src/HOL/Analysis/Extended_Real_Limits.thy
author wenzelm
Mon Mar 25 17:21:26 2019 +0100 (4 weeks ago)
changeset 69981 3dced198b9ec
parent 69722 b5163b2132c5
child 70136 f03a01a18c6e
permissions -rw-r--r--
more strict AFP properties;
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(*  Title:      HOL/Analysis/Extended_Real_Limits.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Robert Himmelmann, TU München
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    Author:     Armin Heller, TU München
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    Author:     Bogdan Grechuk, University of Edinburgh
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*)
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section \<open>Limits on the Extended Real Number Line\<close> (* TO FIX: perhaps put all Nonstandard Analysis related
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topics together? *)
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theory Extended_Real_Limits
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imports
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  Topology_Euclidean_Space
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  "HOL-Library.Extended_Real"
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  "HOL-Library.Extended_Nonnegative_Real"
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  "HOL-Library.Indicator_Function"
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begin
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lemma compact_UNIV:
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  "compact (UNIV :: 'a::{complete_linorder,linorder_topology,second_countable_topology} set)"
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  using compact_complete_linorder
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  by (auto simp: seq_compact_eq_compact[symmetric] seq_compact_def)
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lemma compact_eq_closed:
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  fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
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  shows "compact S \<longleftrightarrow> closed S"
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  using%unimportant closed_Int_compact[of S, OF _ compact_UNIV] compact_imp_closed
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  by auto
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lemma closed_contains_Sup_cl:
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  fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
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  assumes "closed S"
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    and "S \<noteq> {}"
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  shows "Sup S \<in> S"
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proof -
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  from compact_eq_closed[of S] compact_attains_sup[of S] assms
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  obtain s where S: "s \<in> S" "\<forall>t\<in>S. t \<le> s"
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    by auto
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  then have "Sup S = s"
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    by (auto intro!: Sup_eqI)
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  with S show ?thesis
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    by simp
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qed
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lemma closed_contains_Inf_cl:
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  fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
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  assumes "closed S"
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    and "S \<noteq> {}"
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  shows "Inf S \<in> S"
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proof -
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  from compact_eq_closed[of S] compact_attains_inf[of S] assms
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  obtain s where S: "s \<in> S" "\<forall>t\<in>S. s \<le> t"
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    by auto
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  then have "Inf S = s"
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    by (auto intro!: Inf_eqI)
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  with S show ?thesis
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    by simp
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qed
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instance%unimportant enat :: second_countable_topology
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proof
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  show "\<exists>B::enat set set. countable B \<and> open = generate_topology B"
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  proof (intro exI conjI)
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    show "countable (range lessThan \<union> range greaterThan::enat set set)"
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      by auto
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  qed (simp add: open_enat_def)
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qed
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instance%unimportant ereal :: second_countable_topology
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proof (standard, intro exI conjI)
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  let ?B = "(\<Union>r\<in>\<rat>. {{..< r}, {r <..}} :: ereal set set)"
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  show "countable ?B"
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    by (auto intro: countable_rat)
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  show "open = generate_topology ?B"
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  proof (intro ext iffI)
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    fix S :: "ereal set"
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    assume "open S"
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    then show "generate_topology ?B S"
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      unfolding open_generated_order
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    proof induct
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      case (Basis b)
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      then obtain e where "b = {..<e} \<or> b = {e<..}"
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        by auto
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      moreover have "{..<e} = \<Union>{{..<x}|x. x \<in> \<rat> \<and> x < e}" "{e<..} = \<Union>{{x<..}|x. x \<in> \<rat> \<and> e < x}"
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        by (auto dest: ereal_dense3
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                 simp del: ex_simps
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                 simp add: ex_simps[symmetric] conj_commute Rats_def image_iff)
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      ultimately show ?case
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        by (auto intro: generate_topology.intros)
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    qed (auto intro: generate_topology.intros)
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  next
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    fix S
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    assume "generate_topology ?B S"
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    then show "open S"
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      by induct auto
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  qed
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qed
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text \<open>This is a copy from \<open>ereal :: second_countable_topology\<close>. Maybe find a common super class of
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topological spaces where the rational numbers are densely embedded ?\<close>
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instance ennreal :: second_countable_topology
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proof (standard, intro exI conjI)
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  let ?B = "(\<Union>r\<in>\<rat>. {{..< r}, {r <..}} :: ennreal set set)"
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  show "countable ?B"
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    by (auto intro: countable_rat)
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  show "open = generate_topology ?B"
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  proof (intro ext iffI)
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    fix S :: "ennreal set"
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    assume "open S"
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    then show "generate_topology ?B S"
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      unfolding open_generated_order
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    proof induct
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      case (Basis b)
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      then obtain e where "b = {..<e} \<or> b = {e<..}"
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        by auto
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      moreover have "{..<e} = \<Union>{{..<x}|x. x \<in> \<rat> \<and> x < e}" "{e<..} = \<Union>{{x<..}|x. x \<in> \<rat> \<and> e < x}"
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        by (auto dest: ennreal_rat_dense
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                 simp del: ex_simps
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                 simp add: ex_simps[symmetric] conj_commute Rats_def image_iff)
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      ultimately show ?case
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        by (auto intro: generate_topology.intros)
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    qed (auto intro: generate_topology.intros)
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  next
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    fix S
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    assume "generate_topology ?B S"
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    then show "open S"
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      by induct auto
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  qed
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qed
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lemma ereal_open_closed_aux:
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  fixes S :: "ereal set"
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  assumes "open S"
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    and "closed S"
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    and S: "(-\<infinity>) \<notin> S"
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  shows "S = {}"
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proof (rule ccontr)
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  assume "\<not> ?thesis"
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  then have *: "Inf S \<in> S"
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    by (metis assms(2) closed_contains_Inf_cl)
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  {
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    assume "Inf S = -\<infinity>"
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    then have False
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      using * assms(3) by auto
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  }
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  moreover
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  {
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    assume "Inf S = \<infinity>"
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    then have "S = {\<infinity>}"
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      by (metis Inf_eq_PInfty \<open>S \<noteq> {}\<close>)
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    then have False
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      by (metis assms(1) not_open_singleton)
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  }
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  moreover
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  {
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    assume fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>"
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    from ereal_open_cont_interval[OF assms(1) * fin]
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    obtain e where e: "e > 0" "{Inf S - e<..<Inf S + e} \<subseteq> S" .
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    then obtain b where b: "Inf S - e < b" "b < Inf S"
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      using fin ereal_between[of "Inf S" e] dense[of "Inf S - e"]
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      by auto
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    then have "b \<in> {Inf S - e <..< Inf S + e}"
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      using e fin ereal_between[of "Inf S" e]
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      by auto
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    then have "b \<in> S"
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      using e by auto
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    then have False
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      using b by (metis complete_lattice_class.Inf_lower leD)
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  }
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  ultimately show False
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    by auto
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qed
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lemma ereal_open_closed:
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  fixes S :: "ereal set"
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  shows "open S \<and> closed S \<longleftrightarrow> S = {} \<or> S = UNIV"
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proof -
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  {
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    assume lhs: "open S \<and> closed S"
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    {
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      assume "-\<infinity> \<notin> S"
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      then have "S = {}"
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        using lhs ereal_open_closed_aux by auto
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    }
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    moreover
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    {
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      assume "-\<infinity> \<in> S"
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      then have "- S = {}"
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        using lhs ereal_open_closed_aux[of "-S"] by auto
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    }
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    ultimately have "S = {} \<or> S = UNIV"
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      by auto
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  }
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  then show ?thesis
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    by auto
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qed
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lemma ereal_open_atLeast:
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  fixes x :: ereal
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  shows "open {x..} \<longleftrightarrow> x = -\<infinity>"
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proof
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  assume "x = -\<infinity>"
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  then have "{x..} = UNIV"
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    by auto
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  then show "open {x..}"
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    by auto
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next
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  assume "open {x..}"
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  then have "open {x..} \<and> closed {x..}"
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    by auto
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  then have "{x..} = UNIV"
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    unfolding ereal_open_closed by auto
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  then show "x = -\<infinity>"
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    by (simp add: bot_ereal_def atLeast_eq_UNIV_iff)
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qed
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lemma mono_closed_real:
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  fixes S :: "real set"
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  assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
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    and "closed S"
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  shows "S = {} \<or> S = UNIV \<or> (\<exists>a. S = {a..})"
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proof -
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  {
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    assume "S \<noteq> {}"
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    { assume ex: "\<exists>B. \<forall>x\<in>S. B \<le> x"
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      then have *: "\<forall>x\<in>S. Inf S \<le> x"
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        using cInf_lower[of _ S] ex by (metis bdd_below_def)
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      then have "Inf S \<in> S"
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        apply (subst closed_contains_Inf)
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        using ex \<open>S \<noteq> {}\<close> \<open>closed S\<close>
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        apply auto
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        done
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      then have "\<forall>x. Inf S \<le> x \<longleftrightarrow> x \<in> S"
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        using mono[rule_format, of "Inf S"] *
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        by auto
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      then have "S = {Inf S ..}"
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        by auto
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      then have "\<exists>a. S = {a ..}"
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        by auto
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    }
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    moreover
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    {
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      assume "\<not> (\<exists>B. \<forall>x\<in>S. B \<le> x)"
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      then have nex: "\<forall>B. \<exists>x\<in>S. x < B"
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        by (simp add: not_le)
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      {
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        fix y
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        obtain x where "x\<in>S" and "x < y"
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          using nex by auto
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        then have "y \<in> S"
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          using mono[rule_format, of x y] by auto
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      }
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      then have "S = UNIV"
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        by auto
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    }
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    ultimately have "S = UNIV \<or> (\<exists>a. S = {a ..})"
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      by blast
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  }
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  then show ?thesis
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    by blast
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qed
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lemma mono_closed_ereal:
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  fixes S :: "real set"
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  assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
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    and "closed S"
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  shows "\<exists>a. S = {x. a \<le> ereal x}"
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proof -
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  {
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    assume "S = {}"
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    then have ?thesis
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      apply (rule_tac x=PInfty in exI)
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      apply auto
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      done
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  }
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  moreover
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  {
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    assume "S = UNIV"
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    then have ?thesis
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      apply (rule_tac x="-\<infinity>" in exI)
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      apply auto
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      done
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  }
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  moreover
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  {
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    assume "\<exists>a. S = {a ..}"
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    then obtain a where "S = {a ..}"
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      by auto
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    then have ?thesis
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      apply (rule_tac x="ereal a" in exI)
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      apply auto
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      done
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  }
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  ultimately show ?thesis
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    using mono_closed_real[of S] assms by auto
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qed
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lemma Liminf_within:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
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  shows "Liminf (at x within S) f = (SUP e\<in>{0<..}. INF y\<in>(S \<inter> ball x e - {x}). f y)"
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  unfolding Liminf_def eventually_at
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proof (rule SUP_eq, simp_all add: Ball_def Bex_def, safe)
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  fix P d
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  assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
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  then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
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    by (auto simp: zero_less_dist_iff dist_commute)
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  then show "\<exists>r>0. Inf (f ` (Collect P)) \<le> Inf (f ` (S \<inter> ball x r - {x}))"
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    by (intro exI[of _ d] INF_mono conjI \<open>0 < d\<close>) auto
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next
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  fix d :: real
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  assume "0 < d"
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  then show "\<exists>P. (\<exists>d>0. \<forall>xa. xa \<in> S \<longrightarrow> xa \<noteq> x \<and> dist xa x < d \<longrightarrow> P xa) \<and>
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    Inf (f ` (S \<inter> ball x d - {x})) \<le> Inf (f ` (Collect P))"
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    by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
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       (auto intro!: INF_mono exI[of _ d] simp: dist_commute)
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qed
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lemma Limsup_within:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
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  shows "Limsup (at x within S) f = (INF e\<in>{0<..}. SUP y\<in>(S \<inter> ball x e - {x}). f y)"
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  unfolding Limsup_def eventually_at
immler@69681
   323
proof (rule INF_eq, simp_all add: Ball_def Bex_def, safe)
wenzelm@53788
   324
  fix P d
wenzelm@53788
   325
  assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
hoelzl@51340
   326
  then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
hoelzl@51340
   327
    by (auto simp: zero_less_dist_iff dist_commute)
haftmann@69313
   328
  then show "\<exists>r>0. Sup (f ` (S \<inter> ball x r - {x})) \<le> Sup (f ` (Collect P))"
wenzelm@60420
   329
    by (intro exI[of _ d] SUP_mono conjI \<open>0 < d\<close>) auto
hoelzl@51340
   330
next
wenzelm@53788
   331
  fix d :: real
wenzelm@53788
   332
  assume "0 < d"
hoelzl@51641
   333
  then show "\<exists>P. (\<exists>d>0. \<forall>xa. xa \<in> S \<longrightarrow> xa \<noteq> x \<and> dist xa x < d \<longrightarrow> P xa) \<and>
haftmann@69313
   334
    Sup (f ` (Collect P)) \<le> Sup (f ` (S \<inter> ball x d - {x}))"
hoelzl@51340
   335
    by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
hoelzl@51340
   336
       (auto intro!: SUP_mono exI[of _ d] simp: dist_commute)
hoelzl@51340
   337
qed
hoelzl@51340
   338
hoelzl@51340
   339
lemma Liminf_at:
hoelzl@54257
   340
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
haftmann@69260
   341
  shows "Liminf (at x) f = (SUP e\<in>{0<..}. INF y\<in>(ball x e - {x}). f y)"
hoelzl@51340
   342
  using Liminf_within[of x UNIV f] by simp
hoelzl@51340
   343
hoelzl@51340
   344
lemma Limsup_at:
hoelzl@54257
   345
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
haftmann@69260
   346
  shows "Limsup (at x) f = (INF e\<in>{0<..}. SUP y\<in>(ball x e - {x}). f y)"
hoelzl@51340
   347
  using Limsup_within[of x UNIV f] by simp
hoelzl@51340
   348
hoelzl@51340
   349
lemma min_Liminf_at:
wenzelm@53788
   350
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_linorder"
haftmann@69260
   351
  shows "min (f x) (Liminf (at x) f) = (SUP e\<in>{0<..}. INF y\<in>ball x e. f y)"
haftmann@69661
   352
  apply (simp add: inf_min [symmetric] Liminf_at)
hoelzl@51340
   353
  apply (subst inf_commute)
hoelzl@51340
   354
  apply (subst SUP_inf)
haftmann@56166
   355
  apply auto
haftmann@69661
   356
  apply (metis (no_types, lifting) INF_insert centre_in_ball greaterThan_iff image_cong inf_commute insert_Diff)
wenzelm@57865
   357
  done
hoelzl@51340
   358
eberlm@66456
   359
immler@69683
   360
subsection \<open>Extended-Real.thy\<close> (*FIX ME change title *)
eberlm@66456
   361
eberlm@66456
   362
lemma sum_constant_ereal:
eberlm@66456
   363
  fixes a::ereal
eberlm@66456
   364
  shows "(\<Sum>i\<in>I. a) = a * card I"
eberlm@66456
   365
apply (cases "finite I", induct set: finite, simp_all)
eberlm@66456
   366
apply (cases a, auto, metis (no_types, hide_lams) add.commute mult.commute semiring_normalization_rules(3))
eberlm@66456
   367
done
eberlm@66456
   368
eberlm@66456
   369
lemma real_lim_then_eventually_real:
eberlm@66456
   370
  assumes "(u \<longlongrightarrow> ereal l) F"
eberlm@66456
   371
  shows "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F"
eberlm@66456
   372
proof -
eberlm@66456
   373
  have "ereal l \<in> {-\<infinity><..<(\<infinity>::ereal)}" by simp
eberlm@66456
   374
  moreover have "open {-\<infinity><..<(\<infinity>::ereal)}" by simp
eberlm@66456
   375
  ultimately have "eventually (\<lambda>n. u n \<in> {-\<infinity><..<(\<infinity>::ereal)}) F" using assms tendsto_def by blast
eberlm@66456
   376
  moreover have "\<And>x. x \<in> {-\<infinity><..<(\<infinity>::ereal)} \<Longrightarrow> x = ereal(real_of_ereal x)" using ereal_real by auto
eberlm@66456
   377
  ultimately show ?thesis by (metis (mono_tags, lifting) eventually_mono)
eberlm@66456
   378
qed
eberlm@66456
   379
immler@69681
   380
lemma ereal_Inf_cmult:
eberlm@66456
   381
  assumes "c>(0::real)"
eberlm@66456
   382
  shows "Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}"
immler@69681
   383
proof -
eberlm@66456
   384
  have "(\<lambda>x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\<lambda>x::ereal. c * x)`{x::ereal. P x})"
eberlm@66456
   385
    apply (rule mono_bij_Inf)
eberlm@66456
   386
    apply (simp add: assms ereal_mult_left_mono less_imp_le mono_def)
eberlm@66456
   387
    apply (rule bij_betw_byWitness[of _ "\<lambda>x. (x::ereal) / c"], auto simp add: assms ereal_mult_divide)
eberlm@66456
   388
    using assms ereal_divide_eq apply auto
eberlm@66456
   389
    done
eberlm@66456
   390
  then show ?thesis by (simp only: setcompr_eq_image[symmetric])
eberlm@66456
   391
qed
eberlm@66456
   392
eberlm@66456
   393
ak2110@69722
   394
subsubsection%important \<open>Continuity of addition\<close>
eberlm@66456
   395
wenzelm@69566
   396
text \<open>The next few lemmas remove an unnecessary assumption in \<open>tendsto_add_ereal\<close>, culminating
wenzelm@69566
   397
in \<open>tendsto_add_ereal_general\<close> which essentially says that the addition
wenzelm@69566
   398
is continuous on ereal times ereal, except at \<open>(-\<infinity>, \<infinity>)\<close> and \<open>(\<infinity>, -\<infinity>)\<close>.
