author | hoelzl |
Mon, 23 May 2011 19:21:05 +0200 | |
changeset 42950 | 6e5c2a3c69da |
parent 42600 | 604661fb94eb |
child 43138 | 818521a90356 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Extended_Reals.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_Reals |
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imports Complex_Main |
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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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@{text "src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"} |
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*} |
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lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot" |
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proof |
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assume "{x..} = UNIV" |
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show "x = bot" |
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proof (rule ccontr) |
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assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less) |
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then show False using `{x..} = UNIV` by simp |
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qed |
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qed auto |
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lemma SUPR_pair: |
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"(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))" |
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by (rule antisym) (auto intro!: SUP_leI le_SUPI2) |
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lemma INFI_pair: |
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"(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))" |
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by (rule antisym) (auto intro!: le_INFI INF_leI2) |
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subsection {* Definition and basic properties *} |
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datatype extreal = extreal real | PInfty | MInfty |
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notation (xsymbols) |
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PInfty ("\<infinity>") |
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notation (HTML output) |
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PInfty ("\<infinity>") |
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declare [[coercion "extreal :: real \<Rightarrow> extreal"]] |
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instantiation extreal :: uminus |
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begin |
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fun uminus_extreal where |
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"- (extreal r) = extreal (- r)" |
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| "- \<infinity> = MInfty" |
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| "- MInfty = \<infinity>" |
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instance .. |
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end |
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lemma inj_extreal[simp]: "inj_on extreal A" |
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unfolding inj_on_def by auto |
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lemma MInfty_neq_PInfty[simp]: |
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"\<infinity> \<noteq> - \<infinity>" "- \<infinity> \<noteq> \<infinity>" by simp_all |
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lemma MInfty_neq_extreal[simp]: |
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"extreal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> extreal r" by simp_all |
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lemma MInfinity_cases[simp]: |
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"(case - \<infinity> of extreal r \<Rightarrow> f r | \<infinity> \<Rightarrow> y | MInfinity \<Rightarrow> z) = z" |
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by simp |
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lemma extreal_uminus_uminus[simp]: |
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fixes a :: extreal shows "- (- a) = a" |
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by (cases a) simp_all |
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lemma MInfty_eq[simp]: |
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"MInfty = - \<infinity>" by simp |
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declare uminus_extreal.simps(2)[simp del] |
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lemma extreal_cases[case_names real PInf MInf, cases type: extreal]: |
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assumes "\<And>r. x = extreal 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 extreal2_cases = extreal_cases[case_product extreal_cases] |
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lemmas extreal3_cases = extreal2_cases[case_product extreal_cases] |
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lemma extreal_uminus_eq_iff[simp]: |
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fixes a b :: extreal shows "-a = -b \<longleftrightarrow> a = b" |
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by (cases rule: extreal2_cases[of a b]) simp_all |
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function of_extreal :: "extreal \<Rightarrow> real" where |
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"of_extreal (extreal r) = r" | |
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"of_extreal \<infinity> = 0" | |
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"of_extreal (-\<infinity>) = 0" |
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by (auto intro: extreal_cases) |
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termination proof qed (rule wf_empty) |
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defs (overloaded) |
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real_of_extreal_def [code_unfold]: "real \<equiv> of_extreal" |
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lemma real_of_extreal[simp]: |
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"real (- x :: extreal) = - (real x)" |
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"real (extreal r) = r" |
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"real \<infinity> = 0" |
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by (cases x) (simp_all add: real_of_extreal_def) |
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lemma range_extreal[simp]: "range extreal = UNIV - {\<infinity>, -\<infinity>}" |
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proof safe |
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fix x assume "x \<notin> range extreal" "x \<noteq> \<infinity>" |
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then show "x = -\<infinity>" by (cases x) auto |
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qed auto |
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lemma extreal_range_uminus[simp]: "range uminus = (UNIV::extreal set)" |
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proof safe |
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fix x :: extreal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto |
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qed auto |
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instantiation extreal :: number |
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begin |
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definition [simp]: "number_of x = extreal (number_of x)" |
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instance proof qed |
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end |
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instantiation extreal :: abs |
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begin |
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function abs_extreal where |
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"\<bar>extreal r\<bar> = extreal \<bar>r\<bar>" |
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| "\<bar>-\<infinity>\<bar> = \<infinity>" |
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| "\<bar>\<infinity>\<bar> = \<infinity>" |
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by (auto intro: extreal_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!]: "\<lbrakk> \<bar>x\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" |
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by (cases x) auto |
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lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> \<noteq> \<infinity> ; \<And>r. x = extreal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" |
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by (cases x) auto |
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lemma abs_extreal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::extreal\<bar>" |
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by (cases x) auto |
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subsubsection "Addition" |
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instantiation extreal :: comm_monoid_add |
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begin |
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definition "0 = extreal 0" |
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function plus_extreal where |
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"extreal r + extreal p = extreal (r + p)" | |
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"\<infinity> + a = \<infinity>" | |
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"a + \<infinity> = \<infinity>" | |
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"extreal r + -\<infinity> = - \<infinity>" | |
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"-\<infinity> + extreal p = -\<infinity>" | |
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"-\<infinity> + -\<infinity> = -\<infinity>" |
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proof - |
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case (goal1 P x) |
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moreover then obtain a b where "x = (a, b)" by (cases x) auto |
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ultimately show P |
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by (cases rule: extreal2_cases[of a b]) auto |
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qed auto |
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termination proof qed (rule wf_empty) |
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lemma Infty_neq_0[simp]: |
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"\<infinity> \<noteq> 0" "0 \<noteq> \<infinity>" |
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"-\<infinity> \<noteq> 0" "0 \<noteq> -\<infinity>" |
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by (simp_all add: zero_extreal_def) |
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lemma extreal_eq_0[simp]: |
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"extreal r = 0 \<longleftrightarrow> r = 0" |
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"0 = extreal r \<longleftrightarrow> r = 0" |
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unfolding zero_extreal_def by simp_all |
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instance |
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proof |
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fix a :: extreal show "0 + a = a" |
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by (cases a) (simp_all add: zero_extreal_def) |
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fix b :: extreal show "a + b = b + a" |
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by (cases rule: extreal2_cases[of a b]) simp_all |
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fix c :: extreal show "a + b + c = a + (b + c)" |
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by (cases rule: extreal3_cases[of a b c]) simp_all |
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qed |
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end |
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lemma real_of_extreal_0[simp]: "real (0::extreal) = 0" |
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unfolding real_of_extreal_def zero_extreal_def by simp |
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lemma abs_extreal_zero[simp]: "\<bar>0\<bar> = (0::extreal)" |
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unfolding zero_extreal_def abs_extreal.simps by simp |
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lemma extreal_uminus_zero[simp]: |
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"- 0 = (0::extreal)" |
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by (simp add: zero_extreal_def) |
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lemma extreal_uminus_zero_iff[simp]: |
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fixes a :: extreal shows "-a = 0 \<longleftrightarrow> a = 0" |
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by (cases a) simp_all |
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lemma extreal_plus_eq_PInfty[simp]: |
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shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
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by (cases rule: extreal2_cases[of a b]) auto |
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lemma extreal_plus_eq_MInfty[simp]: |
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shows "a + b = -\<infinity> \<longleftrightarrow> |
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(a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>" |
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by (cases rule: extreal2_cases[of a b]) auto |
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lemma extreal_add_cancel_left: |
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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: extreal3_cases[of a b c]) auto |
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lemma extreal_add_cancel_right: |
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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: extreal3_cases[of a b c]) auto |
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lemma extreal_real: |
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"extreal (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_extreal_add: |
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fixes a b :: extreal |
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shows "real (a + b) = (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)" |
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by (cases rule: extreal2_cases[of a b]) auto |
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subsubsection "Linear order on @{typ extreal}" |
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instantiation extreal :: linorder |
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begin |
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function less_extreal where |
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"extreal x < extreal y \<longleftrightarrow> x < y" | |
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" \<infinity> < a \<longleftrightarrow> False" | |
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" a < -\<infinity> \<longleftrightarrow> False" | |
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"extreal x < \<infinity> \<longleftrightarrow> True" | |
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" -\<infinity> < extreal r \<longleftrightarrow> True" | |
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" -\<infinity> < \<infinity> \<longleftrightarrow> True" |
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proof - |
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case (goal1 P x) |
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moreover then obtain a b where "x = (a,b)" by (cases x) auto |
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ultimately show P by (cases rule: extreal2_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::extreal) \<longleftrightarrow> x < y \<or> x = y" |
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lemma extreal_infty_less[simp]: |
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"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 extreal_infty_less_eq[simp]: |
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"\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>" |
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"x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>" |
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by (auto simp add: less_eq_extreal_def) |
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lemma extreal_less[simp]: |
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"extreal r < 0 \<longleftrightarrow> (r < 0)" |
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"0 < extreal r \<longleftrightarrow> (0 < r)" |
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"0 < \<infinity>" |
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"-\<infinity> < 0" |
