1279
    by (fact tendsto_inverse_0)
lp15@59613
  1280
  then show "filtermap inverse at_infinity \<le> at (0::'a)"
lp15@59613
  1281
    apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
lp15@59613
  1282
    apply (rule_tac x="1" in exI, auto)
lp15@59613
  1283
    done
lp15@59613
  1284
next
lp15@59613
  1285
  have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
lp15@59613
  1286
    using filterlim_inverse_at_infinity unfolding filterlim_def
lp15@59613
  1287
    by (rule filtermap_mono)
lp15@59613
  1288
  then show "at (0::'a) \<le> filtermap inverse at_infinity"
lp15@59613
  1289
    by (simp add: filtermap_ident filtermap_filtermap)
lp15@59613
  1290
qed
lp15@59613
  1291
lp15@59613
  1292
lemma lim_at_infinity_0:
haftmann@59867
  1293
  fixes l :: "'a :: {real_normed_field,field}"
lp15@59613
  1294
  shows "(f ---> l) at_infinity \<longleftrightarrow> ((f o inverse) ---> l) (at (0::'a))"
lp15@59613
  1295
by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
lp15@59613
  1296
lp15@59613
  1297
lemma lim_zero_infinity:
haftmann@59867
  1298
  fixes l :: "'a :: {real_normed_field,field}"
lp15@59613
  1299
  shows "((\<lambda>x. f(1 / x)) ---> l) (at (0::'a)) \<Longrightarrow> (f ---> l) at_infinity"
lp15@59613
  1300
by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
lp15@59613
  1301
lp15@59613
  1302
wenzelm@60758
  1303
text \<open>
hoelzl@50324
  1304
hoelzl@50324
  1305
We only show rules for multiplication and addition when the functions are either against a real
hoelzl@50324
  1306
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
hoelzl@50324
  1307
wenzelm@60758
  1308
\<close>
hoelzl@50324
  1309
lp15@60141
  1310
lemma filterlim_tendsto_pos_mult_at_top:
hoelzl@50324
  1311
  assumes f: "(f ---> c) F" and c: "0 < c"
hoelzl@50324
  1312
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1313
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1314
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1315
proof safe
hoelzl@50324
  1316
  fix Z :: real assume "0 < Z"
wenzelm@60758
  1317
  from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
hoelzl@50324
  1318
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
hoelzl@50324
  1319
             simp: dist_real_def abs_real_def split: split_if_asm)
hoelzl@50346
  1320
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
hoelzl@50324
  1321
    unfolding filterlim_at_top by auto
hoelzl@50346
  1322
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1323
  proof eventually_elim
hoelzl@50346
  1324
    fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
wenzelm@60758
  1325
    with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
hoelzl@50346
  1326
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
wenzelm@60758
  1327
    with \<open>0 < c\<close> show "Z \<le> f x * g x"
hoelzl@50324
  1328
       by simp
hoelzl@50324
  1329
  qed
hoelzl@50324
  1330
qed
hoelzl@50324
  1331
lp15@60141
  1332
lemma filterlim_at_top_mult_at_top:
hoelzl@50324
  1333
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1334
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1335
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1336
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1337
proof safe
hoelzl@50324
  1338
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1339
  from f have "eventually (\<lambda>x. 1 \<le> f x) F"
hoelzl@50324
  1340
    unfolding filterlim_at_top by auto
hoelzl@50346
  1341
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1342
    unfolding filterlim_at_top by auto
hoelzl@50346
  1343
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1344
  proof eventually_elim
hoelzl@50346
  1345
    fix x assume "1 \<le> f x" "Z \<le> g x"
wenzelm@60758
  1346
    with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
hoelzl@50346
  1347
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
  1348
    then show "Z \<le> f x * g x"
hoelzl@50324
  1349
       by simp
hoelzl@50324
  1350
  qed
hoelzl@50324
  1351
qed
hoelzl@50324
  1352
hoelzl@50419
  1353
lemma filterlim_tendsto_pos_mult_at_bot:
hoelzl@50419
  1354
  assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
hoelzl@50419
  1355
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@50419
  1356
  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
hoelzl@50419
  1357
  unfolding filterlim_uminus_at_bot by simp
hoelzl@50419
  1358
hoelzl@60182
  1359
lemma filterlim_tendsto_neg_mult_at_bot:
hoelzl@60182
  1360
  assumes c: "(f ---> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
hoelzl@60182
  1361
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@60182
  1362
  using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
hoelzl@60182
  1363
  unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
hoelzl@60182
  1364
hoelzl@56330
  1365
lemma filterlim_pow_at_top:
hoelzl@56330
  1366
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1367
  assumes "0 < n" and f: "LIM x F. f x :> at_top"
hoelzl@56330
  1368
  shows "LIM x F. (f x)^n :: real :> at_top"
wenzelm@60758
  1369
using \<open>0 < n\<close> proof (induct n)
hoelzl@56330
  1370
  case (Suc n) with f show ?case
hoelzl@56330
  1371
    by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
hoelzl@56330
  1372
qed simp
hoelzl@56330
  1373
hoelzl@56330
  1374
lemma filterlim_pow_at_bot_even:
hoelzl@56330
  1375
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1376
  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
  1377
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
hoelzl@56330
  1378
hoelzl@56330
  1379
lemma filterlim_pow_at_bot_odd:
hoelzl@56330
  1380
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1381
  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
  1382
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
hoelzl@56330
  1383
lp15@60141
  1384
lemma filterlim_tendsto_add_at_top:
hoelzl@50324
  1385
  assumes f: "(f ---> c) F"
hoelzl@50324
  1386
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1387
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1388
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1389
proof safe
hoelzl@50324
  1390
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1391
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
hoelzl@50324
  1392
    by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50346
  1393
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
hoelzl@50324
  1394
    unfolding filterlim_at_top by auto
hoelzl@50346
  1395
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1396
    by eventually_elim simp
hoelzl@50324
  1397
qed
hoelzl@50324
  1398
hoelzl@50347
  1399
lemma LIM_at_top_divide:
hoelzl@50347
  1400
  fixes f g :: "'a \<Rightarrow> real"
hoelzl@50347
  1401
  assumes f: "(f ---> a) F" "0 < a"
hoelzl@50347
  1402
  assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
hoelzl@50347
  1403
  shows "LIM x F. f x / g x :> at_top"
hoelzl@50347
  1404
  unfolding divide_inverse
hoelzl@50347
  1405
  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
hoelzl@50347
  1406
lp15@60141
  1407
lemma filterlim_at_top_add_at_top:
hoelzl@50324
  1408
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1409
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1410
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1411
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1412
proof safe
hoelzl@50324
  1413
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1414
  from f have "eventually (\<lambda>x. 0 \<le> f x) F"
hoelzl@50324
  1415
    unfolding filterlim_at_top by auto
hoelzl@50346
  1416
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1417
    unfolding filterlim_at_top by auto
hoelzl@50346
  1418
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1419
    by eventually_elim simp
hoelzl@50324
  1420
qed
hoelzl@50324
  1421
hoelzl@50331
  1422
lemma tendsto_divide_0:
wenzelm@61076
  1423
  fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
hoelzl@50331
  1424
  assumes f: "(f ---> c) F"
hoelzl@50331
  1425
  assumes g: "LIM x F. g x :> at_infinity"
hoelzl@50331
  1426
  shows "((\<lambda>x. f x / g x) ---> 0) F"
hoelzl@50331
  1427
  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
hoelzl@50331
  1428
hoelzl@50331
  1429
lemma linear_plus_1_le_power:
hoelzl@50331
  1430
  fixes x :: real
hoelzl@50331
  1431
  assumes x: "0 \<le> x"
hoelzl@50331
  1432
  shows "real n * x + 1 \<le> (x + 1) ^ n"
hoelzl@50331
  1433
proof (induct n)
hoelzl@50331
  1434
  case (Suc n)
hoelzl@50331
  1435
  have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
lp15@61609
  1436
    by (simp add: field_simps of_nat_Suc x)
hoelzl@50331
  1437
  also have "\<dots> \<le> (x + 1)^Suc n"
hoelzl@50331
  1438
    using Suc x by (simp add: mult_left_mono)
hoelzl@50331
  1439
  finally show ?case .
