src/HOL/Library/Extended_Real.thy

a few new lemmas

+
−(*  Title:      HOL/Library/Extended_Real.thy
+
−    Author:     Johannes Hölzl, TU München
+
−    Author:     Robert Himmelmann, TU München
+
−    Author:     Armin Heller, TU München
+
−    Author:     Bogdan Grechuk, University of Edinburgh
+
−    Author:     Manuel Eberl, TU München
+
−*)
+
−
+
−section \<open>Extended real number line\<close>
+
−
+
−theory Extended_Real
+
−imports Complex_Main Extended_Nat Liminf_Limsup
+
−begin
+
−
+
−text \<open>
+
−  This should be part of \<^theory>\<open>HOL-Library.Extended_Nat\<close> or \<^theory>\<open>HOL-Library.Order_Continuity\<close>, but then the AFP-entry \<open>Jinja_Thread\<close> fails, as it does overload
+
−  certain named from \<^theory>\<open>Complex_Main\<close>.
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−\<close>
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−
+
−lemma incseq_sumI2:
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−  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::ordered_comm_monoid_add"
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−  shows "(\<And>n. n \<in> A \<Longrightarrow> mono (f n)) \<Longrightarrow> mono (\<lambda>i. \<Sum>n\<in>A. f n i)"
+
−  unfolding incseq_def by (auto intro: sum_mono)
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−
+
−lemma incseq_sumI:
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−  fixes f :: "nat \<Rightarrow> 'a::ordered_comm_monoid_add"
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−  assumes "\<And>i. 0 \<le> f i"
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−  shows "incseq (\<lambda>i. sum f {..< i})"
+
−proof (intro incseq_SucI)
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−  fix n
+
−  have "sum f {..< n} + 0 \<le> sum f {..<n} + f n"
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−    using assms by (rule add_left_mono)
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−  then show "sum f {..< n} \<le> sum f {..< Suc n}"
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−    by auto
+
−qed
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−
+
−lemma continuous_at_left_imp_sup_continuous:
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−  fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
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−  assumes "mono f" "\<And>x. continuous (at_left x) f"
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−  shows "sup_continuous f"
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−  unfolding sup_continuous_def
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−proof safe
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−  fix M :: "nat \<Rightarrow> 'a" assume "incseq M" then show "f (SUP i. M i) = (SUP i. f (M i))"
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−    using continuous_at_Sup_mono [OF assms, of "range M"] by (simp add: image_comp)
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−qed
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−
+
−lemma sup_continuous_at_left:
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−  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \<Rightarrow>
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−    'b::{complete_linorder, linorder_topology}"
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−  assumes f: "sup_continuous f"
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−  shows "continuous (at_left x) f"
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−proof cases
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−  assume "x = bot" then show ?thesis
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−    by (simp add: trivial_limit_at_left_bot)
+
−next
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−  assume x: "x \<noteq> bot"
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−  show ?thesis
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−    unfolding continuous_within
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−  proof (intro tendsto_at_left_sequentially[of bot])
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−    fix S :: "nat \<Rightarrow> 'a" assume S: "incseq S" and S_x: "S \<longlonglongrightarrow> x"
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−    from S_x have x_eq: "x = (SUP i. S i)"
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−      by (rule LIMSEQ_unique) (intro LIMSEQ_SUP S)
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−    show "(\<lambda>n. f (S n)) \<longlonglongrightarrow> f x"
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−      unfolding x_eq sup_continuousD[OF f S]
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−      using S sup_continuous_mono[OF f] by (intro LIMSEQ_SUP) (auto simp: mono_def)
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−  qed (insert x, auto simp: bot_less)
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−qed
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−
+
−lemma sup_continuous_iff_at_left:
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−  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \<Rightarrow>
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−    'b::{complete_linorder, linorder_topology}"
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−  shows "sup_continuous f \<longleftrightarrow> (\<forall>x. continuous (at_left x) f) \<and> mono f"
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−  using sup_continuous_at_left[of f] continuous_at_left_imp_sup_continuous[of f]
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−    sup_continuous_mono[of f] by auto
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−
+
−lemma continuous_at_right_imp_inf_continuous:
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−  fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
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−  assumes "mono f" "\<And>x. continuous (at_right x) f"
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−  shows "inf_continuous f"
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−  unfolding inf_continuous_def
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−proof safe
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−  fix M :: "nat \<Rightarrow> 'a" assume "decseq M" then show "f (INF i. M i) = (INF i. f (M i))"
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−    using continuous_at_Inf_mono [OF assms, of "range M"] by (simp add: image_comp)
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−qed
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−
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−lemma inf_continuous_at_right:
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−  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \<Rightarrow>
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−    'b::{complete_linorder, linorder_topology}"
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−  assumes f: "inf_continuous f"
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−  shows "continuous (at_right x) f"
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−proof cases
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−  assume "x = top" then show ?thesis
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−    by (simp add: trivial_limit_at_right_top)
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−next
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−  assume x: "x \<noteq> top"
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−  show ?thesis
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−    unfolding continuous_within
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−  proof (intro tendsto_at_right_sequentially[of _ top])
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−    fix S :: "nat \<Rightarrow> 'a" assume S: "decseq S" and S_x: "S \<longlonglongrightarrow> x"
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−    from S_x have x_eq: "x = (INF i. S i)"
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−      by (rule LIMSEQ_unique) (intro LIMSEQ_INF S)
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−    show "(\<lambda>n. f (S n)) \<longlonglongrightarrow> f x"
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−      unfolding x_eq inf_continuousD[OF f S]
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−      using S inf_continuous_mono[OF f] by (intro LIMSEQ_INF) (auto simp: mono_def antimono_def)
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−  qed (insert x, auto simp: less_top)
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−qed
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−
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−lemma inf_continuous_iff_at_right:
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−  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \<Rightarrow>
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−    'b::{complete_linorder, linorder_topology}"
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−  shows "inf_continuous f \<longleftrightarrow> (\<forall>x. continuous (at_right x) f) \<and> mono f"
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−  using inf_continuous_at_right[of f] continuous_at_right_imp_inf_continuous[of f]
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−    inf_continuous_mono[of f] by auto
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−
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−instantiation enat :: linorder_topology
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−begin
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−
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−definition open_enat :: "enat set \<Rightarrow> bool" where
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−  "open_enat = generate_topology (range lessThan \<union> range greaterThan)"
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−
+
−instance
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−  proof qed (rule open_enat_def)
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−
+
−end
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−
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−lemma open_enat: "open {enat n}"
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−proof (cases n)
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−  case 0
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−  then have "{enat n} = {..< eSuc 0}"
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−    by (auto simp: enat_0)
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−  then show ?thesis
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−    by simp
+
−next
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−  case (Suc n')
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−  then have "{enat n} = {enat n' <..< enat (Suc n)}"
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−    apply auto
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−    apply (case_tac x)
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−    apply auto
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−    done
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−  then show ?thesis
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−    by simp
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−qed
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−
+
−lemma open_enat_iff:
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−  fixes A :: "enat set"
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−  shows "open A \<longleftrightarrow> (\<infinity> \<in> A \<longrightarrow> (\<exists>n::nat. {n <..} \<subseteq> A))"
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−proof safe
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−  assume "\<infinity> \<notin> A"
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−  then have "A = (\<Union>n\<in>{n. enat n \<in> A}. {enat n})"
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−    apply auto
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−    apply (case_tac x)
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−    apply auto
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−    done
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−  moreover have "open \<dots>"
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−    by (auto intro: open_enat)
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−  ultimately show "open A"
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−    by simp
+
−next
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−  fix n assume "{enat n <..} \<subseteq> A"
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−  then have "A = (\<Union>n\<in>{n. enat n \<in> A}. {enat n}) \<union> {enat n <..}"
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−    apply auto
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−    apply (case_tac x)
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−    apply auto
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−    done
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−  moreover have "open \<dots>"
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−    by (intro open_Un open_UN ballI open_enat open_greaterThan)
+
−  ultimately show "open A"
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−    by simp
+
−next
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−  assume "open A" "\<infinity> \<in> A"
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−  then have "generate_topology (range lessThan \<union> range greaterThan) A" "\<infinity> \<in> A"
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−    unfolding open_enat_def by auto
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−  then show "\<exists>n::nat. {n <..} \<subseteq> A"
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−  proof induction
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−    case (Int A B)
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−    then obtain n m where "{enat n<..} \<subseteq> A" "{enat m<..} \<subseteq> B"
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−      by auto
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−    then have "{enat (max n m) <..} \<subseteq> A \<inter> B"
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−      by (auto simp add: subset_eq Ball_def max_def simp flip: enat_ord_code(1))
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−    then show ?case
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−      by auto
+
−  next
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−    case (UN K)
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−    then obtain k where "k \<in> K" "\<infinity> \<in> k"
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−      by auto
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−    with UN.IH[OF this] show ?case
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−      by auto
+
−  qed auto
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−qed
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−
+
−lemma nhds_enat: "nhds x = (if x = \<infinity> then INF i. principal {enat i..} else principal {x})"
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−proof auto
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−  show "nhds \<infinity> = (INF i. principal {enat i..})"
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−    unfolding nhds_def
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−    apply (auto intro!: antisym INF_greatest simp add: open_enat_iff cong: rev_conj_cong)
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−    apply (auto intro!: INF_lower Ioi_le_Ico) []
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−    subgoal for x i
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−      by (auto intro!: INF_lower2[of "Suc i"] simp: subset_eq Ball_def eSuc_enat Suc_ile_eq)
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−    done
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−  show "nhds (enat i) = principal {enat i}" for i
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−    by (simp add: nhds_discrete_open open_enat)
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−qed
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−
+
−instance enat :: topological_comm_monoid_add
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−proof
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−  have [simp]: "enat i \<le> aa \<Longrightarrow> enat i \<le> aa + ba" for aa ba i
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−    by (rule order_trans[OF _ add_mono[of aa aa 0 ba]]) auto
+
−  then have [simp]: "enat i \<le> ba \<Longrightarrow> enat i \<le> aa + ba" for aa ba i
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−    by (metis add.commute)
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−  fix a b :: enat show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
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−    apply (auto simp: nhds_enat filterlim_INF prod_filter_INF1 prod_filter_INF2
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−                      filterlim_principal principal_prod_principal eventually_principal)
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−    subgoal for i
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−      by (auto intro!: eventually_INF1[of i] simp: eventually_principal)
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−    subgoal for j i
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−      by (auto intro!: eventually_INF1[of i] simp: eventually_principal)
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−    subgoal for j i
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−      by (auto intro!: eventually_INF1[of i] simp: eventually_principal)
+
−    done
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−qed
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−
+
−text \<open>
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−  For more lemmas about the extended real numbers see
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−  \<^file>\<open>~~/src/HOL/Analysis/Extended_Real_Limits.thy\<close>.
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−\<close>
+
−
+
−subsection \<open>Definition and basic properties\<close>
+
−
+
−datatype ereal = ereal real | PInfty | MInfty
+
−
+
−lemma ereal_cong: "x = y \<Longrightarrow> ereal x = ereal y" by simp
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−
+
−instantiation ereal :: uminus
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−begin
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−
+
−fun uminus_ereal where
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−  "- (ereal r) = ereal (- r)"
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−| "- PInfty = MInfty"
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−| "- MInfty = PInfty"
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−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation ereal :: infinity
+
−begin
+
−
+
−definition "(\<infinity>::ereal) = PInfty"
+
−instance ..
+
−
+
−end
+
−
+
−declare [[coercion "ereal :: real \<Rightarrow> ereal"]]
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−
+
−lemma ereal_uminus_uminus[simp]:
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−  fixes a :: ereal
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−  shows "- (- a) = a"
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−  by (cases a) simp_all
+
−
+
−lemma
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−  shows PInfty_eq_infinity[simp]: "PInfty = \<infinity>"
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−    and MInfty_eq_minfinity[simp]: "MInfty = - \<infinity>"
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−    and MInfty_neq_PInfty[simp]: "\<infinity> \<noteq> - (\<infinity>::ereal)" "- \<infinity> \<noteq> (\<infinity>::ereal)"
+
−    and MInfty_neq_ereal[simp]: "ereal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> ereal r"
+
−    and PInfty_neq_ereal[simp]: "ereal r \<noteq> \<infinity>" "\<infinity> \<noteq> ereal r"
+
−    and PInfty_cases[simp]: "(case \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = y"
+
−    and MInfty_cases[simp]: "(case - \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = z"
+
−  by (simp_all add: infinity_ereal_def)
+
−
+
−declare
+
−  PInfty_eq_infinity[code_post]
+
−  MInfty_eq_minfinity[code_post]
+
−
+
−lemma [code_unfold]:
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−  "\<infinity> = PInfty"
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−  "- PInfty = MInfty"
+
−  by simp_all
+
−
+
−lemma inj_ereal[simp]: "inj_on ereal A"
+
−  unfolding inj_on_def by auto
+
−
+
−lemma ereal_cases[cases type: ereal]:
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−  obtains (real) r where "x = ereal r"
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−    | (PInf) "x = \<infinity>"
+
−    | (MInf) "x = -\<infinity>"
+
−  by (cases x) auto
+
−
+
−lemmas ereal2_cases = ereal_cases[case_product ereal_cases]
+
−lemmas ereal3_cases = ereal2_cases[case_product ereal_cases]
+
−
+
−lemma ereal_all_split: "\<And>P. (\<forall>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<and> (\<forall>x. P (ereal x)) \<and> P (-\<infinity>)"
+
−  by (metis ereal_cases)
+
−
+
−lemma ereal_ex_split: "\<And>P. (\<exists>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<or> (\<exists>x. P (ereal x)) \<or> P (-\<infinity>)"
+
−  by (metis ereal_cases)
+
−
+
−lemma ereal_uminus_eq_iff[simp]:
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−  fixes a b :: ereal
+
−  shows "-a = -b \<longleftrightarrow> a = b"
+
−  by (cases rule: ereal2_cases[of a b]) simp_all
+
−
+
−function real_of_ereal :: "ereal \<Rightarrow> real" where
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−  "real_of_ereal (ereal r) = r"
+
−| "real_of_ereal \<infinity> = 0"
+
−| "real_of_ereal (-\<infinity>) = 0"
+
−  by (auto intro: ereal_cases)
+
−termination by standard (rule wf_empty)
+
−
+
−lemma real_of_ereal[simp]:
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−  "real_of_ereal (- x :: ereal) = - (real_of_ereal x)"
+
−  by (cases x) simp_all
+
−
+
−lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}"
+
−proof safe
+
−  fix x
+
−  assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
+
−  then show "x = -\<infinity>"
+
−    by (cases x) auto
+
−qed auto
+
−
+
−lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
+
−proof safe
+
−  fix x :: ereal
+
−  show "x \<in> range uminus"
+
−    by (intro image_eqI[of _ _ "-x"]) auto
+
−qed auto
+
−
+
−instantiation ereal :: abs
+
−begin
+
−
+
−function abs_ereal where
+
−  "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
+
−| "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
+
−| "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
+
−by (auto intro: ereal_cases)
+
−termination proof qed (rule wf_empty)
+
−
+
−instance ..
+
−
+
−end
+
−
+
−lemma abs_eq_infinity_cases[elim!]:
+
−  fixes x :: ereal
+
−  assumes "\<bar>x\<bar> = \<infinity>"
+
−  obtains "x = \<infinity>" | "x = -\<infinity>"
+
−  using assms by (cases x) auto
+
−
+
−lemma abs_neq_infinity_cases[elim!]:
+
−  fixes x :: ereal
+
−  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+
−  obtains r where "x = ereal r"
+
−  using assms by (cases x) auto
+
−
+
−lemma abs_ereal_uminus[simp]:
+
−  fixes x :: ereal
+
−  shows "\<bar>- x\<bar> = \<bar>x\<bar>"
+
−  by (cases x) auto
+
−
+
−lemma ereal_infinity_cases:
+
−  fixes a :: ereal
+
−  shows "a \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
+
−  by auto
+
−
+
−subsubsection "Addition"
+
−
+
−instantiation ereal :: "{one,comm_monoid_add,zero_neq_one}"
+
−begin
+
−
+
−definition "0 = ereal 0"
+
−definition "1 = ereal 1"
+
−
+
−function plus_ereal where
+
−  "ereal r + ereal p = ereal (r + p)"
+
−| "\<infinity> + a = (\<infinity>::ereal)"
+
−| "a + \<infinity> = (\<infinity>::ereal)"
+
−| "ereal r + -\<infinity> = - \<infinity>"
+
−| "-\<infinity> + ereal p = -(\<infinity>::ereal)"
+
−| "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
+
−proof goal_cases
+
−  case prems: (1 P x)
+
−  then obtain a b where "x = (a, b)"
+
−    by (cases x) auto
+
−  with prems show P
+
−   by (cases rule: ereal2_cases[of a b]) auto
+
−qed auto
+
−termination by standard (rule wf_empty)
+
−
+
−lemma Infty_neq_0[simp]:
+
−  "(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)"
+
−  "-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)"
+
−  by (simp_all add: zero_ereal_def)
+
−
+
−lemma ereal_eq_0[simp]:
+
−  "ereal r = 0 \<longleftrightarrow> r = 0"
+
−  "0 = ereal r \<longleftrightarrow> r = 0"
+
−  unfolding zero_ereal_def by simp_all
+
−
+
−lemma ereal_eq_1[simp]:
+
−  "ereal r = 1 \<longleftrightarrow> r = 1"
+
−  "1 = ereal r \<longleftrightarrow> r = 1"
+
−  unfolding one_ereal_def by simp_all
+
−
+
−instance
+
−proof
+
−  fix a b c :: ereal
+
−  show "0 + a = a"
+
−    by (cases a) (simp_all add: zero_ereal_def)
+
−  show "a + b = b + a"
+
−    by (cases rule: ereal2_cases[of a b]) simp_all
+
−  show "a + b + c = a + (b + c)"
+
−    by (cases rule: ereal3_cases[of a b c]) simp_all
+
−  show "0 \<noteq> (1::ereal)"
+
−    by (simp add: one_ereal_def zero_ereal_def)
+
−qed
+
−
+
−end
+
−
+
−lemma ereal_0_plus [simp]: "ereal 0 + x = x"
+
−  and plus_ereal_0 [simp]: "x + ereal 0 = x"
+
−by(simp_all flip: zero_ereal_def)
+
−
+
−instance ereal :: numeral ..
+
−
+
−lemma real_of_ereal_0[simp]: "real_of_ereal (0::ereal) = 0"
+
−  unfolding zero_ereal_def by simp
+
−
+
−lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
+
−  unfolding zero_ereal_def abs_ereal.simps by simp
+
−
+
−lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)"
+
−  by (simp add: zero_ereal_def)
+
−
+
−lemma ereal_uminus_zero_iff[simp]:
+
−  fixes a :: ereal
+
−  shows "-a = 0 \<longleftrightarrow> a = 0"
+
−  by (cases a) simp_all
+
−
+
−lemma ereal_plus_eq_PInfty[simp]:
+
−  fixes a b :: ereal
+
−  shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_plus_eq_MInfty[simp]:
+
−  fixes a b :: ereal
+
−  shows "a + b = -\<infinity> \<longleftrightarrow> (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_add_cancel_left:
+
−  fixes a b :: ereal
+
−  assumes "a \<noteq> -\<infinity>"
+
−  shows "a + b = a + c \<longleftrightarrow> a = \<infinity> \<or> b = c"
+
−  using assms by (cases rule: ereal3_cases[of a b c]) auto
+
−
+
−lemma ereal_add_cancel_right:
+
−  fixes a b :: ereal
+
−  assumes "a \<noteq> -\<infinity>"
+
−  shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c"
+
−  using assms by (cases rule: ereal3_cases[of a b c]) auto
+
−
+
−lemma ereal_real: "ereal (real_of_ereal x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
+
−  by (cases x) simp_all
+
−
+
−lemma real_of_ereal_add:
+
−  fixes a b :: ereal
+
−  shows "real_of_ereal (a + b) =
+
−    (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real_of_ereal a + real_of_ereal b else 0)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−
+
−subsubsection "Linear order on \<^typ>\<open>ereal\<close>"
+
−
+
−instantiation ereal :: linorder
+
−begin
+
−
+
−function less_ereal
+
−where
+
−  "   ereal x < ereal y     \<longleftrightarrow> x < y"
+
−| "(\<infinity>::ereal) < a           \<longleftrightarrow> False"
+
−| "         a < -(\<infinity>::ereal) \<longleftrightarrow> False"
+
−| "ereal x    < \<infinity>           \<longleftrightarrow> True"
+
−| "        -\<infinity> < ereal r     \<longleftrightarrow> True"
+
−| "        -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True"
+
−proof goal_cases
+
−  case prems: (1 P x)
+
−  then obtain a b where "x = (a,b)" by (cases x) auto
+
−  with prems show P by (cases rule: ereal2_cases[of a b]) auto
+
−qed simp_all
+
−termination by (relation "{}") simp
+
−
+
−definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"
+
−
+
−lemma ereal_infty_less[simp]:
+
−  fixes x :: ereal
+
−  shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
+
−    "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
+
−  by (cases x, simp_all) (cases x, simp_all)
+
−
+
−lemma ereal_infty_less_eq[simp]:
+
−  fixes x :: ereal
+
−  shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
+
−    and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
+
−  by (auto simp add: less_eq_ereal_def)
+
−
+
−lemma ereal_less[simp]:
+
−  "ereal r < 0 \<longleftrightarrow> (r < 0)"
+
−  "0 < ereal r \<longleftrightarrow> (0 < r)"
+
−  "ereal r < 1 \<longleftrightarrow> (r < 1)"
+
−  "1 < ereal r \<longleftrightarrow> (1 < r)"
+
−  "0 < (\<infinity>::ereal)"
+
−  "-(\<infinity>::ereal) < 0"
+
−  by (simp_all add: zero_ereal_def one_ereal_def)
+
−
+
−lemma ereal_less_eq[simp]:
+
−  "x \<le> (\<infinity>::ereal)"
+
−  "-(\<infinity>::ereal) \<le> x"
+
−  "ereal r \<le> ereal p \<longleftrightarrow> r \<le> p"
+
−  "ereal r \<le> 0 \<longleftrightarrow> r \<le> 0"
+
−  "0 \<le> ereal r \<longleftrightarrow> 0 \<le> r"
+
−  "ereal r \<le> 1 \<longleftrightarrow> r \<le> 1"
+
−  "1 \<le> ereal r \<longleftrightarrow> 1 \<le> r"
+
−  by (auto simp add: less_eq_ereal_def zero_ereal_def one_ereal_def)
+
−
+
−lemma ereal_infty_less_eq2:
+
−  "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)"
+
−  "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)"
+
−  by simp_all
+
−
+
−instance
+
−proof
+
−  fix x y z :: ereal
+
−  show "x \<le> x"
+
−    by (cases x) simp_all
+
−  show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
+
−    by (cases rule: ereal2_cases[of x y]) auto
+
−  show "x \<le> y \<or> y \<le> x "
+
−    by (cases rule: ereal2_cases[of x y]) auto
+
−  {
+
−    assume "x \<le> y" "y \<le> x"
+
−    then show "x = y"
+
−      by (cases rule: ereal2_cases[of x y]) auto
+
−  }
+
−  {
+
−    assume "x \<le> y" "y \<le> z"
+
−    then show "x \<le> z"
+
−      by (cases rule: ereal3_cases[of x y z]) auto
+
−  }
+
−qed
+
−
+
−end
+
−
+
−lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> ereal z < y"
+
−  using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto
+
−
+
−instance ereal :: dense_linorder
+
−  by standard (blast dest: ereal_dense2)
+
−
+
−instance ereal :: ordered_comm_monoid_add
+
−proof
+
−  fix a b c :: ereal
+
−  assume "a \<le> b"
+
−  then show "c + a \<le> c + b"
+
−    by (cases rule: ereal3_cases[of a b c]) auto
+
−qed
+
−
+
−lemma ereal_one_not_less_zero_ereal[simp]: "\<not> 1 < (0::ereal)"
+
−  by (simp add: zero_ereal_def)
+
−
+
−lemma real_of_ereal_positive_mono:
+
−  fixes x y :: ereal
+
−  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real_of_ereal x \<le> real_of_ereal y"
+
−  by (cases rule: ereal2_cases[of x y]) auto
+
−
+
−lemma ereal_MInfty_lessI[intro, simp]:
+
−  fixes a :: ereal
+
−  shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
+
−  by (cases a) auto
+
−
+
−lemma ereal_less_PInfty[intro, simp]:
+
−  fixes a :: ereal
+
−  shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
+
−  by (cases a) auto
+
−
+
−lemma ereal_less_ereal_Ex:
+
−  fixes a b :: ereal
+
−  shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)"
+
−  by (cases x) auto
+
−
+
−lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < ereal (real n))"
+
−proof (cases x)
+
−  case (real r)
+
−  then show ?thesis
+
−    using reals_Archimedean2[of r] by simp
+
−qed simp_all
+
−
+
−lemma ereal_add_strict_mono2:
+
−  fixes a b c d :: ereal
+
−  assumes "a < b"
+
−    and "c < d"
+
−  shows "a + c < b + d"
+
−using assms
+
−apply (cases a)
+
−apply (cases rule: ereal3_cases[of b c d], auto)
+
−apply (cases rule: ereal3_cases[of b c d], auto)
+
−done
+
−
+
−lemma ereal_minus_le_minus[simp]:
+
−  fixes a b :: ereal
+
−  shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_minus_less_minus[simp]:
+
−  fixes a b :: ereal
+
−  shows "- a < - b \<longleftrightarrow> b < a"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_le_real_iff:
+
−  "x \<le> real_of_ereal y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)"
+
−  by (cases y) auto
+
−
+
−lemma real_le_ereal_iff:
+
−  "real_of_ereal y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)"
+
−  by (cases y) auto
+
−
+
−lemma ereal_less_real_iff:
+
−  "x < real_of_ereal y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)"
+
−  by (cases y) auto
+
−
+
−lemma real_less_ereal_iff:
+
−  "real_of_ereal y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
+
−  by (cases y) auto
+
−
+
−text \<open>
+
−  To help with inferences like \<^prop>\<open>a < ereal x \<Longrightarrow> x < y \<Longrightarrow> a < ereal y\<close>,
+
−  where x and y are real.
