src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy
author hoelzl
Tue Nov 05 09:44:57 2013 +0100 (2013-11-05)
changeset 54257 5c7a3b6b05a9
parent 53788 b319a0c8b8a2
child 54258 adfc759263ab
permissions -rw-r--r--
generalize SUP and INF to the syntactic type classes Sup and Inf
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(*  Title:      HOL/Multivariate_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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header {* Limits on the Extended real number line *}
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theory Extended_Real_Limits
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  imports Topology_Euclidean_Space "~~/src/HOL/Library/Extended_Real"
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begin
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lemma convergent_limsup_cl:
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  fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
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  shows "convergent X \<Longrightarrow> limsup X = lim X"
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  by (auto simp: convergent_def limI lim_imp_Limsup)
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lemma lim_increasing_cl:
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  assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<ge> f m"
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  obtains l where "f ----> (l::'a::{complete_linorder,linorder_topology})"
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proof
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  show "f ----> (SUP n. f n)"
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    using assms
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    by (intro increasing_tendsto)
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       (auto simp: SUP_upper eventually_sequentially less_SUP_iff intro: less_le_trans)
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qed
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lemma lim_decreasing_cl:
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  assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<le> f m"
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  obtains l where "f ----> (l::'a::{complete_linorder,linorder_topology})"
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proof
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  show "f ----> (INF n. f n)"
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    using assms
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    by (intro decreasing_tendsto)
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       (auto simp: INF_lower eventually_sequentially INF_less_iff intro: le_less_trans)
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qed
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lemma compact_complete_linorder:
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  fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
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  shows "\<exists>l r. subseq r \<and> (X \<circ> r) ----> l"
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proof -
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  obtain r where "subseq r" and mono: "monoseq (X \<circ> r)"
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    using seq_monosub[of X]
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    unfolding comp_def
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    by auto
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  then have "(\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) m \<le> (X \<circ> r) n) \<or> (\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) n \<le> (X \<circ> r) m)"
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    by (auto simp add: monoseq_def)
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  then obtain l where "(X \<circ> r) ----> l"
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     using lim_increasing_cl[of "X \<circ> r"] lim_decreasing_cl[of "X \<circ> r"]
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     by auto
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  then show ?thesis
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    using `subseq r` by auto
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qed
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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 closed_inter_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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lemma ereal_dense3:
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  fixes x y :: ereal
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  shows "x < y \<Longrightarrow> \<exists>r::rat. x < real_of_rat r \<and> real_of_rat r < y"
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proof (cases x y rule: ereal2_cases, simp_all)
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  fix r q :: real
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  assume "r < q"
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  from Rats_dense_in_real[OF this] show "\<exists>x. r < real_of_rat x \<and> real_of_rat x < q"
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    by (fastforce simp: Rats_def)
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next
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  fix r :: real
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  show "\<exists>x. r < real_of_rat x" "\<exists>x. real_of_rat x < r"
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    using gt_ex[of r] lt_ex[of r] Rats_dense_in_real
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    by (auto simp: Rats_def)
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qed
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instance ereal :: second_countable_topology
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proof (default, 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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lemma continuous_on_ereal[intro, simp]: "continuous_on A ereal"
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  unfolding continuous_on_topological open_ereal_def
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  by auto
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lemma continuous_at_ereal[intro, simp]: "continuous (at x) ereal"
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  using continuous_on_eq_continuous_at[of UNIV]
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  by auto
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lemma continuous_within_ereal[intro, simp]: "x \<in> A \<Longrightarrow> continuous (at x within A) ereal"
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  using continuous_on_eq_continuous_within[of A]
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  by auto
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lemma ereal_open_uminus:
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  fixes S :: "ereal set"
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  assumes "open S"
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  shows "open (uminus ` S)"
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  using `open S`[unfolded open_generated_order]
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proof induct
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  have "range uminus = (UNIV :: ereal set)"
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    by (auto simp: image_iff ereal_uminus_eq_reorder)
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  then show "open (range uminus :: ereal set)"
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    by simp
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qed (auto simp add: image_Union image_Int)
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lemma ereal_uminus_complement:
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  fixes S :: "ereal set"
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  shows "uminus ` (- S) = - uminus ` S"
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  by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)
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lemma ereal_closed_uminus:
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  fixes S :: "ereal set"
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  assumes "closed S"
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  shows "closed (uminus ` S)"
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  using assms
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  unfolding closed_def ereal_uminus_complement[symmetric]
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  by (rule ereal_open_uminus)
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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 `S \<noteq> {}`)
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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: {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_affinity_pos:
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  fixes S :: "ereal set"
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  assumes "open S"
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    and m: "m \<noteq> \<infinity>" "0 < m"
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    and t: "\<bar>t\<bar> \<noteq> \<infinity>"
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  shows "open ((\<lambda>x. m * x + t) ` S)"
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proof -
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  obtain r where r[simp]: "m = ereal r"
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    using m by (cases m) auto
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  obtain p where p[simp]: "t = ereal p"
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    using t by auto
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  have "r \<noteq> 0" "0 < r" and m': "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0"
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    using m by auto
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  from `open S` [THEN ereal_openE] guess l u . note T = this
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  let ?f = "(\<lambda>x. m * x + t)"
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  show ?thesis
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    unfolding open_ereal_def
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  proof (intro conjI impI exI subsetI)
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    have "ereal -` ?f ` S = (\<lambda>x. r * x + p) ` (ereal -` S)"
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    proof safe
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      fix x y
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      assume "ereal y = m * x + t" "x \<in> S"
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      then show "y \<in> (\<lambda>x. r * x + p) ` ereal -` S"
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        using `r \<noteq> 0` by (cases x) (auto intro!: image_eqI[of _ _ "real x"] split: split_if_asm)
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    qed force
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    then show "open (ereal -` ?f ` S)"
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      using open_affinity[OF T(1) `r \<noteq> 0`]
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      by (auto simp: ac_simps)
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  next
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    assume "\<infinity> \<in> ?f`S"
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    with `0 < r` have "\<infinity> \<in> S"
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      by auto
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    fix x
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    assume "x \<in> {ereal (r * l + p)<..}"
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    then have [simp]: "ereal (r * l + p) < x"
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      by auto
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    show "x \<in> ?f`S"
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    proof (rule image_eqI)
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      show "x = m * ((x - t) / m) + t"
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        using m t
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        by (cases rule: ereal3_cases[of m x t]) auto
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      have "ereal l < (x - t) / m"
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        using m t