hoelzl@50331
  1440
qed simp
hoelzl@50331
  1441
hoelzl@50331
  1442
lemma filterlim_realpow_sequentially_gt1:
hoelzl@50331
  1443
  fixes x :: "'a :: real_normed_div_algebra"
hoelzl@50331
  1444
  assumes x[arith]: "1 < norm x"
hoelzl@50331
  1445
  shows "LIM n sequentially. x ^ n :> at_infinity"
hoelzl@50331
  1446
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
hoelzl@50331
  1447
  fix y :: real assume "0 < y"
hoelzl@50331
  1448
  have "0 < norm x - 1" by simp
hoelzl@50331
  1449
  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
hoelzl@50331
  1450
  also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
hoelzl@50331
  1451
  also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
hoelzl@50331
  1452
  also have "\<dots> = norm x ^ N" by simp
hoelzl@50331
  1453
  finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
hoelzl@50331
  1454
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
hoelzl@50331
  1455
  then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
hoelzl@50331
  1456
    unfolding eventually_sequentially
hoelzl@50331
  1457
    by (auto simp: norm_power)
hoelzl@50331
  1458
qed simp
hoelzl@50331
  1459
hoelzl@51471
  1460
wenzelm@60758
  1461
subsection \<open>Limits of Sequences\<close>
hoelzl@51526
  1462
hoelzl@51526
  1463
lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
hoelzl@51526
  1464
  by simp
hoelzl@51526
  1465
hoelzl@51526
  1466
lemma LIMSEQ_iff:
hoelzl@51526
  1467
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1468
  shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
lp15@60017
  1469
unfolding lim_sequentially dist_norm ..
hoelzl@51526
  1470
hoelzl@51526
  1471
lemma LIMSEQ_I:
hoelzl@51526
  1472
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1473
  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
hoelzl@51526
  1474
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1475
hoelzl@51526
  1476
lemma LIMSEQ_D:
hoelzl@51526
  1477
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1478
  shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
hoelzl@51526
  1479
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1480
hoelzl@51526
  1481
lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
hoelzl@51526
  1482
  unfolding tendsto_def eventually_sequentially
haftmann@57512
  1483
  by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
hoelzl@51526
  1484
hoelzl@51526
  1485
lemma Bseq_inverse_lemma:
hoelzl@51526
  1486
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51526
  1487
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
hoelzl@51526
  1488
apply (subst nonzero_norm_inverse, clarsimp)
hoelzl@51526
  1489
apply (erule (1) le_imp_inverse_le)
hoelzl@51526
  1490
done
hoelzl@51526
  1491
hoelzl@51526
  1492
lemma Bseq_inverse:
hoelzl@51526
  1493
  fixes a :: "'a::real_normed_div_algebra"
hoelzl@51526
  1494
  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
hoelzl@51526
  1495
  by (rule Bfun_inverse)
hoelzl@51526
  1496
wenzelm@60758
  1497
text\<open>Transformation of limit.\<close>
lp15@60141
  1498
lp15@60141
  1499
lemma eventually_at2:
lp15@60141
  1500
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
lp15@60141
  1501
  unfolding eventually_at dist_nz by auto
lp15@60141
  1502
lp15@60141
  1503
lemma Lim_transform:
lp15@60141
  1504
  fixes a b :: "'a::real_normed_vector"
lp15@60141
  1505
  shows "\<lbrakk>(g ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (f ---> a) F"
lp15@60141
  1506
  using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
lp15@60141
  1507
lp15@60141
  1508
lemma Lim_transform2:
lp15@60141
  1509
  fixes a b :: "'a::real_normed_vector"
lp15@60141
  1510
  shows "\<lbrakk>(f ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (g ---> a) F"
lp15@60141
  1511
  by (erule Lim_transform) (simp add: tendsto_minus_cancel)
lp15@60141
  1512
lp15@60141
  1513
lemma Lim_transform_eventually:
lp15@60141
  1514
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
lp15@60141
  1515
  apply (rule topological_tendstoI)
lp15@60141
  1516
  apply (drule (2) topological_tendstoD)
lp15@60141
  1517
  apply (erule (1) eventually_elim2, simp)
lp15@60141
  1518
  done
lp15@60141
  1519
lp15@60141
  1520
lemma Lim_transform_within:
lp15@60141
  1521
  assumes "0 < d"
lp15@60141
  1522
    and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
lp15@60141
  1523
    and "(f ---> l) (at x within S)"
lp15@60141
  1524
  shows "(g ---> l) (at x within S)"
lp15@60141
  1525
proof (rule Lim_transform_eventually)
lp15@60141
  1526
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
lp15@60141
  1527
    using assms(1,2) by (auto simp: dist_nz eventually_at)
lp15@60141
  1528
  show "(f ---> l) (at x within S)" by fact
lp15@60141
  1529
qed
lp15@60141
  1530
lp15@60141
  1531
lemma Lim_transform_at:
lp15@60141
  1532
  assumes "0 < d"
lp15@60141
  1533
    and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
lp15@60141
  1534
    and "(f ---> l) (at x)"
lp15@60141
  1535
  shows "(g ---> l) (at x)"
lp15@60141
  1536
  using _ assms(3)
lp15@60141
  1537
proof (rule Lim_transform_eventually)
lp15@60141
  1538
  show "eventually (\<lambda>x. f x = g x) (at x)"
lp15@60141
  1539
    unfolding eventually_at2
lp15@60141
  1540
    using assms(1,2) by auto
lp15@60141
  1541
qed
lp15@60141
  1542
wenzelm@60758
  1543
text\<open>Common case assuming being away from some crucial point like 0.\<close>
hoelzl@51526
  1544
lp15@60141
  1545
lemma Lim_transform_away_within:
lp15@60141
  1546
  fixes a b :: "'a::t1_space"
lp15@60141
  1547
  assumes "a \<noteq> b"
lp15@60141
  1548
    and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
lp15@60141
  1549
    and "(f ---> l) (at a within S)"
lp15@60141
  1550
  shows "(g ---> l) (at a within S)"
lp15@60141
  1551
proof (rule Lim_transform_eventually)
lp15@60141
  1552
  show "(f ---> l) (at a within S)" by fact
lp15@60141
  1553
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
lp15@60141
  1554
    unfolding eventually_at_topological
lp15@60141
  1555
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
lp15@60141
  1556
qed
lp15@60141
  1557
lp15@60141
  1558
lemma Lim_transform_away_at:
lp15@60141
  1559
  fixes a b :: "'a::t1_space"
lp15@60141
  1560
  assumes ab: "a\<noteq>b"
lp15@60141
  1561
    and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
lp15@60141
  1562
    and fl: "(f ---> l) (at a)"
lp15@60141
  1563
  shows "(g ---> l) (at a)"
lp15@60141
  1564
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
lp15@60141
  1565
wenzelm@60758
  1566
text\<open>Alternatively, within an open set.\<close>
