src/HOL/Limits.thy
author immler
Wed May 11 16:13:17 2016 +0200 (2016-05-11)
changeset 63081 5a5beb3dbe7e
parent 63040 eb4ddd18d635
child 63104 9505a883403c
permissions -rw-r--r--
introduced class topological_group between topological_monoid and real_normed_vector
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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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corollary eventually_at_infinity_pos:
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   "eventually p at_infinity \<longleftrightarrow> (\<exists>b. 0 < b \<and> (\<forall>x. norm x \<ge> b \<longrightarrow> p x))"
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apply (simp add: eventually_at_infinity, auto)
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apply (case_tac "b \<le> 0")
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using norm_ge_zero order_trans zero_less_one apply blast
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apply (force simp:)
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done
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lemma at_infinity_eq_at_top_bot:
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  "(at_infinity :: real filter) = sup at_top at_bot"
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  apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
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                   eventually_at_top_linorder eventually_at_bot_linorder)
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  apply safe
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  apply (rule_tac x="b" in exI, simp)
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  apply (rule_tac x="- b" in exI, simp)
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  apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
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  done
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lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
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  unfolding at_infinity_eq_at_top_bot by simp
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lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
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  unfolding at_infinity_eq_at_top_bot by simp
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lemma filterlim_at_top_imp_at_infinity:
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  fixes f :: "_ \<Rightarrow> real"
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  shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
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  by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
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lemma lim_infinity_imp_sequentially:
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  "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
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by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
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subsubsection \<open>Boundedness\<close>
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definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
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  Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
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abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
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  "Bseq X \<equiv> Bfun X sequentially"
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lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
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lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
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  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
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lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
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  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
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lemma Bfun_def:
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  "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
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  unfolding Bfun_metric_def norm_conv_dist
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proof safe
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  fix y K assume K: "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 (force simp del: norm_conv_dist [symmetric])
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lemma BfunI:
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  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
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unfolding Bfun_def
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proof (intro exI conjI allI)
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  show "0 < max K 1" by simp
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next
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  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
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    using K by (rule eventually_mono, simp)
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qed
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lemma BfunE:
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  assumes "Bfun f F"
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  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
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using assms unfolding Bfun_def by blast
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lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
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  unfolding Cauchy_def Bfun_metric_def eventually_sequentially
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  apply (erule_tac x=1 in allE)
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  apply simp
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  apply safe
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  apply (rule_tac x="X M" in exI)
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  apply (rule_tac x=1 in exI)
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  apply (erule_tac x=M in allE)
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  apply simp
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  apply (rule_tac x=M in exI)
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  apply (auto simp: dist_commute)
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  done
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subsubsection \<open>Bounded Sequences\<close>
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lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
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  by (intro BfunI) (auto simp: eventually_sequentially)
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lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
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  by (intro BfunI) (auto simp: eventually_sequentially)
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lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
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  unfolding Bfun_def eventually_sequentially
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proof safe
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  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
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  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
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    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
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       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
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qed auto
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lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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unfolding Bseq_def by auto
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lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
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by (simp add: Bseq_def)
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lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
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by (auto simp add: Bseq_def)
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lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
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proof (elim BseqE, intro bdd_aboveI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
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    by (auto elim!: allE[of _ n])
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qed
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lemma Bseq_bdd_above':
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  "Bseq (X::nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
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proof (elim BseqE, intro bdd_aboveI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "norm (X n) \<le> K"
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    by (auto elim!: allE[of _ n])
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qed
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lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
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proof (elim BseqE, intro bdd_belowI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
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    by (auto elim!: allE[of _ n])
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qed
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lemma Bseq_eventually_mono:
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  assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"
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  shows   "Bseq f"
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proof -
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  from assms(1) obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> norm (g n)"
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    by (auto simp: eventually_at_top_linorder)
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  moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K" by (blast elim!: BseqE)
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  ultimately have "norm (f n) \<le> max K (Max {norm (f n) |n. n < N})" for n
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    apply (cases "n < N")
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    apply (rule max.coboundedI2, rule Max.coboundedI, auto) []
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    apply (rule max.coboundedI1, force intro: order.trans[OF N K])
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    done
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  thus ?thesis by (blast intro: BseqI')
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qed
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lemma lemma_NBseq_def:
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  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
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proof safe
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  fix K :: real
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  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
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  then have "K \<le> real (Suc n)" by auto
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  moreover assume "\<forall>m. norm (X m) \<le> K"
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  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
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    by (blast intro: order_trans)
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  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
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next
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  show "\<And>N. \<forall>n. norm (X n) \<le> real (Suc N) \<Longrightarrow> \<exists>K>0. \<forall>n. norm (X n) \<le> K"
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    using of_nat_0_less_iff by blast
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qed
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text\<open>alternative definition for Bseq\<close>
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lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
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apply (simp add: Bseq_def)
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apply (simp (no_asm) add: lemma_NBseq_def)
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done
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lemma lemma_NBseq_def2:
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     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
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apply (subst lemma_NBseq_def, auto)
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apply (rule_tac x = "Suc N" in exI)
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apply (rule_tac [2] x = N in exI)
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apply (auto simp add: of_nat_Suc)
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 prefer 2 apply (blast intro: order_less_imp_le)
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apply (drule_tac x = n in spec, simp)
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done
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(* yet another definition for Bseq *)
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lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
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by (simp add: Bseq_def lemma_NBseq_def2)
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subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>
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text\<open>alternative formulation for boundedness\<close>
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lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
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apply (unfold Bseq_def, safe)
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apply (rule_tac [2] x = "k + norm x" in exI)
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apply (rule_tac x = K in exI, simp)
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apply (rule exI [where x = 0], auto)
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apply (erule order_less_le_trans, simp)
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apply (drule_tac x=n in spec)
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apply (drule order_trans [OF norm_triangle_ineq2])
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apply simp
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done
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text\<open>alternative formulation for boundedness\<close>
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lemma Bseq_iff3:
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  "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P
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  then obtain K
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    where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
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  from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
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  from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
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    by (auto intro: order_trans norm_triangle_ineq4)
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  then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
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    by simp
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  with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
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next
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  assume ?Q then show ?P by (auto simp add: Bseq_iff2)
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qed
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lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
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apply (simp add: Bseq_def)
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apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
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apply (drule_tac x = n in spec, arith)
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done
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subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>
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lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
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  by (simp add: Bseq_def)
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lemma Bseq_add:
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  assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
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  shows   "Bseq (\<lambda>x. f x + c)"
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proof -
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  from assms obtain K where K: "\<And>x. norm (f x) \<le> K" unfolding Bseq_def by blast
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  {
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    fix x :: nat
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    have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)
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    also have "norm (f x) \<le> K" by (rule K)
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    finally have "norm (f x + c) \<le> K + norm c" by simp
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  }
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  thus ?thesis by (rule BseqI')
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qed
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lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
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  using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto
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lemma Bseq_mult:
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  assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_field)"
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  assumes "Bseq (g :: nat \<Rightarrow> 'a :: real_normed_field)"
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  shows   "Bseq (\<lambda>x. f x * g x)"
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proof -
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  from assms obtain K1 K2 where K: "\<And>x. norm (f x) \<le> K1" "K1 > 0" "\<And>x. norm (g x) \<le> K2" "K2 > 0"
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    unfolding Bseq_def by blast
eberlm@61531
   278
  hence "\<And>x. norm (f x * g x) \<le> K1 * K2" by (auto simp: norm_mult intro!: mult_mono)
eberlm@61531
   279
  thus ?thesis by (rule BseqI')
eberlm@61531
   280
qed
eberlm@61531
   281
eberlm@61531
   282
lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"
eberlm@61531
   283
  unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])
eberlm@61531
   284
eberlm@61531
   285
lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
eberlm@61531
   286
proof
eberlm@61531
   287
  assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"
eberlm@61531
   288
  find_theorems "Bfun (\<lambda>_. ?c) _"
eberlm@61531
   289
  from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)
wenzelm@61799
   290
  with \<open>c \<noteq> 0\<close> show "Bseq f" by (simp add: divide_simps)
eberlm@61531
   291
qed (intro Bseq_mult Bfun_const)
eberlm@61531
   292
eberlm@61531
   293
lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
eberlm@61531
   294
  unfolding Bseq_def by auto
eberlm@61531
   295
eberlm@61531
   296
lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
eberlm@61531
   297
  using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
eberlm@61531
   298
eberlm@61531
   299
lemma increasing_Bseq_subseq_iff:
eberlm@61531
   300
  assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"
eberlm@61531
   301
  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
eberlm@61531
   302
proof
eberlm@61531
   303
  assume "Bseq (\<lambda>x. f (g x))"
eberlm@61531
   304
  then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto
eberlm@61531
   305
  {
eberlm@61531
   306
    fix x :: nat
eberlm@61531
   307
    from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
eberlm@61531
   308
      by (auto simp: filterlim_at_top eventually_at_top_linorder)
eberlm@61531
   309
    hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast
eberlm@61531
   310
    also have "norm (f (g y)) \<le> K" by (rule K)
eberlm@61531
   311
    finally have "norm (f x) \<le> K" .
eberlm@61531
   312
  }
eberlm@61531
   313
  thus "Bseq f" by (rule BseqI')
eberlm@61531
   314
qed (insert Bseq_subseq[of f g], simp_all)
eberlm@61531
   315
eberlm@61531
   316
lemma nonneg_incseq_Bseq_subseq_iff:
eberlm@61531
   317
  assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"
eberlm@61531
   318
  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
eberlm@61531
   319
  using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
eberlm@61531
   320
hoelzl@51531
   321
lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
hoelzl@51531
   322
  apply (simp add: subset_eq)
hoelzl@51531
   323
  apply (rule BseqI'[where K="max (norm a) (norm b)"])
hoelzl@51531
   324
  apply (erule_tac x=n in allE)
hoelzl@51531
   325
  apply auto
hoelzl@51531
   326
  done
hoelzl@51531
   327
hoelzl@51531
   328
lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
hoelzl@51531
   329
  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
hoelzl@51531
   330
hoelzl@51531
   331
lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
hoelzl@51531
   332
  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
hoelzl@51531
   333
wenzelm@60758
   334
subsection \<open>Bounded Monotonic Sequences\<close>
hoelzl@51531
   335
wenzelm@60758
   336
subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
hoelzl@51531
   337
hoelzl@51531
   338
(* TODO: delete *)
hoelzl@51531
   339
(* FIXME: one use in NSA/HSEQ.thy *)
wenzelm@61969
   340
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X \<longlonglongrightarrow> L)"
hoelzl@51531
   341
  apply (rule_tac x="X m" in exI)
hoelzl@51531
   342
  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
hoelzl@51531
   343
  unfolding eventually_sequentially
hoelzl@51531
   344
  apply blast
hoelzl@51531
   345
  done
hoelzl@51531
   346
wenzelm@60758
   347
subsection \<open>Convergence to Zero\<close>
huffman@31349
   348
huffman@44081
   349
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   350
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   351
huffman@31349
   352
lemma ZfunI:
huffman@44195
   353
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   354
  unfolding Zfun_def by simp
huffman@31349
   355
huffman@31349
   356
lemma ZfunD:
huffman@44195
   357
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   358
  unfolding Zfun_def by simp
huffman@31349
   359
huffman@31355
   360
lemma Zfun_ssubst:
huffman@44195
   361
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   362
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   363
huffman@44195
   364
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   365
  unfolding Zfun_def by simp
huffman@31349
   366
huffman@44195
   367
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   368
  unfolding Zfun_def by simp
huffman@31349
   369
huffman@31349
   370
lemma Zfun_imp_Zfun:
huffman@44195
   371
  assumes f: "Zfun f F"
huffman@44195
   372
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   373
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   374
proof (cases)
huffman@31349
   375
  assume K: "0 < K"
huffman@31349
   376
  show ?thesis
huffman@31349
   377
  proof (rule ZfunI)
huffman@31349
   378
    fix r::real assume "0 < r"
nipkow@56541
   379
    hence "0 < r / K" using K by simp
huffman@44195
   380
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
lp15@61649
   381
      using ZfunD [OF f] by blast
huffman@44195
   382
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   383
    proof eventually_elim
noschinl@46887
   384
      case (elim x)
huffman@31487
   385
      hence "norm (f x) * K < r"
huffman@31349
   386
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   387
      thus ?case
noschinl@46887
   388
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   389
    qed
huffman@31349
   390
  qed
huffman@31349
   391
next
huffman@31349
   392
  assume "\<not> 0 < K"
huffman@31349
   393
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   394
  show ?thesis
huffman@31355
   395
  proof (rule ZfunI)
huffman@31355
   396
    fix r :: real
huffman@31355
   397
    assume "0 < r"
huffman@44195
   398
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   399
    proof eventually_elim
noschinl@46887
   400
      case (elim x)
noschinl@46887
   401
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   402
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   403
      finally show ?case
wenzelm@60758
   404
        using \<open>0 < r\<close> by simp
huffman@31355
   405
    qed
huffman@31355
   406
  qed
huffman@31349
   407
qed
huffman@31349
   408
huffman@44195
   409
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   410
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   411
huffman@31349
   412
lemma Zfun_add:
huffman@44195
   413
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   414
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   415
proof (rule ZfunI)
huffman@31349
   416
  fix r::real assume "0 < r"
huffman@31349
   417
  hence r: "0 < r / 2" by simp
huffman@44195
   418
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   419
    using f r by (rule ZfunD)
huffman@31349
   420
  moreover
huffman@44195
   421
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   422
    using g r by (rule ZfunD)
huffman@31349
   423
  ultimately
huffman@44195
   424
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   425
  proof eventually_elim
noschinl@46887
   426
    case (elim x)
huffman@31487
   427
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   428
      by (rule norm_triangle_ineq)
huffman@31349
   429
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   430
      using elim by (rule add_strict_mono)
noschinl@46887
   431
    finally show ?case
huffman@31349
   432
      by simp
huffman@31349
   433
  qed
huffman@31349
   434
qed
huffman@31349
   435
huffman@44195
   436
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   437
  unfolding Zfun_def by simp
huffman@31349
   438
huffman@44195
   439
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
haftmann@54230
   440
  using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
huffman@31349
   441
huffman@31349
   442
lemma (in bounded_linear) Zfun:
huffman@44195
   443
  assumes g: "Zfun g F"
huffman@44195
   444
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   445
proof -
huffman@31349
   446
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
lp15@61649
   447
    using bounded by blast
huffman@44195
   448
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   449
    by simp
huffman@31487
   450
  with g show ?thesis
huffman@31349
   451
    by (rule Zfun_imp_Zfun)
huffman@31349
   452
qed
huffman@31349
   453
huffman@31349
   454
lemma (in bounded_bilinear) Zfun:
huffman@44195
   455
  assumes f: "Zfun f F"
huffman@44195
   456
  assumes g: "Zfun g F"
huffman@44195
   457
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   458
proof (rule ZfunI)
huffman@31349
   459
  fix r::real assume r: "0 < r"
huffman@31349
   460
  obtain K where K: "0 < K"
huffman@31349
   461
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
lp15@61649
   462
    using pos_bounded by blast
huffman@31349
   463
  from K have K': "0 < inverse K"
huffman@31349
   464
    by (rule positive_imp_inverse_positive)
huffman@44195
   465
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   466
    using f r by (rule ZfunD)
huffman@31349
   467
  moreover
huffman@44195
   468
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   469
    using g K' by (rule ZfunD)
huffman@31349
   470
  ultimately
huffman@44195
   471
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   472
  proof eventually_elim
noschinl@46887
   473
    case (elim x)
huffman@31487
   474
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   475
      by (rule norm_le)
huffman@31487
   476
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   477
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   478
    also from K have "r * inverse K * K = r"
huffman@31349
   479
      by simp
noschinl@46887
   480
    finally show ?case .