+
−\<close>
+
−
+
−lemma le_ereal_le: "a \<le> ereal x \<Longrightarrow> x \<le> y \<Longrightarrow> a \<le> ereal y"
+
−  using ereal_less_eq(3) order.trans by blast
+
−
+
−lemma le_ereal_less: "a \<le> ereal x \<Longrightarrow> x < y \<Longrightarrow> a < ereal y"
+
−  by (simp add: le_less_trans)
+
−
+
−lemma less_ereal_le: "a < ereal x \<Longrightarrow> x \<le> y \<Longrightarrow> a < ereal y"
+
−  using ereal_less_ereal_Ex by auto
+
−
+
−lemma ereal_le_le: "ereal y \<le> a \<Longrightarrow> x \<le> y \<Longrightarrow> ereal x \<le> a"
+
−  by (simp add: order_subst2)
+
−
+
−lemma ereal_le_less: "ereal y \<le> a \<Longrightarrow> x < y \<Longrightarrow> ereal x < a"
+
−  by (simp add: dual_order.strict_trans1)
+
−
+
−lemma ereal_less_le: "ereal y < a \<Longrightarrow> x \<le> y \<Longrightarrow> ereal x < a"
+
−  using ereal_less_eq(3) le_less_trans by blast
+
−
+
−lemma real_of_ereal_pos:
+
−  fixes x :: ereal
+
−  shows "0 \<le> x \<Longrightarrow> 0 \<le> real_of_ereal x" by (cases x) auto
+
−
+
−lemmas real_of_ereal_ord_simps =
+
−  ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff
+
−
+
−lemma abs_ereal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x"
+
−  by (cases x) auto
+
−
+
−lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x"
+
−  by (cases x) auto
+
−
+
−lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
+
−  by (cases x) auto
+
−
+
−lemma ereal_abs_leI:
+
−  fixes x y :: ereal
+
−  shows "\<lbrakk> x \<le> y; -x \<le> y \<rbrakk> \<Longrightarrow> \<bar>x\<bar> \<le> y"
+
−by(cases x y rule: ereal2_cases)(simp_all)
+
−
+
−lemma ereal_abs_add:
+
−  fixes a b::ereal
+
−  shows "abs(a+b) \<le> abs a + abs b"
+
−by (cases rule: ereal2_cases[of a b]) (auto)
+
−
+
−lemma real_of_ereal_le_0[simp]: "real_of_ereal (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>"
+
−  by (cases x) auto
+
−
+
−lemma abs_real_of_ereal[simp]: "\<bar>real_of_ereal (x :: ereal)\<bar> = real_of_ereal \<bar>x\<bar>"
+
−  by (cases x) auto
+
−
+
−lemma zero_less_real_of_ereal:
+
−  fixes x :: ereal
+
−  shows "0 < real_of_ereal x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
+
−  by (cases x) auto
+
−
+
−lemma ereal_0_le_uminus_iff[simp]:
+
−  fixes a :: ereal
+
−  shows "0 \<le> - a \<longleftrightarrow> a \<le> 0"
+
−  by (cases rule: ereal2_cases[of a]) auto
+
−
+
−lemma ereal_uminus_le_0_iff[simp]:
+
−  fixes a :: ereal
+
−  shows "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
+
−  by (cases rule: ereal2_cases[of a]) auto
+
−
+
−lemma ereal_add_strict_mono:
+
−  fixes a b c d :: ereal
+
−  assumes "a \<le> b"
+
−    and "0 \<le> a"
+
−    and "a \<noteq> \<infinity>"
+
−    and "c < d"
+
−  shows "a + c < b + d"
+
−  using assms
+
−  by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto
+
−
+
−lemma ereal_less_add:
+
−  fixes a b c :: ereal
+
−  shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
+
−  by (cases rule: ereal2_cases[of b c]) auto
+
−
+
−lemma ereal_add_nonneg_eq_0_iff:
+
−  fixes a b :: ereal
+
−  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
+
−  by (cases a b rule: ereal2_cases) auto
+
−
+
−lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)"
+
−  by auto
+
−
+
−lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)"
+
−  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
+
−
+
−lemma ereal_less_uminus_reorder: "a < - b \<longleftrightarrow> b < - (a::ereal)"
+
−  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
+
−
+
−lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::ereal)"
+
−  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus)
+
−
+
−lemmas ereal_uminus_reorder =
+
−  ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder
+
−
+
−lemma ereal_bot:
+
−  fixes x :: ereal
+
−  assumes "\<And>B. x \<le> ereal B"
+
−  shows "x = - \<infinity>"
+
−proof (cases x)
+
−  case (real r)
+
−  with assms[of "r - 1"] show ?thesis
+
−    by auto
+
−next
+
−  case PInf
+
−  with assms[of 0] show ?thesis
+
−    by auto
+
−next
+
−  case MInf
+
−  then show ?thesis
+
−    by simp
+
−qed
+
−
+
−lemma ereal_top:
+
−  fixes x :: ereal
+
−  assumes "\<And>B. x \<ge> ereal B"
+
−  shows "x = \<infinity>"
+
−proof (cases x)
+
−  case (real r)
+
−  with assms[of "r + 1"] show ?thesis
+
−    by auto
+
−next
+
−  case MInf
+
−  with assms[of 0] show ?thesis
+
−    by auto
+
−next
+
−  case PInf
+
−  then show ?thesis
+
−    by simp
+
−qed
+
−
+
−lemma
+
−  shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)"
+
−    and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)"
+
−  by (simp_all add: min_def max_def)
+
−
+
−lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
+
−  by (auto simp: zero_ereal_def)
+
−
+
−lemma
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  shows ereal_incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
+
−    and ereal_decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
+
−  unfolding decseq_def incseq_def by auto
+
−
+
−lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))"
+
−  unfolding incseq_def by auto
+
−
+
−lemma sum_ereal[simp]: "(\<Sum>x\<in>A. ereal (f x)) = ereal (\<Sum>x\<in>A. f x)"
+
−proof (cases "finite A")
+
−  case True
+
−  then show ?thesis by induct auto
+
−next
+
−  case False
+
−  then show ?thesis by simp
+
−qed
+
−
+
−lemma sum_list_ereal [simp]: "sum_list (map (\<lambda>x. ereal (f x)) xs) = ereal (sum_list (map f xs))"
+
−  by (induction xs) simp_all
+
−
+
−lemma sum_Pinfty:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> finite P \<and> (\<exists>i\<in>P. f i = \<infinity>)"
+
−proof safe
+
−  assume *: "sum f P = \<infinity>"
+
−  show "finite P"
+
−  proof (rule ccontr)
+
−    assume "\<not> finite P"
+
−    with * show False
+
−      by auto
+
−  qed
+
−  show "\<exists>i\<in>P. f i = \<infinity>"
+
−  proof (rule ccontr)
+
−    assume "\<not> ?thesis"
+
−    then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>"
+
−      by auto
+
−    with \<open>finite P\<close> have "sum f P \<noteq> \<infinity>"
+
−      by induct auto
+
−    with * show False
+
−      by auto
+
−  qed
+
−next
+
−  fix i
+
−  assume "finite P" and "i \<in> P" and "f i = \<infinity>"
+
−  then show "sum f P = \<infinity>"
+
−  proof induct
+
−    case (insert x A)
+
−    show ?case using insert by (cases "x = i") auto
+
−  qed simp
+
−qed
+
−
+
−lemma sum_Inf:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "\<bar>sum f A\<bar> = \<infinity> \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
+
−proof
+
−  assume *: "\<bar>sum f A\<bar> = \<infinity>"
+
−  have "finite A"
+
−    by (rule ccontr) (insert *, auto)
+
−  moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
+
−  proof (rule ccontr)
+
−    assume "\<not> ?thesis"
+
−    then have "\<forall>i\<in>A. \<exists>r. f i = ereal r"
+
−      by auto
+
−    from bchoice[OF this] obtain r where "\<forall>x\<in>A. f x = ereal (r x)" ..
+
−    with * show False
+
−      by auto
+
−  qed
+
−  ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
+
−    by auto
+
−next
+
−  assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
+
−  then obtain i where "finite A" "i \<in> A" and "\<bar>f i\<bar> = \<infinity>"
+
−    by auto
+
−  then show "\<bar>sum f A\<bar> = \<infinity>"
+
−  proof induct
+
−    case (insert j A)
+
−    then show ?case
+
−      by (cases rule: ereal3_cases[of "f i" "f j" "sum f A"]) auto
+
−  qed simp
+
−qed
+
−
+
−lemma sum_real_of_ereal:
+
−  fixes f :: "'i \<Rightarrow> ereal"
+
−  assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
+
−  shows "(\<Sum>x\<in>S. real_of_ereal (f x)) = real_of_ereal (sum f S)"
+
−proof -
+
−  have "\<forall>x\<in>S. \<exists>r. f x = ereal r"
+
−  proof
+
−    fix x
+
−    assume "x \<in> S"
+
−    from assms[OF this] show "\<exists>r. f x = ereal r"
+
−      by (cases "f x") auto
+
−  qed
+
−  from bchoice[OF this] obtain r where "\<forall>x\<in>S. f x = ereal (r x)" ..
+
−  then show ?thesis
+
−    by simp
+
−qed
+
−
+
−lemma sum_ereal_0:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes "finite A"
+
−    and "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
+
−  shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
+
−proof
+
−  assume "sum f A = 0" with assms show "\<forall>i\<in>A. f i = 0"
+
−  proof (induction A)
+
−    case (insert a A)
+
−    then have "f a = 0 \<and> (\<Sum>a\<in>A. f a) = 0"
+
−      by (subst ereal_add_nonneg_eq_0_iff[symmetric]) (simp_all add: sum_nonneg)
+
−    with insert show ?case
+
−      by simp
+
−  qed simp
+
−qed auto
+
−
+
−subsubsection "Multiplication"
+
−
+
−instantiation ereal :: "{comm_monoid_mult,sgn}"
+
−begin
+
−
+
−function sgn_ereal :: "ereal \<Rightarrow> ereal" where
+
−  "sgn (ereal r) = ereal (sgn r)"
+
−| "sgn (\<infinity>::ereal) = 1"
+
−| "sgn (-\<infinity>::ereal) = -1"
+
−by (auto intro: ereal_cases)
+
−termination by standard (rule wf_empty)
+
−
+
−function times_ereal where
+
−  "ereal r * ereal p = ereal (r * p)"
+
−| "ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
+
−| "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
+
−| "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
+
−| "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
+
−| "(\<infinity>::ereal) * \<infinity> = \<infinity>"
+
−| "-(\<infinity>::ereal) * \<infinity> = -\<infinity>"
+
−| "(\<infinity>::ereal) * -\<infinity> = -\<infinity>"
+
−| "-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
+
−proof goal_cases
+
−  case prems: (1 P x)
+
−  then obtain a b where "x = (a, b)"
+
−    by (cases x) auto
+
−  with prems show P
+
−    by (cases rule: ereal2_cases[of a b]) auto
+
−qed simp_all
+
−termination by (relation "{}") simp
+
−
+
−instance
+
−proof
+
−  fix a b c :: ereal
+
−  show "1 * a = a"
+
−    by (cases a) (simp_all add: one_ereal_def)
+
−  show "a * b = b * a"
+
−    by (cases rule: ereal2_cases[of a b]) simp_all
+
−  show "a * b * c = a * (b * c)"
+
−    by (cases rule: ereal3_cases[of a b c])
+
−       (simp_all add: zero_ereal_def zero_less_mult_iff)
+
−qed
+
−
+
−end
+
−
+
−lemma [simp]:
+
−  shows ereal_1_times: "ereal 1 * x = x"
+
−  and times_ereal_1: "x * ereal 1 = x"
+
−by(simp_all flip: one_ereal_def)
+
−
+
−lemma one_not_le_zero_ereal[simp]: "\<not> (1 \<le> (0::ereal))"
+
−  by (simp add: one_ereal_def zero_ereal_def)
+
−
+
−lemma real_ereal_1[simp]: "real_of_ereal (1::ereal) = 1"
+
−  unfolding one_ereal_def by simp
+
−
+
−lemma real_of_ereal_le_1:
+
−  fixes a :: ereal
+
−  shows "a \<le> 1 \<Longrightarrow> real_of_ereal a \<le> 1"
+
−  by (cases a) (auto simp: one_ereal_def)
+
−
+
−lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)"
+
−  unfolding one_ereal_def by simp
+
−
+
−lemma ereal_mult_zero[simp]:
+
−  fixes a :: ereal
+
−  shows "a * 0 = 0"
+
−  by (cases a) (simp_all add: zero_ereal_def)
+
−
+
−lemma ereal_zero_mult[simp]:
+
−  fixes a :: ereal
+
−  shows "0 * a = 0"
+
−  by (cases a) (simp_all add: zero_ereal_def)
+
−
+
−lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0"
+
−  by (simp add: zero_ereal_def one_ereal_def)
+
−
+
−lemma ereal_times[simp]:
+
−  "1 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1"
+
−  "1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 1"
+
−  by (auto simp: one_ereal_def)
+
−
+
−lemma ereal_plus_1[simp]:
+
−  "1 + ereal r = ereal (r + 1)"
+
−  "ereal r + 1 = ereal (r + 1)"
+
−  "1 + -(\<infinity>::ereal) = -\<infinity>"
+
−  "-(\<infinity>::ereal) + 1 = -\<infinity>"
+
−  unfolding one_ereal_def by auto
+
−
+
−lemma ereal_zero_times[simp]:
+
−  fixes a b :: ereal
+
−  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_mult_eq_PInfty[simp]:
+
−  "a * b = (\<infinity>::ereal) \<longleftrightarrow>
+
−    (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_mult_eq_MInfty[simp]:
+
−  "a * b = -(\<infinity>::ereal) \<longleftrightarrow>
+
−    (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_abs_mult: "\<bar>x * y :: ereal\<bar> = \<bar>x\<bar> * \<bar>y\<bar>"
+
−  by (cases x y rule: ereal2_cases) (auto simp: abs_mult)
+
−
+
−lemma ereal_0_less_1[simp]: "0 < (1::ereal)"
+
−  by (simp_all add: zero_ereal_def one_ereal_def)
+
−
+
−lemma ereal_mult_minus_left[simp]:
+
−  fixes a b :: ereal
+
−  shows "-a * b = - (a * b)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_mult_minus_right[simp]:
+
−  fixes a b :: ereal
+
−  shows "a * -b = - (a * b)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_mult_infty[simp]:
+
−  "a * (\<infinity>::ereal) = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
+
−  by (cases a) auto
+
−
+
−lemma ereal_infty_mult[simp]:
+
−  "(\<infinity>::ereal) * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
+
−  by (cases a) auto
+
−
+
−lemma ereal_mult_strict_right_mono:
+
−  assumes "a < b"
+
−    and "0 < c"
+
−    and "c < (\<infinity>::ereal)"
+
−  shows "a * c < b * c"
+
−  using assms
+
−  by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff)
+
−
+
−lemma ereal_mult_strict_left_mono:
+
−  "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> c * a < c * b"
+
−  using ereal_mult_strict_right_mono
+
−  by (simp add: mult.commute[of c])
+
−
+
−lemma ereal_mult_right_mono:
+
−  fixes a b c :: ereal
+
−  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
+
−  apply (cases "c = 0")
+
−  apply simp
+
−  apply (cases rule: ereal3_cases[of a b c])
+
−  apply (auto simp: zero_le_mult_iff)
+
−  done
+
−
+
−lemma ereal_mult_left_mono:
+
−  fixes a b c :: ereal
+
−  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
+
−  using ereal_mult_right_mono
+
−  by (simp add: mult.commute[of c])
+
−
+
−lemma ereal_mult_mono:
+
−  fixes a b c d::ereal
+
−  assumes "b \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d"
+
−  shows "a * c \<le> b * d"
+
−by (metis ereal_mult_right_mono mult.commute order_trans assms)
+
−
+
−lemma ereal_mult_mono':
+
−  fixes a b c d::ereal
+
−  assumes "a \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d"
+
−  shows "a * c \<le> b * d"
+
−by (metis ereal_mult_right_mono mult.commute order_trans assms)
+
−
+
−lemma ereal_mult_mono_strict:
+
−  fixes a b c d::ereal
+
−  assumes "b > 0" "c > 0" "a < b" "c < d"
+
−  shows "a * c < b * d"
+
−proof -
+
−  have "c < \<infinity>" using \<open>c < d\<close> by auto
+
−  then have "a * c < b * c" by (metis ereal_mult_strict_left_mono[OF assms(3) assms(2)] mult.commute)
+
−  moreover have "b * c \<le> b * d" using assms(2) assms(4) by (simp add: assms(1) ereal_mult_left_mono less_imp_le)
+
−  ultimately show ?thesis by simp
+
−qed
+
−
+
−lemma ereal_mult_mono_strict':
+
−  fixes a b c d::ereal
+
−  assumes "a > 0" "c > 0" "a < b" "c < d"
+
−  shows "a * c < b * d"
+
−apply (rule ereal_mult_mono_strict, auto simp add: assms) using assms by auto
+
−
+
−lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
+
−  by (simp add: one_ereal_def zero_ereal_def)
+
−
+
−lemma ereal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: ereal)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_right_distrib:
+
−  fixes r a b :: ereal
+
−  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
+
−  by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
+
−
+
−lemma ereal_left_distrib:
+
−  fixes r a b :: ereal
+
−  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
+
−  by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
+
−
+
−lemma ereal_mult_le_0_iff:
+
−  fixes a b :: ereal
+
−  shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
+
−  by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_le_0_iff)
+
−
+
−lemma ereal_zero_le_0_iff:
+
−  fixes a b :: ereal
+
−  shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
+
−  by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
+
−
+
−lemma ereal_mult_less_0_iff:
+
−  fixes a b :: ereal
+
−  shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
+
−  by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_less_0_iff)
+
−
+
−lemma ereal_zero_less_0_iff:
+
−  fixes a b :: ereal
+
−  shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
+
−  by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
+
−
+
−lemma ereal_left_mult_cong:
+
−  fixes a b c :: ereal
+
−  shows  "c = d \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> a * c = b * d"
+
−  by (cases "c = 0") simp_all
+
−
+
−lemma ereal_right_mult_cong:
+
−  fixes a b c :: ereal
+
−  shows "c = d \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> c * a = d * b"
+
−  by (cases "c = 0") simp_all
+
−
+
−lemma ereal_distrib:
+
−  fixes a b c :: ereal
+
−  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
+
−    and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
+
−    and "\<bar>c\<bar> \<noteq> \<infinity>"
+
−  shows "(a + b) * c = a * c + b * c"
+
−  using assms
+
−  by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
−
+
−lemma numeral_eq_ereal [simp]: "numeral w = ereal (numeral w)"
+
−  apply (induct w rule: num_induct)
+
−  apply (simp only: numeral_One one_ereal_def)
+
−  apply (simp only: numeral_inc ereal_plus_1)
+
−  done
+
−
+
−lemma distrib_left_ereal_nn:
+
−  "c \<ge> 0 \<Longrightarrow> (x + y) * ereal c = x * ereal c + y * ereal c"
+
−by(cases x y rule: ereal2_cases)(simp_all add: ring_distribs)
+
−
+
−lemma sum_ereal_right_distrib:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> r * sum f A = (\<Sum>n\<in>A. r * f n)"
+
−  by (induct A rule: infinite_finite_induct)  (auto simp: ereal_right_distrib sum_nonneg)
+
−
+
−lemma sum_ereal_left_distrib:
+
−  "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> sum f A * r = (\<Sum>n\<in>A. f n * r :: ereal)"
+
−  using sum_ereal_right_distrib[of A f r] by (simp add: mult_ac)
+
−
+
−lemma sum_distrib_right_ereal:
+
−  "c \<ge> 0 \<Longrightarrow> sum f A * ereal c = (\<Sum>x\<in>A. f x * c :: ereal)"
+
−by(subst sum_comp_morphism[where h="\<lambda>x. x * ereal c", symmetric])(simp_all add: distrib_left_ereal_nn)
+
−
+
−lemma ereal_le_epsilon:
+
−  fixes x y :: ereal
+
−  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + e"
+
−  shows "x \<le> y"
+
−proof -
+
−  {
+
−    assume a: "\<exists>r. y = ereal r"
+
−    then obtain r where r_def: "y = ereal r"
+
−      by auto
+
−    {
+
−      assume "x = -\<infinity>"
+
−      then have ?thesis by auto
+
−    }
+
−    moreover
+
−    {
+
−      assume "x \<noteq> -\<infinity>"
+
−      then obtain p where p_def: "x = ereal p"
+
−      using a assms[rule_format, of 1]
+
−        by (cases x) auto
+
−      {
+
−        fix e
+
−        have "0 < e \<longrightarrow> p \<le> r + e"
+
−          using assms[rule_format, of "ereal e"] p_def r_def by auto
+
−      }
+
−      then have "p \<le> r"
+
−        apply (subst field_le_epsilon)
+
−        apply auto
+
−        done
+
−      then have ?thesis
+
−        using r_def p_def by auto
+
−    }
+
−    ultimately have ?thesis
+
−      by blast
+
−  }
+
−  moreover
+
−  {
+
−    assume "y = -\<infinity> \<or> y = \<infinity>"
+
−    then have ?thesis
+
−      using assms[rule_format, of 1] by (cases x) auto
+
−  }
+
−  ultimately show ?thesis
+
−    by (cases y) auto
+
−qed
+
−
+
−lemma ereal_le_epsilon2:
+
−  fixes x y :: ereal
+
−  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + ereal e"
+
−  shows "x \<le> y"
+
−proof -
+
−  {
+
−    fix e :: ereal
+
−    assume "e > 0"
+
−    {
+
−      assume "e = \<infinity>"
+
−      then have "x \<le> y + e"
+
−        by auto
+
−    }
+
−    moreover
+
−    {
+
−      assume "e \<noteq> \<infinity>"
+
−      then obtain r where "e = ereal r"
+
−        using \<open>e > 0\<close> by (cases e) auto
+
−      then have "x \<le> y + e"
+
−        using assms[rule_format, of r] \<open>e>0\<close> by auto
+
−    }
+
−    ultimately have "x \<le> y + e"
+
−      by blast
+
−  }
+
−  then show ?thesis
+
−    using ereal_le_epsilon by auto
+
−qed
+
−
+
−lemma ereal_le_real:
+
−  fixes x y :: ereal
+
−  assumes "\<forall>z. x \<le> ereal z \<longrightarrow> y \<le> ereal z"
+
−  shows "y \<le> x"
+
−  by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
+
−
+
−lemma prod_ereal_0:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. f i = 0)"
+
−proof (cases "finite A")
+
−  case True
+
−  then show ?thesis by (induct A) auto
+
−next
+
−  case False
+
−  then show ?thesis by auto
+
−qed
+
−
+
−lemma prod_ereal_pos:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
+
−  shows "0 \<le> (\<Prod>i\<in>I. f i)"
+
−proof (cases "finite I")
+
−  case True
+
−  from this pos show ?thesis
+
−    by induct auto
+
−next
+
−  case False
+
−  then show ?thesis by simp
+
−qed
+
−
+
−lemma prod_PInf:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
+
−  shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
+
−proof (cases "finite I")
+
−  case True
+
−  from this assms show ?thesis
+
−  proof (induct I)
+
−    case (insert i I)
+
−    then have pos: "0 \<le> f i" "0 \<le> prod f I"
+
−      by (auto intro!: prod_ereal_pos)
+
−    from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> prod f I * f i = \<infinity>"
+
−      by auto
+
−    also have "\<dots> \<longleftrightarrow> (prod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> prod f I \<noteq> 0"
+
−      using prod_ereal_pos[of I f] pos
+
−      by (cases rule: ereal2_cases[of "f i" "prod f I"]) auto
+
−    also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)"
+
−      using insert by (auto simp: prod_ereal_0)
+
−    finally show ?case .
+
−  qed simp
+
−next
+
−  case False
+
−  then show ?thesis by simp
+
−qed
+
−
+
−lemma prod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (prod f A)"
+
−proof (cases "finite A")
+
−  case True
+
−  then show ?thesis
+
−    by induct (auto simp: one_ereal_def)
+
−next
+
−  case False
+
−  then show ?thesis
+
−    by (simp add: one_ereal_def)
+
−qed
+
−
+
−
+
−subsubsection \<open>Power\<close>
+
−
+
−lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
+
−  by (induct n) (auto simp: one_ereal_def)
+
−
+
−lemma ereal_power_PInf[simp]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)"
+
−  by (induct n) (auto simp: one_ereal_def)
+
−
+
−lemma ereal_power_uminus[simp]:
+
−  fixes x :: ereal
+
−  shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
+
−  by (induct n) (auto simp: one_ereal_def)
+
−
+
−lemma ereal_power_numeral[simp]:
+
−  "(numeral num :: ereal) ^ n = ereal (numeral num ^ n)"
+
−  by (induct n) (auto simp: one_ereal_def)
+
−
+
−lemma zero_le_power_ereal[simp]:
+
−  fixes a :: ereal
+
−  assumes "0 \<le> a"
+
−  shows "0 \<le> a ^ n"
+
−  using assms by (induct n) (auto simp: ereal_zero_le_0_iff)
+
−
+
−
+
−subsubsection \<open>Subtraction\<close>
+
−
+
−lemma ereal_minus_minus_image[simp]:
+
−  fixes S :: "ereal set"
+
−  shows "uminus ` uminus ` S = S"
+
−  by (auto simp: image_iff)
+
−
+
−lemma ereal_uminus_lessThan[simp]:
+
−  fixes a :: ereal
+
−  shows "uminus ` {..<a} = {-a<..}"
+
−proof -
+
−  {
+
−    fix x
+
−    assume "-a < x"
+
−    then have "- x < - (- a)"
+
−      by (simp del: ereal_uminus_uminus)
+
−    then have "- x < a"
+
−      by simp
+
−  }
+
−  then show ?thesis
+
−    by force
+
−qed
+
−
+
−lemma ereal_uminus_greaterThan[simp]: "uminus ` {(a::ereal)<..} = {..<-a}"
+
−  by (metis ereal_uminus_lessThan ereal_uminus_uminus ereal_minus_minus_image)
+
−
+
−instantiation ereal :: minus
+
−begin
+
−
+
−definition "x - y = x + -(y::ereal)"
+
−instance ..