eberlm@66456
   399
It is much more convenient in many situations, see for instance the proof of
wenzelm@69566
   400
\<open>tendsto_sum_ereal\<close> below.\<close>
eberlm@66456
   401
immler@69681
   402
lemma tendsto_add_ereal_PInf:
eberlm@66456
   403
  fixes y :: ereal
eberlm@66456
   404
  assumes y: "y \<noteq> -\<infinity>"
eberlm@66456
   405
  assumes f: "(f \<longlongrightarrow> \<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
eberlm@66456
   406
  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F"
immler@69681
   407
proof -
eberlm@66456
   408
  have "\<exists>C. eventually (\<lambda>x. g x > ereal C) F"
eberlm@66456
   409
  proof (cases y)
eberlm@66456
   410
    case (real r)
eberlm@66456
   411
    have "y > y-1" using y real by (simp add: ereal_between(1))
eberlm@66456
   412
    then have "eventually (\<lambda>x. g x > y - 1) F" using g y order_tendsto_iff by auto
eberlm@66456
   413
    moreover have "y-1 = ereal(real_of_ereal(y-1))"
eberlm@66456
   414
      by (metis real ereal_eq_1(1) ereal_minus(1) real_of_ereal.simps(1))
eberlm@66456
   415
    ultimately have "eventually (\<lambda>x. g x > ereal(real_of_ereal(y - 1))) F" by simp
eberlm@66456
   416
    then show ?thesis by auto
eberlm@66456
   417
  next
eberlm@66456
   418
    case (PInf)
eberlm@66456
   419
    have "eventually (\<lambda>x. g x > ereal 0) F" using g PInf by (simp add: tendsto_PInfty)
eberlm@66456
   420
    then show ?thesis by auto
eberlm@66456
   421
  qed (simp add: y)
eberlm@66456
   422
  then obtain C::real where ge: "eventually (\<lambda>x. g x > ereal C) F" by auto
eberlm@66456
   423
eberlm@66456
   424
  {
eberlm@66456
   425
    fix M::real
eberlm@66456
   426
    have "eventually (\<lambda>x. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty)
eberlm@66456
   427
    then have "eventually (\<lambda>x. (f x > ereal (M-C)) \<and> (g x > ereal C)) F"
eberlm@66456
   428
      by (auto simp add: ge eventually_conj_iff)
eberlm@66456
   429
    moreover have "\<And>x. ((f x > ereal (M-C)) \<and> (g x > ereal C)) \<Longrightarrow> (f x + g x > ereal M)"
eberlm@66456
   430
      using ereal_add_strict_mono2 by fastforce
eberlm@66456
   431
    ultimately have "eventually (\<lambda>x. f x + g x > ereal M) F" using eventually_mono by force
eberlm@66456
   432
  }
eberlm@66456
   433
  then show ?thesis by (simp add: tendsto_PInfty)
eberlm@66456
   434
qed
eberlm@66456
   435
eberlm@66456
   436
text\<open>One would like to deduce the next lemma from the previous one, but the fact
wenzelm@69566
   437
that \<open>- (x + y)\<close> is in general different from \<open>(- x) + (- y)\<close> in ereal creates difficulties,
eberlm@66456
   438
so it is more efficient to copy the previous proof.\<close>
eberlm@66456
   439
immler@69681
   440
lemma tendsto_add_ereal_MInf:
eberlm@66456
   441
  fixes y :: ereal
eberlm@66456
   442
  assumes y: "y \<noteq> \<infinity>"
eberlm@66456
   443
  assumes f: "(f \<longlongrightarrow> -\<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
eberlm@66456
   444
  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> -\<infinity>) F"
immler@69681
   445
proof -
eberlm@66456
   446
  have "\<exists>C. eventually (\<lambda>x. g x < ereal C) F"
eberlm@66456
   447
  proof (cases y)
eberlm@66456
   448
    case (real r)
eberlm@66456
   449
    have "y < y+1" using y real by (simp add: ereal_between(1))
eberlm@66456
   450
    then have "eventually (\<lambda>x. g x < y + 1) F" using g y order_tendsto_iff by force
eberlm@66456
   451
    moreover have "y+1 = ereal(real_of_ereal (y+1))" by (simp add: real)
eberlm@66456
   452
    ultimately have "eventually (\<lambda>x. g x < ereal(real_of_ereal(y + 1))) F" by simp
eberlm@66456
   453
    then show ?thesis by auto
eberlm@66456
   454
  next
eberlm@66456
   455
    case (MInf)
eberlm@66456
   456
    have "eventually (\<lambda>x. g x < ereal 0) F" using g MInf by (simp add: tendsto_MInfty)
eberlm@66456
   457
    then show ?thesis by auto
eberlm@66456
   458
  qed (simp add: y)
eberlm@66456
   459
  then obtain C::real where ge: "eventually (\<lambda>x. g x < ereal C) F" by auto
eberlm@66456
   460
eberlm@66456
   461
  {
eberlm@66456
   462
    fix M::real
eberlm@66456
   463
    have "eventually (\<lambda>x. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty)
eberlm@66456
   464
    then have "eventually (\<lambda>x. (f x < ereal (M- C)) \<and> (g x < ereal C)) F"
eberlm@66456
   465
      by (auto simp add: ge eventually_conj_iff)
eberlm@66456
   466
    moreover have "\<And>x. ((f x < ereal (M-C)) \<and> (g x < ereal C)) \<Longrightarrow> (f x + g x < ereal M)"
eberlm@66456
   467
      using ereal_add_strict_mono2 by fastforce
eberlm@66456
   468
    ultimately have "eventually (\<lambda>x. f x + g x < ereal M) F" using eventually_mono by force
eberlm@66456
   469
  }
eberlm@66456
   470
  then show ?thesis by (simp add: tendsto_MInfty)
eberlm@66456
   471
qed
eberlm@66456
   472
immler@69681
   473
lemma tendsto_add_ereal_general1:
eberlm@66456
   474
  fixes x y :: ereal
eberlm@66456
   475
  assumes y: "\<bar>y\<bar> \<noteq> \<infinity>"
eberlm@66456
   476
  assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
eberlm@66456
   477
  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
immler@69681
   478
proof (cases x)
eberlm@66456
   479
  case (real r)
eberlm@66456
   480
  have a: "\<bar>x\<bar> \<noteq> \<infinity>" by (simp add: real)
eberlm@66456
   481
  show ?thesis by (rule tendsto_add_ereal[OF a, OF y, OF f, OF g])
eberlm@66456
   482
next
eberlm@66456
   483
  case PInf
eberlm@66456
   484
  then show ?thesis using tendsto_add_ereal_PInf assms by force
eberlm@66456
   485
next
eberlm@66456
   486
  case MInf
eberlm@66456
   487
  then show ?thesis using tendsto_add_ereal_MInf assms
eberlm@66456
   488
    by (metis abs_ereal.simps(3) ereal_MInfty_eq_plus)
eberlm@66456
   489
qed
eberlm@66456
   490
immler@69681
   491
lemma tendsto_add_ereal_general2:
eberlm@66456
   492
  fixes x y :: ereal
eberlm@66456
   493
  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>"
eberlm@66456
   494
      and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
eberlm@66456
   495
  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
immler@69681
   496
proof -
eberlm@66456
   497
  have "((\<lambda>x. g x + f x) \<longlongrightarrow> x + y) F"
eberlm@66456
   498
    using tendsto_add_ereal_general1[OF x, OF g, OF f] add.commute[of "y", of "x"] by simp
eberlm@66456
   499
  moreover have "\<And>x. g x + f x = f x + g x" using add.commute by auto
eberlm@66456
   500
  ultimately show ?thesis by simp
eberlm@66456
   501
qed
eberlm@66456
   502
wenzelm@69566
   503
text \<open>The next lemma says that the addition is continuous on \<open>ereal\<close>, except at
wenzelm@69566
   504
the pairs \<open>(-\<infinity>, \<infinity>)\<close> and \<open>(\<infinity>, -\<infinity>)\<close>.\<close>
eberlm@66456
   505
immler@69681
   506
lemma tendsto_add_ereal_general [tendsto_intros]:
eberlm@66456
   507
  fixes x y :: ereal
eberlm@66456
   508
  assumes "\<not>((x=\<infinity> \<and> y=-\<infinity>) \<or> (x=-\<infinity> \<and> y=\<infinity>))"
eberlm@66456
   509
      and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
eberlm@66456
   510
  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
immler@69681
   511
proof (cases x)
eberlm@66456
   512
  case (real r)
eberlm@66456
   513
  show ?thesis
eberlm@66456
   514
    apply (rule tendsto_add_ereal_general2) using real assms by auto
eberlm@66456
   515
next
eberlm@66456
   516
  case (PInf)
eberlm@66456
   517
  then have "y \<noteq> -\<infinity>" using assms by simp
eberlm@66456
   518
  then show ?thesis using tendsto_add_ereal_PInf PInf assms by auto
eberlm@66456
   519
next
eberlm@66456
   520
  case (MInf)
eberlm@66456
   521
  then have "y \<noteq> \<infinity>" using assms by simp
eberlm@66456
   522
  then show ?thesis using tendsto_add_ereal_MInf MInf f g by (metis ereal_MInfty_eq_plus)
eberlm@66456
   523
qed
eberlm@66456
   524
ak2110@69722
   525
subsubsection%important \<open>Continuity of multiplication\<close>
eberlm@66456
   526
eberlm@66456
   527
text \<open>In the same way as for addition, we prove that the multiplication is continuous on
wenzelm@69566
   528
ereal times ereal, except at \<open>(\<infinity>, 0)\<close> and \<open>(-\<infinity>, 0)\<close> and \<open>(0, \<infinity>)\<close> and \<open>(0, -\<infinity>)\<close>,
eberlm@66456
   529
starting with specific situations.\<close>
eberlm@66456
   530
immler@69681
   531
lemma tendsto_mult_real_ereal:
eberlm@66456
   532
  assumes "(u \<longlongrightarrow> ereal l) F" "(v \<longlongrightarrow> ereal m) F"
eberlm@66456
   533
  shows "((\<lambda>n. u n * v n) \<longlongrightarrow> ereal l * ereal m) F"
immler@69681
   534
proof -
eberlm@66456
   535
  have ureal: "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
eberlm@66456
   536
  then have "((\<lambda>n. ereal(real_of_ereal(u n))) \<longlongrightarrow> ereal l) F" using assms by auto
eberlm@66456
   537
  then have limu: "((\<lambda>n. real_of_ereal(u n)) \<longlongrightarrow> l) F" by auto
eberlm@66456
   538
  have vreal: "eventually (\<lambda>n. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)])
eberlm@66456
   539
  then have "((\<lambda>n. ereal(real_of_ereal(v n))) \<longlongrightarrow> ereal m) F" using assms by auto
eberlm@66456
   540
  then have limv: "((\<lambda>n. real_of_ereal(v n)) \<longlongrightarrow> m) F" by auto
eberlm@66456
   541
eberlm@66456
   542
  {
eberlm@66456
   543
    fix n assume "u n = ereal(real_of_ereal(u n))" "v n = ereal(real_of_ereal(v n))"
eberlm@66456
   544
    then have "ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n" by (metis times_ereal.simps(1))
eberlm@66456
   545
  }
eberlm@66456
   546
  then have *: "eventually (\<lambda>n. ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n) F"
eberlm@66456
   547
    using eventually_elim2[OF ureal vreal] by auto
eberlm@66456
   548
eberlm@66456
   549
  have "((\<lambda>n. real_of_ereal(u n) * real_of_ereal(v n)) \<longlongrightarrow> l * m) F" using tendsto_mult[OF limu limv] by auto
eberlm@66456
   550
  then have "((\<lambda>n. ereal(real_of_ereal(u n)) * real_of_ereal(v n)) \<longlongrightarrow> ereal(l * m)) F" by auto
eberlm@66456
   551
  then show ?thesis using * filterlim_cong by fastforce
eberlm@66456
   552
qed
eberlm@66456
   553
immler@69681
   554
lemma tendsto_mult_ereal_PInf:
eberlm@66456
   555
  fixes f g::"_ \<Rightarrow> ereal"
eberlm@66456
   556
  assumes "(f \<longlongrightarrow> l) F" "l>0" "(g \<longlongrightarrow> \<infinity>) F"
eberlm@66456
   557
  shows "((\<lambda>x. f x * g x) \<longlongrightarrow> \<infinity>) F"
immler@69681
   558
proof -
eberlm@66456
   559
  obtain a::real where "0 < ereal a" "a < l" using assms(2) using ereal_dense2 by blast
eberlm@66456
   560
  have *: "eventually (\<lambda>x. f x > a) F" using \<open>a < l\<close> assms(1) by (simp add: order_tendsto_iff)
eberlm@66456
   561
  {
eberlm@66456
   562
    fix K::real
eberlm@66456
   563
    define M where "M = max K 1"
eberlm@66456
   564
    then have "M > 0" by simp
eberlm@66456
   565
    then have "ereal(M/a) > 0" using \<open>ereal a > 0\<close> by simp
eberlm@66456
   566
    then have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > ereal a * ereal(M/a))"
eberlm@66456
   567
      using ereal_mult_mono_strict'[where ?c = "M/a", OF \<open>0 < ereal a\<close>] by auto
eberlm@66456
   568
    moreover have "ereal a * ereal(M/a) = M" using \<open>ereal a > 0\<close> by simp
eberlm@66456
   569
    ultimately have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > M)" by simp
eberlm@66456
   570
    moreover have "M \<ge> K" unfolding M_def by simp
eberlm@66456
   571
    ultimately have Imp: "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > K)"
eberlm@66456
   572
      using ereal_less_eq(3) le_less_trans by blast
eberlm@66456
   573
eberlm@66456
   574
    have "eventually (\<lambda>x. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty)
eberlm@66456
   575
    then have "eventually (\<lambda>x. (f x > a) \<and> (g x > M/a)) F"
eberlm@66456
   576
      using * by (auto simp add: eventually_conj_iff)
eberlm@66456
   577
    then have "eventually (\<lambda>x. f x * g x > K) F" using eventually_mono Imp by force
eberlm@66456
   578
  }
eberlm@66456
   579
  then show ?thesis by (auto simp add: tendsto_PInfty)
eberlm@66456
   580
qed
eberlm@66456
   581
immler@69681
   582
lemma tendsto_mult_ereal_pos:
eberlm@66456
   583
  fixes f g::"_ \<Rightarrow> ereal"
eberlm@66456
   584
  assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "l>0" "m>0"
eberlm@66456
   585
  shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
immler@69681
   586
proof (cases)
eberlm@66456
   587
  assume *: "l = \<infinity> \<or> m = \<infinity>"
eberlm@66456
   588
  then show ?thesis
eberlm@66456
   589
  proof (cases)
eberlm@66456
   590
    assume "m = \<infinity>"
eberlm@66456
   591
    then show ?thesis using tendsto_mult_ereal_PInf assms by auto
eberlm@66456
   592
  next
eberlm@66456
   593
    assume "\<not>(m = \<infinity>)"
eberlm@66456
   594
    then have "l = \<infinity>" using * by simp
eberlm@66456
   595
    then have "((\<lambda>x. g x * f x) \<longlongrightarrow> l * m) F" using tendsto_mult_ereal_PInf assms by auto
eberlm@66456
   596
    moreover have "\<And>x. g x * f x = f x * g x" using mult.commute by auto
eberlm@66456
   597
    ultimately show ?thesis by simp
eberlm@66456
   598
  qed
eberlm@66456
   599
next
eberlm@66456
   600
  assume "\<not>(l = \<infinity> \<or> m = \<infinity>)"
eberlm@66456
   601
  then have "l < \<infinity>" "m < \<infinity>" by auto
eberlm@66456
   602
  then obtain lr mr where "l = ereal lr" "m = ereal mr"
eberlm@66456
   603
    using \<open>l>0\<close> \<open>m>0\<close> by (metis ereal_cases ereal_less(6) not_less_iff_gr_or_eq)
eberlm@66456
   604
  then show ?thesis using tendsto_mult_real_ereal assms by auto
eberlm@66456
   605
qed
eberlm@66456
   606
eberlm@66456
   607
text \<open>We reduce the general situation to the positive case by multiplying by suitable signs.