hoelzl@51526
  1567
lp15@60141
  1568
lemma Lim_transform_within_open:
lp15@60141
  1569
  assumes "open S" and "a \<in> S"
lp15@60141
  1570
    and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
lp15@60141
  1571
    and "(f ---> l) (at a)"
lp15@60141
  1572
  shows "(g ---> l) (at a)"
lp15@60141
  1573
proof (rule Lim_transform_eventually)
lp15@60141
  1574
  show "eventually (\<lambda>x. f x = g x) (at a)"
lp15@60141
  1575
    unfolding eventually_at_topological
lp15@60141
  1576
    using assms(1,2,3) by auto
lp15@60141
  1577
  show "(f ---> l) (at a)" by fact
lp15@60141
  1578
qed
lp15@60141
  1579
wenzelm@60758
  1580
text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
lp15@60141
  1581
lp15@60141
  1582
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
lp15@60141
  1583
lp15@60141
  1584
lemma Lim_cong_within(*[cong add]*):
lp15@60141
  1585
  assumes "a = b"
lp15@60141
  1586
    and "x = y"
lp15@60141
  1587
    and "S = T"
lp15@60141
  1588
    and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
lp15@60141
  1589
  shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
lp15@60141
  1590
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1591
  using assms by simp
lp15@60141
  1592
lp15@60141
  1593
lemma Lim_cong_at(*[cong add]*):
lp15@60141
  1594
  assumes "a = b" "x = y"
lp15@60141
  1595
    and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
lp15@60141
  1596
  shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
lp15@60141
  1597
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1598
  using assms by simp
wenzelm@60758
  1599
text\<open>An unbounded sequence's inverse tends to 0\<close>
hoelzl@51526
  1600
hoelzl@51526
  1601
lemma LIMSEQ_inverse_zero:
hoelzl@51526
  1602
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
hoelzl@51526
  1603
  apply (rule filterlim_compose[OF tendsto_inverse_0])
hoelzl@51526
  1604
  apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
hoelzl@51526
  1605
  apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
hoelzl@51526
  1606
  done
hoelzl@51526
  1607
wenzelm@60758
  1608
text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
hoelzl@51526
  1609
hoelzl@51526
  1610
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
hoelzl@51526
  1611
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
hoelzl@51526
  1612
            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
hoelzl@51526
  1613
wenzelm@60758
  1614
text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
wenzelm@60758
  1615
infinity is now easily proved\<close>
hoelzl@51526
  1616
hoelzl@51526
  1617
lemma LIMSEQ_inverse_real_of_nat_add:
hoelzl@51526
  1618
     "(%n. r + inverse(real(Suc n))) ----> r"
hoelzl@51526
  1619
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
hoelzl@51526
  1620
hoelzl@51526
  1621
lemma LIMSEQ_inverse_real_of_nat_add_minus:
hoelzl@51526
  1622
     "(%n. r + -inverse(real(Suc n))) ----> r"
hoelzl@51526
  1623
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
hoelzl@51526
  1624
  by auto
hoelzl@51526
  1625
hoelzl@51526
  1626
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
hoelzl@51526
  1627
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
hoelzl@51526
  1628
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
hoelzl@51526
  1629
  by auto
hoelzl@51526
  1630
eberlm@61524
  1631
lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) ---> (0::'a::real_normed_field)) sequentially"
eberlm@61524
  1632
proof (subst lim_sequentially, intro allI impI exI)
eberlm@61524
  1633
  fix e :: real assume e: "e > 0"
eberlm@61524
  1634
  fix n :: nat assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
eberlm@61524
  1635
  have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith
eberlm@61524
  1636
  also note n
eberlm@61524
  1637
  finally show "dist (1 / of_nat n :: 'a) 0 < e" using e 
eberlm@61524
  1638
    by (simp add: divide_simps mult.commute norm_conv_dist[symmetric] norm_divide)
eberlm@61524
  1639
qed
eberlm@61524
  1640
eberlm@61524
  1641
lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) ---> (0::'a::real_normed_field)) sequentially"
eberlm@61524
  1642
  using lim_1_over_n by (simp add: inverse_eq_divide)
eberlm@61524
  1643
eberlm@61524
  1644
lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) ----> 1"
eberlm@61524
  1645
proof (rule Lim_transform_eventually)
eberlm@61524
  1646
  show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
eberlm@61524
  1647
    using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
eberlm@61524
  1648
  have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1 + 0"
eberlm@61524
  1649
    by (intro tendsto_add tendsto_const lim_inverse_n)
eberlm@61524
  1650
  thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1" by simp
eberlm@61524
  1651
qed
eberlm@61524
  1652
eberlm@61524
  1653
lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) ----> 1"
eberlm@61524
  1654
proof (rule Lim_transform_eventually)
eberlm@61524
  1655
  show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) = 
eberlm@61524
  1656
                        of_nat n / of_nat (Suc n)) sequentially"
eberlm@61524
  1657
    using eventually_gt_at_top[of "0::nat"] 
eberlm@61524
  1658
    by eventually_elim (simp add: field_simps del: of_nat_Suc)
eberlm@61524
  1659
  have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> inverse 1"
eberlm@61524
  1660
    by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
eberlm@61524
  1661
  thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> 1" by simp
eberlm@61524
  1662
qed
eberlm@61524
  1663
wenzelm@60758
  1664
subsection \<open>Convergence on sequences\<close>
hoelzl@51526
  1665
eberlm@61531
  1666
lemma convergent_cong:
eberlm@61531
  1667
  assumes "eventually (\<lambda>x. f x = g x) sequentially"
eberlm@61531
  1668
  shows   "convergent f \<longleftrightarrow> convergent g"
eberlm@61531
  1669
  unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)
eberlm@61531
  1670
eberlm@61531
  1671
lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
eberlm@61531
  1672
  by (auto simp: convergent_def LIMSEQ_Suc_iff)
eberlm@61531
  1673
eberlm@61531
  1674
lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
eberlm@61531
  1675
proof (induction m arbitrary: f)
eberlm@61531
  1676
  case (Suc m)
eberlm@61531
  1677
  have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp
eberlm@61531
  1678
  also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)
eberlm@61531
  1679
  also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)
eberlm@61531
  1680
  finally show ?case .