+
−
+
−end
+
−
+
−lemma ereal_minus[simp]:
+
−  "ereal r - ereal p = ereal (r - p)"
+
−  "-\<infinity> - ereal r = -\<infinity>"
+
−  "ereal r - \<infinity> = -\<infinity>"
+
−  "(\<infinity>::ereal) - x = \<infinity>"
+
−  "-(\<infinity>::ereal) - \<infinity> = -\<infinity>"
+
−  "x - -y = x + y"
+
−  "x - 0 = x"
+
−  "0 - x = -x"
+
−  by (simp_all add: minus_ereal_def)
+
−
+
−lemma ereal_x_minus_x[simp]: "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
+
−  by (cases x) simp_all
+
−
+
−lemma ereal_eq_minus_iff:
+
−  fixes x y z :: ereal
+
−  shows "x = z - y \<longleftrightarrow>
+
−    (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
+
−    (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
+
−    (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
+
−    (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
+
−  by (cases rule: ereal3_cases[of x y z]) auto
+
−
+
−lemma ereal_eq_minus:
+
−  fixes x y z :: ereal
+
−  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
+
−  by (auto simp: ereal_eq_minus_iff)
+
−
+
−lemma ereal_less_minus_iff:
+
−  fixes x y z :: ereal
+
−  shows "x < z - y \<longleftrightarrow>
+
−    (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
+
−    (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
+
−    (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
+
−  by (cases rule: ereal3_cases[of x y z]) auto
+
−
+
−lemma ereal_less_minus:
+
−  fixes x y z :: ereal
+
−  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
+
−  by (auto simp: ereal_less_minus_iff)
+
−
+
−lemma ereal_le_minus_iff:
+
−  fixes x y z :: ereal
+
−  shows "x \<le> z - y \<longleftrightarrow> (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
+
−  by (cases rule: ereal3_cases[of x y z]) auto
+
−
+
−lemma ereal_le_minus:
+
−  fixes x y z :: ereal
+
−  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
+
−  by (auto simp: ereal_le_minus_iff)
+
−
+
−lemma ereal_minus_less_iff:
+
−  fixes x y z :: ereal
+
−  shows "x - y < z \<longleftrightarrow> y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
+
−  by (cases rule: ereal3_cases[of x y z]) auto
+
−
+
−lemma ereal_minus_less:
+
−  fixes x y z :: ereal
+
−  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
+
−  by (auto simp: ereal_minus_less_iff)
+
−
+
−lemma ereal_minus_le_iff:
+
−  fixes x y z :: ereal
+
−  shows "x - y \<le> z \<longleftrightarrow>
+
−    (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
+
−    (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
+
−    (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
+
−  by (cases rule: ereal3_cases[of x y z]) auto
+
−
+
−lemma ereal_minus_le:
+
−  fixes x y z :: ereal
+
−  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
+
−  by (auto simp: ereal_minus_le_iff)
+
−
+
−lemma ereal_minus_eq_minus_iff:
+
−  fixes a b c :: ereal
+
−  shows "a - b = a - c \<longleftrightarrow>
+
−    b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
+
−  by (cases rule: ereal3_cases[of a b c]) auto
+
−
+
−lemma ereal_add_le_add_iff:
+
−  fixes a b c :: ereal
+
−  shows "c + a \<le> c + b \<longleftrightarrow>
+
−    a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
+
−  by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
−
+
−lemma ereal_add_le_add_iff2:
+
−  fixes a b c :: ereal
+
−  shows "a + c \<le> b + c \<longleftrightarrow> a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
+
−by(cases rule: ereal3_cases[of a b c])(simp_all add: field_simps)
+
−
+
−lemma ereal_mult_le_mult_iff:
+
−  fixes a b c :: ereal
+
−  shows "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
+
−  by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
+
−
+
−lemma ereal_minus_mono:
+
−  fixes A B C D :: ereal assumes "A \<le> B" "D \<le> C"
+
−  shows "A - C \<le> B - D"
+
−  using assms
+
−  by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all
+
−
+
−lemma ereal_mono_minus_cancel:
+
−  fixes a b c :: ereal
+
−  shows "c - a \<le> c - b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> b \<le> a"
+
−  by (cases a b c rule: ereal3_cases) auto
+
−
+
−lemma real_of_ereal_minus:
+
−  fixes a b :: ereal
+
−  shows "real_of_ereal (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real_of_ereal a - real_of_ereal b)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma real_of_ereal_minus': "\<bar>x\<bar> = \<infinity> \<longleftrightarrow> \<bar>y\<bar> = \<infinity> \<Longrightarrow> real_of_ereal x - real_of_ereal y = real_of_ereal (x - y :: ereal)"
+
−by(subst real_of_ereal_minus) auto
+
−
+
−lemma ereal_diff_positive:
+
−  fixes a b :: ereal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_between:
+
−  fixes x e :: ereal
+
−  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+
−    and "0 < e"
+
−  shows "x - e < x"
+
−    and "x < x + e"
+
−  using assms
+
−  apply (cases x, cases e)
+
−  apply auto
+
−  using assms
+
−  apply (cases x, cases e)
+
−  apply auto
+
−  done
+
−
+
−lemma ereal_minus_eq_PInfty_iff:
+
−  fixes x y :: ereal
+
−  shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
+
−  by (cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ereal_diff_add_eq_diff_diff_swap:
+
−  fixes x y z :: ereal
+
−  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - (y + z) = x - y - z"
+
−by(cases x y z rule: ereal3_cases) simp_all
+
−
+
−lemma ereal_diff_add_assoc2:
+
−  fixes x y z :: ereal
+
−  shows "x + y - z = x - z + y"
+
−by(cases x y z rule: ereal3_cases) simp_all
+
−
+
−lemma ereal_add_uminus_conv_diff: fixes x y z :: ereal shows "- x + y = y - x"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ereal_minus_diff_eq:
+
−  fixes x y :: ereal
+
−  shows "\<lbrakk> x = \<infinity> \<longrightarrow> y \<noteq> \<infinity>; x = -\<infinity> \<longrightarrow> y \<noteq> - \<infinity> \<rbrakk> \<Longrightarrow> - (x - y) = y - x"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ediff_le_self [simp]: "x - y \<le> (x :: enat)"
+
−by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all
+
−
+
−lemma ereal_abs_diff:
+
−  fixes a b::ereal
+
−  shows "abs(a-b) \<le> abs a + abs b"
+
−by (cases rule: ereal2_cases[of a b]) (auto)
+
−
+
−
+
−subsubsection \<open>Division\<close>
+
−
+
−instantiation ereal :: inverse
+
−begin
+
−
+
−function inverse_ereal where
+
−  "inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))"
+
−| "inverse (\<infinity>::ereal) = 0"
+
−| "inverse (-\<infinity>::ereal) = 0"
+
−  by (auto intro: ereal_cases)
+
−termination by (relation "{}") simp
+
−
+
−definition "x div y = x * inverse (y :: ereal)"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−lemma real_of_ereal_inverse[simp]:
+
−  fixes a :: ereal
+
−  shows "real_of_ereal (inverse a) = 1 / real_of_ereal a"
+
−  by (cases a) (auto simp: inverse_eq_divide)
+
−
+
−lemma ereal_inverse[simp]:
+
−  "inverse (0::ereal) = \<infinity>"
+
−  "inverse (1::ereal) = 1"
+
−  by (simp_all add: one_ereal_def zero_ereal_def)
+
−
+
−lemma ereal_divide[simp]:
+
−  "ereal r / ereal p = (if p = 0 then ereal r * \<infinity> else ereal (r / p))"
+
−  unfolding divide_ereal_def by (auto simp: divide_real_def)
+
−
+
−lemma ereal_divide_same[simp]:
+
−  fixes x :: ereal
+
−  shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
+
−  by (cases x) (simp_all add: divide_real_def divide_ereal_def one_ereal_def)
+
−
+
−lemma ereal_inv_inv[simp]:
+
−  fixes x :: ereal
+
−  shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
+
−  by (cases x) auto
+
−
+
−lemma ereal_inverse_minus[simp]:
+
−  fixes x :: ereal
+
−  shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
+
−  by (cases x) simp_all
+
−
+
−lemma ereal_uminus_divide[simp]:
+
−  fixes x y :: ereal
+
−  shows "- x / y = - (x / y)"
+
−  unfolding divide_ereal_def by simp
+
−
+
−lemma ereal_divide_Infty[simp]:
+
−  fixes x :: ereal
+
−  shows "x / \<infinity> = 0" "x / -\<infinity> = 0"
+
−  unfolding divide_ereal_def by simp_all
+
−
+
−lemma ereal_divide_one[simp]: "x / 1 = (x::ereal)"
+
−  unfolding divide_ereal_def by simp
+
−
+
−lemma ereal_divide_ereal[simp]: "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
+
−  unfolding divide_ereal_def by simp
+
−
+
−lemma ereal_inverse_nonneg_iff: "0 \<le> inverse (x :: ereal) \<longleftrightarrow> 0 \<le> x \<or> x = -\<infinity>"
+
−  by (cases x) auto
+
−
+
−lemma inverse_ereal_ge0I: "0 \<le> (x :: ereal) \<Longrightarrow> 0 \<le> inverse x"
+
−by(cases x) simp_all
+
−
+
−lemma zero_le_divide_ereal[simp]:
+
−  fixes a :: ereal
+
−  assumes "0 \<le> a"
+
−    and "0 \<le> b"
+
−  shows "0 \<le> a / b"
+
−  using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff)
+
−
+
−lemma ereal_le_divide_pos:
+
−  fixes x y z :: ereal
+
−  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
+
−  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
+
−
+
−lemma ereal_divide_le_pos:
+
−  fixes x y z :: ereal
+
−  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
+
−  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
+
−
+
−lemma ereal_le_divide_neg:
+
−  fixes x y z :: ereal
+
−  shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
+
−  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
+
−
+
−lemma ereal_divide_le_neg:
+
−  fixes x y z :: ereal
+
−  shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
+
−  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
+
−
+
−lemma ereal_inverse_antimono_strict:
+
−  fixes x y :: ereal
+
−  shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
+
−  by (cases rule: ereal2_cases[of x y]) auto
+
−
+
−lemma ereal_inverse_antimono:
+
−  fixes x y :: ereal
+
−  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> inverse y \<le> inverse x"
+
−  by (cases rule: ereal2_cases[of x y]) auto
+
−
+
−lemma inverse_inverse_Pinfty_iff[simp]:
+
−  fixes x :: ereal
+
−  shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
+
−  by (cases x) auto
+
−
+
−lemma ereal_inverse_eq_0:
+
−  fixes x :: ereal
+
−  shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
+
−  by (cases x) auto
+
−
+
−lemma ereal_0_gt_inverse:
+
−  fixes x :: ereal
+
−  shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
+
−  by (cases x) auto
+
−
+
−lemma ereal_inverse_le_0_iff:
+
−  fixes x :: ereal
+
−  shows "inverse x \<le> 0 \<longleftrightarrow> x < 0 \<or> x = \<infinity>"
+
−  by(cases x) auto
+
−
+
−lemma ereal_divide_eq_0_iff: "x / y = 0 \<longleftrightarrow> x = 0 \<or> \<bar>y :: ereal\<bar> = \<infinity>"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ereal_mult_less_right:
+
−  fixes a b c :: ereal
+
−  assumes "b * a < c * a"
+
−    and "0 < a"
+
−    and "a < \<infinity>"
+
−  shows "b < c"
+
−  using assms
+
−  by (cases rule: ereal3_cases[of a b c])
+
−     (auto split: if_split_asm simp: zero_less_mult_iff zero_le_mult_iff)
+
−
+
−lemma ereal_mult_divide: fixes a b :: ereal shows "0 < b \<Longrightarrow> b < \<infinity> \<Longrightarrow> b * (a / b) = a"
+
−  by (cases a b rule: ereal2_cases) auto
+
−
+
−lemma ereal_power_divide:
+
−  fixes x y :: ereal
+
−  shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n"
+
−  by (cases rule: ereal2_cases [of x y])
+
−     (auto simp: one_ereal_def zero_ereal_def power_divide zero_le_power_eq)
+
−
+
−lemma ereal_le_mult_one_interval:
+
−  fixes x y :: ereal
+
−  assumes y: "y \<noteq> -\<infinity>"
+
−  assumes z: "\<And>z. 0 < z \<Longrightarrow> z < 1 \<Longrightarrow> z * x \<le> y"
+
−  shows "x \<le> y"
+
−proof (cases x)
+
−  case PInf
+
−  with z[of "1 / 2"] show "x \<le> y"
+
−    by (simp add: one_ereal_def)
+
−next
+
−  case (real r)
+
−  note r = this
+
−  show "x \<le> y"
+
−  proof (cases y)
+
−    case (real p)
+
−    note p = this
+
−    have "r \<le> p"
+
−    proof (rule field_le_mult_one_interval)
+
−      fix z :: real
+
−      assume "0 < z" and "z < 1"
+
−      with z[of "ereal z"] show "z * r \<le> p"
+
−        using p r by (auto simp: zero_le_mult_iff one_ereal_def)
+
−    qed
+
−    then show "x \<le> y"
+
−      using p r by simp
+
−  qed (insert y, simp_all)
+
−qed simp
+
−
+
−lemma ereal_divide_right_mono[simp]:
+
−  fixes x y z :: ereal
+
−  assumes "x \<le> y"
+
−    and "0 < z"
+
−  shows "x / z \<le> y / z"
+
−  using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)
+
−
+
−lemma ereal_divide_left_mono[simp]:
+
−  fixes x y z :: ereal
+
−  assumes "y \<le> x"
+
−    and "0 < z"
+
−    and "0 < x * y"
+
−  shows "z / x \<le> z / y"
+
−  using assms
+
−  by (cases x y z rule: ereal3_cases)
+
−     (auto intro: divide_left_mono simp: field_simps zero_less_mult_iff mult_less_0_iff split: if_split_asm)
+
−
+
−lemma ereal_divide_zero_left[simp]:
+
−  fixes a :: ereal
+
−  shows "0 / a = 0"
+
−  by (cases a) (auto simp: zero_ereal_def)
+
−
+
−lemma ereal_times_divide_eq_left[simp]:
+
−  fixes a b c :: ereal
+
−  shows "b / c * a = b * a / c"
+
−  by (cases a b c rule: ereal3_cases) (auto simp: field_simps zero_less_mult_iff mult_less_0_iff)
+
−
+
−lemma ereal_times_divide_eq: "a * (b / c :: ereal) = a * b / c"
+
−  by (cases a b c rule: ereal3_cases)
+
−     (auto simp: field_simps zero_less_mult_iff)
+
−
+
−lemma ereal_inverse_real [simp]: "\<bar>z\<bar> \<noteq> \<infinity> \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> ereal (inverse (real_of_ereal z)) = inverse z"
+
−  by auto
+
−
+
−lemma ereal_inverse_mult:
+
−  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse (a * (b::ereal)) = inverse a * inverse b"
+
−  by (cases a; cases b) auto
+
−
+
−lemma inverse_eq_infinity_iff_eq_zero [simp]:
+
−  "1/(x::ereal) = \<infinity> \<longleftrightarrow> x = 0"
+
−by (simp add: divide_ereal_def)
+
−
+
−lemma ereal_distrib_left:
+
−  fixes a b c :: ereal
+
−  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
+
−    and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
+
−    and "\<bar>c\<bar> \<noteq> \<infinity>"
+
−  shows "c * (a + b) = c * a + c * b"
+
−using assms
+
−by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
−
+
−lemma ereal_distrib_minus_left:
+
−  fixes a b c :: ereal
+
−  assumes "a \<noteq> \<infinity> \<or> b \<noteq> \<infinity>"
+
−    and "a \<noteq> -\<infinity> \<or> b \<noteq> -\<infinity>"
+
−    and "\<bar>c\<bar> \<noteq> \<infinity>"
+
−  shows "c * (a - b) = c * a - c * b"
+
−using assms
+
−by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
−
+
−lemma ereal_distrib_minus_right:
+
−  fixes a b c :: ereal
+
−  assumes "a \<noteq> \<infinity> \<or> b \<noteq> \<infinity>"
+
−    and "a \<noteq> -\<infinity> \<or> b \<noteq> -\<infinity>"
+
−    and "\<bar>c\<bar> \<noteq> \<infinity>"
+
−  shows "(a - b) * c = a * c - b * c"
+
−using assms
+
−by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
−
+
−
+
−subsection "Complete lattice"
+
−
+
−instantiation ereal :: lattice
+
−begin
+
−
+
−definition [simp]: "sup x y = (max x y :: ereal)"
+
−definition [simp]: "inf x y = (min x y :: ereal)"
+
−instance by standard simp_all
+
−
+
−end
+
−
+
−instantiation ereal :: complete_lattice
+
−begin
+
−
+
−definition "bot = (-\<infinity>::ereal)"
+
−definition "top = (\<infinity>::ereal)"
+
−
+
−definition "Sup S = (SOME x :: ereal. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z))"
+
−definition "Inf S = (SOME x :: ereal. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x))"
+
−
+
−lemma ereal_complete_Sup:
+
−  fixes S :: "ereal set"
+
−  shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
+
−proof (cases "\<exists>x. \<forall>a\<in>S. a \<le> ereal x")
+
−  case True
+
−  then obtain y where y: "a \<le> ereal y" if "a\<in>S" for a
+
−    by auto
+
−  then have "\<infinity> \<notin> S"
+
−    by force
+
−  show ?thesis
+
−  proof (cases "S \<noteq> {-\<infinity>} \<and> S \<noteq> {}")
+
−    case True
+
−    with \<open>\<infinity> \<notin> S\<close> obtain x where x: "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
+
−      by auto
+
−    obtain s where s: "\<forall>x\<in>ereal -` S. x \<le> s" "(\<forall>x\<in>ereal -` S. x \<le> z) \<Longrightarrow> s \<le> z" for z
+
−    proof (atomize_elim, rule complete_real)
+
−      show "\<exists>x. x \<in> ereal -` S"
+
−        using x by auto
+
−      show "\<exists>z. \<forall>x\<in>ereal -` S. x \<le> z"
+
−        by (auto dest: y intro!: exI[of _ y])
+
−    qed
+
−    show ?thesis
+
−    proof (safe intro!: exI[of _ "ereal s"])
+
−      fix y
+
−      assume "y \<in> S"
+
−      with s \<open>\<infinity> \<notin> S\<close> show "y \<le> ereal s"
+
−        by (cases y) auto
+
−    next
+
−      fix z
+
−      assume "\<forall>y\<in>S. y \<le> z"
+
−      with \<open>S \<noteq> {-\<infinity>} \<and> S \<noteq> {}\<close> show "ereal s \<le> z"
+
−        by (cases z) (auto intro!: s)
+
−    qed
+
−  next
+
−    case False
+
−    then show ?thesis
+
−      by (auto intro!: exI[of _ "-\<infinity>"])
+
−  qed
+
−next
+
−  case False
+
−  then show ?thesis
+
−    by (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
+
−qed
+
−
+
−lemma ereal_complete_uminus_eq:
+
−  fixes S :: "ereal set"
+
−  shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
+
−     \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
+
−  by simp (metis ereal_minus_le_minus ereal_uminus_uminus)
+
−
+
−lemma ereal_complete_Inf:
+
−  "\<exists>x. (\<forall>y\<in>S::ereal set. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)"
+
−  using ereal_complete_Sup[of "uminus ` S"]
+
−  unfolding ereal_complete_uminus_eq
+
−  by auto
+
−
+
−instance
+
−proof
+
−  show "Sup {} = (bot::ereal)"
+
−    apply (auto simp: bot_ereal_def Sup_ereal_def)
+
−    apply (rule some1_equality)
+
−    apply (metis ereal_bot ereal_less_eq(2))
+
−    apply (metis ereal_less_eq(2))
+
−    done
+
−  show "Inf {} = (top::ereal)"
+
−    apply (auto simp: top_ereal_def Inf_ereal_def)
+
−    apply (rule some1_equality)
+
−    apply (metis ereal_top ereal_less_eq(1))
+
−    apply (metis ereal_less_eq(1))
+
−    done
+
−qed (auto intro: someI2_ex ereal_complete_Sup ereal_complete_Inf
+
−  simp: Sup_ereal_def Inf_ereal_def bot_ereal_def top_ereal_def)
+
−
+
−end
+
−
+
−instance ereal :: complete_linorder ..
