src/HOL/Library/Extended_Real.thy
author hoelzl
Mon, 04 May 2015 17:35:31 +0200
changeset 60172 423273355b55
parent 60060 3630ecde4e7c
child 60352 d46de31a50c4
permissions -rw-r--r--
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
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(*  Title:      HOL/Library/Extended_Real.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Robert Himmelmann, TU München
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    Author:     Armin Heller, TU München
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    Author:     Bogdan Grechuk, University of Edinburgh
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*)
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section {* Extended real number line *}
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theory Extended_Real
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imports Complex_Main Extended_Nat Liminf_Limsup Order_Continuity
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begin
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text {*
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This should be part of @{theory Extended_Nat} or @{theory Order_Continuity}, but then the
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AFP-entry @{text "Jinja_Thread"} fails, as it does overload certain named from @{theory Complex_Main}.
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*}
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lemma continuous_at_left_imp_sup_continuous:
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  fixes f :: "'a \<Rightarrow> 'a::{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
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qed
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lemma sup_continuous_at_left:
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  fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_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)
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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 ----> 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)) ----> 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 \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_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 \<Rightarrow> 'a::{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
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qed
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lemma inf_continuous_at_right:
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  fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_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 ----> 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)) ----> 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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lemma inf_continuous_iff_at_right:
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  fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_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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instantiation enat :: linorder_topology
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begin
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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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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
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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
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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)
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  ultimately show "open A"
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    by simp
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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 enat_ord_code(1)[symmetric] simp del: enat_ord_code(1))
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    then show ?case
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      by auto
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  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
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  qed auto
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qed
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text {*
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For more lemmas about the extended real numbers go to
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  @{file "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"}
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*}
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subsection {* Definition and basic properties *}
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datatype ereal = ereal real | PInfty | MInfty
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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 ..
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end
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instantiation ereal :: infinity
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begin
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definition "(\<infinity>::ereal) = PInfty"
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instance ..
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end
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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
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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)"
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    and MInfty_neq_ereal[simp]: "ereal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> ereal r"
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    and PInfty_neq_ereal[simp]: "ereal r \<noteq> \<infinity>" "\<infinity> \<noteq> ereal r"
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    and PInfty_cases[simp]: "(case \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = y"
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    and MInfty_cases[simp]: "(case - \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = z"
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  by (simp_all add: infinity_ereal_def)
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declare
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  PInfty_eq_infinity[code_post]
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  MInfty_eq_minfinity[code_post]
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lemma [code_unfold]:
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  "\<infinity> = PInfty"
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  "- PInfty = MInfty"
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  by simp_all
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lemma inj_ereal[simp]: "inj_on ereal A"
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  unfolding inj_on_def by auto
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lemma ereal_cases[cases type: ereal]:
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  obtains (real) r where "x = ereal r"
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    | (PInf) "x = \<infinity>"
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    | (MInf) "x = -\<infinity>"
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  using assms by (cases x) auto
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lemmas ereal2_cases = ereal_cases[case_product ereal_cases]
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lemmas ereal3_cases = ereal2_cases[case_product ereal_cases]
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lemma ereal_all_split: "\<And>P. (\<forall>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<and> (\<forall>x. P (ereal x)) \<and> P (-\<infinity>)"
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  by (metis ereal_cases)
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lemma ereal_ex_split: "\<And>P. (\<exists>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<or> (\<exists>x. P (ereal x)) \<or> P (-\<infinity>)"
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  by (metis ereal_cases)
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lemma ereal_uminus_eq_iff[simp]:
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  fixes a b :: ereal
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  shows "-a = -b \<longleftrightarrow> a = b"
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  by (cases rule: ereal2_cases[of a b]) simp_all
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instantiation ereal :: real_of
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begin
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function real_ereal :: "ereal \<Rightarrow> real" where
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  "real_ereal (ereal r) = r"
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| "real_ereal \<infinity> = 0"
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| "real_ereal (-\<infinity>) = 0"
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  by (auto intro: ereal_cases)
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termination by default (rule wf_empty)
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instance ..
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end
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lemma real_of_ereal[simp]:
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  "real (- x :: ereal) = - (real x)"
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  by (cases x) simp_all
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lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}"
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proof safe
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  fix x
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  assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
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  then show "x = -\<infinity>"
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    by (cases x) auto
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qed auto
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lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
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proof safe
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  fix x :: ereal
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  show "x \<in> range uminus"
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    by (intro image_eqI[of _ _ "-x"]) auto
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qed auto
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instantiation ereal :: abs
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begin
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08594daabcd9 tuned proofs;
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function abs_ereal where
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  "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
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| "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
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| "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
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by (auto intro: ereal_cases)
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termination proof qed (rule wf_empty)
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08594daabcd9 tuned proofs;
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instance ..