huffman@31349
   481
  qed
huffman@31349
   482
qed
huffman@31349
   483
huffman@31349
   484
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   485
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   486
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   487
huffman@31349
   488
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   489
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   490
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   491
huffman@44282
   492
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   493
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   494
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   495
wenzelm@61973
   496
lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   497
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   498
wenzelm@61973
   499
lemma tendsto_0_le: "\<lbrakk>(f \<longlongrightarrow> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
wenzelm@61973
   500
                     \<Longrightarrow> (g \<longlongrightarrow> 0) F"
lp15@56366
   501
  by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
lp15@56366
   502
wenzelm@60758
   503
subsubsection \<open>Distance and norms\<close>
huffman@36662
   504
hoelzl@51531
   505
lemma tendsto_dist [tendsto_intros]:
hoelzl@51531
   506
  fixes l m :: "'a :: metric_space"
wenzelm@61973
   507
  assumes f: "(f \<longlongrightarrow> l) F" and g: "(g \<longlongrightarrow> m) F"
wenzelm@61973
   508
  shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
hoelzl@51531
   509
proof (rule tendstoI)
hoelzl@51531
   510
  fix e :: real assume "0 < e"
hoelzl@51531
   511
  hence e2: "0 < e/2" by simp
hoelzl@51531
   512
  from tendstoD [OF f e2] tendstoD [OF g e2]
hoelzl@51531
   513
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
hoelzl@51531
   514
  proof (eventually_elim)
hoelzl@51531
   515
    case (elim x)
hoelzl@51531
   516
    then show "dist (dist (f x) (g x)) (dist l m) < e"
hoelzl@51531
   517
      unfolding dist_real_def
hoelzl@51531
   518
      using dist_triangle2 [of "f x" "g x" "l"]
hoelzl@51531
   519
      using dist_triangle2 [of "g x" "l" "m"]
hoelzl@51531
   520
      using dist_triangle3 [of "l" "m" "f x"]
hoelzl@51531
   521
      using dist_triangle [of "f x" "m" "g x"]
hoelzl@51531
   522
      by arith
hoelzl@51531
   523
  qed
hoelzl@51531
   524
qed
hoelzl@51531
   525
hoelzl@51531
   526
lemma continuous_dist[continuous_intros]:
hoelzl@51531
   527
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   528
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   529
  unfolding continuous_def by (rule tendsto_dist)
hoelzl@51531
   530
hoelzl@56371
   531
lemma continuous_on_dist[continuous_intros]:
hoelzl@51531
   532
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   533
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   534
  unfolding continuous_on_def by (auto intro: tendsto_dist)
hoelzl@51531
   535
huffman@31565
   536
lemma tendsto_norm [tendsto_intros]:
wenzelm@61973
   537
  "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
huffman@44081
   538
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   539
hoelzl@51478
   540
lemma continuous_norm [continuous_intros]:
hoelzl@51478
   541
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
hoelzl@51478
   542
  unfolding continuous_def by (rule tendsto_norm)
hoelzl@51478
   543
hoelzl@56371
   544
lemma continuous_on_norm [continuous_intros]:
hoelzl@51478
   545
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
hoelzl@51478
   546
  unfolding continuous_on_def by (auto intro: tendsto_norm)
hoelzl@51478
   547
huffman@36662
   548
lemma tendsto_norm_zero:
wenzelm@61973
   549
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
huffman@44081
   550
  by (drule tendsto_norm, simp)
huffman@36662
   551
huffman@36662
   552
lemma tendsto_norm_zero_cancel:
wenzelm@61973
   553
  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44081
   554
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   555
huffman@36662
   556
lemma tendsto_norm_zero_iff:
wenzelm@61973
   557
  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44081
   558
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   559
huffman@44194
   560
lemma tendsto_rabs [tendsto_intros]:
wenzelm@61973
   561
  "(f \<longlongrightarrow> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
huffman@44194
   562
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   563
hoelzl@51478
   564
lemma continuous_rabs [continuous_intros]:
hoelzl@51478
   565
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   566
  unfolding real_norm_def[symmetric] by (rule continuous_norm)
hoelzl@51478
   567
hoelzl@56371
   568
lemma continuous_on_rabs [continuous_intros]:
hoelzl@51478
   569
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   570
  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
hoelzl@51478
   571
huffman@44194
   572
lemma tendsto_rabs_zero:
wenzelm@61973
   573
  "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
huffman@44194
   574
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   575
huffman@44194
   576
lemma tendsto_rabs_zero_cancel:
wenzelm@61973
   577
  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44194
   578
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   579
huffman@44194
   580
lemma tendsto_rabs_zero_iff:
wenzelm@61973
   581
  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
huffman@44194
   582
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   583
hoelzl@62368
   584
subsection \<open>Topological Monoid\<close>
hoelzl@62368
   585
hoelzl@62368
   586
class topological_monoid_add = topological_space + monoid_add +
hoelzl@62368
   587
  assumes tendsto_add_Pair: "LIM x (nhds a \<times>\<^sub>F nhds b). fst x + snd x :> nhds (a + b)"
hoelzl@62368
   588
hoelzl@62368
   589
class topological_comm_monoid_add = topological_monoid_add + comm_monoid_add
huffman@44194
   590
huffman@31565
   591
lemma tendsto_add [tendsto_intros]:
hoelzl@62368
   592
  fixes a b :: "'a::topological_monoid_add"
hoelzl@62368
   593
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
hoelzl@62368
   594
  using filterlim_compose[OF tendsto_add_Pair, of "\<lambda>x. (f x, g x)" a b F]
hoelzl@62368
   595
  by (simp add: nhds_prod[symmetric] tendsto_Pair)
huffman@31349
   596
hoelzl@51478
   597
lemma continuous_add [continuous_intros]:
hoelzl@62368
   598
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
hoelzl@51478
   599
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
hoelzl@51478
   600
  unfolding continuous_def by (rule tendsto_add)
hoelzl@51478
   601
hoelzl@56371
   602
lemma continuous_on_add [continuous_intros]:
hoelzl@62368
   603
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
hoelzl@51478
   604
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
hoelzl@51478
   605
  unfolding continuous_on_def by (auto intro: tendsto_add)
hoelzl@51478
   606
huffman@44194
   607
lemma tendsto_add_zero:
hoelzl@62368
   608
  fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
wenzelm@61973
   609
  shows "\<lbrakk>(f \<longlongrightarrow> 0) F; (g \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
huffman@44194
   610
  by (drule (1) tendsto_add, simp)
huffman@44194
   611
hoelzl@62368
   612
lemma tendsto_setsum [tendsto_intros]:
hoelzl@62368
   613
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@62368
   614
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
hoelzl@62368
   615
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
hoelzl@62368
   616
proof (cases "finite S")
hoelzl@62368
   617
  assume "finite S" thus ?thesis using assms
hoelzl@62368
   618
    by (induct, simp, simp add: tendsto_add)
hoelzl@62368
   619
qed simp
hoelzl@62368
   620
hoelzl@62368
   621
lemma continuous_setsum [continuous_intros]:
hoelzl@62368
   622
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@62368
   623
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@62368
   624
  unfolding continuous_def by (rule tendsto_setsum)
hoelzl@62368
   625
hoelzl@62368
   626
lemma continuous_on_setsum [continuous_intros]:
hoelzl@62368
   627
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@62368
   628
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@62368
   629
  unfolding continuous_on_def by (auto intro: tendsto_setsum)
hoelzl@62368
   630
hoelzl@62369
   631
instance nat :: topological_comm_monoid_add
hoelzl@62369
   632
  proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
hoelzl@62369
   633
hoelzl@62369
   634
instance int :: topological_comm_monoid_add
hoelzl@62369
   635
  proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
hoelzl@62369
   636
immler@63081
   637
subsubsection \<open>Topological group\<close>
immler@63081
   638
immler@63081
   639
class topological_group_add = topological_monoid_add + group_add +
immler@63081
   640
  assumes tendsto_uminus_nhds: "(uminus \<longlongrightarrow> - a) (nhds a)"
immler@63081
   641
begin
immler@63081
   642
immler@63081
   643
lemma tendsto_minus [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> -a) F"
immler@63081
   644
  by (rule filterlim_compose[OF tendsto_uminus_nhds])
immler@63081
   645
immler@63081
   646
end
immler@63081
   647
immler@63081
   648
class topological_ab_group_add = topological_group_add + ab_group_add
immler@63081
   649
immler@63081
   650
instance topological_ab_group_add < topological_comm_monoid_add ..
