author | hoelzl |
Wed, 08 Dec 2010 16:15:14 +0100 | |
changeset 41096 | 843c40bbc379 |
parent 41023 | 9118eb4eb8dc |
child 41413 | 64cd30d6b0b8 |
permissions | -rw-r--r-- |
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(* Author: Johannes Hoelzl, TU Muenchen *) |
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header {* A type for positive real numbers with infinity *} |
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theory Positive_Extended_Real |
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imports Complex_Main Nat_Bijection Multivariate_Analysis |
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begin |
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lemma (in complete_lattice) Sup_start: |
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assumes *: "\<And>x. f x \<le> f 0" |
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shows "(SUP n. f n) = f 0" |
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proof (rule antisym) |
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show "f 0 \<le> (SUP n. f n)" by (rule le_SUPI) auto |
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show "(SUP n. f n) \<le> f 0" by (rule SUP_leI[OF *]) |
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qed |
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lemma (in complete_lattice) Inf_start: |
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assumes *: "\<And>x. f 0 \<le> f x" |
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shows "(INF n. f n) = f 0" |
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proof (rule antisym) |
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show "(INF n. f n) \<le> f 0" by (rule INF_leI) simp |
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show "f 0 \<le> (INF n. f n)" by (rule le_INFI[OF *]) |
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qed |
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lemma (in complete_lattice) Sup_mono_offset: |
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fixes f :: "'b :: {ordered_ab_semigroup_add,monoid_add} \<Rightarrow> 'a" |
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assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" and "0 \<le> k" |
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shows "(SUP n . f (k + n)) = (SUP n. f n)" |
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proof (rule antisym) |
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show "(SUP n. f (k + n)) \<le> (SUP n. f n)" |
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by (auto intro!: Sup_mono simp: SUPR_def) |
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{ fix n :: 'b |
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have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono) |
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with * have "f n \<le> f (k + n)" by simp } |
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thus "(SUP n. f n) \<le> (SUP n. f (k + n))" |
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by (auto intro!: Sup_mono exI simp: SUPR_def) |
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qed |
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lemma (in complete_lattice) Sup_mono_offset_Suc: |
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assumes *: "\<And>x. f x \<le> f (Suc x)" |
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shows "(SUP n . f (Suc n)) = (SUP n. f n)" |
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unfolding Suc_eq_plus1 |
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apply (subst add_commute) |
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apply (rule Sup_mono_offset) |
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by (auto intro!: order.lift_Suc_mono_le[of "op \<le>" "op <" f, OF _ *]) default |
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lemma (in complete_lattice) Inf_mono_offset: |
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fixes f :: "'b :: {ordered_ab_semigroup_add,monoid_add} \<Rightarrow> 'a" |
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assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x" and "0 \<le> k" |
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shows "(INF n . f (k + n)) = (INF n. f n)" |
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proof (rule antisym) |
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show "(INF n. f n) \<le> (INF n. f (k + n))" |
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by (auto intro!: Inf_mono simp: INFI_def) |
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{ fix n :: 'b |
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have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono) |
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with * have "f (k + n) \<le> f n" by simp } |
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thus "(INF n. f (k + n)) \<le> (INF n. f n)" |
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by (auto intro!: Inf_mono exI simp: INFI_def) |
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qed |
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lemma (in complete_lattice) isotone_converge: |
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fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y " |
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shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))" |
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proof - |
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have "\<And>n. (SUP m. f (n + m)) = (SUP n. f n)" |
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apply (rule Sup_mono_offset) |
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apply (rule assms) |
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by simp_all |
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moreover |
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{ fix n have "(INF m. f (n + m)) = f n" |
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using Inf_start[of "\<lambda>m. f (n + m)"] assms by simp } |
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ultimately show ?thesis by simp |
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qed |
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lemma (in complete_lattice) antitone_converges: |
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fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x" |
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shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))" |
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proof - |
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have "\<And>n. (INF m. f (n + m)) = (INF n. f n)" |
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apply (rule Inf_mono_offset) |
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apply (rule assms) |
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by simp_all |
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moreover |
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{ fix n have "(SUP m. f (n + m)) = f n" |
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using Sup_start[of "\<lambda>m. f (n + m)"] assms by simp } |
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ultimately show ?thesis by simp |
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qed |
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lemma (in complete_lattice) lim_INF_le_lim_SUP: |
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fixes f :: "nat \<Rightarrow> 'a" |
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shows "(SUP n. INF m. f (n + m)) \<le> (INF n. SUP m. f (n + m))" |
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proof (rule SUP_leI, rule le_INFI) |
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fix i j show "(INF m. f (i + m)) \<le> (SUP m. f (j + m))" |
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proof (cases rule: le_cases) |
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assume "i \<le> j" |
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have "(INF m. f (i + m)) \<le> f (i + (j - i))" by (rule INF_leI) simp |
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also have "\<dots> = f (j + 0)" using `i \<le> j` by auto |
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also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp |
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finally show ?thesis . |
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next |
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assume "j \<le> i" |
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have "(INF m. f (i + m)) \<le> f (i + 0)" by (rule INF_leI) simp |
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also have "\<dots> = f (j + (i - j))" using `j \<le> i` by auto |
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also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp |
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finally show ?thesis . |
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qed |
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qed |
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text {* |
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We introduce the the positive real numbers as needed for measure theory. |
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*} |
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typedef pextreal = "(Some ` {0::real..}) \<union> {None}" |
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by (rule exI[of _ None]) simp |
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subsection "Introduce @{typ pextreal} similar to a datatype" |
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definition "Real x = Abs_pextreal (Some (sup 0 x))" |
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definition "\<omega> = Abs_pextreal None" |
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definition "pextreal_case f i x = (if x = \<omega> then i else f (THE r. 0 \<le> r \<and> x = Real r))" |
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definition "of_pextreal = pextreal_case (\<lambda>x. x) 0" |
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defs (overloaded) |
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real_of_pextreal_def [code_unfold]: "real == of_pextreal" |
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lemma pextreal_Some[simp]: "0 \<le> x \<Longrightarrow> Some x \<in> pextreal" |
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unfolding pextreal_def by simp |
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lemma pextreal_Some_sup[simp]: "Some (sup 0 x) \<in> pextreal" |
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by (simp add: sup_ge1) |
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lemma pextreal_None[simp]: "None \<in> pextreal" |
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unfolding pextreal_def by simp |
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lemma Real_inj[simp]: |
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assumes "0 \<le> x" and "0 \<le> y" |
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shows "Real x = Real y \<longleftrightarrow> x = y" |
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unfolding Real_def assms[THEN sup_absorb2] |
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using assms by (simp add: Abs_pextreal_inject) |
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lemma Real_neq_\<omega>[simp]: |
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"Real x = \<omega> \<longleftrightarrow> False" |
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"\<omega> = Real x \<longleftrightarrow> False" |
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by (simp_all add: Abs_pextreal_inject \<omega>_def Real_def) |
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lemma Real_neg: "x < 0 \<Longrightarrow> Real x = Real 0" |
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unfolding Real_def by (auto simp add: Abs_pextreal_inject intro!: sup_absorb1) |
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lemma pextreal_cases[case_names preal infinite, cases type: pextreal]: |
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assumes preal: "\<And>r. x = Real r \<Longrightarrow> 0 \<le> r \<Longrightarrow> P" and inf: "x = \<omega> \<Longrightarrow> P" |
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shows P |
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proof (cases x rule: pextreal.Abs_pextreal_cases) |
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case (Abs_pextreal y) |
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hence "y = None \<or> (\<exists>x \<ge> 0. y = Some x)" |
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unfolding pextreal_def by auto |
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thus P |
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proof (rule disjE) |
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assume "\<exists>x\<ge>0. y = Some x" then guess x .. |
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thus P by (simp add: preal[of x] Real_def Abs_pextreal(1) sup_absorb2) |
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qed (simp add: \<omega>_def Abs_pextreal(1) inf) |
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qed |
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lemma pextreal_case_\<omega>[simp]: "pextreal_case f i \<omega> = i" |
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unfolding pextreal_case_def by simp |
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lemma pextreal_case_Real[simp]: "pextreal_case f i (Real x) = (if 0 \<le> x then f x else f 0)" |
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proof (cases "0 \<le> x") |
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case True thus ?thesis unfolding pextreal_case_def by (auto intro: theI2) |
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next |
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case False |
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moreover have "(THE r. 0 \<le> r \<and> Real 0 = Real r) = 0" |
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by (auto intro!: the_equality) |
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ultimately show ?thesis unfolding pextreal_case_def by (simp add: Real_neg) |
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qed |
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lemma pextreal_case_cancel[simp]: "pextreal_case (\<lambda>c. i) i x = i" |
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by (cases x) simp_all |
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lemma pextreal_case_split: |
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"P (pextreal_case f i x) = ((x = \<omega> \<longrightarrow> P i) \<and> (\<forall>r\<ge>0. x = Real r \<longrightarrow> P (f r)))" |
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by (cases x) simp_all |
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lemma pextreal_case_split_asm: |
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"P (pextreal_case f i x) = (\<not> (x = \<omega> \<and> \<not> P i \<or> (\<exists>r. r \<ge> 0 \<and> x = Real r \<and> \<not> P (f r))))" |
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by (cases x) auto |
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lemma pextreal_case_cong[cong]: |
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assumes eq: "x = x'" "i = i'" and cong: "\<And>r. 0 \<le> r \<Longrightarrow> f r = f' r" |
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shows "pextreal_case f i x = pextreal_case f' i' x'" |
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unfolding eq using cong by (cases x') simp_all |
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lemma real_Real[simp]: "real (Real x) = (if 0 \<le> x then x else 0)" |
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unfolding real_of_pextreal_def of_pextreal_def by simp |
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lemma Real_real_image: |
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assumes "\<omega> \<notin> A" shows "Real ` real ` A = A" |
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proof safe |
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fix x assume "x \<in> A" |
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hence *: "x = Real (real x)" |
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using `\<omega> \<notin> A` by (cases x) auto |
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show "x \<in> Real ` real ` A" |
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using `x \<in> A` by (subst *) (auto intro!: imageI) |
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next |
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fix x assume "x \<in> A" |