eberlm@61531
  1681
qed simp_all
eberlm@61531
  1682
hoelzl@51526
  1683
lemma convergent_add:
hoelzl@51526
  1684
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1685
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1686
  assumes "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1687
  shows "convergent (\<lambda>n. X n + Y n)"
hoelzl@51526
  1688
  using assms unfolding convergent_def by (fast intro: tendsto_add)
hoelzl@51526
  1689
hoelzl@51526
  1690
lemma convergent_setsum:
hoelzl@51526
  1691
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1692
  assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
hoelzl@51526
  1693
  shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
hoelzl@51526
  1694
proof (cases "finite A")
hoelzl@51526
  1695
  case True from this and assms show ?thesis
hoelzl@51526
  1696
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
hoelzl@51526
  1697
qed (simp add: convergent_const)
hoelzl@51526
  1698
hoelzl@51526
  1699
lemma (in bounded_linear) convergent:
hoelzl@51526
  1700
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1701
  shows "convergent (\<lambda>n. f (X n))"
hoelzl@51526
  1702
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1703
hoelzl@51526
  1704
lemma (in bounded_bilinear) convergent:
hoelzl@51526
  1705
  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1706
  shows "convergent (\<lambda>n. X n ** Y n)"
hoelzl@51526
  1707
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1708
hoelzl@51526
  1709
lemma convergent_minus_iff:
hoelzl@51526
  1710
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1711
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
hoelzl@51526
  1712
apply (simp add: convergent_def)
hoelzl@51526
  1713
apply (auto dest: tendsto_minus)
hoelzl@51526
  1714
apply (drule tendsto_minus, auto)
hoelzl@51526
  1715
done
hoelzl@51526
  1716
eberlm@61531
  1717
lemma convergent_diff:
eberlm@61531
  1718
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
eberlm@61531
  1719
  assumes "convergent (\<lambda>n. X n)"
eberlm@61531
  1720
  assumes "convergent (\<lambda>n. Y n)"
eberlm@61531
  1721
  shows "convergent (\<lambda>n. X n - Y n)"
eberlm@61531
  1722
  using assms unfolding convergent_def by (fast intro: tendsto_diff)
eberlm@61531
  1723
eberlm@61531
  1724
lemma convergent_norm:
eberlm@61531
  1725
  assumes "convergent f"
eberlm@61531
  1726
  shows   "convergent (\<lambda>n. norm (f n))"
eberlm@61531
  1727
proof -
eberlm@61531
  1728
  from assms have "f ----> lim f" by (simp add: convergent_LIMSEQ_iff)
eberlm@61531
  1729
  hence "(\<lambda>n. norm (f n)) ----> norm (lim f)" by (rule tendsto_norm)
eberlm@61531
  1730
  thus ?thesis by (auto simp: convergent_def)
eberlm@61531
  1731
qed
eberlm@61531
  1732
eberlm@61531
  1733
lemma convergent_of_real: 
eberlm@61531
  1734
  "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"
eberlm@61531
  1735
  unfolding convergent_def by (blast intro!: tendsto_of_real)
eberlm@61531
  1736
eberlm@61531
  1737
lemma convergent_add_const_iff: 
eberlm@61531
  1738
  "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1739
proof
eberlm@61531
  1740
  assume "convergent (\<lambda>n. c + f n)"
eberlm@61531
  1741
  from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp
eberlm@61531
  1742
next
eberlm@61531
  1743
  assume "convergent f"
eberlm@61531
  1744
  from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp
eberlm@61531
  1745
qed
eberlm@61531
  1746
eberlm@61531
  1747
lemma convergent_add_const_right_iff: 
eberlm@61531
  1748
  "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1749
  using convergent_add_const_iff[of c f] by (simp add: add_ac)
eberlm@61531
  1750
eberlm@61531
  1751
lemma convergent_diff_const_right_iff: 
eberlm@61531
  1752
  "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1753
  using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)
eberlm@61531
  1754
eberlm@61531
  1755
lemma convergent_mult:
eberlm@61531
  1756
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
eberlm@61531
  1757
  assumes "convergent (\<lambda>n. X n)"
eberlm@61531
  1758
  assumes "convergent (\<lambda>n. Y n)"
eberlm@61531
  1759
  shows "convergent (\<lambda>n. X n * Y n)"
eberlm@61531
  1760
  using assms unfolding convergent_def by (fast intro: tendsto_mult)
eberlm@61531
  1761
eberlm@61531
  1762
lemma convergent_mult_const_iff:
eberlm@61531
  1763
  assumes "c \<noteq> 0"
eberlm@61531
  1764
  shows   "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"
eberlm@61531
  1765
proof
eberlm@61531
  1766
  assume "convergent (\<lambda>n. c * f n)"
eberlm@61531
  1767
  from assms convergent_mult[OF this convergent_const[of "inverse c"]] 
eberlm@61531
  1768
    show "convergent f" by (simp add: field_simps)
eberlm@61531
  1769
next
eberlm@61531
  1770
  assume "convergent f"
eberlm@61531
  1771
  from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp
eberlm@61531
  1772
qed
eberlm@61531
  1773
eberlm@61531
  1774
lemma convergent_mult_const_right_iff:
eberlm@61531
  1775
  assumes "c \<noteq> 0"
eberlm@61531
  1776
  shows   "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"
eberlm@61531
  1777
  using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
eberlm@61531
  1778
eberlm@61531
  1779
lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
eberlm@61531
  1780
  by (simp add: Cauchy_Bseq convergent_Cauchy)
eberlm@61531
  1781
hoelzl@51526
  1782
wenzelm@60758
  1783
text \<open>A monotone sequence converges to its least upper bound.\<close>
hoelzl@51526
  1784
hoelzl@54263
  1785
lemma LIMSEQ_incseq_SUP:
hoelzl@54263
  1786
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1787
  assumes u: "bdd_above (range X)"
hoelzl@54263
  1788
  assumes X: "incseq X"
hoelzl@54263
  1789
  shows "X ----> (SUP i. X i)"
hoelzl@54263
  1790
  by (rule order_tendstoI)
hoelzl@54263
  1791
     (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
hoelzl@51526
  1792
hoelzl@54263
  1793
lemma LIMSEQ_decseq_INF:
hoelzl@54263
  1794
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1795
  assumes u: "bdd_below (range X)"
hoelzl@54263
  1796
  assumes X: "decseq X"
hoelzl@54263
  1797
  shows "X ----> (INF i. X i)"
hoelzl@54263
  1798
  by (rule order_tendstoI)
hoelzl@54263
  1799
     (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
hoelzl@51526
  1800
wenzelm@60758
  1801
text\<open>Main monotonicity theorem\<close>
hoelzl@51526
  1802
hoelzl@51526
  1803
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1804
  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
hoelzl@54263
  1805
hoelzl@54263
  1806
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
  1807
  by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
hoelzl@51526
  1808
eberlm@61531
  1809
lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
eberlm@61531
  1810
  using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
eberlm@61531
  1811
eberlm@61531
  1812
lemma Bseq_monoseq_convergent'_inc:
eberlm@61531
  1813
  "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
  1814
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
eberlm@61531
  1815
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
eberlm@61531
  1816
eberlm@61531
  1817
lemma Bseq_monoseq_convergent'_dec:
eberlm@61531
  1818
  "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
  1819
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
eberlm@61531
  1820
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
eberlm@61531
  1821
hoelzl@51526
  1822
lemma Cauchy_iff:
hoelzl@51526
  1823
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1824
  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
  1825
  unfolding Cauchy_def dist_norm ..