+
−
+
−instance ereal :: linear_continuum
+
−proof
+
−  show "\<exists>a b::ereal. a \<noteq> b"
+
−    using zero_neq_one by blast
+
−qed
+
−
+
−lemma min_PInf [simp]: "min (\<infinity>::ereal) x = x"
+
−by (metis min_top top_ereal_def)
+
−
+
−lemma min_PInf2 [simp]: "min x (\<infinity>::ereal) = x"
+
−by (metis min_top2 top_ereal_def)
+
−
+
−lemma max_PInf [simp]: "max (\<infinity>::ereal) x = \<infinity>"
+
−by (metis max_top top_ereal_def)
+
−
+
−lemma max_PInf2 [simp]: "max x (\<infinity>::ereal) = \<infinity>"
+
−by (metis max_top2 top_ereal_def)
+
−
+
−lemma min_MInf [simp]: "min (-\<infinity>::ereal) x = -\<infinity>"
+
−by (metis min_bot bot_ereal_def)
+
−
+
−lemma min_MInf2 [simp]: "min x (-\<infinity>::ereal) = -\<infinity>"
+
−by (metis min_bot2 bot_ereal_def)
+
−
+
−lemma max_MInf [simp]: "max (-\<infinity>::ereal) x = x"
+
−by (metis max_bot bot_ereal_def)
+
−
+
−lemma max_MInf2 [simp]: "max x (-\<infinity>::ereal) = x"
+
−by (metis max_bot2 bot_ereal_def)
+
−
+
−subsection \<open>Extended real intervals\<close>
+
−
+
−lemma real_greaterThanLessThan_infinity_eq:
+
−  "real_of_ereal ` {N::ereal<..<\<infinity>} =
+
−    (if N = \<infinity> then {} else if N = -\<infinity> then UNIV else {real_of_ereal N<..})"
+
−proof -
+
−  {
+
−    fix x::real
+
−    have "x \<in> real_of_ereal ` {- \<infinity><..<\<infinity>::ereal}"
+
−      by (auto intro!: image_eqI[where x="ereal x"])
+
−  } moreover {
+
−    fix x::ereal
+
−    assume "N \<noteq> - \<infinity>" "N < x" "x \<noteq> \<infinity>"
+
−    then have "real_of_ereal N < real_of_ereal x"
+
−      by (cases N; cases x; simp)
+
−  } moreover {
+
−    fix x::real
+
−    assume "N \<noteq> \<infinity>" "real_of_ereal N < x"
+
−    then have "x \<in> real_of_ereal ` {N<..<\<infinity>}"
+
−      by (cases N) (auto intro!: image_eqI[where x="ereal x"])
+
−  } ultimately show ?thesis by auto
+
−qed
+
−
+
−lemma real_greaterThanLessThan_minus_infinity_eq:
+
−  "real_of_ereal ` {-\<infinity><..<N::ereal} =
+
−    (if N = \<infinity> then UNIV else if N = -\<infinity> then {} else {..<real_of_ereal N})"
+
−proof -
+
−  have "real_of_ereal ` {-\<infinity><..<N::ereal} = uminus ` real_of_ereal ` {-N<..<\<infinity>}"
+
−    by (auto simp: ereal_uminus_less_reorder intro!: image_eqI[where x="-x" for x])
+
−  also note real_greaterThanLessThan_infinity_eq
+
−  finally show ?thesis by (auto intro!: image_eqI[where x="-x" for x])
+
−qed
+
−
+
−lemma real_greaterThanLessThan_inter:
+
−  "real_of_ereal ` {N<..<M::ereal} = real_of_ereal ` {-\<infinity><..<M} \<inter> real_of_ereal ` {N<..<\<infinity>}"
+
−  apply safe
+
−  subgoal by force
+
−  subgoal by force
+
−  subgoal for x y z
+
−    by (cases y; cases z) (auto intro!: image_eqI[where x=z] simp: )
+
−  done
+
−
+
−lemma real_atLeastGreaterThan_eq: "real_of_ereal ` {N<..<M::ereal} =
+
−   (if N = \<infinity> then {} else
+
−   if N = -\<infinity> then
+
−    (if M = \<infinity> then UNIV
+
−    else if M = -\<infinity> then {}
+
−    else {..< real_of_ereal M})
+
−  else if M = - \<infinity> then {}
+
−  else if M = \<infinity> then {real_of_ereal N<..}
+
−  else {real_of_ereal N <..< real_of_ereal M})"
+
−  apply (subst real_greaterThanLessThan_inter)
+
−  apply (subst real_greaterThanLessThan_minus_infinity_eq)
+
−  apply (subst real_greaterThanLessThan_infinity_eq)
+
−  apply auto
+
−  done
+
−
+
−lemma real_image_ereal_ivl:
+
−  fixes a b::ereal
+
−  shows
+
−  "real_of_ereal ` {a<..<b} =
+
−  (if a < b then (if a = - \<infinity> then if b = \<infinity> then UNIV else {..<real_of_ereal b}
+
−  else if b = \<infinity> then {real_of_ereal a<..} else {real_of_ereal a <..< real_of_ereal b}) else {})"
+
−  by (cases a; cases b; simp add: real_atLeastGreaterThan_eq not_less)
+
−
+
−lemma fixes a b c::ereal
+
−  shows not_inftyI: "a < b \<Longrightarrow> b < c \<Longrightarrow> abs b \<noteq> \<infinity>"
+
−  by force
+
−
+
−lemma
+
−  interval_neqs:
+
−  fixes r s t::real
+
−  shows "{r<..<s} \<noteq> {t<..}"
+
−    and "{r<..<s} \<noteq> {..<t}"
+
−    and "{r<..<ra} \<noteq> UNIV"
+
−    and "{r<..} \<noteq> {..<s}"
+
−    and "{r<..} \<noteq> UNIV"
+
−    and "{..<r} \<noteq> UNIV"
+
−    and "{} \<noteq> {r<..}"
+
−    and "{} \<noteq> {..<r}"
+
−  subgoal
+
−    by (metis dual_order.strict_trans greaterThanLessThan_iff greaterThan_iff gt_ex not_le order_refl)
+
−  subgoal
+
−    by (metis (no_types, hide_lams) greaterThanLessThan_empty_iff greaterThanLessThan_iff gt_ex
+
−        lessThan_iff minus_minus neg_less_iff_less not_less order_less_irrefl)
+
−  subgoal by force
+
−  subgoal
+
−    by (metis greaterThanLessThan_empty_iff greaterThanLessThan_eq greaterThan_iff inf.idem
+
−        lessThan_iff lessThan_non_empty less_irrefl not_le)
+
−  subgoal by force
+
−  subgoal by force
+
−  subgoal using greaterThan_non_empty by blast
+
−  subgoal using lessThan_non_empty by blast
+
−  done
+
−
+
−lemma greaterThanLessThan_eq_iff:
+
−  fixes r s t u::real
+
−  shows "({r<..<s} = {t<..<u}) = (r \<ge> s \<and> u \<le> t \<or> r = t \<and> s = u)"
+
−  by (metis cInf_greaterThanLessThan cSup_greaterThanLessThan greaterThanLessThan_empty_iff not_le)
+
−
+
−lemma real_of_ereal_image_greaterThanLessThan_iff:
+
−  "real_of_ereal ` {a <..< b} = real_of_ereal ` {c <..< d} \<longleftrightarrow> (a \<ge> b \<and> c \<ge> d \<or> a = c \<and> b = d)"
+
−  unfolding real_atLeastGreaterThan_eq
+
−  by (cases a; cases b; cases c; cases d;
+
−    simp add: greaterThanLessThan_eq_iff interval_neqs interval_neqs[symmetric])
+
−
+
−lemma uminus_image_real_of_ereal_image_greaterThanLessThan:
+
−  "uminus ` real_of_ereal ` {l <..< u} = real_of_ereal ` {-u <..< -l}"
+
−  by (force simp: algebra_simps ereal_less_uminus_reorder
+
−    ereal_uminus_less_reorder intro: image_eqI[where x="-x" for x])
+
−
+
−lemma add_image_real_of_ereal_image_greaterThanLessThan:
+
−  "(+) c ` real_of_ereal ` {l <..< u} = real_of_ereal ` {c + l <..< c + u}"
+
−  apply safe
+
−  subgoal for x
+
−    using ereal_less_add[of c]
+
−    by (force simp: real_of_ereal_add add.commute)
+
−  subgoal for _ x
+
−    by (force simp: add.commute real_of_ereal_minus ereal_minus_less ereal_less_minus
+
−      intro: image_eqI[where x="x - c"])
+
−  done
+
−
+
−lemma add2_image_real_of_ereal_image_greaterThanLessThan:
+
−  "(\<lambda>x. x + c) ` real_of_ereal ` {l <..< u} = real_of_ereal ` {l + c <..< u + c}"
+
−  using add_image_real_of_ereal_image_greaterThanLessThan[of c l u]
+
−  by (metis add.commute image_cong)
+
−
+
−lemma minus_image_real_of_ereal_image_greaterThanLessThan:
+
−  "(-) c ` real_of_ereal ` {l <..< u} = real_of_ereal ` {c - u <..< c - l}"
+
−  (is "?l = ?r")
+
−proof -
+
−  have "?l = (+) c ` uminus ` real_of_ereal ` {l <..< u}" by auto
+
−  also note uminus_image_real_of_ereal_image_greaterThanLessThan
+
−  also note add_image_real_of_ereal_image_greaterThanLessThan
+
−  finally show ?thesis by (simp add: minus_ereal_def)
+
−qed
+
−
+
−lemma real_ereal_bound_lemma_up:
+
−  assumes "s \<in> real_of_ereal ` {a<..<b}"
+
−  assumes "t \<notin> real_of_ereal ` {a<..<b}"
+
−  assumes "s \<le> t"
+
−  shows "b \<noteq> \<infinity>"
+
−  using assms
+
−  apply (cases b)
+
−  subgoal by force
+
−  subgoal by (metis PInfty_neq_ereal(2) assms dual_order.strict_trans1 ereal_infty_less(1)
+
−    ereal_less_ereal_Ex greaterThanLessThan_empty_iff greaterThanLessThan_iff greaterThan_iff
+
−    image_eqI less_imp_le linordered_field_no_ub not_less order_trans
+
−    real_greaterThanLessThan_infinity_eq real_image_ereal_ivl real_of_ereal.simps(1))
+
−  subgoal by force
+
−  done
+
−
+
−lemma real_ereal_bound_lemma_down:
+
−  assumes "s \<in> real_of_ereal ` {a<..<b}"
+
−  assumes "t \<notin> real_of_ereal ` {a<..<b}"
+
−  assumes "t \<le> s"
+
−  shows "a \<noteq> - \<infinity>"
+
−  using assms
+
−  apply (cases b)
+
−  subgoal
+
−    apply safe
+
−    using assms(1)
+
−    apply (auto simp: real_greaterThanLessThan_minus_infinity_eq)
+
−    done
+
−  subgoal by (auto simp: real_greaterThanLessThan_minus_infinity_eq)
+
−  subgoal by auto
+
−  done
+
−
+
−
+
−subsection "Topological space"
+
−
+
−instantiation ereal :: linear_continuum_topology
+
−begin
+
−
+
−definition "open_ereal" :: "ereal set \<Rightarrow> bool" where
+
−  open_ereal_generated: "open_ereal = generate_topology (range lessThan \<union> range greaterThan)"
+
−
+
−instance
+
−  by standard (simp add: open_ereal_generated)
+
−
+
−end
+
−
+
−lemma continuous_on_ereal[continuous_intros]:
+
−  assumes f: "continuous_on s f" shows "continuous_on s (\<lambda>x. ereal (f x))"
+
−  by (rule continuous_on_compose2 [OF continuous_onI_mono[of ereal UNIV] f]) auto
+
−
+
−lemma tendsto_ereal[tendsto_intros, simp, intro]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. ereal (f x)) \<longlongrightarrow> ereal x) F"
+
−  using isCont_tendsto_compose[of x ereal f F] continuous_on_ereal[of UNIV "\<lambda>x. x"]
+
−  by (simp add: continuous_on_eq_continuous_at)
+
−
+
−lemma tendsto_uminus_ereal[tendsto_intros, simp, intro]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. - f x::ereal) \<longlongrightarrow> - x) F"
+
−  apply (rule tendsto_compose[where g=uminus])
+
−  apply (auto intro!: order_tendstoI simp: eventually_at_topological)
+
−  apply (rule_tac x="{..< -a}" in exI)
+
−  apply (auto split: ereal.split simp: ereal_less_uminus_reorder) []
+
−  apply (rule_tac x="{- a <..}" in exI)
+
−  apply (auto split: ereal.split simp: ereal_uminus_reorder) []
+
−  done
+
−
+
−lemma at_infty_ereal_eq_at_top: "at \<infinity> = filtermap ereal at_top"
+
−  unfolding filter_eq_iff eventually_at_filter eventually_at_top_linorder eventually_filtermap
+
−    top_ereal_def[symmetric]
+
−  apply (subst eventually_nhds_top[of 0])
+
−  apply (auto simp: top_ereal_def less_le ereal_all_split ereal_ex_split)
+
−  apply (metis PInfty_neq_ereal(2) ereal_less_eq(3) ereal_top le_cases order_trans)
+
−  done
+
−
+
−lemma ereal_Lim_uminus: "(f \<longlongrightarrow> f0) net \<longleftrightarrow> ((\<lambda>x. - f x::ereal) \<longlongrightarrow> - f0) net"
+
−  using tendsto_uminus_ereal[of f f0 net] tendsto_uminus_ereal[of "\<lambda>x. - f x" "- f0" net]
+
−  by auto
+
−
+
−lemma ereal_divide_less_iff: "0 < (c::ereal) \<Longrightarrow> c < \<infinity> \<Longrightarrow> a / c < b \<longleftrightarrow> a < b * c"
+
−  by (cases a b c rule: ereal3_cases) (auto simp: field_simps)
+
−
+
−lemma ereal_less_divide_iff: "0 < (c::ereal) \<Longrightarrow> c < \<infinity> \<Longrightarrow> a < b / c \<longleftrightarrow> a * c < b"
+
−  by (cases a b c rule: ereal3_cases) (auto simp: field_simps)
+
−
+
−lemma tendsto_cmult_ereal[tendsto_intros, simp, intro]:
+
−  assumes c: "\<bar>c\<bar> \<noteq> \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
+
−proof -
+
−  { fix c :: ereal assume "0 < c" "c < \<infinity>"
+
−    then have "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
+
−      apply (intro tendsto_compose[OF _ f])
+
−      apply (auto intro!: order_tendstoI simp: eventually_at_topological)
+
−      apply (rule_tac x="{a/c <..}" in exI)
+
−      apply (auto split: ereal.split simp: ereal_divide_less_iff mult.commute) []
+
−      apply (rule_tac x="{..< a/c}" in exI)
+
−      apply (auto split: ereal.split simp: ereal_less_divide_iff mult.commute) []
+
−      done }
+
−  note * = this
+
−
+
−  have "((0 < c \<and> c < \<infinity>) \<or> (-\<infinity> < c \<and> c < 0) \<or> c = 0)"
+
−    using c by (cases c) auto
+
−  then show ?thesis
+
−  proof (elim disjE conjE)
+
−    assume "- \<infinity> < c" "c < 0"
+
−    then have "0 < - c" "- c < \<infinity>"
+
−      by (auto simp: ereal_uminus_reorder ereal_less_uminus_reorder[of 0])
+
−    then have "((\<lambda>x. (- c) * f x) \<longlongrightarrow> (- c) * x) F"
+
−      by (rule *)
+
−    from tendsto_uminus_ereal[OF this] show ?thesis
+
−      by simp
+
−  qed (auto intro!: *)
+
−qed
+
−
+
−lemma tendsto_cmult_ereal_not_0[tendsto_intros, simp, intro]:
+
−  assumes "x \<noteq> 0" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
+
−proof cases
+
−  assume "\<bar>c\<bar> = \<infinity>"
+
−  show ?thesis
+
−  proof (rule filterlim_cong[THEN iffD1, OF refl refl _ tendsto_const])
+
−    have "0 < x \<or> x < 0"
+
−      using \<open>x \<noteq> 0\<close> by (auto simp add: neq_iff)
+
−    then show "eventually (\<lambda>x'. c * x = c * f x') F"
+
−    proof
+
−      assume "0 < x" from order_tendstoD(1)[OF f this] show ?thesis
+
−        by eventually_elim (insert \<open>0<x\<close> \<open>\<bar>c\<bar> = \<infinity>\<close>, auto)
+
−    next
+
−      assume "x < 0" from order_tendstoD(2)[OF f this] show ?thesis
+
−        by eventually_elim (insert \<open>x<0\<close> \<open>\<bar>c\<bar> = \<infinity>\<close>, auto)
+
−    qed
+
−  qed
+
−qed (rule tendsto_cmult_ereal[OF _ f])
+
−
+
−lemma tendsto_cadd_ereal[tendsto_intros, simp, intro]:
+
−  assumes c: "y \<noteq> - \<infinity>" "x \<noteq> - \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y::ereal) \<longlongrightarrow> x + y) F"
+
−  apply (intro tendsto_compose[OF _ f])
+
−  apply (auto intro!: order_tendstoI simp: eventually_at_topological)
+
−  apply (rule_tac x="{a - y <..}" in exI)
+
−  apply (auto split: ereal.split simp: ereal_minus_less_iff c) []
+
−  apply (rule_tac x="{..< a - y}" in exI)
+
−  apply (auto split: ereal.split simp: ereal_less_minus_iff c) []
+
−  done
+
−
+
−lemma tendsto_add_left_ereal[tendsto_intros, simp, intro]:
+
−  assumes c: "\<bar>y\<bar> \<noteq> \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y::ereal) \<longlongrightarrow> x + y) F"
+
−  apply (intro tendsto_compose[OF _ f])
+
−  apply (auto intro!: order_tendstoI simp: eventually_at_topological)
+
−  apply (rule_tac x="{a - y <..}" in exI)
+
−  apply (insert c, auto split: ereal.split simp: ereal_minus_less_iff) []
+
−  apply (rule_tac x="{..< a - y}" in exI)
+
−  apply (auto split: ereal.split simp: ereal_less_minus_iff c) []
+
−  done
+
−
+
−lemma continuous_at_ereal[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. ereal (f x))"
+
−  unfolding continuous_def by auto
+
−
+
−lemma ereal_Sup:
+
−  assumes *: "\<bar>SUP a\<in>A. ereal a\<bar> \<noteq> \<infinity>"
+
−  shows "ereal (Sup A) = (SUP a\<in>A. ereal a)"
+
−proof (rule continuous_at_Sup_mono)
+
−  obtain r where r: "ereal r = (SUP a\<in>A. ereal a)" "A \<noteq> {}"
+
−    using * by (force simp: bot_ereal_def)
+
−  then show "bdd_above A" "A \<noteq> {}"
+
−    by (auto intro!: SUP_upper bdd_aboveI[of _ r] simp flip: ereal_less_eq)
+
−qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ereal)
+
−
+
−lemma ereal_SUP: "\<bar>SUP a\<in>A. ereal (f a)\<bar> \<noteq> \<infinity> \<Longrightarrow> ereal (SUP a\<in>A. f a) = (SUP a\<in>A. ereal (f a))"
+
−  by (simp add: ereal_Sup image_comp)
+
−
+
−lemma ereal_Inf:
+
−  assumes *: "\<bar>INF a\<in>A. ereal a\<bar> \<noteq> \<infinity>"
+
−  shows "ereal (Inf A) = (INF a\<in>A. ereal a)"
+
−proof (rule continuous_at_Inf_mono)
+
−  obtain r where r: "ereal r = (INF a\<in>A. ereal a)" "A \<noteq> {}"
+
−    using * by (force simp: top_ereal_def)
+
−  then show "bdd_below A" "A \<noteq> {}"
+
−    by (auto intro!: INF_lower bdd_belowI[of _ r] simp flip: ereal_less_eq)
+
−qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ereal)
+
−
+
−lemma ereal_Inf':
+
−  assumes *: "bdd_below A" "A \<noteq> {}"
+
−  shows "ereal (Inf A) = (INF a\<in>A. ereal a)"
+
−proof (rule ereal_Inf)
+
−  from * obtain l u where "x \<in> A \<Longrightarrow> l \<le> x" "u \<in> A" for x
+
−    by (auto simp: bdd_below_def)
+
−  then have "l \<le> (INF x\<in>A. ereal x)" "(INF x\<in>A. ereal x) \<le> u"
+
−    by (auto intro!: INF_greatest INF_lower)
+
−  then show "\<bar>INF a\<in>A. ereal a\<bar> \<noteq> \<infinity>"
+
−    by auto
+
−qed
+
−
+
−lemma ereal_INF: "\<bar>INF a\<in>A. ereal (f a)\<bar> \<noteq> \<infinity> \<Longrightarrow> ereal (INF a\<in>A. f a) = (INF a\<in>A. ereal (f a))"
+
−  by (simp add: ereal_Inf image_comp)
+
−
+
−lemma ereal_Sup_uminus_image_eq: "Sup (uminus ` S::ereal set) = - Inf S"
+
−  by (auto intro!: SUP_eqI
+
−           simp: Ball_def[symmetric] ereal_uminus_le_reorder le_Inf_iff
+
−           intro!: complete_lattice_class.Inf_lower2)
+
−
+
−lemma ereal_SUP_uminus_eq:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "(SUP x\<in>S. uminus (f x)) = - (INF x\<in>S. f x)"
+
−  using ereal_Sup_uminus_image_eq [of "f ` S"] by (simp add: image_comp)
+
−
+
−lemma ereal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: ereal set)"
+
−  by (auto intro!: inj_onI)
+
−
+
−lemma ereal_Inf_uminus_image_eq: "Inf (uminus ` S::ereal set) = - Sup S"
+
−  using ereal_Sup_uminus_image_eq[of "uminus ` S"] by simp
+
−
+
−lemma ereal_INF_uminus_eq:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "(INF x\<in>S. - f x) = - (SUP x\<in>S. f x)"
+
−  using ereal_Inf_uminus_image_eq [of "f ` S"] by (simp add: image_comp)
+
−
+
−lemma ereal_SUP_uminus:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "(SUP i \<in> R. - f i) = - (INF i \<in> R. f i)"
+
−  using ereal_Sup_uminus_image_eq[of "f`R"]
+
−  by (simp add: image_image)
+
−
+
−lemma ereal_SUP_not_infty:
+
−  fixes f :: "_ \<Rightarrow> ereal"
+
−  shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>Sup (f ` A)\<bar> \<noteq> \<infinity>"
+
−  using SUP_upper2[of _ A l f] SUP_least[of A f u]
+
−  by (cases "Sup (f ` A)") auto
+
−
+
−lemma ereal_INF_not_infty:
+
−  fixes f :: "_ \<Rightarrow> ereal"
+
−  shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>Inf (f ` A)\<bar> \<noteq> \<infinity>"
+
−  using INF_lower2[of _ A f u] INF_greatest[of A l f]
+
−  by (cases "Inf (f ` A)") auto
+
−
+
−lemma ereal_image_uminus_shift:
+
−  fixes X Y :: "ereal set"
+
−  shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
+
−proof
+
−  assume "uminus ` X = Y"
+
−  then have "uminus ` uminus ` X = uminus ` Y"
+
−    by (simp add: inj_image_eq_iff)
+
−  then show "X = uminus ` Y"
+
−    by (simp add: image_image)
+
−qed (simp add: image_image)
+
−
+
−lemma Sup_eq_MInfty:
+
−  fixes S :: "ereal set"
+
−  shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
+
−  unfolding bot_ereal_def[symmetric] by auto
+
−
+
−lemma Inf_eq_PInfty:
+
−  fixes S :: "ereal set"
+
−  shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
+
−  using Sup_eq_MInfty[of "uminus`S"]
+
−  unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp
+
−
+
−lemma Inf_eq_MInfty:
+
−  fixes S :: "ereal set"
+
−  shows "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>"
+
−  unfolding bot_ereal_def[symmetric] by auto
+
−
+
−lemma Sup_eq_PInfty:
+
−  fixes S :: "ereal set"
+
−  shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>"
+
−  unfolding top_ereal_def[symmetric] by auto
+
−
+
−lemma not_MInfty_nonneg[simp]: "0 \<le> (x::ereal) \<Longrightarrow> x \<noteq> - \<infinity>"
+
−  by auto
+
−
+
−lemma Sup_ereal_close:
+
−  fixes e :: ereal
+
−  assumes "0 < e"
+
−    and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
+
−  shows "\<exists>x\<in>S. Sup S - e < x"
+
−  using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])
+
−
+
−lemma Inf_ereal_close:
+
−  fixes e :: ereal
+
−  assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>"
+
−    and "0 < e"
+
−  shows "\<exists>x\<in>X. x < Inf X + e"
+
−proof (rule Inf_less_iff[THEN iffD1])
+
−  show "Inf X < Inf X + e"
+
−    using assms by (cases e) auto
+
−qed
+
−
+
−lemma SUP_PInfty:
+
−  "(\<And>n::nat. \<exists>i\<in>A. ereal (real n) \<le> f i) \<Longrightarrow> (SUP i\<in>A. f i :: ereal) = \<infinity>"
+
−  unfolding top_ereal_def[symmetric] SUP_eq_top_iff
+
−  by (metis MInfty_neq_PInfty(2) PInfty_neq_ereal(2) less_PInf_Ex_of_nat less_ereal.elims(2) less_le_trans)
+
−
+
−lemma SUP_nat_Infty: "(SUP i. ereal (real i)) = \<infinity>"
+
−  by (rule SUP_PInfty) auto
+
−
+
−lemma SUP_ereal_add_left:
+
−  assumes "I \<noteq> {}" "c \<noteq> -\<infinity>"
+
−  shows "(SUP i\<in>I. f i + c :: ereal) = (SUP i\<in>I. f i) + c"
+
−proof (cases "(SUP i\<in>I. f i) = - \<infinity>")
+
−  case True
+
−  then have "\<And>i. i \<in> I \<Longrightarrow> f i = -\<infinity>"
+
−    unfolding Sup_eq_MInfty by auto
+
−  with True show ?thesis
+
−    by (cases c) (auto simp: \<open>I \<noteq> {}\<close>)
+
−next
+
−  case False
+
−  then show ?thesis
+
−    by (subst continuous_at_Sup_mono[where f="\<lambda>x. x + c"])
+
−      (auto simp: continuous_at_imp_continuous_at_within continuous_at mono_def add_mono \<open>I \<noteq> {}\<close>
+
−      \<open>c \<noteq> -\<infinity>\<close> image_comp)
+
−qed
+
−
+
−lemma SUP_ereal_add_right:
+
−  fixes c :: ereal
+
−  shows "I \<noteq> {} \<Longrightarrow> c \<noteq> -\<infinity> \<Longrightarrow> (SUP i\<in>I. c + f i) = c + (SUP i\<in>I. f i)"
+
−  using SUP_ereal_add_left[of I c f] by (simp add: add.commute)
+
−
+
−lemma SUP_ereal_minus_right:
+
−  assumes "I \<noteq> {}" "c \<noteq> -\<infinity>"
+
−  shows "(SUP i\<in>I. c - f i :: ereal) = c - (INF i\<in>I. f i)"
+
−  using SUP_ereal_add_right[OF assms, of "\<lambda>i. - f i"]
+
−  by (simp add: ereal_SUP_uminus minus_ereal_def)
+
−
+
−lemma SUP_ereal_minus_left:
+
−  assumes "I \<noteq> {}" "c \<noteq> \<infinity>"
+
−  shows "(SUP i\<in>I. f i - c:: ereal) = (SUP i\<in>I. f i) - c"
+
−  using SUP_ereal_add_left[OF \<open>I \<noteq> {}\<close>, of "-c" f] by (simp add: \<open>c \<noteq> \<infinity>\<close> minus_ereal_def)
+
−
+
−lemma INF_ereal_minus_right:
+
−  assumes "I \<noteq> {}" and "\<bar>c\<bar> \<noteq> \<infinity>"
+
−  shows "(INF i\<in>I. c - f i) = c - (SUP i\<in>I. f i::ereal)"
+
−proof -
+
−  { fix b have "(-c) + b = - (c - b)"
+
−      using \<open>\<bar>c\<bar> \<noteq> \<infinity>\<close> by (cases c b rule: ereal2_cases) auto }
+
−  note * = this
+
−  show ?thesis
+
−    using SUP_ereal_add_right[OF \<open>I \<noteq> {}\<close>, of "-c" f] \<open>\<bar>c\<bar> \<noteq> \<infinity>\<close>
+
−    by (auto simp add: * ereal_SUP_uminus_eq)
+
−qed
+
−
+
−lemma SUP_ereal_le_addI:
+
−  fixes f :: "'i \<Rightarrow> ereal"
+
−  assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
+
−  shows "Sup (f ` UNIV) + y \<le> z"
+
−  unfolding SUP_ereal_add_left[OF UNIV_not_empty \<open>y \<noteq> -\<infinity>\<close>, symmetric]
+
−  by (rule SUP_least assms)+
+
−
+
−lemma SUP_combine:
+
−  fixes f :: "'a::semilattice_sup \<Rightarrow> 'a::semilattice_sup \<Rightarrow> 'b::complete_lattice"
+
−  assumes mono: "\<And>a b c d. a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> f a c \<le> f b d"
+
−  shows "(SUP i\<in>UNIV. SUP j\<in>UNIV. f i j) = (SUP i. f i i)"
+
−proof (rule antisym)
+
−  show "(SUP i j. f i j) \<le> (SUP i. f i i)"
+
−    by (rule SUP_least SUP_upper2[where i="sup i j" for i j] UNIV_I mono sup_ge1 sup_ge2)+
+
−  show "(SUP i. f i i) \<le> (SUP i j. f i j)"
+
−    by (rule SUP_least SUP_upper2 UNIV_I mono order_refl)+
+
−qed
+
−
+
−lemma SUP_ereal_add:
+
−  fixes f g :: "nat \<Rightarrow> ereal"
+
−  assumes inc: "incseq f" "incseq g"
+
−    and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
+
−  shows "(SUP i. f i + g i) = Sup (f ` UNIV) + Sup (g ` UNIV)"
+
−  apply (subst SUP_ereal_add_left[symmetric, OF UNIV_not_empty])
+
−  apply (metis SUP_upper UNIV_I assms(4) ereal_infty_less_eq(2))
+
−  apply (subst (2) add.commute)
+
−  apply (subst SUP_ereal_add_left[symmetric, OF UNIV_not_empty assms(3)])
+
−  apply (subst (2) add.commute)
+
−  apply (rule SUP_combine[symmetric] add_mono inc[THEN monoD] | assumption)+
+
−  done
+
−
+
−lemma INF_eq_minf: "(INF i\<in>I. f i::ereal) \<noteq> -\<infinity> \<longleftrightarrow> (\<exists>b>-\<infinity>. \<forall>i\<in>I. b \<le> f i)"
+
−  unfolding bot_ereal_def[symmetric] INF_eq_bot_iff by (auto simp: not_less)
+
−
+
−lemma INF_ereal_add_left:
+
−  assumes "I \<noteq> {}" "c \<noteq> -\<infinity>" "\<And>x. x \<in> I \<Longrightarrow> 0 \<le> f x"
+
−  shows "(INF i\<in>I. f i + c :: ereal) = (INF i\<in>I. f i) + c"
+
−proof -
+
−  have "(INF i\<in>I. f i) \<noteq> -\<infinity>"
+
−    unfolding INF_eq_minf using assms by (intro exI[of _ 0]) auto
+
−  then show ?thesis
+
−    by (subst continuous_at_Inf_mono[where f="\<lambda>x. x + c"])
+
−       (auto simp: mono_def add_mono \<open>I \<noteq> {}\<close> \<open>c \<noteq> -\<infinity>\<close> continuous_at_imp_continuous_at_within
+
−        continuous_at image_comp)
+
−qed
+
−
+
−lemma INF_ereal_add_right:
+
−  assumes "I \<noteq> {}" "c \<noteq> -\<infinity>" "\<And>x. x \<in> I \<Longrightarrow> 0 \<le> f x"
+
−  shows "(INF i\<in>I. c + f i :: ereal) = c + (INF i\<in>I. f i)"
+
−  using INF_ereal_add_left[OF assms] by (simp add: ac_simps)
+
−
+
−lemma INF_ereal_add_directed:
+
−  fixes f g :: "'a \<Rightarrow> ereal"
+
−  assumes nonneg: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> g i"
+
−  assumes directed: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. f i + g j \<ge> f k + g k"
+
−  shows "(INF i\<in>I. f i + g i) = (INF i\<in>I. f i) + (INF i\<in>I. g i)"
+
−proof cases
+
−  assume "I = {}" then show ?thesis
+
−    by (simp add: top_ereal_def)
+
−next
+
−  assume "I \<noteq> {}"
+
−  show ?thesis
+
−  proof (rule antisym)
+
−    show "(INF i\<in>I. f i) + (INF i\<in>I. g i) \<le> (INF i\<in>I. f i + g i)"
+
−      by (rule INF_greatest; intro add_mono INF_lower)
+
−  next
+
−    have "(INF i\<in>I. f i + g i) \<le> (INF i\<in>I. (INF j\<in>I. f i + g j))"
+
−      using directed by (intro INF_greatest) (blast intro: INF_lower2)
+
−    also have "\<dots> = (INF i\<in>I. f i + (INF i\<in>I. g i))"
+
−      using nonneg \<open>I \<noteq> {}\<close> by (auto simp: INF_ereal_add_right)
+
−    also have "\<dots> = (INF i\<in>I. f i) + (INF i\<in>I. g i)"
+
−      using nonneg by (intro INF_ereal_add_left \<open>I \<noteq> {}\<close>) (auto simp: INF_eq_minf intro!: exI[of _ 0])
+
−    finally show "(INF i\<in>I. f i + g i) \<le> (INF i\<in>I. f i) + (INF i\<in>I. g i)" .