eberlm@66456
   608
Unfortunately, as ereal is not a ring, all the neat sign lemmas are not available there. We
eberlm@66456
   609
give the bare minimum we need.\<close>
eberlm@66456
   610
eberlm@66456
   611
lemma ereal_sgn_abs:
eberlm@66456
   612
  fixes l::ereal
eberlm@66456
   613
  shows "sgn(l) * l = abs(l)"
eberlm@66456
   614
apply (cases l) by (auto simp add: sgn_if ereal_less_uminus_reorder)
eberlm@66456
   615
eberlm@66456
   616
lemma sgn_squared_ereal:
eberlm@66456
   617
  assumes "l \<noteq> (0::ereal)"
eberlm@66456
   618
  shows "sgn(l) * sgn(l) = 1"
eberlm@66456
   619
apply (cases l) using assms by (auto simp add: one_ereal_def sgn_if)
eberlm@66456
   620
immler@69681
   621
lemma tendsto_mult_ereal [tendsto_intros]:
eberlm@66456
   622
  fixes f g::"_ \<Rightarrow> ereal"
eberlm@66456
   623
  assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "\<not>((l=0 \<and> abs(m) = \<infinity>) \<or> (m=0 \<and> abs(l) = \<infinity>))"
eberlm@66456
   624
  shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
immler@69681
   625
proof (cases)
eberlm@66456
   626
  assume "l=0 \<or> m=0"
eberlm@66456
   627
  then have "abs(l) \<noteq> \<infinity>" "abs(m) \<noteq> \<infinity>" using assms(3) by auto
eberlm@66456
   628
  then obtain lr mr where "l = ereal lr" "m = ereal mr" by auto
eberlm@66456
   629
  then show ?thesis using tendsto_mult_real_ereal assms by auto
eberlm@66456
   630
next
eberlm@66456
   631
  have sgn_finite: "\<And>a::ereal. abs(sgn a) \<noteq> \<infinity>"
eberlm@66456
   632
    by (metis MInfty_neq_ereal(2) PInfty_neq_ereal(2) abs_eq_infinity_cases ereal_times(1) ereal_times(3) ereal_uminus_eq_reorder sgn_ereal.elims)
eberlm@66456
   633
  then have sgn_finite2: "\<And>a b::ereal. abs(sgn a * sgn b) \<noteq> \<infinity>"
eberlm@66456
   634
    by (metis abs_eq_infinity_cases abs_ereal.simps(2) abs_ereal.simps(3) ereal_mult_eq_MInfty ereal_mult_eq_PInfty)
eberlm@66456
   635
  assume "\<not>(l=0 \<or> m=0)"
eberlm@66456
   636
  then have "l \<noteq> 0" "m \<noteq> 0" by auto
eberlm@66456
   637
  then have "abs(l) > 0" "abs(m) > 0"
eberlm@66456
   638
    by (metis abs_ereal_ge0 abs_ereal_less0 abs_ereal_pos ereal_uminus_uminus ereal_uminus_zero less_le not_less)+
eberlm@66456
   639
  then have "sgn(l) * l > 0" "sgn(m) * m > 0" using ereal_sgn_abs by auto
eberlm@66456
   640
  moreover have "((\<lambda>x. sgn(l) * f x) \<longlongrightarrow> (sgn(l) * l)) F"
eberlm@66456
   641
    by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(1))
eberlm@66456
   642
  moreover have "((\<lambda>x. sgn(m) * g x) \<longlongrightarrow> (sgn(m) * m)) F"
eberlm@66456
   643
    by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(2))
eberlm@66456
   644
  ultimately have *: "((\<lambda>x. (sgn(l) * f x) * (sgn(m) * g x)) \<longlongrightarrow> (sgn(l) * l) * (sgn(m) * m)) F"
eberlm@66456
   645
    using tendsto_mult_ereal_pos by force
eberlm@66456
   646
  have "((\<lambda>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x))) \<longlongrightarrow> (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F"
eberlm@66456
   647
    by (rule tendsto_cmult_ereal, auto simp add: sgn_finite2 *)
eberlm@66456
   648
  moreover have "\<And>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)) = f x * g x"
eberlm@66456
   649
    by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
eberlm@66456
   650
  moreover have "(sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m)) = l * m"
eberlm@66456
   651
    by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
eberlm@66456
   652
  ultimately show ?thesis by auto
eberlm@66456
   653
qed
eberlm@66456
   654
eberlm@66456
   655
lemma tendsto_cmult_ereal_general [tendsto_intros]:
eberlm@66456
   656
  fixes f::"_ \<Rightarrow> ereal" and c::ereal
eberlm@66456
   657
  assumes "(f \<longlongrightarrow> l) F" "\<not> (l=0 \<and> abs(c) = \<infinity>)"
eberlm@66456
   658
  shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
eberlm@66456
   659
by (cases "c = 0", auto simp add: assms tendsto_mult_ereal)
eberlm@66456
   660
eberlm@66456
   661
ak2110@69722
   662
subsubsection%important \<open>Continuity of division\<close>
eberlm@66456
   663
immler@69681
   664
lemma tendsto_inverse_ereal_PInf:
eberlm@66456
   665
  fixes u::"_ \<Rightarrow> ereal"
eberlm@66456
   666
  assumes "(u \<longlongrightarrow> \<infinity>) F"
eberlm@66456
   667
  shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 0) F"
immler@69681
   668
proof -
eberlm@66456
   669
  {
eberlm@66456
   670
    fix e::real assume "e>0"
eberlm@66456
   671
    have "1/e < \<infinity>" by auto
eberlm@66456
   672
    then have "eventually (\<lambda>n. u n > 1/e) F" using assms(1) by (simp add: tendsto_PInfty)
eberlm@66456
   673
    moreover
eberlm@66456
   674
    {
eberlm@66456
   675
      fix z::ereal assume "z>1/e"
eberlm@66456
   676
      then have "z>0" using \<open>e>0\<close> using less_le_trans not_le by fastforce
eberlm@66456
   677
      then have "1/z \<ge> 0" by auto
eberlm@66456
   678
      moreover have "1/z < e" using \<open>e>0\<close> \<open>z>1/e\<close>
eberlm@66456
   679
        apply (cases z) apply auto
eberlm@66456
   680
        by (metis (mono_tags, hide_lams) less_ereal.simps(2) less_ereal.simps(4) divide_less_eq ereal_divide_less_pos ereal_less(4)
eberlm@66456
   681
            ereal_less_eq(4) less_le_trans mult_eq_0_iff not_le not_one_less_zero times_ereal.simps(1))
eberlm@66456
   682
      ultimately have "1/z \<ge> 0" "1/z < e" by auto
eberlm@66456
   683
    }
eberlm@66456
   684
    ultimately have "eventually (\<lambda>n. 1/u n<e) F" "eventually (\<lambda>n. 1/u n\<ge>0) F" by (auto simp add: eventually_mono)
eberlm@66456
   685
  } note * = this
eberlm@66456
   686
  show ?thesis
eberlm@66456
   687
  proof (subst order_tendsto_iff, auto)
eberlm@66456
   688
    fix a::ereal assume "a<0"
eberlm@66456
   689
    then show "eventually (\<lambda>n. 1/u n > a) F" using *(2) eventually_mono less_le_trans linordered_field_no_ub by fastforce
eberlm@66456
   690
  next
eberlm@66456
   691
    fix a::ereal assume "a>0"
eberlm@66456
   692
    then obtain e::real where "e>0" "a>e" using ereal_dense2 ereal_less(2) by blast
eberlm@66456
   693
    then have "eventually (\<lambda>n. 1/u n < e) F" using *(1) by auto
eberlm@66456
   694
    then show "eventually (\<lambda>n. 1/u n < a) F" using \<open>a>e\<close> by (metis (mono_tags, lifting) eventually_mono less_trans)
eberlm@66456
   695
  qed
eberlm@66456
   696
qed
eberlm@66456
   697
eberlm@66456
   698
text \<open>The next lemma deserves to exist by itself, as it is so common and useful.\<close>
eberlm@66456
   699
eberlm@66456
   700
lemma tendsto_inverse_real [tendsto_intros]:
eberlm@66456
   701
  fixes u::"_ \<Rightarrow> real"
eberlm@66456
   702
  shows "(u \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
eberlm@66456
   703
  using tendsto_inverse unfolding inverse_eq_divide .