hoelzl@51526
  1826
hoelzl@51526
  1827
lemma CauchyI:
hoelzl@51526
  1828
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1829
  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
  1830
by (simp add: Cauchy_iff)
hoelzl@51526
  1831
hoelzl@51526
  1832
lemma CauchyD:
hoelzl@51526
  1833
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1834
  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
  1835
by (simp add: Cauchy_iff)
hoelzl@51526
  1836
hoelzl@51526
  1837
lemma incseq_convergent:
hoelzl@51526
  1838
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1839
  assumes "incseq X" and "\<forall>i. X i \<le> B"
hoelzl@51526
  1840
  obtains L where "X ----> L" "\<forall>i. X i \<le> L"
hoelzl@51526
  1841
proof atomize_elim
wenzelm@60758
  1842
  from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
hoelzl@51526
  1843
  obtain L where "X ----> L"
hoelzl@51526
  1844
    by (auto simp: convergent_def monoseq_def incseq_def)
wenzelm@60758
  1845
  with \<open>incseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
hoelzl@51526
  1846
    by (auto intro!: exI[of _ L] incseq_le)
hoelzl@51526
  1847
qed
hoelzl@51526
  1848
hoelzl@51526
  1849
lemma decseq_convergent:
hoelzl@51526
  1850
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1851
  assumes "decseq X" and "\<forall>i. B \<le> X i"
hoelzl@51526
  1852
  obtains L where "X ----> L" "\<forall>i. L \<le> X i"
hoelzl@51526
  1853
proof atomize_elim
wenzelm@60758
  1854
  from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
hoelzl@51526
  1855
  obtain L where "X ----> L"
hoelzl@51526
  1856
    by (auto simp: convergent_def monoseq_def decseq_def)
wenzelm@60758
  1857
  with \<open>decseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
hoelzl@51526
  1858
    by (auto intro!: exI[of _ L] decseq_le)
hoelzl@51526
  1859
qed
hoelzl@51526
  1860
wenzelm@60758
  1861
subsubsection \<open>Cauchy Sequences are Bounded\<close>
hoelzl@51526
  1862
wenzelm@60758
  1863
text\<open>A Cauchy sequence is bounded -- this is the standard
wenzelm@60758
  1864
  proof mechanization rather than the nonstandard proof\<close>
hoelzl@51526
  1865
hoelzl@51526
  1866
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
hoelzl@51526
  1867
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
hoelzl@51526
  1868
apply (clarify, drule spec, drule (1) mp)
hoelzl@51526
  1869
apply (simp only: norm_minus_commute)
hoelzl@51526
  1870
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
hoelzl@51526
  1871
apply simp
hoelzl@51526
  1872
done
hoelzl@51526
  1873
wenzelm@60758
  1874
subsection \<open>Power Sequences\<close>
hoelzl@51526
  1875
wenzelm@60758
  1876
text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
hoelzl@51526
  1877
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
wenzelm@60758
  1878
  also fact that bounded and monotonic sequence converges.\<close>
hoelzl@51526
  1879
hoelzl@51526
  1880
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
hoelzl@51526
  1881
apply (simp add: Bseq_def)
hoelzl@51526
  1882
apply (rule_tac x = 1 in exI)
hoelzl@51526
  1883
apply (simp add: power_abs)
hoelzl@51526
  1884
apply (auto dest: power_mono)
hoelzl@51526
  1885
done
hoelzl@51526
  1886
hoelzl@51526
  1887
lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
hoelzl@51526
  1888
apply (clarify intro!: mono_SucI2)
hoelzl@51526
  1889
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
hoelzl@51526
  1890
done
hoelzl@51526
  1891
hoelzl@51526
  1892
lemma convergent_realpow:
hoelzl@51526
  1893
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
hoelzl@51526
  1894
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
hoelzl@51526
  1895
hoelzl@51526
  1896
lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
hoelzl@51526
  1897
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1898
hoelzl@51526
  1899
lemma LIMSEQ_realpow_zero:
hoelzl@51526
  1900
  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1901
proof cases
hoelzl@51526
  1902
  assume "0 \<le> x" and "x \<noteq> 0"
hoelzl@51526
  1903
  hence x0: "0 < x" by simp
hoelzl@51526
  1904
  assume x1: "x < 1"
hoelzl@51526
  1905
  from x0 x1 have "1 < inverse x"
hoelzl@51526
  1906
    by (rule one_less_inverse)
hoelzl@51526
  1907
  hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
hoelzl@51526
  1908
    by (rule LIMSEQ_inverse_realpow_zero)
hoelzl@51526
  1909
  thus ?thesis by (simp add: power_inverse)
hoelzl@58729
  1910
qed (rule LIMSEQ_imp_Suc, simp)
hoelzl@51526
  1911
hoelzl@51526
  1912
lemma LIMSEQ_power_zero:
hoelzl@51526
  1913
  fixes x :: "'a::{real_normed_algebra_1}"
hoelzl@51526
  1914
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1915
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
hoelzl@51526
  1916
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
hoelzl@51526
  1917
apply (simp add: power_abs norm_power_ineq)
hoelzl@51526
  1918
done
hoelzl@51526
  1919
hoelzl@51526
  1920
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
hoelzl@51526
  1921
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1922
wenzelm@60758
  1923
text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
hoelzl@51526
  1924
hoelzl@51526
  1925
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
hoelzl@51526
  1926
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
hoelzl@51526
  1927
hoelzl@51526
  1928
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
hoelzl@51526
  1929
  by (rule LIMSEQ_power_zero) simp
hoelzl@51526
  1930
hoelzl@51526
  1931
wenzelm@60758
  1932
subsection \<open>Limits of Functions\<close>
hoelzl@51526
  1933
hoelzl@51526
  1934
lemma LIM_eq:
hoelzl@51526
  1935
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1936
  shows "f -- a --> L =
hoelzl@51526
  1937
     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
hoelzl@51526
  1938
by (simp add: LIM_def dist_norm)
hoelzl@51526
  1939
hoelzl@51526
  1940
lemma LIM_I:
hoelzl@51526
  1941
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1942
  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
hoelzl@51526
  1943
      ==> f -- a --> L"
hoelzl@51526
  1944
by (simp add: LIM_eq)
hoelzl@51526
  1945
hoelzl@51526
  1946
lemma LIM_D:
hoelzl@51526
  1947
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1948
  shows "[| f -- a --> L; 0<r |]
hoelzl@51526
  1949
      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
hoelzl@51526
  1950
by (simp add: LIM_eq)
hoelzl@51526
  1951
hoelzl@51526
  1952
lemma LIM_offset:
hoelzl@51526
  1953
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1954
  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
hoelzl@51641
  1955
  unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
hoelzl@51526
  1956
hoelzl@51526
  1957
lemma LIM_offset_zero:
hoelzl@51526
  1958
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1959
  shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
haftmann@57512
  1960
by (drule_tac k="a" in LIM_offset, simp add: add.commute)
hoelzl@51526
  1961
hoelzl@51526
  1962
lemma LIM_offset_zero_cancel:
hoelzl@51526
  1963
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1964
  shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
hoelzl@51526
  1965
by (drule_tac k="- a" in LIM_offset, simp)
hoelzl@51526
  1966
hoelzl@51642
  1967