immler@63081
   651
immler@63081
   652
lemma continuous_minus [continuous_intros]:
immler@63081
   653
  fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
immler@63081
   654
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
immler@63081
   655
  unfolding continuous_def by (rule tendsto_minus)
immler@63081
   656
immler@63081
   657
lemma continuous_on_minus [continuous_intros]:
immler@63081
   658
  fixes f :: "_ \<Rightarrow> 'b::topological_group_add"
immler@63081
   659
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
immler@63081
   660
  unfolding continuous_on_def by (auto intro: tendsto_minus)
hoelzl@62368
   661
immler@63081
   662
lemma tendsto_minus_cancel:
immler@63081
   663
  fixes a :: "'a::topological_group_add"
immler@63081
   664
  shows "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
immler@63081
   665
  by (drule tendsto_minus, simp)
immler@63081
   666
immler@63081
   667
lemma tendsto_minus_cancel_left:
immler@63081
   668
    "(f \<longlongrightarrow> - (y::_::topological_group_add)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
immler@63081
   669
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
immler@63081
   670
  by auto
immler@63081
   671
immler@63081
   672
lemma tendsto_diff [tendsto_intros]:
immler@63081
   673
  fixes a b :: "'a::topological_group_add"
immler@63081
   674
  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
immler@63081
   675
  using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
immler@63081
   676
immler@63081
   677
lemma continuous_diff [continuous_intros]:
immler@63081
   678
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
immler@63081
   679
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
immler@63081
   680
  unfolding continuous_def by (rule tendsto_diff)
immler@63081
   681
immler@63081
   682
lemma continuous_on_diff [continuous_intros]:
immler@63081
   683
  fixes f g :: "_ \<Rightarrow> 'b::topological_group_add"
immler@63081
   684
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
immler@63081
   685
  unfolding continuous_on_def by (auto intro: tendsto_diff)
immler@63081
   686
immler@63081
   687
lemma continuous_on_op_minus: "continuous_on (s::'a::topological_group_add set) (op - x)"
immler@63081
   688
  by (rule continuous_intros | simp)+
immler@63081
   689
immler@63081
   690
instance real_normed_vector < topological_ab_group_add
hoelzl@62368
   691
proof
hoelzl@62368
   692
  fix a b :: 'a show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
hoelzl@62368
   693
    unfolding tendsto_Zfun_iff add_diff_add
hoelzl@62368
   694
    using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
hoelzl@62368
   695
    by (intro Zfun_add)
hoelzl@62368
   696
       (auto simp add: tendsto_Zfun_iff[symmetric] nhds_prod[symmetric] intro!: tendsto_fst)
immler@63081
   697
  show "(uminus \<longlongrightarrow> - a) (nhds a)"
immler@63081
   698
    unfolding tendsto_Zfun_iff minus_diff_minus
immler@63081
   699
    using filterlim_ident[of "nhds a"]
immler@63081
   700
    by (intro Zfun_minus) (simp add: tendsto_Zfun_iff)
hoelzl@62368
   701
qed
hoelzl@62368
   702
hoelzl@50999
   703
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
hoelzl@50999
   704
wenzelm@60758
   705
subsubsection \<open>Linear operators and multiplication\<close>
huffman@44194
   706
lp15@61806
   707
lemma linear_times:
lp15@61806
   708
  fixes c::"'a::real_algebra" shows "linear (\<lambda>x. c * x)"
lp15@61806
   709
  by (auto simp: linearI distrib_left)
lp15@61806
   710
huffman@44282
   711
lemma (in bounded_linear) tendsto:
wenzelm@61973
   712
  "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
huffman@44081
   713
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   714
hoelzl@51478
   715
lemma (in bounded_linear) continuous:
hoelzl@51478
   716
  "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
hoelzl@51478
   717
  using tendsto[of g _ F] by (auto simp: continuous_def)
hoelzl@51478
   718
hoelzl@51478
   719
lemma (in bounded_linear) continuous_on:
hoelzl@51478
   720
  "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
hoelzl@51478
   721
  using tendsto[of g] by (auto simp: continuous_on_def)
hoelzl@51478
   722
huffman@44194
   723
lemma (in bounded_linear) tendsto_zero:
wenzelm@61973
   724
  "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
huffman@44194
   725
  by (drule tendsto, simp only: zero)
huffman@44194
   726
huffman@44282
   727
lemma (in bounded_bilinear) tendsto:
wenzelm@61973
   728
  "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
huffman@44081
   729
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   730
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   731
hoelzl@51478
   732
lemma (in bounded_bilinear) continuous:
hoelzl@51478
   733
  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
hoelzl@51478
   734
  using tendsto[of f _ F g] by (auto simp: continuous_def)
hoelzl@51478
   735
hoelzl@51478
   736
lemma (in bounded_bilinear) continuous_on:
hoelzl@51478
   737
  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
hoelzl@51478
   738
  using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
hoelzl@51478
   739
huffman@44194
   740
lemma (in bounded_bilinear) tendsto_zero:
wenzelm@61973
   741
  assumes f: "(f \<longlongrightarrow> 0) F"
wenzelm@61973
   742
  assumes g: "(g \<longlongrightarrow> 0) F"
wenzelm@61973
   743
  shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
huffman@44194
   744
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   745
huffman@44194
   746
lemma (in bounded_bilinear) tendsto_left_zero:
wenzelm@61973
   747
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) \<longlongrightarrow> 0) F"
huffman@44194
   748
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   749
huffman@44194
   750
lemma (in bounded_bilinear) tendsto_right_zero:
wenzelm@61973
   751
  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) \<longlongrightarrow> 0) F"
huffman@44194
   752
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   753
huffman@44282
   754
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   755
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   756
huffman@44282
   757
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   758
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   759
huffman@44282
   760
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   761
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   762
lp15@61806
   763
lemma tendsto_mult_left:
lp15@61806
   764
  fixes c::"'a::real_normed_algebra"
wenzelm@61973
   765
  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
lp15@61806
   766
by (rule tendsto_mult [OF tendsto_const])
lp15@61806
   767
lp15@61806
   768
lemma tendsto_mult_right:
lp15@61806
   769
  fixes c::"'a::real_normed_algebra"
wenzelm@61973
   770
  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
lp15@61806
   771
by (rule tendsto_mult [OF _ tendsto_const])
lp15@61806
   772
hoelzl@51478
   773
lemmas continuous_of_real [continuous_intros] =
hoelzl@51478
   774
  bounded_linear.continuous [OF bounded_linear_of_real]
hoelzl@51478
   775
hoelzl@51478
   776
lemmas continuous_scaleR [continuous_intros] =
hoelzl@51478
   777
  bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
hoelzl@51478
   778
hoelzl@51478
   779
lemmas continuous_mult [continuous_intros] =
hoelzl@51478
   780
  bounded_bilinear.continuous [OF bounded_bilinear_mult]
hoelzl@51478
   781
hoelzl@56371
   782
lemmas continuous_on_of_real [continuous_intros] =
hoelzl@51478
   783
  bounded_linear.continuous_on [OF bounded_linear_of_real]
hoelzl@51478
   784
hoelzl@56371
   785
lemmas continuous_on_scaleR [continuous_intros] =
hoelzl@51478
   786
  bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
hoelzl@51478
   787
hoelzl@56371
   788
lemmas continuous_on_mult [continuous_intros] =
hoelzl@51478
   789
  bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
hoelzl@51478
   790
huffman@44568
   791
lemmas tendsto_mult_zero =
huffman@44568
   792
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   793
huffman@44568
   794
lemmas tendsto_mult_left_zero =
huffman@44568
   795
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   796
huffman@44568
   797
lemmas tendsto_mult_right_zero =
huffman@44568
   798
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   799
huffman@44194
   800
lemma tendsto_power [tendsto_intros]:
huffman@44194
   801
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
wenzelm@61973
   802
  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
hoelzl@58729
   803
  by (induct n) (simp_all add: tendsto_mult)
huffman@44194
   804
hoelzl@51478
   805
lemma continuous_power [continuous_intros]:
hoelzl@51478
   806
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   807
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
hoelzl@51478
   808
  unfolding continuous_def by (rule tendsto_power)
hoelzl@51478
   809
hoelzl@56371
   810
lemma continuous_on_power [continuous_intros]:
hoelzl@51478
   811
  fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   812
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
hoelzl@51478
   813
  unfolding continuous_on_def by (auto intro: tendsto_power)
hoelzl@51478
   814
huffman@44194
   815
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   816
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
wenzelm@61973
   817
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F"
wenzelm@61973
   818
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   819
proof (cases "finite S")
huffman@44194
   820
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   821
    by (induct, simp, simp add: tendsto_mult)
hoelzl@58729
   822
qed simp
huffman@44194
   823
hoelzl@51478
   824
lemma continuous_setprod [continuous_intros]:
hoelzl@51478
   825
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   826
  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
   827
  unfolding continuous_def by (rule tendsto_setprod)
hoelzl@51478
   828
hoelzl@51478
   829
lemma continuous_on_setprod [continuous_intros]:
hoelzl@51478
   830
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   831
  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
   832
  unfolding continuous_on_def by (auto intro: tendsto_setprod)
hoelzl@51478
   833
eberlm@61531
   834
lemma tendsto_of_real_iff:
wenzelm@61973
   835
  "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
eberlm@61531
   836
  unfolding tendsto_iff by simp
eberlm@61531
   837
eberlm@61531
   838
lemma tendsto_add_const_iff:
wenzelm@61973
   839
  "((\<lambda>x. c + f x :: 'a :: real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
paulson@62087
   840
  using tendsto_add[OF tendsto_const[of c], of f d]
eberlm@61531
   841
        tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
eberlm@61531
   842
eberlm@61531
   843
wenzelm@60758
   844
subsubsection \<open>Inverse and division\<close>
huffman@31355
   845
huffman@31355
   846
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   847
  assumes f: "Zfun f F"
huffman@44195
   848
  assumes g: "Bfun g F"
huffman@44195
   849
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   850
proof -
huffman@31355
   851
  obtain K where K: "0 \<le> K"
huffman@31355
   852
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
lp15@61649
   853
    using nonneg_bounded by blast
huffman@31355
   854
  obtain B where B: "0 < B"
huffman@44195
   855
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   856
    using g by (rule BfunE)
huffman@44195
   857
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   858
  using norm_g proof eventually_elim
noschinl@46887
   859
    case (elim x)
huffman@31487
   860
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   861
      by (rule norm_le)
huffman@31487
   862
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   863
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   864
                mult_nonneg_nonneg K elim)
huffman@31487
   865
    also have "\<dots> = norm (f x) * (B * K)"
haftmann@57512
   866
      by (rule mult.assoc)
huffman@31487
   867
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   868
  qed
huffman@31487
   869
  with f show ?thesis
huffman@31487
   870
    by (rule Zfun_imp_Zfun)
huffman@31355
   871
qed
huffman@31355
   872
huffman@31355
   873
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   874
  assumes f: "Bfun f F"
huffman@44195
   875
  assumes g: "Zfun g F"
huffman@44195
   876
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   877
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   878
huffman@31355
   879
lemma Bfun_inverse_lemma:
huffman@31355
   880
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   881
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   882
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   883
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   884
  done
huffman@31355
   885
huffman@31355
   886
lemma Bfun_inverse:
huffman@31355
   887
  fixes a :: "'a::real_normed_div_algebra"
wenzelm@61973
   888
  assumes f: "(f \<longlongrightarrow> a) F"
huffman@31355
   889
  assumes a: "a \<noteq> 0"
huffman@44195
   890
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   891
proof -
huffman@31355
   892
  from a have "0 < norm a" by simp
huffman@31355
   893
  hence "\<exists>r>0. r < norm a" by (rule dense)
lp15@61649
   894
  then obtain r where r1: "0 < r" and r2: "r < norm a" by blast
huffman@44195
   895
  have "eventually (\<lambda>x. dist (f x) a < r) F"
lp15@61649
   896
    using tendstoD [OF f r1] by blast
huffman@44195
   897
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
   898
  proof eventually_elim
noschinl@46887
   899
    case (elim x)
huffman@31487
   900
    hence 1: "norm (f x - a) < r"
huffman@31355
   901
      by (simp add: dist_norm)
huffman@31487
   902
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   903
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   904
      by (rule nonzero_norm_inverse)
huffman@31355
   905
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   906
    proof (rule le_imp_inverse_le)
huffman@31355
   907
      show "0 < norm a - r" using r2 by simp
huffman@31355
   908
    next
huffman@31487
   909
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   910
        by (rule norm_triangle_ineq2)
huffman@31487
   911
      also have "\<dots> = norm (f x - a)"
huffman@31355
   912
        by (rule norm_minus_commute)
huffman@31355
   913
      also have "\<dots> < r" using 1 .
huffman@31487
   914
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   915
    qed
huffman@31487
   916
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   917
  qed
huffman@31355
   918
  thus ?thesis by (rule BfunI)
huffman@31355
   919
qed
huffman@31355
   920
huffman@31565
   921
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   922
  fixes a :: "'a::real_normed_div_algebra"
wenzelm@61973
   923
  assumes f: "(f \<longlongrightarrow> a) F"
huffman@31355
   924
  assumes a: "a \<noteq> 0"
wenzelm@61973
   925
  shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
huffman@31355
   926
proof -
huffman@31355
   927
  from a have "0 < norm a" by simp
huffman@44195
   928
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   929
    by (rule tendstoD)
huffman@44195
   930
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
lp15@61810
   931
    unfolding dist_norm by (auto elim!: eventually_mono)
huffman@44627
   932
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
   933
    - (inverse (f x) * (f x - a) * inverse a)) F"
lp15@61810
   934
    by (auto elim!: eventually_mono simp: inverse_diff_inverse)
huffman@44627
   935
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   936
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
   937
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
   938
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
   939
  ultimately show ?thesis
huffman@44627
   940
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
   941
qed
huffman@31355
   942
hoelzl@51478
   943
lemma continuous_inverse:
hoelzl@51478
   944
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   945
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   946
  shows "continuous F (\<lambda>x. inverse (f x))"
hoelzl@51478
   947
  using assms unfolding continuous_def by (rule tendsto_inverse)
hoelzl@51478
   948
hoelzl@51478
   949
lemma continuous_at_within_inverse[continuous_intros]:
hoelzl@51478
   950
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   951
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   952
  shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
hoelzl@51478
   953
  using assms unfolding continuous_within by (rule tendsto_inverse)
hoelzl@51478
   954
hoelzl@51478
   955
lemma isCont_inverse[continuous_intros, simp]:
hoelzl@51478
   956
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   957
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   958
  shows "isCont (\<lambda>x. inverse (f x)) a"
hoelzl@51478
   959
  using assms unfolding continuous_at by (rule tendsto_inverse)
hoelzl@51478
   960
hoelzl@56371
   961
lemma continuous_on_inverse[continuous_intros]:
hoelzl@51478
   962
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   963
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   964
  shows "continuous_on s (\<lambda>x. inverse (f x))"
lp15@61649
   965
  using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)
hoelzl@51478
   966
huffman@31565
   967
lemma tendsto_divide [tendsto_intros]:
huffman@31355
   968
  fixes a b :: "'a::real_normed_field"
wenzelm@61973
   969
  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; b \<noteq> 0\<rbrakk>
wenzelm@61973
   970
    \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
huffman@44282
   971
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
   972
hoelzl@51478
   973
lemma continuous_divide:
hoelzl@51478
   974
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   975
  assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   976
  shows "continuous F (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   977
  using assms unfolding continuous_def by (rule tendsto_divide)
hoelzl@51478
   978
hoelzl@51478
   979
lemma continuous_at_within_divide[continuous_intros]:
hoelzl@51478
   980
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   981
  assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
hoelzl@51478
   982
  shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   983
  using assms unfolding continuous_within by (rule tendsto_divide)
hoelzl@51478
   984
hoelzl@51478
   985
lemma isCont_divide[continuous_intros, simp]:
hoelzl@51478
   986
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   987
  assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
hoelzl@51478
   988
  shows "isCont (\<lambda>x. (f x) / g x) a"
hoelzl@51478
   989
  using assms unfolding continuous_at by (rule tendsto_divide)
hoelzl@51478
   990
hoelzl@56371
   991
lemma continuous_on_divide[continuous_intros]:
hoelzl@51478
   992
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   993
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
hoelzl@51478
   994
  shows "continuous_on s (\<lambda>x. (f x) / (g x))"
lp15@61649
   995
  using assms unfolding continuous_on_def by (blast intro: tendsto_divide)
hoelzl@51478
   996
huffman@44194
   997
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
   998
  fixes l :: "'a::real_normed_vector"
wenzelm@61973
   999
  shows "\<lbrakk>(f \<longlongrightarrow> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
huffman@44194
  1000
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
  1001
hoelzl@51478
  1002
lemma continuous_sgn:
hoelzl@51478
  1003
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
  1004
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
  1005
  shows "continuous F (\<lambda>x. sgn (f x))"
hoelzl@51478
  1006
  using assms unfolding continuous_def by (rule tendsto_sgn)
hoelzl@51478
  1007
hoelzl@51478
  1008
lemma continuous_at_within_sgn[continuous_intros]:
hoelzl@51478
  1009
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
  1010
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
  1011
  shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
hoelzl@51478
  1012
  using assms unfolding continuous_within by (rule tendsto_sgn)
hoelzl@51478
  1013
hoelzl@51478
  1014
lemma isCont_sgn[continuous_intros]:
hoelzl@51478
  1015
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
  1016
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
  1017
  shows "isCont (\<lambda>x. sgn (f x)) a"
hoelzl@51478
  1018
  using assms unfolding continuous_at by (rule tendsto_sgn)
hoelzl@51478
  1019
hoelzl@56371
  1020
lemma continuous_on_sgn[continuous_intros]:
hoelzl@51478
  1021
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
  1022
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
  1023
  shows "continuous_on s (\<lambda>x. sgn (f x))"
lp15@61649
  1024
  using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)
hoelzl@51478
  1025
hoelzl@50325
  1026
lemma filterlim_at_infinity:
wenzelm@61076
  1027
  fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"
hoelzl@50325
  1028
  assumes "0 \<le> c"
hoelzl@50325
  1029
  shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
hoelzl@50325
  1030
  unfolding filterlim_iff eventually_at_infinity
hoelzl@50325
  1031
proof safe
hoelzl@50325
  1032
  fix P :: "'a \<Rightarrow> bool" and b
hoelzl@50325
  1033
  assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
hoelzl@50325
  1034
    and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
hoelzl@50325
  1035
  have "max b (c + 1) > c" by auto
hoelzl@50325
  1036
  with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
hoelzl@50325
  1037
    by auto
hoelzl@50325
  1038
  then show "eventually (\<lambda>x. P (f x)) F"
hoelzl@50325
  1039
  proof eventually_elim
hoelzl@50325
  1040
    fix x assume "max b (c + 1) \<le> norm (f x)"
hoelzl@50325
  1041
    with P show "P (f x)" by auto
hoelzl@50325
  1042
  qed
hoelzl@50325
  1043
qed force
hoelzl@50325
  1044
eberlm@61531
  1045
lemma not_tendsto_and_filterlim_at_infinity:
eberlm@61531
  1046
  assumes "F \<noteq> bot"
paulson@62087
  1047
  assumes "(f \<longlongrightarrow> (c :: 'a :: real_normed_vector)) F"
eberlm@61531
  1048
  assumes "filterlim f at_infinity F"
eberlm@61531
  1049
  shows   False
eberlm@61531
  1050
proof -
paulson@62087
  1051
  from tendstoD[OF assms(2), of "1/2"]
eberlm@61531
  1052
    have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp
eberlm@61531
  1053
  moreover from filterlim_at_infinity[of "norm c" f F] assms(3)
eberlm@61531
  1054
    have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
eberlm@61531
  1055
  ultimately have "eventually (\<lambda>x. False) F"
eberlm@61531
  1056
  proof eventually_elim
eberlm@61531
  1057
    fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"
eberlm@61531
  1058
    note B
lp15@62379
  1059
    also have "norm (f x) = dist (f x) 0" by simp
eberlm@61531
  1060
    also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
lp15@62379
  1061
    finally show False using A by simp
eberlm@61531
  1062
  qed
eberlm@61531
  1063
  with assms show False by simp
eberlm@61531
  1064
qed
eberlm@61531
  1065
eberlm@61531
  1066
lemma filterlim_at_infinity_imp_not_convergent:
eberlm@61531
  1067
  assumes "filterlim f at_infinity sequentially"
eberlm@61531
  1068
  shows   "\<not>convergent f"
eberlm@61531
  1069
  by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
eberlm@61531
  1070
     (simp_all add: convergent_LIMSEQ_iff)
eberlm@61531
  1071
eberlm@61531
  1072
lemma filterlim_at_infinity_imp_eventually_ne:
eberlm@61531
  1073
  assumes "filterlim f at_infinity F"
eberlm@61531
  1074
  shows   "eventually (\<lambda>z. f z \<noteq> c) F"
eberlm@61531
  1075
proof -
eberlm@61531
  1076
  have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all
eberlm@61531
  1077
  with filterlim_at_infinity[OF order.refl, of f F] assms
eberlm@61531
  1078
    have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast
eberlm@61531
  1079
  thus ?thesis by eventually_elim auto
eberlm@61531
  1080
qed
eberlm@61531
  1081
paulson@62087
  1082
lemma tendsto_of_nat [tendsto_intros]:
eberlm@61531
  1083
  "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"
eberlm@61531
  1084
proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
wenzelm@63040
  1085
  fix r :: real
wenzelm@63040
  1086
  assume r: "r > 0"
wenzelm@63040
  1087
  define n where "n = nat \<lceil>r\<rceil>"
eberlm@61531
  1088
  from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith
eberlm@61531
  1089
  from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
eberlm@61531
  1090
    by eventually_elim (insert n, simp_all)
eberlm@61531
  1091
qed
eberlm@61531
  1092
eberlm@61531
  1093
wenzelm@60758
  1094
subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
hoelzl@50347
  1095
wenzelm@60758
  1096
text \<open>
hoelzl@50347
  1097
hoelzl@50347
  1098
This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
hoelzl@50347
  1099
@{term "at_right x"} and also @{term "at_right 0"}.