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        by (simp add: ereal_less_divide_pos ereal_less_minus)
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      then show "(x - t) / m \<in> S"
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        using T(2)[OF `\<infinity> \<in> S`] by auto
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    qed
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  next
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    assume "-\<infinity> \<in> ?f ` S"
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    with `0 < r` have "-\<infinity> \<in> S"
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      by auto
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    fix x assume "x \<in> {..<ereal (r * u + p)}"
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    then have [simp]: "x < ereal (r * u + p)"
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      by auto
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    show "x \<in> ?f`S"
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    proof (rule image_eqI)
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      show "x = m * ((x - t) / m) + t"
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        using m t
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        by (cases rule: ereal3_cases[of m x t]) auto
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      have "(x - t)/m < ereal u"
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        using m t
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        by (simp add: ereal_divide_less_pos ereal_minus_less)
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      then show "(x - t)/m \<in> S"
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        using T(3)[OF `-\<infinity> \<in> S`]
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        by auto
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    qed
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   312
  qed
hoelzl@41980
   313
qed
hoelzl@41980
   314
hoelzl@43920
   315
lemma ereal_open_affinity:
hoelzl@43923
   316
  fixes S :: "ereal set"
wenzelm@49664
   317
  assumes "open S"
wenzelm@49664
   318
    and m: "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0"
wenzelm@49664
   319
    and t: "\<bar>t\<bar> \<noteq> \<infinity>"
hoelzl@41980
   320
  shows "open ((\<lambda>x. m * x + t) ` S)"
hoelzl@41980
   321
proof cases
wenzelm@49664
   322
  assume "0 < m"
wenzelm@49664
   323
  then show ?thesis
wenzelm@53788
   324
    using ereal_open_affinity_pos[OF `open S` _ _ t, of m] m
wenzelm@53788
   325
    by auto
hoelzl@41980
   326
next
hoelzl@41980
   327
  assume "\<not> 0 < m" then
wenzelm@53788
   328
  have "0 < -m"
wenzelm@53788
   329
    using `m \<noteq> 0`
wenzelm@53788
   330
    by (cases m) auto
wenzelm@53788
   331
  then have m: "-m \<noteq> \<infinity>" "0 < -m"
wenzelm@53788
   332
    using `\<bar>m\<bar> \<noteq> \<infinity>`
hoelzl@43920
   333
    by (auto simp: ereal_uminus_eq_reorder)
wenzelm@53788
   334
  from ereal_open_affinity_pos[OF ereal_open_uminus[OF `open S`] m t] show ?thesis
wenzelm@53788
   335
    unfolding image_image by simp
hoelzl@41980
   336
qed
hoelzl@41980
   337
hoelzl@43920
   338
lemma ereal_lim_mult:
hoelzl@43920
   339
  fixes X :: "'a \<Rightarrow> ereal"
wenzelm@49664
   340
  assumes lim: "(X ---> L) net"
wenzelm@49664
   341
    and a: "\<bar>a\<bar> \<noteq> \<infinity>"
hoelzl@41980
   342
  shows "((\<lambda>i. a * X i) ---> a * L) net"
hoelzl@41980
   343
proof cases
hoelzl@41980
   344
  assume "a \<noteq> 0"
hoelzl@41980
   345
  show ?thesis
hoelzl@41980
   346
  proof (rule topological_tendstoI)
wenzelm@49664
   347
    fix S
wenzelm@53788
   348
    assume "open S" and "a * L \<in> S"
hoelzl@41980
   349
    have "a * L / a = L"
wenzelm@53788
   350
      using `a \<noteq> 0` a
wenzelm@53788
   351
      by (cases rule: ereal2_cases[of a L]) auto
hoelzl@41980
   352
    then have L: "L \<in> ((\<lambda>x. x / a) ` S)"
wenzelm@53788
   353
      using `a * L \<in> S`
wenzelm@53788
   354
      by (force simp: image_iff)
hoelzl@41980
   355
    moreover have "open ((\<lambda>x. x / a) ` S)"
hoelzl@43920
   356
      using ereal_open_affinity[OF `open S`, of "inverse a" 0] `a \<noteq> 0` a
hoelzl@43920
   357
      by (auto simp: ereal_divide_eq ereal_inverse_eq_0 divide_ereal_def ac_simps)
hoelzl@41980
   358
    note * = lim[THEN topological_tendstoD, OF this L]
wenzelm@53788
   359
    {
wenzelm@53788
   360
      fix x
wenzelm@49664
   361
      from a `a \<noteq> 0` have "a * (x / a) = x"
wenzelm@53788
   362
        by (cases rule: ereal2_cases[of a x]) auto
wenzelm@53788
   363
    }
hoelzl@41980
   364
    note this[simp]
hoelzl@41980
   365
    show "eventually (\<lambda>x. a * X x \<in> S) net"
hoelzl@41980
   366
      by (rule eventually_mono[OF _ *]) auto
hoelzl@41980
   367
  qed
noschinl@44918
   368
qed auto
hoelzl@41980
   369
hoelzl@43920
   370
lemma ereal_lim_uminus:
wenzelm@49664
   371
  fixes X :: "'a \<Rightarrow> ereal"
wenzelm@53788
   372
  shows "((\<lambda>i. - X i) ---> - L) net \<longleftrightarrow> (X ---> L) net"
hoelzl@43920
   373
  using ereal_lim_mult[of X L net "ereal (-1)"]
wenzelm@49664
   374
    ereal_lim_mult[of "(\<lambda>i. - X i)" "-L" net "ereal (-1)"]
hoelzl@41980
   375
  by (auto simp add: algebra_simps)
hoelzl@41980
   376
wenzelm@53788
   377
lemma ereal_open_atLeast:
wenzelm@53788
   378
  fixes x :: ereal
wenzelm@53788
   379
  shows "open {x..} \<longleftrightarrow> x = -\<infinity>"
hoelzl@41980
   380
proof
wenzelm@53788
   381
  assume "x = -\<infinity>"
wenzelm@53788
   382
  then have "{x..} = UNIV"
wenzelm@53788
   383
    by auto
wenzelm@53788
   384
  then show "open {x..}"
wenzelm@53788
   385
    by auto
hoelzl@41980
   386
next
hoelzl@41980
   387
  assume "open {x..}"
wenzelm@53788
   388
  then have "open {x..} \<and> closed {x..}"
wenzelm@53788
   389
    by auto
wenzelm@53788
   390
  then have "{x..} = UNIV"
wenzelm@53788
   391
    unfolding ereal_open_closed by auto
wenzelm@53788
   392
  then show "x = -\<infinity>"
wenzelm@53788
   393
    by (simp add: bot_ereal_def atLeast_eq_UNIV_iff)
hoelzl@41980
   394
qed
hoelzl@41980
   395
wenzelm@53788
   396
lemma open_uminus_iff:
wenzelm@53788
   397
  fixes S :: "ereal set"
wenzelm@53788
   398
  shows "open (uminus ` S) \<longleftrightarrow> open S"
wenzelm@53788
   399
  using ereal_open_uminus[of S] ereal_open_uminus[of "uminus ` S"]
wenzelm@53788
   400
  by auto
hoelzl@41980
   401
hoelzl@43920
   402
lemma ereal_Liminf_uminus:
wenzelm@53788
   403
  fixes f :: "'a \<Rightarrow> ereal"
wenzelm@53788
   404
  shows "Liminf net (\<lambda>x. - (f x)) = - Limsup net f"
hoelzl@43920
   405
  using ereal_Limsup_uminus[of _ "(\<lambda>x. - (f x))"] by auto
hoelzl@41980
   406
hoelzl@43920
   407
lemma ereal_Lim_uminus:
wenzelm@49664
   408
  fixes f :: "'a \<Rightarrow> ereal"
wenzelm@49664
   409
  shows "(f ---> f0) net \<longleftrightarrow> ((\<lambda>x. - f x) ---> - f0) net"
hoelzl@41980
   410
  using
hoelzl@43920
   411
    ereal_lim_mult[of f f0 net "- 1"]
hoelzl@43920
   412
    ereal_lim_mult[of "\<lambda>x. - (f x)" "-f0" net "- 1"]
hoelzl@43920
   413
  by (auto simp: ereal_uminus_reorder)
hoelzl@41980
   414
hoelzl@41980
   415
lemma Liminf_PInfty:
hoelzl@43920
   416
  fixes f :: "'a \<Rightarrow> ereal"
hoelzl@41980
   417
  assumes "\<not> trivial_limit net"
hoelzl@41980
   418
  shows "(f ---> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
wenzelm@53788
   419
  unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
wenzelm@53788
   420
  using Liminf_le_Limsup[OF assms, of f]
wenzelm@53788
   421
  by auto
hoelzl@41980
   422
hoelzl@41980
   423
lemma Limsup_MInfty:
hoelzl@43920
   424
  fixes f :: "'a \<Rightarrow> ereal"
hoelzl@41980
   425
  assumes "\<not> trivial_limit net"
hoelzl@41980
   426
  shows "(f ---> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
wenzelm@53788
   427
  unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
wenzelm@53788
   428
  using Liminf_le_Limsup[OF assms, of f]
wenzelm@53788
   429
  by auto
hoelzl@41980
   430
hoelzl@50104
   431
lemma convergent_ereal:
wenzelm@53788
   432
  fixes X :: "nat \<Rightarrow> 'a :: {complete_linorder,linorder_topology}"
hoelzl@50104
   433
  shows "convergent X \<longleftrightarrow> limsup X = liminf X"
hoelzl@51340
   434
  using tendsto_iff_Liminf_eq_Limsup[of sequentially]
hoelzl@50104
   435
  by (auto simp: convergent_def)
hoelzl@50104
   436
hoelzl@41980
   437
lemma liminf_PInfty:
hoelzl@51351
   438
  fixes X :: "nat \<Rightarrow> ereal"
hoelzl@51351
   439
  shows "X ----> \<infinity> \<longleftrightarrow> liminf X = \<infinity>"
wenzelm@49664
   440
  by (metis Liminf_PInfty trivial_limit_sequentially)
hoelzl@41980
   441
hoelzl@41980
   442
lemma limsup_MInfty:
hoelzl@51351
   443
  fixes X :: "nat \<Rightarrow> ereal"
hoelzl@51351
   444
  shows "X ----> -\<infinity> \<longleftrightarrow> limsup X = -\<infinity>"
wenzelm@49664
   445
  by (metis Limsup_MInfty trivial_limit_sequentially)
hoelzl@41980
   446
hoelzl@43920
   447
lemma ereal_lim_mono:
wenzelm@53788
   448
  fixes X Y :: "nat \<Rightarrow> 'a::linorder_topology"
wenzelm@53788
   449
  assumes "\<And>n. N \<le> n \<Longrightarrow> X n \<le> Y n"
wenzelm@53788
   450
    and "X ----> x"
wenzelm@53788
   451
    and "Y ----> y"
wenzelm@53788
   452
  shows "x \<le> y"
hoelzl@51000
   453
  using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto
hoelzl@41980
   454
hoelzl@43920
   455
lemma incseq_le_ereal:
hoelzl@51351
   456
  fixes X :: "nat \<Rightarrow> 'a::linorder_topology"
wenzelm@53788
   457
  assumes inc: "incseq X"
wenzelm@53788
   458
    and lim: "X ----> L"
hoelzl@41980
   459
  shows "X N \<le> L"
wenzelm@53788
   460
  using inc
wenzelm@53788
   461
  by (intro ereal_lim_mono[of N, OF _ tendsto_const lim]) (simp add: incseq_def)
hoelzl@41980
   462
wenzelm@49664
   463
lemma decseq_ge_ereal:
wenzelm@49664
   464
  assumes dec: "decseq X"
hoelzl@51351
   465
    and lim: "X ----> (L::'a::linorder_topology)"
wenzelm@53788
   466
  shows "X N \<ge> L"
wenzelm@49664
   467
  using dec by (intro ereal_lim_mono[of N, OF _ lim tendsto_const]) (simp add: decseq_def)
hoelzl@41980
   468
hoelzl@41980
   469
lemma bounded_abs:
wenzelm@53788
   470
  fixes a :: real
wenzelm@53788
   471
  assumes "a \<le> x"
wenzelm@53788
   472
    and "x \<le> b"
wenzelm@53788
   473
  shows "abs x \<le> max (abs a) (abs b)"
wenzelm@49664
   474
  by (metis abs_less_iff assms leI le_max_iff_disj
wenzelm@49664
   475
    less_eq_real_def less_le_not_le less_minus_iff minus_minus)
hoelzl@41980
   476
hoelzl@43920
   477
lemma ereal_Sup_lim:
wenzelm@53788
   478
  fixes a :: "'a::{complete_linorder,linorder_topology}"
wenzelm@53788
   479
  assumes "\<And>n. b n \<in> s"
wenzelm@53788
   480
    and "b ----> a"
hoelzl@41980
   481
  shows "a \<le> Sup s"
wenzelm@49664
   482
  by (metis Lim_bounded_ereal assms complete_lattice_class.Sup_upper)
hoelzl@41980
   483
hoelzl@43920
   484
lemma ereal_Inf_lim:
wenzelm@53788
   485
  fixes a :: "'a::{complete_linorder,linorder_topology}"
wenzelm@53788
   486
  assumes "\<And>n. b n \<in> s"
wenzelm@53788
   487
    and "b ----> a"
hoelzl@41980
   488
  shows "Inf s \<le> a"
wenzelm@49664
   489
  by (metis Lim_bounded2_ereal assms complete_lattice_class.Inf_lower)
hoelzl@41980
   490
hoelzl@43920
   491
lemma SUP_Lim_ereal:
wenzelm@53788
   492
  fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
wenzelm@53788
   493
  assumes inc: "incseq X"
wenzelm@53788
   494
    and l: "X ----> l"
wenzelm@53788
   495
  shows "(SUP n. X n) = l"
wenzelm@53788
   496
  using LIMSEQ_SUP[OF inc] tendsto_unique[OF trivial_limit_sequentially l]
wenzelm@53788
   497
  by simp
hoelzl@41980
   498
hoelzl@51351
   499
lemma INF_Lim_ereal:
wenzelm@53788
   500
  fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
wenzelm@53788
   501
  assumes dec: "decseq X"
wenzelm@53788
   502
    and l: "X ----> l"
wenzelm@53788
   503
  shows "(INF n. X n) = l"
wenzelm@53788
   504
  using LIMSEQ_INF[OF dec] tendsto_unique[OF trivial_limit_sequentially l]
wenzelm@53788
   505
  by simp
hoelzl@41980
   506
hoelzl@41980
   507
lemma SUP_eq_LIMSEQ:
hoelzl@41980
   508
  assumes "mono f"
hoelzl@43920
   509
  shows "(SUP n. ereal (f n)) = ereal x \<longleftrightarrow> f ----> x"
hoelzl@41980
   510
proof
hoelzl@43920
   511
  have inc: "incseq (\<lambda>i. ereal (f i))"
hoelzl@41980
   512
    using `mono f` unfolding mono_def incseq_def by auto
wenzelm@53788
   513
  {
wenzelm@53788
   514
    assume "f ----> x"
wenzelm@53788
   515
    then have "(\<lambda>i. ereal (f i)) ----> ereal x"
wenzelm@53788
   516
      by auto
wenzelm@53788
   517
    from SUP_Lim_ereal[OF inc this] show "(SUP n. ereal (f n)) = ereal x" .