lemma LIM_offset_zero_iff:
hoelzl@51642
  1968
  fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
hoelzl@51642
  1969
  shows  "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
hoelzl@51642
  1970
  using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
hoelzl@51642
  1971
hoelzl@51526
  1972
lemma LIM_zero:
hoelzl@51526
  1973
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1974
  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
hoelzl@51526
  1975
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1976
hoelzl@51526
  1977
lemma LIM_zero_cancel:
hoelzl@51526
  1978
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1979
  shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
hoelzl@51526
  1980
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1981
hoelzl@51526
  1982
lemma LIM_zero_iff:
hoelzl@51526
  1983
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1984
  shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
hoelzl@51526
  1985
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1986
hoelzl@51526
  1987
lemma LIM_imp_LIM:
hoelzl@51526
  1988
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1989
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  1990
  assumes f: "f -- a --> l"
hoelzl@51526
  1991
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
hoelzl@51526
  1992
  shows "g -- a --> m"
hoelzl@51526
  1993
  by (rule metric_LIM_imp_LIM [OF f],
hoelzl@51526
  1994
    simp add: dist_norm le)
hoelzl@51526
  1995
hoelzl@51526
  1996
lemma LIM_equal2:
hoelzl@51526
  1997
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1998
  assumes 1: "0 < R"
hoelzl@51526
  1999
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
hoelzl@51526
  2000
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
hoelzl@51526
  2001
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
hoelzl@51526
  2002
hoelzl@51526
  2003
lemma LIM_compose2:
hoelzl@51526
  2004
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  2005
  assumes f: "f -- a --> b"
hoelzl@51526
  2006
  assumes g: "g -- b --> c"
hoelzl@51526
  2007
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
hoelzl@51526
  2008
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51526
  2009
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
hoelzl@51526
  2010
hoelzl@51526
  2011
lemma real_LIM_sandwich_zero:
hoelzl@51526
  2012
  fixes f g :: "'a::topological_space \<Rightarrow> real"
hoelzl@51526
  2013
  assumes f: "f -- a --> 0"
hoelzl@51526
  2014
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
hoelzl@51526
  2015
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
hoelzl@51526
  2016
  shows "g -- a --> 0"
hoelzl@51526
  2017
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
hoelzl@51526
  2018
  fix x assume x: "x \<noteq> a"
hoelzl@51526
  2019
  have "norm (g x - 0) = g x" by (simp add: 1 x)
hoelzl@51526
  2020
  also have "g x \<le> f x" by (rule 2 [OF x])
hoelzl@51526
  2021
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
hoelzl@51526
  2022
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
hoelzl@51526
  2023
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
hoelzl@51526
  2024
qed
hoelzl@51526
  2025
hoelzl@51526
  2026
wenzelm@60758
  2027
subsection \<open>Continuity\<close>
hoelzl@51526
  2028
hoelzl@51526
  2029
lemma LIM_isCont_iff:
hoelzl@51526
  2030
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  2031
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
hoelzl@51526
  2032
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
hoelzl@51526
  2033
hoelzl@51526
  2034
lemma isCont_iff:
hoelzl@51526
  2035
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  2036
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
hoelzl@51526
  2037
by (simp add: isCont_def LIM_isCont_iff)
hoelzl@51526
  2038
hoelzl@51526
  2039
lemma isCont_LIM_compose2:
hoelzl@51526
  2040
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  2041
  assumes f [unfolded isCont_def]: "isCont f a"
hoelzl@51526
  2042
  assumes g: "g -- f a --> l"
hoelzl@51526
  2043
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
hoelzl@51526
  2044
  shows "(\<lambda>x. g (f x)) -- a --> l"
hoelzl@51526
  2045
by (rule LIM_compose2 [OF f g inj])
hoelzl@51526
  2046
hoelzl@51526
  2047
hoelzl@51526
  2048
lemma isCont_norm [simp]:
hoelzl@51526
  2049
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2050
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
hoelzl@51526
  2051
  by (fact continuous_norm)
hoelzl@51526
  2052
hoelzl@51526
  2053
lemma isCont_rabs [simp]:
hoelzl@51526
  2054
  fixes f :: "'a::t2_space \<Rightarrow> real"
hoelzl@51526
  2055
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
hoelzl@51526
  2056
  by (fact continuous_rabs)
hoelzl@51526
  2057
hoelzl@51526
  2058
lemma isCont_add [simp]:
hoelzl@51526
  2059
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2060
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
hoelzl@51526
  2061
  by (fact continuous_add)
hoelzl@51526
  2062
hoelzl@51526
  2063
lemma isCont_minus [simp]:
hoelzl@51526
  2064
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2065
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
hoelzl@51526
  2066
  by (fact continuous_minus)
hoelzl@51526
  2067
hoelzl@51526
  2068
lemma isCont_diff [simp]:
hoelzl@51526
  2069
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2070
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
hoelzl@51526
  2071
  by (fact continuous_diff)
hoelzl@51526
  2072
hoelzl@51526
  2073
lemma isCont_mult [simp]:
hoelzl@51526
  2074
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
hoelzl@51526
  2075
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
hoelzl@51526
  2076
  by (fact continuous_mult)
hoelzl@51526
  2077
hoelzl@51526
  2078
lemma (in bounded_linear) isCont:
hoelzl@51526
  2079
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
hoelzl@51526
  2080
  by (fact continuous)
hoelzl@51526
  2081
hoelzl@51526
  2082
lemma (in bounded_bilinear) isCont:
hoelzl@51526
  2083
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
hoelzl@51526
  2084
  by (fact continuous)
hoelzl@51526
  2085
lp15@60141
  2086
lemmas isCont_scaleR [simp] =
hoelzl@51526
  2087
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
hoelzl@51526
  2088
hoelzl@51526
  2089
lemmas isCont_of_real [simp] =
hoelzl@51526
  2090
  bounded_linear.isCont [OF bounded_linear_of_real]
hoelzl@51526
  2091
hoelzl@51526
  2092
lemma isCont_power [simp]:
hoelzl@51526
  2093
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51526
  2094
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
hoelzl@51526
  2095
  by (fact continuous_power)
hoelzl@51526
  2096
hoelzl@51526
  2097
lemma isCont_setsum [simp]:
hoelzl@51526
  2098
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  2099
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
hoelzl@51526
  2100
  by (auto intro: continuous_setsum)
hoelzl@51526
  2101
wenzelm@60758
  2102
subsection \<open>Uniform Continuity\<close>
hoelzl@51526
  2103
hoelzl@51531
  2104
definition
hoelzl@51531
  2105
  isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
hoelzl@51531
  2106
  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
hoelzl@51531
  2107
hoelzl@51531
  2108
lemma isUCont_isCont: "isUCont f ==> isCont f x"