hoelzl@50347
  1100
wenzelm@60758
  1101
\<close>
hoelzl@50347
  1102
hoelzl@51471
  1103
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
hoelzl@50323
  1104
hoelzl@51641
  1105
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
hoelzl@60721
  1106
  by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
hoelzl@60721
  1107
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@50347
  1108
hoelzl@51641
  1109
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
hoelzl@60721
  1110
  by (rule filtermap_fun_inverse[where g=uminus])
hoelzl@60721
  1111
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@51641
  1112
hoelzl@51641
  1113
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
hoelzl@51641
  1114
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50347
  1115
hoelzl@50347
  1116
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
hoelzl@51641
  1117
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50323
  1118
hoelzl@50347
  1119
lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
hoelzl@50347
  1120
  using filtermap_at_right_shift[of "-a" 0] by simp
hoelzl@50347
  1121
hoelzl@50347
  1122
lemma filterlim_at_right_to_0:
hoelzl@50347
  1123
  "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
hoelzl@50347
  1124
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
hoelzl@50347
  1125
hoelzl@50347
  1126
lemma eventually_at_right_to_0:
hoelzl@50347
  1127
  "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
hoelzl@50347
  1128
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
hoelzl@50347
  1129
hoelzl@51641
  1130
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
hoelzl@51641
  1131
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
  1132
hoelzl@50347
  1133
lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
hoelzl@51641
  1134
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50323
  1135
hoelzl@50347
  1136
lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
hoelzl@51641
  1137
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
  1138
hoelzl@50347
  1139
lemma filterlim_at_left_to_right:
hoelzl@50347
  1140
  "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
hoelzl@50347
  1141
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
hoelzl@50347
  1142
hoelzl@50347
  1143
lemma eventually_at_left_to_right:
hoelzl@50347
  1144
  "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
hoelzl@50347
  1145
  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
hoelzl@50347
  1146
hoelzl@60721
  1147
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@60721
  1148
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@60721
  1149
  by (metis leI minus_less_iff order_less_asym)
hoelzl@60721
  1150
hoelzl@60721
  1151
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@60721
  1152
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@60721
  1153
  by (metis leI less_minus_iff order_less_asym)
hoelzl@60721
  1154
hoelzl@50346
  1155
lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
hoelzl@60721
  1156
  by (rule filtermap_fun_inverse[symmetric, of uminus])
hoelzl@60721
  1157
     (auto intro: filterlim_uminus_at_bot_at_top filterlim_uminus_at_top_at_bot)
hoelzl@50346
  1158
hoelzl@50346
  1159
lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
hoelzl@50346
  1160
  unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
hoelzl@50346
  1161
hoelzl@50346
  1162
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
hoelzl@50346
  1163
  unfolding filterlim_def at_top_mirror filtermap_filtermap ..
hoelzl@50346
  1164
hoelzl@50346
  1165
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
hoelzl@50346
  1166
  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
hoelzl@50346
  1167
hoelzl@50346
  1168
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
hoelzl@50346
  1169
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
hoelzl@50346
  1170
  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
hoelzl@50346
  1171
  by auto
hoelzl@50346
  1172
hoelzl@50346
  1173
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
hoelzl@50346
  1174
  unfolding filterlim_uminus_at_top by simp
hoelzl@50323
  1175
hoelzl@50347
  1176
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
hoelzl@51641
  1177
  unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
hoelzl@50347
  1178
proof safe
hoelzl@50347
  1179
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50347
  1180
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50347
  1181
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@51641
  1182
  then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
lp15@61810
  1183
    by (auto elim!: eventually_mono simp: inverse_eq_divide field_simps)
hoelzl@50347
  1184
qed
hoelzl@50347
  1185
hoelzl@50325
  1186
lemma tendsto_inverse_0:
wenzelm@61076
  1187
  fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
wenzelm@61973
  1188
  shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
hoelzl@50325
  1189
  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
hoelzl@50325
  1190
proof safe
hoelzl@50325
  1191
  fix r :: real assume "0 < r"
hoelzl@50325
  1192
  show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
hoelzl@50325
  1193
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
hoelzl@50325
  1194
    fix x :: 'a
wenzelm@60758
  1195
    from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
hoelzl@50325
  1196
    also assume *: "inverse (r / 2) \<le> norm x"
hoelzl@50325
  1197
    finally show "norm (inverse x) < r"
wenzelm@60758
  1198
      using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
hoelzl@50325
  1199
  qed
hoelzl@50325
  1200
qed
hoelzl@50325
  1201
eberlm@61552
  1202
lemma tendsto_add_filterlim_at_infinity:
wenzelm@61973
  1203
  assumes "(f \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
eberlm@61552
  1204
  assumes "filterlim g at_infinity F"
eberlm@61552
  1205
  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
eberlm@61552
  1206
proof (subst filterlim_at_infinity[OF order_refl], safe)
eberlm@61552
  1207
  fix r :: real assume r: "r > 0"
wenzelm@61973
  1208
  from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F" by (rule tendsto_norm)
eberlm@61552
  1209
  hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
paulson@62087
  1210
  moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all
eberlm@61552
  1211
  with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
eberlm@61552
  1212
    unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all
eberlm@61552
  1213
  ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
eberlm@61552
  1214
  proof eventually_elim
eberlm@61552
  1215
    fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
eberlm@61552
  1216
    from A B have "r \<le> norm (g x) - norm (f x)" by simp
eberlm@61552
  1217
    also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)
eberlm@61552
  1218
    finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)
eberlm@61552
  1219
  qed
eberlm@61552
  1220
qed
eberlm@61552
  1221
eberlm@61552
  1222
lemma tendsto_add_filterlim_at_infinity':
eberlm@61552
  1223
  assumes "filterlim f at_infinity F"
wenzelm@61973
  1224
  assumes "(g \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
eberlm@61552
  1225
  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
eberlm@61552
  1226
  by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
eberlm@61552
  1227
hoelzl@60721
  1228
lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
hoelzl@60721
  1229
  unfolding filterlim_at
hoelzl@60721
  1230
  by (auto simp: eventually_at_top_dense)
hoelzl@60721
  1231
     (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
hoelzl@60721
  1232
hoelzl@60721
  1233
lemma filterlim_inverse_at_top:
wenzelm@61973
  1234
  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@60721
  1235
  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
lp15@61810
  1236
     (simp add: filterlim_def eventually_filtermap eventually_mono at_within_def le_principal)
hoelzl@60721
  1237
hoelzl@60721
  1238
lemma filterlim_inverse_at_bot_neg:
hoelzl@60721
  1239
  "LIM x (at_left (0::real)). inverse x :> at_bot"
hoelzl@60721
  1240
  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
hoelzl@60721
  1241
hoelzl@60721
  1242
lemma filterlim_inverse_at_bot:
wenzelm@61973
  1243
  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@60721
  1244
  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
hoelzl@60721
  1245
  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
hoelzl@60721
  1246
hoelzl@50347
  1247
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
hoelzl@60721
  1248
  by (intro filtermap_fun_inverse[symmetric, where g=inverse])
hoelzl@60721
  1249
     (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)
hoelzl@50347
  1250
hoelzl@50347
  1251
lemma eventually_at_right_to_top:
hoelzl@50347
  1252
  "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
hoelzl@50347
  1253
  unfolding at_right_to_top eventually_filtermap ..
hoelzl@50347
  1254
hoelzl@50347
  1255
lemma filterlim_at_right_to_top:
hoelzl@50347
  1256
  "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
hoelzl@50347
  1257
  unfolding filterlim_def at_right_to_top filtermap_filtermap ..
hoelzl@50347
  1258
hoelzl@50347
  1259
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
hoelzl@50347
  1260
  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
hoelzl@50347
  1261
hoelzl@50347
  1262
lemma eventually_at_top_to_right:
hoelzl@50347
  1263
  "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
hoelzl@50347
  1264
  unfolding at_top_to_right eventually_filtermap ..
hoelzl@50347
  1265
hoelzl@50347
  1266
lemma filterlim_at_top_to_right:
hoelzl@50347
  1267
  "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
hoelzl@50347
  1268
  unfolding filterlim_def at_top_to_right filtermap_filtermap ..
hoelzl@50347
  1269
hoelzl@50325
  1270
lemma filterlim_inverse_at_infinity:
wenzelm@61076
  1271
  fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1272
  shows "filterlim inverse at_infinity (at (0::'a))"
hoelzl@50325
  1273
  unfolding filterlim_at_infinity[OF order_refl]
hoelzl@50325
  1274
proof safe
hoelzl@50325
  1275
  fix r :: real assume "0 < r"
hoelzl@50325
  1276
  then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
hoelzl@50325
  1277
    unfolding eventually_at norm_inverse
hoelzl@50325
  1278
    by (intro exI[of _ "inverse r"])
hoelzl@50325
  1279
       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
hoelzl@50325
  1280
qed
hoelzl@50325
  1281
hoelzl@50325
  1282
lemma filterlim_inverse_at_iff:
wenzelm@61076
  1283
  fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1284
  shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
hoelzl@50325
  1285
  unfolding filterlim_def filtermap_filtermap[symmetric]
hoelzl@50325
  1286
proof
hoelzl@50325
  1287
  assume "filtermap g F \<le> at_infinity"
hoelzl@50325
  1288
  then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
hoelzl@50325
  1289
    by (rule filtermap_mono)
hoelzl@50325
  1290
  also have "\<dots> \<le> at 0"
hoelzl@51641
  1291
    using tendsto_inverse_0[where 'a='b]
hoelzl@51641
  1292
    by (auto intro!: exI[of _ 1]
hoelzl@51641
  1293
             simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
hoelzl@50325
  1294
  finally show "filtermap inverse (filtermap g F) \<le> at 0" .