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   296
end
3fdbc7d5b525 use abs_extreal
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   297
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lemma abs_eq_infinity_cases[elim!]:
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  fixes x :: ereal
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  assumes "\<bar>x\<bar> = \<infinity>"
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  obtains "x = \<infinity>" | "x = -\<infinity>"
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   302
  using assms by (cases x) auto
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3fdbc7d5b525 use abs_extreal
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   303
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   304
lemma abs_neq_infinity_cases[elim!]:
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  fixes x :: ereal
08594daabcd9 tuned proofs;
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   306
  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
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   307
  obtains r where "x = ereal r"
08594daabcd9 tuned proofs;
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   308
  using assms by (cases x) auto
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   309
08594daabcd9 tuned proofs;
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lemma abs_ereal_uminus[simp]:
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  fixes x :: ereal
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   312
  shows "\<bar>- x\<bar> = \<bar>x\<bar>"
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3fdbc7d5b525 use abs_extreal
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   313
  by (cases x) auto
3fdbc7d5b525 use abs_extreal
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   314
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   315
lemma ereal_infinity_cases:
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  fixes a :: ereal
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   317
  shows "a \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
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   318
  by auto
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3fdbc7d5b525 use abs_extreal
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   319
50104
de19856feb54 move theorems to be more generally useable
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   320
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   321
subsubsection "Addition"
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54408
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   323
instantiation ereal :: "{one,comm_monoid_add,zero_neq_one}"
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begin
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   325
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   326
definition "0 = ereal 0"
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   327
definition "1 = ereal 1"
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   328
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   329
function plus_ereal where
53873
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   330
  "ereal r + ereal p = ereal (r + p)"
08594daabcd9 tuned proofs;
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   331
| "\<infinity> + a = (\<infinity>::ereal)"
08594daabcd9 tuned proofs;
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   332
| "a + \<infinity> = (\<infinity>::ereal)"
08594daabcd9 tuned proofs;
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   333
| "ereal r + -\<infinity> = - \<infinity>"
08594daabcd9 tuned proofs;
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   334
| "-\<infinity> + ereal p = -(\<infinity>::ereal)"
08594daabcd9 tuned proofs;
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   335
| "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
41973
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   336
proof -
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   337
  case (goal1 P x)
53873
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   338
  then obtain a b where "x = (a, b)"
08594daabcd9 tuned proofs;
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   339
    by (cases x) auto
53374
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   340
  with goal1 show P
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   341
   by (cases rule: ereal2_cases[of a b]) auto
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qed auto
53374
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   343
termination by default (rule wf_empty)
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   344
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
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   345
lemma Infty_neq_0[simp]:
43923
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   346
  "(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)"
ab93d0190a5d add ereal to typeclass infinity
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   347
  "-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)"
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   348
  by (simp_all add: zero_ereal_def)
41973
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   349
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   350
lemma ereal_eq_0[simp]:
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   351
  "ereal r = 0 \<longleftrightarrow> r = 0"
cedb5cb948fd Rename extreal => ereal
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diff changeset
   352
  "0 = ereal r \<longleftrightarrow> r = 0"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   353
  unfolding zero_ereal_def by simp_all
41973
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   354
54416
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   355
lemma ereal_eq_1[simp]:
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   356
  "ereal r = 1 \<longleftrightarrow> r = 1"
7fb88ed6ff3c better support for enat and ereal conversions
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diff changeset
   357
  "1 = ereal r \<longleftrightarrow> r = 1"
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   358
  unfolding one_ereal_def by simp_all
7fb88ed6ff3c better support for enat and ereal conversions
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diff changeset
   359
41973
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   360
instance
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   361
proof
47082
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parents: 45934
diff changeset
   362
  fix a b c :: ereal
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   363
  show "0 + a = a"
43920
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hoelzl
parents: 43138
diff changeset
   364
    by (cases a) (simp_all add: zero_ereal_def)
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   365
  show "a + b = b + a"
43920
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hoelzl
parents: 43138
diff changeset
   366
    by (cases rule: ereal2_cases[of a b]) simp_all
47082
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wenzelm
parents: 45934
diff changeset
   367
  show "a + b + c = a + (b + c)"
43920
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hoelzl
parents: 43138
diff changeset
   368
    by (cases rule: ereal3_cases[of a b c]) simp_all
54408
67dec4ccaabd equation when indicator function equals 0 or 1
hoelzl
parents: 53873
diff changeset
   369
  show "0 \<noteq> (1::ereal)"
67dec4ccaabd equation when indicator function equals 0 or 1
hoelzl
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diff changeset
   370
    by (simp add: one_ereal_def zero_ereal_def)
41973
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diff changeset
   371
qed
53873
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wenzelm
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diff changeset
   372
41973
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   373
end
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hoelzl
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   374
60060
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Andreas Lochbihler
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   375
lemma ereal_0_plus [simp]: "ereal 0 + x = x"
3630ecde4e7c more lemmas about ereal
Andreas Lochbihler
parents: 59679
diff changeset
   376
  and plus_ereal_0 [simp]: "x + ereal 0 = x"
3630ecde4e7c more lemmas about ereal
Andreas Lochbihler
parents: 59679
diff changeset
   377
by(simp_all add: zero_ereal_def[symmetric])
3630ecde4e7c more lemmas about ereal
Andreas Lochbihler
parents: 59679
diff changeset
   378
51351
dd1dd470690b generalized lemmas in Extended_Real_Limits
hoelzl
parents: 51340
diff changeset
   379
instance ereal :: numeral ..