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thus "Real (real x) \<in> A" |
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using `\<omega> \<notin> A` by (cases x) auto |
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qed |
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lemma real_pextreal_nonneg[simp, intro]: "0 \<le> real (x :: pextreal)" |
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unfolding real_of_pextreal_def of_pextreal_def |
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by (cases x) auto |
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lemma real_\<omega>[simp]: "real \<omega> = 0" |
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unfolding real_of_pextreal_def of_pextreal_def by simp |
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lemma pextreal_noteq_omega_Ex: "X \<noteq> \<omega> \<longleftrightarrow> (\<exists>r\<ge>0. X = Real r)" by (cases X) auto |
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subsection "@{typ pextreal} is a monoid for addition" |
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instantiation pextreal :: comm_monoid_add |
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begin |
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definition "0 = Real 0" |
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definition "x + y = pextreal_case (\<lambda>r. pextreal_case (\<lambda>p. Real (r + p)) \<omega> y) \<omega> x" |
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lemma pextreal_plus[simp]: |
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"Real r + Real p = (if 0 \<le> r then if 0 \<le> p then Real (r + p) else Real r else Real p)" |
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"x + 0 = x" |
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"0 + x = x" |
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"x + \<omega> = \<omega>" |
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"\<omega> + x = \<omega>" |
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by (simp_all add: plus_pextreal_def Real_neg zero_pextreal_def split: pextreal_case_split) |
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lemma \<omega>_neq_0[simp]: |
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"\<omega> = 0 \<longleftrightarrow> False" |
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"0 = \<omega> \<longleftrightarrow> False" |
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by (simp_all add: zero_pextreal_def) |
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lemma Real_eq_0[simp]: |
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"Real r = 0 \<longleftrightarrow> r \<le> 0" |
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"0 = Real r \<longleftrightarrow> r \<le> 0" |
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by (auto simp add: Abs_pextreal_inject zero_pextreal_def Real_def sup_real_def) |
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lemma Real_0[simp]: "Real 0 = 0" by (simp add: zero_pextreal_def) |
38656 | 249 |
|
250 |
instance |
|
251 |
proof |
|
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252 |
fix a :: pextreal |
38656 | 253 |
show "0 + a = a" by (cases a) simp_all |
254 |
||
255 |
fix b show "a + b = b + a" |
|
256 |
by (cases a, cases b) simp_all |
|
257 |
||
258 |
fix c show "a + b + c = a + (b + c)" |
|
259 |
by (cases a, cases b, cases c) simp_all |
|
260 |
qed |
|
261 |
end |
|
262 |
||
41096 | 263 |
lemma Real_minus_abs[simp]: "Real (- \<bar>x\<bar>) = 0" |
264 |
by simp |
|
265 |
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266 |
lemma pextreal_plus_eq_\<omega>[simp]: "(a :: pextreal) + b = \<omega> \<longleftrightarrow> a = \<omega> \<or> b = \<omega>" |
38656 | 267 |
by (cases a, cases b) auto |
268 |
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269 |
lemma pextreal_add_cancel_left: |
38656 | 270 |
"a + b = a + c \<longleftrightarrow> (a = \<omega> \<or> b = c)" |
271 |
by (cases a, cases b, cases c, simp_all, cases c, simp_all) |
|
272 |
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273 |
lemma pextreal_add_cancel_right: |
38656 | 274 |
"b + a = c + a \<longleftrightarrow> (a = \<omega> \<or> b = c)" |
275 |
by (cases a, cases b, cases c, simp_all, cases c, simp_all) |
|
276 |
||
277 |
lemma Real_eq_Real: |
|
278 |
"Real a = Real b \<longleftrightarrow> (a = b \<or> (a \<le> 0 \<and> b \<le> 0))" |
|
279 |
proof (cases "a \<le> 0 \<or> b \<le> 0") |
|
280 |
case False with Real_inj[of a b] show ?thesis by auto |
|
281 |
next |
|
282 |
case True |
|
283 |
thus ?thesis |
|
284 |
proof |
|
285 |
assume "a \<le> 0" |
|
286 |
hence *: "Real a = 0" by simp |
|
287 |
show ?thesis using `a \<le> 0` unfolding * by auto |
|
288 |
next |
|
289 |
assume "b \<le> 0" |
|
290 |
hence *: "Real b = 0" by simp |
|
291 |
show ?thesis using `b \<le> 0` unfolding * by auto |
|
292 |
qed |
|
293 |
qed |
|
294 |
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295 |
lemma real_pextreal_0[simp]: "real (0 :: pextreal) = 0" |
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296 |
unfolding zero_pextreal_def real_Real by simp |
38656 | 297 |
|
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298 |
lemma real_of_pextreal_eq_0: "real X = 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)" |
38656 | 299 |
by (cases X) auto |
300 |
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|
301 |
lemma real_of_pextreal_eq: "real X = real Y \<longleftrightarrow> |
38656 | 302 |
(X = Y \<or> (X = 0 \<and> Y = \<omega>) \<or> (Y = 0 \<and> X = \<omega>))" |
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303 |
by (cases X, cases Y) (auto simp add: real_of_pextreal_eq_0) |
38656 | 304 |
|
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|
305 |
lemma real_of_pextreal_add: "real X + real Y = |
38656 | 306 |
(if X = \<omega> then real Y else if Y = \<omega> then real X else real (X + Y))" |
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307 |
by (auto simp: pextreal_noteq_omega_Ex) |
38656 | 308 |
|
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309 |
subsection "@{typ pextreal} is a monoid for multiplication" |
38656 | 310 |
|
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311 |
instantiation pextreal :: comm_monoid_mult |
38656 | 312 |
begin |
313 |
||
314 |
definition "1 = Real 1" |
|
315 |
definition "x * y = (if x = 0 \<or> y = 0 then 0 else |
|
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316 |
pextreal_case (\<lambda>r. pextreal_case (\<lambda>p. Real (r * p)) \<omega> y) \<omega> x)" |
38656 | 317 |
|
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|
318 |
lemma pextreal_times[simp]: |
38656 | 319 |
"Real r * Real p = (if 0 \<le> r \<and> 0 \<le> p then Real (r * p) else 0)" |
320 |
"\<omega> * x = (if x = 0 then 0 else \<omega>)" |
|
321 |
"x * \<omega> = (if x = 0 then 0 else \<omega>)" |
|
322 |
"0 * x = 0" |
|
323 |
"x * 0 = 0" |
|
324 |
"1 = \<omega> \<longleftrightarrow> False" |
|
325 |
"\<omega> = 1 \<longleftrightarrow> False" |
|
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326 |
by (auto simp add: times_pextreal_def one_pextreal_def) |
38656 | 327 |
|
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328 |
lemma pextreal_one_mult[simp]: |
38656 | 329 |
"Real x + 1 = (if 0 \<le> x then Real (x + 1) else 1)" |
330 |
"1 + Real x = (if 0 \<le> x then Real (1 + x) else 1)" |
|
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331 |
unfolding one_pextreal_def by simp_all |
38656 | 332 |
|
333 |
instance |
|
334 |
proof |
|
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|
335 |
fix a :: pextreal show "1 * a = a" |
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336 |
by (cases a) (simp_all add: one_pextreal_def) |
38656 | 337 |
|
338 |
fix b show "a * b = b * a" |
|
339 |
by (cases a, cases b) (simp_all add: mult_nonneg_nonneg) |
|
340 |
||
341 |
fix c show "a * b * c = a * (b * c)" |
|
342 |
apply (cases a, cases b, cases c) |
|
343 |
apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos) |
|
344 |
apply (cases b, cases c) |
|
345 |
apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos) |
|
346 |
done |
|
347 |
qed |
|
348 |
end |
|
349 |
||
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|
350 |
lemma pextreal_mult_cancel_left: |
38656 | 351 |
"a * b = a * c \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))" |
352 |
by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto) |
|
353 |
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|
354 |
lemma pextreal_mult_cancel_right: |
38656 | 355 |
"b * a = c * a \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))" |
356 |
by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto) |
|
357 |
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|
358 |
lemma Real_1[simp]: "Real 1 = 1" by (simp add: one_pextreal_def) |
38656 | 359 |
|
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|
360 |
lemma real_pextreal_1[simp]: "real (1 :: pextreal) = 1" |
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|
361 |
unfolding one_pextreal_def real_Real by simp |
38656 | 362 |
|
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|
363 |
lemma real_of_pextreal_mult: "real X * real Y = real (X * Y :: pextreal)" |
38656 | 364 |
by (cases X, cases Y) (auto simp: zero_le_mult_iff) |
365 |
||
40874 | 366 |
lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0" |
367 |
shows "Real (x * y) = Real x * Real y" using assms by auto |
|
368 |
||
369 |
lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A" |
|
370 |
proof(cases "finite A") |
|
371 |
case True thus ?thesis using assms |
|
372 |
proof(induct A) case (insert x A) |
|
373 |
have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto |
|
374 |
thus ?case unfolding setprod_insert[OF insert(1-2)] apply- |
|
375 |
apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym]) |
|
376 |
using insert by auto |
|
377 |
qed auto |
|
378 |
qed auto |
|
379 |
||
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|
380 |
subsection "@{typ pextreal} is a linear order" |
38656 | 381 |
|
41023
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|
382 |
instantiation pextreal :: linorder |
38656 | 383 |
begin |
384 |
||
41023
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|
385 |
definition "x < y \<longleftrightarrow> pextreal_case (\<lambda>i. pextreal_case (\<lambda>j. i < j) True y) False x" |
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|
386 |
definition "x \<le> y \<longleftrightarrow> pextreal_case (\<lambda>j. pextreal_case (\<lambda>i. i \<le> j) False x) True y" |
38656 | 387 |
|
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|
388 |
lemma pextreal_less[simp]: |
38656 | 389 |
"Real r < \<omega>" |
390 |
"Real r < Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r < p else 0 < p)" |
|
391 |
"\<omega> < x \<longleftrightarrow> False" |
|
392 |
"0 < \<omega>" |
|
393 |
"0 < Real r \<longleftrightarrow> 0 < r" |
|
394 |
"x < 0 \<longleftrightarrow> False" |
|
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|
395 |
"0 < (1::pextreal)" |
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|
396 |
by (simp_all add: less_pextreal_def zero_pextreal_def one_pextreal_def del: Real_0 Real_1) |
38656 | 397 |
|
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|
398 |
lemma pextreal_less_eq[simp]: |
38656 | 399 |
"x \<le> \<omega>" |
400 |
"Real r \<le> Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r \<le> p else r \<le> 0)" |
|
401 |
"0 \<le> x" |
|
41023
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|
402 |
by (simp_all add: less_eq_pextreal_def zero_pextreal_def del: Real_0) |
38656 | 403 |
|
41023
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|
404 |
lemma pextreal_\<omega>_less_eq[simp]: |
38656 | 405 |
"\<omega> \<le> x \<longleftrightarrow> x = \<omega>" |
41023
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|
406 |
by (cases x) (simp_all add: not_le less_eq_pextreal_def) |
38656 | 407 |
|
41023
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|
408 |
lemma pextreal_less_eq_zero[simp]: |
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|
409 |
"(x::pextreal) \<le> 0 \<longleftrightarrow> x = 0" |
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changeset
|
410 |
by (cases x) (simp_all add: zero_pextreal_def del: Real_0) |
38656 | 411 |
|
412 |
instance |
|
413 |
proof |
|
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|
414 |
fix x :: pextreal |
38656 | 415 |
show "x \<le> x" by (cases x) simp_all |
416 |
fix y |
|
417 |
show "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" |
|
418 |
by (cases x, cases y) auto |
|
419 |
show "x \<le> y \<or> y \<le> x " |
|
420 |
by (cases x, cases y) auto |
|
421 |
{ assume "x \<le> y" "y \<le> x" thus "x = y" |
|
422 |
by (cases x, cases y) auto } |
|
423 |
{ fix z assume "x \<le> y" "y \<le> z" |
|
424 |
thus "x \<le> z" by (cases x, cases y, cases z) auto } |
|
425 |
qed |
|
426 |
end |
|
427 |
||
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|
428 |
lemma pextreal_zero_lessI[intro]: |
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|
429 |
"(a :: pextreal) \<noteq> 0 \<Longrightarrow> 0 < a" |
38656 | 430 |
by (cases a) auto |
431 |
||
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|
432 |
lemma pextreal_less_omegaI[intro, simp]: |
38656 | 433 |
"a \<noteq> \<omega> \<Longrightarrow> a < \<omega>" |
434 |
by (cases a) auto |
|
435 |
||
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changeset
|
436 |
lemma pextreal_plus_eq_0[simp]: "(a :: pextreal) + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" |
38656 | 437 |
by (cases a, cases b) auto |
438 |
||
41023
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|
439 |
lemma pextreal_le_add1[simp, intro]: "n \<le> n + (m::pextreal)" |
38656 | 440 |
by (cases n, cases m) simp_all |
441 |
||
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|
442 |
lemma pextreal_le_add2: "(n::pextreal) + m \<le> k \<Longrightarrow> m \<le> k" |
38656 | 443 |
by (cases n, cases m, cases k) simp_all |
444 |
||
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|
445 |
lemma pextreal_le_add3: "(n::pextreal) + m \<le> k \<Longrightarrow> n \<le> k" |
38656 | 446 |
by (cases n, cases m, cases k) simp_all |
447 |
||
41023
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|
448 |
lemma pextreal_less_\<omega>: "x < \<omega> \<longleftrightarrow> x \<noteq> \<omega>" |
38656 | 449 |
by (cases x) auto |
450 |
||
41023
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|
451 |
lemma pextreal_0_less_mult_iff[simp]: |
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changeset
|
452 |
fixes x y :: pextreal shows "0 < x * y \<longleftrightarrow> 0 < x \<and> 0 < y" |
39092 | 453 |
by (cases x, cases y) (auto simp: zero_less_mult_iff) |
454 |
||
41023
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|
455 |
lemma pextreal_ord_one[simp]: |
40859 | 456 |
"Real p < 1 \<longleftrightarrow> p < 1" |
457 |
"Real p \<le> 1 \<longleftrightarrow> p \<le> 1" |
|
458 |
"1 < Real p \<longleftrightarrow> 1 < p" |
|
459 |
"1 \<le> Real p \<longleftrightarrow> 1 \<le> p" |
|
41023
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|
460 |
by (simp_all add: one_pextreal_def del: Real_1) |
40859 | 461 |
|
41023
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|
462 |
subsection {* @{text "x - y"} on @{typ pextreal} *} |
38656 | 463 |
|
41023
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|
464 |
instantiation pextreal :: minus |
38656 | 465 |
begin |
41023
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|
466 |
definition "x - y = (if y < x then THE d. x = y + d else 0 :: pextreal)" |
38656 | 467 |
|
41023
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|
468 |
lemma minus_pextreal_eq: |
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|
469 |
"(x - y = (z :: pextreal)) \<longleftrightarrow> (if y < x then x = y + z else z = 0)" |
38656 | 470 |
(is "?diff \<longleftrightarrow> ?if") |
471 |
proof |
|
472 |
assume ?diff |
|
473 |
thus ?if |
|
474 |
proof (cases "y < x") |
|
475 |
case True |
|
476 |
then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto |
|
477 |
||
41023
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changeset
|
478 |
show ?thesis unfolding `?diff`[symmetric] if_P[OF True] minus_pextreal_def |
38656 | 479 |
proof (rule theI2[where Q="\<lambda>d. x = y + d"]) |
41023
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changeset
|
480 |
show "x = y + pextreal_case (\<lambda>r. Real (r - real y)) \<omega> x" (is "x = y + ?d") |
38656 | 481 |
using `y < x` p by (cases x) simp_all |
482 |
||
483 |
fix d assume "x = y + d" |
|
484 |
thus "d = ?d" using `y < x` p by (cases d, cases x) simp_all |
|
485 |
qed simp |
|
41023
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changeset
|
486 |
qed (simp add: minus_pextreal_def) |
38656 | 487 |
next |
488 |
assume ?if |
|
489 |
thus ?diff |
|
490 |
proof (cases "y < x") |
|
491 |
case True |
|
492 |