+
−  qed
+
−qed
+
−
+
−lemma INF_ereal_add:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "decseq f" "decseq g"
+
−    and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
+
−  shows "(INF i. f i + g i) = Inf (f ` UNIV) + Inf (g ` UNIV)"
+
−proof -
+
−  have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
+
−    using assms unfolding INF_less_iff by auto
+
−  { fix a b :: ereal assume "a \<noteq> \<infinity>" "b \<noteq> \<infinity>"
+
−    then have "- ((- a) + (- b)) = a + b"
+
−      by (cases a b rule: ereal2_cases) auto }
+
−  note * = this
+
−  have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
+
−    by (simp add: fin *)
+
−  also have "\<dots> = Inf (f ` UNIV) + Inf (g ` UNIV)"
+
−    unfolding ereal_INF_uminus_eq
+
−    using assms INF_less
+
−    by (subst SUP_ereal_add) (auto simp: ereal_SUP_uminus fin *)
+
−  finally show ?thesis .
+
−qed
+
−
+
−lemma SUP_ereal_add_pos:
+
−  fixes f g :: "nat \<Rightarrow> ereal"
+
−  assumes inc: "incseq f" "incseq g"
+
−    and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
+
−  shows "(SUP i. f i + g i) = Sup (f ` UNIV) + Sup (g ` UNIV)"
+
−proof (intro SUP_ereal_add inc)
+
−  fix i
+
−  show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>"
+
−    using pos[of i] by auto
+
−qed
+
−
+
−lemma SUP_ereal_sum:
+
−  fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
+
−  assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
+
−    and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
+
−  shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. Sup ((f n) ` UNIV))"
+
−proof (cases "finite A")
+
−  case True
+
−  then show ?thesis using assms
+
−    by induct (auto simp: incseq_sumI2 sum_nonneg SUP_ereal_add_pos)
+
−next
+
−  case False
+
−  then show ?thesis by simp
+
−qed
+
−
+
−lemma SUP_ereal_mult_left:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes "I \<noteq> {}"
+
−  assumes f: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" and c: "0 \<le> c"
+
−  shows "(SUP i\<in>I. c * f i) = c * (SUP i\<in>I. f i)"
+
−proof (cases "(SUP i \<in> I. f i) = 0")
+
−  case True
+
−  then have "\<And>i. i \<in> I \<Longrightarrow> f i = 0"
+
−    by (metis SUP_upper f antisym)
+
−  with True show ?thesis
+
−    by simp
+
−next
+
−  case False
+
−  then show ?thesis
+
−    by (subst continuous_at_Sup_mono[where f="\<lambda>x. c * x"])
+
−       (auto simp: mono_def continuous_at continuous_at_imp_continuous_at_within \<open>I \<noteq> {}\<close> image_comp
+
−             intro!: ereal_mult_left_mono c)
+
−qed
+
−
+
−lemma countable_approach:
+
−  fixes x :: ereal
+
−  assumes "x \<noteq> -\<infinity>"
+
−  shows "\<exists>f. incseq f \<and> (\<forall>i::nat. f i < x) \<and> (f \<longlonglongrightarrow> x)"
+
−proof (cases x)
+
−  case (real r)
+
−  moreover have "(\<lambda>n. r - inverse (real (Suc n))) \<longlonglongrightarrow> r - 0"
+
−    by (intro tendsto_intros LIMSEQ_inverse_real_of_nat)
+
−  ultimately show ?thesis
+
−    by (intro exI[of _ "\<lambda>n. x - inverse (Suc n)"]) (auto simp: incseq_def)
+
−next
+
−  case PInf with LIMSEQ_SUP[of "\<lambda>n::nat. ereal (real n)"] show ?thesis
+
−    by (intro exI[of _ "\<lambda>n. ereal (real n)"]) (auto simp: incseq_def SUP_nat_Infty)
+
−qed (simp add: assms)
+
−
+
−lemma Sup_countable_SUP:
+
−  assumes "A \<noteq> {}"
+
−  shows "\<exists>f::nat \<Rightarrow> ereal. incseq f \<and> range f \<subseteq> A \<and> Sup A = (SUP i. f i)"
+
−proof cases
+
−  assume "Sup A = -\<infinity>"
+
−  with \<open>A \<noteq> {}\<close> have "A = {-\<infinity>}"
+
−    by (auto simp: Sup_eq_MInfty)
+
−  then show ?thesis
+
−    by (auto intro!: exI[of _ "\<lambda>_. -\<infinity>"] simp: bot_ereal_def)
+
−next
+
−  assume "Sup A \<noteq> -\<infinity>"
+
−  then obtain l where "incseq l" and l: "l i < Sup A" and l_Sup: "l \<longlonglongrightarrow> Sup A" for i :: nat
+
−    by (auto dest: countable_approach)
+
−
+
−  have "\<exists>f. \<forall>n. (f n \<in> A \<and> l n \<le> f n) \<and> (f n \<le> f (Suc n))"
+
−  proof (rule dependent_nat_choice)
+
−    show "\<exists>x. x \<in> A \<and> l 0 \<le> x"
+
−      using l[of 0] by (auto simp: less_Sup_iff)
+
−  next
+
−    fix x n assume "x \<in> A \<and> l n \<le> x"
+
−    moreover from l[of "Suc n"] obtain y where "y \<in> A" "l (Suc n) < y"
+
−      by (auto simp: less_Sup_iff)
+
−    ultimately show "\<exists>y. (y \<in> A \<and> l (Suc n) \<le> y) \<and> x \<le> y"
+
−      by (auto intro!: exI[of _ "max x y"] split: split_max)
+
−  qed
+
−  then guess f .. note f = this
+
−  then have "range f \<subseteq> A" "incseq f"
+
−    by (auto simp: incseq_Suc_iff)
+
−  moreover
+
−  have "(SUP i. f i) = Sup A"
+
−  proof (rule tendsto_unique)
+
−    show "f \<longlonglongrightarrow> (SUP i. f i)"
+
−      by (rule LIMSEQ_SUP \<open>incseq f\<close>)+
+
−    show "f \<longlonglongrightarrow> Sup A"
+
−      using l f
+
−      by (intro tendsto_sandwich[OF _ _ l_Sup tendsto_const])
+
−         (auto simp: Sup_upper)
+
−  qed simp
+
−  ultimately show ?thesis
+
−    by auto
+
−qed
+
−
+
−lemma Inf_countable_INF:
+
−  assumes "A \<noteq> {}" shows "\<exists>f::nat \<Rightarrow> ereal. decseq f \<and> range f \<subseteq> A \<and> Inf A = (INF i. f i)"
+
−proof -
+
−  obtain f where "incseq f" "range f \<subseteq> uminus`A" "Sup (uminus`A) = (SUP i. f i)"
+
−    using Sup_countable_SUP[of "uminus ` A"] \<open>A \<noteq> {}\<close> by auto
+
−  then show ?thesis
+
−    by (intro exI[of _ "\<lambda>x. - f x"])
+
−       (auto simp: ereal_Sup_uminus_image_eq ereal_INF_uminus_eq eq_commute[of "- _"])
+
−qed
+
−
+
−lemma SUP_countable_SUP:
+
−  "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> Sup (g ` A) = Sup (f ` UNIV)"
+
−  using Sup_countable_SUP [of "g`A"] by auto
+
−
+
−subsection "Relation to \<^typ>\<open>enat\<close>"
+
−
+
−definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)"
+
−
+
−declare [[coercion "ereal_of_enat :: enat \<Rightarrow> ereal"]]
+
−declare [[coercion "(\<lambda>n. ereal (real n)) :: nat \<Rightarrow> ereal"]]
+
−
+
−lemma ereal_of_enat_simps[simp]:
+
−  "ereal_of_enat (enat n) = ereal n"
+
−  "ereal_of_enat \<infinity> = \<infinity>"
+
−  by (simp_all add: ereal_of_enat_def)
+
−
+
−lemma ereal_of_enat_le_iff[simp]: "ereal_of_enat m \<le> ereal_of_enat n \<longleftrightarrow> m \<le> n"
+
−  by (cases m n rule: enat2_cases) auto
+
−
+
−lemma ereal_of_enat_less_iff[simp]: "ereal_of_enat m < ereal_of_enat n \<longleftrightarrow> m < n"
+
−  by (cases m n rule: enat2_cases) auto
+
−
+
−lemma numeral_le_ereal_of_enat_iff[simp]: "numeral m \<le> ereal_of_enat n \<longleftrightarrow> numeral m \<le> n"
+
−by (cases n) (auto)
+
−
+
−lemma numeral_less_ereal_of_enat_iff[simp]: "numeral m < ereal_of_enat n \<longleftrightarrow> numeral m < n"
+
−  by (cases n) auto
+
−
+
−lemma ereal_of_enat_ge_zero_cancel_iff[simp]: "0 \<le> ereal_of_enat n \<longleftrightarrow> 0 \<le> n"
+
−  by (cases n) (auto simp flip: enat_0)
+
−
+
−lemma ereal_of_enat_gt_zero_cancel_iff[simp]: "0 < ereal_of_enat n \<longleftrightarrow> 0 < n"
+
−  by (cases n) (auto simp flip: enat_0)
+
−
+
−lemma ereal_of_enat_zero[simp]: "ereal_of_enat 0 = 0"
+
−  by (auto simp flip: enat_0)
+
−
+
−lemma ereal_of_enat_inf[simp]: "ereal_of_enat n = \<infinity> \<longleftrightarrow> n = \<infinity>"
+
−  by (cases n) auto
+
−
+
−lemma ereal_of_enat_add: "ereal_of_enat (m + n) = ereal_of_enat m + ereal_of_enat n"
+
−  by (cases m n rule: enat2_cases) auto
+
−
+
−lemma ereal_of_enat_sub:
+
−  assumes "n \<le> m"
+
−  shows "ereal_of_enat (m - n) = ereal_of_enat m - ereal_of_enat n "
+
−  using assms by (cases m n rule: enat2_cases) auto
+
−
+
−lemma ereal_of_enat_mult:
+
−  "ereal_of_enat (m * n) = ereal_of_enat m * ereal_of_enat n"
+
−  by (cases m n rule: enat2_cases) auto
+
−
+
−lemmas ereal_of_enat_pushin = ereal_of_enat_add ereal_of_enat_sub ereal_of_enat_mult
+
−lemmas ereal_of_enat_pushout = ereal_of_enat_pushin[symmetric]
+
−
+
−lemma ereal_of_enat_nonneg: "ereal_of_enat n \<ge> 0"
+
−by(cases n) simp_all
+
−
+
−lemma ereal_of_enat_Sup:
+
−  assumes "A \<noteq> {}" shows "ereal_of_enat (Sup A) = (SUP a \<in> A. ereal_of_enat a)"
+
−proof (intro antisym mono_Sup)
+
−  show "ereal_of_enat (Sup A) \<le> (SUP a \<in> A. ereal_of_enat a)"
+
−  proof cases
+
−    assume "finite A"
+
−    with \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" "ereal_of_enat (Sup A) = ereal_of_enat a"
+
−      using Max_in[of A] by (auto simp: Sup_enat_def simp del: Max_in)
+
−    then show ?thesis
+
−      by (auto intro: SUP_upper)
+
−  next
+
−    assume "\<not> finite A"
+
−    have [simp]: "(SUP a \<in> A. ereal_of_enat a) = top"
+
−      unfolding SUP_eq_top_iff
+
−    proof safe
+
−      fix x :: ereal assume "x < top"
+
−      then obtain n :: nat where "x < n"
+
−        using less_PInf_Ex_of_nat top_ereal_def by auto
+
−      obtain a where "a \<in> A - enat ` {.. n}"
+
−        by (metis \<open>\<not> finite A\<close> all_not_in_conv finite_Diff2 finite_atMost finite_imageI finite.emptyI)
+
−      then have "a \<in> A" "ereal n \<le> ereal_of_enat a"
+
−        by (auto simp: image_iff Ball_def)
+
−           (metis enat_iless enat_ord_simps(1) ereal_of_enat_less_iff ereal_of_enat_simps(1) less_le not_less)
+
−      with \<open>x < n\<close> show "\<exists>i\<in>A. x < ereal_of_enat i"
+
−        by (auto intro!: bexI[of _ a])
+
−    qed
+
−    show ?thesis
+
−      by simp
+
−  qed
+
−qed (simp add: mono_def)
+
−
+
−lemma ereal_of_enat_SUP:
+
−  "A \<noteq> {} \<Longrightarrow> ereal_of_enat (SUP a\<in>A. f a) = (SUP a \<in> A. ereal_of_enat (f a))"
+
−  by (simp add: ereal_of_enat_Sup image_comp)
+
−
+
−
+
−subsection "Limits on \<^typ>\<open>ereal\<close>"
+
−
+
−lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)"
+
−  unfolding open_ereal_generated
+
−proof (induct rule: generate_topology.induct)
+
−  case (Int A B)
+
−  then obtain x z where "\<infinity> \<in> A \<Longrightarrow> {ereal x <..} \<subseteq> A" "\<infinity> \<in> B \<Longrightarrow> {ereal z <..} \<subseteq> B"
+
−    by auto
+
−  with Int show ?case
+
−    by (intro exI[of _ "max x z"]) fastforce
+
−next
+
−  case (Basis S)
+
−  {
+
−    fix x
+
−    have "x \<noteq> \<infinity> \<Longrightarrow> \<exists>t. x \<le> ereal t"
+
−      by (cases x) auto
+
−  }
+
−  moreover note Basis
+
−  ultimately show ?case
+
−    by (auto split: ereal.split)
+
−qed (fastforce simp add: vimage_Union)+
+
−
+
−lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)"
+
−  unfolding open_ereal_generated
+
−proof (induct rule: generate_topology.induct)
+
−  case (Int A B)
+
−  then obtain x z where "-\<infinity> \<in> A \<Longrightarrow> {..< ereal x} \<subseteq> A" "-\<infinity> \<in> B \<Longrightarrow> {..< ereal z} \<subseteq> B"
+
−    by auto
+
−  with Int show ?case
+
−    by (intro exI[of _ "min x z"]) fastforce
+
−next
+
−  case (Basis S)
+
−  {
+
−    fix x
+
−    have "x \<noteq> - \<infinity> \<Longrightarrow> \<exists>t. ereal t \<le> x"
+
−      by (cases x) auto
+
−  }
+
−  moreover note Basis
+
−  ultimately show ?case
+
−    by (auto split: ereal.split)
+
−qed (fastforce simp add: vimage_Union)+
+
−
+
−lemma open_ereal_vimage: "open S \<Longrightarrow> open (ereal -` S)"
+
−  by (intro open_vimage continuous_intros)
+
−
+
−lemma open_ereal: "open S \<Longrightarrow> open (ereal ` S)"
+
−  unfolding open_generated_order[where 'a=real]
+
−proof (induct rule: generate_topology.induct)
+
−  case (Basis S)
+
−  moreover {
+
−    fix x
+
−    have "ereal ` {..< x} = { -\<infinity> <..< ereal x }"
+
−      apply auto
+
−      apply (case_tac xa)
+
−      apply auto
+
−      done
+
−  }
+
−  moreover {
+
−    fix x
+
−    have "ereal ` {x <..} = { ereal x <..< \<infinity> }"
+
−      apply auto
+
−      apply (case_tac xa)
+
−      apply auto
+
−      done
+
−  }
+
−  ultimately show ?case
+
−     by auto
+
−qed (auto simp add: image_Union image_Int)
+
−
+
−lemma open_image_real_of_ereal:
+
−  fixes X::"ereal set"
+
−  assumes "open X"
+
−  assumes "\<infinity> \<notin> X"
+
−  assumes "-\<infinity> \<notin> X"
+
−  shows "open (real_of_ereal ` X)"
+
−proof -
+
−  have "real_of_ereal ` X = ereal -` X"
+
−    apply safe
+
−    subgoal by (metis assms(2) assms(3) real_of_ereal.elims vimageI)
+
−    subgoal using image_iff by fastforce
+
−    done
+
−  thus ?thesis
+
−    by (auto intro!: open_ereal_vimage assms)
+
−qed
+
−
+
−lemma eventually_finite:
+
−  fixes x :: ereal
+
−  assumes "\<bar>x\<bar> \<noteq> \<infinity>" "(f \<longlongrightarrow> x) F"
+
−  shows "eventually (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>) F"
+
−proof -
+
−  have "(f \<longlongrightarrow> ereal (real_of_ereal x)) F"
+
−    using assms by (cases x) auto
+
−  then have "eventually (\<lambda>x. f x \<in> ereal ` UNIV) F"
+
−    by (rule topological_tendstoD) (auto intro: open_ereal)
+
−  also have "(\<lambda>x. f x \<in> ereal ` UNIV) = (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>)"
+
−    by auto
+
−  finally show ?thesis .
+
−qed
+
−
+
−
+
−lemma open_ereal_def:
+
−  "open A \<longleftrightarrow> open (ereal -` A) \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A)) \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))"
+
−  (is "open A \<longleftrightarrow> ?rhs")
+
−proof
+
−  assume "open A"
+
−  then show ?rhs
+
−    using open_PInfty open_MInfty open_ereal_vimage by auto
+
−next
+
−  assume "?rhs"
+
−  then obtain x y where A: "open (ereal -` A)" "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" "-\<infinity> \<in> A \<Longrightarrow> {..< ereal y} \<subseteq> A"
+
−    by auto
+
−  have *: "A = ereal ` (ereal -` A) \<union> (if \<infinity> \<in> A then {ereal x<..} else {}) \<union> (if -\<infinity> \<in> A then {..< ereal y} else {})"
+
−    using A(2,3) by auto
+
−  from open_ereal[OF A(1)] show "open A"
+
−    by (subst *) (auto simp: open_Un)
+
−qed
+
−
+
−lemma open_PInfty2:
+
−  assumes "open A"
+
−    and "\<infinity> \<in> A"
+
−  obtains x where "{ereal x<..} \<subseteq> A"
+
−  using open_PInfty[OF assms] by auto
+
−
+
−lemma open_MInfty2:
+
−  assumes "open A"
+
−    and "-\<infinity> \<in> A"
+
−  obtains x where "{..<ereal x} \<subseteq> A"
+
−  using open_MInfty[OF assms] by auto
+
−
+
−lemma ereal_openE:
+
−  assumes "open A"
+
−  obtains x y where "open (ereal -` A)"
+
−    and "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A"
+
−    and "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A"
+
−  using assms open_ereal_def by auto
+
−
+
−lemmas open_ereal_lessThan = open_lessThan[where 'a=ereal]
+
−lemmas open_ereal_greaterThan = open_greaterThan[where 'a=ereal]
+
−lemmas ereal_open_greaterThanLessThan = open_greaterThanLessThan[where 'a=ereal]
+
−lemmas closed_ereal_atLeast = closed_atLeast[where 'a=ereal]
+
−lemmas closed_ereal_atMost = closed_atMost[where 'a=ereal]
+
−lemmas closed_ereal_atLeastAtMost = closed_atLeastAtMost[where 'a=ereal]
+
−lemmas closed_ereal_singleton = closed_singleton[where 'a=ereal]
+
−
+
−lemma ereal_open_cont_interval:
+
−  fixes S :: "ereal set"
+
−  assumes "open S"
+
−    and "x \<in> S"
+
−    and "\<bar>x\<bar> \<noteq> \<infinity>"
+
−  obtains e where "e > 0" and "{x-e <..< x+e} \<subseteq> S"
+
−proof -
+
−  from \<open>open S\<close>
+
−  have "open (ereal -` S)"
+
−    by (rule ereal_openE)
+
−  then obtain e where "e > 0" and e: "dist y (real_of_ereal x) < e \<Longrightarrow> ereal y \<in> S" for y
+
−    using assms unfolding open_dist by force
+
−  show thesis
+
−  proof (intro that subsetI)
+
−    show "0 < ereal e"
+
−      using \<open>0 < e\<close> by auto
+
−    fix y
+
−    assume "y \<in> {x - ereal e<..<x + ereal e}"
+
−    with assms obtain t where "y = ereal t" "dist t (real_of_ereal x) < e"
+
−      by (cases y) (auto simp: dist_real_def)
+
−    then show "y \<in> S"
+
−      using e[of t] by auto
+
−  qed
+
−qed
+
−
+
−lemma ereal_open_cont_interval2:
+
−  fixes S :: "ereal set"
+
−  assumes "open S"
+
−    and "x \<in> S"
+
−    and x: "\<bar>x\<bar> \<noteq> \<infinity>"
+
−  obtains a b where "a < x" and "x < b" and "{a <..< b} \<subseteq> S"
+
−proof -
+
−  obtain e where "0 < e" "{x - e<..<x + e} \<subseteq> S"
+
−    using assms by (rule ereal_open_cont_interval)
+
−  with that[of "x - e" "x + e"] ereal_between[OF x, of e]
+
−  show thesis
+
−    by auto
+
−qed
+
−
+
−subsubsection \<open>Convergent sequences\<close>
+
−
+
−lemma lim_real_of_ereal[simp]:
+
−  assumes lim: "(f \<longlongrightarrow> ereal x) net"
+
−  shows "((\<lambda>x. real_of_ereal (f x)) \<longlongrightarrow> x) net"
+
−proof (intro topological_tendstoI)
+
−  fix S
+
−  assume "open S" and "x \<in> S"
+
−  then have S: "open S" "ereal x \<in> ereal ` S"
+
−    by (simp_all add: inj_image_mem_iff)
+
−  show "eventually (\<lambda>x. real_of_ereal (f x) \<in> S) net"
+
−    by (auto intro: eventually_mono [OF lim[THEN topological_tendstoD, OF open_ereal, OF S]])
+
−qed
+
−
+
−lemma lim_ereal[simp]: "((\<lambda>n. ereal (f n)) \<longlongrightarrow> ereal x) net \<longleftrightarrow> (f \<longlongrightarrow> x) net"
+
−  by (auto dest!: lim_real_of_ereal)
+
−
+
−lemma convergent_real_imp_convergent_ereal:
+
−  assumes "convergent a"
+
−  shows "convergent (\<lambda>n. ereal (a n))" and "lim (\<lambda>n. ereal (a n)) = ereal (lim a)"
+
−proof -
+
−  from assms obtain L where L: "a \<longlonglongrightarrow> L" unfolding convergent_def ..
+
−  hence lim: "(\<lambda>n. ereal (a n)) \<longlonglongrightarrow> ereal L" using lim_ereal by auto
+
−  thus "convergent (\<lambda>n. ereal (a n))" unfolding convergent_def ..
+
−  thus "lim (\<lambda>n. ereal (a n)) = ereal (lim a)" using lim L limI by metis
+
−qed
+
−
+
−lemma tendsto_PInfty: "(f \<longlongrightarrow> \<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. ereal r < f x) F)"
+
−proof -
+
−  {
+
−    fix l :: ereal
+
−    assume "\<forall>r. eventually (\<lambda>x. ereal r < f x) F"
+
−    from this[THEN spec, of "real_of_ereal l"] have "l \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. l < f x) F"
+
−      by (cases l) (auto elim: eventually_mono)
+
−  }
+
−  then show ?thesis
+
−    by (auto simp: order_tendsto_iff)
+
−qed
+
−
+
−lemma tendsto_PInfty': "(f \<longlongrightarrow> \<infinity>) F = (\<forall>r>c. eventually (\<lambda>x. ereal r < f x) F)"
+
−proof (subst tendsto_PInfty, intro iffI allI impI)
+
−  assume A: "\<forall>r>c. eventually (\<lambda>x. ereal r < f x) F"
+
−  fix r :: real
+
−  from A have A: "eventually (\<lambda>x. ereal r < f x) F" if "r > c" for r using that by blast
+
−  show "eventually (\<lambda>x. ereal r < f x) F"
+
−  proof (cases "r > c")
+
−    case False
+
−    hence B: "ereal r \<le> ereal (c + 1)" by simp
+
−    have "c < c + 1" by simp
+
−    from A[OF this] show "eventually (\<lambda>x. ereal r < f x) F"
+
−      by eventually_elim (rule le_less_trans[OF B])
+
−  qed (simp add: A)
+
−qed simp
+
−
+
−lemma tendsto_PInfty_eq_at_top:
+
−  "((\<lambda>z. ereal (f z)) \<longlongrightarrow> \<infinity>) F \<longleftrightarrow> (LIM z F. f z :> at_top)"
+
−  unfolding tendsto_PInfty filterlim_at_top_dense by simp
+
−
+
−lemma tendsto_MInfty: "(f \<longlongrightarrow> -\<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. f x < ereal r) F)"
+
−  unfolding tendsto_def
+
−proof safe
+
−  fix S :: "ereal set"
+
−  assume "open S" "-\<infinity> \<in> S"
+
−  from open_MInfty[OF this] obtain B where "{..<ereal B} \<subseteq> S" ..
