src/HOL/Library/Extended_Real.thy
author wenzelm
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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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header {* Extended real number line *}
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theory Extended_Real
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imports Complex_Main Extended_Nat Liminf_Limsup
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begin
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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[case_names real PInf MInf, cases type: ereal]:
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  assumes "\<And>r. x = ereal r \<Longrightarrow> P"
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  assumes "x = \<infinity> \<Longrightarrow> P"
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  assumes "x = -\<infinity> \<Longrightarrow> P"
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  shows P
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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_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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function of_ereal :: "ereal \<Rightarrow> real" where
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  "of_ereal (ereal r) = r"
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| "of_ereal \<infinity> = 0"
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| "of_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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defs (overloaded)
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  real_of_ereal_def [code_unfold]: "real \<equiv> of_ereal"
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lemma real_of_ereal[simp]:
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  "real (- x :: ereal) = - (real x)"
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  "real (ereal r) = r"
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  "real (\<infinity>::ereal) = 0"
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  by (cases x) (simp_all add: real_of_ereal_def)
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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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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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instance ..
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end
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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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  using assms by (cases x) auto
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lemma abs_neq_infinity_cases[elim!]:
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  fixes x :: ereal
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  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
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  obtains r where "x = ereal r"
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  using assms by (cases x) auto
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lemma abs_ereal_uminus[simp]:
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  fixes x :: ereal
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  shows "\<bar>- x\<bar> = \<bar>x\<bar>"
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  by (cases x) auto
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lemma ereal_infinity_cases:
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  fixes a :: ereal
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  shows "a \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
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  by auto
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subsubsection "Addition"
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instantiation ereal :: "{one,comm_monoid_add}"
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begin
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definition "0 = ereal 0"
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definition "1 = ereal 1"
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function plus_ereal where
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  "ereal r + ereal p = ereal (r + p)"
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| "\<infinity> + a = (\<infinity>::ereal)"
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| "a + \<infinity> = (\<infinity>::ereal)"
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| "ereal r + -\<infinity> = - \<infinity>"
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| "-\<infinity> + ereal p = -(\<infinity>::ereal)"
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| "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
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proof -
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  case (goal1 P x)
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  then obtain a b where "x = (a, b)"
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    by (cases x) auto
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  with goal1 show P
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   by (cases rule: ereal2_cases[of a b]) auto
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qed auto
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termination by default (rule wf_empty)
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lemma Infty_neq_0[simp]:
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  "(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)"
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  "-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)"
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  by (simp_all add: zero_ereal_def)
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lemma ereal_eq_0[simp]:
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  "ereal r = 0 \<longleftrightarrow> r = 0"
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  "0 = ereal r \<longleftrightarrow> r = 0"
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  unfolding zero_ereal_def by simp_all
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instance
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proof
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  fix a b c :: ereal
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  show "0 + a = a"
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    by (cases a) (simp_all add: zero_ereal_def)
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  show "a + b = b + a"
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    by (cases rule: ereal2_cases[of a b]) simp_all
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  show "a + b + c = a + (b + c)"
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    by (cases rule: ereal3_cases[of a b c]) simp_all
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qed
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end
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instance ereal :: numeral ..
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lemma real_of_ereal_0[simp]: "real (0::ereal) = 0"
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  unfolding real_of_ereal_def zero_ereal_def by simp
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lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
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  unfolding zero_ereal_def abs_ereal.simps by simp
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lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)"
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  by (simp add: zero_ereal_def)
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lemma ereal_uminus_zero_iff[simp]:
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  fixes a :: ereal
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  shows "-a = 0 \<longleftrightarrow> a = 0"
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  by (cases a) simp_all
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lemma ereal_plus_eq_PInfty[simp]:
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  fixes a b :: ereal
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  shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
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  by (cases rule: ereal2_cases[of a b]) auto
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lemma ereal_plus_eq_MInfty[simp]:
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  fixes a b :: ereal
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  shows "a + b = -\<infinity> \<longleftrightarrow> (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
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  by (cases rule: ereal2_cases[of a b]) auto
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lemma ereal_add_cancel_left:
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  fixes a b :: ereal
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  assumes "a \<noteq> -\<infinity>"
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  shows "a + b = a + c \<longleftrightarrow> a = \<infinity> \<or> b = c"
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  using assms by (cases rule: ereal3_cases[of a b c]) auto
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lemma ereal_add_cancel_right:
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  fixes a b :: ereal
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  assumes "a \<noteq> -\<infinity>"
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  shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c"
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  using assms by (cases rule: ereal3_cases[of a b c]) auto
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lemma ereal_real: "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
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  by (cases x) simp_all
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lemma real_of_ereal_add:
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  fixes a b :: ereal
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  shows "real (a + b) =
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    (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)"
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  by (cases rule: ereal2_cases[of a b]) auto
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subsubsection "Linear order on @{typ ereal}"
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instantiation ereal :: linorder
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begin
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function less_ereal
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where
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  "   ereal x < ereal y     \<longleftrightarrow> x < y"
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| "(\<infinity>::ereal) < a           \<longleftrightarrow> False"
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| "         a < -(\<infinity>::ereal) \<longleftrightarrow> False"
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| "ereal x    < \<infinity>           \<longleftrightarrow> True"
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| "        -\<infinity> < ereal r     \<longleftrightarrow> True"
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| "        -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True"
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proof -
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  case (goal1 P x)