then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto |
|
493 |
||
494 |
from True `?if` have "x = y + z" by simp |
|
495 |
||
41023
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hoelzl
parents:
40874
diff
changeset
|
496 |
show ?thesis unfolding minus_pextreal_def if_P[OF True] unfolding `x = y + z` |
38656 | 497 |
proof (rule the_equality) |
41023
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parents:
40874
diff
changeset
|
498 |
fix d :: pextreal assume "y + z = y + d" |
38656 | 499 |
thus "d = z" using `y < x` p |
500 |
by (cases d, cases z) simp_all |
|
501 |
qed simp |
|
41023
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40874
diff
changeset
|
502 |
qed (simp add: minus_pextreal_def) |
38656 | 503 |
qed |
504 |
||
505 |
instance .. |
|
506 |
end |
|
507 |
||
41023
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hoelzl
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40874
diff
changeset
|
508 |
lemma pextreal_minus[simp]: |
38656 | 509 |
"Real r - Real p = (if 0 \<le> r \<and> p < r then if 0 \<le> p then Real (r - p) else Real r else 0)" |
41023
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hoelzl
parents:
40874
diff
changeset
|
510 |
"(A::pextreal) - A = 0" |
38656 | 511 |
"\<omega> - Real r = \<omega>" |
512 |
"Real r - \<omega> = 0" |
|
513 |
"A - 0 = A" |
|
514 |
"0 - A = 0" |
|
41023
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hoelzl
parents:
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diff
changeset
|
515 |
by (auto simp: minus_pextreal_eq not_less) |
38656 | 516 |
|
41023
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changeset
|
517 |
lemma pextreal_le_epsilon: |
9118eb4eb8dc
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hoelzl
parents:
40874
diff
changeset
|
518 |
fixes x y :: pextreal |
38656 | 519 |
assumes "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e" |
520 |
shows "x \<le> y" |
|
521 |
proof (cases y) |
|
522 |
case (preal r) |
|
523 |
then obtain p where x: "x = Real p" "0 \<le> p" |
|
524 |
using assms[of 1] by (cases x) auto |
|
525 |
{ fix e have "0 < e \<Longrightarrow> p \<le> r + e" |
|
526 |
using assms[of "Real e"] preal x by auto } |
|
527 |
hence "p \<le> r" by (rule field_le_epsilon) |
|
528 |
thus ?thesis using preal x by auto |
|
529 |
qed simp |
|
530 |
||
41023
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40874
diff
changeset
|
531 |
instance pextreal :: "{ordered_comm_semiring, comm_semiring_1}" |
38656 | 532 |
proof |
41023
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hoelzl
parents:
40874
diff
changeset
|
533 |
show "0 \<noteq> (1::pextreal)" unfolding zero_pextreal_def one_pextreal_def |
38656 | 534 |
by (simp del: Real_1 Real_0) |
535 |
||
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
536 |
fix a :: pextreal |
38656 | 537 |
show "0 * a = 0" "a * 0 = 0" by simp_all |
538 |
||
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
539 |
fix b c :: pextreal |
38656 | 540 |
show "(a + b) * c = a * c + b * c" |
541 |
by (cases c, cases a, cases b) |
|
542 |
(auto intro!: arg_cong[where f=Real] simp: field_simps not_le mult_le_0_iff mult_less_0_iff) |
|
543 |
||
544 |
{ assume "a \<le> b" thus "c + a \<le> c + b" |
|
545 |
by (cases c, cases a, cases b) auto } |
|
546 |
||
547 |
assume "a \<le> b" "0 \<le> c" |
|
548 |
thus "c * a \<le> c * b" |
|
549 |
apply (cases c, cases a, cases b) |
|
550 |
by (auto simp: mult_left_mono mult_le_0_iff mult_less_0_iff not_le) |
|
551 |
qed |
|
552 |
||
553 |
lemma mult_\<omega>[simp]: "x * y = \<omega> \<longleftrightarrow> (x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0" |
|
554 |
by (cases x, cases y) auto |
|
555 |
||
556 |
lemma \<omega>_mult[simp]: "(\<omega> = x * y) = ((x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0)" |
|
557 |
by (cases x, cases y) auto |
|
558 |
||
41023
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hoelzl
parents:
40874
diff
changeset
|
559 |
lemma pextreal_mult_0[simp]: "x * y = 0 \<longleftrightarrow> x = 0 \<or> (y::pextreal) = 0" |
38656 | 560 |
by (cases x, cases y) (auto simp: mult_le_0_iff) |
561 |
||
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
562 |
lemma pextreal_mult_cancel: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
563 |
fixes x y z :: pextreal |
38656 | 564 |
assumes "y \<le> z" |
565 |
shows "x * y \<le> x * z" |
|
566 |
using assms |
|
567 |
by (cases x, cases y, cases z) |
|
568 |
(auto simp: mult_le_cancel_left mult_le_0_iff mult_less_0_iff not_le) |
|
569 |
||
570 |
lemma Real_power[simp]: |
|
571 |
"Real x ^ n = (if x \<le> 0 then (if n = 0 then 1 else 0) else Real (x ^ n))" |
|
572 |
by (induct n) auto |
|
573 |
||
574 |
lemma Real_power_\<omega>[simp]: |
|
575 |
"\<omega> ^ n = (if n = 0 then 1 else \<omega>)" |
|
576 |
by (induct n) auto |
|
577 |
||
41023
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hoelzl
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40874
diff
changeset
|
578 |
lemma pextreal_of_nat[simp]: "of_nat m = Real (real m)" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
579 |
by (induct m) (auto simp: real_of_nat_Suc one_pextreal_def simp del: Real_1) |
38656 | 580 |
|
40871 | 581 |
lemma less_\<omega>_Ex_of_nat: "x < \<omega> \<longleftrightarrow> (\<exists>n. x < of_nat n)" |
582 |
proof safe |
|
583 |
assume "x < \<omega>" |
|
584 |
then obtain r where "0 \<le> r" "x = Real r" by (cases x) auto |
|
585 |
moreover obtain n where "r < of_nat n" using ex_less_of_nat by auto |
|
586 |
ultimately show "\<exists>n. x < of_nat n" by (auto simp: real_eq_of_nat) |
|
587 |
qed auto |
|
588 |
||
41023
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hoelzl
parents:
40874
diff
changeset
|
589 |
lemma real_of_pextreal_mono: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
590 |
fixes a b :: pextreal |
38656 | 591 |
assumes "b \<noteq> \<omega>" "a \<le> b" |
592 |
shows "real a \<le> real b" |
|
593 |
using assms by (cases b, cases a) auto |
|
594 |
||
41023
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hoelzl
parents:
40874
diff
changeset
|
595 |
lemma setprod_pextreal_0: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
596 |
"(\<Prod>i\<in>I. f i) = (0::pextreal) \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = 0)" |
40859 | 597 |
proof cases |
598 |
assume "finite I" then show ?thesis |
|
599 |
proof (induct I) |
|
600 |
case (insert i I) |
|
601 |
then show ?case by simp |
|
602 |
qed simp |
|
603 |
qed simp |
|
604 |
||
605 |
lemma setprod_\<omega>: |
|
606 |
"(\<Prod>i\<in>I. f i) = \<omega> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<omega>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)" |
|
607 |
proof cases |
|
608 |
assume "finite I" then show ?thesis |
|
609 |
proof (induct I) |
|
610 |
case (insert i I) then show ?case |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
611 |
by (auto simp: setprod_pextreal_0) |
40859 | 612 |
qed simp |
613 |
qed simp |
|
614 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
615 |
instance pextreal :: "semiring_char_0" |
38656 | 616 |
proof |
617 |
fix m n |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
618 |
show "inj (of_nat::nat\<Rightarrow>pextreal)" by (auto intro!: inj_onI) |
38656 | 619 |
qed |
620 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
621 |
subsection "@{typ pextreal} is a complete lattice" |
38656 | 622 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
623 |
instantiation pextreal :: lattice |
38656 | 624 |
begin |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
625 |
definition [simp]: "sup x y = (max x y :: pextreal)" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
626 |
definition [simp]: "inf x y = (min x y :: pextreal)" |
38656 | 627 |
instance proof qed simp_all |
628 |
end |
|
629 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
630 |
instantiation pextreal :: complete_lattice |
38656 | 631 |
begin |
632 |
||
633 |
definition "bot = Real 0" |
|
634 |
definition "top = \<omega>" |
|
635 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
636 |
definition "Sup S = (LEAST z. \<forall>x\<in>S. x \<le> z :: pextreal)" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
637 |
definition "Inf S = (GREATEST z. \<forall>x\<in>S. z \<le> x :: pextreal)" |
38656 | 638 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
639 |
lemma pextreal_complete_Sup: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
640 |
fixes S :: "pextreal set" assumes "S \<noteq> {}" |
38656 | 641 |
shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)" |
642 |
proof (cases "\<exists>x\<ge>0. \<forall>a\<in>S. a \<le> Real x") |
|
643 |
case False |
|
644 |
hence *: "\<And>x. x\<ge>0 \<Longrightarrow> \<exists>a\<in>S. \<not>a \<le> Real x" by simp |
|
645 |
show ?thesis |
|
646 |
proof (safe intro!: exI[of _ \<omega>]) |
|
647 |
fix y assume **: "\<forall>z\<in>S. z \<le> y" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
648 |
show "\<omega> \<le> y" unfolding pextreal_\<omega>_less_eq |
38656 | 649 |
proof (rule ccontr) |
650 |
assume "y \<noteq> \<omega>" |
|
651 |
then obtain x where [simp]: "y = Real x" and "0 \<le> x" by (cases y) auto |
|
652 |
from *[OF `0 \<le> x`] show False using ** by auto |
|
653 |
qed |
|
654 |
qed simp |
|
655 |
next |
|
656 |
case True then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> Real y" and "0 \<le> y" by auto |
|
657 |
from y[of \<omega>] have "\<omega> \<notin> S" by auto |
|
658 |
||
659 |
with `S \<noteq> {}` obtain x where "x \<in> S" and "x \<noteq> \<omega>" by auto |
|
660 |
||
661 |
have bound: "\<forall>x\<in>real ` S. x \<le> y" |
|
662 |
proof |
|
663 |
fix z assume "z \<in> real ` S" then guess a .. |
|
664 |
with y[of a] `\<omega> \<notin> S` `0 \<le> y` show "z \<le> y" by (cases a) auto |
|
665 |
qed |
|
666 |
with reals_complete2[of "real ` S"] `x \<in> S` |
|
667 |
obtain s where s: "\<forall>y\<in>S. real y \<le> s" "\<forall>z. ((\<forall>y\<in>S. real y \<le> z) \<longrightarrow> s \<le> z)" |
|
668 |
by auto |
|
669 |
||
670 |
show ?thesis |
|
671 |
proof (safe intro!: exI[of _ "Real s"]) |
|
672 |
fix z assume "z \<in> S" thus "z \<le> Real s" |
|
673 |
using s `\<omega> \<notin> S` by (cases z) auto |
|
674 |
next |
|
675 |
fix z assume *: "\<forall>y\<in>S. y \<le> z" |
|
676 |
show "Real s \<le> z" |
|
677 |
proof (cases z) |
|
678 |
case (preal u) |
|
679 |
{ fix v assume "v \<in> S" |
|
680 |
hence "v \<le> Real u" using * preal by auto |
|
681 |
hence "real v \<le> u" using `\<omega> \<notin> S` `0 \<le> u` by (cases v) auto } |
|
682 |
hence "s \<le> u" using s(2) by auto |
|
683 |
thus "Real s \<le> z" using preal by simp |
|
684 |
qed simp |
|
685 |
qed |
|
686 |
qed |
|
687 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
688 |
lemma pextreal_complete_Inf: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
689 |
fixes S :: "pextreal set" assumes "S \<noteq> {}" |
38656 | 690 |
shows "\<exists>x. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)" |
691 |
proof (cases "S = {\<omega>}") |
|
692 |
case True thus ?thesis by (auto intro!: exI[of _ \<omega>]) |
|
693 |
next |
|
694 |
case False with `S \<noteq> {}` have "S - {\<omega>} \<noteq> {}" by auto |
|
695 |
hence not_empty: "\<exists>x. x \<in> uminus ` real ` (S - {\<omega>})" by auto |
|
696 |
have bounds: "\<exists>x. \<forall>y\<in>uminus ` real ` (S - {\<omega>}). y \<le> x" by (auto intro!: exI[of _ 0]) |
|
697 |
from reals_complete2[OF not_empty bounds] |
|
698 |
obtain s where s: "\<And>y. y\<in>S - {\<omega>} \<Longrightarrow> - real y \<le> s" "\<forall>z. ((\<forall>y\<in>S - {\<omega>}. - real y \<le> z) \<longrightarrow> s \<le> z)" |
|
699 |
by auto |
|
700 |
||
701 |
show ?thesis |
|
702 |
proof (safe intro!: exI[of _ "Real (-s)"]) |
|
703 |
fix z assume "z \<in> S" |
|
704 |
show "Real (-s) \<le> z" |
|
705 |
proof (cases z) |
|
706 |
case (preal r) |
|
707 |
with s `z \<in> S` have "z \<in> S - {\<omega>}" by simp |
|
708 |
hence "- r \<le> s" using preal s(1)[of z] by auto |
|
709 |
hence "- s \<le> r" by (subst neg_le_iff_le[symmetric]) simp |
|
710 |
thus ?thesis using preal by simp |
|
711 |
qed simp |
|
712 |
next |
|
713 |
fix z assume *: "\<forall>y\<in>S. z \<le> y" |
|
714 |
show "z \<le> Real (-s)" |
|
715 |
proof (cases z) |
|
716 |
case (preal u) |
|
717 |
{ fix v assume "v \<in> S-{\<omega>}" |
|
718 |
hence "Real u \<le> v" using * preal by auto |
|
719 |
hence "- real v \<le> - u" using `0 \<le> u` `v \<in> S - {\<omega>}` by (cases v) auto } |
|
720 |
hence "u \<le> - s" using s(2) by (subst neg_le_iff_le[symmetric]) auto |
|
721 |
thus "z \<le> Real (-s)" using preal by simp |
|
722 |
next |
|
723 |
case infinite |
|
724 |
with * have "S = {\<omega>}" using `S \<noteq> {}` by auto |
|
725 |
with `S - {\<omega>} \<noteq> {}` show ?thesis by simp |
|
726 |
qed |
|
727 |
qed |
|
728 |
qed |
|
729 |
||
730 |
instance |
|
731 |
proof |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
732 |
fix x :: pextreal and A |
38656 | 733 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
734 |
show "bot \<le> x" by (cases x) (simp_all add: bot_pextreal_def) |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
735 |
show "x \<le> top" by (simp add: top_pextreal_def) |
38656 | 736 |
|
737 |
{ assume "x \<in> A" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
738 |
with pextreal_complete_Sup[of A] |
38656 | 739 |
obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto |
740 |
hence "x \<le> s" using `x \<in> A` by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
741 |
also have "... = Sup A" using s unfolding Sup_pextreal_def |
38656 | 742 |
by (auto intro!: Least_equality[symmetric]) |
743 |
finally show "x \<le> Sup A" . } |
|
744 |
||
745 |
{ assume "x \<in> A" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
746 |
with pextreal_complete_Inf[of A] |
38656 | 747 |
obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
748 |
hence "Inf A = i" unfolding Inf_pextreal_def |
38656 | 749 |
by (auto intro!: Greatest_equality) |
750 |
also have "i \<le> x" using i `x \<in> A` by auto |
|
751 |
finally show "Inf A \<le> x" . } |
|
752 |
||
753 |
{ assume *: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" |
|
754 |
show "Sup A \<le> x" |
|
755 |
proof (cases "A = {}") |
|
756 |
case True |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
757 |
hence "Sup A = 0" unfolding Sup_pextreal_def |
38656 | 758 |
by (auto intro!: Least_equality) |
759 |
thus "Sup A \<le> x" by simp |
|
760 |
next |
|
761 |
case False |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
762 |
with pextreal_complete_Sup[of A] |
38656 | 763 |
obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto |
764 |
hence "Sup A = s" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
765 |
unfolding Sup_pextreal_def by (auto intro!: Least_equality) |
38656 | 766 |
also have "s \<le> x" using * s by auto |
767 |
finally show "Sup A \<le> x" . |
|
768 |
qed } |
|
769 |
||
770 |
{ assume *: "\<And>z. z \<in> A \<Longrightarrow> x \<le> z" |
|
771 |
show "x \<le> Inf A" |
|
772 |
proof (cases "A = {}") |
|
773 |
case True |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
774 |
hence "Inf A = \<omega>" unfolding Inf_pextreal_def |
38656 | 775 |
by (auto intro!: Greatest_equality) |
776 |
thus "x \<le> Inf A" by simp |
|
777 |
next |
|
778 |
case False |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
779 |
with pextreal_complete_Inf[of A] |
38656 | 780 |
obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto |
781 |
have "x \<le> i" using * i by auto |
|
782 |
also have "i = Inf A" using i |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
783 |
unfolding Inf_pextreal_def by (auto intro!: Greatest_equality[symmetric]) |
38656 | 784 |
finally show "x \<le> Inf A" . |
785 |
qed } |
|
786 |
qed |
|
787 |
end |
|
788 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
789 |
lemma Inf_pextreal_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
790 |
fixes z :: pextreal |
38656 | 791 |
shows "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> (\<exists>x\<in>X. x<y) \<longleftrightarrow> Inf X < y" |
792 |
by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear |
|
793 |
order_less_le_trans) |
|
794 |
||
795 |
lemma Inf_greater: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
796 |
fixes z :: pextreal assumes "Inf X < z" |
38656 | 797 |
shows "\<exists>x \<in> X. x < z" |
798 |
proof - |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
799 |
have "X \<noteq> {}" using assms by (auto simp: Inf_empty top_pextreal_def) |
38656 | 800 |
with assms show ?thesis |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
801 |
by (metis Inf_pextreal_iff mem_def not_leE) |
38656 | 802 |
qed |
803 |
||
804 |
lemma Inf_close: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
805 |
fixes e :: pextreal assumes "Inf X \<noteq> \<omega>" "0 < e" |
38656 | 806 |
shows "\<exists>x \<in> X. x < Inf X + e" |
807 |
proof (rule Inf_greater) |
|
808 |
show "Inf X < Inf X + e" using assms |