+
−  moreover
+
−  assume "\<forall>r::real. eventually (\<lambda>z. f z < r) F"
+
−  then have "eventually (\<lambda>z. f z \<in> {..< B}) F"
+
−    by auto
+
−  ultimately show "eventually (\<lambda>z. f z \<in> S) F"
+
−    by (auto elim!: eventually_mono)
+
−next
+
−  fix x
+
−  assume "\<forall>S. open S \<longrightarrow> -\<infinity> \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
+
−  from this[rule_format, of "{..< ereal x}"] show "eventually (\<lambda>y. f y < ereal x) F"
+
−    by auto
+
−qed
+
−
+
−lemma tendsto_MInfty': "(f \<longlongrightarrow> -\<infinity>) F = (\<forall>r<c. eventually (\<lambda>x. ereal r > f x) F)"
+
−proof (subst tendsto_MInfty, intro iffI allI impI)
+
−  assume A: "\<forall>r<c. eventually (\<lambda>x. ereal r > f x) F"
+
−  fix r :: real
+
−  from A have A: "eventually (\<lambda>x. ereal r > f x) F" if "r < c" for r using that by blast
+
−  show "eventually (\<lambda>x. ereal r > f x) F"
+
−  proof (cases "r < c")
+
−    case False
+
−    hence B: "ereal r \<ge> ereal (c - 1)" by simp
+
−    have "c > c - 1" by simp
+
−    from A[OF this] show "eventually (\<lambda>x. ereal r > f x) F"
+
−      by eventually_elim (erule less_le_trans[OF _ B])
+
−  qed (simp add: A)
+
−qed simp
+
−
+
−lemma Lim_PInfty: "f \<longlonglongrightarrow> \<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> ereal B)"
+
−  unfolding tendsto_PInfty eventually_sequentially
+
−proof safe
+
−  fix r
+
−  assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. ereal r \<le> f n"
+
−  then obtain N where "\<forall>n\<ge>N. ereal (r + 1) \<le> f n"
+
−    by blast
+
−  moreover have "ereal r < ereal (r + 1)"
+
−    by auto
+
−  ultimately show "\<exists>N. \<forall>n\<ge>N. ereal r < f n"
+
−    by (blast intro: less_le_trans)
+
−qed (blast intro: less_imp_le)
+
−
+
−lemma Lim_MInfty: "f \<longlonglongrightarrow> -\<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. ereal B \<ge> f n)"
+
−  unfolding tendsto_MInfty eventually_sequentially
+
−proof safe
+
−  fix r
+
−  assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. f n \<le> ereal r"
+
−  then obtain N where "\<forall>n\<ge>N. f n \<le> ereal (r - 1)"
+
−    by blast
+
−  moreover have "ereal (r - 1) < ereal r"
+
−    by auto
+
−  ultimately show "\<exists>N. \<forall>n\<ge>N. f n < ereal r"
+
−    by (blast intro: le_less_trans)
+
−qed (blast intro: less_imp_le)
+
−
+
−lemma Lim_bounded_PInfty: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<And>n. f n \<le> ereal B) \<Longrightarrow> l \<noteq> \<infinity>"
+
−  using LIMSEQ_le_const2[of f l "ereal B"] by auto
+
−
+
−lemma Lim_bounded_MInfty: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<And>n. ereal B \<le> f n) \<Longrightarrow> l \<noteq> -\<infinity>"
+
−  using LIMSEQ_le_const[of f l "ereal B"] by auto
+
−
+
−lemma tendsto_zero_erealI:
+
−  assumes "\<And>e. e > 0 \<Longrightarrow> eventually (\<lambda>x. \<bar>f x\<bar> < ereal e) F"
+
−  shows   "(f \<longlongrightarrow> 0) F"
+
−proof (subst filterlim_cong[OF refl refl])
+
−  from assms[OF zero_less_one] show "eventually (\<lambda>x. f x = ereal (real_of_ereal (f x))) F"
+
−    by eventually_elim (auto simp: ereal_real)
+
−  hence "eventually (\<lambda>x. abs (real_of_ereal (f x)) < e) F" if "e > 0" for e using assms[OF that]
+
−    by eventually_elim (simp add: real_less_ereal_iff that)
+
−  hence "((\<lambda>x. real_of_ereal (f x)) \<longlongrightarrow> 0) F" unfolding tendsto_iff
+
−    by (auto simp: tendsto_iff dist_real_def)
+
−  thus "((\<lambda>x. ereal (real_of_ereal (f x))) \<longlongrightarrow> 0) F" by (simp add: zero_ereal_def)
+
−qed
+
−
+
−lemma Lim_bounded_PInfty2: "f \<longlonglongrightarrow> l \<Longrightarrow> \<forall>n\<ge>N. f n \<le> ereal B \<Longrightarrow> l \<noteq> \<infinity>"
+
−  using LIMSEQ_le_const2[of f l "ereal B"] by fastforce
+
−
+
−lemma real_of_ereal_mult[simp]:
+
−  fixes a b :: ereal
+
−  shows "real_of_ereal (a * b) = real_of_ereal a * real_of_ereal b"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma real_of_ereal_eq_0:
+
−  fixes x :: ereal
+
−  shows "real_of_ereal x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
+
−  by (cases x) auto
+
−
+
−lemma tendsto_ereal_realD:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes "x \<noteq> 0"
+
−    and tendsto: "((\<lambda>x. ereal (real_of_ereal (f x))) \<longlongrightarrow> x) net"
+
−  shows "(f \<longlongrightarrow> x) net"
+
−proof (intro topological_tendstoI)
+
−  fix S
+
−  assume S: "open S" "x \<in> S"
+
−  with \<open>x \<noteq> 0\<close> have "open (S - {0})" "x \<in> S - {0}"
+
−    by auto
+
−  from tendsto[THEN topological_tendstoD, OF this]
+
−  show "eventually (\<lambda>x. f x \<in> S) net"
+
−    by (rule eventually_rev_mp) (auto simp: ereal_real)
+
−qed
+
−
+
−lemma tendsto_ereal_realI:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f \<longlongrightarrow> x) net"
+
−  shows "((\<lambda>x. ereal (real_of_ereal (f x))) \<longlongrightarrow> x) net"
+
−proof (intro topological_tendstoI)
+
−  fix S
+
−  assume "open S" and "x \<in> S"
+
−  with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}"
+
−    by auto
+
−  from tendsto[THEN topological_tendstoD, OF this]
+
−  show "eventually (\<lambda>x. ereal (real_of_ereal (f x)) \<in> S) net"
+
−    by (elim eventually_mono) (auto simp: ereal_real)
+
−qed
+
−
+
−lemma ereal_mult_cancel_left:
+
−  fixes a b c :: ereal
+
−  shows "a * b = a * c \<longleftrightarrow> (\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c"
+
−  by (cases rule: ereal3_cases[of a b c]) (simp_all add: zero_less_mult_iff)
+
−
+
−lemma tendsto_add_ereal:
+
−  fixes x y :: ereal
+
−  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and y: "\<bar>y\<bar> \<noteq> \<infinity>"
+
−  assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
+
−  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
+
−proof -
+
−  from x obtain r where x': "x = ereal r" by (cases x) auto
+
−  with f have "((\<lambda>i. real_of_ereal (f i)) \<longlongrightarrow> r) F" by simp
+
−  moreover
+
−  from y obtain p where y': "y = ereal p" by (cases y) auto
+
−  with g have "((\<lambda>i. real_of_ereal (g i)) \<longlongrightarrow> p) F" by simp
+
−  ultimately have "((\<lambda>i. real_of_ereal (f i) + real_of_ereal (g i)) \<longlongrightarrow> r + p) F"
+
−    by (rule tendsto_add)
+
−  moreover
+
−  from eventually_finite[OF x f] eventually_finite[OF y g]
+
−  have "eventually (\<lambda>x. f x + g x = ereal (real_of_ereal (f x) + real_of_ereal (g x))) F"
+
−    by eventually_elim auto
+
−  ultimately show ?thesis
+
−    by (simp add: x' y' cong: filterlim_cong)
+
−qed
+
−
+
−lemma tendsto_add_ereal_nonneg:
+
−  fixes x y :: "ereal"
+
−  assumes "x \<noteq> -\<infinity>" "y \<noteq> -\<infinity>" "(f \<longlongrightarrow> x) F" "(g \<longlongrightarrow> y) F"
+
−  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
+
−proof cases
+
−  assume "x = \<infinity> \<or> y = \<infinity>"
+
−  moreover
+
−  { fix y :: ereal and f g :: "'a \<Rightarrow> ereal" assume "y \<noteq> -\<infinity>" "(f \<longlongrightarrow> \<infinity>) F" "(g \<longlongrightarrow> y) F"
+
−    then obtain y' where "-\<infinity> < y'" "y' < y"
+
−      using dense[of "-\<infinity>" y] by auto
+
−    have "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F"
+
−    proof (rule tendsto_sandwich)
+
−      have "\<forall>\<^sub>F x in F. y' < g x"
+
−        using order_tendstoD(1)[OF \<open>(g \<longlongrightarrow> y) F\<close> \<open>y' < y\<close>] by auto
+
−      then show "\<forall>\<^sub>F x in F. f x + y' \<le> f x + g x"
+
−        by eventually_elim (auto intro!: add_mono)
+
−      show "\<forall>\<^sub>F n in F. f n + g n \<le> \<infinity>" "((\<lambda>n. \<infinity>) \<longlongrightarrow> \<infinity>) F"
+
−        by auto
+
−      show "((\<lambda>x. f x + y') \<longlongrightarrow> \<infinity>) F"
+
−        using tendsto_cadd_ereal[of y' \<infinity> f F] \<open>(f \<longlongrightarrow> \<infinity>) F\<close> \<open>-\<infinity> < y'\<close> by auto
+
−    qed }
+
−  note this[of y f g] this[of x g f]
+
−  ultimately show ?thesis
+
−    using assms by (auto simp: add_ac)
+
−next
+
−  assume "\<not> (x = \<infinity> \<or> y = \<infinity>)"
+
−  with assms tendsto_add_ereal[of x y f F g]
+
−  show ?thesis
+
−    by auto
+
−qed
+
−
+
−lemma ereal_inj_affinity:
+
−  fixes m t :: ereal
+
−  assumes "\<bar>m\<bar> \<noteq> \<infinity>"
+
−    and "m \<noteq> 0"
+
−    and "\<bar>t\<bar> \<noteq> \<infinity>"
+
−  shows "inj_on (\<lambda>x. m * x + t) A"
+
−  using assms
+
−  by (cases rule: ereal2_cases[of m t])
+
−     (auto intro!: inj_onI simp: ereal_add_cancel_right ereal_mult_cancel_left)
+
−
+
−lemma ereal_PInfty_eq_plus[simp]:
+
−  fixes a b :: ereal
+
−  shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_MInfty_eq_plus[simp]:
+
−  fixes a b :: ereal
+
−  shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_less_divide_pos:
+
−  fixes x y :: ereal
+
−  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
+
−  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
+
−
+
−lemma ereal_divide_less_pos:
+
−  fixes x y z :: ereal
+
−  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
+
−  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
+
−
+
−lemma ereal_divide_eq:
+
−  fixes a b c :: ereal
+
−  shows "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
+
−  by (cases rule: ereal3_cases[of a b c])
+
−     (simp_all add: field_simps)
+
−
+
−lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal) \<noteq> -\<infinity>"
+
−  by (cases a) auto
+
−
+
−lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x"
+
−  by (cases x) auto
+
−
+
−lemma ereal_real':
+
−  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+
−  shows "ereal (real_of_ereal x) = x"
+
−  using assms by auto
+
−
+
−lemma real_ereal_id: "real_of_ereal \<circ> ereal = id"
+
−proof -
+
−  {
+
−    fix x
+
−    have "(real_of_ereal \<circ> ereal) x = id x"
+
−      by auto
+
−  }
+
−  then show ?thesis
+
−    using ext by blast
+
−qed
+
−
+
−lemma open_image_ereal: "open(UNIV-{ \<infinity> , (-\<infinity> :: ereal)})"
+
−  by (metis range_ereal open_ereal open_UNIV)
+
−
+
−lemma ereal_le_distrib:
+
−  fixes a b c :: ereal
+
−  shows "c * (a + b) \<le> c * a + c * b"
+
−  by (cases rule: ereal3_cases[of a b c])
+
−     (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
+
−
+
−lemma ereal_pos_distrib:
+
−  fixes a b c :: ereal
+
−  assumes "0 \<le> c"
+
−    and "c \<noteq> \<infinity>"
+
−  shows "c * (a + b) = c * a + c * b"
+
−  using assms
+
−  by (cases rule: ereal3_cases[of a b c])
+
−    (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
+
−
+
−lemma ereal_LimI_finite:
+
−  fixes x :: ereal
+
−  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+
−    and "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r"
+
−  shows "u \<longlonglongrightarrow> x"
+
−proof (rule topological_tendstoI, unfold eventually_sequentially)
+
−  obtain rx where rx: "x = ereal rx"
+
−    using assms by (cases x) auto
+
−  fix S
+
−  assume "open S" and "x \<in> S"
+
−  then have "open (ereal -` S)"
+
−    unfolding open_ereal_def by auto
+
−  with \<open>x \<in> S\<close> obtain r where "0 < r" and dist: "dist y rx < r \<Longrightarrow> ereal y \<in> S" for y
+
−    unfolding open_dist rx by auto
+
−  then obtain n
+
−    where upper: "u N < x + ereal r"
+
−      and lower: "x < u N + ereal r"
+
−      if "n \<le> N" for N
+
−    using assms(2)[of "ereal r"] by auto
+
−  show "\<exists>N. \<forall>n\<ge>N. u n \<in> S"
+
−  proof (safe intro!: exI[of _ n])
+
−    fix N
+
−    assume "n \<le> N"
+
−    from upper[OF this] lower[OF this] assms \<open>0 < r\<close>
+
−    have "u N \<notin> {\<infinity>,(-\<infinity>)}"
+
−      by auto
+
−    then obtain ra where ra_def: "(u N) = ereal ra"
+
−      by (cases "u N") auto
+
−    then have "rx < ra + r" and "ra < rx + r"
+
−      using rx assms \<open>0 < r\<close> lower[OF \<open>n \<le> N\<close>] upper[OF \<open>n \<le> N\<close>]
+
−      by auto
+
−    then have "dist (real_of_ereal (u N)) rx < r"
+
−      using rx ra_def
+
−      by (auto simp: dist_real_def abs_diff_less_iff field_simps)
+
−    from dist[OF this] show "u N \<in> S"
+
−      using \<open>u N  \<notin> {\<infinity>, -\<infinity>}\<close>
+
−      by (auto simp: ereal_real split: if_split_asm)
+
−  qed
+
−qed
+
−
+
−lemma tendsto_obtains_N:
+
−  assumes "f \<longlonglongrightarrow> f0"
+
−  assumes "open S"
+
−    and "f0 \<in> S"
+
−  obtains N where "\<forall>n\<ge>N. f n \<in> S"
+
−  using assms using tendsto_def
+
−  using lim_explicit[of f f0] assms by auto
+
−
+
−lemma ereal_LimI_finite_iff:
+
−  fixes x :: ereal
+
−  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+
−  shows "u \<longlonglongrightarrow> x \<longleftrightarrow> (\<forall>r. 0 < r \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r))"
+
−  (is "?lhs \<longleftrightarrow> ?rhs")
+
−proof
+
−  assume lim: "u \<longlonglongrightarrow> x"
+
−  {
+
−    fix r :: ereal
+
−    assume "r > 0"
+
−    then obtain N where "\<forall>n\<ge>N. u n \<in> {x - r <..< x + r}"
+
−       apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
+
−       using lim ereal_between[of x r] assms \<open>r > 0\<close>
+
−       apply auto
+
−       done
+
−    then have "\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r"
+
−      using ereal_minus_less[of r x]
+
−      by (cases r) auto
+
−  }
+
−  then show ?rhs
+
−    by auto
+
−next
+
−  assume ?rhs
+
−  then show "u \<longlonglongrightarrow> x"
+
−    using ereal_LimI_finite[of x] assms by auto
+
−qed
+
−
+
−lemma ereal_Limsup_uminus:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "Limsup net (\<lambda>x. - (f x)) = - Liminf net f"
+
−  unfolding Limsup_def Liminf_def ereal_SUP_uminus ereal_INF_uminus_eq ..
+
−
+
−lemma liminf_bounded_iff:
+
−  fixes x :: "nat \<Rightarrow> ereal"
+
−  shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)"
+
−  (is "?lhs \<longleftrightarrow> ?rhs")
+
−  unfolding le_Liminf_iff eventually_sequentially ..
+
−
+
−lemma Liminf_add_le:
+
−  fixes f g :: "_ \<Rightarrow> ereal"
+
−  assumes F: "F \<noteq> bot"
+
−  assumes ev: "eventually (\<lambda>x. 0 \<le> f x) F" "eventually (\<lambda>x. 0 \<le> g x) F"
+
−  shows "Liminf F f + Liminf F g \<le> Liminf F (\<lambda>x. f x + g x)"
+
−  unfolding Liminf_def
+
−proof (subst SUP_ereal_add_left[symmetric])
+
−  let ?F = "{P. eventually P F}"
+
−  let ?INF = "\<lambda>P g. Inf (g ` (Collect P))"
+
−  show "?F \<noteq> {}"
+
−    by (auto intro: eventually_True)
+
−  show "(SUP P\<in>?F. ?INF P g) \<noteq> - \<infinity>"
+
−    unfolding bot_ereal_def[symmetric] SUP_bot_conv INF_eq_bot_iff
+
−    by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def)
+
−  have "(SUP P\<in>?F. ?INF P f + (SUP P\<in>?F. ?INF P g)) \<le> (SUP P\<in>?F. (SUP P'\<in>?F. ?INF P f + ?INF P' g))"
+
−  proof (safe intro!: SUP_mono bexI[of _ "\<lambda>x. P x \<and> 0 \<le> f x" for P])
+
−    fix P let ?P' = "\<lambda>x. P x \<and> 0 \<le> f x"
+
−    assume "eventually P F"
+
−    with ev show "eventually ?P' F"
+
−      by eventually_elim auto
+
−    have "?INF P f + (SUP P\<in>?F. ?INF P g) \<le> ?INF ?P' f + (SUP P\<in>?F. ?INF P g)"
+
−      by (intro add_mono INF_mono) auto
+
−    also have "\<dots> = (SUP P'\<in>?F. ?INF ?P' f + ?INF P' g)"
+
−    proof (rule SUP_ereal_add_right[symmetric])
+
−      show "Inf (f ` {x. P x \<and> 0 \<le> f x}) \<noteq> - \<infinity>"
+
−        unfolding bot_ereal_def[symmetric] INF_eq_bot_iff
+
−        by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def)
+
−    qed fact
+
−    finally show "?INF P f + (SUP P\<in>?F. ?INF P g) \<le> (SUP P'\<in>?F. ?INF ?P' f + ?INF P' g)" .
+
−  qed
+
−  also have "\<dots> \<le> (SUP P\<in>?F. INF x\<in>Collect P. f x + g x)"
+
−  proof (safe intro!: SUP_least)
+
−    fix P Q assume *: "eventually P F" "eventually Q F"
+
−    show "?INF P f + ?INF Q g \<le> (SUP P\<in>?F. INF x\<in>Collect P. f x + g x)"
+
−    proof (rule SUP_upper2)
+
−      show "(\<lambda>x. P x \<and> Q x) \<in> ?F"
+
−        using * by (auto simp: eventually_conj)
+
−      show "?INF P f + ?INF Q g \<le> (INF x\<in>{x. P x \<and> Q x}. f x + g x)"
+
−        by (intro INF_greatest add_mono) (auto intro: INF_lower)
+
−    qed
+
−  qed
+
−  finally show "(SUP P\<in>?F. ?INF P f + (SUP P\<in>?F. ?INF P g)) \<le> (SUP P\<in>?F. INF x\<in>Collect P. f x + g x)" .
+
−qed
+
−
+
−lemma Sup_ereal_mult_right':
+
−  assumes nonempty: "Y \<noteq> {}"
+
−  and x: "x \<ge> 0"
+
−  shows "(SUP i\<in>Y. f i) * ereal x = (SUP i\<in>Y. f i * ereal x)" (is "?lhs = ?rhs")
+
−proof(cases "x = 0")
+
−  case True thus ?thesis by(auto simp add: nonempty zero_ereal_def[symmetric])
+
−next
+
−  case False
+
−  show ?thesis
+
−  proof(rule antisym)
+
−    show "?rhs \<le> ?lhs"
+
−      by(rule SUP_least)(simp add: ereal_mult_right_mono SUP_upper x)
+
−  next
+
−    have "?lhs / ereal x = (SUP i\<in>Y. f i) * (ereal x / ereal x)" by(simp only: ereal_times_divide_eq)
+
−    also have "\<dots> = (SUP i\<in>Y. f i)" using False by simp
+
−    also have "\<dots> \<le> ?rhs / x"
+
−    proof(rule SUP_least)
+
−      fix i
+
−      assume "i \<in> Y"
+
−      have "f i = f i * (ereal x / ereal x)" using False by simp
+
−      also have "\<dots> = f i * x / x" by(simp only: ereal_times_divide_eq)
+
−      also from \<open>i \<in> Y\<close> have "f i * x \<le> ?rhs" by(rule SUP_upper)
+
−      hence "f i * x / x \<le> ?rhs / x" using x False by simp
+
−      finally show "f i \<le> ?rhs / x" .
+
−    qed
+
−    finally have "(?lhs / x) * x \<le> (?rhs / x) * x"
+
−      by(rule ereal_mult_right_mono)(simp add: x)
+
−    also have "\<dots> = ?rhs" using False ereal_divide_eq mult.commute by force
+
−    also have "(?lhs / x) * x = ?lhs" using False ereal_divide_eq mult.commute by force
+
−    finally show "?lhs \<le> ?rhs" .
+
−  qed
+
−qed
+
−
+
−lemma Sup_ereal_mult_left':
+
−  "\<lbrakk> Y \<noteq> {}; x \<ge> 0 \<rbrakk> \<Longrightarrow> ereal x * (SUP i\<in>Y. f i) = (SUP i\<in>Y. ereal x * f i)"
+
−by(subst (1 2) mult.commute)(rule Sup_ereal_mult_right')
+
−
+
−lemma sup_continuous_add[order_continuous_intros]:
+
−  fixes f g :: "'a::complete_lattice \<Rightarrow> ereal"
+
−  assumes nn: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x" and cont: "sup_continuous f" "sup_continuous g"
+
−  shows "sup_continuous (\<lambda>x. f x + g x)"
+
−  unfolding sup_continuous_def
+
−proof safe
+
−  fix M :: "nat \<Rightarrow> 'a" assume "incseq M"
+
−  then show "f (SUP i. M i) + g (SUP i. M i) = (SUP i. f (M i) + g (M i))"
+
−    using SUP_ereal_add_pos[of "\<lambda>i. f (M i)" "\<lambda>i. g (M i)"] nn
+
−      cont[THEN sup_continuous_mono] cont[THEN sup_continuousD]
+
−    by (auto simp: mono_def)
+
−qed
+
−
+
−lemma sup_continuous_mult_right[order_continuous_intros]:
+
−  "0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> sup_continuous f \<Longrightarrow> sup_continuous (\<lambda>x. f x * c :: ereal)"
+
−  by (cases c) (auto simp: sup_continuous_def fun_eq_iff Sup_ereal_mult_right')
+
−
+
−lemma sup_continuous_mult_left[order_continuous_intros]:
+
−  "0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> sup_continuous f \<Longrightarrow> sup_continuous (\<lambda>x. c * f x :: ereal)"
+
−  using sup_continuous_mult_right[of c f] by (simp add: mult_ac)
+
−
+
−lemma sup_continuous_ereal_of_enat[order_continuous_intros]:
+
−  assumes f: "sup_continuous f" shows "sup_continuous (\<lambda>x. ereal_of_enat (f x))"
+
−  by (rule sup_continuous_compose[OF _ f])
+
−     (auto simp: sup_continuous_def ereal_of_enat_SUP)
+
−
+
−subsubsection \<open>Sums\<close>
+
−
+
−lemma sums_ereal_positive:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>i. 0 \<le> f i"
+
−  shows "f sums (SUP n. \<Sum>i<n. f i)"
+
−proof -
+
−  have "incseq (\<lambda>i. \<Sum>j=0..<i. f j)"
+
−    using add_mono[OF _ assms]
+
−    by (auto intro!: incseq_SucI)
+
−  from LIMSEQ_SUP[OF this]
+
−  show ?thesis unfolding sums_def
+
−    by (simp add: atLeast0LessThan)
+
−qed
+
−
+
−lemma summable_ereal_pos:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>i. 0 \<le> f i"
+
−  shows "summable f"
+
−  using sums_ereal_positive[of f, OF assms]
+
−  unfolding summable_def
+
−  by auto
+
−
+
−lemma sums_ereal: "(\<lambda>x. ereal (f x)) sums ereal x \<longleftrightarrow> f sums x"
+
−  unfolding sums_def by simp
+
−
+
−lemma suminf_ereal_eq_SUP:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>i. 0 \<le> f i"
+
−  shows "(\<Sum>x. f x) = (SUP n. \<Sum>i<n. f i)"
+
−  using sums_ereal_positive[of f, OF assms, THEN sums_unique]
+
−  by simp
+
−
+
−lemma suminf_bound:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x"
+
−    and pos: "\<And>n. 0 \<le> f n"
+
−  shows "suminf f \<le> x"
+
−proof (rule Lim_bounded)
+
−  have "summable f" using pos[THEN summable_ereal_pos] .