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  then obtain a b where "x = (a,b)" by (cases x) auto
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  with goal1 show P by (cases rule: ereal2_cases[of a b]) auto
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qed simp_all
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termination by (relation "{}") simp
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definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"
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lemma ereal_infty_less[simp]:
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  fixes x :: ereal
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  shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
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    "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
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  by (cases x, simp_all) (cases x, simp_all)
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lemma ereal_infty_less_eq[simp]:
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  fixes x :: ereal
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  shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
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    and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
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  by (auto simp add: less_eq_ereal_def)
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   286
lemma ereal_less[simp]:
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  "ereal r < 0 \<longleftrightarrow> (r < 0)"
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  "0 < ereal r \<longleftrightarrow> (0 < r)"
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  "0 < (\<infinity>::ereal)"
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  "-(\<infinity>::ereal) < 0"
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  by (simp_all add: zero_ereal_def)
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   293
lemma ereal_less_eq[simp]:
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  "x \<le> (\<infinity>::ereal)"
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  "-(\<infinity>::ereal) \<le> x"
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  "ereal r \<le> ereal p \<longleftrightarrow> r \<le> p"
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  "ereal r \<le> 0 \<longleftrightarrow> r \<le> 0"
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   298
  "0 \<le> ereal r \<longleftrightarrow> 0 \<le> r"
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   299
  by (auto simp add: less_eq_ereal_def zero_ereal_def)
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lemma ereal_infty_less_eq2:
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  "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)"
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  "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)"
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  by simp_all
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instance
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proof
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  fix x y z :: ereal
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  show "x \<le> x"
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    by (cases x) simp_all
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  show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
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    by (cases rule: ereal2_cases[of x y]) auto
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  show "x \<le> y \<or> y \<le> x "
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   314
    by (cases rule: ereal2_cases[of x y]) auto
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  {
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diff changeset
   316
    assume "x \<le> y" "y \<le> x"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   317
    then show "x = y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   318
      by (cases rule: ereal2_cases[of x y]) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   319
  }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   320
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   321
    assume "x \<le> y" "y \<le> z"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   322
    then show "x \<le> z"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   323
      by (cases rule: ereal3_cases[of x y z]) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   324
  }
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   325
qed
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   326
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   327
end
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   328
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   329
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
   330
  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
   331
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 52729
diff changeset
   332
instance ereal :: dense_linorder
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   333
  by default (blast dest: ereal_dense2)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   334
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   335
instance ereal :: ordered_ab_semigroup_add
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   336
proof
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   337
  fix a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   338
  assume "a \<le> b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   339
  then show "c + a \<le> c + b"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   340
    by (cases rule: ereal3_cases[of a b c]) auto
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   341
qed
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   342
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   343
lemma real_of_ereal_positive_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   344
  fixes x y :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   345
  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
   346
  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
   347
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   348
lemma ereal_MInfty_lessI[intro, simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   349
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   350
  shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   351
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   352
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   353
lemma ereal_less_PInfty[intro, simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   354
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   355
  shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   356
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   357
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   358
lemma ereal_less_ereal_Ex:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   359
  fixes a b :: ereal
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   360
  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
   361
  by (cases x) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   362
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   363
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
   364
proof (cases x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   365
  case (real r)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   366
  then show ?thesis
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents: 41979
diff changeset
   367
    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
   368
qed simp_all
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   369
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   370
lemma ereal_add_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   371
  fixes a b c d :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   372
  assumes "a \<le> b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   373
    and "c \<le> d"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   374
  shows "a + c \<le> b + d"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   375
  using assms
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   376
  apply (cases a)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   377
  apply (cases rule: ereal3_cases[of b c d], auto)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   378
  apply (cases rule: ereal3_cases[of b c d], auto)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   379
  done
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   380
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   381
lemma ereal_minus_le_minus[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   382
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   383
  shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   384
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   385
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   386
lemma ereal_minus_less_minus[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   387
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   388
  shows "- a < - b \<longleftrightarrow> b < a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   389
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   390
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   391
lemma ereal_le_real_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   392
  "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
   393
  by (cases y) auto
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 real_le_ereal_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   396
  "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
   397
  by (cases y) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   398
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   399
lemma ereal_less_real_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   400
  "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
   401
  by (cases y) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   402
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   403
lemma real_less_ereal_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   404
  "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
   405
  by (cases y) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   406
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   407
lemma real_of_ereal_pos:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   408
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   409
  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
   410
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   411
lemmas real_of_ereal_ord_simps =
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   412
  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
   413
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   414
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
   415
  by (cases x) auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   416
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   417
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
   418
  by (cases x) auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   419
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   420
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
   421
  by (cases x) auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   422
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   423
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
   424