|
809 |
by (cases "Inf X", cases e) auto |
|
810 |
qed |
|
811 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
812 |
lemma pextreal_SUPI: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
813 |
fixes x :: pextreal |
38656 | 814 |
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x" |
815 |
assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y" |
|
816 |
shows "(SUP i:A. f i) = x" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
817 |
unfolding SUPR_def Sup_pextreal_def |
38656 | 818 |
using assms by (auto intro!: Least_equality) |
819 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
820 |
lemma Sup_pextreal_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
821 |
fixes z :: pextreal |
38656 | 822 |
shows "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> (\<exists>x\<in>X. y<x) \<longleftrightarrow> y < Sup X" |
823 |
by (metis complete_lattice_class.Sup_least complete_lattice_class.Sup_upper less_le_not_le linear |
|
824 |
order_less_le_trans) |
|
825 |
||
826 |
lemma Sup_lesser: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
827 |
fixes z :: pextreal assumes "z < Sup X" |
38656 | 828 |
shows "\<exists>x \<in> X. z < x" |
829 |
proof - |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
830 |
have "X \<noteq> {}" using assms by (auto simp: Sup_empty bot_pextreal_def) |
38656 | 831 |
with assms show ?thesis |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
832 |
by (metis Sup_pextreal_iff mem_def not_leE) |
38656 | 833 |
qed |
834 |
||
835 |
lemma Sup_eq_\<omega>: "\<omega> \<in> S \<Longrightarrow> Sup S = \<omega>" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
836 |
unfolding Sup_pextreal_def |
38656 | 837 |
by (auto intro!: Least_equality) |
838 |
||
839 |
lemma Sup_close: |
|
840 |
assumes "0 < e" and S: "Sup S \<noteq> \<omega>" "S \<noteq> {}" |
|
841 |
shows "\<exists>X\<in>S. Sup S < X + e" |
|
842 |
proof cases |
|
843 |
assume "Sup S = 0" |
|
844 |
moreover obtain X where "X \<in> S" using `S \<noteq> {}` by auto |
|
845 |
ultimately show ?thesis using `0 < e` by (auto intro!: bexI[OF _ `X\<in>S`]) |
|
846 |
next |
|
847 |
assume "Sup S \<noteq> 0" |
|
848 |
have "\<exists>X\<in>S. Sup S - e < X" |
|
849 |
proof (rule Sup_lesser) |
|
850 |
show "Sup S - e < Sup S" using `0 < e` `Sup S \<noteq> 0` `Sup S \<noteq> \<omega>` |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
851 |
by (cases e) (auto simp: pextreal_noteq_omega_Ex) |
38656 | 852 |
qed |
853 |
then guess X .. note X = this |
|
854 |
with `Sup S \<noteq> \<omega>` Sup_eq_\<omega> have "X \<noteq> \<omega>" by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
855 |
thus ?thesis using `Sup S \<noteq> \<omega>` X unfolding pextreal_noteq_omega_Ex |
38656 | 856 |
by (cases e) (auto intro!: bexI[OF _ `X\<in>S`] simp: split: split_if_asm) |
857 |
qed |
|
858 |
||
859 |
lemma Sup_\<omega>: "(SUP i::nat. Real (real i)) = \<omega>" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
860 |
proof (rule pextreal_SUPI) |
38656 | 861 |
fix y assume *: "\<And>i::nat. i \<in> UNIV \<Longrightarrow> Real (real i) \<le> y" |
862 |
thus "\<omega> \<le> y" |
|
863 |
proof (cases y) |
|
864 |
case (preal r) |
|
865 |
then obtain k :: nat where "r < real k" |
|
866 |
using ex_less_of_nat by (auto simp: real_eq_of_nat) |
|
867 |
with *[of k] preal show ?thesis by auto |
|
868 |
qed simp |
|
869 |
qed simp |
|
870 |
||
40871 | 871 |
lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" |
872 |
proof |
|
873 |
assume *: "(SUP i:A. f i) = \<omega>" |
|
874 |
show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric] |
|
875 |
proof (intro allI impI) |
|
876 |
fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i" |
|
877 |
unfolding less_SUP_iff by auto |
|
878 |
qed |
|
879 |
next |
|
880 |
assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i" |
|
881 |
show "(SUP i:A. f i) = \<omega>" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
882 |
proof (rule pextreal_SUPI) |
40871 | 883 |
fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y" |
884 |
show "\<omega> \<le> y" |
|
885 |
proof cases |
|
886 |
assume "y < \<omega>" |
|
887 |
from *[THEN spec, THEN mp, OF this] |
|
888 |
obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto |
|
889 |
with ** show ?thesis by auto |
|
890 |
qed auto |
|
891 |
qed auto |
|
892 |
qed |
|
893 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
894 |
subsubsection {* Equivalence between @{text "f ----> x"} and @{text SUP} on @{typ pextreal} *} |
38656 | 895 |
|
896 |
lemma monoseq_monoI: "mono f \<Longrightarrow> monoseq f" |
|
897 |
unfolding mono_def monoseq_def by auto |
|
898 |
||
899 |
lemma incseq_mono: "mono f \<longleftrightarrow> incseq f" |
|
900 |
unfolding mono_def incseq_def by auto |
|
901 |
||
902 |
lemma SUP_eq_LIMSEQ: |
|
903 |
assumes "mono f" and "\<And>n. 0 \<le> f n" and "0 \<le> x" |
|
904 |
shows "(SUP n. Real (f n)) = Real x \<longleftrightarrow> f ----> x" |
|
905 |
proof |
|
906 |
assume x: "(SUP n. Real (f n)) = Real x" |
|
907 |
{ fix n |
|
908 |
have "Real (f n) \<le> Real x" using x[symmetric] by (auto intro: le_SUPI) |
|
909 |
hence "f n \<le> x" using assms by simp } |
|
910 |
show "f ----> x" |
|
911 |
proof (rule LIMSEQ_I) |
|
912 |
fix r :: real assume "0 < r" |
|
913 |
show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r" |
|
914 |
proof (rule ccontr) |
|
915 |
assume *: "\<not> ?thesis" |
|
916 |
{ fix N |
|
917 |
from * obtain n where "N \<le> n" "r \<le> x - f n" |
|
918 |
using `\<And>n. f n \<le> x` by (auto simp: not_less) |
|
919 |
hence "f N \<le> f n" using `mono f` by (auto dest: monoD) |
|
920 |
hence "f N \<le> x - r" using `r \<le> x - f n` by auto |
|
921 |
hence "Real (f N) \<le> Real (x - r)" and "r \<le> x" using `0 \<le> f N` by auto } |
|
922 |
hence "(SUP n. Real (f n)) \<le> Real (x - r)" |
|
923 |
and "Real (x - r) < Real x" using `0 < r` by (auto intro: SUP_leI) |
|
924 |
hence "(SUP n. Real (f n)) < Real x" by (rule le_less_trans) |
|
925 |
thus False using x by auto |
|
926 |
qed |
|
927 |
qed |
|
928 |
next |
|
929 |
assume "f ----> x" |
|
930 |
show "(SUP n. Real (f n)) = Real x" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
931 |
proof (rule pextreal_SUPI) |
38656 | 932 |
fix n |
933 |
from incseq_le[of f x] `mono f` `f ----> x` |
|
934 |
show "Real (f n) \<le> Real x" using assms incseq_mono by auto |
|
935 |
next |
|
936 |
fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> Real (f n) \<le> y" |
|
937 |
show "Real x \<le> y" |
|
938 |
proof (cases y) |
|
939 |
case (preal r) |
|
940 |
with * have "\<exists>N. \<forall>n\<ge>N. f n \<le> r" using assms by fastsimp |
|
941 |
from LIMSEQ_le_const2[OF `f ----> x` this] |
|
942 |
show "Real x \<le> y" using `0 \<le> x` preal by auto |
|
943 |
qed simp |
|
944 |
qed |
|
945 |
qed |
|
946 |
||
947 |
lemma SUPR_bound: |
|
948 |
assumes "\<forall>N. f N \<le> x" |
|
949 |
shows "(SUP n. f n) \<le> x" |
|
950 |
using assms by (simp add: SUPR_def Sup_le_iff) |
|
951 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
952 |
lemma pextreal_less_eq_diff_eq_sum: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
953 |
fixes x y z :: pextreal |
38656 | 954 |
assumes "y \<le> x" and "x \<noteq> \<omega>" |
955 |
shows "z \<le> x - y \<longleftrightarrow> z + y \<le> x" |
|
956 |
using assms |
|
957 |
apply (cases z, cases y, cases x) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
958 |
by (simp_all add: field_simps minus_pextreal_eq) |
38656 | 959 |
|
960 |
lemma Real_diff_less_omega: "Real r - x < \<omega>" by (cases x) auto |
|
961 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
962 |
subsubsection {* Numbers on @{typ pextreal} *} |
38656 | 963 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
964 |
instantiation pextreal :: number |
38656 | 965 |
begin |
966 |
definition [simp]: "number_of x = Real (number_of x)" |
|
967 |
instance proof qed |
|
968 |
end |
|
969 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
970 |
subsubsection {* Division on @{typ pextreal} *} |
38656 | 971 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
972 |
instantiation pextreal :: inverse |
38656 | 973 |
begin |
974 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
975 |
definition "inverse x = pextreal_case (\<lambda>x. if x = 0 then \<omega> else Real (inverse x)) 0 x" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
976 |
definition [simp]: "x / y = x * inverse (y :: pextreal)" |
38656 | 977 |
|
978 |
instance proof qed |
|
979 |
end |
|
980 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
981 |
lemma pextreal_inverse[simp]: |
38656 | 982 |
"inverse 0 = \<omega>" |
983 |
"inverse (Real x) = (if x \<le> 0 then \<omega> else Real (inverse x))" |
|
984 |
"inverse \<omega> = 0" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
985 |
"inverse (1::pextreal) = 1" |
38656 | 986 |
"inverse (inverse x) = x" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
987 |
by (simp_all add: inverse_pextreal_def one_pextreal_def split: pextreal_case_split del: Real_1) |
38656 | 988 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
989 |
lemma pextreal_inverse_le_eq: |
38656 | 990 |
assumes "x \<noteq> 0" "x \<noteq> \<omega>" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
991 |
shows "y \<le> z / x \<longleftrightarrow> x * y \<le> (z :: pextreal)" |
38656 | 992 |
proof - |
993 |
from assms obtain r where r: "x = Real r" "0 < r" by (cases x) auto |
|
994 |
{ fix p q :: real assume "0 \<le> p" "0 \<le> q" |
|
995 |
have "p \<le> q * inverse r \<longleftrightarrow> p \<le> q / r" by (simp add: divide_inverse) |
|
996 |
also have "... \<longleftrightarrow> p * r \<le> q" using `0 < r` by (auto simp: field_simps) |
|
997 |
finally have "p \<le> q * inverse r \<longleftrightarrow> p * r \<le> q" . } |
|
998 |
with r show ?thesis |
|
999 |
by (cases y, cases z, auto simp: zero_le_mult_iff field_simps) |
|
1000 |
qed |
|
1001 |
||
1002 |
lemma inverse_antimono_strict: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1003 |
fixes x y :: pextreal |
38656 | 1004 |
assumes "x < y" shows "inverse y < inverse x" |
1005 |
using assms by (cases x, cases y) auto |
|
1006 |
||
1007 |
lemma inverse_antimono: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1008 |
fixes x y :: pextreal |
38656 | 1009 |
assumes "x \<le> y" shows "inverse y \<le> inverse x" |
1010 |
using assms by (cases x, cases y) auto |
|
1011 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1012 |
lemma pextreal_inverse_\<omega>_iff[simp]: "inverse x = \<omega> \<longleftrightarrow> x = 0" |
38656 | 1013 |
by (cases x) auto |
1014 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1015 |
subsection "Infinite sum over @{typ pextreal}" |
38656 | 1016 |
|
1017 |
text {* |
|
1018 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1019 |
The infinite sum over @{typ pextreal} has the nice property that it is always |
38656 | 1020 |
defined. |
1021 |
||
1022 |
*} |
|
1023 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1024 |
definition psuminf :: "(nat \<Rightarrow> pextreal) \<Rightarrow> pextreal" (binder "\<Sum>\<^isub>\<infinity>" 10) where |
38656 | 1025 |
"(\<Sum>\<^isub>\<infinity> x. f x) = (SUP n. \<Sum>i<n. f i)" |
1026 |
||
1027 |
subsubsection {* Equivalence between @{text "\<Sum> n. f n"} and @{text "\<Sum>\<^isub>\<infinity> n. f n"} *} |
|
1028 |
||
1029 |
lemma setsum_Real: |
|
1030 |
assumes "\<forall>x\<in>A. 0 \<le> f x" |
|
1031 |
shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)" |
|
1032 |
proof (cases "finite A") |
|
1033 |
case True |
|
1034 |
thus ?thesis using assms |
|
1035 |
proof induct case (insert x s) |
|
1036 |
hence "0 \<le> setsum f s" apply-apply(rule setsum_nonneg) by auto |
|
1037 |
thus ?case using insert by auto |
|
1038 |
qed auto |
|
1039 |
qed simp |
|
1040 |
||
1041 |
lemma setsum_Real': |
|
1042 |
assumes "\<forall>x. 0 \<le> f x" |
|
1043 |
shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)" |
|
1044 |
apply(rule setsum_Real) using assms by auto |
|
1045 |
||
1046 |
lemma setsum_\<omega>: |
|
1047 |
"(\<Sum>x\<in>P. f x) = \<omega> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<omega>))" |
|
1048 |
proof safe |
|
1049 |
assume *: "setsum f P = \<omega>" |
|
1050 |
show "finite P" |
|
1051 |
proof (rule ccontr) assume "infinite P" with * show False by auto qed |
|
1052 |
show "\<exists>i\<in>P. f i = \<omega>" |
|
1053 |
proof (rule ccontr) |
|
1054 |
assume "\<not> ?thesis" hence "\<And>i. i\<in>P \<Longrightarrow> f i \<noteq> \<omega>" by auto |
|
1055 |
from `finite P` this have "setsum f P \<noteq> \<omega>" |
|
1056 |
by induct auto |
|
1057 |
with * show False by auto |
|
1058 |
qed |
|
1059 |
next |
|
1060 |
fix i assume "finite P" "i \<in> P" "f i = \<omega>" |
|
1061 |
thus "setsum f P = \<omega>" |
|
1062 |
proof induct |
|
1063 |
case (insert x A) |
|
1064 |
show ?case using insert by (cases "x = i") auto |
|
1065 |
qed simp |
|
1066 |
qed |
|
1067 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1068 |
lemma real_of_pextreal_setsum: |
38656 | 1069 |
assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> \<omega>" |
1070 |
shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)" |
|
1071 |
proof cases |
|
1072 |
assume "finite S" |
|
1073 |
from this assms show ?thesis |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1074 |
by induct (simp_all add: real_of_pextreal_add setsum_\<omega>) |
38656 | 1075 |
qed simp |
1076 |
||
1077 |
lemma setsum_0: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1078 |
fixes f :: "'a \<Rightarrow> pextreal" assumes "finite A" |
38656 | 1079 |
shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)" |
1080 |
using assms by induct auto |
|
1081 |
||
1082 |
lemma suminf_imp_psuminf: |
|
1083 |
assumes "f sums x" and "\<forall>n. 0 \<le> f n" |
|
1084 |
shows "(\<Sum>\<^isub>\<infinity> x. Real (f x)) = Real x" |
|
1085 |
unfolding psuminf_def setsum_Real'[OF assms(2)] |
|
1086 |
proof (rule SUP_eq_LIMSEQ[THEN iffD2]) |
|
1087 |
show "mono (\<lambda>n. setsum f {..<n})" (is "mono ?S") |
|
1088 |
unfolding mono_iff_le_Suc using assms by simp |
|
1089 |
||
1090 |
{ fix n show "0 \<le> ?S n" |
|
1091 |
using setsum_nonneg[of "{..<n}" f] assms by auto } |
|
1092 |
||
1093 |
thus "0 \<le> x" "?S ----> x" |
|
1094 |
using `f sums x` LIMSEQ_le_const |
|
1095 |
by (auto simp: atLeast0LessThan sums_def) |
|
1096 |
qed |
|
1097 |
||
1098 |
lemma psuminf_equality: |
|
1099 |
assumes "\<And>n. setsum f {..<n} \<le> x" |
|
1100 |
and "\<And>y. y \<noteq> \<omega> \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> y) \<Longrightarrow> x \<le> y" |
|
1101 |
shows "psuminf f = x" |
|
1102 |
unfolding psuminf_def |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1103 |
proof (safe intro!: pextreal_SUPI) |
38656 | 1104 |
fix n show "setsum f {..<n} \<le> x" using assms(1) . |
1105 |
next |
|
1106 |
fix y assume *: "\<forall>n. n \<in> UNIV \<longrightarrow> setsum f {..<n} \<le> y" |
|
1107 |
show "x \<le> y" |
|
1108 |
proof (cases "y = \<omega>") |
|
1109 |
assume "y \<noteq> \<omega>" from assms(2)[OF this] * |
|
1110 |
show "x \<le> y" by auto |
|
1111 |
qed simp |
|
1112 |
qed |
|
1113 |
||
1114 |
lemma psuminf_\<omega>: |
|
1115 |
assumes "f i = \<omega>" |
|
1116 |
shows "(\<Sum>\<^isub>\<infinity> x. f x) = \<omega>" |
|
1117 |
proof (rule psuminf_equality) |
|
1118 |
fix y assume *: "\<And>n. setsum f {..<n} \<le> y" |
|
1119 |
have "setsum f {..<Suc i} = \<omega>" |
|
1120 |
using assms by (simp add: setsum_\<omega>) |
|
1121 |
thus "\<omega> \<le> y" using *[of "Suc i"] by auto |
|
1122 |
qed simp |
|
1123 |
||
1124 |
lemma psuminf_imp_suminf: |
|
1125 |
assumes "(\<Sum>\<^isub>\<infinity> x. f x) \<noteq> \<omega>" |
|
1126 |
shows "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity> x. f x)" |
|
1127 |
proof - |
|
1128 |
have "\<forall>i. \<exists>r. f i = Real r \<and> 0 \<le> r" |
|
1129 |
proof |
|
1130 |
fix i show "\<exists>r. f i = Real r \<and> 0 \<le> r" using psuminf_\<omega> assms by (cases "f i") auto |
|
1131 |
qed |
|
1132 |
from choice[OF this] obtain r where f: "f = (\<lambda>i. Real (r i))" |
|
1133 |
and pos: "\<forall>i. 0 \<le> r i" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1134 |
by (auto simp: fun_eq_iff) |
38656 | 1135 |
hence [simp]: "\<And>i. real (f i) = r i" by auto |
1136 |
||
1137 |
have "mono (\<lambda>n. setsum r {..<n})" (is "mono ?S") |
|
1138 |
unfolding mono_iff_le_Suc using pos by simp |
|
1139 |
||
1140 |
{ fix n have "0 \<le> ?S n" |
|
1141 |
using setsum_nonneg[of "{..<n}" r] pos by auto } |
|
1142 |
||
1143 |
from assms obtain p where *: "(\<Sum>\<^isub>\<infinity> x. f x) = Real p" and "0 \<le> p" |
|
1144 |
by (cases "(\<Sum>\<^isub>\<infinity> x. f x)") auto |
|
1145 |
show ?thesis unfolding * using * pos `0 \<le> p` SUP_eq_LIMSEQ[OF `mono ?S` `\<And>n. 0 \<le> ?S n` `0 \<le> p`] |
|
1146 |
by (simp add: f atLeast0LessThan sums_def psuminf_def setsum_Real'[OF pos] f) |
|
1147 |
qed |
|
1148 |
||