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by (simp_all add: zero_extreal_def) |
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lemma extreal_less_eq[simp]: |
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273 |
"x \<le> \<infinity>" |
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274 |
"-\<infinity> \<le> x" |
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"extreal r \<le> extreal p \<longleftrightarrow> r \<le> p" |
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"extreal r \<le> 0 \<longleftrightarrow> r \<le> 0" |
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"0 \<le> extreal r \<longleftrightarrow> 0 \<le> r" |
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by (auto simp add: less_eq_extreal_def zero_extreal_def) |
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280 |
lemma extreal_infty_less_eq2: |
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281 |
"a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = \<infinity>" |
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282 |
"a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -\<infinity>" |
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283 |
by simp_all |
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285 |
instance |
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286 |
proof |
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287 |
fix x :: extreal show "x \<le> x" |
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288 |
by (cases x) simp_all |
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289 |
fix y :: extreal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
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290 |
by (cases rule: extreal2_cases[of x y]) auto |
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291 |
show "x \<le> y \<or> y \<le> x " |
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by (cases rule: extreal2_cases[of x y]) auto |
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{ assume "x \<le> y" "y \<le> x" then show "x = y" |
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294 |
by (cases rule: extreal2_cases[of x y]) auto } |
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{ fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z" |
|
296 |
by (cases rule: extreal3_cases[of x y z]) auto } |
|
297 |
qed |
|
298 |
end |
|
299 |
||
41978 | 300 |
instance extreal :: ordered_ab_semigroup_add |
301 |
proof |
|
302 |
fix a b c :: extreal assume "a \<le> b" then show "c + a \<le> c + b" |
|
303 |
by (cases rule: extreal3_cases[of a b c]) auto |
|
304 |
qed |
|
305 |
||
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
306 |
lemma real_of_extreal_positive_mono: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
307 |
"\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
308 |
by (cases rule: extreal2_cases[of x y]) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
309 |
|
41973 | 310 |
lemma extreal_MInfty_lessI[intro, simp]: |
311 |
"a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a" |
|
312 |
by (cases a) auto |
|
313 |
||
314 |
lemma extreal_less_PInfty[intro, simp]: |
|
315 |
"a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>" |
|
316 |
by (cases a) auto |
|
317 |
||
318 |
lemma extreal_less_extreal_Ex: |
|
319 |
fixes a b :: extreal |
|
320 |
shows "x < extreal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = extreal p)" |
|
321 |
by (cases x) auto |
|
322 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
323 |
lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < extreal (real n))" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
324 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
325 |
case (real r) then show ?thesis |
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
326 |
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
|
327 |
qed simp_all |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
328 |
|
41973 | 329 |
lemma extreal_add_mono: |
330 |
fixes a b c d :: extreal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d" |
|
331 |
using assms |
|
332 |
apply (cases a) |
|
333 |
apply (cases rule: extreal3_cases[of b c d], auto) |
|
334 |
apply (cases rule: extreal3_cases[of b c d], auto) |
|
335 |
done |
|
336 |
||
337 |
lemma extreal_minus_le_minus[simp]: |
|
338 |
fixes a b :: extreal shows "- a \<le> - b \<longleftrightarrow> b \<le> a" |
|
339 |
by (cases rule: extreal2_cases[of a b]) auto |
|
340 |
||
341 |
lemma extreal_minus_less_minus[simp]: |
|
342 |
fixes a b :: extreal shows "- a < - b \<longleftrightarrow> b < a" |
|
343 |
by (cases rule: extreal2_cases[of a b]) auto |
|
344 |
||
345 |
lemma extreal_le_real_iff: |
|
41976 | 346 |
"x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))" |
41973 | 347 |
by (cases y) auto |
348 |
||
349 |
lemma real_le_extreal_iff: |
|
41976 | 350 |
"real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))" |
41973 | 351 |
by (cases y) auto |
352 |
||
353 |
lemma extreal_less_real_iff: |
|
41976 | 354 |
"x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))" |
41973 | 355 |
by (cases y) auto |
356 |
||
357 |
lemma real_less_extreal_iff: |
|
41976 | 358 |
"real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))" |
41973 | 359 |
by (cases y) auto |
360 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
361 |
lemma real_of_extreal_pos: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
362 |
fixes x :: extreal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
363 |
|
41973 | 364 |
lemmas real_of_extreal_ord_simps = |
365 |
extreal_le_real_iff real_le_extreal_iff extreal_less_real_iff real_less_extreal_iff |
|
366 |
||
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
367 |
lemma abs_extreal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: extreal\<bar> = x" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
368 |
by (cases x) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
369 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
370 |
lemma abs_extreal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: extreal\<bar> = -x" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
371 |
by (cases x) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
372 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
373 |
lemma abs_extreal_pos[simp]: "0 \<le> \<bar>x :: extreal\<bar>" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
374 |
by (cases x) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
375 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
376 |
lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
377 |
by (cases X) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
378 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
379 |
lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
380 |
by (cases X) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
381 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
382 |
lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
383 |
by (cases X) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
384 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
385 |
lemma extreal_0_le_uminus_iff[simp]: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
386 |
fixes a :: extreal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
387 |
by (cases rule: extreal2_cases[of a]) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
388 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
389 |
lemma extreal_uminus_le_0_iff[simp]: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
390 |
fixes a :: extreal shows "-a \<le> 0 \<longleftrightarrow> 0 \<le> a" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
391 |
by (cases rule: extreal2_cases[of a]) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
392 |
|
41973 | 393 |
lemma extreal_dense: |
394 |
fixes x y :: extreal assumes "x < y" |
|
395 |
shows "EX z. x < z & z < y" |
|
396 |
proof - |
|
397 |
{ assume a: "x = (-\<infinity>)" |
|
398 |
{ assume "y = \<infinity>" hence ?thesis using a by (auto intro!: exI[of _ "0"]) } |
|
399 |
moreover |
|
400 |
{ assume "y ~= \<infinity>" |
|
401 |
with `x < y` obtain r where r: "y = extreal r" by (cases y) auto |
|
402 |
hence ?thesis using `x < y` a by (auto intro!: exI[of _ "extreal (r - 1)"]) |
|
403 |
} ultimately have ?thesis by auto |
|
404 |
} |
|
405 |
moreover |
|
406 |
{ assume "x ~= (-\<infinity>)" |
|
407 |
with `x < y` obtain p where p: "x = extreal p" by (cases x) auto |
|
408 |
{ assume "y = \<infinity>" hence ?thesis using `x < y` p |
|
409 |
by (auto intro!: exI[of _ "extreal (p + 1)"]) } |
|
410 |
moreover |
|
411 |
{ assume "y ~= \<infinity>" |
|
412 |
with `x < y` obtain r where r: "y = extreal r" by (cases y) auto |
|
413 |
with p `x < y` have "p < r" by auto |
|
414 |
with dense obtain z where "p < z" "z < r" by auto |
|
415 |
hence ?thesis using r p by (auto intro!: exI[of _ "extreal z"]) |
|
416 |
} ultimately have ?thesis by auto |
|
417 |
} ultimately show ?thesis by auto |
|
418 |
qed |
|
419 |
||
420 |
lemma extreal_dense2: |
|
421 |
fixes x y :: extreal assumes "x < y" |
|
422 |
shows "EX z. x < extreal z & extreal z < y" |
|
423 |
by (metis extreal_dense[OF `x < y`] extreal_cases less_extreal.simps(2,3)) |
|
424 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
425 |
lemma extreal_add_strict_mono: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
426 |
fixes a b c d :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
427 |
assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
428 |
shows "a + c < b + d" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
429 |
using assms by (cases rule: extreal3_cases[case_product extreal_cases, of a b c d]) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
430 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
431 |
lemma extreal_less_add: "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
432 |
by (cases rule: extreal2_cases[of b c]) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
433 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
434 |
lemma extreal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::extreal)" by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
435 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
436 |
lemma extreal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::extreal)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
437 |
by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_less_minus) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
438 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
439 |
lemma extreal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::extreal)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
440 |
by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_le_minus) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
441 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
442 |
lemmas extreal_uminus_reorder = |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
443 |
extreal_uminus_eq_reorder extreal_uminus_less_reorder extreal_uminus_le_reorder |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
444 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
445 |
lemma extreal_bot: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
446 |
fixes x :: extreal assumes "\<And>B. x \<le> extreal B" shows "x = - \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
447 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
448 |
case (real r) with assms[of "r - 1"] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
449 |
next case PInf with assms[of 0] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
450 |
next case MInf then show ?thesis by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
451 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
452 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
453 |
lemma extreal_top: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
454 |
fixes x :: extreal assumes "\<And>B. x \<ge> extreal B" shows "x = \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
455 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
456 |
case (real r) with assms[of "r + 1"] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
457 |
next case MInf with assms[of 0] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
458 |
next case PInf then show ?thesis by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
459 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
460 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
461 |
lemma |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
462 |
shows extreal_max[simp]: "extreal (max x y) = max (extreal x) (extreal y)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
463 |
and extreal_min[simp]: "extreal (min x y) = min (extreal x) (extreal y)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
464 |
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
|
465 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
466 |
lemma extreal_max_0: "max 0 (extreal r) = extreal (max 0 r)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
467 |
by (auto simp: zero_extreal_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
468 |
|
41978 | 469 |
lemma |
470 |
fixes f :: "nat \<Rightarrow> extreal" |
|
471 |
shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f" |
|
472 |
and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f" |
|
473 |
unfolding decseq_def incseq_def by auto |
|
474 |
||
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
475 |
lemma incseq_extreal: "incseq f \<Longrightarrow> incseq (\<lambda>x. extreal (f x))" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
476 |
unfolding incseq_def by auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
477 |
|
41978 | 478 |
lemma extreal_add_nonneg_nonneg: |
479 |
fixes a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b" |
|
480 |
using add_mono[of 0 a 0 b] by simp |
|
481 |
||
482 |
lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)" |
|
483 |
by auto |
|
484 |
||
485 |
lemma incseq_setsumI: |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
486 |
fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}" |
41978 | 487 |
assumes "\<And>i. 0 \<le> f i" |
488 |
shows "incseq (\<lambda>i. setsum f {..< i})" |
|
489 |
proof (intro incseq_SucI) |
|
490 |
fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n" |
|
491 |
using assms by (rule add_left_mono) |
|
492 |
then show "setsum f {..< n} \<le> setsum f {..< Suc n}" |
|
493 |
by auto |
|
494 |
qed |
|
495 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
496 |
lemma incseq_setsumI2: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
497 |
fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
498 |
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
|
499 |
shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
500 |
using assms unfolding incseq_def by (auto intro: setsum_mono) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
501 |
|
41973 | 502 |
subsubsection "Multiplication" |
503 |
||
41976 | 504 |
instantiation extreal :: "{comm_monoid_mult, sgn}" |
41973 | 505 |
begin |
506 |
||
507 |
definition "1 = extreal 1" |
|
508 |
||
41976 | 509 |
function sgn_extreal where |
510 |
"sgn (extreal r) = extreal (sgn r)" |
|
511 |
| "sgn \<infinity> = 1" |
|
512 |
| "sgn (-\<infinity>) = -1" |
|
513 |
by (auto intro: extreal_cases) |
|
514 |
termination proof qed (rule wf_empty) |
|
515 |
||
41973 | 516 |
function times_extreal where |
517 |
"extreal r * extreal p = extreal (r * p)" | |
|
518 |
"extreal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" | |
|
519 |
"\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" | |
|
520 |
"extreal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" | |
|
521 |
"-\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" | |
|
522 |
"\<infinity> * \<infinity> = \<infinity>" | |
|
523 |
"-\<infinity> * \<infinity> = -\<infinity>" | |
|
524 |
"\<infinity> * -\<infinity> = -\<infinity>" | |
|
525 |
"-\<infinity> * -\<infinity> = \<infinity>" |
|
526 |
proof - |
|
527 |
case (goal1 P x) |
|
528 |
moreover then obtain a b where "x = (a, b)" by (cases x) auto |
|
529 |
ultimately show P by (cases rule: extreal2_cases[of a b]) auto |
|
530 |
qed simp_all |
|
531 |
termination by (relation "{}") simp |
|
532 |
||
533 |
instance |
|
534 |
proof |
|
535 |
fix a :: extreal show "1 * a = a" |
|
536 |
by (cases a) (simp_all add: one_extreal_def) |
|
537 |
fix b :: extreal show "a * b = b * a" |
|
538 |
by (cases rule: extreal2_cases[of a b]) simp_all |
|
539 |
fix c :: extreal show "a * b * c = a * (b * c)" |
|
540 |
by (cases rule: extreal3_cases[of a b c]) |
|
541 |
(simp_all add: zero_extreal_def zero_less_mult_iff) |
|
542 |
qed |
|
543 |
end |
|
544 |
||
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
545 |
lemma real_of_extreal_le_1: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
546 |
fixes a :: extreal shows "a \<le> 1 \<Longrightarrow> real a \<le> 1" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
547 |
by (cases a) (auto simp: one_extreal_def) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
548 |
|
41976 | 549 |
lemma abs_extreal_one[simp]: "\<bar>1\<bar> = (1::extreal)" |
550 |
unfolding one_extreal_def by simp |
|
551 |
||
41973 | 552 |
lemma extreal_mult_zero[simp]: |
553 |
fixes a :: extreal shows "a * 0 = 0" |
|
554 |
by (cases a) (simp_all add: zero_extreal_def) |
|
555 |
||
556 |
lemma extreal_zero_mult[simp]: |
|
557 |
fixes a :: extreal shows "0 * a = 0" |
|
558 |