  by (cases x) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   425
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   426
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
   427
  by (cases x) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   428
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   429
lemma zero_less_real_of_ereal:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   430
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   431
  shows "0 < real x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   432
  by (cases x) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   433
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   434
lemma ereal_0_le_uminus_iff[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   435
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   436
  shows "0 \<le> - a \<longleftrightarrow> a \<le> 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   437
  by (cases rule: ereal2_cases[of a]) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   438
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   439
lemma ereal_uminus_le_0_iff[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   440
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   441
  shows "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   442
  by (cases rule: ereal2_cases[of a]) auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   443
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   444
lemma ereal_add_strict_mono:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   445
  fixes a b c d :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   446
  assumes "a = b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   447
    and "0 \<le> a"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   448
    and "a \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   449
    and "c < d"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   450
  shows "a + c < b + d"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   451
  using assms
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   452
  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
   453
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   454
lemma ereal_less_add:
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   455
  fixes a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   456
  shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   457
  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
   458
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   459
lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   460
  by auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   461
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   462
lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   463
  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
   464
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   465
lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   466
  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
   467
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   468
lemmas ereal_uminus_reorder =
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   469
  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
   470
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   471
lemma ereal_bot:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   472
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   473
  assumes "\<And>B. x \<le> ereal B"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   474
  shows "x = - \<infinity>"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   475
proof (cases x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   476
  case (real r)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   477
  with assms[of "r - 1"] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   478
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   479
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   480
  case PInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   481
  with assms[of 0] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   482
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   483
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   484
  case MInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   485
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   486
    by simp
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   487
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   488
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   489
lemma ereal_top:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   490
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   491
  assumes "\<And>B. x \<ge> ereal B"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   492
  shows "x = \<infinity>"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   493
proof (cases x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   494
  case (real r)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   495
  with assms[of "r + 1"] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   496
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   497
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   498
  case MInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   499
  with assms[of 0] show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   500
    by auto
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   501
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   502
  case PInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   503
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   504
    by simp
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   505
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   506
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   507
lemma
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   508
  shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   509
    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
   510
  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
   511
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   512
lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   513
  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
   514
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   515
lemma
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   516
  fixes f :: "nat \<Rightarrow> ereal"
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   517
  shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   518
  and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   519
  unfolding decseq_def incseq_def by auto
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   520
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   521
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
   522
  unfolding incseq_def by auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   523
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   524
lemma ereal_add_nonneg_nonneg:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   525
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   526
  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
   527
  using add_mono[of 0 a 0 b] by simp
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   528
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   529
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
   530
  by auto
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   531
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   532
lemma incseq_setsumI:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   533
  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
   534
  assumes "\<And>i. 0 \<le> f i"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   535
  shows "incseq (\<lambda>i. setsum f {..< i})"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   536
proof (intro incseq_SucI)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   537
  fix n
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   538
  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
   539
    using assms by (rule add_left_mono)
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   540
  then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   541
    by auto
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   542
qed
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   543
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   544
lemma incseq_setsumI2:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   545
  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
   546
  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
   547
  shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   548
  using assms
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   549
  unfolding incseq_def by (auto intro: setsum_mono)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   550
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   551
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   552
subsubsection "Multiplication"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   553
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   554
instantiation ereal :: "{comm_monoid_mult,sgn}"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   555
begin
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   556
51351
dd1dd470690b generalized lemmas in Extended_Real_Limits
hoelzl
parents: 51340
diff changeset
   557
function sgn_ereal :: "ereal \<Rightarrow> ereal" where
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   558
  "sgn (ereal r) = ereal (sgn r)"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   559
| "sgn (\<infinity>::ereal) = 1"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   560
| "sgn (-\<infinity>::ereal) = -1"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   561
by (auto intro: ereal_cases)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   562
termination by default (rule wf_empty)
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
   563
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   564
function times_ereal where
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   565
  "ereal r * ereal p = ereal (r * p)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   566
| "ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   567
| "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   568
| "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   569
| "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   570
| "(\<infinity>::ereal) * \<infinity> = \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   571
| "-(\<infinity>::ereal) * \<infinity> = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   572
| "(\<infinity>::ereal) * -\<infinity> = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   573
| "-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   574
proof -
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   575
  case (goal1 P x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   576
  then obtain a b where "x = (a, b)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   577
    by (cases x) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   578
  with goal1 show P
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   579
    by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   580
qed simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   581
termination by (relation "{}") simp
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   582
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   583
instance
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   584
proof
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   585