1149 |
lemma psuminf_bound: |
|
1150 |
assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x" |
|
1151 |
shows "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x" |
|
1152 |
using assms by (simp add: psuminf_def SUPR_def Sup_le_iff) |
|
1153 |
||
1154 |
lemma psuminf_bound_add: |
|
1155 |
assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x" |
|
1156 |
shows "(\<Sum>\<^isub>\<infinity> n. f n) + y \<le> x" |
|
1157 |
proof (cases "x = \<omega>") |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1158 |
have "y \<le> x" using assms by (auto intro: pextreal_le_add2) |
38656 | 1159 |
assume "x \<noteq> \<omega>" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1160 |
note move_y = pextreal_less_eq_diff_eq_sum[OF `y \<le> x` this] |
38656 | 1161 |
|
1162 |
have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y" using assms by (simp add: move_y) |
|
1163 |
hence "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x - y" by (rule psuminf_bound) |
|
1164 |
thus ?thesis by (simp add: move_y) |
|
1165 |
qed simp |
|
1166 |
||
1167 |
lemma psuminf_finite: |
|
1168 |
assumes "\<forall>N\<ge>n. f N = 0" |
|
1169 |
shows "(\<Sum>\<^isub>\<infinity> n. f n) = (\<Sum>N<n. f N)" |
|
1170 |
proof (rule psuminf_equality) |
|
1171 |
fix N |
|
1172 |
show "setsum f {..<N} \<le> setsum f {..<n}" |
|
1173 |
proof (cases rule: linorder_cases) |
|
1174 |
assume "N < n" thus ?thesis by (auto intro!: setsum_mono3) |
|
1175 |
next |
|
1176 |
assume "n < N" |
|
1177 |
hence *: "{..<N} = {..<n} \<union> {n..<N}" by auto |
|
1178 |
moreover have "setsum f {n..<N} = 0" |
|
1179 |
using assms by (auto intro!: setsum_0') |
|
1180 |
ultimately show ?thesis unfolding * |
|
1181 |
by (subst setsum_Un_disjoint) auto |
|
1182 |
qed simp |
|
1183 |
qed simp |
|
1184 |
||
1185 |
lemma psuminf_upper: |
|
1186 |
shows "(\<Sum>n<N. f n) \<le> (\<Sum>\<^isub>\<infinity> n. f n)" |
|
1187 |
unfolding psuminf_def SUPR_def |
|
1188 |
by (auto intro: complete_lattice_class.Sup_upper image_eqI) |
|
1189 |
||
1190 |
lemma psuminf_le: |
|
1191 |
assumes "\<And>N. f N \<le> g N" |
|
1192 |
shows "psuminf f \<le> psuminf g" |
|
1193 |
proof (safe intro!: psuminf_bound) |
|
1194 |
fix n |
|
1195 |
have "setsum f {..<n} \<le> setsum g {..<n}" using assms by (auto intro: setsum_mono) |
|
1196 |
also have "... \<le> psuminf g" by (rule psuminf_upper) |
|
1197 |
finally show "setsum f {..<n} \<le> psuminf g" . |
|
1198 |
qed |
|
1199 |
||
1200 |
lemma psuminf_const[simp]: "psuminf (\<lambda>n. c) = (if c = 0 then 0 else \<omega>)" (is "_ = ?if") |
|
1201 |
proof (rule psuminf_equality) |
|
1202 |
fix y assume *: "\<And>n :: nat. (\<Sum>n<n. c) \<le> y" and "y \<noteq> \<omega>" |
|
1203 |
then obtain r p where |
|
1204 |
y: "y = Real r" "0 \<le> r" and |
|
1205 |
c: "c = Real p" "0 \<le> p" |
|
1206 |
using *[of 1] by (cases c, cases y) auto |
|
1207 |
show "(if c = 0 then 0 else \<omega>) \<le> y" |
|
1208 |
proof (cases "p = 0") |
|
1209 |
assume "p = 0" with c show ?thesis by simp |
|
1210 |
next |
|
1211 |
assume "p \<noteq> 0" |
|
1212 |
with * c y have **: "\<And>n :: nat. real n \<le> r / p" |
|
1213 |
by (auto simp: zero_le_mult_iff field_simps) |
|
1214 |
from ex_less_of_nat[of "r / p"] guess n .. |
|
1215 |
with **[of n] show ?thesis by (simp add: real_eq_of_nat) |
|
1216 |
qed |
|
1217 |
qed (cases "c = 0", simp_all) |
|
1218 |
||
1219 |
lemma psuminf_add[simp]: "psuminf (\<lambda>n. f n + g n) = psuminf f + psuminf g" |
|
1220 |
proof (rule psuminf_equality) |
|
1221 |
fix n |
|
1222 |
from psuminf_upper[of f n] psuminf_upper[of g n] |
|
1223 |
show "(\<Sum>n<n. f n + g n) \<le> psuminf f + psuminf g" |
|
1224 |
by (auto simp add: setsum_addf intro!: add_mono) |
|
1225 |
next |
|
1226 |
fix y assume *: "\<And>n. (\<Sum>n<n. f n + g n) \<le> y" and "y \<noteq> \<omega>" |
|
1227 |
{ fix n m |
|
1228 |
have **: "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y" |
|
1229 |
proof (cases rule: linorder_le_cases) |
|
1230 |
assume "n \<le> m" |
|
1231 |
hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<m. f n) + (\<Sum>n<m. g n)" |
|
1232 |
by (auto intro!: add_right_mono setsum_mono3) |
|
1233 |
also have "... \<le> y" |
|
1234 |
using * by (simp add: setsum_addf) |
|
1235 |
finally show ?thesis . |
|
1236 |
next |
|
1237 |
assume "m \<le> n" |
|
1238 |
hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<n. f n) + (\<Sum>n<n. g n)" |
|
1239 |
by (auto intro!: add_left_mono setsum_mono3) |
|
1240 |
also have "... \<le> y" |
|
1241 |
using * by (simp add: setsum_addf) |
|
1242 |
finally show ?thesis . |
|
1243 |
qed } |
|
1244 |
hence "\<And>m. \<forall>n. (\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y" by simp |
|
1245 |
from psuminf_bound_add[OF this] |
|
1246 |
have "\<forall>m. (\<Sum>n<m. g n) + psuminf f \<le> y" by (simp add: ac_simps) |
|
1247 |
from psuminf_bound_add[OF this] |
|
1248 |
show "psuminf f + psuminf g \<le> y" by (simp add: ac_simps) |
|
1249 |
qed |
|
1250 |
||
1251 |
lemma psuminf_0: "psuminf f = 0 \<longleftrightarrow> (\<forall>i. f i = 0)" |
|
1252 |
proof safe |
|
1253 |
assume "\<forall>i. f i = 0" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1254 |
hence "f = (\<lambda>i. 0)" by (simp add: fun_eq_iff) |
38656 | 1255 |
thus "psuminf f = 0" using psuminf_const by simp |
1256 |
next |
|
1257 |
fix i assume "psuminf f = 0" |
|
1258 |
hence "(\<Sum>n<Suc i. f n) = 0" using psuminf_upper[of f "Suc i"] by simp |
|
1259 |
thus "f i = 0" by simp |
|
1260 |
qed |
|
1261 |
||
1262 |
lemma psuminf_cmult_right[simp]: "psuminf (\<lambda>n. c * f n) = c * psuminf f" |
|
1263 |
proof (rule psuminf_equality) |
|
1264 |
fix n show "(\<Sum>n<n. c * f n) \<le> c * psuminf f" |
|
1265 |
by (auto simp: setsum_right_distrib[symmetric] intro: mult_left_mono psuminf_upper) |
|
1266 |
next |
|
1267 |
fix y |
|
1268 |
assume "\<And>n. (\<Sum>n<n. c * f n) \<le> y" |
|
1269 |
hence *: "\<And>n. c * (\<Sum>n<n. f n) \<le> y" by (auto simp add: setsum_right_distrib) |
|
1270 |
thus "c * psuminf f \<le> y" |
|
1271 |
proof (cases "c = \<omega> \<or> c = 0") |
|
1272 |
assume "c = \<omega> \<or> c = 0" |
|
1273 |
thus ?thesis |
|
1274 |
using * by (fastsimp simp add: psuminf_0 setsum_0 split: split_if_asm) |
|
1275 |
next |
|
1276 |
assume "\<not> (c = \<omega> \<or> c = 0)" |
|
1277 |
hence "c \<noteq> 0" "c \<noteq> \<omega>" by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1278 |
note rewrite_div = pextreal_inverse_le_eq[OF this, of _ y] |
38656 | 1279 |
hence "\<forall>n. (\<Sum>n<n. f n) \<le> y / c" using * by simp |
1280 |
hence "psuminf f \<le> y / c" by (rule psuminf_bound) |
|
1281 |
thus ?thesis using rewrite_div by simp |
|
1282 |
qed |
|
1283 |
qed |
|
1284 |
||
1285 |
lemma psuminf_cmult_left[simp]: "psuminf (\<lambda>n. f n * c) = psuminf f * c" |
|
1286 |
using psuminf_cmult_right[of c f] by (simp add: ac_simps) |
|
1287 |
||
1288 |
lemma psuminf_half_series: "psuminf (\<lambda>n. (1/2)^Suc n) = 1" |
|
1289 |
using suminf_imp_psuminf[OF power_half_series] by auto |
|
1290 |
||
1291 |
lemma setsum_pinfsum: "(\<Sum>\<^isub>\<infinity> n. \<Sum>m\<in>A. f n m) = (\<Sum>m\<in>A. (\<Sum>\<^isub>\<infinity> n. f n m))" |
|
1292 |
proof (cases "finite A") |
|
1293 |
assume "finite A" |
|
1294 |
thus ?thesis by induct simp_all |
|
1295 |
qed simp |
|
1296 |
||
1297 |
lemma psuminf_reindex: |
|
1298 |
fixes f:: "nat \<Rightarrow> nat" assumes "bij f" |
|
1299 |
shows "psuminf (g \<circ> f) = psuminf g" |
|
1300 |
proof - |
|
1301 |
have [intro, simp]: "\<And>A. inj_on f A" using `bij f` unfolding bij_def by (auto intro: subset_inj_on) |
|
1302 |
have f[intro, simp]: "\<And>x. f (inv f x) = x" |
|
1303 |
using `bij f` unfolding bij_def by (auto intro: surj_f_inv_f) |
|
1304 |
show ?thesis |
|
1305 |
proof (rule psuminf_equality) |
|
1306 |
fix n |
|
1307 |
have "setsum (g \<circ> f) {..<n} = setsum g (f ` {..<n})" |
|
1308 |
by (simp add: setsum_reindex) |
|
1309 |
also have "\<dots> \<le> setsum g {..Max (f ` {..<n})}" |
|
1310 |
by (rule setsum_mono3) auto |
|
1311 |
also have "\<dots> \<le> psuminf g" unfolding lessThan_Suc_atMost[symmetric] by (rule psuminf_upper) |
|
1312 |
finally show "setsum (g \<circ> f) {..<n} \<le> psuminf g" . |
|
1313 |
next |
|
1314 |
fix y assume *: "\<And>n. setsum (g \<circ> f) {..<n} \<le> y" |
|
1315 |
show "psuminf g \<le> y" |
|
1316 |
proof (safe intro!: psuminf_bound) |
|
1317 |
fix N |
|
1318 |
have "setsum g {..<N} \<le> setsum g (f ` {..Max (inv f ` {..<N})})" |
|
1319 |
by (rule setsum_mono3) (auto intro!: image_eqI[where f="f", OF f[symmetric]]) |
|
1320 |
also have "\<dots> = setsum (g \<circ> f) {..Max (inv f ` {..<N})}" |
|
1321 |
by (simp add: setsum_reindex) |
|
1322 |
also have "\<dots> \<le> y" unfolding lessThan_Suc_atMost[symmetric] by (rule *) |
|
1323 |
finally show "setsum g {..<N} \<le> y" . |
|
1324 |
qed |
|
1325 |
qed |
|
1326 |
qed |
|
1327 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1328 |
lemma pextreal_mult_less_right: |
38656 | 1329 |
assumes "b * a < c * a" "0 < a" "a < \<omega>" |
1330 |
shows "b < c" |
|
1331 |
using assms |
|
1332 |
by (cases a, cases b, cases c) (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff) |
|
1333 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1334 |
lemma pextreal_\<omega>_eq_plus[simp]: "\<omega> = a + b \<longleftrightarrow> (a = \<omega> \<or> b = \<omega>)" |
38656 | 1335 |
by (cases a, cases b) auto |
1336 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1337 |
lemma pextreal_of_nat_le_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1338 |
"(of_nat k :: pextreal) \<le> of_nat m \<longleftrightarrow> k \<le> m" by auto |
38656 | 1339 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1340 |
lemma pextreal_of_nat_less_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1341 |
"(of_nat k :: pextreal) < of_nat m \<longleftrightarrow> k < m" by auto |
38656 | 1342 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1343 |
lemma pextreal_bound_add: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1344 |
assumes "\<forall>N. f N + y \<le> (x::pextreal)" |
38656 | 1345 |
shows "(SUP n. f n) + y \<le> x" |
1346 |
proof (cases "x = \<omega>") |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1347 |
have "y \<le> x" using assms by (auto intro: pextreal_le_add2) |
38656 | 1348 |
assume "x \<noteq> \<omega>" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1349 |
note move_y = pextreal_less_eq_diff_eq_sum[OF `y \<le> x` this] |
38656 | 1350 |
|
1351 |
have "\<forall>N. f N \<le> x - y" using assms by (simp add: move_y) |
|
1352 |
hence "(SUP n. f n) \<le> x - y" by (rule SUPR_bound) |
|
1353 |
thus ?thesis by (simp add: move_y) |
|
1354 |
qed simp |
|
1355 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1356 |
lemma SUPR_pextreal_add: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1357 |
fixes f g :: "nat \<Rightarrow> pextreal" |
38656 | 1358 |
assumes f: "\<forall>n. f n \<le> f (Suc n)" and g: "\<forall>n. g n \<le> g (Suc n)" |
1359 |
shows "(SUP n. f n + g n) = (SUP n. f n) + (SUP n. g n)" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1360 |
proof (rule pextreal_SUPI) |
38656 | 1361 |
fix n :: nat from le_SUPI[of n UNIV f] le_SUPI[of n UNIV g] |
1362 |
show "f n + g n \<le> (SUP n. f n) + (SUP n. g n)" |
|
1363 |
by (auto intro!: add_mono) |
|
1364 |
next |
|
1365 |
fix y assume *: "\<And>n. n \<in> UNIV \<Longrightarrow> f n + g n \<le> y" |
|
1366 |
{ fix n m |
|
1367 |
have "f n + g m \<le> y" |
|
1368 |
proof (cases rule: linorder_le_cases) |
|
1369 |
assume "n \<le> m" |
|
1370 |
hence "f n + g m \<le> f m + g m" |
|
1371 |
using f lift_Suc_mono_le by (auto intro!: add_right_mono) |
|
1372 |
also have "\<dots> \<le> y" using * by simp |
|
1373 |
finally show ?thesis . |
|
1374 |
next |
|
1375 |
assume "m \<le> n" |
|
1376 |
hence "f n + g m \<le> f n + g n" |
|
1377 |
using g lift_Suc_mono_le by (auto intro!: add_left_mono) |
|
1378 |
also have "\<dots> \<le> y" using * by simp |
|
1379 |
finally show ?thesis . |
|
1380 |
qed } |
|
1381 |
hence "\<And>m. \<forall>n. f n + g m \<le> y" by simp |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1382 |
from pextreal_bound_add[OF this] |
38656 | 1383 |
have "\<forall>m. (g m) + (SUP n. f n) \<le> y" by (simp add: ac_simps) |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1384 |
from pextreal_bound_add[OF this] |
38656 | 1385 |
show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps) |
1386 |
qed |
|
1387 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1388 |
lemma SUPR_pextreal_setsum: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1389 |
fixes f :: "'x \<Rightarrow> nat \<Rightarrow> pextreal" |
38656 | 1390 |
assumes "\<And>i. i \<in> P \<Longrightarrow> \<forall>n. f i n \<le> f i (Suc n)" |
1391 |
shows "(SUP n. \<Sum>i\<in>P. f i n) = (\<Sum>i\<in>P. SUP n. f i n)" |
|
1392 |
proof cases |
|
1393 |
assume "finite P" from this assms show ?thesis |
|
1394 |
proof induct |
|
1395 |
case (insert i P) |
|
1396 |
thus ?case |
|
1397 |
apply simp |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1398 |
apply (subst SUPR_pextreal_add) |
38656 | 1399 |
by (auto intro!: setsum_mono) |
1400 |
qed simp |
|
1401 |
qed simp |
|
1402 |
||
40871 | 1403 |
lemma psuminf_SUP_eq: |
1404 |
assumes "\<And>n i. f n i \<le> f (Suc n) i" |
|
1405 |
shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)" |
|
1406 |
proof - |
|
1407 |
{ fix n :: nat |
|
1408 |
have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1409 |
using assms by (auto intro!: SUPR_pextreal_setsum[symmetric]) } |
40871 | 1410 |
note * = this |
1411 |
show ?thesis |
|
1412 |
unfolding psuminf_def |
|
1413 |
unfolding * |
|
40872 | 1414 |
apply (subst SUP_commute) .. |
40871 | 1415 |
qed |
1416 |
||
1417 |
lemma psuminf_commute: |
|
1418 |
shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)" |
|
1419 |
proof - |
|
1420 |
have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1421 |
apply (subst SUPR_pextreal_setsum) |
40871 | 1422 |
by auto |
1423 |
also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)" |
|
40872 | 1424 |
apply (subst SUP_commute) |
40871 | 1425 |
apply (subst setsum_commute) |
1426 |
by auto |
|
1427 |
also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1428 |
apply (subst SUPR_pextreal_setsum) |
40871 | 1429 |
by auto |
1430 |
finally show ?thesis |
|
1431 |
unfolding psuminf_def by auto |
|
1432 |
qed |
|
1433 |
||
40872 | 1434 |
lemma psuminf_2dimen: |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1435 |
fixes f:: "nat * nat \<Rightarrow> pextreal" |
40872 | 1436 |
assumes fsums: "\<And>m. g m = (\<Sum>\<^isub>\<infinity> n. f (m,n))" |
1437 |
shows "psuminf (f \<circ> prod_decode) = psuminf g" |
|
1438 |
proof (rule psuminf_equality) |
|
1439 |
fix n :: nat |
|
1440 |
let ?P = "prod_decode ` {..<n}" |
|
1441 |
have "setsum (f \<circ> prod_decode) {..<n} = setsum f ?P" |
|
1442 |
by (auto simp: setsum_reindex inj_prod_decode) |
|
1443 |
also have "\<dots> \<le> setsum f ({..Max (fst ` ?P)} \<times> {..Max (snd ` ?P)})" |
|
1444 |
proof (safe intro!: setsum_mono3 Max_ge image_eqI) |
|
1445 |
fix a b x assume "(a, b) = prod_decode x" |
|
1446 |
from this[symmetric] show "a = fst (prod_decode x)" "b = snd (prod_decode x)" |
|
1447 |
by simp_all |
|
1448 |
qed simp_all |
|
1449 |
also have "\<dots> = (\<Sum>m\<le>Max (fst ` ?P). (\<Sum>n\<le>Max (snd ` ?P). f (m,n)))" |
|
1450 |
unfolding setsum_cartesian_product by simp |
|
1451 |
also have "\<dots> \<le> (\<Sum>m\<le>Max (fst ` ?P). g m)" |
|
1452 |
by (auto intro!: setsum_mono psuminf_upper simp del: setsum_lessThan_Suc |
|
1453 |
simp: fsums lessThan_Suc_atMost[symmetric]) |
|
1454 |
also have "\<dots> \<le> psuminf g" |
|
1455 |
by (auto intro!: psuminf_upper simp del: setsum_lessThan_Suc |
|
1456 |
simp: lessThan_Suc_atMost[symmetric]) |
|
1457 |
finally show "setsum (f \<circ> prod_decode) {..<n} \<le> psuminf g" . |
|
1458 |
next |
|
1459 |
fix y assume *: "\<And>n. setsum (f \<circ> prod_decode) {..<n} \<le> y" |
|
1460 |
have g: "g = (\<lambda>m. \<Sum>\<^isub>\<infinity> n. f (m,n))" unfolding fsums[symmetric] .. |
|
1461 |
show "psuminf g \<le> y" unfolding g |
|
1462 |
proof (rule psuminf_bound, unfold setsum_pinfsum[symmetric], safe intro!: psuminf_bound) |
|
1463 |
fix N M :: nat |
|
1464 |
let ?P = "{..<N} \<times> {..<M}" |
|
1465 |
let ?M = "Max (prod_encode ` ?P)" |
|
1466 |
have "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> (\<Sum>(m, n)\<in>?P. f (m, n))" |
|
1467 |
unfolding setsum_commute[of _ _ "{..<M}"] unfolding setsum_cartesian_product .. |
|
1468 |
also have "\<dots> \<le> (\<Sum>(m,n)\<in>(prod_decode ` {..?M}). f (m, n))" |
|
1469 |
by (auto intro!: setsum_mono3 image_eqI[where f=prod_decode, OF prod_encode_inverse[symmetric]]) |
|
1470 |
also have "\<dots> \<le> y" using *[of "Suc ?M"] |
|
1471 |
by (simp add: lessThan_Suc_atMost[symmetric] setsum_reindex |
|
1472 |
inj_prod_decode del: setsum_lessThan_Suc) |
|
1473 |