dd1dd470690b generalized lemmas in Extended_Real_Limits
hoelzl
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diff changeset
   380
43920
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parents: 43138
diff changeset
   381
lemma real_of_ereal_0[simp]: "real (0::ereal) = 0"
58042
ffa9e39763e3 introduce real_of typeclass for real :: 'a => real
hoelzl
parents: 57512
diff changeset
   382
  unfolding zero_ereal_def by simp
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   383
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   384
lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   385
  unfolding zero_ereal_def abs_ereal.simps by simp
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
   386
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   387
lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   388
  by (simp add: zero_ereal_def)
41973
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hoelzl
parents:
diff changeset
   389
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   390
lemma ereal_uminus_zero_iff[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   391
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   392
  shows "-a = 0 \<longleftrightarrow> a = 0"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   393
  by (cases a) simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   394
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   395
lemma ereal_plus_eq_PInfty[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   396
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   397
  shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   398
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   399
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   400
lemma ereal_plus_eq_MInfty[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   401
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   402
  shows "a + b = -\<infinity> \<longleftrightarrow> (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   403
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   404
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   405
lemma ereal_add_cancel_left:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   406
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   407
  assumes "a \<noteq> -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   408
  shows "a + b = a + c \<longleftrightarrow> a = \<infinity> \<or> b = c"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   409
  using assms by (cases rule: ereal3_cases[of a b c]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   410
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   411
lemma ereal_add_cancel_right:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   412
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   413
  assumes "a \<noteq> -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   414
  shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   415
  using assms by (cases rule: ereal3_cases[of a b c]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   416
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   417
lemma ereal_real: "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   418
  by (cases x) simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   419
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   420
lemma real_of_ereal_add:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   421
  fixes a b :: ereal
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   422
  shows "real (a + b) =
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   423
    (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   424
  by (cases rule: ereal2_cases[of a b]) auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   425
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   426
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   427
subsubsection "Linear order on @{typ ereal}"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   428
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   429
instantiation ereal :: linorder
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   430
begin
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   431
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   432
function less_ereal
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   433
where
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   434
  "   ereal x < ereal y     \<longleftrightarrow> x < y"
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   435
| "(\<infinity>::ereal) < a           \<longleftrightarrow> False"
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   436
| "         a < -(\<infinity>::ereal) \<longleftrightarrow> False"
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   437
| "ereal x    < \<infinity>           \<longleftrightarrow> True"
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   438
| "        -\<infinity> < ereal r     \<longleftrightarrow> True"
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   439
| "        -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   440
proof -
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   441
  case (goal1 P x)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   442
  then obtain a b where "x = (a,b)" by (cases x) auto
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   443
  with goal1 show P by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   444
qed simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   445
termination by (relation "{}") simp
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   446
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   447
definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   448
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   449
lemma ereal_infty_less[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   450
  fixes x :: ereal
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   451
  shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   452
    "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   453
  by (cases x, simp_all) (cases x, simp_all)
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   454
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   455
lemma ereal_infty_less_eq[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   456
  fixes x :: ereal
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   457
  shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   458
    and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   459
  by (auto simp add: less_eq_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   460
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   461
lemma ereal_less[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   462
  "ereal r < 0 \<longleftrightarrow> (r < 0)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   463
  "0 < ereal r \<longleftrightarrow> (0 < r)"
54416
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   464
  "ereal r < 1 \<longleftrightarrow> (r < 1)"
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   465
  "1 < ereal r \<longleftrightarrow> (1 < r)"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   466
  "0 < (\<infinity>::ereal)"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   467
  "-(\<infinity>::ereal) < 0"
54416
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   468
  by (simp_all add: zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   469
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   470
lemma ereal_less_eq[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   471
  "x \<le> (\<infinity>::ereal)"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   472
  "-(\<infinity>::ereal) \<le> x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   473
  "ereal r \<le> ereal p \<longleftrightarrow> r \<le> p"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   474
  "ereal r \<le> 0 \<longleftrightarrow> r \<le> 0"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   475
  "0 \<le> ereal r \<longleftrightarrow> 0 \<le> r"
54416
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   476
  "ereal r \<le> 1 \<longleftrightarrow> r \<le> 1"
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   477
  "1 \<le> ereal r \<longleftrightarrow> 1 \<le> r"
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   478
  by (auto simp add: less_eq_ereal_def zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   479
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   480
lemma ereal_infty_less_eq2:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   481
  "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   482
  "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   483
  by simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   484
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   485
instance
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   486
proof
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   487
  fix x y z :: ereal
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   488
  show "x \<le> x"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   489
    by (cases x) simp_all
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   490
  show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   491
    by (cases rule: ereal2_cases[of x y]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   492
  show "x \<le> y \<or> y \<le> x "
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   493
    by (cases rule: ereal2_cases[of x y]) auto
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   494
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   495
    assume "x \<le> y" "y \<le> x"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   496
    then show "x = y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   497
      by (cases rule: ereal2_cases[of x y]) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   498
  }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   499
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   500
    assume "x \<le> y" "y \<le> z"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   501
    then show "x \<le> z"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   502
      by (cases rule: ereal3_cases[of x y z]) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   503
  }
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   504
qed
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   505
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   506
end
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   507
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   508
lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> ereal z < y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   509
  using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   510
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 52729
diff changeset
   511
instance ereal :: dense_linorder
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   512
  by default (blast dest: ereal_dense2)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   513
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   514
instance ereal :: ordered_ab_semigroup_add
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   515
proof
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   516
  fix a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   517
  assume "a \<le> b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   518
  then show "c + a \<le> c + b"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   519
    by (cases rule: ereal3_cases[of a b c]) auto
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   520