  fix a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   586
  show "1 * a = a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   587
    by (cases a) (simp_all add: one_ereal_def)
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   588
  show "a * b = b * a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   589
    by (cases rule: ereal2_cases[of a b]) simp_all
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   590
  show "a * b * c = a * (b * c)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   591
    by (cases rule: ereal3_cases[of a b c])
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   592
       (simp_all add: zero_ereal_def zero_less_mult_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   593
qed
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   594
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   595
end
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   596
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   597
lemma real_ereal_1[simp]: "real (1::ereal) = 1"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   598
  unfolding one_ereal_def by simp
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   599
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   600
lemma real_of_ereal_le_1:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   601
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   602
  shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   603
  by (cases a) (auto simp: one_ereal_def)
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   604
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   605
lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   606
  unfolding one_ereal_def by simp
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
   607
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   608
lemma ereal_mult_zero[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   609
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   610
  shows "a * 0 = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   611
  by (cases a) (simp_all add: zero_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   612
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   613
lemma ereal_zero_mult[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 * a = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   616
  by (cases a) (simp_all add: zero_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   617
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   618
lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   619
  by (simp add: zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   620
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   621
lemma ereal_zero_m1[simp]: "1 \<noteq> (0::ereal)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   622
  by (simp add: zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   623
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   624
lemma ereal_times_0[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   625
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   626
  shows "0 * x = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   627
  by (cases x) (auto simp: zero_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   628
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   629
lemma ereal_times[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   630
  "1 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   631
  "1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 1"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   632
  by (auto simp add: times_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   633
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   634
lemma ereal_plus_1[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   635
  "1 + ereal r = ereal (r + 1)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   636
  "ereal r + 1 = ereal (r + 1)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   637
  "1 + -(\<infinity>::ereal) = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   638
  "-(\<infinity>::ereal) + 1 = -\<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   639
  unfolding one_ereal_def by auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   640
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   641
lemma ereal_zero_times[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   642
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   643
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   644
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   645
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   646
lemma ereal_mult_eq_PInfty[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   647
  "a * b = (\<infinity>::ereal) \<longleftrightarrow>
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   648
    (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
   649
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   650
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   651
lemma ereal_mult_eq_MInfty[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   652
  "a * b = -(\<infinity>::ereal) \<longleftrightarrow>
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   653
    (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
   654
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   655
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   656
lemma ereal_0_less_1[simp]: "0 < (1::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   657
  by (simp_all add: zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   658
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   659
lemma ereal_zero_one[simp]: "0 \<noteq> (1::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   660
  by (simp_all add: zero_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   661
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   662
lemma ereal_mult_minus_left[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   663
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   664
  shows "-a * b = - (a * b)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   665
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   666
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   667
lemma ereal_mult_minus_right[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   668
  fixes a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   669
  shows "a * -b = - (a * b)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   670
  by (cases rule: ereal2_cases[of a b]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   671
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   672
lemma ereal_mult_infty[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   673
  "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
   674
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   675
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   676
lemma ereal_infty_mult[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   677
  "(\<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
   678
  by (cases a) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   679
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   680
lemma ereal_mult_strict_right_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   681
  assumes "a < b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   682
    and "0 < c"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   683
    and "c < (\<infinity>::ereal)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   684
  shows "a * c < b * c"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   685
  using assms
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   686
  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
   687
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   688
lemma ereal_mult_strict_left_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   689
  "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> c * a < c * b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   690
  using ereal_mult_strict_right_mono
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   691
  by (simp add: mult_commute[of c])
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   692
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   693
lemma ereal_mult_right_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   694
  fixes a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   695
  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
   696
  using assms
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   697
  apply (cases "c = 0")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   698
  apply simp
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   699
  apply (cases rule: ereal3_cases[of a b c])
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   700
  apply (auto simp: zero_le_mult_iff)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   701
  done
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   702
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   703
lemma ereal_mult_left_mono:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   704
  fixes a b c :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   705
  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   706
  using ereal_mult_right_mono
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   707
  by (simp add: mult_commute[of c])
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   708
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   709
lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   710
  by (simp add: one_ereal_def zero_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   711
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   712
lemma ereal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   713
  by (cases rule: ereal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   714
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   715
lemma ereal_right_distrib:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   716
  fixes r a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   717
  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
   718
  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
   719
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   720
lemma ereal_left_distrib:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   721
  fixes r a b :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   722
  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
   723
  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
   724
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   725
lemma ereal_mult_le_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   726
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   727
  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
   728
  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
   729
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   730
lemma ereal_zero_le_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   731
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   732
  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
   733
  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
   734
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   735
lemma ereal_mult_less_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   736
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   737
  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
   738
  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
   739
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   740
lemma ereal_zero_less_0_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   741