eberlm@66456
   704
immler@69681
   705
lemma tendsto_inverse_ereal [tendsto_intros]:
eberlm@66456
   706
  fixes u::"_ \<Rightarrow> ereal"
eberlm@66456
   707
  assumes "(u \<longlongrightarrow> l) F" "l \<noteq> 0"
eberlm@66456
   708
  shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
immler@69681
   709
proof (cases l)
eberlm@66456
   710
  case (real r)
eberlm@66456
   711
  then have "r \<noteq> 0" using assms(2) by auto
eberlm@66456
   712
  then have "1/l = ereal(1/r)" using real by (simp add: one_ereal_def)
eberlm@66456
   713
  define v where "v = (\<lambda>n. real_of_ereal(u n))"
eberlm@66456
   714
  have ureal: "eventually (\<lambda>n. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto
eberlm@66456
   715
  then have "((\<lambda>n. ereal(v n)) \<longlongrightarrow> ereal r) F" using assms real v_def by auto
eberlm@66456
   716
  then have *: "((\<lambda>n. v n) \<longlongrightarrow> r) F" by auto
eberlm@66456
   717
  then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 1/r) F" using \<open>r \<noteq> 0\<close> tendsto_inverse_real by auto
eberlm@66456
   718
  then have lim: "((\<lambda>n. ereal(1/v n)) \<longlongrightarrow> 1/l) F" using \<open>1/l = ereal(1/r)\<close> by auto
eberlm@66456
   719
eberlm@66456
   720
  have "r \<in> -{0}" "open (-{(0::real)})" using \<open>r \<noteq> 0\<close> by auto
eberlm@66456
   721
  then have "eventually (\<lambda>n. v n \<in> -{0}) F" using * using topological_tendstoD by blast
eberlm@66456
   722
  then have "eventually (\<lambda>n. v n \<noteq> 0) F" by auto
eberlm@66456
   723
  moreover
eberlm@66456
   724
  {
eberlm@66456
   725
    fix n assume H: "v n \<noteq> 0" "u n = ereal(v n)"
eberlm@66456
   726
    then have "ereal(1/v n) = 1/ereal(v n)" by (simp add: one_ereal_def)
eberlm@66456
   727
    then have "ereal(1/v n) = 1/u n" using H(2) by simp
eberlm@66456
   728
  }
eberlm@66456
   729
  ultimately have "eventually (\<lambda>n. ereal(1/v n) = 1/u n) F" using ureal eventually_elim2 by force
eberlm@66456
   730
  with Lim_transform_eventually[OF this lim] show ?thesis by simp
eberlm@66456
   731
next
eberlm@66456
   732
  case (PInf)
eberlm@66456
   733
  then have "1/l = 0" by auto
eberlm@66456
   734
  then show ?thesis using tendsto_inverse_ereal_PInf assms PInf by auto
eberlm@66456
   735
next
eberlm@66456
   736
  case (MInf)
eberlm@66456
   737
  then have "1/l = 0" by auto
eberlm@66456
   738
  have "1/z = -1/ -z" if "z < 0" for z::ereal
eberlm@66456
   739
    apply (cases z) using divide_ereal_def \<open> z < 0 \<close> by auto
eberlm@66456
   740
  moreover have "eventually (\<lambda>n. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def)
eberlm@66456
   741
  ultimately have *: "eventually (\<lambda>n. -1/-u n = 1/u n) F" by (simp add: eventually_mono)
eberlm@66456
   742
eberlm@66456
   743
  define v where "v = (\<lambda>n. - u n)"
eberlm@66456
   744
  have "(v \<longlongrightarrow> \<infinity>) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce
eberlm@66456
   745
  then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 0) F" using tendsto_inverse_ereal_PInf by auto
eberlm@66456
   746
  then have "((\<lambda>n. -1/v n) \<longlongrightarrow> 0) F" using tendsto_uminus_ereal by fastforce
eberlm@66456
   747
  then show ?thesis unfolding v_def using Lim_transform_eventually[OF *] \<open> 1/l = 0 \<close> by auto
eberlm@66456
   748
qed
eberlm@66456
   749
immler@69681
   750
lemma tendsto_divide_ereal [tendsto_intros]:
eberlm@66456
   751
  fixes f g::"_ \<Rightarrow> ereal"
eberlm@66456
   752
  assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "m \<noteq> 0" "\<not>(abs(l) = \<infinity> \<and> abs(m) = \<infinity>)"
eberlm@66456
   753
  shows "((\<lambda>x. f x / g x) \<longlongrightarrow> l / m) F"
immler@69681
   754
proof -
eberlm@66456
   755
  define h where "h = (\<lambda>x. 1/ g x)"
eberlm@66456
   756
  have *: "(h \<longlongrightarrow> 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
eberlm@66456
   757
  have "((\<lambda>x. f x * h x) \<longlongrightarrow> l * (1/m)) F"
eberlm@66456
   758
    apply (rule tendsto_mult_ereal[OF assms(1) *]) using assms(3) assms(4) by (auto simp add: divide_ereal_def)
eberlm@66456
   759
  moreover have "f x * h x = f x / g x" for x unfolding h_def by (simp add: divide_ereal_def)
eberlm@66456
   760
  moreover have "l * (1/m) = l/m" by (simp add: divide_ereal_def)
eberlm@66456
   761
  ultimately show ?thesis unfolding h_def using Lim_transform_eventually by auto
eberlm@66456
   762
qed
eberlm@66456
   763
eberlm@66456
   764
immler@69683
   765
subsubsection \<open>Further limits\<close>
eberlm@66456
   766
immler@67727
   767
text \<open>The assumptions of @{thm tendsto_diff_ereal} are too strong, we weaken them here.\<close>
immler@67727
   768
immler@69681
   769
lemma tendsto_diff_ereal_general [tendsto_intros]:
immler@67727
   770
  fixes u v::"'a \<Rightarrow> ereal"
immler@67727
   771
  assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>((l = \<infinity> \<and> m = \<infinity>) \<or> (l = -\<infinity> \<and> m = -\<infinity>))"
immler@67727
   772
  shows "((\<lambda>n. u n - v n) \<longlongrightarrow> l - m) F"
immler@69681
   773
proof -
immler@67727
   774
  have "((\<lambda>n. u n + (-v n)) \<longlongrightarrow> l + (-m)) F"
immler@67727
   775
    apply (intro tendsto_intros assms) using assms by (auto simp add: ereal_uminus_eq_reorder)
immler@67727
   776
  then show ?thesis by (simp add: minus_ereal_def)
immler@67727
   777
qed
immler@67727
   778
eberlm@66456
   779
lemma id_nat_ereal_tendsto_PInf [tendsto_intros]:
eberlm@66456
   780
  "(\<lambda> n::nat. real n) \<longlonglongrightarrow> \<infinity>"
eberlm@66456
   781
by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top)
eberlm@66456
   782
immler@69681
   783
lemma tendsto_at_top_pseudo_inverse [tendsto_intros]:
eberlm@66456
   784
  fixes u::"nat \<Rightarrow> nat"
eberlm@66456
   785
  assumes "LIM n sequentially. u n :> at_top"
eberlm@66456
   786
  shows "LIM n sequentially. Inf {N. u N \<ge> n} :> at_top"
immler@69681
   787
proof -
eberlm@66456
   788
  {
eberlm@66456
   789
    fix C::nat
eberlm@66456
   790
    define M where "M = Max {u n| n. n \<le> C}+1"
eberlm@66456
   791
    {
eberlm@66456
   792
      fix n assume "n \<ge> M"
eberlm@66456
   793
      have "eventually (\<lambda>N. u N \<ge> n) sequentially" using assms
eberlm@66456
   794
        by (simp add: filterlim_at_top)
eberlm@66456
   795
      then have *: "{N. u N \<ge> n} \<noteq> {}" by force
eberlm@66456
   796
eberlm@66456
   797
      have "N > C" if "u N \<ge> n" for N
eberlm@66456
   798
      proof (rule ccontr)
eberlm@66456
   799
        assume "\<not>(N > C)"
eberlm@66456
   800
        have "u N \<le> Max {u n| n. n \<le> C}"
eberlm@66456
   801
          apply (rule Max_ge) using \<open>\<not>(N > C)\<close> by auto
eberlm@66456
   802
        then show False using \<open>u N \<ge> n\<close> \<open>n \<ge> M\<close> unfolding M_def by auto
eberlm@66456
   803
      qed
eberlm@66456
   804
      then have **: "{N. u N \<ge> n} \<subseteq> {C..}" by fastforce
eberlm@66456
   805
      have "Inf {N. u N \<ge> n} \<ge> C"
eberlm@66456
   806
        by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq)
eberlm@66456
   807
    }
eberlm@66456
   808
    then have "eventually (\<lambda>n. Inf {N. u N \<ge> n} \<ge> C) sequentially"
eberlm@66456
   809
      using eventually_sequentially by auto
eberlm@66456
   810
  }
eberlm@66456
   811
  then show ?thesis using filterlim_at_top by auto
eberlm@66456
   812
qed
eberlm@66456
   813
immler@69681
   814
lemma pseudo_inverse_finite_set:
eberlm@66456
   815
  fixes u::"nat \<Rightarrow> nat"
eberlm@66456
   816
  assumes "LIM n sequentially. u n :> at_top"
eberlm@66456
   817
  shows "finite {N. u N \<le> n}"
immler@69681
   818
proof -
eberlm@66456
   819
  fix n
eberlm@66456
   820
  have "eventually (\<lambda>N. u N \<ge> n+1) sequentially" using assms
eberlm@66456
   821
    by (simp add: filterlim_at_top)
eberlm@66456
   822
  then obtain N1 where N1: "\<And>N. N \<ge> N1 \<Longrightarrow> u N \<ge> n + 1"
eberlm@66456
   823
    using eventually_sequentially by auto
eberlm@66456
   824
  have "{N. u N \<le> n} \<subseteq> {..<N1}"
eberlm@66456
   825
    apply auto using N1 by (metis Suc_eq_plus1 not_less not_less_eq_eq)
eberlm@66456
   826
  then show "finite {N. u N \<le> n}" by (simp add: finite_subset)
eberlm@66456
   827
qed
eberlm@66456
   828
eberlm@66456
   829
lemma tendsto_at_top_pseudo_inverse2 [tendsto_intros]:
eberlm@66456
   830
  fixes u::"nat \<Rightarrow> nat"
eberlm@66456
   831
  assumes "LIM n sequentially. u n :> at_top"
eberlm@66456
   832
  shows "LIM n sequentially. Max {N. u N \<le> n} :> at_top"
eberlm@66456
   833
proof -
eberlm@66456
   834
  {
eberlm@66456
   835
    fix N0::nat
eberlm@66456
   836
    have "N0 \<le> Max {N. u N \<le> n}" if "n \<ge> u N0" for n
eberlm@66456
   837
      apply (rule Max.coboundedI) using pseudo_inverse_finite_set[OF assms] that by auto
eberlm@66456
   838
    then have "eventually (\<lambda>n. N0 \<le> Max {N. u N \<le> n}) sequentially"
eberlm@66456
   839
      using eventually_sequentially by blast
eberlm@66456
   840
  }
eberlm@66456
   841
  then show ?thesis using filterlim_at_top by auto
eberlm@66456
   842
qed
eberlm@66456
   843
eberlm@66456
   844
lemma ereal_truncation_top [tendsto_intros]:
eberlm@66456
   845
  fixes x::ereal
eberlm@66456
   846
  shows "(\<lambda>n::nat. min x n) \<longlonglongrightarrow> x"
eberlm@66456
   847
proof (cases x)
eberlm@66456
   848
  case (real r)
eberlm@66456
   849
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
eberlm@66456
   850
  then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
eberlm@66456
   851
  then have "eventually (\<lambda>n. min x n = x) sequentially" using eventually_at_top_linorder by blast
eberlm@66456
   852
  then show ?thesis by (simp add: Lim_eventually)
eberlm@66456
   853
next
eberlm@66456
   854
  case (PInf)
eberlm@66456
   855
  then have "min x n = n" for n::nat by (auto simp add: min_def)
eberlm@66456
   856
  then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
eberlm@66456
   857
next
eberlm@66456
   858
  case (MInf)
eberlm@66456
   859
  then have "min x n = x" for n::nat by (auto simp add: min_def)