finally show "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> y" . |
|
1474 |
qed |
|
1475 |
qed |
|
1476 |
||
38656 | 1477 |
lemma Real_max: |
1478 |
assumes "x \<ge> 0" "y \<ge> 0" |
|
1479 |
shows "Real (max x y) = max (Real x) (Real y)" |
|
1480 |
using assms unfolding max_def by (auto simp add:not_le) |
|
1481 |
||
1482 |
lemma Real_real: "Real (real x) = (if x = \<omega> then 0 else x)" |
|
1483 |
using assms by (cases x) auto |
|
1484 |
||
1485 |
lemma inj_on_real: "inj_on real (UNIV - {\<omega>})" |
|
1486 |
proof (rule inj_onI) |
|
1487 |
fix x y assume mem: "x \<in> UNIV - {\<omega>}" "y \<in> UNIV - {\<omega>}" and "real x = real y" |
|
1488 |
thus "x = y" by (cases x, cases y) auto |
|
1489 |
qed |
|
1490 |
||
1491 |
lemma inj_on_Real: "inj_on Real {0..}" |
|
1492 |
by (auto intro!: inj_onI) |
|
1493 |
||
1494 |
lemma range_Real[simp]: "range Real = UNIV - {\<omega>}" |
|
1495 |
proof safe |
|
1496 |
fix x assume "x \<notin> range Real" |
|
1497 |
thus "x = \<omega>" by (cases x) auto |
|
1498 |
qed auto |
|
1499 |
||
1500 |
lemma image_Real[simp]: "Real ` {0..} = UNIV - {\<omega>}" |
|
1501 |
proof safe |
|
1502 |
fix x assume "x \<notin> Real ` {0..}" |
|
1503 |
thus "x = \<omega>" by (cases x) auto |
|
1504 |
qed auto |
|
1505 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1506 |
lemma pextreal_SUP_cmult: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1507 |
fixes f :: "'a \<Rightarrow> pextreal" |
38656 | 1508 |
shows "(SUP i : R. z * f i) = z * (SUP i : R. f i)" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1509 |
proof (rule pextreal_SUPI) |
38656 | 1510 |
fix i assume "i \<in> R" |
1511 |
from le_SUPI[OF this] |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1512 |
show "z * f i \<le> z * (SUP i:R. f i)" by (rule pextreal_mult_cancel) |
38656 | 1513 |
next |
1514 |
fix y assume "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y" |
|
1515 |
hence *: "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y" by auto |
|
1516 |
show "z * (SUP i:R. f i) \<le> y" |
|
1517 |
proof (cases "\<forall>i\<in>R. f i = 0") |
|
1518 |
case True |
|
1519 |
show ?thesis |
|
1520 |
proof cases |
|
1521 |
assume "R \<noteq> {}" hence "f ` R = {0}" using True by auto |
|
1522 |
thus ?thesis by (simp add: SUPR_def) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1523 |
qed (simp add: SUPR_def Sup_empty bot_pextreal_def) |
38656 | 1524 |
next |
1525 |
case False then obtain i where i: "i \<in> R" and f0: "f i \<noteq> 0" by auto |
|
1526 |
show ?thesis |
|
1527 |
proof (cases "z = 0 \<or> z = \<omega>") |
|
1528 |
case True with f0 *[OF i] show ?thesis by auto |
|
1529 |
next |
|
1530 |
case False hence z: "z \<noteq> 0" "z \<noteq> \<omega>" by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1531 |
note div = pextreal_inverse_le_eq[OF this, symmetric] |
38656 | 1532 |
hence "\<And>i. i\<in>R \<Longrightarrow> f i \<le> y / z" using * by auto |
1533 |
thus ?thesis unfolding div SUP_le_iff by simp |
|
1534 |
qed |
|
1535 |
qed |
|
1536 |
qed |
|
1537 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1538 |
instantiation pextreal :: topological_space |
38656 | 1539 |
begin |
1540 |
||
1541 |
definition "open A \<longleftrightarrow> |
|
1542 |
(\<exists>T. open T \<and> (Real ` (T\<inter>{0..}) = A - {\<omega>})) \<and> (\<omega> \<in> A \<longrightarrow> (\<exists>x\<ge>0. {Real x <..} \<subseteq> A))" |
|
1543 |
||
1544 |
lemma open_omega: "open A \<Longrightarrow> \<omega> \<in> A \<Longrightarrow> (\<exists>x\<ge>0. {Real x<..} \<subseteq> A)" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1545 |
unfolding open_pextreal_def by auto |
38656 | 1546 |
|
1547 |
lemma open_omegaD: assumes "open A" "\<omega> \<in> A" obtains x where "x\<ge>0" "{Real x<..} \<subseteq> A" |
|
1548 |
using open_omega[OF assms] by auto |
|
1549 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1550 |
lemma pextreal_openE: assumes "open A" obtains A' x where |
38656 | 1551 |
"open A'" "Real ` (A' \<inter> {0..}) = A - {\<omega>}" |
1552 |
"x \<ge> 0" "\<omega> \<in> A \<Longrightarrow> {Real x<..} \<subseteq> A" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1553 |
using assms open_pextreal_def by auto |
38656 | 1554 |
|
1555 |
instance |
|
1556 |
proof |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1557 |
let ?U = "UNIV::pextreal set" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1558 |
show "open ?U" unfolding open_pextreal_def |
38656 | 1559 |
by (auto intro!: exI[of _ "UNIV"] exI[of _ 0]) |
1560 |
next |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1561 |
fix S T::"pextreal set" assume "open S" and "open T" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1562 |
from `open S`[THEN pextreal_openE] guess S' xS . note S' = this |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1563 |
from `open T`[THEN pextreal_openE] guess T' xT . note T' = this |
38656 | 1564 |
|
1565 |
from S'(1-3) T'(1-3) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1566 |
show "open (S \<inter> T)" unfolding open_pextreal_def |
38656 | 1567 |
proof (safe intro!: exI[of _ "S' \<inter> T'"] exI[of _ "max xS xT"]) |
1568 |
fix x assume *: "Real (max xS xT) < x" and "\<omega> \<in> S" "\<omega> \<in> T" |
|
1569 |
from `\<omega> \<in> S`[THEN S'(4)] * show "x \<in> S" |
|
1570 |
by (cases x, auto simp: max_def split: split_if_asm) |
|
1571 |
from `\<omega> \<in> T`[THEN T'(4)] * show "x \<in> T" |
|
1572 |
by (cases x, auto simp: max_def split: split_if_asm) |
|
1573 |
next |
|
1574 |
fix x assume x: "x \<notin> Real ` (S' \<inter> T' \<inter> {0..})" |
|
1575 |
have *: "S' \<inter> T' \<inter> {0..} = (S' \<inter> {0..}) \<inter> (T' \<inter> {0..})" by auto |
|
1576 |
assume "x \<in> T" "x \<in> S" |
|
1577 |
with S'(2) T'(2) show "x = \<omega>" |
|
1578 |
using x[unfolded *] inj_on_image_Int[OF inj_on_Real] by auto |
|
1579 |
qed auto |
|
1580 |
next |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1581 |
fix K assume openK: "\<forall>S \<in> K. open (S:: pextreal set)" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1582 |
hence "\<forall>S\<in>K. \<exists>T. open T \<and> Real ` (T \<inter> {0..}) = S - {\<omega>}" by (auto simp: open_pextreal_def) |
38656 | 1583 |
from bchoice[OF this] guess T .. note T = this[rule_format] |
1584 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1585 |
show "open (\<Union>K)" unfolding open_pextreal_def |
38656 | 1586 |
proof (safe intro!: exI[of _ "\<Union>(T ` K)"]) |
1587 |
fix x S assume "0 \<le> x" "x \<in> T S" "S \<in> K" |
|
1588 |
with T[OF `S \<in> K`] show "Real x \<in> \<Union>K" by auto |
|
1589 |
next |
|
1590 |
fix x S assume x: "x \<notin> Real ` (\<Union>T ` K \<inter> {0..})" "S \<in> K" "x \<in> S" |
|
1591 |
hence "x \<notin> Real ` (T S \<inter> {0..})" |
|
1592 |
by (auto simp: image_UN UN_simps[symmetric] simp del: UN_simps) |
|
1593 |
thus "x = \<omega>" using T[OF `S \<in> K`] `x \<in> S` by auto |
|
1594 |
next |
|
1595 |
fix S assume "\<omega> \<in> S" "S \<in> K" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1596 |
from openK[rule_format, OF `S \<in> K`, THEN pextreal_openE] guess S' x . |
38656 | 1597 |
from this(3, 4) `\<omega> \<in> S` |
1598 |
show "\<exists>x\<ge>0. {Real x<..} \<subseteq> \<Union>K" |
|
1599 |
by (auto intro!: exI[of _ x] bexI[OF _ `S \<in> K`]) |
|
1600 |
next |
|
1601 |
from T[THEN conjunct1] show "open (\<Union>T`K)" by auto |
|
1602 |
qed auto |
|
1603 |
qed |
|
1604 |
end |
|
1605 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1606 |
lemma open_pextreal_lessThan[simp]: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1607 |
"open {..< a :: pextreal}" |
38656 | 1608 |
proof (cases a) |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1609 |
case (preal x) thus ?thesis unfolding open_pextreal_def |
38656 | 1610 |
proof (safe intro!: exI[of _ "{..< x}"]) |
1611 |
fix y assume "y < Real x" |
|
1612 |
moreover assume "y \<notin> Real ` ({..<x} \<inter> {0..})" |
|
1613 |
ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto |
|
1614 |
thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm) |
|
1615 |
qed auto |
|
1616 |
next |
|
1617 |
case infinite thus ?thesis |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1618 |
unfolding open_pextreal_def by (auto intro!: exI[of _ UNIV]) |
38656 | 1619 |
qed |
1620 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1621 |
lemma open_pextreal_greaterThan[simp]: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1622 |
"open {a :: pextreal <..}" |
38656 | 1623 |
proof (cases a) |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1624 |
case (preal x) thus ?thesis unfolding open_pextreal_def |
38656 | 1625 |
proof (safe intro!: exI[of _ "{x <..}"]) |
1626 |
fix y assume "Real x < y" |
|
1627 |
moreover assume "y \<notin> Real ` ({x<..} \<inter> {0..})" |
|
1628 |
ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto |
|
1629 |
thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm) |
|
1630 |
qed auto |
|
1631 |
next |
|
1632 |
case infinite thus ?thesis |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1633 |
unfolding open_pextreal_def by (auto intro!: exI[of _ "{}"]) |
38656 | 1634 |
qed |
1635 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1636 |
lemma pextreal_open_greaterThanLessThan[simp]: "open {a::pextreal <..< b}" |
38656 | 1637 |
unfolding greaterThanLessThan_def by auto |
1638 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1639 |
lemma closed_pextreal_atLeast[simp, intro]: "closed {a :: pextreal ..}" |
38656 | 1640 |
proof - |
1641 |
have "- {a ..} = {..< a}" by auto |
|
1642 |
then show "closed {a ..}" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1643 |
unfolding closed_def using open_pextreal_lessThan by auto |
38656 | 1644 |
qed |
1645 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1646 |
lemma closed_pextreal_atMost[simp, intro]: "closed {.. b :: pextreal}" |
38656 | 1647 |
proof - |
1648 |
have "- {.. b} = {b <..}" by auto |
|
1649 |
then show "closed {.. b}" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1650 |
unfolding closed_def using open_pextreal_greaterThan by auto |
38656 | 1651 |
qed |
1652 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1653 |
lemma closed_pextreal_atLeastAtMost[simp, intro]: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1654 |
shows "closed {a :: pextreal .. b}" |
38656 | 1655 |
unfolding atLeastAtMost_def by auto |
1656 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1657 |
lemma pextreal_dense: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1658 |
fixes x y :: pextreal assumes "x < y" |
38656 | 1659 |
shows "\<exists>z. x < z \<and> z < y" |
1660 |
proof - |
|
1661 |
from `x < y` obtain p where p: "x = Real p" "0 \<le> p" by (cases x) auto |
|
1662 |
show ?thesis |
|
1663 |
proof (cases y) |
|
1664 |
case (preal r) with p `x < y` have "p < r" by auto |
|
1665 |
with dense obtain z where "p < z" "z < r" by auto |
|
1666 |
thus ?thesis using preal p by (auto intro!: exI[of _ "Real z"]) |
|
1667 |
next |
|
1668 |
case infinite thus ?thesis using `x < y` p |
|
1669 |
by (auto intro!: exI[of _ "Real p + 1"]) |
|
1670 |
qed |
|
1671 |
qed |
|
1672 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1673 |
instance pextreal :: t2_space |
38656 | 1674 |
proof |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1675 |
fix x y :: pextreal assume "x \<noteq> y" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1676 |
let "?P x (y::pextreal)" = "\<exists> U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" |
38656 | 1677 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1678 |
{ fix x y :: pextreal assume "x < y" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1679 |
from pextreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto |
38656 | 1680 |
have "?P x y" |
1681 |
apply (rule exI[of _ "{..<z}"]) |
|
1682 |
apply (rule exI[of _ "{z<..}"]) |
|
1683 |
using z by auto } |
|
1684 |
note * = this |
|
1685 |
||
1686 |
from `x \<noteq> y` |
|
1687 |
show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" |
|
1688 |
proof (cases rule: linorder_cases) |
|
1689 |
assume "x = y" with `x \<noteq> y` show ?thesis by simp |
|
1690 |
next assume "x < y" from *[OF this] show ?thesis by auto |
|
1691 |
next assume "y < x" from *[OF this] show ?thesis by auto |
|
1692 |
qed |
|
1693 |
qed |
|
1694 |
||
1695 |
definition (in complete_lattice) isoton :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<up>" 50) where |
|
1696 |
"A \<up> X \<longleftrightarrow> (\<forall>i. A i \<le> A (Suc i)) \<and> (SUP i. A i) = X" |
|
1697 |
||
1698 |
definition (in complete_lattice) antiton (infix "\<down>" 50) where |
|
1699 |
"A \<down> X \<longleftrightarrow> (\<forall>i. A i \<ge> A (Suc i)) \<and> (INF i. A i) = X" |
|
1700 |
||
40859 | 1701 |
lemma isotoneI[intro?]: "\<lbrakk> \<And>i. f i \<le> f (Suc i) ; (SUP i. f i) = F \<rbrakk> \<Longrightarrow> f \<up> F" |
1702 |
unfolding isoton_def by auto |
|
1703 |
||
1704 |
lemma (in complete_lattice) isotonD[dest]: |
|
1705 |
assumes "A \<up> X" shows "A i \<le> A (Suc i)" "(SUP i. A i) = X" |
|
1706 |
using assms unfolding isoton_def by auto |
|
1707 |
||
1708 |
lemma isotonD'[dest]: |
|
1709 |
assumes "(A::_=>_) \<up> X" shows "A i x \<le> A (Suc i) x" "(SUP i. A i) = X" |
|
1710 |
using assms unfolding isoton_def le_fun_def by auto |
|
1711 |
||
1712 |
lemma isoton_mono_le: |
|
1713 |
assumes "f \<up> x" "i \<le> j" |
|
1714 |
shows "f i \<le> f j" |
|
1715 |
using `f \<up> x`[THEN isotonD(1)] lift_Suc_mono_le[of f, OF _ `i \<le> j`] by auto |
|
1716 |
||
1717 |
lemma isoton_const: |
|
1718 |
shows "(\<lambda> i. c) \<up> c" |
|
1719 |
unfolding isoton_def by auto |
|
38656 | 1720 |
|
1721 |
lemma isoton_cmult_right: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1722 |
assumes "f \<up> (x::pextreal)" |
38656 | 1723 |
shows "(\<lambda>i. c * f i) \<up> (c * x)" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1724 |
using assms unfolding isoton_def pextreal_SUP_cmult |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1725 |
by (auto intro: pextreal_mult_cancel) |
38656 | 1726 |
|
1727 |
lemma isoton_cmult_left: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1728 |
"f \<up> (x::pextreal) \<Longrightarrow> (\<lambda>i. f i * c) \<up> (x * c)" |
38656 | 1729 |
by (subst (1 2) mult_commute) (rule isoton_cmult_right) |
1730 |
||
1731 |
lemma isoton_add: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1732 |
assumes "f \<up> (x::pextreal)" and "g \<up> y" |
38656 | 1733 |
shows "(\<lambda>i. f i + g i) \<up> (x + y)" |
1734 |
using assms unfolding isoton_def |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1735 |
by (auto intro: pextreal_mult_cancel add_mono simp: SUPR_pextreal_add) |
38656 | 1736 |
|
1737 |
lemma isoton_fun_expand: |
|
1738 |
"f \<up> x \<longleftrightarrow> (\<forall>i. (\<lambda>j. f j i) \<up> (x i))" |
|
1739 |
proof - |
|
1740 |
have "\<And>i. {y. \<exists>f'\<in>range f. y = f' i} = range (\<lambda>j. f j i)" |
|
1741 |
by auto |
|
1742 |
with assms show ?thesis |
|
1743 |
by (auto simp add: isoton_def le_fun_def Sup_fun_def SUPR_def) |
|
1744 |
qed |
|
1745 |
||
1746 |
lemma isoton_indicator: |
|
1747 |
assumes "f \<up> g" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1748 |
shows "(\<lambda>i x. f i x * indicator A x) \<up> (\<lambda>x. g x * indicator A x :: pextreal)" |
38656 | 1749 |
using assms unfolding isoton_fun_expand by (auto intro!: isoton_cmult_left) |
1750 |
||
40859 | 1751 |
lemma isoton_setsum: |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1752 |
fixes f :: "'a \<Rightarrow> nat \<Rightarrow> pextreal" |
40859 | 1753 |
assumes "finite A" "A \<noteq> {}" |
1754 |
assumes "\<And> x. x \<in> A \<Longrightarrow> f x \<up> y x" |
|
1755 |
shows "(\<lambda> i. (\<Sum> x \<in> A. f x i)) \<up> (\<Sum> x \<in> A. y x)" |
|
1756 |
using assms |
|
1757 |
proof (induct A rule:finite_ne_induct) |
|
1758 |
case singleton thus ?case by auto |
|
1759 |
next |
|
1760 |
case (insert a A) note asms = this |
|
1761 |
hence *: "(\<lambda> i. \<Sum> x \<in> A. f x i) \<up> (\<Sum> x \<in> A. y x)" by auto |
|
1762 |
have **: "(\<lambda> i. f a i) \<up> y a" using asms by simp |
|
1763 |
have "(\<lambda> i. f a i + (\<Sum> x \<in> A. f x i)) \<up> (y a + (\<Sum> x \<in> A. y x))" |
|
1764 |
using * ** isoton_add by auto |
|
1765 |
thus "(\<lambda> i. \<Sum> x \<in> insert a A. f x i) \<up> (\<Sum> x \<in> insert a A. y x)" |
|
1766 |
using asms by fastsimp |
|
1767 |
qed |
|
38656 | 1768 |
|
1769 |
lemma isoton_Sup: |
|
1770 |
assumes "f \<up> u" |
|
1771 |
shows "f i \<le> u" |
|
1772 |
using le_SUPI[of i UNIV f] assms |
|
1773 |
unfolding isoton_def by auto |
|
1774 |
||
1775 |
lemma isoton_mono: |
|
1776 |
assumes iso: "x \<up> a" "y \<up> b" and *: "\<And>n. x n \<le> y (N n)" |
|
1777 |
shows "a \<le> b" |
|
1778 |
proof - |
|
1779 |
from iso have "a = (SUP n. x n)" "b = (SUP n. y n)" |
|
1780 |
unfolding isoton_def by auto |
|
1781 |