  fixes a b :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   742
  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
   743
  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
   744
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   745
lemma ereal_left_mult_cong:
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   746
  fixes a b c :: ereal
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   747
  shows "(c \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> c * a = c * b"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   748
  by (cases "c = 0") simp_all
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   749
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   750
lemma ereal_right_mult_cong:
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   751
  fixes a b c :: ereal
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   752
  shows "(c \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> a * c = b * c"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   753
  by (cases "c = 0") simp_all
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
   754
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   755
lemma ereal_distrib:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   756
  fixes a b c :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   757
  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   758
    and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   759
    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
   760
  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
   761
  using assms
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   762
  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
   763
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   764
lemma numeral_eq_ereal [simp]: "numeral w = ereal (numeral w)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   765
  apply (induct w rule: num_induct)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   766
  apply (simp only: numeral_One one_ereal_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   767
  apply (simp only: numeral_inc ereal_plus_1)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   768
  done
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   769
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   770
lemma ereal_le_epsilon:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   771
  fixes x y :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   772
  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   773
  shows "x \<le> y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   774
proof -
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   775
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   776
    assume a: "\<exists>r. y = ereal r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   777
    then obtain r where r_def: "y = ereal r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   778
      by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   779
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   780
      assume "x = -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   781
      then have ?thesis by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   782
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   783
    moreover
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   784
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   785
      assume "x \<noteq> -\<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   786
      then obtain p where p_def: "x = ereal p"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   787
      using a assms[rule_format, of 1]
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   788
        by (cases x) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   789
      {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   790
        fix e
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   791
        have "0 < e \<longrightarrow> p \<le> r + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   792
          using assms[rule_format, of "ereal e"] p_def r_def by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   793
      }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   794
      then have "p \<le> r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   795
        apply (subst field_le_epsilon)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   796
        apply auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   797
        done
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   798
      then have ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   799
        using r_def p_def by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   800
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   801
    ultimately have ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   802
      by blast
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   803
  }
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   804
  moreover
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   805
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   806
    assume "y = -\<infinity> | y = \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   807
    then have ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   808
      using assms[rule_format, of 1] by (cases x) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   809
  }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   810
  ultimately show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   811
    by (cases y) auto
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   812
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   813
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   814
lemma ereal_le_epsilon2:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   815
  fixes x y :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   816
  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + ereal e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   817
  shows "x \<le> y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   818
proof -
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   819
  {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   820
    fix e :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   821
    assume "e > 0"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   822
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   823
      assume "e = \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   824
      then have "x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   825
        by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   826
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   827
    moreover
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   828
    {
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   829
      assume "e \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   830
      then obtain r where "e = ereal r"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   831
        using `e > 0` by (cases e) auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   832
      then have "x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   833
        using assms[rule_format, of r] `e>0` by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   834
    }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   835
    ultimately have "x \<le> y + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   836
      by blast
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   837
  }
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   838
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   839
    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
   840
qed
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   841
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   842
lemma ereal_le_real:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   843
  fixes x y :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   844
  assumes "\<forall>z. x \<le> ereal z \<longrightarrow> y \<le> ereal z"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   845
  shows "y \<le> x"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   846
  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
   847
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   848
lemma setprod_ereal_0:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   849
  fixes f :: "'a \<Rightarrow> ereal"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   850
  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
   851
proof (cases "finite A")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   852
  case True
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   853
  then show ?thesis by (induct A) auto
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   854
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   855
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   856
  then show ?thesis by auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   857
qed
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   858
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   859
lemma setprod_ereal_pos:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   860
  fixes f :: "'a \<Rightarrow> ereal"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   861
  assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   862
  shows "0 \<le> (\<Prod>i\<in>I. f i)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   863
proof (cases "finite I")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   864
  case True
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   865
  from this pos show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   866
    by induct auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   867
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   868
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   869
  then show ?thesis by simp
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   870
qed
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   871
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   872
lemma setprod_PInf:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   873
  fixes f :: "'a \<Rightarrow> ereal"
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   874
  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
   875
  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
   876
proof (cases "finite I")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   877
  case True
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   878
  from this assms show ?thesis
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   879
  proof (induct I)
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   880
    case (insert i I)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   881
    then have pos: "0 \<le> f i" "0 \<le> setprod f I"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   882
      by (auto intro!: setprod_ereal_pos)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   883
    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
   884
      by auto
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   885
    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
   886
      using setprod_ereal_pos[of I f] pos
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   887
      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
   888
    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
   889
      using insert by (auto simp: setprod_ereal_0)
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   890
    finally show ?case .