eberlm@66456
   860
  then show ?thesis by auto
eberlm@66456
   861
qed
eberlm@66456
   862
immler@69681
   863
lemma ereal_truncation_real_top [tendsto_intros]:
eberlm@66456
   864
  fixes x::ereal
eberlm@66456
   865
  assumes "x \<noteq> - \<infinity>"
eberlm@66456
   866
  shows "(\<lambda>n::nat. real_of_ereal(min x n)) \<longlonglongrightarrow> x"
immler@69681
   867
proof (cases x)
eberlm@66456
   868
  case (real r)
eberlm@66456
   869
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
eberlm@66456
   870
  then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
eberlm@66456
   871
  then have "real_of_ereal(min x n) = r" if "n \<ge> K" for n using real that by auto
eberlm@66456
   872
  then have "eventually (\<lambda>n. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast
eberlm@66456
   873
  then have "(\<lambda>n. real_of_ereal(min x n)) \<longlonglongrightarrow> r" by (simp add: Lim_eventually)
eberlm@66456
   874
  then show ?thesis using real by auto
eberlm@66456
   875
next
eberlm@66456
   876
  case (PInf)
eberlm@66456
   877
  then have "real_of_ereal(min x n) = n" for n::nat by (auto simp add: min_def)
eberlm@66456
   878
  then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
eberlm@66456
   879
qed (simp add: assms)
eberlm@66456
   880
immler@69681
   881
lemma ereal_truncation_bottom [tendsto_intros]:
eberlm@66456
   882
  fixes x::ereal
eberlm@66456
   883
  shows "(\<lambda>n::nat. max x (- real n)) \<longlonglongrightarrow> x"
immler@69681
   884
proof (cases x)
eberlm@66456
   885
  case (real r)
eberlm@66456
   886
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
eberlm@66456
   887
  then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
eberlm@66456
   888
  then have "eventually (\<lambda>n. max x (-real n) = x) sequentially" using eventually_at_top_linorder by blast
eberlm@66456
   889
  then show ?thesis by (simp add: Lim_eventually)
eberlm@66456
   890
next
eberlm@66456
   891
  case (MInf)
eberlm@66456
   892
  then have "max x (-real n) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
eberlm@66456
   893
  moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
eberlm@66456
   894
    using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
eberlm@66456
   895
  ultimately show ?thesis using MInf by auto
eberlm@66456
   896
next
eberlm@66456
   897
  case (PInf)
eberlm@66456
   898
  then have "max x (-real n) = x" for n::nat by (auto simp add: max_def)
eberlm@66456
   899
  then show ?thesis by auto
eberlm@66456
   900
qed
eberlm@66456
   901
immler@69681
   902
lemma ereal_truncation_real_bottom [tendsto_intros]:
eberlm@66456
   903
  fixes x::ereal
eberlm@66456
   904
  assumes "x \<noteq> \<infinity>"
eberlm@66456
   905
  shows "(\<lambda>n::nat. real_of_ereal(max x (- real n))) \<longlonglongrightarrow> x"
immler@69681
   906
proof (cases x)
eberlm@66456
   907
  case (real r)
eberlm@66456
   908
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
eberlm@66456
   909
  then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
eberlm@66456
   910
  then have "real_of_ereal(max x (-real n)) = r" if "n \<ge> K" for n using real that by auto
eberlm@66456
   911
  then have "eventually (\<lambda>n. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast
eberlm@66456
   912
  then have "(\<lambda>n. real_of_ereal(max x (-real n))) \<longlonglongrightarrow> r" by (simp add: Lim_eventually)
eberlm@66456
   913
  then show ?thesis using real by auto
eberlm@66456
   914
next
eberlm@66456
   915
  case (MInf)
eberlm@66456
   916
  then have "real_of_ereal(max x (-real n)) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
eberlm@66456
   917
  moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
eberlm@66456
   918
    using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
eberlm@66456
   919
  ultimately show ?thesis using MInf by auto
eberlm@66456
   920
qed (simp add: assms)
eberlm@66456
   921
wenzelm@69566
   922
text \<open>the next one is copied from \<open>tendsto_sum\<close>.\<close>
eberlm@66456
   923
lemma tendsto_sum_ereal [tendsto_intros]:
eberlm@66456
   924
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> ereal"
eberlm@66456
   925
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
eberlm@66456
   926
          "\<And>i. abs(a i) \<noteq> \<infinity>"
eberlm@66456
   927
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
eberlm@66456
   928
proof (cases "finite S")
eberlm@66456
   929
  assume "finite S" then show ?thesis using assms
eberlm@66456
   930
    by (induct, simp, simp add: tendsto_add_ereal_general2 assms)
eberlm@66456
   931
qed(simp)
eberlm@66456
   932
eberlm@66456
   933
immler@69681
   934
lemma continuous_ereal_abs:
immler@67727
   935
  "continuous_on (UNIV::ereal set) abs"
immler@69681
   936
proof -
immler@67727
   937
  have "continuous_on ({..0} \<union> {(0::ereal)..}) abs"
immler@67727
   938
    apply (rule continuous_on_closed_Un, auto)
immler@67727
   939
    apply (rule iffD1[OF continuous_on_cong, of "{..0}" _ "\<lambda>x. -x"])
immler@67727
   940
    using less_eq_ereal_def apply (auto simp add: continuous_uminus_ereal)
immler@67727
   941
    apply (rule iffD1[OF continuous_on_cong, of "{0..}" _ "\<lambda>x. x"])
immler@67727
   942
      apply (auto simp add: continuous_on_id)
immler@67727
   943
    done
immler@67727
   944
  moreover have "(UNIV::ereal set) = {..0} \<union> {(0::ereal)..}" by auto
immler@67727
   945
  ultimately show ?thesis by auto
immler@67727
   946
qed
immler@67727
   947
immler@67727
   948
lemmas continuous_on_compose_ereal_abs[continuous_intros] =
immler@67727
   949
  continuous_on_compose2[OF continuous_ereal_abs _ subset_UNIV]
immler@67727
   950
immler@67727
   951
lemma tendsto_abs_ereal [tendsto_intros]:
immler@67727
   952
  assumes "(u \<longlongrightarrow> (l::ereal)) F"
immler@67727
   953
  shows "((\<lambda>n. abs(u n)) \<longlongrightarrow> abs l) F"
immler@67727
   954
using continuous_ereal_abs assms by (metis UNIV_I continuous_on tendsto_compose)
immler@67727
   955
immler@67727
   956
lemma ereal_minus_real_tendsto_MInf [tendsto_intros]:
immler@67727
   957
  "(\<lambda>x. ereal (- real x)) \<longlonglongrightarrow> - \<infinity>"
immler@67727
   958
by (subst uminus_ereal.simps(1)[symmetric], intro tendsto_intros)
immler@67727
   959
immler@67727
   960
immler@69683
   961
subsection \<open>Extended-Nonnegative-Real.thy\<close> (*FIX title *)
immler@67727
   962
immler@67727
   963
lemma tendsto_diff_ennreal_general [tendsto_intros]:
immler@67727
   964
  fixes u v::"'a \<Rightarrow> ennreal"
immler@67727
   965
  assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>(l = \<infinity> \<and> m = \<infinity>)"
immler@67727
   966
  shows "((\<lambda>n. u n - v n) \<longlongrightarrow> l - m) F"
immler@67727
   967
proof -
immler@67727
   968
  have "((\<lambda>n. e2ennreal(enn2ereal(u n) - enn2ereal(v n))) \<longlongrightarrow> e2ennreal(enn2ereal l - enn2ereal m)) F"
immler@67727
   969
    apply (intro tendsto_intros) using assms by  auto
immler@67727
   970
  then show ?thesis by auto
immler@67727
   971
qed
immler@67727
   972
immler@69681
   973
lemma tendsto_mult_ennreal [tendsto_intros]:
immler@67727
   974
  fixes l m::ennreal
immler@67727
   975
  assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>((l = 0 \<and> m = \<infinity>) \<or> (l = \<infinity> \<and> m = 0))"
immler@67727
   976
  shows "((\<lambda>n. u n * v n) \<longlongrightarrow> l * m) F"
immler@69681
   977
proof -
immler@67727
   978
  have "((\<lambda>n. e2ennreal(enn2ereal (u n) * enn2ereal (v n))) \<longlongrightarrow> e2ennreal(enn2ereal l * enn2ereal m)) F"
immler@67727
   979
    apply (intro tendsto_intros) using assms apply auto
immler@67727
   980
    using enn2ereal_inject zero_ennreal.rep_eq by fastforce+
immler@67727
   981
  moreover have "e2ennreal(enn2ereal (u n) * enn2ereal (v n)) = u n * v n" for n
immler@67727
   982
    by (subst times_ennreal.abs_eq[symmetric], auto simp add: eq_onp_same_args)
immler@67727
   983
  moreover have "e2ennreal(enn2ereal l * enn2ereal m)  = l * m"
immler@67727
   984
    by (subst times_ennreal.abs_eq[symmetric], auto simp add: eq_onp_same_args)
immler@67727
   985
  ultimately show ?thesis
immler@67727
   986
    by auto
immler@67727
   987
qed
immler@67727
   988
immler@67727
   989
immler@69683
   990
subsection \<open>monoset\<close> (*FIX ME title *)
hoelzl@51340
   991
immler@69681
   992
definition (in order) mono_set:
hoelzl@51340
   993
  "mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
hoelzl@51340
   994
hoelzl@51340
   995
lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
hoelzl@51340
   996
lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto
hoelzl@51340
   997
lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
hoelzl@51340
   998
lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto
hoelzl@51340
   999
immler@69681
  1000
lemma (in complete_linorder) mono_set_iff:
hoelzl@51340
  1001
  fixes S :: "'a set"
hoelzl@51340
  1002
  defines "a \<equiv> Inf S"
wenzelm@53788
  1003
  shows "mono_set S \<longleftrightarrow> S = {a <..} \<or> S = {a..}" (is "_ = ?c")
immler@69681
  1004
proof
hoelzl@51340
  1005
  assume "mono_set S"
wenzelm@53788
  1006
  then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S"
wenzelm@53788
  1007
    by (auto simp: mono_set)
hoelzl@51340
  1008
  show ?c
hoelzl@51340
  1009
  proof cases
hoelzl@51340
  1010
    assume "a \<in> S"
hoelzl@51340
  1011
    show ?c
wenzelm@60420
  1012
      using mono[OF _ \<open>a \<in> S\<close>]
hoelzl@51340
  1013
      by (auto intro: Inf_lower simp: a_def)
hoelzl@51340
  1014
  next
hoelzl@51340
  1015
    assume "a \<notin> S"
hoelzl@51340
  1016
    have "S = {a <..}"
hoelzl@51340
  1017
    proof safe
hoelzl@51340
  1018
      fix x assume "x \<in> S"
wenzelm@53788
  1019
      then have "a \<le> x"
wenzelm@53788
  1020
        unfolding a_def by (rule Inf_lower)
wenzelm@53788
  1021
      then show "a < x"
wenzelm@60420
  1022
        using \<open>x \<in> S\<close> \<open>a \<notin> S\<close> by (cases "a = x") auto
hoelzl@51340
  1023
    next
hoelzl@51340
  1024
      fix x assume "a < x"
wenzelm@53788
  1025
      then obtain y where "y < x" "y \<in> S"
wenzelm@53788
  1026
        unfolding a_def Inf_less_iff ..