by (cases a) (simp_all add: zero_extreal_def) |
|
559 |
||
560 |
lemma extreal_m1_less_0[simp]: |
|
561 |
"-(1::extreal) < 0" |
|
562 |
by (simp add: zero_extreal_def one_extreal_def) |
|
563 |
||
564 |
lemma extreal_zero_m1[simp]: |
|
565 |
"1 \<noteq> (0::extreal)" |
|
566 |
by (simp add: zero_extreal_def one_extreal_def) |
|
567 |
||
568 |
lemma extreal_times_0[simp]: |
|
569 |
fixes x :: extreal shows "0 * x = 0" |
|
570 |
by (cases x) (auto simp: zero_extreal_def) |
|
571 |
||
572 |
lemma extreal_times[simp]: |
|
573 |
"1 \<noteq> \<infinity>" "\<infinity> \<noteq> 1" |
|
574 |
"1 \<noteq> -\<infinity>" "-\<infinity> \<noteq> 1" |
|
575 |
by (auto simp add: times_extreal_def one_extreal_def) |
|
576 |
||
577 |
lemma extreal_plus_1[simp]: |
|
578 |
"1 + extreal r = extreal (r + 1)" "extreal r + 1 = extreal (r + 1)" |
|
579 |
"1 + -\<infinity> = -\<infinity>" "-\<infinity> + 1 = -\<infinity>" |
|
580 |
unfolding one_extreal_def by auto |
|
581 |
||
582 |
lemma extreal_zero_times[simp]: |
|
583 |
fixes a b :: extreal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
|
584 |
by (cases rule: extreal2_cases[of a b]) auto |
|
585 |
||
586 |
lemma extreal_mult_eq_PInfty[simp]: |
|
587 |
shows "a * b = \<infinity> \<longleftrightarrow> |
|
588 |
(a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)" |
|
589 |
by (cases rule: extreal2_cases[of a b]) auto |
|
590 |
||
591 |
lemma extreal_mult_eq_MInfty[simp]: |
|
592 |
shows "a * b = -\<infinity> \<longleftrightarrow> |
|
593 |
(a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)" |
|
594 |
by (cases rule: extreal2_cases[of a b]) auto |
|
595 |
||
596 |
lemma extreal_0_less_1[simp]: "0 < (1::extreal)" |
|
597 |
by (simp_all add: zero_extreal_def one_extreal_def) |
|
598 |
||
599 |
lemma extreal_zero_one[simp]: "0 \<noteq> (1::extreal)" |
|
600 |
by (simp_all add: zero_extreal_def one_extreal_def) |
|
601 |
||
602 |
lemma extreal_mult_minus_left[simp]: |
|
603 |
fixes a b :: extreal shows "-a * b = - (a * b)" |
|
604 |
by (cases rule: extreal2_cases[of a b]) auto |
|
605 |
||
606 |
lemma extreal_mult_minus_right[simp]: |
|
607 |
fixes a b :: extreal shows "a * -b = - (a * b)" |
|
608 |
by (cases rule: extreal2_cases[of a b]) auto |
|
609 |
||
610 |
lemma extreal_mult_infty[simp]: |
|
611 |
"a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
|
612 |
by (cases a) auto |
|
613 |
||
614 |
lemma extreal_infty_mult[simp]: |
|
615 |
"\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
|
616 |
by (cases a) auto |
|
617 |
||
618 |
lemma extreal_mult_strict_right_mono: |
|
619 |
assumes "a < b" and "0 < c" "c < \<infinity>" |
|
620 |
shows "a * c < b * c" |
|
621 |
using assms |
|
622 |
by (cases rule: extreal3_cases[of a b c]) |
|
623 |
(auto simp: zero_le_mult_iff extreal_less_PInfty) |
|
624 |
||
625 |
lemma extreal_mult_strict_left_mono: |
|
626 |
"\<lbrakk> a < b ; 0 < c ; c < \<infinity>\<rbrakk> \<Longrightarrow> c * a < c * b" |
|
627 |
using extreal_mult_strict_right_mono by (simp add: mult_commute[of c]) |
|
628 |
||
629 |
lemma extreal_mult_right_mono: |
|
630 |
fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c" |
|
631 |
using assms |
|
632 |
apply (cases "c = 0") apply simp |
|
633 |
by (cases rule: extreal3_cases[of a b c]) |
|
634 |
(auto simp: zero_le_mult_iff extreal_less_PInfty) |
|
635 |
||
636 |
lemma extreal_mult_left_mono: |
|
637 |
fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b" |
|
638 |
using extreal_mult_right_mono by (simp add: mult_commute[of c]) |
|
639 |
||
41978 | 640 |
lemma zero_less_one_extreal[simp]: "0 \<le> (1::extreal)" |
641 |
by (simp add: one_extreal_def zero_extreal_def) |
|
642 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
643 |
lemma extreal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: extreal)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
644 |
by (cases rule: extreal2_cases[of a b]) (auto simp: mult_nonneg_nonneg) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
645 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
646 |
lemma extreal_right_distrib: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
647 |
fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
648 |
by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
649 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
650 |
lemma extreal_left_distrib: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
651 |
fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
652 |
by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
653 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
654 |
lemma extreal_mult_le_0_iff: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
655 |
fixes a b :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
656 |
shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
657 |
by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_le_0_iff) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
658 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
659 |
lemma extreal_zero_le_0_iff: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
660 |
fixes a b :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
661 |
shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
662 |
by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_le_mult_iff) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
663 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
664 |
lemma extreal_mult_less_0_iff: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
665 |
fixes a b :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
666 |
shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
667 |
by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_less_0_iff) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
668 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
669 |
lemma extreal_zero_less_0_iff: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
670 |
fixes a b :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
671 |
shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
672 |
by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_less_mult_iff) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
673 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
674 |
lemma extreal_distrib: |
41978 | 675 |
fixes a b c :: extreal |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
676 |
assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
677 |
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
|
678 |
using assms |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
679 |
by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
680 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
681 |
lemma extreal_le_epsilon: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
682 |
fixes x y :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
683 |
assumes "ALL e. 0 < e --> x <= y + e" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
684 |
shows "x <= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
685 |
proof- |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
686 |
{ assume a: "EX r. y = extreal r" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
687 |
from this obtain r where r_def: "y = extreal r" by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
688 |
{ assume "x=(-\<infinity>)" hence ?thesis by auto } |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
689 |
moreover |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
690 |
{ assume "~(x=(-\<infinity>))" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
691 |
from this obtain p where p_def: "x = extreal p" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
692 |
using a assms[rule_format, of 1] by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
693 |
{ fix e have "0 < e --> p <= r + e" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
694 |
using assms[rule_format, of "extreal e"] p_def r_def by auto } |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
695 |
hence "p <= r" apply (subst field_le_epsilon) by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
696 |
hence ?thesis using r_def p_def by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
697 |
} ultimately have ?thesis by blast |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
698 |
} |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
699 |
moreover |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
700 |
{ assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
701 |
using assms[rule_format, of 1] by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
702 |
} ultimately show ?thesis by (cases y) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
703 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
704 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
705 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
706 |
lemma extreal_le_epsilon2: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
707 |
fixes x y :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
708 |
assumes "ALL e. 0 < e --> x <= y + extreal e" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
709 |
shows "x <= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
710 |
proof- |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
711 |
{ fix e :: extreal assume "e>0" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
712 |
{ assume "e=\<infinity>" hence "x<=y+e" by auto } |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
713 |
moreover |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
714 |
{ assume "e~=\<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
715 |
from this obtain r where "e = extreal r" using `e>0` apply (cases e) by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
716 |
hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
717 |
} ultimately have "x<=y+e" by blast |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
718 |
} from this show ?thesis using extreal_le_epsilon by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
719 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
720 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
721 |
lemma extreal_le_real: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
722 |
fixes x y :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
723 |
assumes "ALL z. x <= extreal z --> y <= extreal z" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
724 |
shows "y <= x" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
725 |
by (metis assms extreal.exhaust extreal_bot extreal_less_eq(1) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
726 |
extreal_less_eq(2) order_refl uminus_extreal.simps(2)) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
727 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
728 |
lemma extreal_le_extreal: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
729 |
fixes x y :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
730 |
assumes "\<And>B. B < x \<Longrightarrow> B <= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
731 |
shows "x <= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
732 |
by (metis assms extreal_dense leD linorder_le_less_linear) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
733 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
734 |
lemma extreal_ge_extreal: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
735 |
fixes x y :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
736 |
assumes "ALL B. B>x --> B >= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
737 |
shows "x >= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
738 |
by (metis assms extreal_dense leD linorder_le_less_linear) |
41978 | 739 |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
740 |
lemma setprod_extreal_0: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
741 |
fixes f :: "'a \<Rightarrow> extreal" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
742 |
shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
743 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
744 |
assume "finite A" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
745 |
then show ?thesis by (induct A) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
746 |
qed auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
747 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
748 |
lemma setprod_extreal_pos: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
749 |
fixes f :: "'a \<Rightarrow> extreal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
750 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
751 |
assume "finite I" from this pos show ?thesis by induct auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
752 |
qed simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
753 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
754 |
lemma setprod_PInf: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
755 |
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
|
756 |
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)" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
757 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
758 |
assume "finite I" from this assms show ?thesis |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
759 |
proof (induct I) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
760 |
case (insert i I) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
761 |
then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_extreal_pos) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
762 |
from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
763 |
also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
764 |
using setprod_extreal_pos[of I f] pos |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
765 |
by (cases rule: extreal2_cases[of "f i" "setprod f I"]) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
766 |
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)" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
767 |
using insert by (auto simp: setprod_extreal_0) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
768 |
finally show ?case . |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
769 |
qed simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
770 |
qed simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
771 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
772 |
lemma setprod_extreal: "(\<Prod>i\<in>A. extreal (f i)) = extreal (setprod f A)" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
773 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
774 |
assume "finite A" then show ?thesis |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
775 |
by induct (auto simp: one_extreal_def) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
776 |
qed (simp add: one_extreal_def) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
777 |
|
41978 | 778 |
subsubsection {* Power *} |
779 |
||
780 |
instantiation extreal :: power |
|
781 |
begin |
|
782 |
primrec power_extreal where |
|
783 |
"power_extreal x 0 = 1" | |
|
784 |
"power_extreal x (Suc n) = x * x ^ n" |
|
785 |
instance .. |
|
786 |
end |
|
787 |
||
788 |
lemma extreal_power[simp]: "(extreal x) ^ n = extreal (x^n)" |
|
789 |
by (induct n) (auto simp: one_extreal_def) |
|
790 |
||
791 |
lemma extreal_power_PInf[simp]: "\<infinity> ^ n = (if n = 0 then 1 else \<infinity>)" |
|
792 |
by (induct n) (auto simp: one_extreal_def) |
|
793 |
||
794 |
lemma extreal_power_uminus[simp]: |
|
795 |
fixes x :: extreal |
|
796 |
shows "(- x) ^ n = (if even n then x ^ n else - (x^n))" |
|
797 |
by (induct n) (auto simp: one_extreal_def) |
|
798 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
799 |
lemma extreal_power_number_of[simp]: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
800 |
"(number_of num :: extreal) ^ n = extreal (number_of num ^ n)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
801 |
by (induct n) (auto simp: one_extreal_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
802 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
803 |
lemma zero_le_power_extreal[simp]: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
804 |
fixes a :: extreal assumes "0 \<le> a" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
805 |
shows "0 \<le> a ^ n" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
806 |
using assms by (induct n) (auto simp: extreal_zero_le_0_iff) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
807 |
|
41973 | 808 |
subsubsection {* Subtraction *} |
809 |
||
810 |
lemma extreal_minus_minus_image[simp]: |
|
811 |
fixes S :: "extreal set" |
|
812 |
shows "uminus ` uminus ` S = S" |
|
813 |
by (auto simp: image_iff) |
|
814 |
||
815 |
lemma extreal_uminus_lessThan[simp]: |
|
816 |
fixes a :: extreal shows "uminus ` {..<a} = {-a<..}" |
|
817 |
proof (safe intro!: image_eqI) |
|
818 |
fix x assume "-a < x" |
|
819 |
then have "- x < - (- a)" by (simp del: extreal_uminus_uminus) |
|
820 |
then show "- x < a" by simp |
|
821 |
qed auto |
|
822 |
||
823 |
lemma extreal_uminus_greaterThan[simp]: |
|
824 |
"uminus ` {(a::extreal)<..} = {..<-a}" |
|
825 |
by (metis extreal_uminus_lessThan extreal_uminus_uminus |
|
826 |
extreal_minus_minus_image) |
|
827 |
||
828 |
instantiation extreal :: minus |
|
829 |
begin |
|
830 |
definition "x - y = x + -(y::extreal)" |
|
831 |
instance .. |
|
832 |
end |
|
833 |
||
834 |
lemma extreal_minus[simp]: |
|
835 |
"extreal r - extreal p = extreal (r - p)" |
|
836 |
"-\<infinity> - extreal r = -\<infinity>" |
|
837 |
"extreal r - \<infinity> = -\<infinity>" |