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   891
  qed simp
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   892
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   893
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   894
  then show ?thesis by simp
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   895
qed
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 setprod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (setprod f A)"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   898
proof (cases "finite A")
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   899
  case True
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   900
  then show ?thesis
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   901
    by induct (auto simp: one_ereal_def)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   902
next
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   903
  case False
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   904
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   905
    by (simp add: one_ereal_def)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   906
qed
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   907
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
   908
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   909
subsubsection {* Power *}
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   910
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   911
lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   912
  by (induct n) (auto simp: one_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   913
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   914
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
   915
  by (induct n) (auto simp: one_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   916
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   917
lemma ereal_power_uminus[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   918
  fixes x :: ereal
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   919
  shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   920
  by (induct n) (auto simp: one_ereal_def)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
   921
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   922
lemma ereal_power_numeral[simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 47082
diff changeset
   923
  "(numeral num :: ereal) ^ n = ereal (numeral num ^ n)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   924
  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
   925
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   926
lemma zero_le_power_ereal[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   927
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   928
  assumes "0 \<le> a"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
   929
  shows "0 \<le> a ^ n"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   930
  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
   931
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   932
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   933
subsubsection {* Subtraction *}
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_minus_minus_image[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   936
  fixes S :: "ereal set"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   937
  shows "uminus ` uminus ` S = S"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   938
  by (auto simp: image_iff)
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   939
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   940
lemma ereal_uminus_lessThan[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   941
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   942
  shows "uminus ` {..<a} = {-a<..}"
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   943
proof -
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   944
  {
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   945
    fix x
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   946
    assume "-a < x"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   947
    then have "- x < - (- a)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   948
      by (simp del: ereal_uminus_uminus)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   949
    then have "- x < a"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   950
      by simp
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   951
  }
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   952
  then show ?thesis
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   953
    by (auto intro!: image_eqI)
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
   954
qed
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   955
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   956
lemma ereal_uminus_greaterThan[simp]: "uminus ` {(a::ereal)<..} = {..<-a}"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   957
  by (metis ereal_uminus_lessThan ereal_uminus_uminus ereal_minus_minus_image)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   958
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   959
instantiation ereal :: minus
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   960
begin
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   961
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   962
definition "x - y = x + -(y::ereal)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   963
instance ..
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   964
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   965
end
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   966
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   967
lemma ereal_minus[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   968
  "ereal r - ereal p = ereal (r - p)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   969
  "-\<infinity> - ereal r = -\<infinity>"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   970
  "ereal r - \<infinity> = -\<infinity>"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   971
  "(\<infinity>::ereal) - x = \<infinity>"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   972
  "-(\<infinity>::ereal) - \<infinity> = -\<infinity>"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   973
  "x - -y = x + y"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   974
  "x - 0 = x"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   975
  "0 - x = -x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   976
  by (simp_all add: minus_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   977
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
   978
lemma ereal_x_minus_x[simp]: "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   979
  by (cases x) simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   980
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   981
lemma ereal_eq_minus_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   982
  fixes x y z :: ereal
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   983
  shows "x = z - y \<longleftrightarrow>
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
   984
    (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   985
    (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   986
    (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   987
    (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   988
  by (cases rule: ereal3_cases[of x y z]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   989
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   990
lemma ereal_eq_minus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   991
  fixes x y z :: ereal
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
   992
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   993
  by (auto simp: ereal_eq_minus_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   994
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   995
lemma ereal_less_minus_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
   996
  fixes x y z :: ereal
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   997
  shows "x < z - y \<longleftrightarrow>
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   998