wenzelm@53788
  1027
      with mono[of y x] show "x \<in> S"
wenzelm@53788
  1028
        by auto
hoelzl@51340
  1029
    qed
hoelzl@51340
  1030
    then show ?c ..
hoelzl@51340
  1031
  qed
hoelzl@51340
  1032
qed auto
hoelzl@51340
  1033
hoelzl@51340
  1034
lemma ereal_open_mono_set:
hoelzl@51340
  1035
  fixes S :: "ereal set"
wenzelm@53788
  1036
  shows "open S \<and> mono_set S \<longleftrightarrow> S = UNIV \<or> S = {Inf S <..}"
hoelzl@51340
  1037
  by (metis Inf_UNIV atLeast_eq_UNIV_iff ereal_open_atLeast
hoelzl@51340
  1038
    ereal_open_closed mono_set_iff open_ereal_greaterThan)
hoelzl@51340
  1039
hoelzl@51340
  1040
lemma ereal_closed_mono_set:
hoelzl@51340
  1041
  fixes S :: "ereal set"
wenzelm@53788
  1042
  shows "closed S \<and> mono_set S \<longleftrightarrow> S = {} \<or> S = {Inf S ..}"
hoelzl@51340
  1043
  by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_ereal_atLeast
hoelzl@51340
  1044
    ereal_open_closed mono_empty mono_set_iff open_ereal_greaterThan)
hoelzl@51340
  1045
immler@69681
  1046
lemma ereal_Liminf_Sup_monoset:
wenzelm@53788
  1047
  fixes f :: "'a \<Rightarrow> ereal"
hoelzl@51340
  1048
  shows "Liminf net f =
hoelzl@51340
  1049
    Sup {l. \<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
hoelzl@51340
  1050
    (is "_ = Sup ?A")
immler@69681
  1051
proof (safe intro!: Liminf_eqI complete_lattice_class.Sup_upper complete_lattice_class.Sup_least)
wenzelm@53788
  1052
  fix P
wenzelm@53788
  1053
  assume P: "eventually P net"
wenzelm@53788
  1054
  fix S
haftmann@69313
  1055
  assume S: "mono_set S" "Inf (f ` (Collect P)) \<in> S"
wenzelm@53788
  1056
  {
wenzelm@53788
  1057
    fix x
wenzelm@53788
  1058
    assume "P x"
haftmann@69313
  1059
    then have "Inf (f ` (Collect P)) \<le> f x"
hoelzl@51340
  1060
      by (intro complete_lattice_class.INF_lower) simp
hoelzl@51340
  1061
    with S have "f x \<in> S"
wenzelm@53788
  1062
      by (simp add: mono_set)
wenzelm@53788
  1063
  }
hoelzl@51340
  1064
  with P show "eventually (\<lambda>x. f x \<in> S) net"
lp15@61810
  1065
    by (auto elim: eventually_mono)
hoelzl@51340
  1066
next
hoelzl@51340
  1067
  fix y l
hoelzl@51340
  1068
  assume S: "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net"
haftmann@69313
  1069
  assume P: "\<forall>P. eventually P net \<longrightarrow> Inf (f ` (Collect P)) \<le> y"
hoelzl@51340
  1070
  show "l \<le> y"
hoelzl@51340
  1071
  proof (rule dense_le)
wenzelm@53788
  1072
    fix B
wenzelm@53788
  1073
    assume "B < l"
hoelzl@51340
  1074
    then have "eventually (\<lambda>x. f x \<in> {B <..}) net"
hoelzl@51340
  1075
      by (intro S[rule_format]) auto
haftmann@69313
  1076
    then have "Inf (f ` {x. B < f x}) \<le> y"
hoelzl@51340
  1077
      using P by auto
haftmann@69313
  1078
    moreover have "B \<le> Inf (f ` {x. B < f x})"
hoelzl@51340
  1079
      by (intro INF_greatest) auto
hoelzl@51340
  1080
    ultimately show "B \<le> y"
hoelzl@51340
  1081
      by simp
hoelzl@51340
  1082
  qed
hoelzl@51340
  1083
qed
hoelzl@51340
  1084
immler@69681
  1085
lemma ereal_Limsup_Inf_monoset:
wenzelm@53788
  1086
  fixes f :: "'a \<Rightarrow> ereal"
hoelzl@51340
  1087
  shows "Limsup net f =
hoelzl@51340
  1088
    Inf {l. \<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
hoelzl@51340
  1089
    (is "_ = Inf ?A")
immler@69681
  1090
proof (safe intro!: Limsup_eqI complete_lattice_class.Inf_lower complete_lattice_class.Inf_greatest)
wenzelm@53788
  1091
  fix P
wenzelm@53788
  1092
  assume P: "eventually P net"
wenzelm@53788
  1093
  fix S
haftmann@69313
  1094
  assume S: "mono_set (uminus`S)" "Sup (f ` (Collect P)) \<in> S"
wenzelm@53788
  1095
  {
wenzelm@53788
  1096
    fix x
wenzelm@53788
  1097
    assume "P x"
haftmann@69313
  1098
    then have "f x \<le> Sup (f ` (Collect P))"
hoelzl@51340
  1099
      by (intro complete_lattice_class.SUP_upper) simp
haftmann@69313
  1100
    with S(1)[unfolded mono_set, rule_format, of "- Sup (f ` (Collect P))" "- f x"] S(2)
hoelzl@51340
  1101
    have "f x \<in> S"
hoelzl@51340
  1102
      by (simp add: inj_image_mem_iff) }
hoelzl@51340
  1103
  with P show "eventually (\<lambda>x. f x \<in> S) net"
lp15@61810
  1104
    by (auto elim: eventually_mono)
hoelzl@51340
  1105
next
hoelzl@51340
  1106
  fix y l
hoelzl@51340
  1107
  assume S: "\<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net"
haftmann@69313
  1108
  assume P: "\<forall>P. eventually P net \<longrightarrow> y \<le> Sup (f ` (Collect P))"
hoelzl@51340
  1109
  show "y \<le> l"
hoelzl@51340
  1110
  proof (rule dense_ge)
wenzelm@53788
  1111
    fix B
wenzelm@53788
  1112
    assume "l < B"
hoelzl@51340
  1113
    then have "eventually (\<lambda>x. f x \<in> {..< B}) net"
hoelzl@51340
  1114
      by (intro S[rule_format]) auto
haftmann@69313
  1115
    then have "y \<le> Sup (f ` {x. f x < B})"
hoelzl@51340
  1116
      using P by auto
haftmann@69313
  1117
    moreover have "Sup (f ` {x. f x < B}) \<le> B"
hoelzl@51340
  1118
      by (intro SUP_least) auto
hoelzl@51340
  1119
    ultimately show "y \<le> B"
hoelzl@51340
  1120
      by simp
hoelzl@51340
  1121
  qed
hoelzl@51340
  1122
qed
hoelzl@51340
  1123
immler@69681
  1124
lemma liminf_bounded_open:
hoelzl@51340
  1125
  fixes x :: "nat \<Rightarrow> ereal"
hoelzl@51340
  1126
  shows "x0 \<le> liminf x \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> x0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. x n \<in> S))"
hoelzl@51340
  1127
  (is "_ \<longleftrightarrow> ?P x0")
immler@69681
  1128
proof
hoelzl@51340
  1129
  assume "?P x0"
hoelzl@51340
  1130
  then show "x0 \<le> liminf x"
hoelzl@51340
  1131
    unfolding ereal_Liminf_Sup_monoset eventually_sequentially
hoelzl@51340
  1132
    by (intro complete_lattice_class.Sup_upper) auto
hoelzl@51340
  1133
next
hoelzl@51340
  1134
  assume "x0 \<le> liminf x"
wenzelm@53788
  1135
  {
wenzelm@53788
  1136
    fix S :: "ereal set"
wenzelm@53788
  1137
    assume om: "open S" "mono_set S" "x0 \<in> S"
wenzelm@53788
  1138
    {
wenzelm@53788
  1139
      assume "S = UNIV"
wenzelm@53788
  1140
      then have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
wenzelm@53788
  1141
        by auto
wenzelm@53788
  1142
    }
hoelzl@51340
  1143
    moreover
wenzelm@53788
  1144
    {
wenzelm@53788
  1145
      assume "S \<noteq> UNIV"
wenzelm@53788
  1146
      then obtain B where B: "S = {B<..}"
wenzelm@53788
  1147
        using om ereal_open_mono_set by auto
wenzelm@53788
  1148
      then have "B < x0"
wenzelm@53788
  1149
        using om by auto
wenzelm@53788
  1150
      then have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
wenzelm@53788
  1151
        unfolding B
wenzelm@60420
  1152
        using \<open>x0 \<le> liminf x\<close> liminf_bounded_iff
wenzelm@53788
  1153
        by auto
hoelzl@51340
  1154
    }
wenzelm@53788
  1155
    ultimately have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
wenzelm@53788
  1156
      by auto
hoelzl@51340
  1157
  }
wenzelm@53788
  1158
  then show "?P x0"
wenzelm@53788
  1159
    by auto
hoelzl@51340
  1160
qed
hoelzl@51340
  1161
immler@69681
  1162
lemma limsup_finite_then_bounded:
eberlm@66456
  1163
  fixes u::"nat \<Rightarrow> real"
eberlm@66456
  1164
  assumes "limsup u < \<infinity>"
eberlm@66456
  1165
  shows "\<exists>C. \<forall>n. u n \<le> C"
immler@69681
  1166
proof -
eberlm@66456
  1167
  obtain C where C: "limsup u < C" "C < \<infinity>" using assms ereal_dense2 by blast
eberlm@66456
  1168
  then have "C = ereal(real_of_ereal C)" using ereal_real by force
eberlm@66456
  1169
  have "eventually (\<lambda>n. u n < C) sequentially" using C(1) unfolding Limsup_def
eberlm@66456
  1170
    apply (auto simp add: INF_less_iff)
eberlm@66456
  1171
    using SUP_lessD eventually_mono by fastforce
eberlm@66456
  1172
  then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n < C" using eventually_sequentially by auto
eberlm@66456
  1173
  define D where "D = max (real_of_ereal C) (Max {u n |n. n \<le> N})"
eberlm@66456
  1174
  have "\<And>n. u n \<le> D"
eberlm@66456
  1175
  proof -
eberlm@66456
  1176
    fix n show "u n \<le> D"
eberlm@66456
  1177
    proof (cases)
eberlm@66456
  1178
      assume *: "n \<le> N"
eberlm@66456
  1179
      have "u n \<le> Max {u n |n. n \<le> N}" by (rule Max_ge, auto simp add: *)
eberlm@66456
  1180
      then show "u n \<le> D" unfolding D_def by linarith
eberlm@66456
  1181
    next
eberlm@66456
  1182
      assume "\<not>(n \<le> N)"
eberlm@66456
  1183
      then have "n \<ge> N" by simp
eberlm@66456
  1184
      then have "u n < C" using N by auto
eberlm@66456
  1185
      then have "u n < real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce
eberlm@66456
  1186
      then show "u n \<le> D" unfolding D_def by linarith
eberlm@66456
  1187
    qed
eberlm@66456
  1188
  qed
eberlm@66456
  1189
  then show ?thesis by blast
eberlm@66456
  1190
qed
eberlm@66456
  1191
eberlm@66456
  1192
lemma liminf_finite_then_bounded_below:
eberlm@66456
  1193
  fixes u::"nat \<Rightarrow> real"
eberlm@66456
  1194
  assumes "liminf u > -\<infinity>"
eberlm@66456
  1195
  shows "\<exists>C. \<forall>n. u n \<ge> C"
eberlm@66456
  1196
proof -
eberlm@66456
  1197
  obtain C where C: "liminf u > C" "C > -\<infinity>" using assms using ereal_dense2 by blast
eberlm@66456
  1198
  then have "C = ereal(real_of_ereal C)" using ereal_real by force
eberlm@66456
  1199
  have "eventually (\<lambda>n. u n > C) sequentially" using C(1) unfolding Liminf_def
eberlm@66456
  1200
    apply (auto simp add: less_SUP_iff)
eberlm@66456
  1201
    using eventually_elim2 less_INF_D by fastforce
eberlm@66456
  1202
  then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n > C" using eventually_sequentially by auto
eberlm@66456
  1203
  define D where "D = min (real_of_ereal C) (Min {u n |n. n \<le> N})"
eberlm@66456
  1204
  have "\<And>n. u n \<ge> D"
eberlm@66456
  1205
  proof -
eberlm@66456
  1206
    fix n show "u n \<ge> D"
eberlm@66456
  1207
    proof (cases)
eberlm@66456
  1208
      assume *: "n \<le> N"
eberlm@66456
  1209
      have "u n \<ge> Min {u n |n. n \<le> N}" by (rule Min_le, auto simp add: *)
eberlm@66456
  1210
      then show "u n \<ge> D" unfolding D_def by linarith
eberlm@66456
  1211
    next
eberlm@66456
  1212
      assume "\<not>(n \<le> N)"
eberlm@66456
  1213
      then have "n \<ge> N" by simp
eberlm@66456
  1214
      then have "u n > C" using N by auto
eberlm@66456
  1215
      then have "u n > real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce
eberlm@66456
  1216
      then show "u n \<ge> D" unfolding D_def by linarith
eberlm@66456
  1217
    qed
eberlm@66456
  1218
  qed
eberlm@66456
  1219
  then show ?thesis by blast
eberlm@66456
  1220
qed
eberlm@66456
  1221
eberlm@66456
  1222
lemma liminf_upper_bound:
eberlm@66456
  1223
  fixes u:: "nat \<Rightarrow> ereal"
eberlm@66456
  1224
  assumes "liminf u < l"
eberlm@66456
  1225
  shows "\<exists>N>k. u N < l"
eberlm@66456
  1226
by (metis assms gt_ex less_le_trans liminf_bounded_iff not_less)
eberlm@66456
  1227
eberlm@66456
  1228
lemma limsup_shift:
eberlm@66456
  1229
  "limsup (\<lambda>n. u (n+1)) = limsup u"
eberlm@66456
  1230
proof -
haftmann@69260
  1231
  have "(SUP m\<in>{n+1..}. u m) = (SUP m\<in>{n..}. u (m + 1))" for n
eberlm@66456
  1232
    apply (rule SUP_eq) using Suc_le_D by auto
haftmann@69260
  1233
  then have a: "(INF n. SUP m\<in>{n..}. u (m + 1)) = (INF n. (SUP m\<in>{n+1..}. u m))" by auto
haftmann@69260
  1234
  have b: "(INF n. (SUP m\<in>{n+1..}. u m)) = (INF n\<in>{1..}. (SUP m\<in>{n..}. u m))"
eberlm@66456
  1235
    apply (rule INF_eq) using Suc_le_D by auto
haftmann@69260
  1236
  have "(INF n\<in>{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat \<Rightarrow> 'a"
eberlm@66456
  1237
    apply (rule INF_eq) using \<open>decseq v\<close> decseq_Suc_iff by auto
haftmann@69260
  1238
  moreover have "decseq (\<lambda>n. (SUP m\<in>{n..}. u m))" by (simp add: SUP_subset_mono decseq_def)
haftmann@69260
  1239
  ultimately have c: "(INF n\<in>{1..}. (SUP m\<in>{n..}. u m)) = (INF n. (SUP m\<in>{n..}. u m))" by simp
haftmann@69313
  1240
  have "(INF n. Sup (u ` {n..})) = (INF n. SUP m\<in>{n..}. u (m + 1))" using a b c by simp
eberlm@66456
  1241
  then show ?thesis by (auto cong: limsup_INF_SUP)
eberlm@66456
  1242
qed
eberlm@66456
  1243
eberlm@66456
  1244
lemma limsup_shift_k:
eberlm@66456
  1245
  "limsup (\<lambda>n. u (n+k)) = limsup u"
eberlm@66456
  1246
proof (induction k)
eberlm@66456
  1247
  case (Suc k)
eberlm@66456
  1248
  have "limsup (\<lambda>n. u (n+k+1)) = limsup (\<lambda>n. u (n+k))" using limsup_shift[where ?u="\<lambda>n. u(n+k)"] by simp
eberlm@66456
  1249
  then show ?case using Suc.IH by simp
eberlm@66456
  1250
qed (auto)
eberlm@66456
  1251
eberlm@66456
  1252
lemma liminf_shift:
eberlm@66456
  1253
  "liminf (\<lambda>n. u (n+1)) = liminf u"
eberlm@66456
  1254
proof -
haftmann@69260
  1255
  have "(INF m\<in>{n+1..}. u m) = (INF m\<in>{n..}. u (m + 1))" for n
eberlm@66456
  1256
    apply (rule INF_eq) using Suc_le_D by (auto)
haftmann@69260
  1257
  then have a: "(SUP n. INF m\<in>{n..}. u (m + 1)) = (SUP n. (INF m\<in>{n+1..}. u m))" by auto
haftmann@69260
  1258
  have b: "(SUP n. (INF m\<in>{n+1..}. u m)) = (SUP n\<in>{1..}. (INF m\<in>{n..}. u m))"
eberlm@66456
  1259
    apply (rule SUP_eq) using Suc_le_D by (auto)
haftmann@69260
  1260
  have "(SUP n\<in>{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat \<Rightarrow> 'a"
eberlm@66456
  1261
    apply (rule SUP_eq) using \<open>incseq v\<close> incseq_Suc_iff by auto
haftmann@69260
  1262
  moreover have "incseq (\<lambda>n. (INF m\<in>{n..}. u m))" by (simp add: INF_superset_mono mono_def)
haftmann@69260
  1263
  ultimately have c: "(SUP n\<in>{1..}. (INF m\<in>{n..}. u m)) = (SUP n. (INF m\<in>{n..}. u m))" by simp
haftmann@69313
  1264
  have "(SUP n. Inf (u ` {n..})) = (SUP n. INF m\<in>{n..}. u (m + 1))" using a b c by simp
eberlm@66456
  1265
  then show ?thesis by (auto cong: liminf_SUP_INF)
eberlm@66456
  1266
qed
eberlm@66456
  1267
eberlm@66456
  1268
lemma liminf_shift_k:
eberlm@66456
  1269
  "liminf (\<lambda>n. u (n+k)) = liminf u"
eberlm@66456
  1270
proof (induction k)
eberlm@66456
  1271
  case (Suc k)
eberlm@66456
  1272
  have "liminf (\<lambda>n. u (n+k+1)) = liminf (\<lambda>n. u (n+k))" using liminf_shift[where ?u="\<lambda>n. u(n+k)"] by simp
eberlm@66456
  1273
  then show ?case using Suc.IH by simp
eberlm@66456
  1274
qed (auto)
eberlm@66456
  1275
immler@69681
  1276
lemma Limsup_obtain:
eberlm@66456
  1277
  fixes u::"_ \<Rightarrow> 'a :: complete_linorder"
eberlm@66456
  1278
  assumes "Limsup F u > c"
eberlm@66456
  1279
  shows "\<exists>i. u i > c"
immler@69681
  1280
proof -
haftmann@69260
  1281
  have "(INF P\<in>{P. eventually P F}. SUP x\<in>{x. P x}. u x) > c" using assms by (simp add: Limsup_def)
eberlm@66456
  1282
  then show ?thesis by (metis eventually_True mem_Collect_eq less_INF_D less_SUP_iff)
eberlm@66456
  1283
qed
eberlm@66456
  1284
eberlm@66456
  1285
text \<open>The next lemma is extremely useful, as it often makes it possible to reduce statements
eberlm@66456
  1286
about limsups to statements about limits.\<close>
eberlm@66456
  1287
immler@69681
  1288
lemma limsup_subseq_lim:
eberlm@66456
  1289
  fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
eberlm@66456
  1290
  shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> limsup u"
immler@69681
  1291
proof (cases)
eberlm@66456
  1292
  assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<le> u p"
eberlm@66456
  1293
  then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<le> u (r n)) \<and> r n < r (Suc n)"
eberlm@66456
  1294
    by (intro dependent_nat_choice) (auto simp: conj_commute)
eberlm@66456
  1295
  then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<le> u (r n)"
eberlm@66456
  1296
    by (auto simp: strict_mono_Suc_iff)
haftmann@69260
  1297
  define umax where "umax = (\<lambda>n. (SUP m\<in>{n..}. u m))"
eberlm@66456
  1298
  have "decseq umax" unfolding umax_def by (simp add: SUP_subset_mono antimono_def)
eberlm@66456
  1299
  then have "umax \<longlonglongrightarrow> limsup u" unfolding umax_def by (metis LIMSEQ_INF limsup_INF_SUP)
eberlm@66456
  1300