|
838 |
"\<infinity> - x = \<infinity>" |
|
839 |
"-\<infinity> - \<infinity> = -\<infinity>" |
|
840 |
"x - -y = x + y" |
|
841 |
"x - 0 = x" |
|
842 |
"0 - x = -x" |
|
843 |
by (simp_all add: minus_extreal_def) |
|
844 |
||
845 |
lemma extreal_x_minus_x[simp]: |
|
41976 | 846 |
"x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0)" |
41973 | 847 |
by (cases x) simp_all |
848 |
||
849 |
lemma extreal_eq_minus_iff: |
|
850 |
fixes x y z :: extreal |
|
851 |
shows "x = z - y \<longleftrightarrow> |
|
41976 | 852 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and> |
41973 | 853 |
(y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and> |
854 |
(y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and> |
|
855 |
(y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)" |
|
856 |
by (cases rule: extreal3_cases[of x y z]) auto |
|
857 |
||
858 |
lemma extreal_eq_minus: |
|
859 |
fixes x y z :: extreal |
|
41976 | 860 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z" |
861 |
by (auto simp: extreal_eq_minus_iff) |
|
41973 | 862 |
|
863 |
lemma extreal_less_minus_iff: |
|
864 |
fixes x y z :: extreal |
|
865 |
shows "x < z - y \<longleftrightarrow> |
|
866 |
(y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and> |
|
867 |
(y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and> |
|
41976 | 868 |
(\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)" |
41973 | 869 |
by (cases rule: extreal3_cases[of x y z]) auto |
870 |
||
871 |
lemma extreal_less_minus: |
|
872 |
fixes x y z :: extreal |
|
41976 | 873 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z" |
874 |
by (auto simp: extreal_less_minus_iff) |
|
41973 | 875 |
|
876 |
lemma extreal_le_minus_iff: |
|
877 |
fixes x y z :: extreal |
|
878 |
shows "x \<le> z - y \<longleftrightarrow> |
|
879 |
(y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> |
|
41976 | 880 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)" |
41973 | 881 |
by (cases rule: extreal3_cases[of x y z]) auto |
882 |
||
883 |
lemma extreal_le_minus: |
|
884 |
fixes x y z :: extreal |
|
41976 | 885 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z" |
886 |
by (auto simp: extreal_le_minus_iff) |
|
41973 | 887 |
|
888 |
lemma extreal_minus_less_iff: |
|
889 |
fixes x y z :: extreal |
|
890 |
shows "x - y < z \<longleftrightarrow> |
|
891 |
y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> |
|
892 |
(y \<noteq> \<infinity> \<longrightarrow> x < z + y)" |
|
893 |
by (cases rule: extreal3_cases[of x y z]) auto |
|
894 |
||
895 |
lemma extreal_minus_less: |
|
896 |
fixes x y z :: extreal |
|
41976 | 897 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y" |
898 |
by (auto simp: extreal_minus_less_iff) |
|
41973 | 899 |
|
900 |
lemma extreal_minus_le_iff: |
|
901 |
fixes x y z :: extreal |
|
902 |
shows "x - y \<le> z \<longleftrightarrow> |
|
903 |
(y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
904 |
(y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
41976 | 905 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)" |
41973 | 906 |
by (cases rule: extreal3_cases[of x y z]) auto |
907 |
||
908 |
lemma extreal_minus_le: |
|
909 |
fixes x y z :: extreal |
|
41976 | 910 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y" |
911 |
by (auto simp: extreal_minus_le_iff) |
|
41973 | 912 |
|
913 |
lemma extreal_minus_eq_minus_iff: |
|
914 |
fixes a b c :: extreal |
|
915 |
shows "a - b = a - c \<longleftrightarrow> |
|
916 |
b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)" |
|
917 |
by (cases rule: extreal3_cases[of a b c]) auto |
|
918 |
||
919 |
lemma extreal_add_le_add_iff: |
|
920 |
"c + a \<le> c + b \<longleftrightarrow> |
|
921 |
a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)" |
|
922 |
by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps) |
|
923 |
||
924 |
lemma extreal_mult_le_mult_iff: |
|
41976 | 925 |
"\<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)" |
41973 | 926 |
by (cases rule: extreal3_cases[of a b c]) (simp_all add: mult_le_cancel_left) |
927 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
928 |
lemma extreal_minus_mono: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
929 |
fixes A B C D :: extreal assumes "A \<le> B" "D \<le> C" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
930 |
shows "A - C \<le> B - D" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
931 |
using assms |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
932 |
by (cases rule: extreal3_cases[case_product extreal_cases, of A B C D]) simp_all |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
933 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
934 |
lemma real_of_extreal_minus: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
935 |
"real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
936 |
by (cases rule: extreal2_cases[of a b]) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
937 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
938 |
lemma extreal_diff_positive: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
939 |
fixes a b :: extreal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
940 |
by (cases rule: extreal2_cases[of a b]) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
941 |
|
41973 | 942 |
lemma extreal_between: |
943 |
fixes x e :: extreal |
|
41976 | 944 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e" |
41973 | 945 |
shows "x - e < x" "x < x + e" |
946 |
using assms apply (cases x, cases e) apply auto |
|
947 |
using assms by (cases x, cases e) auto |
|
948 |
||
949 |
subsubsection {* Division *} |
|
950 |
||
951 |
instantiation extreal :: inverse |
|
952 |
begin |
|
953 |
||
954 |
function inverse_extreal where |
|
955 |
"inverse (extreal r) = (if r = 0 then \<infinity> else extreal (inverse r))" | |
|
956 |
"inverse \<infinity> = 0" | |
|
957 |
"inverse (-\<infinity>) = 0" |
|
958 |
by (auto intro: extreal_cases) |
|
959 |
termination by (relation "{}") simp |
|
960 |
||
961 |
definition "x / y = x * inverse (y :: extreal)" |
|
962 |
||
963 |
instance proof qed |
|
964 |
end |
|
965 |
||
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
966 |
lemma real_of_extreal_inverse[simp]: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
967 |
fixes a :: extreal |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
968 |
shows "real (inverse a) = 1 / real a" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
969 |
by (cases a) (auto simp: inverse_eq_divide) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
970 |
|
41973 | 971 |
lemma extreal_inverse[simp]: |
972 |
"inverse 0 = \<infinity>" |
|
973 |
"inverse (1::extreal) = 1" |
|
974 |
by (simp_all add: one_extreal_def zero_extreal_def) |
|
975 |
||
976 |
lemma extreal_divide[simp]: |
|
977 |
"extreal r / extreal p = (if p = 0 then extreal r * \<infinity> else extreal (r / p))" |
|
978 |
unfolding divide_extreal_def by (auto simp: divide_real_def) |
|
979 |
||
980 |
lemma extreal_divide_same[simp]: |
|
41976 | 981 |
"x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)" |
41973 | 982 |
by (cases x) |
983 |
(simp_all add: divide_real_def divide_extreal_def one_extreal_def) |
|
984 |
||
985 |
lemma extreal_inv_inv[simp]: |
|
986 |
"inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)" |
|
987 |
by (cases x) auto |
|
988 |
||
989 |
lemma extreal_inverse_minus[simp]: |
|
990 |
"inverse (- x) = (if x = 0 then \<infinity> else -inverse x)" |
|
991 |
by (cases x) simp_all |
|
992 |
||
993 |
lemma extreal_uminus_divide[simp]: |
|
994 |
fixes x y :: extreal shows "- x / y = - (x / y)" |
|
995 |
unfolding divide_extreal_def by simp |
|
996 |
||
997 |
lemma extreal_divide_Infty[simp]: |
|
998 |
"x / \<infinity> = 0" "x / -\<infinity> = 0" |
|
999 |
unfolding divide_extreal_def by simp_all |
|
1000 |
||
1001 |
lemma extreal_divide_one[simp]: |
|
1002 |
"x / 1 = (x::extreal)" |
|
1003 |
unfolding divide_extreal_def by simp |
|
1004 |
||
1005 |
lemma extreal_divide_extreal[simp]: |
|
1006 |
"\<infinity> / extreal r = (if 0 \<le> r then \<infinity> else -\<infinity>)" |
|
1007 |
unfolding divide_extreal_def by simp |
|
1008 |
||
41978 | 1009 |
lemma zero_le_divide_extreal[simp]: |
1010 |
fixes a :: extreal assumes "0 \<le> a" "0 \<le> b" |
|
1011 |
shows "0 \<le> a / b" |
|
1012 |
using assms by (cases rule: extreal2_cases[of a b]) (auto simp: zero_le_divide_iff) |
|
1013 |
||
41973 | 1014 |
lemma extreal_le_divide_pos: |
1015 |
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z" |
|
1016 |
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1017 |
||
1018 |
lemma extreal_divide_le_pos: |
|
1019 |
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y" |
|
1020 |
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1021 |
||
1022 |
lemma extreal_le_divide_neg: |
|
1023 |
"x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y" |
|
1024 |
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1025 |
||
1026 |
lemma extreal_divide_le_neg: |
|
1027 |
"x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z" |
|
1028 |
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1029 |
||
1030 |
lemma extreal_inverse_antimono_strict: |
|
1031 |
fixes x y :: extreal |
|
1032 |
shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x" |
|
1033 |
by (cases rule: extreal2_cases[of x y]) auto |
|
1034 |
||
1035 |
lemma extreal_inverse_antimono: |
|
1036 |
fixes x y :: extreal |
|
1037 |
shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x" |
|
1038 |
by (cases rule: extreal2_cases[of x y]) auto |
|
1039 |
||
1040 |
lemma inverse_inverse_Pinfty_iff[simp]: |
|
1041 |
"inverse x = \<infinity> \<longleftrightarrow> x = 0" |
|
1042 |
by (cases x) auto |
|
1043 |
||
1044 |
lemma extreal_inverse_eq_0: |
|
1045 |
"inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>" |
|
1046 |
by (cases x) auto |
|
1047 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1048 |
lemma extreal_0_gt_inverse: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1049 |
fixes x :: extreal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1050 |
by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1051 |
|
41973 | 1052 |
lemma extreal_mult_less_right: |
1053 |
assumes "b * a < c * a" "0 < a" "a < \<infinity>" |
|
1054 |
shows "b < c" |
|
1055 |
using assms |
|
1056 |
by (cases rule: extreal3_cases[of a b c]) |
|
1057 |
(auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff) |
|
1058 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1059 |
lemma extreal_power_divide: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1060 |
"y \<noteq> 0 \<Longrightarrow> (x / y :: extreal) ^ n = x^n / y^n" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1061 |
by (cases rule: extreal2_cases[of x y]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1062 |
(auto simp: one_extreal_def zero_extreal_def power_divide not_le |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1063 |
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
|
1064 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1065 |
lemma extreal_le_mult_one_interval: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1066 |
fixes x y :: extreal |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1067 |
assumes y: "y \<noteq> -\<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1068 |
assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1069 |
shows "x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1070 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1071 |
case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_extreal_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1072 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1073 |
case (real r) note r = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1074 |
show "x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1075 |
proof (cases y) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1076 |
case (real p) note p = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1077 |
have "r \<le> p" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1078 |
proof (rule field_le_mult_one_interval) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1079 |
fix z :: real assume "0 < z" and "z < 1" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1080 |
with z[of "extreal z"] |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1081 |
show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_extreal_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1082 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1083 |
then show "x \<le> y" using p r by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1084 |
qed (insert y, simp_all) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1085 |
qed simp |
41978 | 1086 |
|
41973 | 1087 |
subsection "Complete lattice" |
1088 |
||
1089 |
instantiation extreal :: lattice |
|
1090 |
begin |
|
1091 |
definition [simp]: "sup x y = (max x y :: extreal)" |
|
1092 |
definition [simp]: "inf x y = (min x y :: extreal)" |
|
1093 |
instance proof qed simp_all |
|
1094 |
end |
|
1095 |
||
1096 |
instantiation extreal :: complete_lattice |
|
1097 |
begin |
|
1098 |
||
41976 | 1099 |
definition "bot = -\<infinity>" |
41973 | 1100 |
definition "top = \<infinity>" |
1101 |
||
1102 |
definition "Sup S = (LEAST z. ALL x:S. x <= z :: extreal)" |
|
1103 |
definition "Inf S = (GREATEST z. ALL x:S. z <= x :: extreal)" |
|
1104 |
||
1105 |
lemma extreal_complete_Sup: |
|
1106 |
fixes S :: "extreal set" assumes "S \<noteq> {}" |
|
1107 |
shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)" |
|
1108 |
proof cases |
|
1109 |
assume "\<exists>x. \<forall>a\<in>S. a \<le> extreal x" |
|
1110 |
then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> extreal y" by auto |
|
1111 |
then have "\<infinity> \<notin> S" by force |
|
1112 |
show ?thesis |
|
1113 |
proof cases |
|
1114 |
assume "S = {-\<infinity>}" |
|
1115 |
then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"]) |
|
1116 |
next |
|
1117 |
assume "S \<noteq> {-\<infinity>}" |
|
1118 |
with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto |
|
1119 |
with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y" |
|
1120 |
by (auto simp: real_of_extreal_ord_simps) |
|
1121 |
with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}` |
|
1122 |
obtain s where s: |
|
1123 |
"\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z" |
|
1124 |
by auto |
|
1125 |
show ?thesis |
|
1126 |
proof (safe intro!: exI[of _ "extreal s"]) |
|
1127 |
fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> extreal s" |
|
1128 |
proof (cases z) |
|
1129 |
case (real r) |
|
1130 |
then show ?thesis |
|
1131 |
using s(1)[rule_format, of z] `z \<in> S` `z = extreal r` by auto |
|
1132 |
qed auto |
|
1133 |
next |
|
1134 |
fix z assume *: "\<forall>y\<in>S. y \<le> z" |
|
1135 |
with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "extreal s \<le> z" |
|
1136 |
proof (cases z) |
|
1137 |
case (real u) |
|
1138 |
with * have "s \<le> u" |
|
1139 |
by (intro s(2)[of u]) (auto simp: real_of_extreal_ord_simps) |
|
1140 |
then show ?thesis using real by simp |
|
1141 |
qed auto |
|
1142 |
qed |
|
1143 |
qed |
|
1144 |
next |
|
1145 |
assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> extreal x)" |
|
1146 |
show ?thesis |
|
1147 |
proof (safe intro!: exI[of _ \<infinity>]) |
|
1148 |
fix y assume **: "\<forall>z\<in>S. z \<le> y" |
|
1149 |
with * show "\<infinity> \<le> y" |
|
1150 |
proof (cases y) |
|
1151 |
case MInf with * ** show ?thesis by (force simp: not_le) |
|
1152 |
qed auto |
|
1153 |
qed simp |
|
1154 |
qed |
|
1155 |
||
1156 |
lemma extreal_complete_Inf: |
|
1157 |
fixes S :: "extreal set" assumes "S ~= {}" |
|
1158 |
shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)" |
|
1159 |
proof- |
|
1160 |
def S1 == "uminus ` S" |
|
1161 |
hence "S1 ~= {}" using assms by auto |
|
1162 |
from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)" |
|
1163 |
using extreal_complete_Sup[of S1] by auto |
|
1164 |
{ fix z assume "ALL y:S. z <= y" |
|
1165 |
hence "ALL y:S1. y <= -z" unfolding S1_def by auto |
|
1166 |
hence "x <= -z" using x_def by auto |
|
1167 |
hence "z <= -x" |
|
1168 |
apply (subst extreal_uminus_uminus[symmetric]) |
|
1169 |
unfolding extreal_minus_le_minus . } |
|
1170 |
moreover have "(ALL y:S. -x <= y)" |
|
1171 |
using x_def unfolding S1_def |
|
1172 |
apply simp |
|
1173 |
apply (subst (3) extreal_uminus_uminus[symmetric]) |
|
1174 |
unfolding extreal_minus_le_minus by simp |
|
1175 |
ultimately show ?thesis by auto |
|
1176 |
qed |
|
1177 |
||
1178 |
lemma extreal_complete_uminus_eq: |
|
1179 |
fixes S :: "extreal set" |
|
1180 |
shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z) |
|
1181 |
\<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)" |
|
1182 |
by simp (metis extreal_minus_le_minus extreal_uminus_uminus) |
|
1183 |
||
1184 |
lemma extreal_Sup_uminus_image_eq: |
|
1185 |
fixes S :: "extreal set" |
|
1186 |
shows "Sup (uminus ` S) = - Inf S" |
|
1187 |
proof cases |
|
1188 |
assume "S = {}" |
|
1189 |
moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::extreal)" |
|
1190 |
by (rule the_equality) (auto intro!: extreal_bot) |
|
1191 |
moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::extreal)" |
|
1192 |
by (rule some_equality) (auto intro!: extreal_top) |
|
1193 |
ultimately show ?thesis unfolding Inf_extreal_def Sup_extreal_def |
|
1194 |
Least_def Greatest_def GreatestM_def by simp |
|
1195 |
next |
|
1196 |
assume "S \<noteq> {}" |