wenzelm@53788
   518
  next
wenzelm@53788
   519
    assume "(SUP n. ereal (f n)) = ereal x"
wenzelm@53788
   520
    with LIMSEQ_SUP[OF inc] show "f ----> x" by auto
wenzelm@53788
   521
  }
hoelzl@41980
   522
qed
hoelzl@41980
   523
hoelzl@43920
   524
lemma liminf_ereal_cminus:
wenzelm@49664
   525
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
   526
  assumes "c \<noteq> -\<infinity>"
hoelzl@42950
   527
  shows "liminf (\<lambda>x. c - f x) = c - limsup f"
hoelzl@42950
   528
proof (cases c)
wenzelm@49664
   529
  case PInf
wenzelm@53788
   530
  then show ?thesis
wenzelm@53788
   531
    by (simp add: Liminf_const)
hoelzl@42950
   532
next
wenzelm@49664
   533
  case (real r)
wenzelm@49664
   534
  then show ?thesis
hoelzl@42950
   535
    unfolding liminf_SUPR_INFI limsup_INFI_SUPR
hoelzl@43920
   536
    apply (subst INFI_ereal_cminus)
hoelzl@42950
   537
    apply auto
hoelzl@43920
   538
    apply (subst SUPR_ereal_cminus)
hoelzl@42950
   539
    apply auto
hoelzl@42950
   540
    done
hoelzl@42950
   541
qed (insert `c \<noteq> -\<infinity>`, simp)
hoelzl@42950
   542
wenzelm@49664
   543
hoelzl@41980
   544
subsubsection {* Continuity *}
hoelzl@41980
   545
hoelzl@43920
   546
lemma continuous_at_of_ereal:
hoelzl@43920
   547
  fixes x0 :: ereal
hoelzl@41980
   548
  assumes "\<bar>x0\<bar> \<noteq> \<infinity>"
hoelzl@41980
   549
  shows "continuous (at x0) real"
wenzelm@49664
   550
proof -
wenzelm@53788
   551
  {
wenzelm@53788
   552
    fix T
wenzelm@53788
   553
    assume T: "open T" "real x0 \<in> T"
wenzelm@53788
   554
    def S \<equiv> "ereal ` T"
wenzelm@53788
   555
    then have "ereal (real x0) \<in> S"
wenzelm@53788
   556
      using T by auto
wenzelm@53788
   557
    then have "x0 \<in> S"
wenzelm@53788
   558
      using assms ereal_real by auto
wenzelm@53788
   559
    moreover have "open S"
wenzelm@53788
   560
      using open_ereal S_def T by auto
wenzelm@53788
   561
    moreover have "\<forall>y\<in>S. real y \<in> T"
wenzelm@53788
   562
      using S_def T by auto
wenzelm@53788
   563
    ultimately have "\<exists>S. x0 \<in> S \<and> open S \<and> (\<forall>y\<in>S. real y \<in> T)"
wenzelm@53788
   564
      by auto
wenzelm@49664
   565
  }
wenzelm@53788
   566
  then show ?thesis
wenzelm@53788
   567
    unfolding continuous_at_open by blast
hoelzl@41980
   568
qed
hoelzl@41980
   569
hoelzl@43920
   570
lemma continuous_at_iff_ereal:
wenzelm@53788
   571
  fixes f :: "'a::t2_space \<Rightarrow> real"
wenzelm@53788
   572
  shows "continuous (at x0) f \<longleftrightarrow> continuous (at x0) (ereal \<circ> f)"
wenzelm@49664
   573
proof -
wenzelm@53788
   574
  {
wenzelm@53788
   575
    assume "continuous (at x0) f"
wenzelm@53788
   576
    then have "continuous (at x0) (ereal \<circ> f)"
wenzelm@53788
   577
      using continuous_at_ereal continuous_at_compose[of x0 f ereal]
wenzelm@53788
   578
      by auto
wenzelm@49664
   579
  }
wenzelm@49664
   580
  moreover
wenzelm@53788
   581
  {
wenzelm@53788
   582
    assume "continuous (at x0) (ereal \<circ> f)"
wenzelm@53788
   583
    then have "continuous (at x0) (real \<circ> (ereal \<circ> f))"
wenzelm@53788
   584
      using continuous_at_of_ereal
wenzelm@53788
   585
      by (intro continuous_at_compose[of x0 "ereal \<circ> f"]) auto
wenzelm@53788
   586
    moreover have "real \<circ> (ereal \<circ> f) = f"
wenzelm@53788
   587
      using real_ereal_id by (simp add: o_assoc)
wenzelm@53788
   588
    ultimately have "continuous (at x0) f"
wenzelm@53788
   589
      by auto
wenzelm@53788
   590
  }
wenzelm@53788
   591
  ultimately show ?thesis
wenzelm@53788
   592
    by auto
hoelzl@41980
   593
qed
hoelzl@41980
   594
hoelzl@41980
   595
hoelzl@43920
   596
lemma continuous_on_iff_ereal:
wenzelm@49664
   597
  fixes f :: "'a::t2_space => real"
wenzelm@53788
   598
  assumes "open A"
wenzelm@53788
   599
  shows "continuous_on A f \<longleftrightarrow> continuous_on A (ereal \<circ> f)"
wenzelm@53788
   600
  using continuous_at_iff_ereal assms
wenzelm@53788
   601
  by (auto simp add: continuous_on_eq_continuous_at cong del: continuous_on_cong)
hoelzl@41980
   602
wenzelm@53788
   603
lemma continuous_on_real: "continuous_on (UNIV - {\<infinity>, -\<infinity>::ereal}) real"
wenzelm@53788
   604
  using continuous_at_of_ereal continuous_on_eq_continuous_at open_image_ereal
wenzelm@53788
   605
  by auto
hoelzl@41980
   606
hoelzl@41980
   607
lemma continuous_on_iff_real:
wenzelm@53788
   608
  fixes f :: "'a::t2_space \<Rightarrow> ereal"
hoelzl@41980
   609
  assumes "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
hoelzl@41980
   610
  shows "continuous_on A f \<longleftrightarrow> continuous_on A (real \<circ> f)"
wenzelm@49664
   611
proof -
wenzelm@53788
   612
  have "f ` A \<subseteq> UNIV - {\<infinity>, -\<infinity>}"
wenzelm@53788
   613
    using assms by force
wenzelm@49664
   614
  then have *: "continuous_on (f ` A) real"
wenzelm@49664
   615
    using continuous_on_real by (simp add: continuous_on_subset)
wenzelm@53788
   616
  have **: "continuous_on ((real \<circ> f) ` A) ereal"
wenzelm@53788
   617
    using continuous_on_ereal continuous_on_subset[of "UNIV" "ereal" "(real \<circ> f) ` A"]
wenzelm@53788
   618
    by blast
wenzelm@53788
   619
  {
wenzelm@53788
   620
    assume "continuous_on A f"
wenzelm@53788
   621
    then have "continuous_on A (real \<circ> f)"
wenzelm@49664
   622
      apply (subst continuous_on_compose)
wenzelm@53788
   623
      using *
wenzelm@53788
   624
      apply auto
wenzelm@49664
   625
      done
wenzelm@49664
   626
  }
wenzelm@49664
   627
  moreover
wenzelm@53788
   628
  {
wenzelm@53788
   629
    assume "continuous_on A (real \<circ> f)"
wenzelm@53788
   630
    then have "continuous_on A (ereal \<circ> (real \<circ> f))"
wenzelm@49664
   631
      apply (subst continuous_on_compose)
wenzelm@53788
   632
      using **
wenzelm@53788
   633
      apply auto
wenzelm@49664
   634
      done
wenzelm@49664
   635
    then have "continuous_on A f"
wenzelm@53788
   636
      apply (subst continuous_on_eq[of A "ereal \<circ> (real \<circ> f)" f])
wenzelm@53788
   637
      using assms ereal_real
wenzelm@53788
   638
      apply auto
wenzelm@49664
   639
      done
wenzelm@49664
   640
  }
wenzelm@53788
   641
  ultimately show ?thesis
wenzelm@53788
   642
    by auto
hoelzl@41980
   643
qed
hoelzl@41980
   644
hoelzl@41980
   645
lemma continuous_at_const:
wenzelm@53788
   646
  fixes f :: "'a::t2_space \<Rightarrow> ereal"
wenzelm@53788
   647
  assumes "\<forall>x. f x = C"
wenzelm@53788
   648
  shows "\<forall>x. continuous (at x) f"
wenzelm@53788
   649
  unfolding continuous_at_open
wenzelm@53788
   650
  using assms t1_space
wenzelm@53788
   651
  by auto
hoelzl@41980
   652
hoelzl@41980
   653
lemma mono_closed_real:
hoelzl@41980
   654
  fixes S :: "real set"
wenzelm@53788
   655
  assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
wenzelm@49664
   656
    and "closed S"
wenzelm@53788
   657
  shows "S = {} \<or> S = UNIV \<or> (\<exists>a. S = {a..})"
wenzelm@49664
   658
proof -
wenzelm@53788
   659
  {
wenzelm@53788
   660
    assume "S \<noteq> {}"
wenzelm@53788
   661
    { assume ex: "\<exists>B. \<forall>x\<in>S. B \<le> x"
wenzelm@53788
   662
      then have *: "\<forall>x\<in>S. Inf S \<le> x"
wenzelm@53788
   663
        using cInf_lower_EX[of _ S] ex by metis