  then have *: "(umax o r) \<longlonglongrightarrow> limsup u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>strict_mono r\<close>)
eberlm@66456
  1301
  have "\<And>n. umax(r n) = u(r n)" unfolding umax_def using mono
eberlm@66456
  1302
    by (metis SUP_le_iff antisym atLeast_def mem_Collect_eq order_refl)
eberlm@66456
  1303
  then have "umax o r = u o r" unfolding o_def by simp
eberlm@66456
  1304
  then have "(u o r) \<longlonglongrightarrow> limsup u" using * by simp
eberlm@66456
  1305
  then show ?thesis using \<open>strict_mono r\<close> by blast
eberlm@66456
  1306
next
eberlm@66456
  1307
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<le> u p))"
eberlm@66456
  1308
  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p < u m" by (force simp: not_le le_less)
eberlm@66456
  1309
  have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))"
eberlm@66456
  1310
  proof (rule dependent_nat_choice)
eberlm@66456
  1311
    fix x assume "N < x"
eberlm@66456
  1312
    then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
eberlm@66456
  1313
    have "Max {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Max_in) using a by (auto)
eberlm@66456
  1314
    then obtain p where "p \<in> {N<..x}" and upmax: "u p = Max{u i |i. i \<in> {N<..x}}" by auto
eberlm@66456
  1315
    define U where "U = {m. m > p \<and> u p < u m}"
eberlm@66456
  1316
    have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto
eberlm@66456
  1317
    define y where "y = Inf U"
eberlm@66456
  1318
    then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1)
eberlm@66456
  1319
    have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<le> u p"
eberlm@66456
  1320
    proof -
eberlm@66456
  1321
      fix i assume "i \<in> {N<..x}"
eberlm@66456
  1322
      then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
eberlm@66456
  1323
      then show "u i \<le> u p" using upmax by simp
eberlm@66456
  1324
    qed
eberlm@66456
  1325
    moreover have "u p < u y" using \<open>y \<in> U\<close> U_def by auto
eberlm@66456
  1326
    ultimately have "y \<notin> {N<..x}" using not_le by blast
eberlm@66456
  1327
    moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto
eberlm@66456
  1328
    ultimately have "y > x" by auto
eberlm@66456
  1329
eberlm@66456
  1330
    have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<le> u y"
eberlm@66456
  1331
    proof -
eberlm@66456
  1332
      fix i assume "i \<in> {N<..y}" show "u i \<le> u y"
eberlm@66456
  1333
      proof (cases)
eberlm@66456
  1334
        assume "i = y"
eberlm@66456
  1335
        then show ?thesis by simp
eberlm@66456
  1336
      next
eberlm@66456
  1337
        assume "\<not>(i=y)"
eberlm@66456
  1338
        then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp
eberlm@66456
  1339
        have "u i \<le> u p"
eberlm@66456
  1340
        proof (cases)
eberlm@66456
  1341
          assume "i \<le> x"
eberlm@66456
  1342
          then have "i \<in> {N<..x}" using i by simp
eberlm@66456
  1343
          then show ?thesis using a by simp
eberlm@66456
  1344
        next
eberlm@66456
  1345
          assume "\<not>(i \<le> x)"
eberlm@66456
  1346
          then have "i > x" by simp
eberlm@66456
  1347
          then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp
eberlm@66456
  1348
          have "i < Inf U" using i y_def by simp
eberlm@66456
  1349
          then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
eberlm@66456
  1350
          then show ?thesis using U_def * by auto
eberlm@66456
  1351
        qed
eberlm@66456
  1352
        then show "u i \<le> u y" using \<open>u p < u y\<close> by auto
eberlm@66456
  1353
      qed
eberlm@66456
  1354
    qed
eberlm@66456
  1355
    then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto
eberlm@66456
  1356
    then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" by auto
eberlm@66456
  1357
  qed (auto)
eberlm@66456
  1358
  then obtain r where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))" by auto
eberlm@66456
  1359
  have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
eberlm@66456
  1360
  have "incseq (u o r)" unfolding o_def using r by (simp add: incseq_SucI order.strict_implies_order)
eberlm@66456
  1361
  then have "(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)" using LIMSEQ_SUP by blast
eberlm@66456
  1362
  then have "limsup (u o r) = (SUP n. (u o r) n)" by (simp add: lim_imp_Limsup)
eberlm@66456
  1363
  moreover have "limsup (u o r) \<le> limsup u" using \<open>strict_mono r\<close> by (simp add: limsup_subseq_mono)
eberlm@66456
  1364
  ultimately have "(SUP n. (u o r) n) \<le> limsup u" by simp
eberlm@66456
  1365
eberlm@66456
  1366
  {
eberlm@66456
  1367
    fix i assume i: "i \<in> {N<..}"
eberlm@66456
  1368
    obtain n where "i < r (Suc n)" using \<open>strict_mono r\<close> using Suc_le_eq seq_suble by blast
eberlm@66456
  1369
    then have "i \<in> {N<..r(Suc n)}" using i by simp
eberlm@66456
  1370
    then have "u i \<le> u (r(Suc n))" using r by simp
eberlm@66456
  1371
    then have "u i \<le> (SUP n. (u o r) n)" unfolding o_def by (meson SUP_upper2 UNIV_I)
eberlm@66456
  1372
  }
haftmann@69260
  1373
  then have "(SUP i\<in>{N<..}. u i) \<le> (SUP n. (u o r) n)" using SUP_least by blast
eberlm@66456
  1374
  then have "limsup u \<le> (SUP n. (u o r) n)" unfolding Limsup_def
eberlm@66456
  1375
    by (metis (mono_tags, lifting) INF_lower2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
eberlm@66456
  1376
  then have "limsup u = (SUP n. (u o r) n)" using \<open>(SUP n. (u o r) n) \<le> limsup u\<close> by simp
eberlm@66456
  1377
  then have "(u o r) \<longlonglongrightarrow> limsup u" using \<open>(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)\<close> by simp
eberlm@66456
  1378
  then show ?thesis using \<open>strict_mono r\<close> by auto
eberlm@66456
  1379
qed
eberlm@66456
  1380
immler@69681
  1381
lemma liminf_subseq_lim:
eberlm@66456
  1382
  fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
eberlm@66456
  1383
  shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> liminf u"
immler@69681
  1384
proof (cases)
eberlm@66456
  1385
  assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<ge> u p"
eberlm@66456
  1386
  then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<ge> u (r n)) \<and> r n < r (Suc n)"
eberlm@66456
  1387
    by (intro dependent_nat_choice) (auto simp: conj_commute)
eberlm@66456
  1388
  then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<ge> u (r n)"
eberlm@66456
  1389
    by (auto simp: strict_mono_Suc_iff)
haftmann@69260
  1390
  define umin where "umin = (\<lambda>n. (INF m\<in>{n..}. u m))"
eberlm@66456
  1391
  have "incseq umin" unfolding umin_def by (simp add: INF_superset_mono incseq_def)
eberlm@66456
  1392
  then have "umin \<longlonglongrightarrow> liminf u" unfolding umin_def by (metis LIMSEQ_SUP liminf_SUP_INF)
eberlm@66456
  1393
  then have *: "(umin o r) \<longlonglongrightarrow> liminf u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>strict_mono r\<close>)
eberlm@66456
  1394
  have "\<And>n. umin(r n) = u(r n)" unfolding umin_def using mono
eberlm@66456
  1395
    by (metis le_INF_iff antisym atLeast_def mem_Collect_eq order_refl)
eberlm@66456
  1396
  then have "umin o r = u o r" unfolding o_def by simp
eberlm@66456
  1397
  then have "(u o r) \<longlonglongrightarrow> liminf u" using * by simp
eberlm@66456
  1398
  then show ?thesis using \<open>strict_mono r\<close> by blast
eberlm@66456
  1399
next
eberlm@66456
  1400
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<ge> u p))"
eberlm@66456
  1401
  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p > u m" by (force simp: not_le le_less)
eberlm@66456
  1402
  have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))"
eberlm@66456
  1403
  proof (rule dependent_nat_choice)
eberlm@66456
  1404
    fix x assume "N < x"
eberlm@66456
  1405
    then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
eberlm@66456
  1406
    have "Min {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Min_in) using a by (auto)
eberlm@66456
  1407
    then obtain p where "p \<in> {N<..x}" and upmin: "u p = Min{u i |i. i \<in> {N<..x}}" by auto
eberlm@66456
  1408
    define U where "U = {m. m > p \<and> u p > u m}"
eberlm@66456
  1409
    have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto
eberlm@66456
  1410
    define y where "y = Inf U"
eberlm@66456
  1411
    then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1)
eberlm@66456
  1412
    have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<ge> u p"
eberlm@66456
  1413
    proof -
eberlm@66456
  1414
      fix i assume "i \<in> {N<..x}"
eberlm@66456
  1415
      then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
eberlm@66456
  1416
      then show "u i \<ge> u p" using upmin by simp
eberlm@66456
  1417
    qed
eberlm@66456
  1418
    moreover have "u p > u y" using \<open>y \<in> U\<close> U_def by auto
eberlm@66456
  1419
    ultimately have "y \<notin> {N<..x}" using not_le by blast
eberlm@66456
  1420
    moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto
eberlm@66456
  1421
    ultimately have "y > x" by auto
eberlm@66456
  1422
eberlm@66456
  1423
    have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<ge> u y"
eberlm@66456
  1424
    proof -
eberlm@66456
  1425
      fix i assume "i \<in> {N<..y}" show "u i \<ge> u y"
eberlm@66456
  1426
      proof (cases)
eberlm@66456
  1427
        assume "i = y"
eberlm@66456
  1428
        then show ?thesis by simp
eberlm@66456
  1429
      next
eberlm@66456
  1430
        assume "\<not>(i=y)"
eberlm@66456
  1431
        then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp
eberlm@66456
  1432
        have "u i \<ge> u p"
eberlm@66456
  1433
        proof (cases)
eberlm@66456
  1434
          assume "i \<le> x"
eberlm@66456
  1435
          then have "i \<in> {N<..x}" using i by simp
eberlm@66456
  1436
          then show ?thesis using a by simp
eberlm@66456
  1437
        next
eberlm@66456
  1438
          assume "\<not>(i \<le> x)"
eberlm@66456
  1439
          then have "i > x" by simp
eberlm@66456
  1440
          then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp
eberlm@66456
  1441
          have "i < Inf U" using i y_def by simp
eberlm@66456
  1442
          then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
eberlm@66456
  1443
          then show ?thesis using U_def * by auto
eberlm@66456
  1444
        qed
eberlm@66456
  1445
        then show "u i \<ge> u y" using \<open>u p > u y\<close> by auto
eberlm@66456
  1446
      qed
eberlm@66456
  1447
    qed
eberlm@66456
  1448
    then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto
eberlm@66456
  1449
    then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" by auto
eberlm@66456
  1450
  qed (auto)
eberlm@66456
  1451
  then obtain r :: "nat \<Rightarrow> nat" 
eberlm@66456
  1452
    where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))" by auto
eberlm@66456
  1453
  have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
eberlm@66456
  1454
  have "decseq (u o r)" unfolding o_def using r by (simp add: decseq_SucI order.strict_implies_order)
eberlm@66456
  1455
  then have "(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)" using LIMSEQ_INF by blast
eberlm@66456
  1456
  then have "liminf (u o r) = (INF n. (u o r) n)" by (simp add: lim_imp_Liminf)
eberlm@66456
  1457
  moreover have "liminf (u o r) \<ge> liminf u" using \<open>strict_mono r\<close> by (simp add: liminf_subseq_mono)
eberlm@66456
  1458
  ultimately have "(INF n. (u o r) n) \<ge> liminf u" by simp
eberlm@66456
  1459
eberlm@66456
  1460
  {
eberlm@66456
  1461
    fix i assume i: "i \<in> {N<..}"
eberlm@66456
  1462
    obtain n where "i < r (Suc n)" using \<open>strict_mono r\<close> using Suc_le_eq seq_suble by blast
eberlm@66456
  1463
    then have "i \<in> {N<..r(Suc n)}" using i by simp
eberlm@66456
  1464
    then have "u i \<ge> u (r(Suc n))" using r by simp
eberlm@66456
  1465
    then have "u i \<ge> (INF n. (u o r) n)" unfolding o_def by (meson INF_lower2 UNIV_I)
eberlm@66456
  1466
  }
haftmann@69260
  1467
  then have "(INF i\<in>{N<..}. u i) \<ge> (INF n. (u o r) n)" using INF_greatest by blast
eberlm@66456
  1468
  then have "liminf u \<ge> (INF n. (u o r) n)" unfolding Liminf_def
eberlm@66456
  1469
    by (metis (mono_tags, lifting) SUP_upper2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
eberlm@66456
  1470
  then have "liminf u = (INF n. (u o r) n)" using \<open>(INF n. (u o r) n) \<ge> liminf u\<close> by simp
eberlm@66456
  1471
  then have "(u o r) \<longlonglongrightarrow> liminf u" using \<open>(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)\<close> by simp
eberlm@66456
  1472
  then show ?thesis using \<open>strict_mono r\<close> by auto
eberlm@66456
  1473
qed
eberlm@66456
  1474
eberlm@66456
  1475
text \<open>The following statement about limsups is reduced to a statement about limits using
wenzelm@69566
  1476
subsequences thanks to \<open>limsup_subseq_lim\<close>. The statement for limits follows for instance from
wenzelm@69566
  1477
\<open>tendsto_add_ereal_general\<close>.\<close>
eberlm@66456
  1478
immler@69681
  1479
lemma ereal_limsup_add_mono:
eberlm@66456
  1480
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1481
  shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
immler@69681
  1482
proof (cases)
eberlm@66456
  1483
  assume "(limsup u = \<infinity>) \<or> (limsup v = \<infinity>)"
eberlm@66456
  1484
  then have "limsup u + limsup v = \<infinity>" by simp
eberlm@66456
  1485
  then show ?thesis by auto
eberlm@66456
  1486
next
eberlm@66456
  1487
  assume "\<not>((limsup u = \<infinity>) \<or> (limsup v = \<infinity>))"
eberlm@66456
  1488
  then have "limsup u < \<infinity>" "limsup v < \<infinity>" by auto
eberlm@66456
  1489
eberlm@66456
  1490
  define w where "w = (\<lambda>n. u n + v n)"
eberlm@66456
  1491
  obtain r where r: "strict_mono r" "(w o r) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
eberlm@66456
  1492
  obtain s where s: "strict_mono s" "(u o r o s) \<longlonglongrightarrow> limsup (u o r)" using limsup_subseq_lim by auto
eberlm@66456
  1493
  obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
eberlm@66456
  1494
eberlm@66456
  1495
  define a where "a = r o s o t"
eberlm@66456
  1496
  have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
eberlm@66456
  1497
  have l:"(w o a) \<longlonglongrightarrow> limsup w"
eberlm@66456
  1498
         "(u o a) \<longlonglongrightarrow> limsup (u o r)"
eberlm@66456
  1499
         "(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
eberlm@66456
  1500
  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
eberlm@66456
  1501
  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
eberlm@66456
  1502
  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
eberlm@66456
  1503
  done
eberlm@66456
  1504
eberlm@66456
  1505
  have "limsup (u o r) \<le> limsup u" by (simp add: limsup_subseq_mono r(1))
eberlm@66456
  1506
  then have a: "limsup (u o r) \<noteq> \<infinity>" using \<open>limsup u < \<infinity>\<close> by auto
eberlm@66456
  1507
  have "limsup (v o r o s) \<le> limsup v" 
eberlm@66456
  1508
    by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
eberlm@66456
  1509
  then have b: "limsup (v o r o s) \<noteq> \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto
eberlm@66456
  1510
eberlm@66456
  1511
  have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)"
eberlm@66456
  1512
    using l tendsto_add_ereal_general a b by fastforce
eberlm@66456
  1513
  moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
eberlm@66456
  1514
  ultimately have "(w o a) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)" by simp
eberlm@66456
  1515
  then have "limsup w = limsup (u o r) + limsup (v o r o s)" using l(1) LIMSEQ_unique by blast
eberlm@66456
  1516
  then have "limsup w \<le> limsup u + limsup v"
nipkow@68752
  1517
    using \<open>limsup (u o r) \<le> limsup u\<close> \<open>limsup (v o r o s) \<le> limsup v\<close> add_mono by simp
eberlm@66456
  1518
  then show ?thesis unfolding w_def by simp
eberlm@66456
  1519
qed
eberlm@66456
  1520
wenzelm@69566
  1521
text \<open>There is an asymmetry between liminfs and limsups in \<open>ereal\<close>, as \<open>\<infinity> + (-\<infinity>) = \<infinity>\<close>.