|
1197 |
with extreal_complete_Sup[of "uminus`S"] |
|
1198 |
obtain x where x: "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)" |
|
1199 |
unfolding extreal_complete_uminus_eq by auto |
|
1200 |
show "Sup (uminus ` S) = - Inf S" |
|
1201 |
unfolding Inf_extreal_def Greatest_def GreatestM_def |
|
1202 |
proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"]) |
|
1203 |
show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)" |
|
1204 |
using x . |
|
1205 |
fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')" |
|
1206 |
then have "(\<forall>y\<in>uminus`S. y \<le> - x') \<and> (\<forall>y. (\<forall>z\<in>uminus`S. z \<le> y) \<longrightarrow> - x' \<le> y)" |
|
1207 |
unfolding extreal_complete_uminus_eq by simp |
|
1208 |
then show "Sup (uminus ` S) = -x'" |
|
1209 |
unfolding Sup_extreal_def extreal_uminus_eq_iff |
|
1210 |
by (intro Least_equality) auto |
|
1211 |
qed |
|
1212 |
qed |
|
1213 |
||
1214 |
instance |
|
1215 |
proof |
|
1216 |
{ fix x :: extreal and A |
|
1217 |
show "bot <= x" by (cases x) (simp_all add: bot_extreal_def) |
|
1218 |
show "x <= top" by (simp add: top_extreal_def) } |
|
1219 |
||
1220 |
{ fix x :: extreal and A assume "x : A" |
|
1221 |
with extreal_complete_Sup[of A] |
|
1222 |
obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto |
|
1223 |
hence "x <= s" using `x : A` by auto |
|
1224 |
also have "... = Sup A" using s unfolding Sup_extreal_def |
|
1225 |
by (auto intro!: Least_equality[symmetric]) |
|
1226 |
finally show "x <= Sup A" . } |
|
1227 |
note le_Sup = this |
|
1228 |
||
1229 |
{ fix x :: extreal and A assume *: "!!z. (z : A ==> z <= x)" |
|
1230 |
show "Sup A <= x" |
|
1231 |
proof (cases "A = {}") |
|
1232 |
case True |
|
1233 |
hence "Sup A = -\<infinity>" unfolding Sup_extreal_def |
|
1234 |
by (auto intro!: Least_equality) |
|
1235 |
thus "Sup A <= x" by simp |
|
1236 |
next |
|
1237 |
case False |
|
1238 |
with extreal_complete_Sup[of A] |
|
1239 |
obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto |
|
1240 |
hence "Sup A = s" |
|
1241 |
unfolding Sup_extreal_def by (auto intro!: Least_equality) |
|
1242 |
also have "s <= x" using * s by auto |
|
1243 |
finally show "Sup A <= x" . |
|
1244 |
qed } |
|
1245 |
note Sup_le = this |
|
1246 |
||
1247 |
{ fix x :: extreal and A assume "x \<in> A" |
|
1248 |
with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x" |
|
1249 |
unfolding extreal_Sup_uminus_image_eq by simp } |
|
1250 |
||
1251 |
{ fix x :: extreal and A assume *: "!!z. (z : A ==> x <= z)" |
|
1252 |
with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A" |
|
1253 |
unfolding extreal_Sup_uminus_image_eq by force } |
|
1254 |
qed |
|
1255 |
end |
|
1256 |
||
1257 |
lemma extreal_SUPR_uminus: |
|
1258 |
fixes f :: "'a => extreal" |
|
1259 |
shows "(SUP i : R. -(f i)) = -(INF i : R. f i)" |
|
1260 |
unfolding SUPR_def INFI_def |
|
1261 |
using extreal_Sup_uminus_image_eq[of "f`R"] |
|
1262 |
by (simp add: image_image) |
|
1263 |
||
1264 |
lemma extreal_INFI_uminus: |
|
1265 |
fixes f :: "'a => extreal" |
|
1266 |
shows "(INF i : R. -(f i)) = -(SUP i : R. f i)" |
|
1267 |
using extreal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp |
|
1268 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1269 |
lemma extreal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::extreal set)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1270 |
using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1271 |
|
41973 | 1272 |
lemma extreal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: extreal set)" |
1273 |
by (auto intro!: inj_onI) |
|
1274 |
||
1275 |
lemma extreal_image_uminus_shift: |
|
1276 |
fixes X Y :: "extreal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y" |
|
1277 |
proof |
|
1278 |
assume "uminus ` X = Y" |
|
1279 |
then have "uminus ` uminus ` X = uminus ` Y" |
|
1280 |
by (simp add: inj_image_eq_iff) |
|
1281 |
then show "X = uminus ` Y" by (simp add: image_image) |
|
1282 |
qed (simp add: image_image) |
|
1283 |
||
1284 |
lemma Inf_extreal_iff: |
|
1285 |
fixes z :: extreal |
|
1286 |
shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y" |
|
1287 |
by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear |
|
1288 |
order_less_le_trans) |
|
1289 |
||
1290 |
lemma Sup_eq_MInfty: |
|
1291 |
fixes S :: "extreal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}" |
|
1292 |
proof |
|
1293 |
assume a: "Sup S = -\<infinity>" |
|
1294 |
with complete_lattice_class.Sup_upper[of _ S] |
|
1295 |
show "S={} \<or> S={-\<infinity>}" by auto |
|
1296 |
next |
|
1297 |
assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>" |
|
1298 |
unfolding Sup_extreal_def by (auto intro!: Least_equality) |
|
1299 |
qed |
|
1300 |
||
1301 |
lemma Inf_eq_PInfty: |
|
1302 |
fixes S :: "extreal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}" |
|
1303 |
using Sup_eq_MInfty[of "uminus`S"] |
|
1304 |
unfolding extreal_Sup_uminus_image_eq extreal_image_uminus_shift by simp |
|
1305 |
||
1306 |
lemma Inf_eq_MInfty: "-\<infinity> : S ==> Inf S = -\<infinity>" |
|
1307 |
unfolding Inf_extreal_def |
|
1308 |
by (auto intro!: Greatest_equality) |
|
1309 |
||
1310 |
lemma Sup_eq_PInfty: "\<infinity> : S ==> Sup S = \<infinity>" |
|
1311 |
unfolding Sup_extreal_def |
|
1312 |
by (auto intro!: Least_equality) |
|
1313 |
||
1314 |
lemma extreal_SUPI: |
|
1315 |
fixes x :: extreal |
|
1316 |
assumes "!!i. i : A ==> f i <= x" |
|
1317 |
assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y" |
|
1318 |
shows "(SUP i:A. f i) = x" |
|
1319 |
unfolding SUPR_def Sup_extreal_def |
|
1320 |
using assms by (auto intro!: Least_equality) |
|
1321 |
||
1322 |
lemma extreal_INFI: |
|
1323 |
fixes x :: extreal |
|
1324 |
assumes "!!i. i : A ==> f i >= x" |
|
1325 |
assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y" |
|
1326 |
shows "(INF i:A. f i) = x" |
|
1327 |
unfolding INFI_def Inf_extreal_def |
|
1328 |
using assms by (auto intro!: Greatest_equality) |
|
1329 |
||
1330 |
lemma Sup_extreal_close: |
|
1331 |
fixes e :: extreal |
|
41976 | 1332 |
assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}" |
41973 | 1333 |
shows "\<exists>x\<in>S. Sup S - e < x" |
41976 | 1334 |
using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1]) |
41973 | 1335 |
|
1336 |
lemma Inf_extreal_close: |
|
41976 | 1337 |
fixes e :: extreal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e" |
41973 | 1338 |
shows "\<exists>x\<in>X. x < Inf X + e" |
1339 |
proof (rule Inf_less_iff[THEN iffD1]) |
|
1340 |
show "Inf X < Inf X + e" using assms |
|
41976 | 1341 |
by (cases e) auto |
41973 | 1342 |
qed |
1343 |
||
1344 |
lemma Sup_eq_top_iff: |
|
1345 |
fixes A :: "'a::{complete_lattice, linorder} set" |
|
1346 |
shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)" |
|
1347 |
proof |
|
1348 |
assume *: "Sup A = top" |
|
1349 |
show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric] |
|
1350 |
proof (intro allI impI) |
|
1351 |
fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i" |
|
1352 |
unfolding less_Sup_iff by auto |
|
1353 |
qed |
|
1354 |
next |
|
1355 |
assume *: "\<forall>x<top. \<exists>i\<in>A. x < i" |
|
1356 |
show "Sup A = top" |
|
1357 |
proof (rule ccontr) |
|
1358 |
assume "Sup A \<noteq> top" |
|
1359 |
with top_greatest[of "Sup A"] |
|
1360 |
have "Sup A < top" unfolding le_less by auto |
|
1361 |
then have "Sup A < Sup A" |
|
1362 |
using * unfolding less_Sup_iff by auto |
|
1363 |
then show False by auto |
|
1364 |
qed |
|
1365 |
qed |
|
1366 |
||
1367 |
lemma SUP_eq_top_iff: |
|
1368 |
fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}" |
|
1369 |
shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)" |
|
1370 |
unfolding SUPR_def Sup_eq_top_iff by auto |
|
1371 |
||
1372 |
lemma SUP_nat_Infty: "(SUP i::nat. extreal (real i)) = \<infinity>" |
|
1373 |
proof - |
|
1374 |
{ fix x assume "x \<noteq> \<infinity>" |
|
1375 |
then have "\<exists>k::nat. x < extreal (real k)" |
|
1376 |
proof (cases x) |
|
1377 |
case MInf then show ?thesis by (intro exI[of _ 0]) auto |
|
1378 |
next |
|
1379 |
case (real r) |
|
1380 |
moreover obtain k :: nat where "r < real k" |
|
1381 |
using ex_less_of_nat by (auto simp: real_eq_of_nat) |
|
1382 |
ultimately show ?thesis by auto |
|
1383 |
qed simp } |
|
1384 |
then show ?thesis |
|
1385 |
using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. extreal (real n)"] |
|
1386 |
by (auto simp: top_extreal_def) |
|
1387 |
qed |
|
1388 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1389 |
lemma extreal_le_Sup: |
41973 | 1390 |
fixes x :: extreal |
1391 |
shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))" |
|
1392 |
(is "?lhs <-> ?rhs") |
|
1393 |
proof- |
|
1394 |
{ assume "?rhs" |
|
1395 |
{ assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le) |
|
1396 |
from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using extreal_dense by auto |
|
1397 |
from this obtain i where "i : A & y <= f i" using `?rhs` by auto |
|
1398 |
hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto |
|
1399 |
hence False using y_def by auto |
|
1400 |
} hence "?lhs" by auto |
|
1401 |
} |
|
1402 |
moreover |
|
1403 |
{ assume "?lhs" hence "?rhs" |
|
1404 |
by (metis Collect_def Collect_mem_eq SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff |
|
1405 |
inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8)) |
|
1406 |
} ultimately show ?thesis by auto |
|
1407 |
qed |
|
1408 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1409 |
lemma extreal_Inf_le: |
41973 | 1410 |
fixes x :: extreal |
1411 |
shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))" |
|
1412 |
(is "?lhs <-> ?rhs") |
|
1413 |
proof- |
|
1414 |
{ assume "?rhs" |
|
1415 |
{ assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le) |
|
1416 |
from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using extreal_dense by auto |
|
1417 |
from this obtain i where "i : A & f i <= y" using `?rhs` by auto |
|
1418 |
hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto |
|
1419 |
hence False using y_def by auto |
|
1420 |
} hence "?lhs" by auto |
|
1421 |
} |
|
1422 |
moreover |
|
1423 |
{ assume "?lhs" hence "?rhs" |
|
1424 |
by (metis Collect_def Collect_mem_eq le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff |
|
1425 |
inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8)) |
|
1426 |
} ultimately show ?thesis by auto |
|
1427 |
qed |
|
1428 |
||
1429 |
lemma Inf_less: |
|
1430 |
fixes x :: extreal |
|
1431 |
assumes "(INF i:A. f i) < x" |
|
1432 |
shows "EX i. i : A & f i <= x" |
|
1433 |
proof(rule ccontr) |
|
1434 |
assume "~ (EX i. i : A & f i <= x)" |
|
1435 |
hence "ALL i:A. f i > x" by auto |
|
1436 |
hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto |
|
1437 |
thus False using assms by auto |
|
1438 |
qed |
|
1439 |
||
1440 |
lemma same_INF: |
|
1441 |
assumes "ALL e:A. f e = g e" |
|
1442 |
shows "(INF e:A. f e) = (INF e:A. g e)" |
|
1443 |
proof- |
|
1444 |
have "f ` A = g ` A" unfolding image_def using assms by auto |
|
1445 |
thus ?thesis unfolding INFI_def by auto |
|
1446 |
qed |
|
1447 |
||
1448 |
lemma same_SUP: |
|
1449 |
assumes "ALL e:A. f e = g e" |
|
1450 |
shows "(SUP e:A. f e) = (SUP e:A. g e)" |
|
1451 |
proof- |
|
1452 |
have "f ` A = g ` A" unfolding image_def using assms by auto |
|
1453 |
thus ?thesis unfolding SUPR_def by auto |
|
1454 |
qed |
|
1455 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1456 |
lemma SUPR_eq: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1457 |
assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1458 |
assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1459 |
shows "(SUP i:A. f i) = (SUP j:B. g j)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1460 |
proof (intro antisym) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1461 |
show "(SUP i:A. f i) \<le> (SUP j:B. g j)" |
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1462 |
using assms by (metis SUP_leI le_SUPI2) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1463 |
show "(SUP i:B. g i) \<le> (SUP j:A. f j)" |
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1464 |
using assms by (metis SUP_leI le_SUPI2) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1465 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1466 |
|
41978 | 1467 |
lemma SUP_extreal_le_addI: |
1468 |
assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>" |
|
1469 |
shows "SUPR UNIV f + y \<le> z" |
|
1470 |
proof (cases y) |
|
1471 |
case (real r) |
|
1472 |
then have "\<And>i. f i \<le> z - y" using assms by (simp add: extreal_le_minus_iff) |
|
1473 |
then have "SUPR UNIV f \<le> z - y" by (rule SUP_leI) |
|
1474 |
then show ?thesis using real by (simp add: extreal_le_minus_iff) |
|
1475 |
qed (insert assms, auto) |
|
1476 |
||
1477 |
lemma SUPR_extreal_add: |
|
1478 |
fixes f g :: "nat \<Rightarrow> extreal" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1479 |
assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>" |
41978 | 1480 |
shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g" |
1481 |
proof (rule extreal_SUPI) |
|
1482 |
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y" |
|
1483 |
have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos |
|
1484 |
unfolding SUPR_def Sup_eq_MInfty by (auto dest: image_eqD) |
|
1485 |
{ fix j |
|
1486 |
{ fix i |
|
1487 |
have "f i + g j \<le> f i + g (max i j)" |
|
1488 |
using `incseq g`[THEN incseqD] by (rule add_left_mono) auto |
|
1489 |
also have "\<dots> \<le> f (max i j) + g (max i j)" |
|
1490 |
using `incseq f`[THEN incseqD] by (rule add_right_mono) auto |
|
1491 |
also have "\<dots> \<le> y" using * by auto |
|
1492 |
finally have "f i + g j \<le> y" . } |
|
1493 |
then have "SUPR UNIV f + g j \<le> y" |
|
1494 |
using assms(4)[of j] by (intro SUP_extreal_le_addI) auto |
|
1495 |
then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) } |
|
1496 |
then have "SUPR UNIV g + SUPR UNIV f \<le> y" |
|
1497 |
using f by (rule SUP_extreal_le_addI) |
|
1498 |
then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps) |
|
1499 |
qed (auto intro!: add_mono le_SUPI) |
|
1500 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1501 |
lemma SUPR_extreal_add_pos: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1502 |
fixes f g :: "nat \<Rightarrow> extreal" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1503 |
assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1504 |
shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1505 |
proof (intro SUPR_extreal_add inc) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1506 |
fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1507 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1508 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1509 |
lemma SUPR_extreal_setsum: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1510 |
fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> extreal" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1511 |
assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1512 |
shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1513 |
proof cases |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1514 |
assume "finite A" then show ?thesis using assms |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1515 |
by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_extreal_add_pos) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1516 |
qed simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1517 |
|
41978 | 1518 |
lemma SUPR_extreal_cmult: |
1519 |
fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c" |
|
1520 |
shows "(SUP i. c * f i) = c * SUPR UNIV f" |
|
1521 |
proof (rule extreal_SUPI) |
|
1522 |
fix i have "f i \<le> SUPR UNIV f" by (rule le_SUPI) auto |
|
1523 |
then show "c * f i \<le> c * SUPR UNIV f" |
|
1524 |
using `0 \<le> c` by (rule extreal_mult_left_mono) |
|
1525 |
next |
|
1526 |
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y" |
|
1527 |
show "c * SUPR UNIV f \<le> y" |
|
1528 |
proof cases |
|
1529 |
assume c: "0 < c \<and> c \<noteq> \<infinity>" |
|
1530 |
with * have "SUPR UNIV f \<le> y / c" |
|
1531 |
by (intro SUP_leI) (auto simp: extreal_le_divide_pos) |
|
1532 |
with c show ?thesis |
|
1533 |
by (auto simp: extreal_le_divide_pos) |
|
1534 |
next |
|
1535 |
{ assume "c = \<infinity>" have ?thesis |
|
1536 |
proof cases |
|
1537 |
assume "\<forall>i. f i = 0" |
|
1538 |
moreover then have "range f = {0}" by auto |
|
1539 |
ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def) |
|
1540 |
next |
|
1541 |
assume "\<not> (\<forall>i. f i = 0)" |
|
1542 |
then obtain i where "f i \<noteq> 0" by auto |
|
1543 |
with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm) |
|
1544 |
qed } |
|
1545 |
moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)" |
|
1546 |
ultimately show ?thesis using * `0 \<le> c` by auto |
|
1547 |
qed |
|
1548 |
qed |
|
1549 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1550 |
lemma SUP_PInfty: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1551 |
fixes f :: "'a \<Rightarrow> extreal" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1552 |
assumes "\<And>n::nat. \<exists>i\<in>A. extreal (real n) \<le> f i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1553 |
shows "(SUP i:A. f i) = \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1554 |
unfolding SUPR_def Sup_eq_top_iff[where 'a=extreal, unfolded top_extreal_def] |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1555 |