wenzelm@53788
   664
      then have "Inf S \<in> S"
wenzelm@53788
   665
        apply (subst closed_contains_Inf)
wenzelm@53788
   666
        using ex `S \<noteq> {}` `closed S`
wenzelm@53788
   667
        apply auto
wenzelm@53788
   668
        done
wenzelm@53788
   669
      then have "\<forall>x. Inf S \<le> x \<longleftrightarrow> x \<in> S"
wenzelm@53788
   670
        using mono[rule_format, of "Inf S"] *
wenzelm@53788
   671
        by auto
wenzelm@53788
   672
      then have "S = {Inf S ..}"
wenzelm@53788
   673
        by auto
wenzelm@53788
   674
      then have "\<exists>a. S = {a ..}"
wenzelm@53788
   675
        by auto
wenzelm@49664
   676
    }
wenzelm@49664
   677
    moreover
wenzelm@53788
   678
    {
wenzelm@53788
   679
      assume "\<not> (\<exists>B. \<forall>x\<in>S. B \<le> x)"
wenzelm@53788
   680
      then have nex: "\<forall>B. \<exists>x\<in>S. x < B"
wenzelm@53788
   681
        by (simp add: not_le)
wenzelm@53788
   682
      {
wenzelm@53788
   683
        fix y
wenzelm@53788
   684
        obtain x where "x\<in>S" and "x < y"
wenzelm@53788
   685
          using nex by auto
wenzelm@53788
   686
        then have "y \<in> S"
wenzelm@53788
   687
          using mono[rule_format, of x y] by auto
wenzelm@53788
   688
      }
wenzelm@53788
   689
      then have "S = UNIV"
wenzelm@53788
   690
        by auto
wenzelm@49664
   691
    }
wenzelm@53788
   692
    ultimately have "S = UNIV \<or> (\<exists>a. S = {a ..})"
wenzelm@53788
   693
      by blast
wenzelm@53788
   694
  }
wenzelm@53788
   695
  then show ?thesis
wenzelm@53788
   696
    by blast
hoelzl@41980
   697
qed
hoelzl@41980
   698
hoelzl@43920
   699
lemma mono_closed_ereal:
hoelzl@41980
   700
  fixes S :: "real set"
wenzelm@53788
   701
  assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
wenzelm@49664
   702
    and "closed S"
wenzelm@53788
   703
  shows "\<exists>a. S = {x. a \<le> ereal x}"
wenzelm@49664
   704
proof -
wenzelm@53788
   705
  {
wenzelm@53788
   706
    assume "S = {}"
wenzelm@53788
   707
    then have ?thesis
wenzelm@53788
   708
      apply (rule_tac x=PInfty in exI)
wenzelm@53788
   709
      apply auto
wenzelm@53788
   710
      done
wenzelm@53788
   711
  }
wenzelm@49664
   712
  moreover
wenzelm@53788
   713
  {
wenzelm@53788
   714
    assume "S = UNIV"
wenzelm@53788
   715
    then have ?thesis
wenzelm@53788
   716
      apply (rule_tac x="-\<infinity>" in exI)
wenzelm@53788
   717
      apply auto
wenzelm@53788
   718
      done
wenzelm@53788
   719
  }
wenzelm@49664
   720
  moreover
wenzelm@53788
   721
  {
wenzelm@53788
   722
    assume "\<exists>a. S = {a ..}"
wenzelm@53788
   723
    then obtain a where "S = {a ..}"
wenzelm@53788
   724
      by auto
wenzelm@53788
   725
    then have ?thesis
wenzelm@53788
   726
      apply (rule_tac x="ereal a" in exI)
wenzelm@53788
   727
      apply auto
wenzelm@53788
   728
      done
wenzelm@49664
   729
  }
wenzelm@53788
   730
  ultimately show ?thesis
wenzelm@53788
   731
    using mono_closed_real[of S] assms by auto
hoelzl@41980
   732
qed
hoelzl@41980
   733
wenzelm@53788
   734
hoelzl@41980
   735
subsection {* Sums *}
hoelzl@41980
   736
wenzelm@49664
   737
lemma setsum_ereal[simp]: "(\<Sum>x\<in>A. ereal (f x)) = ereal (\<Sum>x\<in>A. f x)"
wenzelm@53788
   738
proof (cases "finite A")
wenzelm@53788
   739
  case True
wenzelm@49664
   740
  then show ?thesis by induct auto
wenzelm@53788
   741
next
wenzelm@53788
   742
  case False
wenzelm@53788
   743
  then show ?thesis by simp
wenzelm@53788
   744
qed
hoelzl@41980
   745
hoelzl@43923
   746
lemma setsum_Pinfty:
hoelzl@43923
   747
  fixes f :: "'a \<Rightarrow> ereal"
wenzelm@53788
   748
  shows "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> finite P \<and> (\<exists>i\<in>P. f i = \<infinity>)"
hoelzl@41980
   749
proof safe
hoelzl@41980
   750
  assume *: "setsum f P = \<infinity>"
hoelzl@41980
   751
  show "finite P"
wenzelm@53788
   752
  proof (rule ccontr)
wenzelm@53788
   753
    assume "infinite P"
wenzelm@53788
   754
    with * show False
wenzelm@53788
   755
      by auto
wenzelm@53788
   756
  qed
hoelzl@41980
   757
  show "\<exists>i\<in>P. f i = \<infinity>"
hoelzl@41980
   758
  proof (rule ccontr)
wenzelm@53788
   759
    assume "\<not> ?thesis"
wenzelm@53788
   760
    then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>"
wenzelm@53788
   761
      by auto
wenzelm@53788
   762
    with `finite P` have "setsum f P \<noteq> \<infinity>"
hoelzl@41980
   763
      by induct auto
wenzelm@53788
   764
    with * show False
wenzelm@53788
   765
      by auto
hoelzl@41980
   766
  qed
hoelzl@41980
   767
next
wenzelm@53788
   768
  fix i
wenzelm@53788
   769
  assume "finite P" and "i \<in> P" and "f i = \<infinity>"
wenzelm@49664
   770
  then show "setsum f P = \<infinity>"
hoelzl@41980
   771
  proof induct
hoelzl@41980
   772
    case (insert x A)
hoelzl@41980
   773
    show ?case using insert by (cases "x = i") auto
hoelzl@41980
   774
  qed simp
hoelzl@41980
   775
qed
hoelzl@41980
   776
hoelzl@41980
   777
lemma setsum_Inf:
hoelzl@43923
   778
  fixes f :: "'a \<Rightarrow> ereal"
wenzelm@53788
   779
  shows "\<bar>setsum f A\<bar> = \<infinity> \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
hoelzl@41980
   780
proof
hoelzl@41980
   781
  assume *: "\<bar>setsum f A\<bar> = \<infinity>"
wenzelm@53788
   782
  have "finite A"
wenzelm@53788
   783
    by (rule ccontr) (insert *, auto)
hoelzl@41980
   784
  moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
hoelzl@41980
   785
  proof (rule ccontr)
wenzelm@53788
   786
    assume "\<not> ?thesis"
wenzelm@53788
   787
    then have "\<forall>i\<in>A. \<exists>r. f i = ereal r"
wenzelm@53788
   788
      by auto
wenzelm@53788
   789
    from bchoice[OF this] obtain r where "\<forall>x\<in>A. f x = ereal (r x)" ..
wenzelm@53788
   790
    with * show False
wenzelm@53788
   791
      by auto
hoelzl@41980
   792
  qed
wenzelm@53788
   793
  ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
wenzelm@53788
   794
    by auto
hoelzl@41980
   795
next
hoelzl@41980
   796
  assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
wenzelm@53788
   797
  then obtain i where "finite A" "i \<in> A" and "\<bar>f i\<bar> = \<infinity>"
wenzelm@53788
   798
    by auto
hoelzl@41980
   799
  then show "\<bar>setsum f A\<bar> = \<infinity>"
hoelzl@41980
   800
  proof induct
wenzelm@53788
   801
    case (insert j A)
wenzelm@53788
   802
    then show ?case
hoelzl@43920
   803
      by (cases rule: ereal3_cases[of "f i" "f j" "setsum f A"]) auto
hoelzl@41980
   804
  qed simp
hoelzl@41980
   805
qed
hoelzl@41980
   806
hoelzl@43920
   807
lemma setsum_real_of_ereal:
hoelzl@43923
   808
  fixes f :: "'i \<Rightarrow> ereal"
hoelzl@41980
   809
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
hoelzl@41980
   810
  shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
hoelzl@41980
   811
proof -
hoelzl@43920
   812
  have "\<forall>x\<in>S. \<exists>r. f x = ereal r"
hoelzl@41980
   813
  proof
wenzelm@53788
   814
    fix x
wenzelm@53788
   815
    assume "x \<in> S"
wenzelm@53788
   816
    from assms[OF this] show "\<exists>r. f x = ereal r"
wenzelm@53788
   817
      by (cases "f x") auto
hoelzl@41980
   818
  qed
wenzelm@53788
   819
  from bchoice[OF this] obtain r where "\<forall>x\<in>S. f x = ereal (r x)" ..