qed
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   521
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   522
lemma real_of_ereal_positive_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   523
  fixes x y :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   524
  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real x \<le> real y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   525
  by (cases rule: ereal2_cases[of x y]) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   526
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   527
lemma ereal_MInfty_lessI[intro, simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   528
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   529
  shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   530
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   531
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   532
lemma ereal_less_PInfty[intro, simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   533
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   534
  shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   535
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   536
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   537
lemma ereal_less_ereal_Ex:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   538
  fixes a b :: ereal
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   539
  shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   540
  by (cases x) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   541
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   542
lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < ereal (real n))"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   543
proof (cases x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   544
  case (real r)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   545
  then show ?thesis
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents: 41979
diff changeset
   546
    using reals_Archimedean2[of r] by simp
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   547
qed simp_all
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   548
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   549
lemma ereal_add_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   550
  fixes a b c d :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   551
  assumes "a \<le> b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   552
    and "c \<le> d"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   553
  shows "a + c \<le> b + d"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   554
  using assms
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   555
  apply (cases a)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   556
  apply (cases rule: ereal3_cases[of b c d], auto)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   557
  apply (cases rule: ereal3_cases[of b c d], auto)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   558
  done
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   559
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   560
lemma ereal_minus_le_minus[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   561
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   562
  shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   563
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   564
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   565
lemma ereal_minus_less_minus[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   566
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   567
  shows "- a < - b \<longleftrightarrow> b < a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   568
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   569
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   570
lemma ereal_le_real_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   571
  "x \<le> real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   572
  by (cases y) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   573
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   574
lemma real_le_ereal_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   575
  "real y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   576
  by (cases y) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   577
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   578
lemma ereal_less_real_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   579
  "x < real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   580
  by (cases y) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   581
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   582
lemma real_less_ereal_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   583
  "real y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   584
  by (cases y) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   585
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   586
lemma real_of_ereal_pos:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   587
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   588
  shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   589
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   590
lemmas real_of_ereal_ord_simps =
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   591
  ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   592
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   593
lemma abs_ereal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x"
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   594
  by (cases x) auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   595
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   596
lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x"
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   597
  by (cases x) auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   598
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   599
lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   600
  by (cases x) auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   601
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   602
lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   603
  by (cases x) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   604
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   605
lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   606
  by (cases x) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   607
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   608
lemma zero_less_real_of_ereal:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   609
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   610
  shows "0 < real x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   611
  by (cases x) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   612
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   613
lemma ereal_0_le_uminus_iff[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   614
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   615
  shows "0 \<le> - a \<longleftrightarrow> a \<le> 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   616
  by (cases rule: ereal2_cases[of a]) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   617
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   618
lemma ereal_uminus_le_0_iff[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   619
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   620
  shows "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   621
  by (cases rule: ereal2_cases[of a]) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   622
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   623
lemma ereal_add_strict_mono:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   624
  fixes a b c d :: ereal
56993
e5366291d6aa introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents: 56927
diff changeset
   625
  assumes "a \<le> b"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   626
    and "0 \<le> a"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   627
    and "a \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   628
    and "c < d"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   629
  shows "a + c < b + d"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   630
  using assms
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   631
  by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   632
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   633
lemma ereal_less_add:
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   634
  fixes a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   635
  shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   636
  by (cases rule: ereal2_cases[of b c]) auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   637
54416
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   638
lemma ereal_add_nonneg_eq_0_iff:
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   639
  fixes a b :: ereal
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   640
  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   641
  by (cases a b rule: ereal2_cases) auto
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   642
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   643
lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   644
  by auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   645
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   646
lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   647
  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   648
59452
2538b2c51769 ereal: tuned proofs concerning continuity and suprema
hoelzl
parents: 59425
diff changeset
   649
lemma ereal_less_uminus_reorder: "a < - b \<longleftrightarrow> b < - (a::ereal)"
2538b2c51769 ereal: tuned proofs concerning continuity and suprema
hoelzl
parents: 59425
diff changeset
   650
  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
2538b2c51769 ereal: tuned proofs concerning continuity and suprema
hoelzl
parents: 59425
diff changeset
   651
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   652
lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   653
  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   654
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   655
lemmas ereal_uminus_reorder =
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   656
  ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   657
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   658
lemma ereal_bot:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   659
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   660
  assumes "\<And>B. x \<le> ereal B"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   661
  shows "x = - \<infinity>"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   662
proof (cases x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   663
  case (real r)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   664
  with assms[of "r - 1"] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   665
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   666
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   667
  case PInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   668
  with assms[of 0] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   669
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   670
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   671
  case MInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   672
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   673
    by simp
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   674