hoelzl@50325
  1295
next
hoelzl@50325
  1296
  assume "filtermap inverse (filtermap g F) \<le> at 0"
hoelzl@50325
  1297
  then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
hoelzl@50325
  1298
    by (rule filtermap_mono)
hoelzl@50325
  1299
  with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
hoelzl@50325
  1300
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
hoelzl@50325
  1301
qed
hoelzl@50325
  1302
eberlm@61531
  1303
lemma tendsto_mult_filterlim_at_infinity:
wenzelm@61973
  1304
  assumes "F \<noteq> bot" "(f \<longlongrightarrow> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
eberlm@61531
  1305
  assumes "filterlim g at_infinity F"
eberlm@61531
  1306
  shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"
eberlm@61531
  1307
proof -
wenzelm@61973
  1308
  have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
eberlm@61531
  1309
    by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
eberlm@61531
  1310
  hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
eberlm@61531
  1311
    unfolding filterlim_at using assms
eberlm@61531
  1312
    by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
eberlm@61531
  1313
  thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
eberlm@61531
  1314
qed
eberlm@61531
  1315
wenzelm@61973
  1316
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
hoelzl@51641
  1317
 by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
hoelzl@50419
  1318
eberlm@61531
  1319
lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
eberlm@61531
  1320
  by (rule filterlim_subseq) (auto simp: subseq_def)
eberlm@61531
  1321
eberlm@61531
  1322
lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"
eberlm@61531
  1323
  by (rule filterlim_subseq) (auto simp: subseq_def)
lp15@59613
  1324
lp15@59613
  1325
lemma at_to_infinity:
wenzelm@61076
  1326
  fixes x :: "'a :: {real_normed_field,field}"
lp15@59613
  1327
  shows "(at (0::'a)) = filtermap inverse at_infinity"
lp15@59613
  1328
proof (rule antisym)
wenzelm@61973
  1329
  have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
lp15@59613
  1330
    by (fact tendsto_inverse_0)
lp15@59613
  1331
  then show "filtermap inverse at_infinity \<le> at (0::'a)"
lp15@59613
  1332
    apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
lp15@59613
  1333
    apply (rule_tac x="1" in exI, auto)
lp15@59613
  1334
    done
lp15@59613
  1335
next
lp15@59613
  1336
  have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
lp15@59613
  1337
    using filterlim_inverse_at_infinity unfolding filterlim_def
lp15@59613
  1338
    by (rule filtermap_mono)
lp15@59613
  1339
  then show "at (0::'a) \<le> filtermap inverse at_infinity"
lp15@59613
  1340
    by (simp add: filtermap_ident filtermap_filtermap)
lp15@59613
  1341
qed
lp15@59613
  1342
lp15@59613
  1343
lemma lim_at_infinity_0:
haftmann@59867
  1344
  fixes l :: "'a :: {real_normed_field,field}"
wenzelm@61973
  1345
  shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f o inverse) \<longlongrightarrow> l) (at (0::'a))"
lp15@59613
  1346
by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
lp15@59613
  1347
lp15@59613
  1348
lemma lim_zero_infinity:
haftmann@59867
  1349
  fixes l :: "'a :: {real_normed_field,field}"
wenzelm@61973
  1350
  shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
lp15@59613
  1351
by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
lp15@59613
  1352
lp15@59613
  1353
wenzelm@60758
  1354
text \<open>
hoelzl@50324
  1355
hoelzl@50324
  1356
We only show rules for multiplication and addition when the functions are either against a real
hoelzl@50324
  1357
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
hoelzl@50324
  1358
wenzelm@60758
  1359
\<close>
hoelzl@50324
  1360
lp15@60141
  1361
lemma filterlim_tendsto_pos_mult_at_top:
wenzelm@61973
  1362
  assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c"
hoelzl@50324
  1363
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1364
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1365
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1366
proof safe
hoelzl@50324
  1367
  fix Z :: real assume "0 < Z"
wenzelm@60758
  1368
  from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
lp15@61810
  1369
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
nipkow@62390
  1370
             simp: dist_real_def abs_real_def split: if_split_asm)
hoelzl@50346
  1371
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
hoelzl@50324
  1372
    unfolding filterlim_at_top by auto
hoelzl@50346
  1373
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1374
  proof eventually_elim
hoelzl@50346
  1375
    fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
wenzelm@60758
  1376
    with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
hoelzl@50346
  1377
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
wenzelm@60758
  1378
    with \<open>0 < c\<close> show "Z \<le> f x * g x"
hoelzl@50324
  1379
       by simp
hoelzl@50324
  1380
  qed
hoelzl@50324
  1381
qed
hoelzl@50324
  1382
lp15@60141
  1383
lemma filterlim_at_top_mult_at_top:
hoelzl@50324
  1384
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1385
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1386
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1387
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1388
proof safe
hoelzl@50324
  1389
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1390
  from f have "eventually (\<lambda>x. 1 \<le> f x) F"
hoelzl@50324
  1391
    unfolding filterlim_at_top by auto
hoelzl@50346
  1392
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1393
    unfolding filterlim_at_top by auto
hoelzl@50346
  1394
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1395
  proof eventually_elim
hoelzl@50346
  1396
    fix x assume "1 \<le> f x" "Z \<le> g x"
wenzelm@60758
  1397
    with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
hoelzl@50346
  1398
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
  1399
    then show "Z \<le> f x * g x"
hoelzl@50324
  1400
       by simp
hoelzl@50324
  1401
  qed
hoelzl@50324
  1402
qed
hoelzl@50324
  1403
hoelzl@50419
  1404
lemma filterlim_tendsto_pos_mult_at_bot:
wenzelm@61973
  1405
  assumes "(f \<longlongrightarrow> c) F" "0 < (c::real)" "filterlim g at_bot F"
hoelzl@50419
  1406
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@50419
  1407
  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
hoelzl@50419
  1408
  unfolding filterlim_uminus_at_bot by simp
hoelzl@50419
  1409
hoelzl@60182
  1410
lemma filterlim_tendsto_neg_mult_at_bot:
wenzelm@61973
  1411
  assumes c: "(f \<longlongrightarrow> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
hoelzl@60182
  1412
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@60182
  1413
  using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
hoelzl@60182
  1414
  unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
hoelzl@60182
  1415
hoelzl@56330
  1416
lemma filterlim_pow_at_top:
hoelzl@56330
  1417
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1418
  assumes "0 < n" and f: "LIM x F. f x :> at_top"
hoelzl@56330
  1419
  shows "LIM x F. (f x)^n :: real :> at_top"
wenzelm@60758
  1420
using \<open>0 < n\<close> proof (induct n)
hoelzl@56330
  1421
  case (Suc n) with f show ?case
hoelzl@56330
  1422
    by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
hoelzl@56330
  1423
qed simp
hoelzl@56330
  1424
hoelzl@56330
  1425
lemma filterlim_pow_at_bot_even:
hoelzl@56330
  1426
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1427
  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
  1428
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
hoelzl@56330
  1429
hoelzl@56330
  1430
lemma filterlim_pow_at_bot_odd:
hoelzl@56330
  1431
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1432
  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
  1433
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
hoelzl@56330
  1434
lp15@60141
  1435
lemma filterlim_tendsto_add_at_top:
wenzelm@61973
  1436
  assumes f: "(f \<longlongrightarrow> c) F"
hoelzl@50324
  1437
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1438
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1439
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1440
proof safe
hoelzl@50324
  1441
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1442
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
lp15@61810
  1443
    by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
hoelzl@50346
  1444
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
hoelzl@50324
  1445
    unfolding filterlim_at_top by auto
hoelzl@50346
  1446
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1447
    by eventually_elim simp
hoelzl@50324
  1448
qed
hoelzl@50324
  1449
hoelzl@50347
  1450
lemma LIM_at_top_divide:
hoelzl@50347
  1451
  fixes f g :: "'a \<Rightarrow> real"
wenzelm@61973
  1452
  assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
wenzelm@61973
  1453
  assumes g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
hoelzl@50347
  1454
  shows "LIM x F. f x / g x :> at_top"
hoelzl@50347
  1455
  unfolding divide_inverse
hoelzl@50347
  1456
  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
hoelzl@50347
  1457
lp15@60141
  1458
lemma filterlim_at_top_add_at_top:
hoelzl@50324
  1459
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1460
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1461
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1462
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1463
proof safe
hoelzl@50324
  1464
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1465
  from f have "eventually (\<lambda>x. 0 \<le> f x) F"
hoelzl@50324
  1466
    unfolding filterlim_at_top by auto
hoelzl@50346
  1467
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1468
    unfolding filterlim_at_top by auto
hoelzl@50346
  1469
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1470
    by eventually_elim simp
hoelzl@50324
  1471
qed
hoelzl@50324
  1472
hoelzl@50331
  1473
lemma tendsto_divide_0:
wenzelm@61076
  1474
  fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
wenzelm@61973
  1475
  assumes f: "(f \<longlongrightarrow> c) F"
hoelzl@50331
  1476
  assumes g: "LIM x F. g x :> at_infinity"
wenzelm@61973
  1477
  shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
hoelzl@50331
  1478
  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
hoelzl@50331
  1479
hoelzl@50331
  1480
lemma linear_plus_1_le_power:
hoelzl@50331
  1481
  fixes x :: real
hoelzl@50331
  1482
  assumes x: "0 \<le> x"
hoelzl@50331
  1483
  shows "real n * x + 1 \<le> (x + 1) ^ n"
hoelzl@50331
  1484
proof (induct n)
hoelzl@50331
  1485
  case (Suc n)
hoelzl@50331
  1486
  have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
lp15@61609
  1487
    by (simp add: field_simps of_nat_Suc x)
hoelzl@50331
  1488
  also have "\<dots> \<le> (x + 1)^Suc n"
hoelzl@50331
  1489
    using Suc x by (simp add: mult_left_mono)
hoelzl@50331
  1490
  finally show ?case .
hoelzl@50331
  1491
qed simp
hoelzl@50331
  1492
hoelzl@50331
  1493
lemma filterlim_realpow_sequentially_gt1:
hoelzl@50331
  1494
  fixes x :: "'a :: real_normed_div_algebra"
hoelzl@50331
  1495
  assumes x[arith]: "1 < norm x"
hoelzl@50331
  1496
  shows "LIM n sequentially. x ^ n :> at_infinity"
hoelzl@50331
  1497
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
hoelzl@50331
  1498
  fix y :: real assume "0 < y"
hoelzl@50331
  1499
  have "0 < norm x - 1" by simp
hoelzl@50331
  1500
  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
hoelzl@50331
  1501
  also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
hoelzl@50331
  1502
  also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
hoelzl@50331
  1503
  also have "\<dots> = norm x ^ N" by simp
hoelzl@50331
  1504
  finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
hoelzl@50331
  1505
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
hoelzl@50331
  1506
  then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
hoelzl@50331
  1507
    unfolding eventually_sequentially
hoelzl@50331
  1508
    by (auto simp: norm_power)
hoelzl@50331
  1509
qed simp
hoelzl@50331
  1510
hoelzl@51471
  1511
wenzelm@60758
  1512
subsection \<open>Limits of Sequences\<close>
hoelzl@51526
  1513
hoelzl@62368
  1514
lemma [trans]: "X = Y \<Longrightarrow> Y \<longlonglongrightarrow> z \<Longrightarrow> X \<longlonglongrightarrow> z"
hoelzl@51526
  1515
  by simp
hoelzl@51526
  1516
hoelzl@51526
  1517
lemma LIMSEQ_iff:
hoelzl@51526
  1518
  fixes L :: "'a::real_normed_vector"
wenzelm@61969
  1519
  shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
lp15@60017
  1520
unfolding lim_sequentially dist_norm ..