+
−  then show "(\<lambda>N. \<Sum>n<N. f n) \<longlonglongrightarrow> suminf f"
+
−    by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
+
−  show "\<forall>n\<ge>0. sum f {..<n} \<le> x"
+
−    using assms by auto
+
−qed
+
−
+
−lemma suminf_bound_add:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x"
+
−    and pos: "\<And>n. 0 \<le> f n"
+
−    and "y \<noteq> -\<infinity>"
+
−  shows "suminf f + y \<le> x"
+
−proof (cases y)
+
−  case (real r)
+
−  then have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y"
+
−    using assms by (simp add: ereal_le_minus)
+
−  then have "(\<Sum> n. f n) \<le> x - y"
+
−    using pos by (rule suminf_bound)
+
−  then show "(\<Sum> n. f n) + y \<le> x"
+
−    using assms real by (simp add: ereal_le_minus)
+
−qed (insert assms, auto)
+
−
+
−lemma suminf_upper:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>n. 0 \<le> f n"
+
−  shows "(\<Sum>n<N. f n) \<le> (\<Sum>n. f n)"
+
−  unfolding suminf_ereal_eq_SUP [OF assms]
+
−  by (auto intro: complete_lattice_class.SUP_upper)
+
−
+
−lemma suminf_0_le:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>n. 0 \<le> f n"
+
−  shows "0 \<le> (\<Sum>n. f n)"
+
−  using suminf_upper[of f 0, OF assms]
+
−  by simp
+
−
+
−lemma suminf_le_pos:
+
−  fixes f g :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>N. f N \<le> g N"
+
−    and "\<And>N. 0 \<le> f N"
+
−  shows "suminf f \<le> suminf g"
+
−proof (safe intro!: suminf_bound)
+
−  fix n
+
−  {
+
−    fix N
+
−    have "0 \<le> g N"
+
−      using assms(2,1)[of N] by auto
+
−  }
+
−  have "sum f {..<n} \<le> sum g {..<n}"
+
−    using assms by (auto intro: sum_mono)
+
−  also have "\<dots> \<le> suminf g"
+
−    using \<open>\<And>N. 0 \<le> g N\<close>
+
−    by (rule suminf_upper)
+
−  finally show "sum f {..<n} \<le> suminf g" .
+
−qed (rule assms(2))
+
−
+
−lemma suminf_half_series_ereal: "(\<Sum>n. (1/2 :: ereal) ^ Suc n) = 1"
+
−  using sums_ereal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]
+
−  by (simp add: one_ereal_def)
+
−
+
−lemma suminf_add_ereal:
+
−  fixes f g :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>i. 0 \<le> f i"
+
−    and "\<And>i. 0 \<le> g i"
+
−  shows "(\<Sum>i. f i + g i) = suminf f + suminf g"
+
−  apply (subst (1 2 3) suminf_ereal_eq_SUP)
+
−  unfolding sum.distrib
+
−  apply (intro assms add_nonneg_nonneg SUP_ereal_add_pos incseq_sumI sum_nonneg ballI)+
+
−  done
+
−
+
−lemma suminf_cmult_ereal:
+
−  fixes f g :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>i. 0 \<le> f i"
+
−    and "0 \<le> a"
+
−  shows "(\<Sum>i. a * f i) = a * suminf f"
+
−  by (auto simp: sum_ereal_right_distrib[symmetric] assms
+
−       ereal_zero_le_0_iff sum_nonneg suminf_ereal_eq_SUP
+
−       intro!: SUP_ereal_mult_left)
+
−
+
−lemma suminf_PInfty:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "\<And>i. 0 \<le> f i"
+
−    and "suminf f \<noteq> \<infinity>"
+
−  shows "f i \<noteq> \<infinity>"
+
−proof -
+
−  from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
+
−  have "(\<Sum>i<Suc i. f i) \<noteq> \<infinity>"
+
−    by auto
+
−  then show ?thesis
+
−    unfolding sum_Pinfty by simp
+
−qed
+
−
+
−lemma suminf_PInfty_fun:
+
−  assumes "\<And>i. 0 \<le> f i"
+
−    and "suminf f \<noteq> \<infinity>"
+
−  shows "\<exists>f'. f = (\<lambda>x. ereal (f' x))"
+
−proof -
+
−  have "\<forall>i. \<exists>r. f i = ereal r"
+
−  proof
+
−    fix i
+
−    show "\<exists>r. f i = ereal r"
+
−      using suminf_PInfty[OF assms] assms(1)[of i]
+
−      by (cases "f i") auto
+
−  qed
+
−  from choice[OF this] show ?thesis
+
−    by auto
+
−qed
+
−
+
−lemma summable_ereal:
+
−  assumes "\<And>i. 0 \<le> f i"
+
−    and "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>"
+
−  shows "summable f"
+
−proof -
+
−  have "0 \<le> (\<Sum>i. ereal (f i))"
+
−    using assms by (intro suminf_0_le) auto
+
−  with assms obtain r where r: "(\<Sum>i. ereal (f i)) = ereal r"
+
−    by (cases "\<Sum>i. ereal (f i)") auto
+
−  from summable_ereal_pos[of "\<lambda>x. ereal (f x)"]
+
−  have "summable (\<lambda>x. ereal (f x))"
+
−    using assms by auto
+
−  from summable_sums[OF this]
+
−  have "(\<lambda>x. ereal (f x)) sums (\<Sum>x. ereal (f x))"
+
−    by auto
+
−  then show "summable f"
+
−    unfolding r sums_ereal summable_def ..
+
−qed
+
−
+
−lemma suminf_ereal:
+
−  assumes "\<And>i. 0 \<le> f i"
+
−    and "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>"
+
−  shows "(\<Sum>i. ereal (f i)) = ereal (suminf f)"
+
−proof (rule sums_unique[symmetric])
+
−  from summable_ereal[OF assms]
+
−  show "(\<lambda>x. ereal (f x)) sums (ereal (suminf f))"
+
−    unfolding sums_ereal
+
−    using assms
+
−    by (intro summable_sums summable_ereal)
+
−qed
+
−
+
−lemma suminf_ereal_minus:
+
−  fixes f g :: "nat \<Rightarrow> ereal"
+
−  assumes ord: "\<And>i. g i \<le> f i" "\<And>i. 0 \<le> g i"
+
−    and fin: "suminf f \<noteq> \<infinity>" "suminf g \<noteq> \<infinity>"
+
−  shows "(\<Sum>i. f i - g i) = suminf f - suminf g"
+
−proof -
+
−  {
+
−    fix i
+
−    have "0 \<le> f i"
+
−      using ord[of i] by auto
+
−  }
+
−  moreover
+
−  from suminf_PInfty_fun[OF \<open>\<And>i. 0 \<le> f i\<close> fin(1)] obtain f' where [simp]: "f = (\<lambda>x. ereal (f' x))" ..
+
−  from suminf_PInfty_fun[OF \<open>\<And>i. 0 \<le> g i\<close> fin(2)] obtain g' where [simp]: "g = (\<lambda>x. ereal (g' x))" ..
+
−  {
+
−    fix i
+
−    have "0 \<le> f i - g i"
+
−      using ord[of i] by (auto simp: ereal_le_minus_iff)
+
−  }
+
−  moreover
+
−  have "suminf (\<lambda>i. f i - g i) \<le> suminf f"
+
−    using assms by (auto intro!: suminf_le_pos simp: field_simps)
+
−  then have "suminf (\<lambda>i. f i - g i) \<noteq> \<infinity>"
+
−    using fin by auto
+
−  ultimately show ?thesis
+
−    using assms \<open>\<And>i. 0 \<le> f i\<close>
+
−    apply simp
+
−    apply (subst (1 2 3) suminf_ereal)
+
−    apply (auto intro!: suminf_diff[symmetric] summable_ereal)
+
−    done
+
−qed
+
−
+
−lemma suminf_ereal_PInf [simp]: "(\<Sum>x. \<infinity>::ereal) = \<infinity>"
+
−proof -
+
−  have "(\<Sum>i<Suc 0. \<infinity>) \<le> (\<Sum>x. \<infinity>::ereal)"
+
−    by (rule suminf_upper) auto
+
−  then show ?thesis
+
−    by simp
+
−qed
+
−
+
−lemma summable_real_of_ereal:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes f: "\<And>i. 0 \<le> f i"
+
−    and fin: "(\<Sum>i. f i) \<noteq> \<infinity>"
+
−  shows "summable (\<lambda>i. real_of_ereal (f i))"
+
−proof (rule summable_def[THEN iffD2])
+
−  have "0 \<le> (\<Sum>i. f i)"
+
−    using assms by (auto intro: suminf_0_le)
+
−  with fin obtain r where r: "ereal r = (\<Sum>i. f i)"
+
−    by (cases "(\<Sum>i. f i)") auto
+
−  {
+
−    fix i
+
−    have "f i \<noteq> \<infinity>"
+
−      using f by (intro suminf_PInfty[OF _ fin]) auto
+
−    then have "\<bar>f i\<bar> \<noteq> \<infinity>"
+
−      using f[of i] by auto
+
−  }
+
−  note fin = this
+
−  have "(\<lambda>i. ereal (real_of_ereal (f i))) sums (\<Sum>i. ereal (real_of_ereal (f i)))"
+
−    using f
+
−    by (auto intro!: summable_ereal_pos simp: ereal_le_real_iff zero_ereal_def)
+
−  also have "\<dots> = ereal r"
+
−    using fin r by (auto simp: ereal_real)
+
−  finally show "\<exists>r. (\<lambda>i. real_of_ereal (f i)) sums r"
+
−    by (auto simp: sums_ereal)
+
−qed
+
−
+
−lemma suminf_SUP_eq:
+
−  fixes f :: "nat \<Rightarrow> nat \<Rightarrow> ereal"
+
−  assumes "\<And>i. incseq (\<lambda>n. f n i)"
+
−    and "\<And>n i. 0 \<le> f n i"
+
−  shows "(\<Sum>i. SUP n. f n i) = (SUP n. \<Sum>i. f n i)"
+
−proof -
+
−  {
+
−    fix n :: nat
+
−    have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
+
−      using assms
+
−      by (auto intro!: SUP_ereal_sum [symmetric])
+
−  }
+
−  note * = this
+
−  show ?thesis
+
−    using assms
+
−    apply (subst (1 2) suminf_ereal_eq_SUP)
+
−    unfolding *
+
−    apply (auto intro!: SUP_upper2)
+
−    apply (subst SUP_commute)
+
−    apply rule
+
−    done
+
−qed
+
−
+
−lemma suminf_sum_ereal:
+
−  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> ereal"
+
−  assumes nonneg: "\<And>i a. a \<in> A \<Longrightarrow> 0 \<le> f i a"
+
−  shows "(\<Sum>i. \<Sum>a\<in>A. f i a) = (\<Sum>a\<in>A. \<Sum>i. f i a)"
+
−proof (cases "finite A")
+
−  case True
+
−  then show ?thesis
+
−    using nonneg
+
−    by induct (simp_all add: suminf_add_ereal sum_nonneg)
+
−next
+
−  case False
+
−  then show ?thesis by simp
+
−qed
+
−
+
−lemma suminf_ereal_eq_0:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes nneg: "\<And>i. 0 \<le> f i"
+
−  shows "(\<Sum>i. f i) = 0 \<longleftrightarrow> (\<forall>i. f i = 0)"
+
−proof
+
−  assume "(\<Sum>i. f i) = 0"
+
−  {
+
−    fix i
+
−    assume "f i \<noteq> 0"
+
−    with nneg have "0 < f i"
+
−      by (auto simp: less_le)
+
−    also have "f i = (\<Sum>j. if j = i then f i else 0)"
+
−      by (subst suminf_finite[where N="{i}"]) auto
+
−    also have "\<dots> \<le> (\<Sum>i. f i)"
+
−      using nneg
+
−      by (auto intro!: suminf_le_pos)
+
−    finally have False
+
−      using \<open>(\<Sum>i. f i) = 0\<close> by auto
+
−  }
+
−  then show "\<forall>i. f i = 0"
+
−    by auto
+
−qed simp
+
−
+
−lemma suminf_ereal_offset_le:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes f: "\<And>i. 0 \<le> f i"
+
−  shows "(\<Sum>i. f (i + k)) \<le> suminf f"
+
−proof -
+
−  have "(\<lambda>n. \<Sum>i<n. f (i + k)) \<longlonglongrightarrow> (\<Sum>i. f (i + k))"
+
−    using summable_sums[OF summable_ereal_pos]
+
−    by (simp add: sums_def atLeast0LessThan f)
+
−  moreover have "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> (\<Sum>i. f i)"
+
−    using summable_sums[OF summable_ereal_pos]
+
−    by (simp add: sums_def atLeast0LessThan f)
+
−  then have "(\<lambda>n. \<Sum>i<n + k. f i) \<longlonglongrightarrow> (\<Sum>i. f i)"
+
−    by (rule LIMSEQ_ignore_initial_segment)
+
−  ultimately show ?thesis
+
−  proof (rule LIMSEQ_le, safe intro!: exI[of _ k])
+
−    fix n assume "k \<le> n"
+
−    have "(\<Sum>i<n. f (i + k)) = (\<Sum>i<n. (f \<circ> plus k) i)"
+
−      by (simp add: ac_simps)
+
−    also have "\<dots> = (\<Sum>i\<in>(plus k) ` {..<n}. f i)"
+
−      by (rule sum.reindex [symmetric]) simp
+
−    also have "\<dots> \<le> sum f {..<n + k}"
+
−      by (intro sum_mono2) (auto simp: f)
+
−    finally show "(\<Sum>i<n. f (i + k)) \<le> sum f {..<n + k}" .
+
−  qed
+
−qed
+
−
+
−lemma sums_suminf_ereal: "f sums x \<Longrightarrow> (\<Sum>i. ereal (f i)) = ereal x"
+
−  by (metis sums_ereal sums_unique)
+
−
+
−lemma suminf_ereal': "summable f \<Longrightarrow> (\<Sum>i. ereal (f i)) = ereal (\<Sum>i. f i)"
+
−  by (metis sums_ereal sums_unique summable_def)
+
−
+
−lemma suminf_ereal_finite: "summable f \<Longrightarrow> (\<Sum>i. ereal (f i)) \<noteq> \<infinity>"
+
−  by (auto simp: summable_def simp flip: sums_ereal sums_unique)
+
−
+
−lemma suminf_ereal_finite_neg:
+
−  assumes "summable f"
+
−  shows "(\<Sum>x. ereal (f x)) \<noteq> -\<infinity>"
+
−proof-
+
−  from assms obtain x where "f sums x" by blast
+
−  hence "(\<lambda>x. ereal (f x)) sums ereal x" by (simp add: sums_ereal)
+
−  from sums_unique[OF this] have "(\<Sum>x. ereal (f x)) = ereal x" ..
+
−  thus "(\<Sum>x. ereal (f x)) \<noteq> -\<infinity>" by simp_all
+
−qed
+
−
+
−lemma SUP_ereal_add_directed:
+
−  fixes f g :: "'a \<Rightarrow> ereal"
+
−  assumes nonneg: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> g i"
+
−  assumes directed: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. f i + g j \<le> f k + g k"
+
−  shows "(SUP i\<in>I. f i + g i) = (SUP i\<in>I. f i) + (SUP i\<in>I. g i)"
+
−proof cases
+
−  assume "I = {}" then show ?thesis
+
−    by (simp add: bot_ereal_def)
+
−next
+
−  assume "I \<noteq> {}"
+
−  show ?thesis
+
−  proof (rule antisym)
+
−    show "(SUP i\<in>I. f i + g i) \<le> (SUP i\<in>I. f i) + (SUP i\<in>I. g i)"
+
−      by (rule SUP_least; intro add_mono SUP_upper)
+
−  next
+
−    have "bot < (SUP i\<in>I. g i)"
+
−      using \<open>I \<noteq> {}\<close> nonneg(2) by (auto simp: bot_ereal_def less_SUP_iff)
+
−    then have "(SUP i\<in>I. f i) + (SUP i\<in>I. g i) = (SUP i\<in>I. f i + (SUP i\<in>I. g i))"
+
−      by (intro SUP_ereal_add_left[symmetric] \<open>I \<noteq> {}\<close>) auto
+
−    also have "\<dots> = (SUP i\<in>I. (SUP j\<in>I. f i + g j))"
+
−      using nonneg(1) \<open>I \<noteq> {}\<close> by (simp add: SUP_ereal_add_right)
+
−    also have "\<dots> \<le> (SUP i\<in>I. f i + g i)"
+
−      using directed by (intro SUP_least) (blast intro: SUP_upper2)
+
−    finally show "(SUP i\<in>I. f i) + (SUP i\<in>I. g i) \<le> (SUP i\<in>I. f i + g i)" .
+
−  qed
+
−qed
+
−
+
−lemma SUP_ereal_sum_directed:
+
−  fixes f g :: "'a \<Rightarrow> 'b \<Rightarrow> ereal"
+
−  assumes "I \<noteq> {}"
+
−  assumes directed: "\<And>N i j. N \<subseteq> A \<Longrightarrow> i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. \<forall>n\<in>N. f n i \<le> f n k \<and> f n j \<le> f n k"
+
−  assumes nonneg: "\<And>n i. i \<in> I \<Longrightarrow> n \<in> A \<Longrightarrow> 0 \<le> f n i"
+
−  shows "(SUP i\<in>I. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUP i\<in>I. f n i)"
+
−proof -
+
−  have "N \<subseteq> A \<Longrightarrow> (SUP i\<in>I. \<Sum>n\<in>N. f n i) = (\<Sum>n\<in>N. SUP i\<in>I. f n i)" for N
+
−  proof (induction N rule: infinite_finite_induct)
+
−    case (insert n N)
+
−    moreover have "(SUP i\<in>I. f n i + (\<Sum>l\<in>N. f l i)) = (SUP i\<in>I. f n i) + (SUP i\<in>I. \<Sum>l\<in>N. f l i)"
+
−    proof (rule SUP_ereal_add_directed)
+
−      fix i assume "i \<in> I" then show "0 \<le> f n i" "0 \<le> (\<Sum>l\<in>N. f l i)"
+
−        using insert by (auto intro!: sum_nonneg nonneg)
+
−    next
+
−      fix i j assume "i \<in> I" "j \<in> I"
+
−      from directed[OF \<open>insert n N \<subseteq> A\<close> this] guess k ..
+
−      then show "\<exists>k\<in>I. f n i + (\<Sum>l\<in>N. f l j) \<le> f n k + (\<Sum>l\<in>N. f l k)"
+
−        by (intro bexI[of _ k]) (auto intro!: add_mono sum_mono)
+
−    qed
+
−    ultimately show ?case
+
−      by simp
+
−  qed (simp_all add: SUP_constant \<open>I \<noteq> {}\<close>)
+
−  from this[of A] show ?thesis by simp
+
−qed
+
−
+
−lemma suminf_SUP_eq_directed:
+
−  fixes f :: "_ \<Rightarrow> nat \<Rightarrow> ereal"
+
−  assumes "I \<noteq> {}"
+
−  assumes directed: "\<And>N i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> finite N \<Longrightarrow> \<exists>k\<in>I. \<forall>n\<in>N. f i n \<le> f k n \<and> f j n \<le> f k n"
+
−  assumes nonneg: "\<And>n i. 0 \<le> f n i"
+
−  shows "(\<Sum>i. SUP n\<in>I. f n i) = (SUP n\<in>I. \<Sum>i. f n i)"
+
−proof (subst (1 2) suminf_ereal_eq_SUP)
+
−  show "\<And>n i. 0 \<le> f n i" "\<And>i. 0 \<le> (SUP n\<in>I. f n i)"
+
−    using \<open>I \<noteq> {}\<close> nonneg by (auto intro: SUP_upper2)
+
−  show "(SUP n. \<Sum>i<n. SUP n\<in>I. f n i) = (SUP n\<in>I. SUP j. \<Sum>i<j. f n i)"
+
−    apply (subst SUP_commute)
+
−    apply (subst SUP_ereal_sum_directed)
+
−    apply (auto intro!: assms simp: finite_subset)
+
−    done
+
−qed
+
−
+
−lemma ereal_dense3:
+
−  fixes x y :: ereal
+
−  shows "x < y \<Longrightarrow> \<exists>r::rat. x < real_of_rat r \<and> real_of_rat r < y"
+
−proof (cases x y rule: ereal2_cases, simp_all)
+
−  fix r q :: real
+
−  assume "r < q"
+
−  from Rats_dense_in_real[OF this] show "\<exists>x. r < real_of_rat x \<and> real_of_rat x < q"
+
−    by (fastforce simp: Rats_def)
+
−next
+
−  fix r :: real
+
−  show "\<exists>x. r < real_of_rat x" "\<exists>x. real_of_rat x < r"
+
−    using gt_ex[of r] lt_ex[of r] Rats_dense_in_real
+
−    by (auto simp: Rats_def)
+
−qed
+
−
+
−lemma continuous_within_ereal[intro, simp]: "x \<in> A \<Longrightarrow> continuous (at x within A) ereal"
+
−  using continuous_on_eq_continuous_within[of A ereal]
+
−  by (auto intro: continuous_on_ereal continuous_on_id)
+
−
+
−lemma ereal_open_uminus:
+
−  fixes S :: "ereal set"
+
−  assumes "open S"
+
−  shows "open (uminus ` S)"
+
−  using \<open>open S\<close>[unfolded open_generated_order]
+
−proof induct
+
−  have "range uminus = (UNIV :: ereal set)"
+
−    by (auto simp: image_iff ereal_uminus_eq_reorder)
+
−  then show "open (range uminus :: ereal set)"
+
−    by simp
+
−qed (auto simp add: image_Union image_Int)
+
−
+
−lemma ereal_uminus_complement:
+
−  fixes S :: "ereal set"
+
−  shows "uminus ` (- S) = - uminus ` S"
+
−  by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)
+
−
+
−lemma ereal_closed_uminus:
+
−  fixes S :: "ereal set"
+
−  assumes "closed S"
+
−  shows "closed (uminus ` S)"
+
−  using assms
+
−  unfolding closed_def ereal_uminus_complement[symmetric]
+
−  by (rule ereal_open_uminus)
+
−
+
−lemma ereal_open_affinity_pos:
+
−  fixes S :: "ereal set"
+
−  assumes "open S"
+
−    and m: "m \<noteq> \<infinity>" "0 < m"
+
−    and t: "\<bar>t\<bar> \<noteq> \<infinity>"
+
−  shows "open ((\<lambda>x. m * x + t) ` S)"
+
−proof -
+
−  have "open ((\<lambda>x. inverse m * (x + -t)) -` S)"
+
−    using m t
+
−    apply (intro open_vimage \<open>open S\<close>)
+
−    apply (intro continuous_at_imp_continuous_on ballI tendsto_cmult_ereal continuous_at[THEN iffD2]
+
−                 tendsto_ident_at tendsto_add_left_ereal)
+
−    apply auto
+
−    done
+
−  also have "(\<lambda>x. inverse m * (x + -t)) -` S = (\<lambda>x. (x - t) / m) -` S"
+
−    using m t by (auto simp: divide_ereal_def mult.commute minus_ereal_def
+
−                       simp flip: uminus_ereal.simps)
+
−  also have "(\<lambda>x. (x - t) / m) -` S = (\<lambda>x. m * x + t) ` S"
+
−    using m t
+
−    by (simp add: set_eq_iff image_iff)
+
−       (metis abs_ereal_less0 abs_ereal_uminus ereal_divide_eq ereal_eq_minus ereal_minus(7,8)
+
−              ereal_minus_less_minus ereal_mult_eq_PInfty ereal_uminus_uminus ereal_zero_mult)
+
−  finally show ?thesis .
+
−qed
+
−
+
−lemma ereal_open_affinity:
+
−  fixes S :: "ereal set"
+
−  assumes "open S"
+
−    and m: "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0"
+
−    and t: "\<bar>t\<bar> \<noteq> \<infinity>"
+
−  shows "open ((\<lambda>x. m * x + t) ` S)"
+
−proof cases
+
−  assume "0 < m"
+
−  then show ?thesis
+
−    using ereal_open_affinity_pos[OF \<open>open S\<close> _ _ t, of m] m
+
−    by auto
+
−next
+
−  assume "\<not> 0 < m" then
+
−  have "0 < -m"
+
−    using \<open>m \<noteq> 0\<close>
+
−    by (cases m) auto
+
−  then have m: "-m \<noteq> \<infinity>" "0 < -m"
+
−    using \<open>\<bar>m\<bar> \<noteq> \<infinity>\<close>
+
−    by (auto simp: ereal_uminus_eq_reorder)
+
−  from ereal_open_affinity_pos[OF ereal_open_uminus[OF \<open>open S\<close>] m t] show ?thesis
+
−    unfolding image_image by simp
+
−qed
+
−
+
−lemma open_uminus_iff:
+
−  fixes S :: "ereal set"
+
−  shows "open (uminus ` S) \<longleftrightarrow> open S"
+
−  using ereal_open_uminus[of S] ereal_open_uminus[of "uminus ` S"]
+
−  by auto
+
−
+
−lemma ereal_Liminf_uminus:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  shows "Liminf net (\<lambda>x. - (f x)) = - Limsup net f"
+
−  using ereal_Limsup_uminus[of _ "(\<lambda>x. - (f x))"] by auto
+
−
+
−lemma Liminf_PInfty:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes "\<not> trivial_limit net"
+
−  shows "(f \<longlongrightarrow> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
+
−  unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
+
−  using Liminf_le_Limsup[OF assms, of f]
+
−  by auto
+
−
+
−lemma Limsup_MInfty:
+
−  fixes f :: "'a \<Rightarrow> ereal"
+
−  assumes "\<not> trivial_limit net"
+
−  shows "(f \<longlongrightarrow> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
+
−  unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
+
−  using Liminf_le_Limsup[OF assms, of f]
+
−  by auto
+
−
+
−lemma convergent_ereal: \<comment> \<open>RENAME\<close>
+
−  fixes X :: "nat \<Rightarrow> 'a :: {complete_linorder,linorder_topology}"
+
−  shows "convergent X \<longleftrightarrow> limsup X = liminf X"
+
−  using tendsto_iff_Liminf_eq_Limsup[of sequentially]
+
−  by (auto simp: convergent_def)
+
−
+
−lemma limsup_le_liminf_real:
+
−  fixes X :: "nat \<Rightarrow> real" and L :: real
+
−  assumes 1: "limsup X \<le> L" and 2: "L \<le> liminf X"
+
−  shows "X \<longlonglongrightarrow> L"
+
−proof -
+
−  from 1 2 have "limsup X \<le> liminf X" by auto
+
−  hence 3: "limsup X = liminf X"
+
−    apply (subst eq_iff, rule conjI)
+
−    by (rule Liminf_le_Limsup, auto)
+
−  hence 4: "convergent (\<lambda>n. ereal (X n))"
+
−    by (subst convergent_ereal)
+
−  hence "limsup X = lim (\<lambda>n. ereal(X n))"
+
−    by (rule convergent_limsup_cl)
+
−  also from 1 2 3 have "limsup X = L" by auto
+
−  finally have "lim (\<lambda>n. ereal(X n)) = L" ..