wenzelm@53788
   820
  then show ?thesis
wenzelm@53788
   821
    by simp
hoelzl@41980
   822
qed
hoelzl@41980
   823
hoelzl@43920
   824
lemma setsum_ereal_0:
wenzelm@53788
   825
  fixes f :: "'a \<Rightarrow> ereal"
wenzelm@53788
   826
  assumes "finite A"
wenzelm@53788
   827
    and "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
hoelzl@41980
   828
  shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
hoelzl@41980
   829
proof
hoelzl@41980
   830
  assume *: "(\<Sum>x\<in>A. f x) = 0"
wenzelm@53788
   831
  then have "(\<Sum>x\<in>A. f x) \<noteq> \<infinity>"
wenzelm@53788
   832
    by auto
wenzelm@53788
   833
  then have "\<forall>i\<in>A. \<bar>f i\<bar> \<noteq> \<infinity>"
wenzelm@53788
   834
    using assms by (force simp: setsum_Pinfty)
wenzelm@53788
   835
  then have "\<forall>i\<in>A. \<exists>r. f i = ereal r"
wenzelm@53788
   836
    by auto
hoelzl@41980
   837
  from bchoice[OF this] * assms show "\<forall>i\<in>A. f i = 0"
hoelzl@41980
   838
    using setsum_nonneg_eq_0_iff[of A "\<lambda>i. real (f i)"] by auto
hoelzl@41980
   839
qed (rule setsum_0')
hoelzl@41980
   840
hoelzl@43920
   841
lemma setsum_ereal_right_distrib:
wenzelm@49664
   842
  fixes f :: "'a \<Rightarrow> ereal"
wenzelm@49664
   843
  assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
hoelzl@41980
   844
  shows "r * setsum f A = (\<Sum>n\<in>A. r * f n)"
hoelzl@41980
   845
proof cases
wenzelm@49664
   846
  assume "finite A"
wenzelm@49664
   847
  then show ?thesis using assms
hoelzl@43920
   848
    by induct (auto simp: ereal_right_distrib setsum_nonneg)
hoelzl@41980
   849
qed simp
hoelzl@41980
   850
hoelzl@43920
   851
lemma sums_ereal_positive:
wenzelm@49664
   852
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
   853
  assumes "\<And>i. 0 \<le> f i"
wenzelm@49664
   854
  shows "f sums (SUP n. \<Sum>i<n. f i)"
hoelzl@41980
   855
proof -
hoelzl@41980
   856
  have "incseq (\<lambda>i. \<Sum>j=0..<i. f j)"
wenzelm@53788
   857
    using ereal_add_mono[OF _ assms]
wenzelm@53788
   858
    by (auto intro!: incseq_SucI)
hoelzl@51000
   859
  from LIMSEQ_SUP[OF this]
wenzelm@53788
   860
  show ?thesis unfolding sums_def
wenzelm@53788
   861
    by (simp add: atLeast0LessThan)
hoelzl@41980
   862
qed
hoelzl@41980
   863
hoelzl@43920
   864
lemma summable_ereal_pos:
wenzelm@49664
   865
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
   866
  assumes "\<And>i. 0 \<le> f i"
wenzelm@49664
   867
  shows "summable f"
wenzelm@53788
   868
  using sums_ereal_positive[of f, OF assms]
wenzelm@53788
   869
  unfolding summable_def
wenzelm@53788
   870
  by auto
hoelzl@41980
   871
hoelzl@43920
   872
lemma suminf_ereal_eq_SUPR:
wenzelm@49664
   873
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
   874
  assumes "\<And>i. 0 \<le> f i"
hoelzl@41980
   875
  shows "(\<Sum>x. f x) = (SUP n. \<Sum>i<n. f i)"
wenzelm@53788
   876
  using sums_ereal_positive[of f, OF assms, THEN sums_unique]
wenzelm@53788
   877
  by simp
hoelzl@41980
   878
wenzelm@49664
   879
lemma sums_ereal: "(\<lambda>x. ereal (f x)) sums ereal x \<longleftrightarrow> f sums x"
hoelzl@41980
   880
  unfolding sums_def by simp
hoelzl@41980
   881
hoelzl@41980
   882
lemma suminf_bound:
hoelzl@43920
   883
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@53788
   884
  assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x"
wenzelm@53788
   885
    and pos: "\<And>n. 0 \<le> f n"
hoelzl@41980
   886
  shows "suminf f \<le> x"
hoelzl@43920
   887
proof (rule Lim_bounded_ereal)
hoelzl@43920
   888
  have "summable f" using pos[THEN summable_ereal_pos] .
hoelzl@41980
   889
  then show "(\<lambda>N. \<Sum>n<N. f n) ----> suminf f"
hoelzl@41980
   890
    by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
hoelzl@41980
   891
  show "\<forall>n\<ge>0. setsum f {..<n} \<le> x"
hoelzl@41980
   892
    using assms by auto
hoelzl@41980
   893
qed
hoelzl@41980
   894
hoelzl@41980
   895
lemma suminf_bound_add:
hoelzl@43920
   896
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
   897
  assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x"
wenzelm@49664
   898
    and pos: "\<And>n. 0 \<le> f n"
wenzelm@49664
   899
    and "y \<noteq> -\<infinity>"
hoelzl@41980
   900
  shows "suminf f + y \<le> x"
hoelzl@41980
   901
proof (cases y)
wenzelm@49664
   902
  case (real r)
wenzelm@49664
   903
  then have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y"
hoelzl@43920
   904
    using assms by (simp add: ereal_le_minus)
wenzelm@53788
   905
  then have "(\<Sum> n. f n) \<le> x - y"
wenzelm@53788
   906
    using pos by (rule suminf_bound)
hoelzl@41980
   907
  then show "(\<Sum> n. f n) + y \<le> x"
hoelzl@43920
   908
    using assms real by (simp add: ereal_le_minus)
hoelzl@41980
   909
qed (insert assms, auto)
hoelzl@41980
   910
hoelzl@41980
   911
lemma suminf_upper:
wenzelm@49664
   912
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
   913
  assumes "\<And>n. 0 \<le> f n"
hoelzl@41980
   914
  shows "(\<Sum>n<N. f n) \<le> (\<Sum>n. f n)"
hoelzl@44928
   915
  unfolding suminf_ereal_eq_SUPR[OF assms] SUP_def
huffman@45031
   916
  by (auto intro: complete_lattice_class.Sup_upper)
hoelzl@41980
   917
hoelzl@41980
   918
lemma suminf_0_le:
wenzelm@49664
   919
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
   920
  assumes "\<And>n. 0 \<le> f n"
hoelzl@41980
   921
  shows "0 \<le> (\<Sum>n. f n)"
wenzelm@53788
   922
  using suminf_upper[of f 0, OF assms]
wenzelm@53788
   923
  by simp
hoelzl@41980
   924
hoelzl@41980
   925
lemma suminf_le_pos:
hoelzl@43920
   926
  fixes f g :: "nat \<Rightarrow> ereal"
wenzelm@53788
   927
  assumes "\<And>N. f N \<le> g N"
wenzelm@53788
   928
    and "\<And>N. 0 \<le> f N"
hoelzl@41980
   929
  shows "suminf f \<le> suminf g"
hoelzl@41980
   930
proof (safe intro!: suminf_bound)
wenzelm@49664
   931
  fix n
wenzelm@53788
   932
  {
wenzelm@53788
   933
    fix N
wenzelm@53788
   934
    have "0 \<le> g N"
wenzelm@53788
   935
      using assms(2,1)[of N] by auto
wenzelm@53788
   936
  }
wenzelm@49664
   937
  have "setsum f {..<n} \<le> setsum g {..<n}"
wenzelm@49664
   938
    using assms by (auto intro: setsum_mono)
wenzelm@53788
   939
  also have "\<dots> \<le> suminf g"
wenzelm@53788
   940
    using `\<And>N. 0 \<le> g N`
wenzelm@53788
   941
    by (rule suminf_upper)
hoelzl@41980
   942
  finally show "setsum f {..<n} \<le> suminf g" .