with * show ?thesis by (auto intro!: SUP_mono) |
|
1782 |
qed |
|
1783 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1784 |
lemma pextreal_le_mult_one_interval: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1785 |
fixes x y :: pextreal |
38656 | 1786 |
assumes "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" |
1787 |
shows "x \<le> y" |
|
1788 |
proof (cases x, cases y) |
|
1789 |
assume "x = \<omega>" |
|
1790 |
with assms[of "1 / 2"] |
|
1791 |
show "x \<le> y" by simp |
|
1792 |
next |
|
1793 |
fix r p assume *: "y = Real p" "x = Real r" and **: "0 \<le> r" "0 \<le> p" |
|
1794 |
have "r \<le> p" |
|
1795 |
proof (rule field_le_mult_one_interval) |
|
1796 |
fix z :: real assume "0 < z" and "z < 1" |
|
1797 |
with assms[of "Real z"] |
|
1798 |
show "z * r \<le> p" using ** * by (auto simp: zero_le_mult_iff) |
|
1799 |
qed |
|
1800 |
thus "x \<le> y" using ** * by simp |
|
1801 |
qed simp |
|
1802 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1803 |
lemma pextreal_greater_0[intro]: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1804 |
fixes a :: pextreal |
38656 | 1805 |
assumes "a \<noteq> 0" |
1806 |
shows "a > 0" |
|
1807 |
using assms apply (cases a) by auto |
|
1808 |
||
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
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diff
changeset
|
1809 |
lemma pextreal_mult_strict_right_mono: |
38656 | 1810 |
assumes "a < b" and "0 < c" "c < \<omega>" |
1811 |
shows "a * c < b * c" |
|
1812 |
using assms |
|
1813 |
by (cases a, cases b, cases c) |
|
41023
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it is known as the extended reals, not the infinite reals
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parents:
40874
diff
changeset
|
1814 |
(auto simp: zero_le_mult_iff pextreal_less_\<omega>) |
38656 | 1815 |
|
41023
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parents:
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diff
changeset
|
1816 |
lemma minus_pextreal_eq2: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1817 |
fixes x y z :: pextreal |
38656 | 1818 |
assumes "y \<le> x" and "y \<noteq> \<omega>" shows "z = x - y \<longleftrightarrow> z + y = x" |
1819 |
using assms |
|
1820 |
apply (subst eq_commute) |
|
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1821 |
apply (subst minus_pextreal_eq) |
38656 | 1822 |
by (cases x, cases z, auto simp add: ac_simps not_less) |
1823 |
||
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
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diff
changeset
|
1824 |
lemma pextreal_diff_eq_diff_imp_eq: |
38656 | 1825 |
assumes "a \<noteq> \<omega>" "b \<le> a" "c \<le> a" |
1826 |
assumes "a - b = a - c" |
|
1827 |
shows "b = c" |
|
1828 |
using assms |
|
1829 |
by (cases a, cases b, cases c) (auto split: split_if_asm) |
|
1830 |
||
41023
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parents:
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diff
changeset
|
1831 |
lemma pextreal_inverse_eq_0: "inverse x = 0 \<longleftrightarrow> x = \<omega>" |
38656 | 1832 |
by (cases x) auto |
1833 |
||
41023
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hoelzl
parents:
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diff
changeset
|
1834 |
lemma pextreal_mult_inverse: |
38656 | 1835 |
"\<lbrakk> x \<noteq> \<omega> ; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x * inverse x = 1" |
1836 |
by (cases x) auto |
|
1837 |
||
41023
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it is known as the extended reals, not the infinite reals
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parents:
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diff
changeset
|
1838 |
lemma pextreal_zero_less_diff_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1839 |
fixes a b :: pextreal shows "0 < a - b \<longleftrightarrow> b < a" |
38656 | 1840 |
apply (cases a, cases b) |
41023
9118eb4eb8dc
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hoelzl
parents:
40874
diff
changeset
|
1841 |
apply (auto simp: pextreal_noteq_omega_Ex pextreal_less_\<omega>) |
38656 | 1842 |
apply (cases b) |
1843 |
by auto |
|
1844 |
||
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1845 |
lemma pextreal_less_Real_Ex: |
9118eb4eb8dc
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hoelzl
parents:
40874
diff
changeset
|
1846 |
fixes a b :: pextreal shows "x < Real r \<longleftrightarrow> (\<exists>p\<ge>0. p < r \<and> x = Real p)" |
38656 | 1847 |
by (cases x) auto |
1848 |
||
1849 |
lemma open_Real: assumes "open S" shows "open (Real ` ({0..} \<inter> S))" |
|
41023
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hoelzl
parents:
40874
diff
changeset
|
1850 |
unfolding open_pextreal_def apply(rule,rule,rule,rule assms) by auto |
38656 | 1851 |
|
41023
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parents:
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diff
changeset
|
1852 |
lemma pextreal_zero_le_diff: |
9118eb4eb8dc
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hoelzl
parents:
40874
diff
changeset
|
1853 |
fixes a b :: pextreal shows "a - b = 0 \<longleftrightarrow> a \<le> b" |
38656 | 1854 |
by (cases a, cases b, simp_all, cases b, auto) |
1855 |
||
1856 |
lemma lim_Real[simp]: assumes "\<forall>n. f n \<ge> 0" "m\<ge>0" |
|
1857 |
shows "(\<lambda>n. Real (f n)) ----> Real m \<longleftrightarrow> (\<lambda>n. f n) ----> m" (is "?l = ?r") |
|
1858 |
proof assume ?l show ?r unfolding Lim_sequentially |
|
1859 |
proof safe fix e::real assume e:"e>0" |
|
1860 |
note open_ball[of m e] note open_Real[OF this] |
|
1861 |
note * = `?l`[unfolded tendsto_def,rule_format,OF this] |
|
1862 |
have "eventually (\<lambda>x. Real (f x) \<in> Real ` ({0..} \<inter> ball m e)) sequentially" |
|
1863 |
apply(rule *) unfolding image_iff using assms(2) e by auto |
|
1864 |
thus "\<exists>N. \<forall>n\<ge>N. dist (f n) m < e" unfolding eventually_sequentially |
|
1865 |
apply safe apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe) |
|
1866 |
proof- fix n x assume "Real (f n) = Real x" "0 \<le> x" |
|
1867 |
hence *:"f n = x" using assms(1) by auto |
|
1868 |
assume "x \<in> ball m e" thus "dist (f n) m < e" unfolding * |
|
1869 |
by (auto simp add:dist_commute) |
|
1870 |
qed qed |
|
1871 |
next assume ?r show ?l unfolding tendsto_def eventually_sequentially |
|
1872 |
proof safe fix S assume S:"open S" "Real m \<in> S" |
|
41023
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parents:
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diff
changeset
|
1873 |
guess T y using S(1) apply-apply(erule pextreal_openE) . note T=this |
38656 | 1874 |
have "m\<in>real ` (S - {\<omega>})" unfolding image_iff |
1875 |
apply(rule_tac x="Real m" in bexI) using assms(2) S(2) by auto |
|
1876 |
hence "m \<in> T" unfolding T(2)[THEN sym] by auto |
|
1877 |
from `?r`[unfolded tendsto_def eventually_sequentially,rule_format,OF T(1) this] |
|
1878 |
guess N .. note N=this[rule_format] |
|
1879 |
show "\<exists>N. \<forall>n\<ge>N. Real (f n) \<in> S" apply(rule_tac x=N in exI) |
|
1880 |
proof safe fix n assume n:"N\<le>n" |
|
1881 |
have "f n \<in> real ` (S - {\<omega>})" using N[OF n] assms unfolding T(2)[THEN sym] |
|
1882 |
unfolding image_iff apply-apply(rule_tac x="Real (f n)" in bexI) |
|
1883 |
unfolding real_Real by auto |
|
1884 |
then guess x unfolding image_iff .. note x=this |
|
1885 |
show "Real (f n) \<in> S" unfolding x apply(subst Real_real) using x by auto |
|
1886 |
qed |
|
1887 |
qed |
|
1888 |
qed |
|
1889 |
||
41023
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parents:
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diff
changeset
|
1890 |
lemma pextreal_INFI: |
9118eb4eb8dc
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parents:
40874
diff
changeset
|
1891 |
fixes x :: pextreal |
38656 | 1892 |
assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i" |
1893 |
assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> f i) \<Longrightarrow> y \<le> x" |
|
1894 |
shows "(INF i:A. f i) = x" |
|
41023
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parents:
40874
diff
changeset
|
1895 |
unfolding INFI_def Inf_pextreal_def |
38656 | 1896 |
using assms by (auto intro!: Greatest_equality) |
1897 |
||
41023
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parents:
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diff
changeset
|
1898 |
lemma real_of_pextreal_less:"x < y \<Longrightarrow> y\<noteq>\<omega> \<Longrightarrow> real x < real y" |
38656 | 1899 |
proof- case goal1 |
1900 |
have *:"y = Real (real y)" "x = Real (real x)" using goal1 Real_real by auto |
|
1901 |
show ?case using goal1 apply- apply(subst(asm) *(1))apply(subst(asm) *(2)) |
|
41023
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hoelzl
parents:
40874
diff
changeset
|
1902 |
unfolding pextreal_less by auto |
38656 | 1903 |
qed |
1904 |
||
1905 |
lemma not_less_omega[simp]:"\<not> x < \<omega> \<longleftrightarrow> x = \<omega>" |
|
41023
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hoelzl
parents:
40874
diff
changeset
|
1906 |
by (metis antisym_conv3 pextreal_less(3)) |
38656 | 1907 |
|
1908 |
lemma Real_real': assumes "x\<noteq>\<omega>" shows "Real (real x) = x" |
|
1909 |
proof- have *:"(THE r. 0 \<le> r \<and> x = Real r) = real x" |
|
1910 |
apply(rule the_equality) using assms unfolding Real_real by auto |
|
1911 |
have "Real (THE r. 0 \<le> r \<and> x = Real r) = x" unfolding * |
|
1912 |
using assms unfolding Real_real by auto |
|
41023
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hoelzl
parents:
40874
diff
changeset
|
1913 |
thus ?thesis unfolding real_of_pextreal_def of_pextreal_def |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1914 |
unfolding pextreal_case_def using assms by auto |
38656 | 1915 |
qed |
1916 |
||
1917 |
lemma Real_less_plus_one:"Real x < Real (max (x + 1) 1)" |
|
41023
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it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1918 |
unfolding pextreal_less by auto |
38656 | 1919 |
|
1920 |
lemma Lim_omega: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r") |
|
1921 |
proof assume ?r show ?l apply(rule topological_tendstoI) |
|
1922 |
unfolding eventually_sequentially |
|
1923 |
proof- fix S assume "open S" "\<omega> \<in> S" |
|
1924 |
from open_omega[OF this] guess B .. note B=this |
|
1925 |
from `?r`[rule_format,of "(max B 0)+1"] guess N .. note N=this |
|
1926 |
show "\<exists>N. \<forall>n\<ge>N. f n \<in> S" apply(rule_tac x=N in exI) |
|
1927 |
proof safe case goal1 |
|
1928 |
have "Real B < Real ((max B 0) + 1)" by auto |
|
1929 |
also have "... \<le> f n" using goal1 N by auto |
|
1930 |
finally show ?case using B by fastsimp |
|
1931 |
qed |
|
1932 |
qed |
|
1933 |
next assume ?l show ?r |
|
1934 |
proof fix B::real have "open {Real B<..}" "\<omega> \<in> {Real B<..}" by auto |
|
1935 |
from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially] |
|
1936 |
guess N .. note N=this |
|
1937 |
show "\<exists>N. \<forall>n\<ge>N. Real B \<le> f n" apply(rule_tac x=N in exI) using N by auto |
|
1938 |
qed |
|
1939 |
qed |
|
1940 |
||
1941 |
lemma Lim_bounded_omgea: assumes lim:"f ----> l" and "\<And>n. f n \<le> Real B" shows "l \<noteq> \<omega>" |
|
1942 |
proof(rule ccontr,unfold not_not) let ?B = "max (B + 1) 1" assume as:"l=\<omega>" |
|
1943 |
from lim[unfolded this Lim_omega,rule_format,of "?B"] |
|
1944 |
guess N .. note N=this[rule_format,OF le_refl] |
|
1945 |
hence "Real ?B \<le> Real B" using assms(2)[of N] by(rule order_trans) |
|
1946 |
hence "Real ?B < Real ?B" using Real_less_plus_one[of B] by(rule le_less_trans) |
|
1947 |
thus False by auto |
|
1948 |
qed |
|
1949 |
||
41023
9118eb4eb8dc
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hoelzl
parents:
40874
diff
changeset
|
1950 |
lemma incseq_le_pextreal: assumes inc: "\<And>n m. n\<ge>m \<Longrightarrow> X n \<ge> X m" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1951 |
and lim: "X ----> (L::pextreal)" shows "X n \<le> L" |
38656 | 1952 |
proof(cases "L = \<omega>") |
1953 |
case False have "\<forall>n. X n \<noteq> \<omega>" |
|
1954 |
proof(rule ccontr,unfold not_all not_not,safe) |
|
1955 |
case goal1 hence "\<forall>n\<ge>x. X n = \<omega>" using inc[of x] by auto |
|
1956 |
hence "X ----> \<omega>" unfolding tendsto_def eventually_sequentially |
|
1957 |
apply safe apply(rule_tac x=x in exI) by auto |
|
1958 |
note Lim_unique[OF trivial_limit_sequentially this lim] |
|
1959 |
with False show False by auto |
|
1960 |
qed note * =this[rule_format] |
|
1961 |
||
1962 |
have **:"\<forall>m n. m \<le> n \<longrightarrow> Real (real (X m)) \<le> Real (real (X n))" |
|
1963 |
unfolding Real_real using * inc by auto |
|
1964 |
have "real (X n) \<le> real L" apply-apply(rule incseq_le) defer |
|
1965 |
apply(subst lim_Real[THEN sym]) apply(rule,rule,rule) |
|
1966 |
unfolding Real_real'[OF *] Real_real'[OF False] |
|
1967 |
unfolding incseq_def using ** lim by auto |
|
1968 |
hence "Real (real (X n)) \<le> Real (real L)" by auto |
|
1969 |
thus ?thesis unfolding Real_real using * False by auto |
|
1970 |
qed auto |
|
1971 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1972 |
lemma SUP_Lim_pextreal: assumes "\<And>n m. n\<ge>m \<Longrightarrow> f n \<ge> f m" "f ----> l" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1973 |
shows "(SUP n. f n) = (l::pextreal)" unfolding SUPR_def Sup_pextreal_def |
38656 | 1974 |
proof (safe intro!: Least_equality) |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1975 |
fix n::nat show "f n \<le> l" apply(rule incseq_le_pextreal) |
38656 | 1976 |
using assms by auto |
1977 |
next fix y assume y:"\<forall>x\<in>range f. x \<le> y" show "l \<le> y" |
|
1978 |
proof(rule ccontr,cases "y=\<omega>",unfold not_le) |
|
1979 |
case False assume as:"y < l" |
|
1980 |
have l:"l \<noteq> \<omega>" apply(rule Lim_bounded_omgea[OF assms(2), of "real y"]) |
|
1981 |
using False y unfolding Real_real by auto |
|
1982 |
||
1983 |
have yl:"real y < real l" using as apply- |
|
1984 |
apply(subst(asm) Real_real'[THEN sym,OF `y\<noteq>\<omega>`]) |
|
1985 |
apply(subst(asm) Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
1986 |
unfolding pextreal_less apply(subst(asm) if_P) by auto |
38656 | 1987 |
hence "y + (y - l) * Real (1 / 2) < l" apply- |
1988 |
apply(subst Real_real'[THEN sym,OF `y\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `y\<noteq>\<omega>`]) |
|
1989 |
apply(subst Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) by auto |
|
1990 |
hence *:"l \<in> {y + (y - l) / 2<..}" by auto |
|
1991 |
have "open {y + (y-l)/2 <..}" by auto |
|
1992 |
note topological_tendstoD[OF assms(2) this *] |
|
1993 |
from this[unfolded eventually_sequentially] guess N .. note this[rule_format, of N] |
|
1994 |
hence "y + (y - l) * Real (1 / 2) < y" using y[rule_format,of "f N"] by auto |
|
1995 |
hence "Real (real y) + (Real (real y) - Real (real l)) * Real (1 / 2) < Real (real y)" |
|
1996 |
unfolding Real_real using `y\<noteq>\<omega>` `l\<noteq>\<omega>` by auto |
|
1997 |
thus False using yl by auto |
|
1998 |
qed auto |
|
1999 |
qed |
|
2000 |
||
2001 |
lemma Real_max':"Real x = Real (max x 0)" |
|
2002 |
proof(cases "x < 0") case True |
|
2003 |
hence *:"max x 0 = 0" by auto |
|
2004 |
show ?thesis unfolding * using True by auto |
|
2005 |
qed auto |
|
2006 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2007 |
lemma lim_pextreal_increasing: assumes "\<forall>n m. n\<ge>m \<longrightarrow> f n \<ge> f m" |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2008 |
obtains l where "f ----> (l::pextreal)" |
38656 | 2009 |
proof(cases "\<exists>B. \<forall>n. f n < Real B") |
2010 |
case False thus thesis apply- apply(rule that[of \<omega>]) unfolding Lim_omega not_ex not_all |
|
2011 |
apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe) |
|
2012 |
apply(rule order_trans[OF _ assms[rule_format]]) by auto |
|
2013 |
next case True then guess B .. note B = this[rule_format] |
|
2014 |
hence *:"\<And>n. f n < \<omega>" apply-apply(rule less_le_trans,assumption) by auto |
|
2015 |
have *:"\<And>n. f n \<noteq> \<omega>" proof- case goal1 show ?case using *[of n] by auto qed |
|
2016 |
have B':"\<And>n. real (f n) \<le> max 0 B" proof- case goal1 thus ?case |
|
2017 |
using B[of n] apply-apply(subst(asm) Real_real'[THEN sym]) defer |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2018 |
apply(subst(asm)(2) Real_max') unfolding pextreal_less apply(subst(asm) if_P) using *[of n] by auto |
38656 | 2019 |
qed |
2020 |
have "\<exists>l. (\<lambda>n. real (f n)) ----> l" apply(rule Topology_Euclidean_Space.bounded_increasing_convergent) |
|
2021 |
proof safe show "bounded {real (f n) |n. True}" |
|
2022 |
unfolding bounded_def apply(rule_tac x=0 in exI,rule_tac x="max 0 B" in exI) |
|
2023 |
using B' unfolding dist_norm by auto |
|
2024 |
fix n::nat have "Real (real (f n)) \<le> Real (real (f (Suc n)))" |
|
2025 |