    (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
   999
    (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
  1000
    (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1001
  by (cases rule: ereal3_cases[of x y z]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1002
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1003
lemma ereal_less_minus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1004
  fixes x y z :: ereal
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
  1005
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1006
  by (auto simp: ereal_less_minus_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1007
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1008
lemma ereal_le_minus_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1009
  fixes x y z :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1010
  shows "x \<le> z - y \<longleftrightarrow> (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1011
  by (cases rule: ereal3_cases[of x y z]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1012
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1013
lemma ereal_le_minus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1014
  fixes x y z :: ereal
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
  1015
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1016
  by (auto simp: ereal_le_minus_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1017
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1018
lemma ereal_minus_less_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1019
  fixes x y z :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1020
  shows "x - y < z \<longleftrightarrow> y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1021
  by (cases rule: ereal3_cases[of x y z]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1022
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1023
lemma ereal_minus_less:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1024
  fixes x y z :: ereal
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
  1025
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1026
  by (auto simp: ereal_minus_less_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1027
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1028
lemma ereal_minus_le_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1029
  fixes x y z :: ereal
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1030
  shows "x - y \<le> z \<longleftrightarrow>
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1031
    (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1032
    (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
  1033
    (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1034
  by (cases rule: ereal3_cases[of x y z]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1035
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1036
lemma ereal_minus_le:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1037
  fixes x y z :: ereal
41976
3fdbc7d5b525 use abs_extreal
hoelzl
parents: 41975
diff changeset
  1038
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1039
  by (auto simp: ereal_minus_le_iff)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1040
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1041
lemma ereal_minus_eq_minus_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1042
  fixes a b c :: ereal
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1043
  shows "a - b = a - c \<longleftrightarrow>
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1044
    b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1045
  by (cases rule: ereal3_cases[of a b c]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1046
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1047
lemma ereal_add_le_add_iff:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1048
  fixes a b c :: ereal
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1049
  shows "c + a \<le> c + b \<longleftrightarrow>
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1050
    a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1051
  by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1052
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1053
lemma ereal_mult_le_mult_iff:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1054
  fixes a b c :: ereal
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1055
  shows "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1056
  by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1057
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1058
lemma ereal_minus_mono:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1059
  fixes A B C D :: ereal assumes "A \<le> B" "D \<le> C"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1060
  shows "A - C \<le> B - D"
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1061
  using assms
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1062
  by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1063
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1064
lemma real_of_ereal_minus:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1065
  fixes a b :: ereal
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1066
  shows "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1067
  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
  1068
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1069
lemma ereal_diff_positive:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1070
  fixes a b :: ereal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1071
  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
  1072
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1073
lemma ereal_between:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1074
  fixes x e :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1075
  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1076
    and "0 < e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1077
  shows "x - e < x"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1078
    and "x < x + e"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1079
  using assms
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1080
  apply (cases x, cases e)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1081
  apply auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1082
  using assms
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1083
  apply (cases x, cases e)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1084
  apply auto
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1085
  done
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1086
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1087
lemma ereal_minus_eq_PInfty_iff:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1088
  fixes x y :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1089
  shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1090
  by (cases x y rule: ereal2_cases) simp_all
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 47108
diff changeset
  1091
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1092
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1093
subsubsection {* Division *}
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1094
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1095
instantiation ereal :: inverse
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1096
begin
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1097
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1098
function inverse_ereal where
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1099
  "inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1100
| "inverse (\<infinity>::ereal) = 0"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1101
| "inverse (-\<infinity>::ereal) = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1102
  by (auto intro: ereal_cases)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1103
termination by (relation "{}") simp
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1104
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1105
definition "x / y = x * inverse (y :: ereal)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1106
47082
737d7bc8e50f tuned proofs;
wenzelm
parents: 45934
diff changeset
  1107
instance ..