hoelzl@41980
   943
qed (rule assms(2))
hoelzl@41980
   944
wenzelm@53788
   945
lemma suminf_half_series_ereal: "(\<Sum>n. (1/2 :: ereal) ^ Suc n) = 1"
hoelzl@43920
   946
  using sums_ereal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]
hoelzl@43920
   947
  by (simp add: one_ereal_def)
hoelzl@41980
   948
hoelzl@43920
   949
lemma suminf_add_ereal:
hoelzl@43920
   950
  fixes f g :: "nat \<Rightarrow> ereal"
wenzelm@53788
   951
  assumes "\<And>i. 0 \<le> f i"
wenzelm@53788
   952
    and "\<And>i. 0 \<le> g i"
hoelzl@41980
   953
  shows "(\<Sum>i. f i + g i) = suminf f + suminf g"
hoelzl@43920
   954
  apply (subst (1 2 3) suminf_ereal_eq_SUPR)
hoelzl@41980
   955
  unfolding setsum_addf
wenzelm@49664
   956
  apply (intro assms ereal_add_nonneg_nonneg SUPR_ereal_add_pos incseq_setsumI setsum_nonneg ballI)+
wenzelm@49664
   957
  done
hoelzl@41980
   958
hoelzl@43920
   959
lemma suminf_cmult_ereal:
hoelzl@43920
   960
  fixes f g :: "nat \<Rightarrow> ereal"
wenzelm@53788
   961
  assumes "\<And>i. 0 \<le> f i"
wenzelm@53788
   962
    and "0 \<le> a"
hoelzl@41980
   963
  shows "(\<Sum>i. a * f i) = a * suminf f"
hoelzl@43920
   964
  by (auto simp: setsum_ereal_right_distrib[symmetric] assms
wenzelm@53788
   965
       ereal_zero_le_0_iff setsum_nonneg suminf_ereal_eq_SUPR
wenzelm@53788
   966
       intro!: SUPR_ereal_cmult )
hoelzl@41980
   967
hoelzl@41980
   968
lemma suminf_PInfty:
hoelzl@43923
   969
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@53788
   970
  assumes "\<And>i. 0 \<le> f i"
wenzelm@53788
   971
    and "suminf f \<noteq> \<infinity>"
hoelzl@41980
   972
  shows "f i \<noteq> \<infinity>"
hoelzl@41980
   973
proof -
hoelzl@41980
   974
  from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
wenzelm@53788
   975
  have "(\<Sum>i<Suc i. f i) \<noteq> \<infinity>"
wenzelm@53788
   976
    by auto
wenzelm@53788
   977
  then show ?thesis
wenzelm@53788
   978
    unfolding setsum_Pinfty by simp
hoelzl@41980
   979
qed
hoelzl@41980
   980
hoelzl@41980
   981
lemma suminf_PInfty_fun:
wenzelm@53788
   982
  assumes "\<And>i. 0 \<le> f i"
wenzelm@53788
   983
    and "suminf f \<noteq> \<infinity>"
hoelzl@43920
   984
  shows "\<exists>f'. f = (\<lambda>x. ereal (f' x))"
hoelzl@41980
   985
proof -
hoelzl@43920
   986
  have "\<forall>i. \<exists>r. f i = ereal r"
hoelzl@41980
   987
  proof
wenzelm@53788
   988
    fix i
wenzelm@53788
   989
    show "\<exists>r. f i = ereal r"
wenzelm@53788
   990
      using suminf_PInfty[OF assms] assms(1)[of i]
wenzelm@53788
   991
      by (cases "f i") auto
hoelzl@41980
   992
  qed
wenzelm@53788
   993
  from choice[OF this] show ?thesis
wenzelm@53788
   994
    by auto
hoelzl@41980
   995
qed
hoelzl@41980
   996
hoelzl@43920
   997
lemma summable_ereal:
wenzelm@53788
   998
  assumes "\<And>i. 0 \<le> f i"
wenzelm@53788
   999
    and "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>"
hoelzl@41980
  1000
  shows "summable f"
hoelzl@41980
  1001
proof -
hoelzl@43920
  1002
  have "0 \<le> (\<Sum>i. ereal (f i))"
hoelzl@41980
  1003
    using assms by (intro suminf_0_le) auto
hoelzl@43920
  1004
  with assms obtain r where r: "(\<Sum>i. ereal (f i)) = ereal r"
hoelzl@43920
  1005
    by (cases "\<Sum>i. ereal (f i)") auto
hoelzl@43920
  1006
  from summable_ereal_pos[of "\<lambda>x. ereal (f x)"]
wenzelm@53788
  1007
  have "summable (\<lambda>x. ereal (f x))"
wenzelm@53788
  1008
    using assms by auto
hoelzl@41980
  1009
  from summable_sums[OF this]
wenzelm@53788
  1010
  have "(\<lambda>x. ereal (f x)) sums (\<Sum>x. ereal (f x))"
wenzelm@53788
  1011
    by auto
hoelzl@41980
  1012
  then show "summable f"
hoelzl@43920
  1013
    unfolding r sums_ereal summable_def ..
hoelzl@41980
  1014
qed
hoelzl@41980
  1015
hoelzl@43920
  1016
lemma suminf_ereal:
wenzelm@53788
  1017
  assumes "\<And>i. 0 \<le> f i"
wenzelm@53788
  1018
    and "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>"
hoelzl@43920
  1019
  shows "(\<Sum>i. ereal (f i)) = ereal (suminf f)"
hoelzl@41980
  1020
proof (rule sums_unique[symmetric])
hoelzl@43920
  1021
  from summable_ereal[OF assms]
hoelzl@43920
  1022
  show "(\<lambda>x. ereal (f x)) sums (ereal (suminf f))"
wenzelm@53788
  1023
    unfolding sums_ereal
wenzelm@53788
  1024
    using assms
wenzelm@53788
  1025
    by (intro summable_sums summable_ereal)
hoelzl@41980
  1026
qed
hoelzl@41980
  1027
hoelzl@43920
  1028
lemma suminf_ereal_minus:
hoelzl@43920
  1029
  fixes f g :: "nat \<Rightarrow> ereal"
wenzelm@53788
  1030
  assumes ord: "\<And>i. g i \<le> f i" "\<And>i. 0 \<le> g i"
wenzelm@53788
  1031
    and fin: "suminf f \<noteq> \<infinity>" "suminf g \<noteq> \<infinity>"
hoelzl@41980
  1032
  shows "(\<Sum>i. f i - g i) = suminf f - suminf g"
hoelzl@41980
  1033
proof -
wenzelm@53788
  1034
  {
wenzelm@53788
  1035
    fix i
wenzelm@53788
  1036
    have "0 \<le> f i"
wenzelm@53788
  1037
      using ord[of i] by auto
wenzelm@53788
  1038
  }
hoelzl@41980
  1039
  moreover
wenzelm@53788
  1040
  from suminf_PInfty_fun[OF `\<And>i. 0 \<le> f i` fin(1)] obtain f' where [simp]: "f = (\<lambda>x. ereal (f' x))" ..
wenzelm@53788
  1041
  from suminf_PInfty_fun[OF `\<And>i. 0 \<le> g i` fin(2)] obtain g' where [simp]: "g = (\<lambda>x. ereal (g' x))" ..
wenzelm@53788
  1042
  {
wenzelm@53788
  1043
    fix i
wenzelm@53788
  1044
    have "0 \<le> f i - g i"
wenzelm@53788
  1045
      using ord[of i] by (auto simp: ereal_le_minus_iff)
wenzelm@53788
  1046
  }
hoelzl@41980
  1047
  moreover
hoelzl@41980
  1048
  have "suminf (\<lambda>i. f i - g i) \<le> suminf f"
hoelzl@41980
  1049
    using assms by (auto intro!: suminf_le_pos simp: field_simps)
wenzelm@53788
  1050
  then have "suminf (\<lambda>i. f i - g i) \<noteq> \<infinity>"
wenzelm@53788
  1051
    using fin by auto
wenzelm@53788
  1052
  ultimately show ?thesis
wenzelm@53788
  1053
    using assms `\<And>i. 0 \<le> f i`
hoelzl@41980
  1054
    apply simp
wenzelm@49664
  1055
    apply (subst (1 2 3) suminf_ereal)
wenzelm@49664
  1056
    apply (auto intro!: suminf_diff[symmetric] summable_ereal)
wenzelm@49664
  1057
    done
hoelzl@41980
  1058
qed
hoelzl@41980
  1059
wenzelm@49664
  1060
lemma suminf_ereal_PInf [simp]: "(\<Sum>x. \<infinity>::ereal) = \<infinity>"
hoelzl@41980
  1061
proof -
wenzelm@53788
  1062
  have "(\<Sum>i<Suc 0. \<infinity>) \<le> (\<Sum>x. \<infinity>::ereal)"
wenzelm@53788
  1063
    by (rule suminf_upper) auto
wenzelm@53788
  1064
  then show ?thesis
wenzelm@53788
  1065
    by simp
hoelzl@41980
  1066
qed
hoelzl@41980
  1067
hoelzl@43920
  1068
lemma summable_real_of_ereal:
hoelzl@43923
  1069
  fixes f :: "nat \<Rightarrow> ereal"
wenzelm@49664
  1070
  assumes f: "\<And>i. 0 \<le> f i"
wenzelm@49664
  1071
    and fin: "(\<Sum>i. f i) \<noteq> \<infinity>"
hoelzl@41980
  1072
  shows "summable (\<lambda>i. real (f i))"
hoelzl@41980
  1073
proof (rule summable_def[THEN iffD2])
wenzelm@53788
  1074
  have "0 \<le> (\<Sum>i. f i)"
wenzelm@53788
  1075
    using assms by (auto intro: suminf_0_le)
wenzelm@53788
  1076
  with fin obtain r where r: "ereal r = (\<Sum>i. f i)"
wenzelm@53788
  1077
    by (cases "(\<Sum>i. f i)") auto
wenzelm@53788
  1078
  {
wenzelm@53788
  1079
    fix i
wenzelm@53788
  1080
    have "f i \<noteq> \<infinity>"
wenzelm@53788
  1081
      using f by (intro suminf_PInfty[OF _ fin]) auto
wenzelm@53788
  1082
    then have "\<bar>f i\<bar> \<noteq> \<infinity>"
wenzelm@53788
  1083
      using f[of i] by auto
wenzelm@53788
  1084
  }
hoelzl@41980
  1085
  note fin = this
hoelzl@43920
  1086
  have "(\<lambda>i. ereal (real (f i))) sums (\<Sum>i. ereal (real (f i)))"
wenzelm@53788
  1087
    using f
wenzelm@53788
  1088
    by (auto intro!: summable_ereal_pos summable_sums simp: ereal_le_real_iff zero_ereal_def)
wenzelm@53788
  1089
  also have "\<dots> = ereal r"
wenzelm@53788
  1090
    using fin r by (auto simp: ereal_real)
wenzelm@53788
  1091
  finally show "\<exists>r. (\<lambda>i. real (f i)) sums r"
wenzelm@53788
  1092
    by (auto simp: sums_ereal)
hoelzl@41980
  1093
qed
hoelzl@41980
  1094
hoelzl@42950
  1095
lemma suminf_SUP_eq:
hoelzl@43920
  1096
  fixes f :: "nat \<Rightarrow> nat \<Rightarrow> ereal"
wenzelm@53788
  1097
  assumes "\<And>i. incseq (\<lambda>n. f n i)"
wenzelm@53788
  1098
    and "\<And>n i. 0 \<le> f n i"