apply simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1556 |
proof safe |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1557 |
fix x assume "x \<noteq> \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1558 |
show "\<exists>i\<in>A. x < f i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1559 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1560 |
case PInf with `x \<noteq> \<infinity>` show ?thesis by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1561 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1562 |
case MInf with assms[of "0"] show ?thesis by force |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1563 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1564 |
case (real r) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1565 |
with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < extreal (real n)" by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1566 |
moreover from assms[of n] guess i .. |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1567 |
ultimately show ?thesis |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1568 |
by (auto intro!: bexI[of _ i]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1569 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1570 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1571 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1572 |
lemma Sup_countable_SUPR: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1573 |
assumes "A \<noteq> {}" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1574 |
shows "\<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1575 |
proof (cases "Sup A") |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1576 |
case (real r) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1577 |
have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1578 |
proof |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1579 |
fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / extreal (real n) < x" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1580 |
using assms real by (intro Sup_extreal_close) (auto simp: one_extreal_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1581 |
then guess x .. |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1582 |
then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1583 |
by (auto intro!: exI[of _ x] simp: extreal_minus_less_iff) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1584 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1585 |
from choice[OF this] guess f .. note f = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1586 |
have "SUPR UNIV f = Sup A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1587 |
proof (rule extreal_SUPI) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1588 |
fix i show "f i \<le> Sup A" using f |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1589 |
by (auto intro!: complete_lattice_class.Sup_upper) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1590 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1591 |
fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1592 |
show "Sup A \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1593 |
proof (rule extreal_le_epsilon, intro allI impI) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1594 |
fix e :: extreal assume "0 < e" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1595 |
show "Sup A \<le> y + e" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1596 |
proof (cases e) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1597 |
case (real r) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1598 |
hence "0 < r" using `0 < e` by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1599 |
then obtain n ::nat where *: "1 / real n < r" "0 < n" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1600 |
using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1601 |
have "Sup A \<le> f n + 1 / extreal (real n)" using f[THEN spec, of n] by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1602 |
also have "1 / extreal (real n) \<le> e" using real * by (auto simp: one_extreal_def ) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1603 |
with bound have "f n + 1 / extreal (real n) \<le> y + e" by (rule add_mono) simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1604 |
finally show "Sup A \<le> y + e" . |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1605 |
qed (insert `0 < e`, auto) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1606 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1607 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1608 |
with f show ?thesis by (auto intro!: exI[of _ f]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1609 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1610 |
case PInf |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1611 |
from `A \<noteq> {}` obtain x where "x \<in> A" by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1612 |
show ?thesis |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1613 |
proof cases |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1614 |
assume "\<infinity> \<in> A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1615 |
moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1616 |
ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1617 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1618 |
assume "\<infinity> \<notin> A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1619 |
have "\<exists>x\<in>A. 0 \<le> x" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1620 |
by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least extreal_infty_less_eq2 linorder_linear) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1621 |
then obtain x where "x \<in> A" "0 \<le> x" by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1622 |
have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + extreal (real n) \<le> f" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1623 |
proof (rule ccontr) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1624 |
assume "\<not> ?thesis" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1625 |
then have "\<exists>n::nat. Sup A \<le> x + extreal (real n)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1626 |
by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1627 |
then show False using `x \<in> A` `\<infinity> \<notin> A` PInf |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1628 |
by(cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1629 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1630 |
from choice[OF this] guess f .. note f = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1631 |
have "SUPR UNIV f = \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1632 |
proof (rule SUP_PInfty) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1633 |
fix n :: nat show "\<exists>i\<in>UNIV. extreal (real n) \<le> f i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1634 |
using f[THEN spec, of n] `0 \<le> x` |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1635 |
by (cases rule: extreal2_cases[of "f n" x]) (auto intro!: exI[of _ n]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1636 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1637 |
then show ?thesis using f PInf by (auto intro!: exI[of _ f]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1638 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1639 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1640 |
case MInf |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1641 |
with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1642 |
then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1643 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1644 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1645 |
lemma SUPR_countable_SUPR: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1646 |
"A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1647 |
using Sup_countable_SUPR[of "g`A"] by (auto simp: SUPR_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1648 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1649 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1650 |
lemma Sup_extreal_cadd: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1651 |
fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1652 |
shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1653 |
proof (rule antisym) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1654 |
have *: "\<And>a::extreal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1655 |
by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1656 |
then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" . |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1657 |
show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1658 |
proof (cases a) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1659 |
case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1660 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1661 |
case (real r) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1662 |
then have **: "op + (- a) ` op + a ` A = A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1663 |
by (auto simp: image_iff ac_simps zero_extreal_def[symmetric]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1664 |
from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding ** |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1665 |
by (cases rule: extreal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1666 |
qed (insert `a \<noteq> -\<infinity>`, auto) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1667 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1668 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1669 |
lemma Sup_extreal_cminus: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1670 |
fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1671 |
shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1672 |
using Sup_extreal_cadd[of "uminus ` A" a] assms |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1673 |
by (simp add: comp_def image_image minus_extreal_def |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1674 |
extreal_Sup_uminus_image_eq) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1675 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1676 |
lemma SUPR_extreal_cminus: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1677 |
fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1678 |
shows "(SUP x:A. a - f x) = a - (INF x:A. f x)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1679 |
using Sup_extreal_cminus[of "f`A" a] assms |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1680 |
unfolding SUPR_def INFI_def image_image by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1681 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1682 |
lemma Inf_extreal_cminus: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1683 |
fixes A :: "extreal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1684 |
shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1685 |
proof - |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1686 |
{ fix x have "-a - -x = -(a - x)" using assms by (cases x) auto } |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1687 |
moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1688 |
by (auto simp: image_image) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1689 |
ultimately show ?thesis |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1690 |
using Sup_extreal_cminus[of "uminus ` A" "-a"] assms |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1691 |
by (auto simp add: extreal_Sup_uminus_image_eq extreal_Inf_uminus_image_eq) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1692 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1693 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1694 |
lemma INFI_extreal_cminus: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1695 |
fixes A assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1696 |
shows "(INF x:A. a - f x) = a - (SUP x:A. f x)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1697 |
using Inf_extreal_cminus[of "f`A" a] assms |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1698 |
unfolding SUPR_def INFI_def image_image |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1699 |
by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1700 |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1701 |
lemma uminus_extreal_add_uminus_uminus: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1702 |
fixes a b :: extreal shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1703 |
by (cases rule: extreal2_cases[of a b]) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1704 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1705 |
lemma INFI_extreal_add: |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1706 |
assumes "decseq f" "decseq g" and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1707 |
shows "(INF i. f i + g i) = INFI UNIV f + INFI UNIV g" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1708 |
proof - |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1709 |
have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1710 |
using assms unfolding INF_less_iff by auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1711 |
{ fix i from fin[of i] have "- ((- f i) + (- g i)) = f i + g i" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1712 |
by (rule uminus_extreal_add_uminus_uminus) } |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1713 |
then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1714 |
by simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1715 |
also have "\<dots> = INFI UNIV f + INFI UNIV g" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1716 |
unfolding extreal_INFI_uminus |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1717 |
using assms INF_less |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1718 |
by (subst SUPR_extreal_add) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1719 |
(auto simp: extreal_SUPR_uminus intro!: uminus_extreal_add_uminus_uminus) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1720 |
finally show ?thesis . |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1721 |
qed |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1722 |
|
41973 | 1723 |
subsection "Limits on @{typ extreal}" |
1724 |
||
1725 |
subsubsection "Topological space" |
|
1726 |
||
1727 |
instantiation extreal :: topological_space |
|
1728 |
begin |
|
1729 |
||
41975 | 1730 |
definition "open A \<longleftrightarrow> open (extreal -` A) |
41973 | 1731 |
\<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {extreal x <..} \<subseteq> A)) |
1732 |
\<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A))" |
|
1733 |
||
41975 | 1734 |
lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {extreal x<..} \<subseteq> A)" |
41973 | 1735 |
unfolding open_extreal_def by auto |
1736 |
||
41975 | 1737 |
lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A)" |
41973 | 1738 |
unfolding open_extreal_def by auto |
1739 |
||
41975 | 1740 |
lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{extreal x<..} \<subseteq> A" |
41973 | 1741 |
using open_PInfty[OF assms] by auto |
1742 |
||
41975 | 1743 |
lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<extreal x} \<subseteq> A" |
41973 | 1744 |
using open_MInfty[OF assms] by auto |
1745 |
||
41975 | 1746 |
lemma extreal_openE: assumes "open A" obtains x y where |
1747 |
"open (extreal -` A)" |
|
1748 |
"\<infinity> \<in> A \<Longrightarrow> {extreal x<..} \<subseteq> A" |
|
1749 |
"-\<infinity> \<in> A \<Longrightarrow> {..<extreal y} \<subseteq> A" |
|
41973 | 1750 |
using assms open_extreal_def by auto |
1751 |
||
1752 |
instance |
|
1753 |
proof |
|
1754 |
let ?U = "UNIV::extreal set" |
|
1755 |
show "open ?U" unfolding open_extreal_def |
|
41975 | 1756 |
by (auto intro!: exI[of _ 0]) |
41973 | 1757 |
next |
1758 |
fix S T::"extreal set" assume "open S" and "open T" |
|
41975 | 1759 |
from `open S`[THEN extreal_openE] guess xS yS . |
1760 |
moreover from `open T`[THEN extreal_openE] guess xT yT . |
|
1761 |
ultimately have |
|
1762 |
"open (extreal -` (S \<inter> T))" |
|
1763 |
"\<infinity> \<in> S \<inter> T \<Longrightarrow> {extreal (max xS xT) <..} \<subseteq> S \<inter> T" |
|
1764 |
"-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< extreal (min yS yT)} \<subseteq> S \<inter> T" |
|
1765 |
by auto |
|
1766 |
then show "open (S Int T)" unfolding open_extreal_def by blast |
|
41973 | 1767 |
next |
41975 | 1768 |
fix K :: "extreal set set" assume "\<forall>S\<in>K. open S" |
1769 |
then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (extreal -` S) \<and> |
|
1770 |
(\<infinity> \<in> S \<longrightarrow> {extreal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< extreal y} \<subseteq> S)" |
|
1771 |
by (auto simp: open_extreal_def) |
|
1772 |
then show "open (Union K)" unfolding open_extreal_def |
|
1773 |
proof (intro conjI impI) |
|
1774 |
show "open (extreal -` \<Union>K)" |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1775 |
using *[THEN choice] by (auto simp: vimage_Union) |
41975 | 1776 |
qed ((metis UnionE Union_upper subset_trans *)+) |
41973 | 1777 |
qed |
1778 |
end |
|
1779 |
||
41976 | 1780 |
lemma open_extreal: "open S \<Longrightarrow> open (extreal ` S)" |
1781 |
by (auto simp: inj_vimage_image_eq open_extreal_def) |
|
1782 |
||
1783 |
lemma open_extreal_vimage: "open S \<Longrightarrow> open (extreal -` S)" |