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   675
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   676
lemma ereal_top:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   677
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   678
  assumes "\<And>B. x \<ge> ereal B"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   679
  shows "x = \<infinity>"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   680
proof (cases x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   681
  case (real r)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   682
  with assms[of "r + 1"] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   683
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   684
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   685
  case MInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   686
  with assms[of 0] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   687
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   688
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   689
  case PInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   690
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   691
    by simp
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   692
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   693
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   694
lemma
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   695
  shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   696
    and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   697
  by (simp_all add: min_def max_def)
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   698
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   699
lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   700
  by (auto simp: zero_ereal_def)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   701
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   702
lemma
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   703
  fixes f :: "nat \<Rightarrow> ereal"
54416
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   704
  shows ereal_incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   705
    and ereal_decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   706
  unfolding decseq_def incseq_def by auto
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   707
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   708
lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))"
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   709
  unfolding incseq_def by auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   710
56537
01caba82e1d2 made ereal_add_nonneg_nonneg a simp rule
nipkow
parents: 56536
diff changeset
   711
lemma ereal_add_nonneg_nonneg[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   712
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   713
  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   714
  using add_mono[of 0 a 0 b] by simp
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   715
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   716
lemma image_eqD: "f ` A = B \<Longrightarrow> \<forall>x\<in>A. f x \<in> B"
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   717
  by auto
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   718
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   719
lemma incseq_setsumI:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   720
  fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}"
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   721
  assumes "\<And>i. 0 \<le> f i"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   722
  shows "incseq (\<lambda>i. setsum f {..< i})"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   723
proof (intro incseq_SucI)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   724
  fix n
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   725
  have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   726
    using assms by (rule add_left_mono)
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   727
  then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   728
    by auto
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   729
qed
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   730
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   731
lemma incseq_setsumI2:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   732
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   733
  assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   734
  shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   735
  using assms
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   736
  unfolding incseq_def by (auto intro: setsum_mono)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   737
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   738
lemma setsum_ereal[simp]: "(\<Sum>x\<in>A. ereal (f x)) = ereal (\<Sum>x\<in>A. f x)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   739
proof (cases "finite A")
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   740
  case True
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   741
  then show ?thesis by induct auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   742
next
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   743
  case False
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   744
  then show ?thesis by simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   745
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   746
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   747
lemma setsum_Pinfty:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   748
  fixes f :: "'a \<Rightarrow> ereal"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   749
  shows "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> finite P \<and> (\<exists>i\<in>P. f i = \<infinity>)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   750
proof safe
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   751
  assume *: "setsum f P = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   752
  show "finite P"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   753
  proof (rule ccontr)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   754
    assume "\<not> finite P"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   755
    with * show False
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   756
      by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   757
  qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   758
  show "\<exists>i\<in>P. f i = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   759
  proof (rule ccontr)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   760
    assume "\<not> ?thesis"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   761
    then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   762
      by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   763
    with `finite P` have "setsum f P \<noteq> \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   764
      by induct auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   765
    with * show False
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   766
      by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   767
  qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   768
next
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   769
  fix i
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   770
  assume "finite P" and "i \<in> P" and "f i = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   771
  then show "setsum f P = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   772
  proof induct
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   773
    case (insert x A)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   774
    show ?case using insert by (cases "x = i") auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   775
  qed simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   776
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   777
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   778
lemma setsum_Inf:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   779
  fixes f :: "'a \<Rightarrow> ereal"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   780
  shows "\<bar>setsum f A\<bar> = \<infinity> \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   781
proof
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   782
  assume *: "\<bar>setsum f A\<bar> = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   783
  have "finite A"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   784
    by (rule ccontr) (insert *, auto)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   785
  moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   786
  proof (rule ccontr)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   787
    assume "\<not> ?thesis"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   788
    then have "\<forall>i\<in>A. \<exists>r. f i = ereal r"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   789
      by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   790
    from bchoice[OF this] obtain r where "\<forall>x\<in>A. f x = ereal (r x)" ..
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   791
    with * show False
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   792
      by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   793
  qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   794
  ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   795
    by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   796
next
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   797
  assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   798
  then obtain i where "finite A" "i \<in> A" and "\<bar>f i\<bar> = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   799
    by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   800
  then show "\<bar>setsum f A\<bar> = \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   801
  proof induct
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   802
    case (insert j A)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   803
    then show ?case
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   804
      by (cases rule: ereal3_cases[of "f i" "f j" "setsum f A"]) auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   805
  qed simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   806
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   807
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   808
lemma setsum_real_of_ereal:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   809
  fixes f :: "'i \<Rightarrow> ereal"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   810
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   811
  shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   812
proof -
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   813
  have "\<forall>x\<in>S. \<exists>r. f x = ereal r"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   814
  proof
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   815
    fix x
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   816
    assume "x \<in> S"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   817
    from assms[OF this] show "\<exists>r. f x = ereal r"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   818
      by (cases "f x") auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   819
  qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   820
  from bchoice[OF this] obtain r where "\<forall>x\<in>S. f x = ereal (r x)" ..