hoelzl@51526
  1521
hoelzl@51526
  1522
lemma LIMSEQ_I:
hoelzl@51526
  1523
  fixes L :: "'a::real_normed_vector"
wenzelm@61969
  1524
  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
hoelzl@51526
  1525
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1526
hoelzl@51526
  1527
lemma LIMSEQ_D:
hoelzl@51526
  1528
  fixes L :: "'a::real_normed_vector"
wenzelm@61969
  1529
  shows "\<lbrakk>X \<longlonglongrightarrow> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
hoelzl@51526
  1530
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1531
wenzelm@61969
  1532
lemma LIMSEQ_linear: "\<lbrakk> X \<longlonglongrightarrow> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
hoelzl@51526
  1533
  unfolding tendsto_def eventually_sequentially
haftmann@57512
  1534
  by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
hoelzl@51526
  1535
hoelzl@51526
  1536
lemma Bseq_inverse_lemma:
hoelzl@51526
  1537
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51526
  1538
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
hoelzl@51526
  1539
apply (subst nonzero_norm_inverse, clarsimp)
hoelzl@51526
  1540
apply (erule (1) le_imp_inverse_le)
hoelzl@51526
  1541
done
hoelzl@51526
  1542
hoelzl@51526
  1543
lemma Bseq_inverse:
hoelzl@51526
  1544
  fixes a :: "'a::real_normed_div_algebra"
wenzelm@61969
  1545
  shows "\<lbrakk>X \<longlonglongrightarrow> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
hoelzl@51526
  1546
  by (rule Bfun_inverse)
hoelzl@51526
  1547
wenzelm@60758
  1548
text\<open>Transformation of limit.\<close>
lp15@60141
  1549
lp15@60141
  1550
lemma Lim_transform:
lp15@60141
  1551
  fixes a b :: "'a::real_normed_vector"
wenzelm@61973
  1552
  shows "\<lbrakk>(g \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> a) F"
lp15@60141
  1553
  using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
lp15@60141
  1554
lp15@60141
  1555
lemma Lim_transform2:
lp15@60141
  1556
  fixes a b :: "'a::real_normed_vector"
wenzelm@61973
  1557
  shows "\<lbrakk>(f \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> a) F"
lp15@60141
  1558
  by (erule Lim_transform) (simp add: tendsto_minus_cancel)
lp15@60141
  1559
lp15@62379
  1560
proposition Lim_transform_eq:
lp15@62379
  1561
  fixes a :: "'a::real_normed_vector"
lp15@62379
  1562
  shows "((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F \<longleftrightarrow> (g \<longlongrightarrow> a) F"
lp15@62379
  1563
using Lim_transform Lim_transform2 by blast
lp15@62379
  1564
lp15@60141
  1565
lemma Lim_transform_eventually:
wenzelm@61973
  1566
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> (g \<longlongrightarrow> l) net"
lp15@60141
  1567
  apply (rule topological_tendstoI)
lp15@60141
  1568
  apply (drule (2) topological_tendstoD)
lp15@60141
  1569
  apply (erule (1) eventually_elim2, simp)
lp15@60141
  1570
  done
lp15@60141
  1571
lp15@60141
  1572
lemma Lim_transform_within:
paulson@62087
  1573
  assumes "(f \<longlongrightarrow> l) (at x within S)"
paulson@62087
  1574
    and "0 < d"
paulson@62087
  1575
    and "\<And>x'. \<lbrakk>x'\<in>S; 0 < dist x' x; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
wenzelm@61973
  1576
  shows "(g \<longlongrightarrow> l) (at x within S)"
lp15@60141
  1577
proof (rule Lim_transform_eventually)
lp15@60141
  1578
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
paulson@62087
  1579
    using assms by (auto simp: eventually_at)
wenzelm@61973
  1580
  show "(f \<longlongrightarrow> l) (at x within S)" by fact
lp15@60141
  1581
qed
lp15@60141
  1582
wenzelm@60758
  1583
text\<open>Common case assuming being away from some crucial point like 0.\<close>
hoelzl@51526
  1584
lp15@60141
  1585
lemma Lim_transform_away_within:
lp15@60141
  1586
  fixes a b :: "'a::t1_space"
lp15@60141
  1587
  assumes "a \<noteq> b"
lp15@60141
  1588
    and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
wenzelm@61973
  1589
    and "(f \<longlongrightarrow> l) (at a within S)"
wenzelm@61973
  1590
  shows "(g \<longlongrightarrow> l) (at a within S)"
lp15@60141
  1591
proof (rule Lim_transform_eventually)
wenzelm@61973
  1592
  show "(f \<longlongrightarrow> l) (at a within S)" by fact
lp15@60141
  1593
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
lp15@60141
  1594
    unfolding eventually_at_topological
lp15@60141
  1595
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
lp15@60141
  1596
qed
lp15@60141
  1597
lp15@60141
  1598
lemma Lim_transform_away_at:
lp15@60141
  1599
  fixes a b :: "'a::t1_space"
lp15@60141
  1600
  assumes ab: "a\<noteq>b"
lp15@60141
  1601
    and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
wenzelm@61973
  1602
    and fl: "(f \<longlongrightarrow> l) (at a)"
wenzelm@61973
  1603
  shows "(g \<longlongrightarrow> l) (at a)"
lp15@60141
  1604
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
lp15@60141
  1605
wenzelm@60758
  1606
text\<open>Alternatively, within an open set.\<close>
hoelzl@51526
  1607
lp15@60141
  1608
lemma Lim_transform_within_open:
paulson@62087
  1609
  assumes "(f \<longlongrightarrow> l) (at a within T)"
paulson@62087
  1610
    and "open s" and "a \<in> s"
paulson@62087
  1611
    and "\<And>x. \<lbrakk>x\<in>s; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
paulson@62087
  1612
  shows "(g \<longlongrightarrow> l) (at a within T)"
lp15@60141
  1613
proof (rule Lim_transform_eventually)
paulson@62087
  1614
  show "eventually (\<lambda>x. f x = g x) (at a within T)"
lp15@60141
  1615
    unfolding eventually_at_topological
paulson@62087
  1616
    using assms by auto
paulson@62087
  1617
  show "(f \<longlongrightarrow> l) (at a within T)" by fact
lp15@60141
  1618
qed
lp15@60141
  1619
wenzelm@60758
  1620
text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
lp15@60141
  1621
lp15@60141
  1622
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
lp15@60141
  1623
lp15@60141
  1624
lemma Lim_cong_within(*[cong add]*):
lp15@60141
  1625
  assumes "a = b"
lp15@60141
  1626
    and "x = y"
lp15@60141
  1627
    and "S = T"
lp15@60141
  1628
    and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
wenzelm@61973
  1629
  shows "(f \<longlongrightarrow> x) (at a within S) \<longleftrightarrow> (g \<longlongrightarrow> y) (at b within T)"
lp15@60141
  1630
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1631
  using assms by simp
lp15@60141
  1632
lp15@60141
  1633
lemma Lim_cong_at(*[cong add]*):
lp15@60141
  1634
  assumes "a = b" "x = y"
lp15@60141
  1635
    and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
wenzelm@61973
  1636
  shows "((\<lambda>x. f x) \<longlongrightarrow> x) (at a) \<longleftrightarrow> ((g \<longlongrightarrow> y) (at a))"
lp15@60141
  1637
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1638
  using assms by simp
wenzelm@60758
  1639
text\<open>An unbounded sequence's inverse tends to 0\<close>
hoelzl@51526
  1640
hoelzl@51526
  1641
lemma LIMSEQ_inverse_zero:
wenzelm@61969
  1642
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
hoelzl@51526
  1643
  apply (rule filterlim_compose[OF tendsto_inverse_0])
hoelzl@51526
  1644
  apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
hoelzl@51526
  1645
  apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
hoelzl@51526
  1646
  done
hoelzl@51526
  1647
wenzelm@60758
  1648
text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
hoelzl@51526
  1649
wenzelm@61969
  1650
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) \<longlonglongrightarrow> 0"
hoelzl@51526
  1651
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
hoelzl@51526
  1652
            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
hoelzl@51526
  1653
wenzelm@60758
  1654
text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
wenzelm@60758
  1655
infinity is now easily proved\<close>
hoelzl@51526
  1656
hoelzl@51526
  1657
lemma LIMSEQ_inverse_real_of_nat_add:
wenzelm@61969
  1658
     "(%n. r + inverse(real(Suc n))) \<longlonglongrightarrow> r"
hoelzl@51526
  1659
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
hoelzl@51526
  1660
hoelzl@51526
  1661
lemma LIMSEQ_inverse_real_of_nat_add_minus:
wenzelm@61969
  1662
     "(%n. r + -inverse(real(Suc n))) \<longlonglongrightarrow> r"
hoelzl@51526
  1663
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
hoelzl@51526
  1664
  by auto
hoelzl@51526
  1665
hoelzl@51526
  1666
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
wenzelm@61969
  1667
     "(%n. r*( 1 + -inverse(real(Suc n)))) \<longlonglongrightarrow> r"
hoelzl@51526
  1668
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
hoelzl@51526
  1669
  by auto
hoelzl@51526
  1670
wenzelm@61973
  1671
lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
eberlm@61524
  1672
  using lim_1_over_n by (simp add: inverse_eq_divide)
eberlm@61524
  1673
wenzelm@61969
  1674
lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
eberlm@61524
  1675
proof (rule Lim_transform_eventually)
eberlm@61524
  1676
  show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
eberlm@61524
  1677
    using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
wenzelm@61969
  1678
  have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
eberlm@61524
  1679
    by (intro tendsto_add tendsto_const lim_inverse_n)
wenzelm@61969
  1680
  thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1" by simp
eberlm@61524
  1681
qed
eberlm@61524
  1682
wenzelm@61969
  1683
lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
eberlm@61524
  1684
proof (rule Lim_transform_eventually)
paulson@62087
  1685
  show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) =
eberlm@61524
  1686
                        of_nat n / of_nat (Suc n)) sequentially"
paulson@62087
  1687
    using eventually_gt_at_top[of "0::nat"]
eberlm@61524
  1688
    by eventually_elim (simp add: field_simps del: of_nat_Suc)
wenzelm@61969
  1689
  have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
eberlm@61524
  1690
    by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
wenzelm@61969
  1691
  thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1" by simp
eberlm@61524
  1692
qed
eberlm@61524
  1693
wenzelm@60758
  1694
subsection \<open>Convergence on sequences\<close>
hoelzl@51526
  1695
eberlm@61531
  1696
lemma convergent_cong:
eberlm@61531
  1697
  assumes "eventually (\<lambda>x. f x = g x) sequentially"
eberlm@61531
  1698
  shows   "convergent f \<longleftrightarrow> convergent g"
eberlm@61531
  1699
  unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)
eberlm@61531
  1700
eberlm@61531
  1701
lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
eberlm@61531
  1702
  by (auto simp: convergent_def LIMSEQ_Suc_iff)
eberlm@61531
  1703
eberlm@61531
  1704
lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
eberlm@61531
  1705
proof (induction m arbitrary: f)
eberlm@61531
  1706
  case (Suc m)
eberlm@61531
  1707
  have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp
eberlm@61531
  1708
  also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)
eberlm@61531
  1709
  also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)
eberlm@61531
  1710
  finally show ?case .
eberlm@61531
  1711
qed simp_all
eberlm@61531
  1712
hoelzl@51526
  1713
lemma convergent_add:
hoelzl@51526
  1714
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1715
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1716
  assumes "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1717
  shows "convergent (\<lambda>n. X n + Y n)"
lp15@61649
  1718
  using assms unfolding convergent_def by (blast intro: tendsto_add)
hoelzl@51526
  1719
hoelzl@51526
  1720
lemma convergent_setsum:
hoelzl@51526
  1721
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1722
  assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
hoelzl@51526
  1723
  shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
hoelzl@51526
  1724
proof (cases "finite A")
hoelzl@51526
  1725
  case True from this and assms show ?thesis
hoelzl@51526
  1726
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
hoelzl@51526
  1727
qed (simp add: convergent_const)
hoelzl@51526
  1728
hoelzl@51526
  1729
lemma (in bounded_linear) convergent:
hoelzl@51526
  1730
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1731
  shows "convergent (\<lambda>n. f (X n))"
lp15@61649
  1732
  using assms unfolding convergent_def by (blast intro: tendsto)
hoelzl@51526
  1733
hoelzl@51526
  1734
lemma (in bounded_bilinear) convergent:
hoelzl@51526
  1735
  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1736
  shows "convergent (\<lambda>n. X n ** Y n)"
lp15@61649
  1737
  using assms unfolding convergent_def by (blast intro: tendsto)
hoelzl@51526
  1738
hoelzl@51526
  1739
lemma convergent_minus_iff:
hoelzl@51526
  1740
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1741
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
hoelzl@51526
  1742
apply (simp add: convergent_def)
hoelzl@51526
  1743
apply (auto dest: tendsto_minus)
hoelzl@51526
  1744
apply (drule tendsto_minus, auto)
hoelzl@51526
  1745
done
hoelzl@51526
  1746
eberlm@61531
  1747
lemma convergent_diff:
eberlm@61531
  1748
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
eberlm@61531
  1749
  assumes "convergent (\<lambda>n. X n)"
eberlm@61531
  1750
  assumes "convergent (\<lambda>n. Y n)"
eberlm@61531
  1751
  shows "convergent (\<lambda>n. X n - Y n)"
lp15@61649
  1752
  using assms unfolding convergent_def by (blast intro: tendsto_diff)
eberlm@61531
  1753
eberlm@61531
  1754
lemma convergent_norm:
eberlm@61531
  1755
  assumes "convergent f"
eberlm@61531
  1756
  shows   "convergent (\<lambda>n. norm (f n))"
eberlm@61531
  1757
proof -
wenzelm@61969
  1758
  from assms have "f \<longlonglongrightarrow> lim f" by (simp add: convergent_LIMSEQ_iff)
wenzelm@61969
  1759
  hence "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)" by (rule tendsto_norm)
eberlm@61531
  1760
  thus ?thesis by (auto simp: convergent_def)
eberlm@61531
  1761
qed
eberlm@61531
  1762
paulson@62087
  1763
lemma convergent_of_real:
eberlm@61531
  1764
  "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"
eberlm@61531
  1765
  unfolding convergent_def by (blast intro!: tendsto_of_real)
eberlm@61531
  1766
paulson@62087
  1767
lemma convergent_add_const_iff:
eberlm@61531
  1768
  "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1769
proof
eberlm@61531
  1770
  assume "convergent (\<lambda>n. c + f n)"
eberlm@61531
  1771
  from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp
eberlm@61531
  1772
next
eberlm@61531
  1773
  assume "convergent f"
eberlm@61531
  1774
  from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp
eberlm@61531
  1775
qed
eberlm@61531
  1776
paulson@62087
  1777
lemma convergent_add_const_right_iff:
eberlm@61531
  1778
  "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1779
  using convergent_add_const_iff[of c f] by (simp add: add_ac)
eberlm@61531
  1780
paulson@62087
  1781
lemma convergent_diff_const_right_iff:
eberlm@61531
  1782
  "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
eberlm@61531
  1783
  using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)
eberlm@61531
  1784
eberlm@61531
  1785
lemma convergent_mult:
eberlm@61531
  1786
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
eberlm@61531
  1787
  assumes "convergent (\<lambda>n. X n)"
eberlm@61531
  1788
  assumes "convergent (\<lambda>n. Y n)"
eberlm@61531
  1789
  shows "convergent (\<lambda>n. X n * Y n)"
lp15@61649
  1790
  using assms unfolding convergent_def by (blast intro: tendsto_mult)
eberlm@61531
  1791
eberlm@61531
  1792
lemma convergent_mult_const_iff:
eberlm@61531
  1793
  assumes "c \<noteq> 0"
eberlm@61531
  1794
  shows   "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"
eberlm@61531
  1795
proof
eberlm@61531
  1796
  assume "convergent (\<lambda>n. c * f n)"
paulson@62087
  1797
  from assms convergent_mult[OF this convergent_const[of "inverse c"]]
eberlm@61531
  1798
    show "convergent f" by (simp add: field_simps)
eberlm@61531
  1799
next
eberlm@61531
  1800
  assume "convergent f"
eberlm@61531
  1801
  from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp
eberlm@61531
  1802
qed
eberlm@61531
  1803
eberlm@61531
  1804
lemma convergent_mult_const_right_iff:
eberlm@61531
  1805
  assumes "c \<noteq> 0"
eberlm@61531
  1806
  shows   "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"
eberlm@61531
  1807
  using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
eberlm@61531
  1808
eberlm@61531
  1809
lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
eberlm@61531
  1810
  by (simp add: Cauchy_Bseq convergent_Cauchy)
eberlm@61531
  1811
hoelzl@51526
  1812
wenzelm@60758
  1813
text \<open>A monotone sequence converges to its least upper bound.\<close>
hoelzl@51526
  1814
hoelzl@54263
  1815
lemma LIMSEQ_incseq_SUP:
hoelzl@54263
  1816
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1817
  assumes u: "bdd_above (range X)"
hoelzl@54263
  1818
  assumes X: "incseq X"
wenzelm@61969
  1819
  shows "X \<longlonglongrightarrow> (SUP i. X i)"
hoelzl@54263
  1820
  by (rule order_tendstoI)
hoelzl@54263
  1821
     (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
hoelzl@51526
  1822
hoelzl@54263
  1823
lemma LIMSEQ_decseq_INF:
hoelzl@54263
  1824
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1825
  assumes u: "bdd_below (range X)"
hoelzl@54263
  1826
  assumes X: "decseq X"
wenzelm@61969
  1827
  shows "X \<longlonglongrightarrow> (INF i. X i)"
hoelzl@54263
  1828
  by (rule order_tendstoI)
hoelzl@54263
  1829
     (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
hoelzl@51526
  1830
wenzelm@60758
  1831
text\<open>Main monotonicity theorem\<close>
hoelzl@51526
  1832
hoelzl@51526
  1833
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1834
  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
hoelzl@54263
  1835
hoelzl@54263
  1836
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
  1837
  by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
hoelzl@51526
  1838
eberlm@61531
  1839
lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
eberlm@61531
  1840
  using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
eberlm@61531
  1841
eberlm@61531
  1842
lemma Bseq_monoseq_convergent'_inc:
eberlm@61531
  1843
  "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
  1844
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
eberlm@61531
  1845
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
eberlm@61531
  1846
eberlm@61531
  1847
lemma Bseq_monoseq_convergent'_dec:
eberlm@61531
  1848
  "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
  1849
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
eberlm@61531
  1850
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
eberlm@61531
  1851
hoelzl@51526
  1852
lemma Cauchy_iff:
hoelzl@51526
  1853
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1854
  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
  1855
  unfolding Cauchy_def dist_norm ..