|
1784 |
unfolding open_extreal_def by auto |
|
1785 |
||
41975 | 1786 |
lemma open_extreal_lessThan[intro, simp]: "open {..< a :: extreal}" |
1787 |
proof - |
|
1788 |
have "\<And>x. extreal -` {..<extreal x} = {..< x}" |
|
1789 |
"extreal -` {..< \<infinity>} = UNIV" "extreal -` {..< -\<infinity>} = {}" by auto |
|
1790 |
then show ?thesis by (cases a) (auto simp: open_extreal_def) |
|
1791 |
qed |
|
1792 |
||
1793 |
lemma open_extreal_greaterThan[intro, simp]: |
|
41973 | 1794 |
"open {a :: extreal <..}" |
41975 | 1795 |
proof - |
1796 |
have "\<And>x. extreal -` {extreal x<..} = {x<..}" |
|
1797 |
"extreal -` {\<infinity><..} = {}" "extreal -` {-\<infinity><..} = UNIV" by auto |
|
1798 |
then show ?thesis by (cases a) (auto simp: open_extreal_def) |
|
1799 |
qed |
|
1800 |
||
1801 |
lemma extreal_open_greaterThanLessThan[intro, simp]: "open {a::extreal <..< b}" |
|
41973 | 1802 |
unfolding greaterThanLessThan_def by auto |
1803 |
||
1804 |
lemma closed_extreal_atLeast[simp, intro]: "closed {a :: extreal ..}" |
|
1805 |
proof - |
|
1806 |
have "- {a ..} = {..< a}" by auto |
|
1807 |
then show "closed {a ..}" |
|
1808 |
unfolding closed_def using open_extreal_lessThan by auto |
|
1809 |
qed |
|
1810 |
||
1811 |
lemma closed_extreal_atMost[simp, intro]: "closed {.. b :: extreal}" |
|
1812 |
proof - |
|
1813 |
have "- {.. b} = {b <..}" by auto |
|
1814 |
then show "closed {.. b}" |
|
1815 |
unfolding closed_def using open_extreal_greaterThan by auto |
|
1816 |
qed |
|
1817 |
||
1818 |
lemma closed_extreal_atLeastAtMost[simp, intro]: |
|
1819 |
shows "closed {a :: extreal .. b}" |
|
1820 |
unfolding atLeastAtMost_def by auto |
|
1821 |
||
1822 |
lemma closed_extreal_singleton: |
|
1823 |
"closed {a :: extreal}" |
|
1824 |
by (metis atLeastAtMost_singleton closed_extreal_atLeastAtMost) |
|
1825 |
||
1826 |
lemma extreal_open_cont_interval: |
|
41976 | 1827 |
assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 1828 |
obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S" |
1829 |
proof- |
|
41975 | 1830 |
from `open S` have "open (extreal -` S)" by (rule extreal_openE) |
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1831 |
then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> extreal y \<in> S" |
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1832 |
using assms unfolding open_dist by force |
41975 | 1833 |
show thesis |
1834 |
proof (intro that subsetI) |
|
1835 |
show "0 < extreal e" using `0 < e` by auto |
|
1836 |
fix y assume "y \<in> {x - extreal e<..<x + extreal e}" |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1837 |
with assms obtain t where "y = extreal t" "dist t (real x) < e" |
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1838 |
apply (cases y) by (auto simp: dist_real_def) |
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1839 |
then show "y \<in> S" using e[of t] by auto |
41975 | 1840 |
qed |
41973 | 1841 |
qed |
1842 |
||
1843 |
lemma extreal_open_cont_interval2: |
|
41976 | 1844 |
assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 1845 |
obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S" |
1846 |
proof- |
|
1847 |
guess e using extreal_open_cont_interval[OF assms] . |
|
1848 |
with that[of "x-e" "x+e"] extreal_between[OF x, of e] |
|
1849 |
show thesis by auto |
|
1850 |
qed |
|
1851 |
||
1852 |
instance extreal :: t2_space |
|
1853 |
proof |
|
1854 |
fix x y :: extreal assume "x ~= y" |
|
1855 |
let "?P x (y::extreal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}" |
|
1856 |
||
1857 |
{ fix x y :: extreal assume "x < y" |
|
1858 |
from extreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto |
|
1859 |
have "?P x y" |
|
1860 |
apply (rule exI[of _ "{..<z}"]) |
|
1861 |
apply (rule exI[of _ "{z<..}"]) |
|
1862 |
using z by auto } |
|
1863 |
note * = this |
|
1864 |
||
1865 |
from `x ~= y` |
|
1866 |
show "EX U V. open U & open V & x : U & y : V & U Int V = {}" |
|
1867 |
proof (cases rule: linorder_cases) |
|
1868 |
assume "x = y" with `x ~= y` show ?thesis by simp |
|
1869 |
next assume "x < y" from *[OF this] show ?thesis by auto |
|
1870 |
next assume "y < x" from *[OF this] show ?thesis by auto |
|
1871 |
qed |
|
1872 |
qed |
|
1873 |
||
1874 |
subsubsection {* Convergent sequences *} |
|
1875 |
||
1876 |
lemma lim_extreal[simp]: |
|
1877 |
"((\<lambda>n. extreal (f n)) ---> extreal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r") |
|
1878 |
proof (intro iffI topological_tendstoI) |
|
1879 |
fix S assume "?l" "open S" "x \<in> S" |
|
1880 |
then show "eventually (\<lambda>x. f x \<in> S) net" |
|
1881 |
using `?l`[THEN topological_tendstoD, OF open_extreal, OF `open S`] |
|
1882 |
by (simp add: inj_image_mem_iff) |
|
1883 |
next |
|
1884 |
fix S assume "?r" "open S" "extreal x \<in> S" |
|
1885 |
show "eventually (\<lambda>x. extreal (f x) \<in> S) net" |
|
41975 | 1886 |
using `?r`[THEN topological_tendstoD, OF open_extreal_vimage, OF `open S`] |
1887 |
using `extreal x \<in> S` by auto |
|
41973 | 1888 |
qed |
1889 |
||
1890 |
lemma lim_real_of_extreal[simp]: |
|
1891 |
assumes lim: "(f ---> extreal x) net" |
|
1892 |
shows "((\<lambda>x. real (f x)) ---> x) net" |
|
1893 |
proof (intro topological_tendstoI) |
|
1894 |
fix S assume "open S" "x \<in> S" |
|
1895 |
then have S: "open S" "extreal x \<in> extreal ` S" |
|
1896 |
by (simp_all add: inj_image_mem_iff) |
|
1897 |
have "\<forall>x. f x \<in> extreal ` S \<longrightarrow> real (f x) \<in> S" by auto |
|
1898 |
from this lim[THEN topological_tendstoD, OF open_extreal, OF S] |
|
1899 |
show "eventually (\<lambda>x. real (f x) \<in> S) net" |
|
1900 |
by (rule eventually_mono) |
|
1901 |
qed |
|
1902 |
||
1903 |
lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= extreal B)" (is "?l = ?r") |
|
1904 |
proof assume ?r show ?l apply(rule topological_tendstoI) |
|
1905 |
unfolding eventually_sequentially |
|
1906 |
proof- fix S assume "open S" "\<infinity> : S" |
|
1907 |
from open_PInfty[OF this] guess B .. note B=this |
|
1908 |
from `?r`[rule_format,of "B+1"] guess N .. note N=this |
|
1909 |
show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI) |
|
1910 |
proof safe case goal1 |
|
1911 |
have "extreal B < extreal (B + 1)" by auto |
|
1912 |
also have "... <= f n" using goal1 N by auto |
|
1913 |
finally show ?case using B by fastsimp |
|
1914 |
qed |
|
1915 |
qed |
|
1916 |
next assume ?l show ?r |
|
1917 |
proof fix B::real have "open {extreal B<..}" "\<infinity> : {extreal B<..}" by auto |
|
1918 |
from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially] |
|
1919 |
guess N .. note N=this |
|
1920 |
show "EX N. ALL n>=N. extreal B <= f n" apply(rule_tac x=N in exI) using N by auto |
|
1921 |
qed |
|
1922 |
qed |
|
1923 |
||
1924 |
||
1925 |
lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= extreal B)" (is "?l = ?r") |
|
1926 |
proof assume ?r show ?l apply(rule topological_tendstoI) |
|
1927 |
unfolding eventually_sequentially |
|
1928 |
proof- fix S assume "open S" "(-\<infinity>) : S" |
|
1929 |
from open_MInfty[OF this] guess B .. note B=this |
|
1930 |
from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this |
|
1931 |
show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI) |
|
1932 |
proof safe case goal1 |
|
1933 |
have "extreal (B - 1) >= f n" using goal1 N by auto |
|
1934 |
also have "... < extreal B" by auto |
|
1935 |
finally show ?case using B by fastsimp |
|
1936 |
qed |
|
1937 |
qed |
|
1938 |
next assume ?l show ?r |
|
1939 |
proof fix B::real have "open {..<extreal B}" "(-\<infinity>) : {..<extreal B}" by auto |
|
1940 |
from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially] |
|
1941 |
guess N .. note N=this |
|
1942 |
show "EX N. ALL n>=N. extreal B >= f n" apply(rule_tac x=N in exI) using N by auto |
|
1943 |
qed |
|
1944 |
qed |
|
1945 |
||
1946 |
||
1947 |
lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= extreal B" shows "l ~= \<infinity>" |
|
1948 |
proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>" |
|
1949 |
from lim[unfolded this Lim_PInfty,rule_format,of "?B"] |
|
1950 |
guess N .. note N=this[rule_format,OF le_refl] |
|
1951 |
hence "extreal ?B <= extreal B" using assms(2)[of N] by(rule order_trans) |
|
1952 |
hence "extreal ?B < extreal ?B" apply (rule le_less_trans) by auto |
|
1953 |
thus False by auto |
|
1954 |
qed |
|
1955 |
||
1956 |
||
1957 |
lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= extreal B" shows "l ~= (-\<infinity>)" |
|
1958 |
proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)" |
|
1959 |
from lim[unfolded this Lim_MInfty,rule_format,of "?B"] |
|
1960 |
guess N .. note N=this[rule_format,OF le_refl] |
|
1961 |
hence "extreal B <= extreal ?B" using assms(2)[of N] order_trans[of "extreal B" "f N" "extreal(B - 1)"] by blast |
|
1962 |
thus False by auto |
|
1963 |
qed |
|
1964 |
||
1965 |
||
1966 |
lemma tendsto_explicit: |
|
1967 |
"f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))" |
|
1968 |
unfolding tendsto_def eventually_sequentially by auto |
|
1969 |
||
1970 |
||
1971 |
lemma tendsto_obtains_N: |
|
1972 |
assumes "f ----> f0" |
|
1973 |
assumes "open S" "f0 : S" |
|
1974 |
obtains N where "ALL n>=N. f n : S" |
|
1975 |
using tendsto_explicit[of f f0] assms by auto |
|
1976 |
||
1977 |
||
1978 |
lemma tail_same_limit: |
|
1979 |
fixes X Y N |
|
1980 |
assumes "X ----> L" "ALL n>=N. X n = Y n" |
|
1981 |
shows "Y ----> L" |
|
1982 |
proof- |
|
1983 |
{ fix S assume "open S" and "L:S" |
|
1984 |
from this obtain N1 where "ALL n>=N1. X n : S" |
|
1985 |
using assms unfolding tendsto_def eventually_sequentially by auto |
|
1986 |
hence "ALL n>=max N N1. Y n : S" using assms by auto |
|
1987 |
hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto |
|
1988 |
} |
|
1989 |
thus ?thesis using tendsto_explicit by auto |
|
1990 |
qed |
|
1991 |
||
1992 |
||
1993 |
lemma Lim_bounded_PInfty2: |
|
1994 |
assumes lim:"f ----> l" and "ALL n>=N. f n <= extreal B" |
|
1995 |
shows "l ~= \<infinity>" |
|
1996 |
proof- |
|
1997 |
def g == "(%n. if n>=N then f n else extreal B)" |
|
1998 |
hence "g ----> l" using tail_same_limit[of f l N g] lim by auto |
|
1999 |
moreover have "!!n. g n <= extreal B" using g_def assms by auto |
|
2000 |
ultimately show ?thesis using Lim_bounded_PInfty by auto |
|
2001 |
qed |
|
2002 |
||
2003 |
lemma Lim_bounded_extreal: |
|
2004 |
assumes lim:"f ----> (l :: extreal)" |
|
2005 |
and "ALL n>=M. f n <= C" |
|
2006 |
shows "l<=C" |
|
2007 |
proof- |
|
2008 |
{ assume "l=(-\<infinity>)" hence ?thesis by auto } |
|
2009 |
moreover |
|
2010 |
{ assume "~(l=(-\<infinity>))" |
|
2011 |
{ assume "C=\<infinity>" hence ?thesis by auto } |
|
2012 |
moreover |
|
2013 |
{ assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto |
|
2014 |
hence "l=(-\<infinity>)" using assms |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
2015 |
tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto |
41973 | 2016 |
hence ?thesis by auto } |
2017 |
moreover |
|
2018 |
{ assume "EX B. C = extreal B" |
|
2019 |
from this obtain B where B_def: "C=extreal B" by auto |
|
2020 |
hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto |
|
2021 |
from this obtain m where m_def: "extreal m=l" using `~(l=(-\<infinity>))` by (cases l) auto |
|
2022 |
from this obtain N where N_def: "ALL n>=N. f n : {extreal(m - 1) <..< extreal(m+1)}" |
|
2023 |
apply (subst tendsto_obtains_N[of f l "{extreal(m - 1) <..< extreal(m+1)}"]) using assms by auto |
|
2024 |
{ fix n assume "n>=N" |
|
2025 |
hence "EX r. extreal r = f n" using N_def by (cases "f n") auto |
|
2026 |
} from this obtain g where g_def: "ALL n>=N. extreal (g n) = f n" by metis |
|
2027 |
hence "(%n. extreal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto |
|
2028 |
hence *: "(%n. g n) ----> m" using m_def by auto |
|
2029 |
{ fix n assume "n>=max N M" |
|
2030 |
hence "extreal (g n) <= extreal B" using assms g_def B_def by auto |
|
2031 |
hence "g n <= B" by auto |
|
2032 |
} hence "EX N. ALL n>=N. g n <= B" by blast |
|
2033 |
hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto |
|
2034 |
hence ?thesis using m_def B_def by auto |
|
2035 |
} ultimately have ?thesis by (cases C) auto |
|
2036 |
} ultimately show ?thesis by blast |
|
2037 |
qed |
|
2038 |
||
2039 |
lemma real_of_extreal_mult[simp]: |
|
2040 |
fixes a b :: extreal shows "real (a * b) = real a * real b" |
|
2041 |
by (cases rule: extreal2_cases[of a b]) auto |
|
2042 |
||
2043 |
lemma real_of_extreal_eq_0: |
|
2044 |
"real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0" |
|
2045 |
by (cases x) auto |
|
2046 |
||
2047 |
lemma tendsto_extreal_realD: |
|
2048 |
fixes f :: "'a \<Rightarrow> extreal" |
|
2049 |
assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. extreal (real (f x))) ---> x) net" |
|
2050 |
shows "(f ---> x) net" |
|
2051 |
proof (intro topological_tendstoI) |
|
2052 |
fix S assume S: "open S" "x \<in> S" |
|
2053 |
with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto |
|
2054 |
from tendsto[THEN topological_tendstoD, OF this] |
|
2055 |
show "eventually (\<lambda>x. f x \<in> S) net" |
|
2056 |
by (rule eventually_rev_mp) (auto simp: extreal_real real_of_extreal_0) |
|
2057 |
qed |
|
2058 |
||
2059 |
lemma tendsto_extreal_realI: |
|
2060 |
fixes f :: "'a \<Rightarrow> extreal" |
|
41976 | 2061 |
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net" |
41973 | 2062 |
shows "((\<lambda>x. extreal (real (f x))) ---> x) net" |
2063 |
proof (intro topological_tendstoI) |
|
2064 |
fix S assume "open S" "x \<in> S" |
|
2065 |
with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto |
|
2066 |
from tendsto[THEN topological_tendstoD, OF this] |
|
2067 |
show "eventually (\<lambda>x. extreal (real (f x)) \<in> S) net" |
|
2068 |
by (elim eventually_elim1) (auto simp: extreal_real) |
|
2069 |
qed |
|
2070 |
||
2071 |
lemma extreal_mult_cancel_left: |
|
2072 |
fixes a b c :: extreal shows "a * b = a * c \<longleftrightarrow> |
|
41976 | 2073 |
((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)" |
41973 | 2074 |
by (cases rule: extreal3_cases[of a b c]) |
2075 |
(simp_all add: zero_less_mult_iff) |
|
2076 |
||
2077 |
lemma extreal_inj_affinity: |
|
41976 | 2078 |
assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>" |
41973 | 2079 |
shows "inj_on (\<lambda>x. m * x + t) A" |
2080 |
using assms |
|
2081 |
by (cases rule: extreal2_cases[of m t]) |
|
2082 |
(auto intro!: inj_onI simp: extreal_add_cancel_right extreal_mult_cancel_left) |
|
2083 |
||
2084 |
lemma extreal_PInfty_eq_plus[simp]: |
|
2085 |
shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
|
2086 |
by (cases rule: extreal2_cases[of a b]) auto |
|
2087 |
||
2088 |
lemma extreal_MInfty_eq_plus[simp]: |
|
2089 |
shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)" |
|
2090 |
by (cases rule: extreal2_cases[of a b]) auto |
|
2091 |
||
2092 |
lemma extreal_less_divide_pos: |
|
2093 |
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z" |
|
2094 |
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
2095 |
||
2096 |
lemma extreal_divide_less_pos: |
|
2097 |
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z" |
|
2098 |
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
2099 |
||
2100 |
lemma extreal_divide_eq: |
|
41976 | 2101 |
"b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c" |
41973 | 2102 |
by (cases rule: extreal3_cases[of a b c]) |
2103 |
(simp_all add: field_simps) |
|
2104 |
||
2105 |
lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>" |
|
2106 |
by (cases a) auto |
|
2107 |
||
2108 |
lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x" |
|
2109 |
by (cases x) auto |
|
2110 |
||
2111 |
lemma extreal_LimI_finite: |
|
41976 | 2112 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 2113 |
assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r" |
2114 |
shows "u ----> x" |
|
2115 |
proof (rule topological_tendstoI, unfold eventually_sequentially) |
|
2116 |
obtain rx where rx_def: "x=extreal rx" using assms by (cases x) auto |
|
2117 |
fix S assume "open S" "x : S" |
|
41975 | 2118 |
then have "open (extreal -` S)" unfolding open_extreal_def by auto |
2119 |
with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> extreal y \<in> S" |
|
2120 |
unfolding open_real_def rx_def by auto |
|
41973 | 2121 |
then obtain n where |
2122 |
upper: "!!N. n <= N ==> u N < x + extreal r" and |
|
41976 | 2123 |
lower: "!!N. n <= N ==> x < u N + extreal r" using assms(2)[of "extreal r"] by auto |
41973 | 2124 |
show "EX N. ALL n>=N. u n : S" |
2125 |
proof (safe intro!: exI[of _ n]) |
|
2126 |
fix N assume "n <= N" |
|
2127 |
from upper[OF this] lower[OF this] assms `0 < r` |
|
2128 |
have "u N ~: {\<infinity>,(-\<infinity>)}" by auto |
|
2129 |
from this obtain ra where ra_def: "(u N) = extreal ra" by (cases "u N") auto |
|
2130 |
hence "rx < ra + r" and "ra < rx + r" |
|
2131 |
using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto |
|
41975 | 2132 |
hence "dist (real (u N)) rx < r" |
41973 | 2133 |
using rx_def ra_def |
2134 |
by (auto simp: dist_real_def abs_diff_less_iff field_simps) |
|
41976 | 2135 |
from dist[OF this] show "u N : S" using `u N ~: {\<infinity>, -\<infinity>}` |
2136 |
by (auto simp: extreal_real split: split_if_asm) |
|
41973 | 2137 |
qed |
2138 |
qed |
|
2139 |
||
2140 |
lemma extreal_LimI_finite_iff: |
|
41976 | 2141 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 2142 |
shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))" |
2143 |