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   821
  then show ?thesis
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   822
    by simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   823
qed
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   824
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   825
lemma setsum_ereal_0:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   826
  fixes f :: "'a \<Rightarrow> ereal"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   827
  assumes "finite A"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   828
    and "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   829
  shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   830
proof
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   831
  assume "setsum f A = 0" with assms show "\<forall>i\<in>A. f i = 0"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   832
  proof (induction A)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   833
    case (insert a A)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   834
    then have "f a = 0 \<and> (\<Sum>a\<in>A. f a) = 0"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   835
      by (subst ereal_add_nonneg_eq_0_iff[symmetric]) (simp_all add: setsum_nonneg)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   836
    with insert show ?case
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   837
      by simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   838
  qed simp
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   839
qed auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   840
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   841
subsubsection "Multiplication"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   842
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   843
instantiation ereal :: "{comm_monoid_mult,sgn}"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   844
begin
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   845
51351
dd1dd470690b generalized lemmas in Extended_Real_Limits
hoelzl
parents: 51340
diff changeset
   846
function sgn_ereal :: "ereal \<Rightarrow> ereal" where
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   847
  "sgn (ereal r) = ereal (sgn r)"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   848
| "sgn (\<infinity>::ereal) = 1"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   849
| "sgn (-\<infinity>::ereal) = -1"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   850
by (auto intro: ereal_cases)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   851
termination by default (rule wf_empty)
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
   852
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   853
function times_ereal where
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   854
  "ereal r * ereal p = ereal (r * p)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   855
| "ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   856
| "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   857
| "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   858
| "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   859
| "(\<infinity>::ereal) * \<infinity> = \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   860
| "-(\<infinity>::ereal) * \<infinity> = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   861
| "(\<infinity>::ereal) * -\<infinity> = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   862
| "-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   863
proof -
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   864
  case (goal1 P x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   865
  then obtain a b where "x = (a, b)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   866
    by (cases x) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   867
  with goal1 show P
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   868
    by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   869
qed simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   870
termination by (relation "{}") simp
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   871
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   872
instance
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   873
proof
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   874
  fix a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   875
  show "1 * a = a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   876
    by (cases a) (simp_all add: one_ereal_def)
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   877
  show "a * b = b * a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   878
    by (cases rule: ereal2_cases[of a b]) simp_all
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   879
  show "a * b * c = a * (b * c)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   880
    by (cases rule: ereal3_cases[of a b c])
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   881
       (simp_all add: zero_ereal_def zero_less_mult_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   882
qed
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   883
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   884
end
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   885
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   886
lemma one_not_le_zero_ereal[simp]: "\<not> (1 \<le> (0::ereal))"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   887
  by (simp add: one_ereal_def zero_ereal_def)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
   888
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   889
lemma real_ereal_1[simp]: "real (1::ereal) = 1"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   890
  unfolding one_ereal_def by simp
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   891
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   892
lemma real_of_ereal_le_1:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   893
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   894
  shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   895
  by (cases a) (auto simp: one_ereal_def)
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   896
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   897
lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   898
  unfolding one_ereal_def by simp
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
   899
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   900
lemma ereal_mult_zero[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   901
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   902
  shows "a * 0 = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   903
  by (cases a) (simp_all add: zero_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   904
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   905
lemma ereal_zero_mult[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   906
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   907
  shows "0 * a = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   908
  by (cases a) (simp_all add: zero_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   909
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   910
lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   911
  by (simp add: zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   912
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   913
lemma ereal_times[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   914
  "1 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   915
  "1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 1"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   916
  by (auto simp add: times_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   917
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   918
lemma ereal_plus_1[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   919
  "1 + ereal r = ereal (r + 1)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   920
  "ereal r + 1 = ereal (r + 1)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   921
  "1 + -(\<infinity>::ereal) = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   922
  "-(\<infinity>::ereal) + 1 = -\<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   923
  unfolding one_ereal_def by auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   924
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   925
lemma ereal_zero_times[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   926
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   927
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   928
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   929
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   930
lemma ereal_mult_eq_PInfty[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   931
  "a * b = (\<infinity>::ereal) \<longleftrightarrow>
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   932
    (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   933
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   934
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   935
lemma ereal_mult_eq_MInfty[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   936
  "a * b = -(\<infinity>::ereal) \<longleftrightarrow>
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   937
    (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   938
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   939
54416
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   940
lemma ereal_abs_mult: "\<bar>x * y :: ereal\<bar> = \<bar>x\<bar> * \<bar>y\<bar>"
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   941
  by (cases x y rule: ereal2_cases) (auto simp: abs_mult)
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
   942
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   943
lemma ereal_0_less_1[simp]: "0 < (1::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   944
  by (simp_all add: zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   945
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   946
lemma ereal_mult_minus_left[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   947
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   948
  shows "-a * b = - (a * b)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   949
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   950
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   951
lemma ereal_mult_minus_right[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   952
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   953
  shows "a * -b = - (a * b)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   954
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   955
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   956
lemma ereal_mult_infty[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   957
  "a * (\<infinity>::ereal) = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   958
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   959
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   960
lemma ereal_infty_mult[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   961
  "(\<infinity>::ereal) * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   962
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   963