hoelzl@42950
  1099
  shows "(\<Sum>i. SUP n. f n i) = (SUP n. \<Sum>i. f n i)"
hoelzl@42950
  1100
proof -
wenzelm@53788
  1101
  {
wenzelm@53788
  1102
    fix n :: nat
hoelzl@42950
  1103
    have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
wenzelm@53788
  1104
      using assms
wenzelm@53788
  1105
      by (auto intro!: SUPR_ereal_setsum[symmetric])
wenzelm@53788
  1106
  }
hoelzl@42950
  1107
  note * = this
wenzelm@53788
  1108
  show ?thesis
wenzelm@53788
  1109
    using assms
hoelzl@43920
  1110
    apply (subst (1 2) suminf_ereal_eq_SUPR)
hoelzl@42950
  1111
    unfolding *
hoelzl@44928
  1112
    apply (auto intro!: SUP_upper2)
wenzelm@49664
  1113
    apply (subst SUP_commute)
wenzelm@49664
  1114
    apply rule
wenzelm@49664
  1115
    done
hoelzl@42950
  1116
qed
hoelzl@42950
  1117
hoelzl@47761
  1118
lemma suminf_setsum_ereal:
hoelzl@47761
  1119
  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> ereal"
hoelzl@47761
  1120
  assumes nonneg: "\<And>i a. a \<in> A \<Longrightarrow> 0 \<le> f i a"
hoelzl@47761
  1121
  shows "(\<Sum>i. \<Sum>a\<in>A. f i a) = (\<Sum>a\<in>A. \<Sum>i. f i a)"
wenzelm@53788
  1122
proof (cases "finite A")
wenzelm@53788
  1123
  case True
wenzelm@53788
  1124
  then show ?thesis
wenzelm@53788
  1125
    using nonneg
hoelzl@47761
  1126
    by induct (simp_all add: suminf_add_ereal setsum_nonneg)
wenzelm@53788
  1127
next
wenzelm@53788
  1128
  case False
wenzelm@53788
  1129
  then show ?thesis by simp
wenzelm@53788
  1130
qed
hoelzl@47761
  1131
hoelzl@50104
  1132
lemma suminf_ereal_eq_0:
hoelzl@50104
  1133
  fixes f :: "nat \<Rightarrow> ereal"
hoelzl@50104
  1134
  assumes nneg: "\<And>i. 0 \<le> f i"
hoelzl@50104
  1135
  shows "(\<Sum>i. f i) = 0 \<longleftrightarrow> (\<forall>i. f i = 0)"
hoelzl@50104
  1136
proof
hoelzl@50104
  1137
  assume "(\<Sum>i. f i) = 0"
wenzelm@53788
  1138
  {
wenzelm@53788
  1139
    fix i
wenzelm@53788
  1140
    assume "f i \<noteq> 0"
wenzelm@53788
  1141
    with nneg have "0 < f i"
wenzelm@53788
  1142
      by (auto simp: less_le)
hoelzl@50104
  1143
    also have "f i = (\<Sum>j. if j = i then f i else 0)"
hoelzl@50104
  1144
      by (subst suminf_finite[where N="{i}"]) auto
hoelzl@50104
  1145
    also have "\<dots> \<le> (\<Sum>i. f i)"
wenzelm@53788
  1146
      using nneg
wenzelm@53788
  1147
      by (auto intro!: suminf_le_pos)
wenzelm@53788
  1148
    finally have False
wenzelm@53788
  1149
      using `(\<Sum>i. f i) = 0` by auto
wenzelm@53788
  1150
  }
wenzelm@53788
  1151
  then show "\<forall>i. f i = 0"
wenzelm@53788
  1152
    by auto
hoelzl@50104
  1153
qed simp
hoelzl@50104
  1154
hoelzl@51340
  1155
lemma Liminf_within:
hoelzl@51340
  1156
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
hoelzl@51340
  1157
  shows "Liminf (at x within S) f = (SUP e:{0<..}. INF y:(S \<inter> ball x e - {x}). f y)"
hoelzl@51641
  1158
  unfolding Liminf_def eventually_at
hoelzl@51340
  1159
proof (rule SUPR_eq, simp_all add: Ball_def Bex_def, safe)
wenzelm@53788
  1160
  fix P d
wenzelm@53788
  1161
  assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
hoelzl@51340
  1162
  then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
hoelzl@51340
  1163
    by (auto simp: zero_less_dist_iff dist_commute)
hoelzl@51340
  1164
  then show "\<exists>r>0. INFI (Collect P) f \<le> INFI (S \<inter> ball x r - {x}) f"
hoelzl@51340
  1165
    by (intro exI[of _ d] INF_mono conjI `0 < d`) auto
hoelzl@51340
  1166
next
wenzelm@53788
  1167
  fix d :: real
wenzelm@53788
  1168
  assume "0 < d"
hoelzl@51641
  1169
  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>
hoelzl@51340
  1170
    INFI (S \<inter> ball x d - {x}) f \<le> INFI (Collect P) f"
hoelzl@51340
  1171
    by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
hoelzl@51340
  1172
       (auto intro!: INF_mono exI[of _ d] simp: dist_commute)
hoelzl@51340
  1173
qed
hoelzl@51340
  1174
hoelzl@51340
  1175
lemma Limsup_within:
wenzelm@53788
  1176
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
hoelzl@51340
  1177
  shows "Limsup (at x within S) f = (INF e:{0<..}. SUP y:(S \<inter> ball x e - {x}). f y)"
hoelzl@51641
  1178
  unfolding Limsup_def eventually_at
hoelzl@51340
  1179
proof (rule INFI_eq, simp_all add: Ball_def Bex_def, safe)
wenzelm@53788
  1180
  fix P d
wenzelm@53788
  1181
  assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
hoelzl@51340
  1182
  then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
hoelzl@51340
  1183
    by (auto simp: zero_less_dist_iff dist_commute)
hoelzl@51340
  1184
  then show "\<exists>r>0. SUPR (S \<inter> ball x r - {x}) f \<le> SUPR (Collect P) f"
hoelzl@51340
  1185
    by (intro exI[of _ d] SUP_mono conjI `0 < d`) auto
hoelzl@51340
  1186
next
wenzelm@53788
  1187
  fix d :: real
wenzelm@53788
  1188
  assume "0 < d"
hoelzl@51641
  1189
  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>
hoelzl@51340
  1190
    SUPR (Collect P) f \<le> SUPR (S \<inter> ball x d - {x}) f"
hoelzl@51340
  1191
    by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
hoelzl@51340
  1192
       (auto intro!: SUP_mono exI[of _ d] simp: dist_commute)
hoelzl@51340
  1193
qed
hoelzl@51340
  1194
hoelzl@51340
  1195
lemma Liminf_at:
hoelzl@54257
  1196
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
hoelzl@51340
  1197
  shows "Liminf (at x) f = (SUP e:{0<..}. INF y:(ball x e - {x}). f y)"
hoelzl@51340
  1198
  using Liminf_within[of x UNIV f] by simp
hoelzl@51340
  1199
hoelzl@51340
  1200
lemma Limsup_at:
hoelzl@54257
  1201
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
hoelzl@51340
  1202
  shows "Limsup (at x) f = (INF e:{0<..}. SUP y:(ball x e - {x}). f y)"
hoelzl@51340
  1203
  using Limsup_within[of x UNIV f] by simp
hoelzl@51340
  1204
hoelzl@51340
  1205
lemma min_Liminf_at:
wenzelm@53788
  1206
  fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_linorder"
hoelzl@51340
  1207
  shows "min (f x) (Liminf (at x) f) = (SUP e:{0<..}. INF y:ball x e. f y)"
hoelzl@51340
  1208
  unfolding inf_min[symmetric] Liminf_at
hoelzl@51340
  1209
  apply (subst inf_commute)
hoelzl@51340
  1210
  apply (subst SUP_inf)
hoelzl@51340
  1211
  apply (intro SUP_cong[OF refl])
hoelzl@51340
  1212
  apply (cut_tac A="ball x b - {x}" and B="{x}" and M=f in INF_union)
hoelzl@51340
  1213
  apply (simp add: INF_def del: inf_ereal_def)
hoelzl@51340
  1214
  done
hoelzl@51340
  1215
wenzelm@53788
  1216
hoelzl@51340
  1217
subsection {* monoset *}
hoelzl@51340
  1218
hoelzl@51340
  1219
definition (in order) mono_set:
hoelzl@51340
  1220
  "mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
hoelzl@51340
  1221
hoelzl@51340
  1222
lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
hoelzl@51340
  1223
lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto
hoelzl@51340
  1224
lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
hoelzl@51340
  1225
lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto
hoelzl@51340
  1226
hoelzl@51340
  1227
lemma (in complete_linorder) mono_set_iff:
hoelzl@51340
  1228
  fixes S :: "'a set"
hoelzl@51340
  1229
  defines "a \<equiv> Inf S"
wenzelm@53788
  1230
  shows "mono_set S \<longleftrightarrow> S = {a <..} \<or> S = {a..}" (is "_ = ?c")
hoelzl@51340
  1231
proof
hoelzl@51340
  1232
  assume "mono_set S"
wenzelm@53788
  1233
  then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S"
wenzelm@53788
  1234
    by (auto simp: mono_set)
hoelzl@51340
  1235
  show ?c
hoelzl@51340
  1236
  proof cases
hoelzl@51340
  1237
    assume "a \<in> S"
hoelzl@51340
  1238
    show ?c
hoelzl@51340
  1239
      using mono[OF _ `a \<in> S`]
hoelzl@51340
  1240
      by (auto intro: Inf_lower simp: a_def)
hoelzl@51340
  1241
  next
hoelzl@51340
  1242
    assume "a \<notin> S"
hoelzl@51340
  1243
    have "S = {a <..}"
hoelzl@51340
  1244
    proof safe
hoelzl@51340
  1245
      fix x assume "x \<in> S"
wenzelm@53788
  1246
      then have "a \<le> x"
wenzelm@53788
  1247
        unfolding a_def by (rule Inf_lower)
wenzelm@53788
  1248
      then show "a < x"
wenzelm@53788
  1249
        using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
hoelzl@51340
  1250
    next
hoelzl@51340
  1251
      fix x assume "a < x"
wenzelm@53788
  1252
      then obtain y where "y < x" "y \<in> S"
wenzelm@53788
  1253
        unfolding a_def Inf_less_iff ..