hoelzl@51526
  1856
hoelzl@51526
  1857
lemma CauchyI:
hoelzl@51526
  1858
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1859
  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
  1860
by (simp add: Cauchy_iff)
hoelzl@51526
  1861
hoelzl@51526
  1862
lemma CauchyD:
hoelzl@51526
  1863
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1864
  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
  1865
by (simp add: Cauchy_iff)
hoelzl@51526
  1866
hoelzl@51526
  1867
lemma incseq_convergent:
hoelzl@51526
  1868
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1869
  assumes "incseq X" and "\<forall>i. X i \<le> B"
wenzelm@61969
  1870
  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
hoelzl@51526
  1871
proof atomize_elim
wenzelm@60758
  1872
  from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
wenzelm@61969
  1873
  obtain L where "X \<longlonglongrightarrow> L"
hoelzl@51526
  1874
    by (auto simp: convergent_def monoseq_def incseq_def)
wenzelm@61969
  1875
  with \<open>incseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. X i \<le> L)"
hoelzl@51526
  1876
    by (auto intro!: exI[of _ L] incseq_le)
hoelzl@51526
  1877
qed
hoelzl@51526
  1878
hoelzl@51526
  1879
lemma decseq_convergent:
hoelzl@51526
  1880
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1881
  assumes "decseq X" and "\<forall>i. B \<le> X i"
wenzelm@61969
  1882
  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
hoelzl@51526
  1883
proof atomize_elim
wenzelm@60758
  1884
  from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
wenzelm@61969
  1885
  obtain L where "X \<longlonglongrightarrow> L"
hoelzl@51526
  1886
    by (auto simp: convergent_def monoseq_def decseq_def)
wenzelm@61969
  1887
  with \<open>decseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. L \<le> X i)"
hoelzl@51526
  1888
    by (auto intro!: exI[of _ L] decseq_le)
hoelzl@51526
  1889
qed
hoelzl@51526
  1890
wenzelm@60758
  1891
subsubsection \<open>Cauchy Sequences are Bounded\<close>
hoelzl@51526
  1892
wenzelm@60758
  1893
text\<open>A Cauchy sequence is bounded -- this is the standard
wenzelm@60758
  1894
  proof mechanization rather than the nonstandard proof\<close>
hoelzl@51526
  1895
hoelzl@51526
  1896
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
hoelzl@51526
  1897
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
hoelzl@51526
  1898
apply (clarify, drule spec, drule (1) mp)
hoelzl@51526
  1899
apply (simp only: norm_minus_commute)
hoelzl@51526
  1900
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
hoelzl@51526
  1901
apply simp
hoelzl@51526
  1902
done
hoelzl@51526
  1903
wenzelm@60758
  1904
subsection \<open>Power Sequences\<close>
hoelzl@51526
  1905
wenzelm@60758
  1906
text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
hoelzl@51526
  1907
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
wenzelm@60758
  1908
  also fact that bounded and monotonic sequence converges.\<close>
hoelzl@51526
  1909
hoelzl@51526
  1910
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
hoelzl@51526
  1911
apply (simp add: Bseq_def)
hoelzl@51526
  1912
apply (rule_tac x = 1 in exI)
hoelzl@51526
  1913
apply (simp add: power_abs)
hoelzl@51526
  1914
apply (auto dest: power_mono)
hoelzl@51526
  1915
done
hoelzl@51526
  1916
hoelzl@51526
  1917
lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
hoelzl@51526
  1918
apply (clarify intro!: mono_SucI2)
hoelzl@51526
  1919
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
hoelzl@51526
  1920
done
hoelzl@51526
  1921
hoelzl@51526
  1922
lemma convergent_realpow:
hoelzl@51526
  1923
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
hoelzl@51526
  1924
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
hoelzl@51526
  1925
wenzelm@61969
  1926
lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
hoelzl@51526
  1927
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1928
hoelzl@51526
  1929
lemma LIMSEQ_realpow_zero:
wenzelm@61969
  1930
  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
hoelzl@51526
  1931
proof cases
hoelzl@51526
  1932
  assume "0 \<le> x" and "x \<noteq> 0"
hoelzl@51526
  1933
  hence x0: "0 < x" by simp
hoelzl@51526
  1934
  assume x1: "x < 1"
hoelzl@51526
  1935
  from x0 x1 have "1 < inverse x"
hoelzl@51526
  1936
    by (rule one_less_inverse)
wenzelm@61969
  1937
  hence "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
hoelzl@51526
  1938
    by (rule LIMSEQ_inverse_realpow_zero)
hoelzl@51526
  1939
  thus ?thesis by (simp add: power_inverse)
hoelzl@58729
  1940
qed (rule LIMSEQ_imp_Suc, simp)
hoelzl@51526
  1941
hoelzl@51526
  1942
lemma LIMSEQ_power_zero:
hoelzl@51526
  1943
  fixes x :: "'a::{real_normed_algebra_1}"
wenzelm@61969
  1944
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
hoelzl@51526
  1945
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
hoelzl@51526
  1946
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
hoelzl@51526
  1947
apply (simp add: power_abs norm_power_ineq)
hoelzl@51526
  1948
done
hoelzl@51526
  1949
wenzelm@61969
  1950
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
hoelzl@51526
  1951
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1952
wenzelm@60758
  1953
text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
hoelzl@51526
  1954
wenzelm@61969
  1955
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
hoelzl@51526
  1956
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
hoelzl@51526
  1957
wenzelm@61969
  1958
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) \<longlonglongrightarrow> 0"
hoelzl@51526
  1959
  by (rule LIMSEQ_power_zero) simp
hoelzl@51526
  1960
hoelzl@51526
  1961
wenzelm@60758
  1962
subsection \<open>Limits of Functions\<close>
hoelzl@51526
  1963
hoelzl@51526
  1964
lemma LIM_eq:
hoelzl@51526
  1965
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
wenzelm@61976
  1966
  shows "f \<midarrow>a\<rightarrow> L =
hoelzl@51526
  1967
     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
hoelzl@51526
  1968
by (simp add: LIM_def dist_norm)
hoelzl@51526
  1969
hoelzl@51526
  1970
lemma LIM_I:
hoelzl@51526
  1971
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1972
  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
wenzelm@61976
  1973
      ==> f \<midarrow>a\<rightarrow> L"
hoelzl@51526
  1974
by (simp add: LIM_eq)
hoelzl@51526
  1975
hoelzl@51526
  1976
lemma LIM_D:
hoelzl@51526
  1977
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
wenzelm@61976
  1978
  shows "[| f \<midarrow>a\<rightarrow> L; 0<r |]
hoelzl@51526
  1979
      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
hoelzl@51526
  1980
by (simp add: LIM_eq)
hoelzl@51526
  1981
hoelzl@51526
  1982
lemma LIM_offset:
hoelzl@51526
  1983
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  1984
  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
hoelzl@51641
  1985
  unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
hoelzl@51526
  1986
hoelzl@51526
  1987
lemma LIM_offset_zero:
hoelzl@51526
  1988
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  1989
  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
haftmann@57512
  1990
by (drule_tac k="a" in LIM_offset, simp add: add.commute)
hoelzl@51526
  1991
hoelzl@51526
  1992
lemma LIM_offset_zero_cancel:
hoelzl@51526
  1993
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  1994
  shows "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
hoelzl@51526
  1995
by (drule_tac k="- a" in LIM_offset, simp)
hoelzl@51526
  1996
hoelzl@51642
  1997
lemma LIM_offset_zero_iff:
hoelzl@51642
  1998
  fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
wenzelm@61976
  1999
  shows  "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
hoelzl@51642
  2000
  using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
hoelzl@51642
  2001
hoelzl@51526
  2002
lemma LIM_zero:
hoelzl@51526
  2003
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@61973
  2004
  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
hoelzl@51526
  2005
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  2006
hoelzl@51526
  2007
lemma LIM_zero_cancel:
hoelzl@51526
  2008
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@61973
  2009
  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
hoelzl@51526
  2010
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  2011
hoelzl@51526
  2012
lemma LIM_zero_iff:
hoelzl@51526
  2013
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
wenzelm@61973
  2014
  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
hoelzl@51526
  2015
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  2016
hoelzl@51526
  2017
lemma LIM_imp_LIM:
hoelzl@51526
  2018
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2019
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
wenzelm@61976
  2020
  assumes f: "f \<midarrow>a\<rightarrow> l"
hoelzl@51526
  2021
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
wenzelm@61976
  2022
  shows "g \<midarrow>a\<rightarrow> m"
hoelzl@51526
  2023
  by (rule metric_LIM_imp_LIM [OF f],
hoelzl@51526
  2024
    simp add: dist_norm le)
hoelzl@51526
  2025
hoelzl@51526
  2026
lemma LIM_equal2:
hoelzl@51526
  2027
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  2028
  assumes 1: "0 < R"
hoelzl@51526
  2029
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
wenzelm@61976
  2030
  shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l"
hoelzl@51526
  2031
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
hoelzl@51526
  2032
hoelzl@51526
  2033
lemma LIM_compose2:
hoelzl@51526
  2034
  fixes a :: "'a::real_normed_vector"
wenzelm@61976
  2035
  assumes f: "f \<midarrow>a\<rightarrow> b"
wenzelm@61976
  2036
  assumes g: "g \<midarrow>b\<rightarrow> c"
hoelzl@51526
  2037
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
wenzelm@61976
  2038
  shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
hoelzl@51526
  2039
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
hoelzl@51526
  2040
hoelzl@51526
  2041
lemma real_LIM_sandwich_zero:
hoelzl@51526
  2042
  fixes f g :: "'a::topological_space \<Rightarrow> real"
wenzelm@61976
  2043
  assumes f: "f \<midarrow>a\<rightarrow> 0"
hoelzl@51526
  2044
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
hoelzl@51526
  2045
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
wenzelm@61976
  2046
  shows "g \<midarrow>a\<rightarrow> 0"
hoelzl@51526
  2047
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
hoelzl@51526
  2048
  fix x assume x: "x \<noteq> a"
hoelzl@51526
  2049
  have "norm (g x - 0) = g x" by (simp add: 1 x)
hoelzl@51526
  2050
  also have "g x \<le> f x" by (rule 2 [OF x])
hoelzl@51526
  2051
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
hoelzl@51526
  2052
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
hoelzl@51526
  2053
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
hoelzl@51526
  2054
qed
hoelzl@51526
  2055
hoelzl@51526
  2056
wenzelm@60758
  2057
subsection \<open>Continuity\<close>
hoelzl@51526
  2058
hoelzl@51526
  2059
lemma LIM_isCont_iff:
hoelzl@51526
  2060
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
wenzelm@61976
  2061
  shows "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
hoelzl@51526
  2062
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
hoelzl@51526
  2063
hoelzl@51526
  2064
lemma isCont_iff:
hoelzl@51526
  2065
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
wenzelm@61976
  2066
  shows "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
hoelzl@51526
  2067
by (simp add: isCont_def LIM_isCont_iff)
hoelzl@51526
  2068
hoelzl@51526
  2069
lemma isCont_LIM_compose2:
hoelzl@51526
  2070
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  2071
  assumes f [unfolded isCont_def]: "isCont f a"
wenzelm@61976
  2072
  assumes g: "g \<midarrow>f a\<rightarrow> l"
hoelzl@51526
  2073
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
wenzelm@61976
  2074
  shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
hoelzl@51526
  2075
by (rule LIM_compose2 [OF f g inj])
hoelzl@51526
  2076
hoelzl@51526
  2077
hoelzl@51526
  2078
lemma isCont_norm [simp]:
hoelzl@51526
  2079
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2080
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
hoelzl@51526
  2081
  by (fact continuous_norm)
hoelzl@51526
  2082
hoelzl@51526
  2083
lemma isCont_rabs [simp]:
hoelzl@51526
  2084
  fixes f :: "'a::t2_space \<Rightarrow> real"
hoelzl@51526
  2085
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
hoelzl@51526
  2086
  by (fact continuous_rabs)
hoelzl@51526
  2087
hoelzl@51526
  2088
lemma isCont_add [simp]:
hoelzl@62368
  2089
  fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
hoelzl@51526
  2090
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
hoelzl@51526
  2091
  by (fact continuous_add)
hoelzl@51526
  2092
hoelzl@51526
  2093
lemma isCont_minus [simp]:
hoelzl@51526
  2094
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2095
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
hoelzl@51526
  2096
  by (fact continuous_minus)
hoelzl@51526
  2097
hoelzl@51526
  2098
lemma isCont_diff [simp]:
hoelzl@51526
  2099
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  2100
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
hoelzl@51526
  2101
  by (fact continuous_diff)
hoelzl@51526
  2102
hoelzl@51526
  2103
lemma isCont_mult [simp]:
hoelzl@51526
  2104
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
hoelzl@51526
  2105
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
hoelzl@51526
  2106
  by (fact continuous_mult)
hoelzl@51526
  2107
hoelzl@51526
  2108
lemma (in bounded_linear) isCont:
hoelzl@51526
  2109
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
hoelzl@51526
  2110
  by (fact continuous)
hoelzl@51526
  2111
hoelzl@51526
  2112
lemma (in bounded_bilinear) isCont:
hoelzl@51526
  2113
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
hoelzl@51526
  2114
  by (fact continuous)
hoelzl@51526
  2115
lp15@60141
  2116
lemmas isCont_scaleR [simp] =
hoelzl@51526
  2117
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
hoelzl@51526
  2118
hoelzl@51526
  2119
lemmas isCont_of_real [simp] =
hoelzl@51526
  2120
  bounded_linear.isCont [OF bounded_linear_of_real]
hoelzl@51526
  2121
hoelzl@51526
  2122
lemma isCont_power [simp]:
hoelzl@51526
  2123
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51526
  2124
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
hoelzl@51526
  2125
  by (fact continuous_power)
hoelzl@51526
  2126
hoelzl@51526
  2127
lemma isCont_setsum [simp]:
hoelzl@62368
  2128
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
hoelzl@51526
  2129
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
hoelzl@51526
  2130
  by (auto intro: continuous_setsum)
hoelzl@51526
  2131
wenzelm@60758
  2132
subsection \<open>Uniform Continuity\<close>
hoelzl@51526
  2133
hoelzl@51531
  2134
definition
hoelzl@51531
  2135
  isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
hoelzl@51531
  2136
  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
hoelzl@51531
  2137
hoelzl@51531
  2138
lemma isUCont_isCont: "isUCont f ==> isCont f x"
hoelzl@51531
  2139
by (simp add: isUCont_def isCont_def LIM_def, force)
hoelzl@51531
  2140
hoelzl@51531
  2141
lemma isUCont_Cauchy:
hoelzl@51531
  2142
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51531
  2143
unfolding isUCont_def
hoelzl@51531
  2144
apply (rule metric_CauchyI)
hoelzl@51531
  2145
apply (drule_tac x=e in spec, safe)
hoelzl@51531
  2146
apply (drule_tac e=s in metric_CauchyD, safe)
hoelzl@51531
  2147
apply (rule_tac x=M in exI, simp)
hoelzl@51531
  2148
done
hoelzl@51531
  2149
hoelzl@51526
  2150
lemma (in bounded_linear) isUCont: "isUCont f"
hoelzl@51526
  2151
unfolding isUCont_def dist_norm
hoelzl@51526
  2152
proof (intro allI impI)
hoelzl@51526
  2153
  fix r::real assume r: "0 < r"
hoelzl@51526
  2154
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
lp15@61649
  2155
    using pos_bounded by blast
hoelzl@51526
  2156
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
hoelzl@51526
  2157
  proof (rule exI, safe)
nipkow@56541
  2158
    from r K show "0 < r / K" by simp
hoelzl@51526
  2159
  next
hoelzl@51526
  2160
    fix x y :: 'a
hoelzl@51526
  2161
    assume xy: "norm (x - y) < r / K"
hoelzl@51526
  2162
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
hoelzl@51526
  2163
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
hoelzl@51526
  2164
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
hoelzl@51526
  2165
    finally show "norm (f x - f y) < r" .