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1108
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1109
end
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1110
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1111
lemma real_of_ereal_inverse[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1112
  fixes a :: ereal
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1113
  shows "real (inverse a) = 1 / real a"
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1114
  by (cases a) (auto simp: inverse_eq_divide)
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42600
diff changeset
  1115
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1116
lemma ereal_inverse[simp]:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1117
  "inverse (0::ereal) = \<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1118
  "inverse (1::ereal) = 1"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1119
  by (simp_all add: one_ereal_def zero_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1120
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1121
lemma ereal_divide[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1122
  "ereal r / ereal p = (if p = 0 then ereal r * \<infinity> else ereal (r / p))"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1123
  unfolding divide_ereal_def by (auto simp: divide_real_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1124
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1125
lemma ereal_divide_same[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1126
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1127
  shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1128
  by (cases x) (simp_all add: divide_real_def divide_ereal_def one_ereal_def)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1129
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1130
lemma ereal_inv_inv[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1131
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1132
  shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1133
  by (cases x) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1134
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1135
lemma ereal_inverse_minus[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1136
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1137
  shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1138
  by (cases x) simp_all
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1139
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1140
lemma ereal_uminus_divide[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1141
  fixes x y :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1142
  shows "- x / y = - (x / y)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1143
  unfolding divide_ereal_def by simp
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1144
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1145
lemma ereal_divide_Infty[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1146
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1147
  shows "x / \<infinity> = 0" "x / -\<infinity> = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1148
  unfolding divide_ereal_def by simp_all
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1149
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1150
lemma ereal_divide_one[simp]: "x / 1 = (x::ereal)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1151
  unfolding divide_ereal_def by simp
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1152
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1153
lemma ereal_divide_ereal[simp]: "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1154
  unfolding divide_ereal_def by simp
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1155
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1156
lemma zero_le_divide_ereal[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1157
  fixes a :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1158
  assumes "0 \<le> a"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1159
    and "0 \<le> b"
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1160
  shows "0 \<le> a / b"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1161
  using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff)
41978
656298577a33 add infinite sums and power on extreal
hoelzl
parents: 41977
diff changeset
  1162
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1163
lemma ereal_le_divide_pos:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1164
  fixes x y z :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1165
  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1166
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1167
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1168
lemma ereal_divide_le_pos:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1169
  fixes x y z :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1170
  shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1171
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1172
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1173
lemma ereal_le_divide_neg:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1174
  fixes x y z :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1175
  shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1176
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1177
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1178
lemma ereal_divide_le_neg:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1179
  fixes x y z :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1180
  shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1181
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1182
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1183
lemma ereal_inverse_antimono_strict:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1184
  fixes x y :: ereal
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1185
  shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1186
  by (cases rule: ereal2_cases[of x y]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1187
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1188
lemma ereal_inverse_antimono:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1189
  fixes x y :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1190
  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> inverse y \<le> inverse x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1191
  by (cases rule: ereal2_cases[of x y]) auto
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1192
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1193
lemma inverse_inverse_Pinfty_iff[simp]:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1194
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1195
  shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1196
  by (cases x) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1197
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1198
lemma ereal_inverse_eq_0:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1199
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1200
  shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1201
  by (cases x) auto
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1202
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1203
lemma ereal_0_gt_inverse:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1204
  fixes x :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1205
  shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1206
  by (cases x) auto
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1207
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1208
lemma ereal_mult_less_right:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1209
  fixes a b c :: ereal
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1210
  assumes "b * a < c * a"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1211
    and "0 < a"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1212
    and "a < \<infinity>"
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1213
  shows "b < c"
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1214
  using assms
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1215
  by (cases rule: ereal3_cases[of a b c])
41973
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1216
     (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
15927c040731 add Extended_Reals from AFP/Lower_Semicontinuous
hoelzl
parents:
diff changeset
  1217
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1218
lemma ereal_power_divide:
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1219
  fixes x y :: ereal
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1220
  shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1221
  by (cases rule: ereal2_cases[of x y])
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1222
     (auto simp: one_ereal_def zero_ereal_def power_divide not_le
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1223
                 power_less_zero_eq zero_le_power_iff)
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1224
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1225
lemma ereal_le_mult_one_interval:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43138
diff changeset
  1226
  fixes x y :: ereal
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1227
  assumes y: "y \<noteq> -\<infinity>"
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1228
  assumes z: "\<And>z. 0 < z \<Longrightarrow> z < 1 \<Longrightarrow> z * x \<le> y"
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1229
  shows "x \<le> y"
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1230
proof (cases x)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1231
  case PInf
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1232
  with z[of "1 / 2"] show "x \<le> y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1233
    by (simp add: one_ereal_def)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1234
next
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1235
  case (real r)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1236
  note r = this
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1237
  show "x \<le> y"
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1238
  proof (cases y)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1239
    case (real p)
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1240
    note p = this
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1241
    have "r \<le> p"
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1242
    proof (rule field_le_mult_one_interval)
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1243
      fix z :: real
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1244
      assume "0 < z" and "z < 1"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1245
      with z[of "ereal z"] show "z * r \<le> p"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1246
        using p r by (auto simp: zero_le_mult_iff one_ereal_def)
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1247
    qed
53873
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1248
    then show "x \<le> y"
08594daabcd9 tuned proofs;
wenzelm
parents: 53381
diff changeset
  1249
      using p r by simp
41979
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978
diff changeset
  1250
  qed (insert y, simp_all)
b10ec1f5e9d5 lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents: 41978