wenzelm@53788
  1254
      with mono[of y x] show "x \<in> S"
wenzelm@53788
  1255
        by auto
hoelzl@51340
  1256
    qed
hoelzl@51340
  1257
    then show ?c ..
hoelzl@51340
  1258
  qed
hoelzl@51340
  1259
qed auto
hoelzl@51340
  1260
hoelzl@51340
  1261
lemma ereal_open_mono_set:
hoelzl@51340
  1262
  fixes S :: "ereal set"
wenzelm@53788
  1263
  shows "open S \<and> mono_set S \<longleftrightarrow> S = UNIV \<or> S = {Inf S <..}"
hoelzl@51340
  1264
  by (metis Inf_UNIV atLeast_eq_UNIV_iff ereal_open_atLeast
hoelzl@51340
  1265
    ereal_open_closed mono_set_iff open_ereal_greaterThan)
hoelzl@51340
  1266
hoelzl@51340
  1267
lemma ereal_closed_mono_set:
hoelzl@51340
  1268
  fixes S :: "ereal set"
wenzelm@53788
  1269
  shows "closed S \<and> mono_set S \<longleftrightarrow> S = {} \<or> S = {Inf S ..}"
hoelzl@51340
  1270
  by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_ereal_atLeast
hoelzl@51340
  1271
    ereal_open_closed mono_empty mono_set_iff open_ereal_greaterThan)
hoelzl@51340
  1272
hoelzl@51340
  1273
lemma ereal_Liminf_Sup_monoset:
wenzelm@53788
  1274
  fixes f :: "'a \<Rightarrow> ereal"
hoelzl@51340
  1275
  shows "Liminf net f =
hoelzl@51340
  1276
    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
  1277
    (is "_ = Sup ?A")
hoelzl@51340
  1278
proof (safe intro!: Liminf_eqI complete_lattice_class.Sup_upper complete_lattice_class.Sup_least)
wenzelm@53788
  1279
  fix P
wenzelm@53788
  1280
  assume P: "eventually P net"
wenzelm@53788
  1281
  fix S
wenzelm@53788
  1282
  assume S: "mono_set S" "INFI (Collect P) f \<in> S"
wenzelm@53788
  1283
  {
wenzelm@53788
  1284
    fix x
wenzelm@53788
  1285
    assume "P x"
hoelzl@51340
  1286
    then have "INFI (Collect P) f \<le> f x"
hoelzl@51340
  1287
      by (intro complete_lattice_class.INF_lower) simp
hoelzl@51340
  1288
    with S have "f x \<in> S"
wenzelm@53788
  1289
      by (simp add: mono_set)
wenzelm@53788
  1290
  }
hoelzl@51340
  1291
  with P show "eventually (\<lambda>x. f x \<in> S) net"
hoelzl@51340
  1292
    by (auto elim: eventually_elim1)
hoelzl@51340
  1293
next
hoelzl@51340
  1294
  fix y l
hoelzl@51340
  1295
  assume S: "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net"
hoelzl@51340
  1296
  assume P: "\<forall>P. eventually P net \<longrightarrow> INFI (Collect P) f \<le> y"
hoelzl@51340
  1297
  show "l \<le> y"
hoelzl@51340
  1298
  proof (rule dense_le)
wenzelm@53788
  1299
    fix B
wenzelm@53788
  1300
    assume "B < l"
hoelzl@51340
  1301
    then have "eventually (\<lambda>x. f x \<in> {B <..}) net"
hoelzl@51340
  1302
      by (intro S[rule_format]) auto
hoelzl@51340
  1303
    then have "INFI {x. B < f x} f \<le> y"
hoelzl@51340
  1304
      using P by auto
hoelzl@51340
  1305
    moreover have "B \<le> INFI {x. B < f x} f"
hoelzl@51340
  1306
      by (intro INF_greatest) auto
hoelzl@51340
  1307
    ultimately show "B \<le> y"
hoelzl@51340
  1308
      by simp
hoelzl@51340
  1309
  qed
hoelzl@51340
  1310
qed
hoelzl@51340
  1311
hoelzl@51340
  1312
lemma ereal_Limsup_Inf_monoset:
wenzelm@53788
  1313
  fixes f :: "'a \<Rightarrow> ereal"
hoelzl@51340
  1314
  shows "Limsup net f =
hoelzl@51340
  1315
    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
  1316
    (is "_ = Inf ?A")
hoelzl@51340
  1317
proof (safe intro!: Limsup_eqI complete_lattice_class.Inf_lower complete_lattice_class.Inf_greatest)
wenzelm@53788
  1318
  fix P
wenzelm@53788
  1319
  assume P: "eventually P net"
wenzelm@53788
  1320
  fix S
wenzelm@53788
  1321
  assume S: "mono_set (uminus`S)" "SUPR (Collect P) f \<in> S"
wenzelm@53788
  1322
  {
wenzelm@53788
  1323
    fix x
wenzelm@53788
  1324
    assume "P x"
hoelzl@51340
  1325
    then have "f x \<le> SUPR (Collect P) f"
hoelzl@51340
  1326
      by (intro complete_lattice_class.SUP_upper) simp
hoelzl@51340
  1327
    with S(1)[unfolded mono_set, rule_format, of "- SUPR (Collect P) f" "- f x"] S(2)
hoelzl@51340
  1328
    have "f x \<in> S"
hoelzl@51340
  1329
      by (simp add: inj_image_mem_iff) }
hoelzl@51340
  1330
  with P show "eventually (\<lambda>x. f x \<in> S) net"
hoelzl@51340
  1331
    by (auto elim: eventually_elim1)
hoelzl@51340
  1332
next
hoelzl@51340
  1333
  fix y l
hoelzl@51340
  1334
  assume S: "\<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net"
hoelzl@51340
  1335
  assume P: "\<forall>P. eventually P net \<longrightarrow> y \<le> SUPR (Collect P) f"
hoelzl@51340
  1336
  show "y \<le> l"
hoelzl@51340
  1337
  proof (rule dense_ge)
wenzelm@53788
  1338
    fix B
wenzelm@53788
  1339
    assume "l < B"
hoelzl@51340
  1340
    then have "eventually (\<lambda>x. f x \<in> {..< B}) net"
hoelzl@51340
  1341
      by (intro S[rule_format]) auto
hoelzl@51340
  1342
    then have "y \<le> SUPR {x. f x < B} f"
hoelzl@51340
  1343
      using P by auto
hoelzl@51340
  1344
    moreover have "SUPR {x. f x < B} f \<le> B"
hoelzl@51340
  1345
      by (intro SUP_least) auto
hoelzl@51340
  1346
    ultimately show "y \<le> B"
hoelzl@51340
  1347
      by simp
hoelzl@51340
  1348
  qed
hoelzl@51340
  1349
qed
hoelzl@51340
  1350
hoelzl@51340
  1351
lemma liminf_bounded_open:
hoelzl@51340
  1352
  fixes x :: "nat \<Rightarrow> ereal"
hoelzl@51340
  1353
  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
  1354
  (is "_ \<longleftrightarrow> ?P x0")
hoelzl@51340
  1355
proof
hoelzl@51340
  1356
  assume "?P x0"
hoelzl@51340
  1357
  then show "x0 \<le> liminf x"
hoelzl@51340
  1358
    unfolding ereal_Liminf_Sup_monoset eventually_sequentially
hoelzl@51340
  1359
    by (intro complete_lattice_class.Sup_upper) auto
hoelzl@51340
  1360
next
hoelzl@51340
  1361
  assume "x0 \<le> liminf x"
wenzelm@53788
  1362
  {
wenzelm@53788
  1363
    fix S :: "ereal set"
wenzelm@53788
  1364
    assume om: "open S" "mono_set S" "x0 \<in> S"
wenzelm@53788
  1365
    {
wenzelm@53788
  1366
      assume "S = UNIV"
wenzelm@53788
  1367
      then have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
wenzelm@53788
  1368
        by auto
wenzelm@53788
  1369
    }
hoelzl@51340
  1370
    moreover
wenzelm@53788
  1371
    {
wenzelm@53788
  1372
      assume "S \<noteq> UNIV"
wenzelm@53788
  1373
      then obtain B where B: "S = {B<..}"
wenzelm@53788
  1374
        using om ereal_open_mono_set by auto
wenzelm@53788
  1375
      then have "B < x0"
wenzelm@53788
  1376
        using om by auto
wenzelm@53788
  1377
      then have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
wenzelm@53788
  1378
        unfolding B
wenzelm@53788
  1379
        using `x0 \<le> liminf x` liminf_bounded_iff
wenzelm@53788
  1380
        by auto
hoelzl@51340
  1381
    }
wenzelm@53788
  1382
    ultimately have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
wenzelm@53788
  1383
      by auto
hoelzl@51340
  1384
  }
wenzelm@53788
  1385
  then show "?P x0"
wenzelm@53788
  1386
    by auto
hoelzl@51340
  1387
qed
hoelzl@51340
  1388
huffman@44125
  1389
end