hoelzl@51526
  2166
  qed
hoelzl@51526
  2167
qed
hoelzl@51526
  2168
hoelzl@51526
  2169
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51526
  2170
by (rule isUCont [THEN isUCont_Cauchy])
hoelzl@51526
  2171
lp15@60141
  2172
lemma LIM_less_bound:
hoelzl@51526
  2173
  fixes f :: "real \<Rightarrow> real"
hoelzl@51526
  2174
  assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
hoelzl@51526
  2175
  shows "0 \<le> f x"
hoelzl@51526
  2176
proof (rule tendsto_le_const)
wenzelm@61973
  2177
  show "(f \<longlongrightarrow> f x) (at_left x)"
wenzelm@60758
  2178
    using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
hoelzl@51526
  2179
  show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
hoelzl@51641
  2180
    using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
hoelzl@51526
  2181
qed simp
hoelzl@51471
  2182
hoelzl@51529
  2183
wenzelm@60758
  2184
subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
hoelzl@51529
  2185
hoelzl@51529
  2186
lemma nested_sequence_unique:
wenzelm@61969
  2187
  assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0"
wenzelm@61969
  2188
  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f \<longlonglongrightarrow> l) \<and> ((\<forall>n. l \<le> g n) \<and> g \<longlonglongrightarrow> l)"
hoelzl@51529
  2189
proof -
hoelzl@51529
  2190
  have "incseq f" unfolding incseq_Suc_iff by fact
hoelzl@51529
  2191
  have "decseq g" unfolding decseq_Suc_iff by fact
hoelzl@51529
  2192
hoelzl@51529
  2193
  { fix n
wenzelm@60758
  2194
    from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
wenzelm@60758
  2195
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
wenzelm@61969
  2196
  then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
wenzelm@60758
  2197
    using incseq_convergent[OF \<open>incseq f\<close>] by auto
hoelzl@51529
  2198
  moreover
hoelzl@51529
  2199
  { fix n
wenzelm@60758
  2200
    from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
wenzelm@60758
  2201
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
wenzelm@61969
  2202
  then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
wenzelm@60758
  2203
    using decseq_convergent[OF \<open>decseq g\<close>] by auto
wenzelm@61969
  2204
  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
hoelzl@51529
  2205
  ultimately show ?thesis by auto
hoelzl@51529
  2206
qed
hoelzl@51529
  2207
hoelzl@51529
  2208
lemma Bolzano[consumes 1, case_names trans local]:
hoelzl@51529
  2209
  fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
hoelzl@51529
  2210
  assumes [arith]: "a \<le> b"
hoelzl@51529
  2211
  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
  2212
  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
  2213
  shows "P a b"
hoelzl@51529
  2214
proof -
wenzelm@63040
  2215
  define bisect where "bisect =
wenzelm@63040
  2216
    rec_nat (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
wenzelm@63040
  2217
  define l u where "l n = fst (bisect n)" and "u n = snd (bisect n)" for n
hoelzl@51529
  2218
  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
  2219
    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
  2220
    by (simp_all add: l_def u_def bisect_def split: prod.split)
hoelzl@51529
  2221
hoelzl@51529
  2222
  { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
hoelzl@51529
  2223
wenzelm@61969
  2224
  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
hoelzl@51529
  2225
  proof (safe intro!: nested_sequence_unique)
hoelzl@51529
  2226
    fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
hoelzl@51529
  2227
  next
hoelzl@51529
  2228
    { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
wenzelm@61969
  2229
    then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0" by (simp add: LIMSEQ_divide_realpow_zero)
hoelzl@51529
  2230
  qed fact
wenzelm@61969
  2231
  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x" by auto
hoelzl@51529
  2232
  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
  2233
    using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
hoelzl@51529
  2234
hoelzl@51529
  2235
  show "P a b"
hoelzl@51529
  2236
  proof (rule ccontr)
lp15@60141
  2237
    assume "\<not> P a b"
hoelzl@51529
  2238
    { fix n have "\<not> P (l n) (u n)"
hoelzl@51529
  2239
      proof (induct n)
hoelzl@51529
  2240
        case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
wenzelm@60758
  2241
      qed (simp add: \<open>\<not> P a b\<close>) }
hoelzl@51529
  2242
    moreover
hoelzl@51529
  2243
    { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
wenzelm@61969
  2244
        using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  2245
      moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
wenzelm@61969
  2246
        using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  2247
      ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
hoelzl@51529
  2248
      proof eventually_elim
hoelzl@51529
  2249
        fix n assume "x - d / 2 < l n" "u n < x + d / 2"
hoelzl@51529
  2250
        from add_strict_mono[OF this] have "u n - l n < d" by simp
hoelzl@51529
  2251
        with x show "P (l n) (u n)" by (rule d)
hoelzl@51529
  2252
      qed }
hoelzl@51529
  2253
    ultimately show False by simp
hoelzl@51529
  2254
  qed
hoelzl@51529
  2255
qed
hoelzl@51529
  2256
hoelzl@51529
  2257
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
hoelzl@51529
  2258
proof (cases "a \<le> b", rule compactI)
hoelzl@51529
  2259
  fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
wenzelm@63040
  2260
  define T where "T = {a .. b}"
hoelzl@51529
  2261
  from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
hoelzl@51529
  2262
  proof (induct rule: Bolzano)
hoelzl@51529
  2263
    case (trans a b c)
hoelzl@51529
  2264
    then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
hoelzl@51529
  2265
    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
  2266
      by (auto simp: *)
hoelzl@51529
  2267
    with trans show ?case
hoelzl@51529
  2268
      unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
hoelzl@51529
  2269
  next
hoelzl@51529
  2270
    case (local x)
hoelzl@51529
  2271
    then have "x \<in> \<Union>C" using C by auto
hoelzl@51529
  2272
    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
hoelzl@51529
  2273
    then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
hoelzl@62101
  2274
      by (auto simp: open_dist dist_real_def subset_eq Ball_def abs_less_iff)
wenzelm@60758
  2275
    with \<open>c \<in> C\<close> show ?case
hoelzl@51529
  2276
      by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
hoelzl@51529
  2277
  qed
hoelzl@51529
  2278
qed simp
hoelzl@51529
  2279
hoelzl@51529
  2280
hoelzl@57447
  2281
lemma continuous_image_closed_interval:
hoelzl@57447
  2282
  fixes a b and f :: "real \<Rightarrow> real"
hoelzl@57447
  2283
  defines "S \<equiv> {a..b}"
hoelzl@57447
  2284
  assumes "a \<le> b" and f: "continuous_on S f"
hoelzl@57447
  2285
  shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
hoelzl@57447
  2286
proof -
hoelzl@57447
  2287
  have S: "compact S" "S \<noteq> {}"
wenzelm@60758
  2288
    using \<open>a \<le> b\<close> by (auto simp: S_def)
hoelzl@57447
  2289
  obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
hoelzl@57447
  2290
    using continuous_attains_sup[OF S f] by auto
hoelzl@57447
  2291
  moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
hoelzl@57447
  2292
    using continuous_attains_inf[OF S f] by auto
hoelzl@57447
  2293
  moreover have "connected (f`S)"
hoelzl@57447
  2294
    using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
hoelzl@57447
  2295
  ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
hoelzl@57447
  2296
    by (auto simp: connected_iff_interval)
hoelzl@57447
  2297
  then show ?thesis
hoelzl@57447
  2298
    by auto
hoelzl@57447
  2299
qed
hoelzl@57447
  2300
lp15@60974
  2301
lemma open_Collect_positive:
lp15@60974
  2302
 fixes f :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  2303
 assumes f: "continuous_on s f"
lp15@60974
  2304
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
lp15@60974
  2305
 using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
lp15@60974
  2306
 by (auto simp: Int_def field_simps)
lp15@60974
  2307
lp15@60974
  2308
lemma open_Collect_less_Int:
lp15@60974
  2309
 fixes f g :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  2310
 assumes f: "continuous_on s f" and g: "continuous_on s g"
lp15@60974
  2311
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
lp15@60974
  2312
 using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
lp15@60974
  2313
lp15@60974
  2314
wenzelm@60758
  2315
subsection \<open>Boundedness of continuous functions\<close>
hoelzl@51529
  2316
wenzelm@60758
  2317
text\<open>By bisection, function continuous on closed interval is bounded above\<close>
hoelzl@51529
  2318
hoelzl@51529
  2319
lemma isCont_eq_Ub:
hoelzl@51529
  2320
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2321
  shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2322
    \<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
  2323
  using continuous_attains_sup[of "{a .. b}" f]
hoelzl@51529
  2324
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2325
hoelzl@51529
  2326
lemma isCont_eq_Lb:
hoelzl@51529
  2327
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2328
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2329
    \<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
  2330
  using continuous_attains_inf[of "{a .. b}" f]
hoelzl@51529
  2331
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2332
hoelzl@51529
  2333
lemma isCont_bounded:
hoelzl@51529
  2334
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2335
  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
  2336
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2337
hoelzl@51529
  2338
lemma isCont_has_Ub:
hoelzl@51529
  2339
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2340
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2341
    \<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
  2342
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2343
hoelzl@51529
  2344
(*HOL style here: object-level formulations*)
hoelzl@51529
  2345
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
hoelzl@51529
  2346
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2347
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2348
  by (blast intro: IVT)
hoelzl@51529
  2349
hoelzl@51529
  2350
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
hoelzl@51529
  2351
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2352
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2353
  by (blast intro: IVT2)
hoelzl@51529
  2354
hoelzl@51529
  2355
lemma isCont_Lb_Ub:
hoelzl@51529
  2356
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  2357
  assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
lp15@60141
  2358
  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
  2359
               (\<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
  2360
proof -
hoelzl@51529
  2361
  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
  2362
    using isCont_eq_Ub[OF assms] by auto
hoelzl@51529
  2363
  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
  2364
    using isCont_eq_Lb[OF assms] by auto
hoelzl@51529
  2365
  show ?thesis
hoelzl@51529
  2366
    using IVT[of f L _ M] IVT2[of f L _ M] M L assms
hoelzl@51529
  2367
    apply (rule_tac x="f L" in exI)
hoelzl@51529
  2368
    apply (rule_tac x="f M" in exI)
hoelzl@51529
  2369
    apply (cases "L \<le> M")
hoelzl@51529
  2370
    apply (simp, metis order_trans)
hoelzl@51529
  2371
    apply (simp, metis order_trans)
hoelzl@51529
  2372
    done
hoelzl@51529
  2373
qed
hoelzl@51529
  2374
hoelzl@51529
  2375
wenzelm@60758
  2376
text\<open>Continuity of inverse function\<close>
hoelzl@51529
  2377
hoelzl@51529
  2378
lemma isCont_inverse_function:
hoelzl@51529
  2379
  fixes f g :: "real \<Rightarrow> real"
hoelzl@51529
  2380
  assumes d: "0 < d"
hoelzl@51529
  2381
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
hoelzl@51529
  2382
      and cont: "\<forall>z. \<bar