(is "?lhs <-> ?rhs") |
|
41976 | 2144 |
proof |
2145 |
assume lim: "u ----> x" |
|
41973 | 2146 |
{ fix r assume "(r::extreal)>0" |
2147 |
from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}" |
|
2148 |
apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"]) |
|
2149 |
using lim extreal_between[of x r] assms `r>0` by auto |
|
2150 |
hence "EX N. ALL n>=N. u n < x + r & x < u n + r" |
|
2151 |
using extreal_minus_less[of r x] by (cases r) auto |
|
41976 | 2152 |
} then show "?rhs" by auto |
2153 |
next |
|
2154 |
assume ?rhs then show "u ----> x" |
|
2155 |
using extreal_LimI_finite[of x] assms by auto |
|
41973 | 2156 |
qed |
2157 |
||
2158 |
||
2159 |
subsubsection {* @{text Liminf} and @{text Limsup} *} |
|
2160 |
||
2161 |
definition |
|
2162 |
"Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)" |
|
2163 |
||
2164 |
definition |
|
2165 |
"Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)" |
|
2166 |
||
2167 |
lemma Liminf_Sup: |
|
2168 |
fixes f :: "'a => 'b::{complete_lattice, linorder}" |
|
2169 |
shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}" |
|
2170 |
by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def) |
|
2171 |
||
2172 |
lemma Limsup_Inf: |
|
2173 |
fixes f :: "'a => 'b::{complete_lattice, linorder}" |
|
2174 |
shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}" |
|
2175 |
by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def) |
|
2176 |
||
2177 |
lemma extreal_SupI: |
|
2178 |
fixes x :: extreal |
|
2179 |
assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" |
|
2180 |
assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y" |
|
2181 |
shows "Sup A = x" |
|
2182 |
unfolding Sup_extreal_def |
|
2183 |
using assms by (auto intro!: Least_equality) |
|
2184 |
||
2185 |
lemma extreal_InfI: |
|
2186 |
fixes x :: extreal |
|
2187 |
assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i" |
|
2188 |
assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x" |
|
2189 |
shows "Inf A = x" |
|
2190 |
unfolding Inf_extreal_def |
|
2191 |
using assms by (auto intro!: Greatest_equality) |
|
2192 |
||
2193 |
lemma Limsup_const: |
|
2194 |
fixes c :: "'a::{complete_lattice, linorder}" |
|
2195 |
assumes ntriv: "\<not> trivial_limit net" |
|
2196 |
shows "Limsup net (\<lambda>x. c) = c" |
|
2197 |
unfolding Limsup_Inf |
|
2198 |
proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower) |
|
2199 |
fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net" |
|
2200 |
show "c \<le> x" |
|
2201 |
proof (rule ccontr) |
|
2202 |
assume "\<not> c \<le> x" then have "x < c" by auto |
|
2203 |
then show False using ntriv * by (auto simp: trivial_limit_def) |
|
2204 |
qed |
|
2205 |
qed auto |
|
2206 |
||
2207 |
lemma Liminf_const: |
|
2208 |
fixes c :: "'a::{complete_lattice, linorder}" |
|
2209 |
assumes ntriv: "\<not> trivial_limit net" |
|
2210 |
shows "Liminf net (\<lambda>x. c) = c" |
|
2211 |
unfolding Liminf_Sup |
|
2212 |
proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper) |
|
2213 |
fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net" |
|
2214 |
show "x \<le> c" |
|
2215 |
proof (rule ccontr) |
|
2216 |
assume "\<not> x \<le> c" then have "c < x" by auto |
|
2217 |
then show False using ntriv * by (auto simp: trivial_limit_def) |
|
2218 |
qed |
|
2219 |
qed auto |
|
2220 |
||
2221 |
lemma mono_set: |
|
2222 |
fixes S :: "('a::order) set" |
|
2223 |
shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)" |
|
2224 |
by (auto simp: mono_def mem_def) |
|
2225 |
||
2226 |
lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto |
|
2227 |
lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto |
|
2228 |
lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto |
|
2229 |
lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto |
|
2230 |
||
2231 |
lemma mono_set_iff: |
|
2232 |
fixes S :: "'a::{linorder,complete_lattice} set" |
|
2233 |
defines "a \<equiv> Inf S" |
|
2234 |
shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c") |
|
2235 |
proof |
|
2236 |
assume "mono S" |
|
2237 |
then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set) |
|
2238 |
show ?c |
|
2239 |
proof cases |
|
2240 |
assume "a \<in> S" |
|
2241 |
show ?c |
|
2242 |
using mono[OF _ `a \<in> S`] |
|
2243 |
by (auto intro: complete_lattice_class.Inf_lower simp: a_def) |
|
2244 |
next |
|
2245 |
assume "a \<notin> S" |
|
2246 |
have "S = {a <..}" |
|
2247 |
proof safe |
|
2248 |
fix x assume "x \<in> S" |
|
2249 |
then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower) |
|
2250 |
then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto |
|
2251 |
next |
|
2252 |
fix x assume "a < x" |
|
2253 |
then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff .. |
|
2254 |
with mono[of y x] show "x \<in> S" by auto |
|
2255 |
qed |
|
2256 |
then show ?c .. |
|
2257 |
qed |
|
2258 |
qed auto |
|
2259 |
||
2260 |
lemma lim_imp_Liminf: |
|
2261 |
fixes f :: "'a \<Rightarrow> extreal" |
|
2262 |
assumes ntriv: "\<not> trivial_limit net" |
|
2263 |
assumes lim: "(f ---> f0) net" |
|
2264 |
shows "Liminf net f = f0" |
|
2265 |
unfolding Liminf_Sup |
|
2266 |
proof (safe intro!: extreal_SupI) |
|
2267 |
fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net" |
|
2268 |
show "y \<le> f0" |
|
2269 |
proof (rule extreal_le_extreal) |
|
2270 |
fix B assume "B < y" |
|
2271 |
{ assume "f0 < B" |
|
2272 |
then have "eventually (\<lambda>x. f x < B \<and> B < f x) net" |
|
2273 |
using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`] |
|
2274 |
by (auto intro: eventually_conj) |
|
2275 |
also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff) |
|
2276 |
finally have False using ntriv[unfolded trivial_limit_def] by auto |
|
2277 |
} then show "B \<le> f0" by (metis linorder_le_less_linear) |
|
2278 |
qed |
|
2279 |
next |
|
2280 |
fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y" |
|
2281 |
show "f0 \<le> y" |
|
2282 |
proof (safe intro!: *[rule_format]) |
|
2283 |
fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net" |
|
2284 |
using lim[THEN topological_tendstoD, of "{y <..}"] by auto |
|
2285 |
qed |
|
2286 |
qed |
|
2287 |
||
2288 |
lemma extreal_Liminf_le_Limsup: |
|
2289 |
fixes f :: "'a \<Rightarrow> extreal" |
|
2290 |
assumes ntriv: "\<not> trivial_limit net" |
|
2291 |
shows "Liminf net f \<le> Limsup net f" |
|
2292 |
unfolding Limsup_Inf Liminf_Sup |
|
2293 |
proof (safe intro!: complete_lattice_class.Inf_greatest complete_lattice_class.Sup_least) |
|
2294 |
fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net" |
|
2295 |
show "u \<le> v" |
|
2296 |
proof (rule ccontr) |
|
2297 |
assume "\<not> u \<le> v" |
|
2298 |
then obtain t where "t < u" "v < t" |
|
2299 |
using extreal_dense[of v u] by (auto simp: not_le) |
|
2300 |
then have "eventually (\<lambda>x. t < f x \<and> f x < t) net" |
|
2301 |
using * by (auto intro: eventually_conj) |
|
2302 |
also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff) |
|
2303 |
finally show False using ntriv by (auto simp: trivial_limit_def) |
|
2304 |
qed |
|
2305 |
qed |
|
2306 |
||
2307 |
lemma Liminf_mono: |
|
2308 |
fixes f g :: "'a => extreal" |
|
2309 |
assumes ev: "eventually (\<lambda>x. f x \<le> g x) net" |
|
2310 |
shows "Liminf net f \<le> Liminf net g" |
|
2311 |
unfolding Liminf_Sup |
|
2312 |
proof (safe intro!: Sup_mono bexI) |
|
2313 |
fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a" |
|
2314 |
then have "eventually (\<lambda>x. y < f x) net" by auto |
|
2315 |
then show "eventually (\<lambda>x. y < g x) net" |
|
2316 |
by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto) |
|
2317 |
qed simp |
|
2318 |
||
2319 |
lemma Liminf_eq: |
|
2320 |
fixes f g :: "'a \<Rightarrow> extreal" |
|
2321 |
assumes "eventually (\<lambda>x. f x = g x) net" |
|
2322 |
shows "Liminf net f = Liminf net g" |
|
2323 |
by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto |
|
2324 |
||
2325 |
lemma Liminf_mono_all: |
|
2326 |
fixes f g :: "'a \<Rightarrow> extreal" |
|
2327 |
assumes "\<And>x. f x \<le> g x" |
|
2328 |
shows "Liminf net f \<le> Liminf net g" |
|
2329 |
using assms by (intro Liminf_mono always_eventually) auto |
|
2330 |
||
2331 |
lemma Limsup_mono: |
|
2332 |
fixes f g :: "'a \<Rightarrow> extreal" |
|
2333 |
assumes ev: "eventually (\<lambda>x. f x \<le> g x) net" |
|
2334 |
shows "Limsup net f \<le> Limsup net g" |
|
2335 |
unfolding Limsup_Inf |
|
2336 |
proof (safe intro!: Inf_mono bexI) |
|
2337 |
fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y" |
|
2338 |
then have "eventually (\<lambda>x. g x < y) net" by auto |
|
2339 |
then show "eventually (\<lambda>x. f x < y) net" |
|
2340 |
by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto) |
|
2341 |
qed simp |
|
2342 |
||
2343 |
lemma Limsup_mono_all: |
|
2344 |
fixes f g :: "'a \<Rightarrow> extreal" |
|
2345 |
assumes "\<And>x. f x \<le> g x" |
|
2346 |
shows "Limsup net f \<le> Limsup net g" |
|
2347 |
using assms by (intro Limsup_mono always_eventually) auto |
|
2348 |
||
2349 |
lemma Limsup_eq: |
|
2350 |
fixes f g :: "'a \<Rightarrow> extreal" |
|
2351 |
assumes "eventually (\<lambda>x. f x = g x) net" |
|
2352 |
shows "Limsup net f = Limsup net g" |
|
2353 |
by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto |
|
2354 |
||
2355 |
abbreviation "liminf \<equiv> Liminf sequentially" |
|
2356 |
||
2357 |
abbreviation "limsup \<equiv> Limsup sequentially" |
|
2358 |
||
2359 |
lemma (in complete_lattice) less_INFD: |
|
2360 |
assumes "y < INFI A f"" i \<in> A" shows "y < f i" |
|
2361 |
proof - |
|
2362 |
note `y < INFI A f` |
|
2363 |
also have "INFI A f \<le> f i" using `i \<in> A` by (rule INF_leI) |
|
2364 |
finally show "y < f i" . |
|
2365 |
qed |
|
2366 |
||
2367 |
lemma liminf_SUPR_INFI: |
|
2368 |
fixes f :: "nat \<Rightarrow> extreal" |
|
2369 |
shows "liminf f = (SUP n. INF m:{n..}. f m)" |
|
2370 |
unfolding Liminf_Sup eventually_sequentially |
|
2371 |
proof (safe intro!: antisym complete_lattice_class.Sup_least) |
|
2372 |
fix x assume *: "\<forall>y<x. \<exists>N. \<forall>n\<ge>N. y < f n" show "x \<le> (SUP n. INF m:{n..}. f m)" |
|
2373 |
proof (rule extreal_le_extreal) |
|
2374 |
fix y assume "y < x" |
|
2375 |
with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto |
|
2376 |
then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff) |
|
2377 |
also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto |
|
2378 |
finally show "y \<le> (SUP n. INF m:{n..}. f m)" . |
|
2379 |
qed |
|
2380 |
next |
|
2381 |
show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}" |
|
2382 |
proof (unfold SUPR_def, safe intro!: Sup_mono bexI) |
|
2383 |
fix y n assume "y < INFI {n..} f" |
|
2384 |
from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto |
|
2385 |
qed (rule order_refl) |
|
2386 |
qed |
|
2387 |
||
2388 |
lemma tail_same_limsup: |
|
2389 |
fixes X Y :: "nat => extreal" |
|
2390 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n" |
|
2391 |
shows "limsup X = limsup Y" |
|
2392 |
using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto |
|
2393 |
||
2394 |
lemma tail_same_liminf: |
|
2395 |
fixes X Y :: "nat => extreal" |
|
2396 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n" |
|
2397 |
shows "liminf X = liminf Y" |
|
2398 |
using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto |
|
2399 |
||
2400 |
lemma liminf_mono: |
|
2401 |
fixes X Y :: "nat \<Rightarrow> extreal" |
|
2402 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n" |
|
2403 |
shows "liminf X \<le> liminf Y" |
|
2404 |
using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto |
|
2405 |
||
2406 |
lemma limsup_mono: |
|
2407 |
fixes X Y :: "nat => extreal" |
|
2408 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n" |
|
2409 |
shows "limsup X \<le> limsup Y" |
|
2410 |
using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto |
|
2411 |
||
2412 |
declare trivial_limit_sequentially[simp] |
|
2413 |
||
41978 | 2414 |
lemma |
2415 |
fixes X :: "nat \<Rightarrow> extreal" |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
2416 |
shows extreal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X" |
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
2417 |
and extreal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X" |
41978 | 2418 |
unfolding incseq_def decseq_def by auto |
2419 |
||
41973 | 2420 |
lemma liminf_bounded: |
2421 |
fixes X Y :: "nat \<Rightarrow> extreal" |
|
2422 |
assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n" |
|
2423 |
shows "C \<le> liminf X" |
|
2424 |
using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp |
|
2425 |
||
2426 |
lemma limsup_bounded: |
|
2427 |
fixes X Y :: "nat => extreal" |
|
2428 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C" |
|
2429 |
shows "limsup X \<le> C" |
|
2430 |
using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp |
|
2431 |
||
2432 |
lemma liminf_bounded_iff: |
|
2433 |
fixes x :: "nat \<Rightarrow> extreal" |
|
2434 |
shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs") |
|
2435 |
proof safe |
|
2436 |
fix B assume "B < C" "C \<le> liminf x" |
|
2437 |
then have "B < liminf x" by auto |
|
2438 |
then obtain N where "B < (INF m:{N..}. x m)" |
|
2439 |
unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto |
|
2440 |
from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto |
|
2441 |
next |
|
2442 |
assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n" |
|
2443 |
{ fix B assume "B<C" |
|
2444 |
then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto |
|
2445 |
hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto |
|
2446 |
also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp |
|
2447 |
finally have "B \<le> liminf x" . |
|
2448 |
} then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear) |
|
2449 |
qed |
|
2450 |
||
2451 |
lemma liminf_subseq_mono: |
|
2452 |
fixes X :: "nat \<Rightarrow> extreal" |
|
2453 |
assumes "subseq r" |
|
2454 |
shows "liminf X \<le> liminf (X \<circ> r) " |
|
2455 |
proof- |
|
2456 |
have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)" |
|
2457 |
proof (safe intro!: INF_mono) |
|
2458 |
fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m" |
|
2459 |
using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto |
|
2460 |
qed |
|
2461 |
then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def) |
|
2462 |
qed |
|
2463 |
||
41976 | 2464 |
lemma extreal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "extreal (real x) = x" |
2465 |
using assms by auto |
|
41973 | 2466 |
|
41978 | 2467 |
lemma extreal_le_extreal_bounded: |
2468 |
fixes x y z :: extreal |
|
2469 |
assumes "z \<le> y" |
|
2470 |
assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y" |
|
2471 |
shows "x \<le> y" |
|
2472 |
proof (rule extreal_le_extreal) |
|
2473 |
fix B assume "B < x" |
|
2474 |
show "B \<le> y" |
|
2475 |
proof cases |
|
2476 |
assume "z < B" from *[OF this `B < x`] show "B \<le> y" . |
|
41976 | 2477 |
next |
41978 | 2478 |
assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto |
41976 | 2479 |
qed |
41973 | 2480 |
qed |
2481 |
||
41978 | 2482 |
lemma fixes x y :: extreal |
2483 |
shows Sup_atMost[simp]: "Sup {.. y} = y" |
|
2484 |
and Sup_lessThan[simp]: "Sup {..< y} = y" |
|
2485 |
and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y" |
|
2486 |
and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y" |
|
2487 |
and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y" |
|
2488 |
by (auto simp: Sup_extreal_def intro!: Least_equality |
|
2489 |
intro: extreal_le_extreal extreal_le_extreal_bounded[of x]) |
|
2490 |
||
2491 |
lemma Sup_greaterThanlessThan[simp]: |
|
2492 |
fixes x y :: extreal assumes "x < y" shows "Sup { x <..< y} = y" |
|
2493 |
unfolding Sup_extreal_def |
|
2494 |
proof (intro Least_equality extreal_le_extreal_bounded[of _ _ y]) |
|
2495 |
fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z" |
|
2496 |
from extreal_dense[OF `x < y`] guess w .. note w = this |
|
2497 |
with z[THEN bspec, of w] show "x \<le> z" by auto |
|
2498 |
qed auto |
|
2499 |
||
41973 | 2500 |
lemma real_extreal_id: "real o extreal = id" |
2501 |
proof- |
|
2502 |
{ fix x have "(real o extreal) x = id x" by auto } |
|
2503 |
from this show ?thesis using ext by blast |
|
2504 |
qed |
|
2505 |
||
2506 |
lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})" |
|
2507 |
by (metis range_extreal open_extreal open_UNIV) |
|
2508 |
||
2509 |
lemma extreal_le_distrib: |
|
2510 |
fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b" |
|
2511 |
by (cases rule: extreal3_cases[of a b c]) |
|
2512 |
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
|
2513 |
||
2514 |
lemma extreal_pos_distrib: |
|
2515 |
fixes a b c :: extreal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b" |
|
2516 |
using assms by (cases rule: extreal3_cases[of a b c]) |
|
2517 |
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
|
2518 |
||
2519 |
lemma extreal_pos_le_distrib: |
|
2520 |
fixes a b c :: extreal |
|
2521 |
assumes "c>=0" |
|
2522 |
shows "c * (a + b) <= c * a + c * b" |
|
2523 |
using assms by (cases rule: extreal3_cases[of a b c]) |
|
2524 |
(auto simp add: field_simps) |
|
2525 |
||
2526 |
lemma extreal_max_mono: |
|
2527 |
"[| (a::extreal) <= b; c <= d |] ==> max a c <= max b d" |
|
2528 |
by (metis sup_extreal_def sup_mono) |
|
2529 |
||
2530 |
||
2531 |
lemma extreal_max_least: |
|
2532 |
"[| (a::extreal) <= x; c <= x |] ==> max a c <= x" |
|
2533 |
by (metis sup_extreal_def sup_least) |
|
2534 |
||
2535 |
end |