hoelzl@51531
  2109
by (simp add: isUCont_def isCont_def LIM_def, force)
hoelzl@51531
  2110
hoelzl@51531
  2111
lemma isUCont_Cauchy:
hoelzl@51531
  2112
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51531
  2113
unfolding isUCont_def
hoelzl@51531
  2114
apply (rule metric_CauchyI)
hoelzl@51531
  2115
apply (drule_tac x=e in spec, safe)
hoelzl@51531
  2116
apply (drule_tac e=s in metric_CauchyD, safe)
hoelzl@51531
  2117
apply (rule_tac x=M in exI, simp)
hoelzl@51531
  2118
done
hoelzl@51531
  2119
hoelzl@51526
  2120
lemma (in bounded_linear) isUCont: "isUCont f"
hoelzl@51526
  2121
unfolding isUCont_def dist_norm
hoelzl@51526
  2122
proof (intro allI impI)
hoelzl@51526
  2123
  fix r::real assume r: "0 < r"
hoelzl@51526
  2124
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
hoelzl@51526
  2125
    using pos_bounded by fast
hoelzl@51526
  2126
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
hoelzl@51526
  2127
  proof (rule exI, safe)
nipkow@56541
  2128
    from r K show "0 < r / K" by simp
hoelzl@51526
  2129
  next
hoelzl@51526
  2130
    fix x y :: 'a
hoelzl@51526
  2131
    assume xy: "norm (x - y) < r / K"
hoelzl@51526
  2132
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
hoelzl@51526
  2133
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
hoelzl@51526
  2134
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
hoelzl@51526
  2135
    finally show "norm (f x - f y) < r" .
hoelzl@51526
  2136
  qed
hoelzl@51526
  2137
qed
hoelzl@51526
  2138
hoelzl@51526
  2139
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51526
  2140
by (rule isUCont [THEN isUCont_Cauchy])
hoelzl@51526
  2141
lp15@60141
  2142
lemma LIM_less_bound:
hoelzl@51526
  2143
  fixes f :: "real \<Rightarrow> real"
hoelzl@51526
  2144
  assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
hoelzl@51526
  2145
  shows "0 \<le> f x"
hoelzl@51526
  2146
proof (rule tendsto_le_const)
hoelzl@51526
  2147
  show "(f ---> f x) (at_left x)"
wenzelm@60758
  2148
    using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
hoelzl@51526
  2149
  show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
hoelzl@51641
  2150
    using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
hoelzl@51526
  2151
qed simp
hoelzl@51471
  2152
hoelzl@51529
  2153
wenzelm@60758
  2154
subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
hoelzl@51529
  2155
hoelzl@51529
  2156
lemma nested_sequence_unique:
hoelzl@51529
  2157
  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) ----> 0"
hoelzl@51529
  2158
  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
hoelzl@51529
  2159
proof -
hoelzl@51529
  2160
  have "incseq f" unfolding incseq_Suc_iff by fact
hoelzl@51529
  2161
  have "decseq g" unfolding decseq_Suc_iff by fact
hoelzl@51529
  2162
hoelzl@51529
  2163
  { fix n
wenzelm@60758
  2164
    from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
wenzelm@60758
  2165
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
hoelzl@51529
  2166
  then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
wenzelm@60758
  2167
    using incseq_convergent[OF \<open>incseq f\<close>] by auto
hoelzl@51529
  2168
  moreover
hoelzl@51529
  2169
  { fix n
wenzelm@60758
  2170
    from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
wenzelm@60758
  2171
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
hoelzl@51529
  2172
  then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
wenzelm@60758
  2173
    using decseq_convergent[OF \<open>decseq g\<close>] by auto
wenzelm@60758
  2174
  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f ----> u\<close> \<open>g ----> l\<close>]]
hoelzl@51529
  2175
  ultimately show ?thesis by auto
hoelzl@51529
  2176
qed
hoelzl@51529
  2177
hoelzl@51529
  2178
lemma Bolzano[consumes 1, case_names trans local]:
hoelzl@51529
  2179
  fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
hoelzl@51529
  2180
  assumes [arith]: "a \<le> b"
hoelzl@51529
  2181
  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
  2182
  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
  2183
  shows "P a b"
hoelzl@51529
  2184
proof -
blanchet@55415
  2185
  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
  2186
  def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
hoelzl@51529
  2187
  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
  2188
    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
  2189
    by (simp_all add: l_def u_def bisect_def split: prod.split)
hoelzl@51529
  2190
hoelzl@51529
  2191
  { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
hoelzl@51529
  2192
hoelzl@51529
  2193
  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
hoelzl@51529
  2194
  proof (safe intro!: nested_sequence_unique)
hoelzl@51529
  2195
    fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
hoelzl@51529
  2196
  next
hoelzl@51529
  2197
    { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
hoelzl@51529
  2198
    then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
hoelzl@51529
  2199
  qed fact
hoelzl@51529
  2200
  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
hoelzl@51529
  2201
  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
  2202
    using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
hoelzl@51529
  2203
hoelzl@51529
  2204
  show "P a b"
hoelzl@51529
  2205
  proof (rule ccontr)
lp15@60141
  2206
    assume "\<not> P a b"
hoelzl@51529
  2207
    { fix n have "\<not> P (l n) (u n)"
hoelzl@51529
  2208
      proof (induct n)
hoelzl@51529
  2209
        case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
wenzelm@60758
  2210
      qed (simp add: \<open>\<not> P a b\<close>) }
hoelzl@51529
  2211
    moreover
hoelzl@51529
  2212
    { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
wenzelm@60758
  2213
        using \<open>0 < d\<close> \<open>l ----> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  2214
      moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
wenzelm@60758
  2215
        using \<open>0 < d\<close> \<open>u ----> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  2216
      ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
hoelzl@51529
  2217
      proof eventually_elim
hoelzl@51529
  2218
        fix n assume "x - d / 2 < l n" "u n < x + d / 2"
hoelzl@51529
  2219
        from add_strict_mono[OF this] have "u n - l n < d" by simp
hoelzl@51529
  2220
        with x show "P (l n) (u n)" by (rule d)
hoelzl@51529
  2221
      qed }
hoelzl@51529
  2222
    ultimately show False by simp
hoelzl@51529
  2223
  qed
hoelzl@51529
  2224
qed
hoelzl@51529
  2225
hoelzl@51529
  2226
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
hoelzl@51529
  2227
proof (cases "a \<le> b", rule compactI)
hoelzl@51529
  2228
  fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
hoelzl@51529
  2229
  def T == "{a .. b}"
hoelzl@51529
  2230
  from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
hoelzl@51529
  2231
  proof (induct rule: Bolzano)
hoelzl@51529
  2232
    case (trans a b c)
hoelzl@51529
  2233
    then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
hoelzl@51529
  2234
    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
  2235
      by (auto simp: *)
hoelzl@51529
  2236
    with trans show ?case
hoelzl@51529
  2237
      unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
hoelzl@51529
  2238
  next
hoelzl@51529
  2239
    case (local x)
hoelzl@51529
  2240
    then have "x \<in> \<Union>C" using C by auto
hoelzl@51529
  2241
    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
hoelzl@51529
  2242
    then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
hoelzl@51529
  2243
      by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
wenzelm@60758
  2244
    with \<open>c \<in> C\<close> show ?case
hoelzl@51529
  2245
      by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
hoelzl@51529
  2246
  qed
hoelzl@51529
  2247
qed simp
hoelzl@51529
  2248
hoelzl@51529
  2249
hoelzl@57447
  2250
lemma continuous_image_closed_interval:
hoelzl@57447
  2251
  fixes a b and f :: "real \<Rightarrow> real"
hoelzl@57447
  2252
  defines "S \<equiv> {a..b}"
hoelzl@57447
  2253
  assumes "a \<le> b" and f: "continuous_on S f"
hoelzl@57447
  2254
  shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
hoelzl@57447
  2255
proof -
hoelzl@57447
  2256
  have S: "compact S" "S \<noteq> {}"
wenzelm@60758
  2257
    using \<open>a \<le> b\<close> by (auto simp: S_def)
hoelzl@57447
  2258
  obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
hoelzl@57447
  2259
    using continuous_attains_sup[OF S f] by auto
hoelzl@57447
  2260
  moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
hoelzl@57447
  2261
    using continuous_attains_inf[OF S f] by auto
hoelzl@57447
  2262
  moreover have "connected (f`S)"
hoelzl@57447
  2263
    using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
hoelzl@57447
  2264
  ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
hoelzl@57447
  2265
    by (auto simp: connected_iff_interval)
hoelzl@57447
  2266
  then show ?thesis
hoelzl@57447
  2267
    by auto
hoelzl@57447
  2268
qed
hoelzl@57447
  2269
lp15@60974
  2270
lemma open_Collect_positive:
lp15@60974
  2271
 fixes f :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  2272
 assumes f: "continuous_on s f"
lp15@60974
  2273
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
lp15@60974
  2274
 using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
lp15@60974
  2275
 by (auto simp: Int_def field_simps)
lp15@60974
  2276
lp15@60974
  2277
lemma open_Collect_less_Int:
lp15@60974
  2278
 fixes f g :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  2279
 assumes f: "continuous_on s f" and g: "continuous_on s g"
lp15@60974
  2280
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
lp15@60974
  2281
 using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
lp15@60974
  2282
lp15@60974
  2283
wenzelm@60758
  2284
subsection \<open>Boundedness of continuous functions\<close>
hoelzl@51529
  2285
wenzelm@60758
  2286
text\<open>By bisection, function continuous on closed interval is bounded above\<close>
hoelzl@51529
  2287
hoelzl@51529
  2288
lemma isCont_eq_Ub:
hoelzl@51529
  2289
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2290
  shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2291
    \<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
  2292
  using continuous_attains_sup[of "{a .. b}" f]
hoelzl@51529
  2293
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2294
hoelzl@51529
  2295
lemma isCont_eq_Lb:
hoelzl@51529
  2296
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2297
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2298
    \<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
  2299
  using continuous_attains_inf[of "{a .. b}" f]
hoelzl@51529
  2300
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2301
hoelzl@51529
  2302
lemma isCont_bounded:
hoelzl@51529
  2303
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2304
  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
  2305
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2306
hoelzl@51529
  2307
lemma isCont_has_Ub:
hoelzl@51529
  2308
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2309
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2310
    \<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
  2311
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2312
hoelzl@51529
  2313
(*HOL style here: object-level formulations*)
hoelzl@51529
  2314
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
hoelzl@51529
  2315
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2316
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2317
  by (blast intro: IVT)
hoelzl@51529
  2318
hoelzl@51529
  2319
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
hoelzl@51529
  2320
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2321
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2322
  by (blast intro: IVT2)
hoelzl@51529
  2323
hoelzl@51529
  2324
lemma isCont_Lb_Ub:
hoelzl@51529
  2325
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  2326
  assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
lp15@60141
  2327
  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
  2328
               (\<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
  2329
proof -
hoelzl@51529
  2330
  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
  2331
    using isCont_eq_Ub[OF assms] by auto
hoelzl@51529
  2332
  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
  2333
    using isCont_eq_Lb[OF assms] by auto
hoelzl@51529
  2334
  show ?thesis
hoelzl@51529
  2335
    using IVT[of f L _ M] IVT2[of f L _ M] M L assms
hoelzl@51529
  2336
    apply (rule_tac x="f L" in exI)
hoelzl@51529
  2337
    apply (rule_tac x="f M" in exI)
hoelzl@51529
  2338
    apply (cases "L \<le> M")
hoelzl@51529
  2339
    apply (simp, metis order_trans)
hoelzl@51529
  2340
    apply (simp, metis order_trans)
hoelzl@51529
  2341
    done
hoelzl@51529
  2342
qed
hoelzl@51529
  2343
hoelzl@51529
  2344
wenzelm@60758
  2345
text\<open>Continuity of inverse function\<close>
hoelzl@51529
  2346
hoelzl@51529
  2347
lemma isCont_inverse_function:
hoelzl@51529
  2348
  fixes f g :: "real \<Rightarrow> real"
hoelzl@51529
  2349
  assumes d: "0 < d"
hoelzl@51529
  2350
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
hoelzl@51529
  2351
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
hoelzl@51529
  2352
  shows "isCont g (f x)"
hoelzl@51529
  2353
proof -
hoelzl@51529
  2354
  let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
hoelzl@51529
  2355
hoelzl@51529
  2356
  have f: "continuous_on ?D f"
hoelzl@51529
  2357
    using cont by (intro continuous_at_imp_continuous_on ballI) auto
hoelzl@51529
  2358
  then have g: "continuous_on (f`?D) g"
hoelzl@51529
  2359
    using inj by (intro continuous_on_inv) auto
hoelzl@51529
  2360
hoelzl@51529
  2361
  from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
hoelzl@51529
  2362
    by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
hoelzl@51529
  2363
  with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
hoelzl@51529
  2364
    by (rule continuous_on_subset)
hoelzl@51529
  2365
  moreover
hoelzl@51529
  2366
  have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
hoelzl@51529
  2367
    using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
hoelzl@51529
  2368
  then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
hoelzl@51529
  2369
    by auto
hoelzl@51529
  2370
  ultimately
hoelzl@51529
  2371
  show ?thesis