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   964
lemma ereal_mult_strict_right_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   965
  assumes "a < b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   966
    and "0 < c"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   967
    and "c < (\<infinity>::ereal)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   968
  shows "a * c < b * c"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   969
  using assms
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   970
  by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   971
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   972
lemma ereal_mult_strict_left_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   973
  "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> c * a < c * b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   974
  using ereal_mult_strict_right_mono
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57447
diff changeset
   975
  by (simp add: mult.commute[of c])
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   976
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   977
lemma ereal_mult_right_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   978
  fixes a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   979
  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   980
  using assms
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   981
  apply (cases "c = 0")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   982
  apply simp
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   983
  apply (cases rule: ereal3_cases[of a b c])
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   984
  apply (auto simp: zero_le_mult_iff)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   985
  done
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   986
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   987
lemma ereal_mult_left_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   988
  fixes a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   989
  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   990
  using ereal_mult_right_mono
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57447
diff changeset
   991
  by (simp add: mult.commute[of c])
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   992
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   993
lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   994
  by (simp add: one_ereal_def zero_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   995
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   996
lemma ereal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: ereal)"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56248
diff changeset
   997
  by (cases rule: ereal2_cases[of a b]) auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   998
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   999
lemma ereal_right_distrib:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1000
  fixes r a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1001
  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1002
  by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1003
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1004
lemma ereal_left_distrib:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1005
  fixes r a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1006
  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1007
  by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1008
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1009
lemma ereal_mult_le_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1010
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1011
  shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1012
  by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_le_0_iff)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1013
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1014
lemma ereal_zero_le_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1015
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1016
  shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1017
  by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1018
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1019
lemma ereal_mult_less_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1020
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1021
  shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1022
  by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_less_0_iff)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1023
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1024
lemma ereal_zero_less_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1025
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1026
  shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1027
  by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1028
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1029
lemma ereal_left_mult_cong:
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1030
  fixes a b c :: ereal
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
  1031
  shows  "c = d \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> a * c = b * d"
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1032
  by (cases "c = 0") simp_all
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1033
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
  1034
lemma ereal_right_mult_cong: 
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
  1035
  fixes a b c :: ereal
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
  1036
  shows "c = d \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> c * a = d * b"
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
  1037
  by (cases "c = 0") simp_all
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1038
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1039
lemma ereal_distrib:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1040
  fixes a b c :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1041
  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1042
    and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1043
    and "\<bar>c\<bar> \<noteq> \<infinity>"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1044
  shows "(a + b) * c = a * c + b * c"
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1045
  using assms
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1046
  by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1047
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1048
lemma numeral_eq_ereal [simp]: "numeral w = ereal (numeral w)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1049
  apply (induct w rule: num_induct)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1050
  apply (simp only: numeral_One one_ereal_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1051
  apply (simp only: numeral_inc ereal_plus_1)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1052
  done
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1053
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
  1054
lemma setsum_ereal_right_distrib:
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
  1055
  fixes f :: "'a \<Rightarrow> ereal"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
  1056
  shows "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> r * setsum f A = (\<Sum>n\<in>A. r * f n)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
  1057
  by (induct A rule: infinite_finite_induct)  (auto simp: ereal_right_distrib setsum_nonneg)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58881
diff changeset
  1058
59002
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
  1059
lemma setsum_ereal_left_distrib:
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
  1060
  "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> setsum f A * r = (\<Sum>n\<in>A. f n * r :: ereal)"
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
  1061
  using setsum_ereal_right_distrib[of A f r] by (simp add: mult_ac)
2c8b2fb54b88 cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents: 59000
diff changeset
  1062
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1063
lemma ereal_le_epsilon:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1064
  fixes x y :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1065
  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1066
  shows "x \<le> y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1067
proof -
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1068
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1069
    assume a: "\<exists>r. y = ereal r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1070
    then obtain r where r_def: "y = ereal r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1071
      by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1072
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1073
      assume "x = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1074
      then have ?thesis by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1075
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1076
    moreover
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1077
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1078
      assume "x \<noteq> -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1079
      then obtain p where p_def: "x = ereal p"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1080
      using a assms[rule_format, of 1]
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1081
        by (cases x) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1082
      {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1083
        fix e
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1084
        have "0 < e \<longrightarrow> p \<le> r + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1085
          using assms[rule_format, of "ereal e"] p_def r_def by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1086
      }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1087
      then have "p \<le> r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1088
        apply (subst field_le_epsilon)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1089
        apply auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1090
        done
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1091
      then have ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1092
        using r_def p_def by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1093
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1094
    ultimately have ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1095
      by blast
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1096
  }
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1097
  moreover
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1098
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1099
    assume "y = -\<infinity> | y = \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1100
    then have ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1101
      using assms[rule_format, of 1] by (cases x) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1102
  }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1103
  ultimately show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1104