eberlm@66456
  1522
This explains why there are more assumptions in the next lemma dealing with liminfs that in the
eberlm@66456
  1523
previous one about limsups.\<close>
eberlm@66456
  1524
immler@69681
  1525
lemma ereal_liminf_add_mono:
eberlm@66456
  1526
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1527
  assumes "\<not>((liminf u = \<infinity> \<and> liminf v = -\<infinity>) \<or> (liminf u = -\<infinity> \<and> liminf v = \<infinity>))"
eberlm@66456
  1528
  shows "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
immler@69681
  1529
proof (cases)
eberlm@66456
  1530
  assume "(liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>)"
eberlm@66456
  1531
  then have *: "liminf u + liminf v = -\<infinity>" using assms by auto
eberlm@66456
  1532
  show ?thesis by (simp add: *)
eberlm@66456
  1533
next
eberlm@66456
  1534
  assume "\<not>((liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>))"
eberlm@66456
  1535
  then have "liminf u > -\<infinity>" "liminf v > -\<infinity>" by auto
eberlm@66456
  1536
eberlm@66456
  1537
  define w where "w = (\<lambda>n. u n + v n)"
eberlm@66456
  1538
  obtain r where r: "strict_mono r" "(w o r) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
eberlm@66456
  1539
  obtain s where s: "strict_mono s" "(u o r o s) \<longlonglongrightarrow> liminf (u o r)" using liminf_subseq_lim by auto
eberlm@66456
  1540
  obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> liminf (v o r o s)" using liminf_subseq_lim by auto
eberlm@66456
  1541
eberlm@66456
  1542
  define a where "a = r o s o t"
eberlm@66456
  1543
  have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
eberlm@66456
  1544
  have l:"(w o a) \<longlonglongrightarrow> liminf w"
eberlm@66456
  1545
         "(u o a) \<longlonglongrightarrow> liminf (u o r)"
eberlm@66456
  1546
         "(v o a) \<longlonglongrightarrow> liminf (v o r o s)"
eberlm@66456
  1547
  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
eberlm@66456
  1548
  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
eberlm@66456
  1549
  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
eberlm@66456
  1550
  done
eberlm@66456
  1551
eberlm@66456
  1552
  have "liminf (u o r) \<ge> liminf u" by (simp add: liminf_subseq_mono r(1))
eberlm@66456
  1553
  then have a: "liminf (u o r) \<noteq> -\<infinity>" using \<open>liminf u > -\<infinity>\<close> by auto
eberlm@66456
  1554
  have "liminf (v o r o s) \<ge> liminf v" 
eberlm@66456
  1555
    by (simp add: comp_assoc liminf_subseq_mono r(1) s(1) strict_mono_o)
eberlm@66456
  1556
  then have b: "liminf (v o r o s) \<noteq> -\<infinity>" using \<open>liminf v > -\<infinity>\<close> by auto
eberlm@66456
  1557
eberlm@66456
  1558
  have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)"
eberlm@66456
  1559
    using l tendsto_add_ereal_general a b by fastforce
eberlm@66456
  1560
  moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
eberlm@66456
  1561
  ultimately have "(w o a) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)" by simp
eberlm@66456
  1562
  then have "liminf w = liminf (u o r) + liminf (v o r o s)" using l(1) LIMSEQ_unique by blast
eberlm@66456
  1563
  then have "liminf w \<ge> liminf u + liminf v"
nipkow@68752
  1564
    using \<open>liminf (u o r) \<ge> liminf u\<close> \<open>liminf (v o r o s) \<ge> liminf v\<close> add_mono by simp
eberlm@66456
  1565
  then show ?thesis unfolding w_def by simp
eberlm@66456
  1566
qed
eberlm@66456
  1567
immler@69681
  1568
lemma ereal_limsup_lim_add:
eberlm@66456
  1569
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1570
  assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
eberlm@66456
  1571
  shows "limsup (\<lambda>n. u n + v n) = a + limsup v"
immler@69681
  1572
proof -
eberlm@66456
  1573
  have "limsup u = a" using assms(1) using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
eberlm@66456
  1574
  have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
eberlm@66456
  1575
  then have "limsup (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
eberlm@66456
  1576
eberlm@66456
  1577
  have "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
eberlm@66456
  1578
    by (rule ereal_limsup_add_mono)
eberlm@66456
  1579
  then have up: "limsup (\<lambda>n. u n + v n) \<le> a + limsup v" using \<open>limsup u = a\<close> by simp
eberlm@66456
  1580
eberlm@66456
  1581
  have a: "limsup (\<lambda>n. (u n + v n) + (-u n)) \<le> limsup (\<lambda>n. u n + v n) + limsup (\<lambda>n. -u n)"
eberlm@66456
  1582
    by (rule ereal_limsup_add_mono)
eberlm@66456
  1583
  have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
eberlm@66456
  1584
    real_lim_then_eventually_real by auto
eberlm@66456
  1585
  moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
eberlm@66456
  1586
    by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
eberlm@66456
  1587
  ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
eberlm@66456
  1588
    by (metis (mono_tags, lifting) eventually_mono)
eberlm@66456
  1589
  moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
eberlm@66456
  1590
    by (metis add.commute add.left_commute add.left_neutral)
eberlm@66456
  1591
  ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
eberlm@66456
  1592
    using eventually_mono by force
eberlm@66456
  1593
  then have "limsup v = limsup (\<lambda>n. u n + v n + (-u n))" using Limsup_eq by force
eberlm@66456
  1594
  then have "limsup v \<le> limsup (\<lambda>n. u n + v n) -a" using a \<open>limsup (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def)
eberlm@66456
  1595
  then have "limsup (\<lambda>n. u n + v n) \<ge> a + limsup v" using assms(2) by (metis add.commute ereal_le_minus)
eberlm@66456
  1596
  then show ?thesis using up by simp
eberlm@66456
  1597
qed
eberlm@66456
  1598
immler@69681
  1599
lemma ereal_limsup_lim_mult:
eberlm@66456
  1600
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1601
  assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
eberlm@66456
  1602
  shows "limsup (\<lambda>n. u n * v n) = a * limsup v"
immler@69681
  1603
proof -
eberlm@66456
  1604
  define w where "w = (\<lambda>n. u n * v n)"
eberlm@66456
  1605
  obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> limsup v" using limsup_subseq_lim by auto
eberlm@66456
  1606
  have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
eberlm@66456
  1607
  with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * limsup v" using assms(2) assms(3) by auto
eberlm@66456
  1608
  moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
eberlm@66456
  1609
  ultimately have "(w o r) \<longlonglongrightarrow> a * limsup v" unfolding w_def by presburger
eberlm@66456
  1610
  then have "limsup (w o r) = a * limsup v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
eberlm@66456
  1611
  then have I: "limsup w \<ge> a * limsup v" by (metis limsup_subseq_mono r(1))
eberlm@66456
  1612
eberlm@66456
  1613
  obtain s where s: "strict_mono s" "(w o s) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
eberlm@66456
  1614
  have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
eberlm@66456
  1615
  have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
eberlm@66456
  1616
  moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
eberlm@66456
  1617
  moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
eberlm@66456
  1618
    unfolding w_def using that by (auto simp add: ereal_divide_eq)
eberlm@66456
  1619
  ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
eberlm@66456
  1620
  moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (limsup w) / a"
eberlm@66456
  1621
    apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
eberlm@66456
  1622
  ultimately have "(v o s) \<longlonglongrightarrow> (limsup w) / a" using Lim_transform_eventually by fastforce
eberlm@66456
  1623
  then have "limsup (v o s) = (limsup w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
eberlm@66456
  1624
  then have "limsup v \<ge> (limsup w) / a" by (metis limsup_subseq_mono s(1))
eberlm@66456
  1625
  then have "a * limsup v \<ge> limsup w" using assms(2) assms(3) by (simp add: ereal_divide_le_pos)
eberlm@66456
  1626
  then show ?thesis using I unfolding w_def by auto
eberlm@66456
  1627
qed
eberlm@66456
  1628
immler@69681
  1629
lemma ereal_liminf_lim_mult:
eberlm@66456
  1630
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1631
  assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
eberlm@66456
  1632
  shows "liminf (\<lambda>n. u n * v n) = a * liminf v"
immler@69681
  1633
proof -
eberlm@66456
  1634
  define w where "w = (\<lambda>n. u n * v n)"
eberlm@66456
  1635
  obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> liminf v" using liminf_subseq_lim by auto
eberlm@66456
  1636
  have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
eberlm@66456
  1637
  with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * liminf v" using assms(2) assms(3) by auto
eberlm@66456
  1638
  moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
eberlm@66456
  1639
  ultimately have "(w o r) \<longlonglongrightarrow> a * liminf v" unfolding w_def by presburger
eberlm@66456
  1640
  then have "liminf (w o r) = a * liminf v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
eberlm@66456
  1641
  then have I: "liminf w \<le> a * liminf v" by (metis liminf_subseq_mono r(1))
eberlm@66456
  1642
eberlm@66456
  1643
  obtain s where s: "strict_mono s" "(w o s) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
eberlm@66456
  1644
  have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
eberlm@66456
  1645
  have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
eberlm@66456
  1646
  moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
eberlm@66456
  1647
  moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
eberlm@66456
  1648
    unfolding w_def using that by (auto simp add: ereal_divide_eq)
eberlm@66456
  1649
  ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
eberlm@66456
  1650
  moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (liminf w) / a"
eberlm@66456
  1651
    apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
eberlm@66456
  1652
  ultimately have "(v o s) \<longlonglongrightarrow> (liminf w) / a" using Lim_transform_eventually by fastforce
eberlm@66456
  1653
  then have "liminf (v o s) = (liminf w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
eberlm@66456
  1654
  then have "liminf v \<le> (liminf w) / a" by (metis liminf_subseq_mono s(1))
eberlm@66456
  1655
  then have "a * liminf v \<le> liminf w" using assms(2) assms(3) by (simp add: ereal_le_divide_pos)
eberlm@66456
  1656
  then show ?thesis using I unfolding w_def by auto
eberlm@66456
  1657
qed
eberlm@66456
  1658
immler@69681
  1659
lemma ereal_liminf_lim_add:
eberlm@66456
  1660
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1661
  assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
eberlm@66456
  1662
  shows "liminf (\<lambda>n. u n + v n) = a + liminf v"
immler@69681
  1663
proof -
eberlm@66456
  1664
  have "liminf u = a" using assms(1) tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
eberlm@66456
  1665
  then have *: "abs(liminf u) \<noteq> \<infinity>" using assms(2) by auto
eberlm@66456
  1666
  have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
eberlm@66456
  1667
  then have "liminf (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
eberlm@66456
  1668
  then have **: "abs(liminf (\<lambda>n. -u n)) \<noteq> \<infinity>" using assms(2) by auto
eberlm@66456
  1669
eberlm@66456
  1670
  have "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
eberlm@66456
  1671
    apply (rule ereal_liminf_add_mono) using * by auto
eberlm@66456
  1672
  then have up: "liminf (\<lambda>n. u n + v n) \<ge> a + liminf v" using \<open>liminf u = a\<close> by simp
eberlm@66456
  1673
eberlm@66456
  1674
  have a: "liminf (\<lambda>n. (u n + v n) + (-u n)) \<ge> liminf (\<lambda>n. u n + v n) + liminf (\<lambda>n. -u n)"
eberlm@66456
  1675
    apply (rule ereal_liminf_add_mono) using ** by auto
eberlm@66456
  1676
  have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
eberlm@66456
  1677
    real_lim_then_eventually_real by auto
eberlm@66456
  1678
  moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
eberlm@66456
  1679
    by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
eberlm@66456
  1680
  ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
eberlm@66456
  1681
    by (metis (mono_tags, lifting) eventually_mono)
eberlm@66456
  1682
  moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
eberlm@66456
  1683
    by (metis add.commute add.left_commute add.left_neutral)
eberlm@66456
  1684
  ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
eberlm@66456
  1685
    using eventually_mono by force
eberlm@66456
  1686
  then have "liminf v = liminf (\<lambda>n. u n + v n + (-u n))" using Liminf_eq by force
eberlm@66456
  1687
  then have "liminf v \<ge> liminf (\<lambda>n. u n + v n) -a" using a \<open>liminf (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def)
eberlm@66456
  1688
  then have "liminf (\<lambda>n. u n + v n) \<le> a + liminf v" using assms(2) by (metis add.commute ereal_minus_le)
eberlm@66456
  1689
  then show ?thesis using up by simp
eberlm@66456
  1690
qed
eberlm@66456
  1691
immler@69681
  1692
lemma ereal_liminf_limsup_add:
eberlm@66456
  1693
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1694
  shows "liminf (\<lambda>n. u n + v n) \<le> liminf u + limsup v"
immler@69681
  1695
proof (cases)
eberlm@66456
  1696
  assume "limsup v = \<infinity> \<or> liminf u = \<infinity>"
eberlm@66456
  1697
  then show ?thesis by auto
eberlm@66456
  1698
next
eberlm@66456
  1699
  assume "\<not>(limsup v = \<infinity> \<or> liminf u = \<infinity>)"
eberlm@66456
  1700
  then have "limsup v < \<infinity>" "liminf u < \<infinity>" by auto
eberlm@66456
  1701
eberlm@66456
  1702
  define w where "w = (\<lambda>n. u n + v n)"
eberlm@66456
  1703
  obtain r where r: "strict_mono r" "(u o r) \<longlonglongrightarrow> liminf u" using liminf_subseq_lim by auto
eberlm@66456
  1704
  obtain s where s: "strict_mono s" "(w o r o s) \<longlonglongrightarrow> liminf (w o r)" using liminf_subseq_lim by auto
eberlm@66456
  1705
  obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
eberlm@66456
  1706
eberlm@66456
  1707
  define a where "a = r o s o t"
eberlm@66456
  1708
  have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
eberlm@66456
  1709
  have l:"(u o a) \<longlonglongrightarrow> liminf u"
eberlm@66456
  1710
         "(w o a) \<longlonglongrightarrow> liminf (w o r)"
eberlm@66456
  1711
         "(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
eberlm@66456
  1712
  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
eberlm@66456
  1713
  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
eberlm@66456
  1714
  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
eberlm@66456
  1715
  done
eberlm@66456
  1716
eberlm@66456
  1717
  have "liminf (w o r) \<ge> liminf w" by (simp add: liminf_subseq_mono r(1))
eberlm@66456
  1718
  have "limsup (v o r o s) \<le> limsup v" 
eberlm@66456
  1719
    by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
eberlm@66456
  1720
  then have b: "limsup (v o r o s) < \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto
eberlm@66456
  1721
eberlm@66456
  1722
  have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf u + limsup (v o r o s)"
eberlm@66456
  1723
    apply (rule tendsto_add_ereal_general) using b \<open>liminf u < \<infinity>\<close> l(1) l(3) by force+
eberlm@66456
  1724
  moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
eberlm@66456
  1725
  ultimately have "(w o a) \<longlonglongrightarrow> liminf u + limsup (v o r o s)" by simp
eberlm@66456
  1726
  then have "liminf (w o r) = liminf u + limsup (v o r o s)" using l(2) using LIMSEQ_unique by blast
eberlm@66456
  1727
  then have "liminf w \<le> liminf u + limsup v"
eberlm@66456
  1728
    using \<open>liminf (w o r) \<ge> liminf w\<close> \<open>limsup (v o r o s) \<le> limsup v\<close>
eberlm@66456
  1729
    by (metis add_mono_thms_linordered_semiring(2) le_less_trans not_less)
eberlm@66456
  1730
  then show ?thesis unfolding w_def by simp
eberlm@66456
  1731
qed
eberlm@66456
  1732
eberlm@66456
  1733
lemma ereal_liminf_limsup_minus:
eberlm@66456
  1734
  fixes u v::"nat \<Rightarrow> ereal"
eberlm@66456
  1735
  shows "liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
eberlm@66456
  1736
  unfolding minus_ereal_def
eberlm@66456
  1737
  apply (subst add.commute)
eberlm@66456
  1738
  apply (rule order_trans[OF ereal_liminf_limsup_add])
eberlm@66456
  1739
  using ereal_Limsup_uminus[of sequentially "\<lambda>n. - v n"]
eberlm@66456
  1740
  apply (simp add: add.commute)
eberlm@66456
  1741
  done
eberlm@66456
  1742
eberlm@66456
  1743
immler@69681
  1744
lemma liminf_minus_ennreal:
eberlm@66456
  1745
  fixes u v::"nat \<Rightarrow> ennreal"
eberlm@66456
  1746
  shows "(\<And>n. v n \<le> u n) \<Longrightarrow> liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
eberlm@66456
  1747
  unfolding liminf_SUP_INF limsup_INF_SUP
eberlm@66456
  1748
  including ennreal.lifting
immler@69681
  1749
proof (transfer, clarsimp)
eberlm@66456
  1750
  fix v u :: "nat \<Rightarrow> ereal" assume *: "\<forall>x. 0 \<le> v x" "\<forall>x. 0 \<le> u x" "\<And>n. v n \<le> u n"
eberlm@66456
  1751
  moreover have "0 \<le> limsup u - limsup v"
eberlm@66456
  1752
    using * by (intro ereal_diff_positive Limsup_mono always_eventually) simp
haftmann@69313
  1753
  moreover have "0 \<le> Sup (u ` {x..})" for x
eberlm@66456
  1754
    using * by (intro SUP_upper2[of x]) auto
haftmann@69313
  1755
  moreover have "0 \<le> Sup (v ` {x..})" for x
eberlm@66456
  1756
    using * by (intro SUP_upper2[of x]) auto
haftmann@69260
  1757
  ultimately show "(SUP n. INF n\<in>{n..}. max 0 (u n - v n))
haftmann@69313
  1758
            \<le> max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))"
eberlm@66456
  1759
    by (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus)
eberlm@66456
  1760
qed
eberlm@66456
  1761
immler@69683
  1762
subsection "Relate extended reals and the indicator function"
hoelzl@57446
  1763
hoelzl@59000
  1764
lemma ereal_indicator_le_0: "(indicator S x::ereal) \<le> 0 \<longleftrightarrow> x \<notin> S"
hoelzl@59000
  1765
  by (auto split: split_indicator simp: one_ereal_def)
hoelzl@59000
  1766
hoelzl@57446
  1767
lemma ereal_indicator: "ereal (indicator A x) = indicator A x"
hoelzl@57446
  1768
  by (auto simp: indicator_def one_ereal_def)
hoelzl@57446
  1769
hoelzl@57446
  1770
lemma ereal_mult_indicator: "ereal (x * indicator A y) = ereal x * indicator A y"
hoelzl@57446
  1771
  by (simp split: split_indicator)
hoelzl@57446
  1772
hoelzl@57446
  1773
lemma ereal_indicator_mult: "ereal (indicator A y * x) = indicator A y * ereal x"
hoelzl@57446
  1774
  by (simp split: split_indicator)
hoelzl@57446
  1775
hoelzl@57446
  1776
lemma ereal_indicator_nonneg[simp, intro]: "0 \<le> (indicator A x ::ereal)"
hoelzl@57446
  1777
  unfolding indicator_def by auto
hoelzl@57446
  1778
hoelzl@59425
  1779
lemma indicator_inter_arith_ereal: "indicator A x * indicator B x = (indicator (A \<inter> B) x :: ereal)"
hoelzl@59425
  1780
  by (simp split: split_indicator)
hoelzl@59425
  1781
huffman@44125
  1782
end