+
−  hence "(\<lambda>n. ereal (X n)) \<longlonglongrightarrow> L"
+
−    apply (elim subst)
+
−    by (subst convergent_LIMSEQ_iff [symmetric], rule 4)
+
−  thus ?thesis by simp
+
−qed
+
−
+
−lemma liminf_PInfty:
+
−  fixes X :: "nat \<Rightarrow> ereal"
+
−  shows "X \<longlonglongrightarrow> \<infinity> \<longleftrightarrow> liminf X = \<infinity>"
+
−  by (metis Liminf_PInfty trivial_limit_sequentially)
+
−
+
−lemma limsup_MInfty:
+
−  fixes X :: "nat \<Rightarrow> ereal"
+
−  shows "X \<longlonglongrightarrow> -\<infinity> \<longleftrightarrow> limsup X = -\<infinity>"
+
−  by (metis Limsup_MInfty trivial_limit_sequentially)
+
−
+
−lemma SUP_eq_LIMSEQ:
+
−  assumes "mono f"
+
−  shows "(SUP n. ereal (f n)) = ereal x \<longleftrightarrow> f \<longlonglongrightarrow> x"
+
−proof
+
−  have inc: "incseq (\<lambda>i. ereal (f i))"
+
−    using \<open>mono f\<close> unfolding mono_def incseq_def by auto
+
−  {
+
−    assume "f \<longlonglongrightarrow> x"
+
−    then have "(\<lambda>i. ereal (f i)) \<longlonglongrightarrow> ereal x"
+
−      by auto
+
−    from SUP_Lim[OF inc this] show "(SUP n. ereal (f n)) = ereal x" .
+
−  next
+
−    assume "(SUP n. ereal (f n)) = ereal x"
+
−    with LIMSEQ_SUP[OF inc] show "f \<longlonglongrightarrow> x" by auto
+
−  }
+
−qed
+
−
+
−lemma liminf_ereal_cminus:
+
−  fixes f :: "nat \<Rightarrow> ereal"
+
−  assumes "c \<noteq> -\<infinity>"
+
−  shows "liminf (\<lambda>x. c - f x) = c - limsup f"
+
−proof (cases c)
+
−  case PInf
+
−  then show ?thesis
+
−    by (simp add: Liminf_const)
+
−next
+
−  case (real r)
+
−  then show ?thesis
+
−    unfolding liminf_SUP_INF limsup_INF_SUP
+
−    apply (subst INF_ereal_minus_right)
+
−    apply auto
+
−    apply (subst SUP_ereal_minus_right)
+
−    apply auto
+
−    done
+
−qed (insert \<open>c \<noteq> -\<infinity>\<close>, simp)
+
−
+
−
+
−subsubsection \<open>Continuity\<close>
+
−
+
−lemma continuous_at_of_ereal:
+
−  "\<bar>x0 :: ereal\<bar> \<noteq> \<infinity> \<Longrightarrow> continuous (at x0) real_of_ereal"
+
−  unfolding continuous_at
+
−  by (rule lim_real_of_ereal) (simp add: ereal_real)
+
−
+
−lemma nhds_ereal: "nhds (ereal r) = filtermap ereal (nhds r)"
+
−  by (simp add: filtermap_nhds_open_map open_ereal continuous_at_of_ereal)
+
−
+
−lemma at_ereal: "at (ereal r) = filtermap ereal (at r)"
+
−  by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)
+
−
+
−lemma at_left_ereal: "at_left (ereal r) = filtermap ereal (at_left r)"
+
−  by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)
+
−
+
−lemma at_right_ereal: "at_right (ereal r) = filtermap ereal (at_right r)"
+
−  by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)
+
−
+
−lemma
+
−  shows at_left_PInf: "at_left \<infinity> = filtermap ereal at_top"
+
−    and at_right_MInf: "at_right (-\<infinity>) = filtermap ereal at_bot"
+
−  unfolding filter_eq_iff eventually_filtermap eventually_at_top_dense eventually_at_bot_dense
+
−    eventually_at_left[OF ereal_less(5)] eventually_at_right[OF ereal_less(6)]
+
−  by (auto simp add: ereal_all_split ereal_ex_split)
+
−
+
−lemma ereal_tendsto_simps1:
+
−  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_left (ereal x)) \<longleftrightarrow> (f \<longlongrightarrow> y) (at_left x)"
+
−  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_right (ereal x)) \<longleftrightarrow> (f \<longlongrightarrow> y) (at_right x)"
+
−  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_left (\<infinity>::ereal)) \<longleftrightarrow> (f \<longlongrightarrow> y) at_top"
+
−  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_right (-\<infinity>::ereal)) \<longleftrightarrow> (f \<longlongrightarrow> y) at_bot"
+
−  unfolding tendsto_compose_filtermap at_left_ereal at_right_ereal at_left_PInf at_right_MInf
+
−  by (auto simp: filtermap_filtermap filtermap_ident)
+
−
+
−lemma ereal_tendsto_simps2:
+
−  "((ereal \<circ> f) \<longlongrightarrow> ereal a) F \<longleftrightarrow> (f \<longlongrightarrow> a) F"
+
−  "((ereal \<circ> f) \<longlongrightarrow> \<infinity>) F \<longleftrightarrow> (LIM x F. f x :> at_top)"
+
−  "((ereal \<circ> f) \<longlongrightarrow> -\<infinity>) F \<longleftrightarrow> (LIM x F. f x :> at_bot)"
+
−  unfolding tendsto_PInfty filterlim_at_top_dense tendsto_MInfty filterlim_at_bot_dense
+
−  using lim_ereal by (simp_all add: comp_def)
+
−
+
−lemma inverse_infty_ereal_tendsto_0: "inverse \<midarrow>\<infinity>\<rightarrow> (0::ereal)"
+
−proof -
+
−  have **: "((\<lambda>x. ereal (inverse x)) \<longlongrightarrow> ereal 0) at_infinity"
+
−    by (intro tendsto_intros tendsto_inverse_0)
+
−
+
−  show ?thesis
+
−    by (simp add: at_infty_ereal_eq_at_top tendsto_compose_filtermap[symmetric] comp_def)
+
−       (auto simp: eventually_at_top_linorder exI[of _ 1] zero_ereal_def at_top_le_at_infinity
+
−             intro!: filterlim_mono_eventually[OF **])
+
−qed
+
−
+
−lemma inverse_ereal_tendsto_pos:
+
−  fixes x :: ereal assumes "0 < x"
+
−  shows "inverse \<midarrow>x\<rightarrow> inverse x"
+
−proof (cases x)
+
−  case (real r)
+
−  with \<open>0 < x\<close> have **: "(\<lambda>x. ereal (inverse x)) \<midarrow>r\<rightarrow> ereal (inverse r)"
+
−    by (auto intro!: tendsto_inverse)
+
−  from real \<open>0 < x\<close> show ?thesis
+
−    by (auto simp: at_ereal tendsto_compose_filtermap[symmetric] eventually_at_filter
+
−             intro!: Lim_transform_eventually[OF **] t1_space_nhds)
+
−qed (insert \<open>0 < x\<close>, auto intro!: inverse_infty_ereal_tendsto_0)
+
−
+
−lemma inverse_ereal_tendsto_at_right_0: "(inverse \<longlongrightarrow> \<infinity>) (at_right (0::ereal))"
+
−  unfolding tendsto_compose_filtermap[symmetric] at_right_ereal zero_ereal_def
+
−  by (subst filterlim_cong[OF refl refl, where g="\<lambda>x. ereal (inverse x)"])
+
−     (auto simp: eventually_at_filter tendsto_PInfty_eq_at_top filterlim_inverse_at_top_right)
+
−
+
−lemmas ereal_tendsto_simps = ereal_tendsto_simps1 ereal_tendsto_simps2
+
−
+
−lemma continuous_at_iff_ereal:
+
−  fixes f :: "'a::t2_space \<Rightarrow> real"
+
−  shows "continuous (at x0 within s) f \<longleftrightarrow> continuous (at x0 within s) (ereal \<circ> f)"
+
−  unfolding continuous_within comp_def lim_ereal ..
+
−
+
−lemma continuous_on_iff_ereal:
+
−  fixes f :: "'a::t2_space => real"
+
−  assumes "open A"
+
−  shows "continuous_on A f \<longleftrightarrow> continuous_on A (ereal \<circ> f)"
+
−  unfolding continuous_on_def comp_def lim_ereal ..
+
−
+
−lemma continuous_on_real: "continuous_on (UNIV - {\<infinity>, -\<infinity>::ereal}) real_of_ereal"
+
−  using continuous_at_of_ereal continuous_on_eq_continuous_at open_image_ereal
+
−  by auto
+
−
+
−lemma continuous_on_iff_real:
+
−  fixes f :: "'a::t2_space \<Rightarrow> ereal"
+
−  assumes *: "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
+
−  shows "continuous_on A f \<longleftrightarrow> continuous_on A (real_of_ereal \<circ> f)"
+
−proof -
+
−  have "f ` A \<subseteq> UNIV - {\<infinity>, -\<infinity>}"
+
−    using assms by force
+
−  then have *: "continuous_on (f ` A) real_of_ereal"
+
−    using continuous_on_real by (simp add: continuous_on_subset)
+
−  have **: "continuous_on ((real_of_ereal \<circ> f) ` A) ereal"
+
−    by (intro continuous_on_ereal continuous_on_id)
+
−  {
+
−    assume "continuous_on A f"
+
−    then have "continuous_on A (real_of_ereal \<circ> f)"
+
−      apply (subst continuous_on_compose)
+
−      using *
+
−      apply auto
+
−      done
+
−  }
+
−  moreover
+
−  {
+
−    assume "continuous_on A (real_of_ereal \<circ> f)"
+
−    then have "continuous_on A (ereal \<circ> (real_of_ereal \<circ> f))"
+
−      apply (subst continuous_on_compose)
+
−      using **
+
−      apply auto
+
−      done
+
−    then have "continuous_on A f"
+
−      apply (subst continuous_on_cong[of _ A _ "ereal \<circ> (real_of_ereal \<circ> f)"])
+
−      using assms ereal_real
+
−      apply auto
+
−      done
+
−  }
+
−  ultimately show ?thesis
+
−    by auto
+
−qed
+
−
+
−lemma continuous_uminus_ereal [continuous_intros]: "continuous_on (A :: ereal set) uminus"
+
−  unfolding continuous_on_def
+
−  by (intro ballI tendsto_uminus_ereal[of "\<lambda>x. x::ereal"]) simp
+
−
+
−lemma ereal_uminus_atMost [simp]: "uminus ` {..(a::ereal)} = {-a..}"
+
−proof (intro equalityI subsetI)
+
−  fix x :: ereal assume "x \<in> {-a..}"
+
−  hence "-(-x) \<in> uminus ` {..a}" by (intro imageI) (simp add: ereal_uminus_le_reorder)
+
−  thus "x \<in> uminus ` {..a}" by simp
+
−qed auto
+
−
+
−lemma continuous_on_inverse_ereal [continuous_intros]:
+
−  "continuous_on {0::ereal ..} inverse"
+
−  unfolding continuous_on_def
+
−proof clarsimp
+
−  fix x :: ereal assume "0 \<le> x"
+
−  moreover have "at 0 within {0 ..} = at_right (0::ereal)"
+
−    by (auto simp: filter_eq_iff eventually_at_filter le_less)
+
−  moreover have "at x within {0 ..} = at x" if "0 < x"
+
−    using that by (intro at_within_nhd[of _ "{0<..}"]) auto
+
−  ultimately show "(inverse \<longlongrightarrow> inverse x) (at x within {0..})"
+
−    by (auto simp: le_less inverse_ereal_tendsto_at_right_0 inverse_ereal_tendsto_pos)
+
−qed
+
−
+
−lemma continuous_inverse_ereal_nonpos: "continuous_on ({..<0} :: ereal set) inverse"
+
−proof (subst continuous_on_cong[OF refl])
+
−  have "continuous_on {(0::ereal)<..} inverse"
+
−    by (rule continuous_on_subset[OF continuous_on_inverse_ereal]) auto
+
−  thus "continuous_on {..<(0::ereal)} (uminus \<circ> inverse \<circ> uminus)"
+
−    by (intro continuous_intros) simp_all
+
−qed simp
+
−
+
−lemma tendsto_inverse_ereal:
+
−  assumes "(f \<longlongrightarrow> (c :: ereal)) F"
+
−  assumes "eventually (\<lambda>x. f x \<ge> 0) F"
+
−  shows   "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse c) F"
+
−  by (cases "F = bot")
+
−     (auto intro!: tendsto_lowerbound assms
+
−                   continuous_on_tendsto_compose[OF continuous_on_inverse_ereal])
+
−
+
−
+
−subsubsection \<open>liminf and limsup\<close>
+
−
+
−lemma Limsup_ereal_mult_right:
+
−  assumes "F \<noteq> bot" "(c::real) \<ge> 0"
+
−  shows   "Limsup F (\<lambda>n. f n * ereal c) = Limsup F f * ereal c"
+
−proof (rule Limsup_compose_continuous_mono)
+
−  from assms show "continuous_on UNIV (\<lambda>a. a * ereal c)"
+
−    using tendsto_cmult_ereal[of "ereal c" "\<lambda>x. x" ]
+
−    by (force simp: continuous_on_def mult_ac)
+
−qed (insert assms, auto simp: mono_def ereal_mult_right_mono)
+
−
+
−lemma Liminf_ereal_mult_right:
+
−  assumes "F \<noteq> bot" "(c::real) \<ge> 0"
+
−  shows   "Liminf F (\<lambda>n. f n * ereal c) = Liminf F f * ereal c"
+
−proof (rule Liminf_compose_continuous_mono)
+
−  from assms show "continuous_on UNIV (\<lambda>a. a * ereal c)"
+
−    using tendsto_cmult_ereal[of "ereal c" "\<lambda>x. x" ]
+
−    by (force simp: continuous_on_def mult_ac)
+
−qed (insert assms, auto simp: mono_def ereal_mult_right_mono)
+
−
+
−lemma Liminf_ereal_mult_left:
+
−  assumes "F \<noteq> bot" "(c::real) \<ge> 0"
+
−    shows "Liminf F (\<lambda>n. ereal c * f n) = ereal c * Liminf F f"
+
−using Liminf_ereal_mult_right[OF assms] by (subst (1 2) mult.commute)
+
−
+
−lemma Limsup_ereal_mult_left:
+
−  assumes "F \<noteq> bot" "(c::real) \<ge> 0"
+
−  shows   "Limsup F (\<lambda>n. ereal c * f n) = ereal c * Limsup F f"
+
−  using Limsup_ereal_mult_right[OF assms] by (subst (1 2) mult.commute)
+
−
+
−lemma limsup_ereal_mult_right:
+
−  "(c::real) \<ge> 0 \<Longrightarrow> limsup (\<lambda>n. f n * ereal c) = limsup f * ereal c"
+
−  by (rule Limsup_ereal_mult_right) simp_all
+
−
+
−lemma limsup_ereal_mult_left:
+
−  "(c::real) \<ge> 0 \<Longrightarrow> limsup (\<lambda>n. ereal c * f n) = ereal c * limsup f"
+
−  by (subst (1 2) mult.commute, rule limsup_ereal_mult_right) simp_all
+
−
+
−lemma Limsup_add_ereal_right:
+
−  "F \<noteq> bot \<Longrightarrow> abs c \<noteq> \<infinity> \<Longrightarrow> Limsup F (\<lambda>n. g n + (c :: ereal)) = Limsup F g + c"
+
−  by (rule Limsup_compose_continuous_mono) (auto simp: mono_def add_mono continuous_on_def)
+
−
+
−lemma Limsup_add_ereal_left:
+
−  "F \<noteq> bot \<Longrightarrow> abs c \<noteq> \<infinity> \<Longrightarrow> Limsup F (\<lambda>n. (c :: ereal) + g n) = c + Limsup F g"
+
−  by (subst (1 2) add.commute) (rule Limsup_add_ereal_right)
+
−
+
−lemma Liminf_add_ereal_right:
+
−  "F \<noteq> bot \<Longrightarrow> abs c \<noteq> \<infinity> \<Longrightarrow> Liminf F (\<lambda>n. g n + (c :: ereal)) = Liminf F g + c"
+
−  by (rule Liminf_compose_continuous_mono) (auto simp: mono_def add_mono continuous_on_def)
+
−
+
−lemma Liminf_add_ereal_left:
+
−  "F \<noteq> bot \<Longrightarrow> abs c \<noteq> \<infinity> \<Longrightarrow> Liminf F (\<lambda>n. (c :: ereal) + g n) = c + Liminf F g"
+
−  by (subst (1 2) add.commute) (rule Liminf_add_ereal_right)
+
−
+
−lemma
+
−  assumes "F \<noteq> bot"
+
−  assumes nonneg: "eventually (\<lambda>x. f x \<ge> (0::ereal)) F"
+
−  shows   Liminf_inverse_ereal: "Liminf F (\<lambda>x. inverse (f x)) = inverse (Limsup F f)"
+
−  and     Limsup_inverse_ereal: "Limsup F (\<lambda>x. inverse (f x)) = inverse (Liminf F f)"
+
−proof -
+
−  define inv where [abs_def]: "inv x = (if x \<le> 0 then \<infinity> else inverse x)" for x :: ereal
+
−  have "continuous_on ({..0} \<union> {0..}) inv" unfolding inv_def
+
−    by (intro continuous_on_If) (auto intro!: continuous_intros)
+
−  also have "{..0} \<union> {0..} = (UNIV :: ereal set)" by auto
+
−  finally have cont: "continuous_on UNIV inv" .
+
−  have antimono: "antimono inv" unfolding inv_def antimono_def
+
−    by (auto intro!: ereal_inverse_antimono)
+
−
+
−  have "Liminf F (\<lambda>x. inverse (f x)) = Liminf F (\<lambda>x. inv (f x))" using nonneg
+
−    by (auto intro!: Liminf_eq elim!: eventually_mono simp: inv_def)
+
−  also have "... = inv (Limsup F f)"
+
−    by (simp add: assms(1) Liminf_compose_continuous_antimono[OF cont antimono])
+
−  also from assms have "Limsup F f \<ge> 0" by (intro le_Limsup) simp_all
+
−  hence "inv (Limsup F f) = inverse (Limsup F f)" by (simp add: inv_def)
+
−  finally show "Liminf F (\<lambda>x. inverse (f x)) = inverse (Limsup F f)" .
+
−
+
−  have "Limsup F (\<lambda>x. inverse (f x)) = Limsup F (\<lambda>x. inv (f x))" using nonneg
+
−    by (auto intro!: Limsup_eq elim!: eventually_mono simp: inv_def)
+
−  also have "... = inv (Liminf F f)"
+
−    by (simp add: assms(1) Limsup_compose_continuous_antimono[OF cont antimono])
+
−  also from assms have "Liminf F f \<ge> 0" by (intro Liminf_bounded) simp_all
+
−  hence "inv (Liminf F f) = inverse (Liminf F f)" by (simp add: inv_def)
+
−  finally show "Limsup F (\<lambda>x. inverse (f x)) = inverse (Liminf F f)" .
+
−qed
+
−
+
−lemma ereal_diff_le_mono_left: "\<lbrakk> x \<le> z; 0 \<le> y \<rbrakk> \<Longrightarrow> x - y \<le> (z :: ereal)"
+
−by(cases x y z rule: ereal3_cases) simp_all
+
−
+
−lemma neg_0_less_iff_less_erea [simp]: "0 < - a \<longleftrightarrow> (a :: ereal) < 0"
+
−by(cases a) simp_all
+
−
+
−lemma not_infty_ereal: "\<bar>x\<bar> \<noteq> \<infinity> \<longleftrightarrow> (\<exists>x'. x = ereal x')"
+
−by(cases x) simp_all
+
−
+
−lemma neq_PInf_trans: fixes x y :: ereal shows "\<lbrakk> y \<noteq> \<infinity>; x \<le> y \<rbrakk> \<Longrightarrow> x \<noteq> \<infinity>"
+
−by auto
+
−
+
−lemma mult_2_ereal: "ereal 2 * x = x + x"
+
−by(cases x) simp_all
+
−
+
−lemma ereal_diff_le_self: "0 \<le> y \<Longrightarrow> x - y \<le> (x :: ereal)"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ereal_le_add_self: "0 \<le> y \<Longrightarrow> x \<le> x + (y :: ereal)"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ereal_le_add_self2: "0 \<le> y \<Longrightarrow> x \<le> y + (x :: ereal)"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ereal_le_add_mono1: "\<lbrakk> x \<le> y; 0 \<le> (z :: ereal) \<rbrakk> \<Longrightarrow> x \<le> y + z"
+
−using add_mono by fastforce
+
−
+
−lemma ereal_le_add_mono2: "\<lbrakk> x \<le> z; 0 \<le> (y :: ereal) \<rbrakk> \<Longrightarrow> x \<le> y + z"
+
−using add_mono by fastforce
+
−
+
−lemma ereal_diff_nonpos:
+
−  fixes a b :: ereal shows "\<lbrakk> a \<le> b; a = \<infinity> \<Longrightarrow> b \<noteq> \<infinity>; a = -\<infinity> \<Longrightarrow> b \<noteq> -\<infinity> \<rbrakk> \<Longrightarrow> a - b \<le> 0"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma minus_ereal_0 [simp]: "x - ereal 0 = x"
+
−by(simp flip: zero_ereal_def)
+
−
+
−lemma ereal_diff_eq_0_iff: fixes a b :: ereal
+
−  shows "(\<bar>a\<bar> = \<infinity> \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity>) \<Longrightarrow> a - b = 0 \<longleftrightarrow> a = b"
+
−by(cases a b rule: ereal2_cases) simp_all
+
−
+
−lemma SUP_ereal_eq_0_iff_nonneg:
+
−  fixes f :: "_ \<Rightarrow> ereal" and A
+
−  assumes nonneg: "\<forall>x\<in>A. f x \<ge> 0"
+
−  and A:"A \<noteq> {}"
+
−  shows "(SUP x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)" (is "?lhs \<longleftrightarrow> ?rhs")
+
−proof(intro iffI ballI)
+
−  fix x
+
−  assume "?lhs" "x \<in> A"
+
−  from \<open>x \<in> A\<close> have "f x \<le> (SUP x\<in>A. f x)" by(rule SUP_upper)
+
−  with \<open>?lhs\<close> show "f x = 0" using nonneg \<open>x \<in> A\<close> by auto
+
−qed (simp add: A)
+
−
+
−lemma ereal_divide_le_posI:
+
−  fixes x y z :: ereal
+
−  shows "x > 0 \<Longrightarrow> z \<noteq> - \<infinity> \<Longrightarrow> z \<le> x * y \<Longrightarrow> z / x \<le> y"
+
−by (cases rule: ereal3_cases[of x y z])(auto simp: field_simps split: if_split_asm)
+
−
+
−lemma add_diff_eq_ereal: fixes x y z :: ereal
+
−  shows "x + (y - z) = x + y - z"
+
−by(cases x y z rule: ereal3_cases) simp_all
+
−
+
−lemma ereal_diff_gr0:
+
−  fixes a b :: ereal shows "a < b \<Longrightarrow> 0 < b - a"
+
−  by (cases rule: ereal2_cases[of a b]) auto
+
−
+
−lemma ereal_minus_minus: fixes x y z :: ereal shows
+
−  "(\<bar>y\<bar> = \<infinity> \<Longrightarrow> \<bar>z\<bar> \<noteq> \<infinity>) \<Longrightarrow> x - (y - z) = x + z - y"
+
−by(cases x y z rule: ereal3_cases) simp_all
+
−
+
−lemma diff_add_eq_ereal: fixes a b c :: ereal shows "a - b + c = a + c - b"
+
−by(cases a b c rule: ereal3_cases) simp_all
+
−
+
−lemma diff_diff_commute_ereal: fixes x y z :: ereal shows "x - y - z = x - z - y"
+
−by(cases x y z rule: ereal3_cases) simp_all
+
−
+
−lemma ereal_diff_eq_MInfty_iff: fixes x y :: ereal shows "x - y = -\<infinity> \<longleftrightarrow> x = -\<infinity> \<and> y \<noteq> -\<infinity> \<or> y = \<infinity> \<and> \<bar>x\<bar> \<noteq> \<infinity>"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma ereal_diff_add_inverse: fixes x y :: ereal shows "\<bar>x\<bar> \<noteq> \<infinity> \<Longrightarrow> x + y - x = y"
+
−by(cases x y rule: ereal2_cases) simp_all
+
−
+
−lemma tendsto_diff_ereal:
+
−  fixes x y :: ereal
+
−  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and y: "\<bar>y\<bar> \<noteq> \<infinity>"
+
−  assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
+
−  shows "((\<lambda>x. f x - g x) \<longlongrightarrow> x - y) F"
+
−proof -
+
−  from x obtain r where x': "x = ereal r" by (cases x) auto
+
−  with f have "((\<lambda>i. real_of_ereal (f i)) \<longlongrightarrow> r) F" by simp
+
−  moreover
+
−  from y obtain p where y': "y = ereal p" by (cases y) auto
+
−  with g have "((\<lambda>i. real_of_ereal (g i)) \<longlongrightarrow> p) F" by simp
+
−  ultimately have "((\<lambda>i. real_of_ereal (f i) - real_of_ereal (g i)) \<longlongrightarrow> r - p) F"
+
−    by (rule tendsto_diff)
+
−  moreover
+
−  from eventually_finite[OF x f] eventually_finite[OF y g]
+
−  have "eventually (\<lambda>x. f x - g x = ereal (real_of_ereal (f x) - real_of_ereal (g x))) F"
+
−    by eventually_elim auto
+
−  ultimately show ?thesis
+
−    by (simp add: x' y' cong: filterlim_cong)
+
−qed
+
−
+
−lemma continuous_on_diff_ereal:
+
−  "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>) \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> \<bar>g x\<bar> \<noteq> \<infinity>) \<Longrightarrow> continuous_on A (\<lambda>z. f z - g z::ereal)"
+
−  apply (auto simp: continuous_on_def)
+
−  apply (intro tendsto_diff_ereal)
+
−  apply metis+
+
−  done
+
−
+
−
+
−subsubsection \<open>Tests for code generator\<close>
+
−
+
−text \<open>A small list of simple arithmetic expressions.\<close>
+
−
+
−value "- \<infinity> :: ereal"
+
−value "\<bar>-\<infinity>\<bar> :: ereal"
+
−value "4 + 5 / 4 - ereal 2 :: ereal"
+
−value "ereal 3 < \<infinity>"
+
−value "real_of_ereal (\<infinity>::ereal) = 0"
+
−
+
−end