    by (cases y) auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1105
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1106
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1107
lemma ereal_le_epsilon2:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1108
  fixes x y :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1109
  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + ereal e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1110
  shows "x \<le> y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1111
proof -
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1112
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1113
    fix e :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1114
    assume "e > 0"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1115
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1116
      assume "e = \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1117
      then have "x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1118
        by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1119
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1120
    moreover
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1121
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1122
      assume "e \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1123
      then obtain r where "e = ereal r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1124
        using `e > 0` by (cases e) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1125
      then have "x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1126
        using assms[rule_format, of r] `e>0` by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1127
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1128
    ultimately have "x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1129
      by blast
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1130
  }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1131
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1132
    using ereal_le_epsilon by auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1133
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1134
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1135
lemma ereal_le_real:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1136
  fixes x y :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1137
  assumes "\<forall>z. x \<le> ereal z \<longrightarrow> y \<le> ereal z"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1138
  shows "y \<le> x"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1139
  by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1140
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1141
lemma setprod_ereal_0:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1142
  fixes f :: "'a \<Rightarrow> ereal"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1143
  shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. f i = 0)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1144
proof (cases "finite A")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1145
  case True
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1146
  then show ?thesis by (induct A) auto
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1147
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1148
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1149
  then show ?thesis by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1150
qed
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1151
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1152
lemma setprod_ereal_pos:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1153
  fixes f :: "'a \<Rightarrow> ereal"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1154
  assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1155
  shows "0 \<le> (\<Prod>i\<in>I. f i)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1156
proof (cases "finite I")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1157
  case True
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1158
  from this pos show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1159
    by induct auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1160
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1161
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1162
  then show ?thesis by simp
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1163
qed
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1164
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1165
lemma setprod_PInf:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1166
  fixes f :: "'a \<Rightarrow> ereal"
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1167
  assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1168
  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)"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1169
proof (cases "finite I")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1170
  case True
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1171
  from this assms show ?thesis
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1172
  proof (induct I)
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1173
    case (insert i I)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1174
    then have pos: "0 \<le> f i" "0 \<le> setprod f I"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1175
      by (auto intro!: setprod_ereal_pos)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1176
    from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1177
      by auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1178
    also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1179
      using setprod_ereal_pos[of I f] pos
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1180
      by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1181
    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)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1182
      using insert by (auto simp: setprod_ereal_0)
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1183
    finally show ?case .
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1184
  qed simp
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1185
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1186
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1187
  then show ?thesis by simp
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1188
qed
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1189
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1190
lemma setprod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (setprod f A)"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1191
proof (cases "finite A")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1192
  case True
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1193
  then show ?thesis
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1194
    by induct (auto simp: one_ereal_def)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1195
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1196
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1197
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1198
    by (simp add: one_ereal_def)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1199
qed
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1200
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1201
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1202
subsubsection {* Power *}
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1203
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1204
lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1205
  by (induct n) (auto simp: one_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1206
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1207
lemma ereal_power_PInf[simp]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1208
  by (induct n) (auto simp: one_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1209
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1210
lemma ereal_power_uminus[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1211
  fixes x :: ereal
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1212
  shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1213
  by (induct n) (auto simp: one_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1214
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1215
lemma ereal_power_numeral[simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
  1216
  "(numeral num :: ereal) ^ n = ereal (numeral num ^ n)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1217
  by (induct n) (auto simp: one_ereal_def)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1218
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1219
lemma zero_le_power_ereal[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1220
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1221
  assumes "0 \<le> a"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1222
  shows "0 \<le> a ^ n"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1223
  using assms by (induct n) (auto simp: ereal_zero_le_0_iff)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1224
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1225
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1226
subsubsection {* Subtraction *}
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1227
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1228
lemma ereal_minus_minus_image[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1229
  fixes S :: "ereal set"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1230
  shows "uminus ` uminus ` S = S"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1231
  by (auto simp: image_iff)
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1232
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1233
lemma ereal_uminus_lessThan[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1234
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1235
  shows "uminus ` {..<a} = {-a<..}"
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
  1236
proof -
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
  1237
  {
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1238
    fix x
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1239
    assume "-a < x"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1240
    then have "- x < - (- a)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1241
      by (simp del: ereal_uminus_uminus)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1242
    then have "- x < a"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1243
      by simp
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
  1244
  }
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1245
  then show ?thesis
54416
7fb88ed6ff3c better support for enat and ereal conversions
hoelzl
parents: 54408
diff changeset
  1246
    by force
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
  1247
qed
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1248
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1249
lemma ereal_uminus_greaterThan[simp]: "uminus ` {(a::ereal)<..} = {..<-a}"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1250
  by (metis ereal_uminus_lessThan ereal_uminus_uminus ereal_minus_minus