using assms[rule_format,of n "Suc n"] apply(subst Real_real)+ |
|
2026 |
using *[of n] *[of "Suc n"] by fastsimp |
|
2027 |
thus "real (f n) \<le> real (f (Suc n))" by auto |
|
2028 |
qed then guess l .. note l=this |
|
2029 |
have "0 \<le> l" apply(rule LIMSEQ_le_const[OF l]) |
|
2030 |
by(rule_tac x=0 in exI,auto) |
|
2031 |
||
2032 |
thus ?thesis apply-apply(rule that[of "Real l"]) |
|
2033 |
using l apply-apply(subst(asm) lim_Real[THEN sym]) prefer 3 |
|
2034 |
unfolding Real_real using * by auto |
|
2035 |
qed |
|
2036 |
||
2037 |
lemma setsum_neq_omega: assumes "finite s" "\<And>x. x \<in> s \<Longrightarrow> f x \<noteq> \<omega>" |
|
2038 |
shows "setsum f s \<noteq> \<omega>" using assms |
|
2039 |
proof induct case (insert x s) |
|
2040 |
show ?case unfolding setsum.insert[OF insert(1-2)] |
|
2041 |
using insert by auto |
|
2042 |
qed auto |
|
2043 |
||
2044 |
||
2045 |
lemma real_Real': "0 \<le> x \<Longrightarrow> real (Real x) = x" |
|
2046 |
unfolding real_Real by auto |
|
2047 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2048 |
lemma real_pextreal_pos[intro]: |
38656 | 2049 |
assumes "x \<noteq> 0" "x \<noteq> \<omega>" |
2050 |
shows "real x > 0" |
|
2051 |
apply(subst real_Real'[THEN sym,of 0]) defer |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2052 |
apply(rule real_of_pextreal_less) using assms by auto |
38656 | 2053 |
|
2054 |
lemma Lim_omega_gt: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n > Real B)" (is "?l = ?r") |
|
2055 |
proof assume ?l thus ?r unfolding Lim_omega apply safe |
|
2056 |
apply(erule_tac x="max B 0 +1" in allE,safe) |
|
2057 |
apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe) |
|
2058 |
apply(rule_tac y="Real (max B 0 + 1)" in less_le_trans) by auto |
|
2059 |
next assume ?r thus ?l unfolding Lim_omega apply safe |
|
2060 |
apply(erule_tac x=B in allE,safe) apply(rule_tac x=N in exI,safe) by auto |
|
2061 |
qed |
|
2062 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2063 |
lemma pextreal_minus_le_cancel: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2064 |
fixes a b c :: pextreal |
38656 | 2065 |
assumes "b \<le> a" |
2066 |
shows "c - a \<le> c - b" |
|
2067 |
using assms by (cases a, cases b, cases c, simp, simp, simp, cases b, cases c, simp_all) |
|
2068 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2069 |
lemma pextreal_minus_\<omega>[simp]: "x - \<omega> = 0" by (cases x) simp_all |
38656 | 2070 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2071 |
lemma pextreal_minus_mono[intro]: "a - x \<le> (a::pextreal)" |
38656 | 2072 |
proof- have "a - x \<le> a - 0" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2073 |
apply(rule pextreal_minus_le_cancel) by auto |
38656 | 2074 |
thus ?thesis by auto |
2075 |
qed |
|
2076 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2077 |
lemma pextreal_minus_eq_\<omega>[simp]: "x - y = \<omega> \<longleftrightarrow> (x = \<omega> \<and> y \<noteq> \<omega>)" |
38656 | 2078 |
by (cases x, cases y) (auto, cases y, auto) |
2079 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2080 |
lemma pextreal_less_minus_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2081 |
fixes a b c :: pextreal |
38656 | 2082 |
shows "a < b - c \<longleftrightarrow> c + a < b" |
2083 |
by (cases c, cases a, cases b, auto) |
|
2084 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2085 |
lemma pextreal_minus_less_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2086 |
fixes a b c :: pextreal shows "a - c < b \<longleftrightarrow> (0 < b \<and> (c \<noteq> \<omega> \<longrightarrow> a < b + c))" |
38656 | 2087 |
by (cases c, cases a, cases b, auto) |
2088 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2089 |
lemma pextreal_le_minus_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2090 |
fixes a b c :: pextreal |
38656 | 2091 |
shows "a \<le> c - b \<longleftrightarrow> ((c \<le> b \<longrightarrow> a = 0) \<and> (b < c \<longrightarrow> a + b \<le> c))" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2092 |
by (cases a, cases c, cases b, auto simp: pextreal_noteq_omega_Ex) |
38656 | 2093 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2094 |
lemma pextreal_minus_le_iff: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2095 |
fixes a b c :: pextreal |
38656 | 2096 |
shows "a - c \<le> b \<longleftrightarrow> (c \<le> a \<longrightarrow> a \<le> b + c)" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2097 |
by (cases a, cases c, cases b, auto simp: pextreal_noteq_omega_Ex) |
38656 | 2098 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2099 |
lemmas pextreal_minus_order = pextreal_minus_le_iff pextreal_minus_less_iff pextreal_le_minus_iff pextreal_less_minus_iff |
38656 | 2100 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2101 |
lemma pextreal_minus_strict_mono: |
38656 | 2102 |
assumes "a > 0" "x > 0" "a\<noteq>\<omega>" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2103 |
shows "a - x < (a::pextreal)" |
38656 | 2104 |
using assms by(cases x, cases a, auto) |
2105 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2106 |
lemma pextreal_minus': |
38656 | 2107 |
"Real r - Real p = (if 0 \<le> r \<and> p \<le> r then if 0 \<le> p then Real (r - p) else Real r else 0)" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2108 |
by (auto simp: minus_pextreal_eq not_less) |
38656 | 2109 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2110 |
lemma pextreal_minus_plus: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2111 |
"x \<le> (a::pextreal) \<Longrightarrow> a - x + x = a" |
38656 | 2112 |
by (cases a, cases x) auto |
2113 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2114 |
lemma pextreal_cancel_plus_minus: "b \<noteq> \<omega> \<Longrightarrow> a + b - b = a" |
38656 | 2115 |
by (cases a, cases b) auto |
2116 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2117 |
lemma pextreal_minus_le_cancel_right: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2118 |
fixes a b c :: pextreal |
38656 | 2119 |
assumes "a \<le> b" "c \<le> a" |
2120 |
shows "a - c \<le> b - c" |
|
2121 |
using assms by (cases a, cases b, cases c, auto, cases c, auto) |
|
2122 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2123 |
lemma real_of_pextreal_setsum': |
38656 | 2124 |
assumes "\<forall>x \<in> S. f x \<noteq> \<omega>" |
2125 |
shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)" |
|
2126 |
proof cases |
|
2127 |
assume "finite S" |
|
2128 |
from this assms show ?thesis |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2129 |
by induct (simp_all add: real_of_pextreal_add setsum_\<omega>) |
38656 | 2130 |
qed simp |
2131 |
||
2132 |
lemma Lim_omega_pos: "f ----> \<omega> \<longleftrightarrow> (\<forall>B>0. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r") |
|
2133 |
unfolding Lim_omega apply safe defer |
|
2134 |
apply(erule_tac x="max 1 B" in allE) apply safe defer |
|
2135 |
apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe) |
|
2136 |
apply(rule_tac y="Real (max 1 B)" in order_trans) by auto |
|
2137 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2138 |
lemma pextreal_LimI_finite: |
38656 | 2139 |
assumes "x \<noteq> \<omega>" "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r" |
2140 |
shows "u ----> x" |
|
2141 |
proof (rule topological_tendstoI, unfold eventually_sequentially) |
|
2142 |
fix S assume "open S" "x \<in> S" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2143 |
then obtain A where "open A" and A_eq: "Real ` (A \<inter> {0..}) = S - {\<omega>}" by (auto elim!: pextreal_openE) |
38656 | 2144 |
then have "x \<in> Real ` (A \<inter> {0..})" using `x \<in> S` `x \<noteq> \<omega>` by auto |
2145 |
then have "real x \<in> A" by auto |
|
2146 |
then obtain r where "0 < r" and dist: "\<And>y. dist y (real x) < r \<Longrightarrow> y \<in> A" |
|
2147 |
using `open A` unfolding open_real_def by auto |
|
2148 |
then obtain n where |
|
2149 |
upper: "\<And>N. n \<le> N \<Longrightarrow> u N < x + Real r" and |
|
2150 |
lower: "\<And>N. n \<le> N \<Longrightarrow> x < u N + Real r" using assms(2)[of "Real r"] by auto |
|
2151 |
show "\<exists>N. \<forall>n\<ge>N. u n \<in> S" |
|
2152 |
proof (safe intro!: exI[of _ n]) |
|
2153 |
fix N assume "n \<le> N" |
|
2154 |
from upper[OF this] `x \<noteq> \<omega>` `0 < r` |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2155 |
have "u N \<noteq> \<omega>" by (force simp: pextreal_noteq_omega_Ex) |
38656 | 2156 |
with `x \<noteq> \<omega>` `0 < r` lower[OF `n \<le> N`] upper[OF `n \<le> N`] |
2157 |
have "dist (real (u N)) (real x) < r" "u N \<noteq> \<omega>" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2158 |
by (auto simp: pextreal_noteq_omega_Ex dist_real_def abs_diff_less_iff field_simps) |
38656 | 2159 |
from dist[OF this(1)] |
2160 |
have "u N \<in> Real ` (A \<inter> {0..})" using `u N \<noteq> \<omega>` |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2161 |
by (auto intro!: image_eqI[of _ _ "real (u N)"] simp: pextreal_noteq_omega_Ex Real_real) |
38656 | 2162 |
thus "u N \<in> S" using A_eq by simp |
2163 |
qed |
|
2164 |
qed |
|
2165 |
||
2166 |
lemma real_Real_max:"real (Real x) = max x 0" |
|
2167 |
unfolding real_Real by auto |
|
2168 |
||
2169 |
lemma Sup_lim: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2170 |
assumes "\<forall>n. b n \<in> s" "b ----> (a::pextreal)" |
38656 | 2171 |
shows "a \<le> Sup s" |
2172 |
proof(rule ccontr,unfold not_le) |
|
2173 |
assume as:"Sup s < a" hence om:"Sup s \<noteq> \<omega>" by auto |
|
2174 |
have s:"s \<noteq> {}" using assms by auto |
|
2175 |
{ presume *:"\<forall>n. b n < a \<Longrightarrow> False" |
|
2176 |
show False apply(cases,rule *,assumption,unfold not_all not_less) |
|
2177 |
proof- case goal1 then guess n .. note n=this |
|
2178 |
thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]] |
|
2179 |
using as by auto |
|
2180 |
qed |
|
2181 |
} assume b:"\<forall>n. b n < a" |
|
2182 |
show False |
|
2183 |
proof(cases "a = \<omega>") |
|
2184 |
case False have *:"a - Sup s > 0" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2185 |
using False as by(auto simp: pextreal_zero_le_diff) |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2186 |
have "(a - Sup s) / 2 \<le> a / 2" unfolding divide_pextreal_def |
38656 | 2187 |
apply(rule mult_right_mono) by auto |
2188 |
also have "... = Real (real (a / 2))" apply(rule Real_real'[THEN sym]) |
|
2189 |
using False by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2190 |
also have "... < Real (real a)" unfolding pextreal_less using as False |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2191 |
by(auto simp add: real_of_pextreal_mult[THEN sym]) |
38656 | 2192 |
also have "... = a" apply(rule Real_real') using False by auto |
2193 |
finally have asup:"a > (a - Sup s) / 2" . |
|
2194 |
have "\<exists>n. a - b n < (a - Sup s) / 2" |
|
2195 |
proof(rule ccontr,unfold not_ex not_less) |
|
2196 |
case goal1 |
|
2197 |
have "(a - Sup s) * Real (1 / 2) > 0" |
|
2198 |
using * by auto |
|
2199 |
hence "a - (a - Sup s) * Real (1 / 2) < a" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2200 |
apply-apply(rule pextreal_minus_strict_mono) |
38656 | 2201 |
using False * by auto |
2202 |
hence *:"a \<in> {a - (a - Sup s) / 2<..}"using asup by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2203 |
note topological_tendstoD[OF assms(2) open_pextreal_greaterThan,OF *] |
38656 | 2204 |
from this[unfolded eventually_sequentially] guess n .. |
2205 |
note n = this[rule_format,of n] |
|
2206 |
have "b n + (a - Sup s) / 2 \<le> a" |
|
2207 |
using add_right_mono[OF goal1[rule_format,of n],of "b n"] |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2208 |
unfolding pextreal_minus_plus[OF less_imp_le[OF b[rule_format]]] |
38656 | 2209 |
by(auto simp: add_commute) |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2210 |
hence "b n \<le> a - (a - Sup s) / 2" unfolding pextreal_le_minus_iff |
38656 | 2211 |
using asup by auto |
2212 |
hence "b n \<notin> {a - (a - Sup s) / 2<..}" by auto |
|
2213 |
thus False using n by auto |
|
2214 |
qed |
|
2215 |
then guess n .. note n = this |
|
2216 |
have "Sup s < a - (a - Sup s) / 2" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2217 |
using False as om by (cases a) (auto simp: pextreal_noteq_omega_Ex field_simps) |
38656 | 2218 |
also have "... \<le> b n" |
2219 |
proof- note add_right_mono[OF less_imp_le[OF n],of "b n"] |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2220 |
note this[unfolded pextreal_minus_plus[OF less_imp_le[OF b[rule_format]]]] |
38656 | 2221 |
hence "a - (a - Sup s) / 2 \<le> (a - Sup s) / 2 + b n - (a - Sup s) / 2" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2222 |
apply(rule pextreal_minus_le_cancel_right) using asup by auto |
38656 | 2223 |
also have "... = b n + (a - Sup s) / 2 - (a - Sup s) / 2" |
2224 |
by(auto simp add: add_commute) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2225 |
also have "... = b n" apply(subst pextreal_cancel_plus_minus) |
38656 | 2226 |
proof(rule ccontr,unfold not_not) case goal1 |
2227 |
show ?case using asup unfolding goal1 by auto |
|
2228 |
qed auto |
|
2229 |
finally show ?thesis . |
|
2230 |
qed |
|
2231 |
finally show False |
|
2232 |
using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]] by auto |
|
2233 |
next case True |
|
2234 |
from assms(2)[unfolded True Lim_omega_gt,rule_format,of "real (Sup s)"] |
|
2235 |
guess N .. note N = this[rule_format,of N] |
|
2236 |
thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of N]] |
|
2237 |
unfolding Real_real using om by auto |
|
2238 |
qed qed |
|
2239 |
||
2240 |
lemma Sup_mono_lim: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2241 |
assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> b ----> (a::pextreal)" |
38656 | 2242 |
shows "Sup A \<le> Sup B" |
2243 |
unfolding Sup_le_iff apply(rule) apply(drule assms[rule_format]) apply safe |
|
2244 |
apply(rule_tac b=b in Sup_lim) by auto |
|
2245 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2246 |
lemma pextreal_less_add: |
38656 | 2247 |
assumes "x \<noteq> \<omega>" "a < b" |
2248 |
shows "x + a < x + b" |
|
2249 |
using assms by (cases a, cases b, cases x) auto |
|
2250 |
||
2251 |
lemma SUPR_lim: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2252 |
assumes "\<forall>n. b n \<in> B" "(\<lambda>n. f (b n)) ----> (f a::pextreal)" |
38656 | 2253 |
shows "f a \<le> SUPR B f" |
2254 |
unfolding SUPR_def apply(rule Sup_lim[of "\<lambda>n. f (b n)"]) |
|
2255 |
using assms by auto |
|
2256 |
||
2257 |
lemma SUP_\<omega>_imp: |
|
2258 |
assumes "(SUP i. f i) = \<omega>" |
|
2259 |
shows "\<exists>i. Real x < f i" |
|
2260 |
proof (rule ccontr) |
|
2261 |
assume "\<not> ?thesis" hence "\<And>i. f i \<le> Real x" by (simp add: not_less) |
|
2262 |
hence "(SUP i. f i) \<le> Real x" unfolding SUP_le_iff by auto |
|
2263 |
with assms show False by auto |
|
2264 |
qed |
|
2265 |
||
2266 |
lemma SUPR_mono_lim: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2267 |
assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> (\<lambda>n. f (b n)) ----> (f a::pextreal)" |
38656 | 2268 |
shows "SUPR A f \<le> SUPR B f" |
2269 |
unfolding SUPR_def apply(rule Sup_mono_lim) |
|
2270 |
apply safe apply(drule assms[rule_format],safe) |
|
2271 |
apply(rule_tac x="\<lambda>n. f (b n)" in exI) by auto |
|
2272 |
||
2273 |
lemma real_0_imp_eq_0: |
|
2274 |
assumes "x \<noteq> \<omega>" "real x = 0" |
|
2275 |
shows "x = 0" |
|
2276 |
using assms by (cases x) auto |
|
2277 |
||
2278 |
lemma SUPR_mono: |
|
2279 |
assumes "\<forall>a\<in>A. \<exists>b\<in>B. f b \<ge> f a" |
|
2280 |
shows "SUPR A f \<le> SUPR B f" |
|
2281 |
unfolding SUPR_def apply(rule Sup_mono) |
|
2282 |
using assms by auto |
|
2283 |
||
2284 |
lemma less_add_Real: |
|
2285 |
fixes x :: real |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2286 |
fixes a b :: pextreal |
38656 | 2287 |
assumes "x \<ge> 0" "a < b" |
2288 |
shows "a + Real x < b + Real x" |
|
2289 |
using assms by (cases a, cases b) auto |
|
2290 |
||
2291 |
lemma le_add_Real: |
|
2292 |
fixes x :: real |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2293 |
fixes a b :: pextreal |
38656 | 2294 |
assumes "x \<ge> 0" "a \<le> b" |
2295 |
shows "a + Real x \<le> b + Real x" |
|
2296 |
using assms by (cases a, cases b) auto |
|
2297 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2298 |
lemma le_imp_less_pextreal: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2299 |
fixes x :: pextreal |
38656 | 2300 |
assumes "x > 0" "a + x \<le> b" "a \<noteq> \<omega>" |
2301 |
shows "a < b" |
|
2302 |
using assms by (cases x, cases a, cases b) auto |
|
2303 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2304 |
lemma pextreal_INF_minus: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2305 |
fixes f :: "nat \<Rightarrow> pextreal" |
38656 | 2306 |
assumes "c \<noteq> \<omega>" |
2307 |
shows "(INF i. c - f i) = c - (SUP i. f i)" |
|
2308 |
proof (cases "SUP i. f i") |
|
2309 |
case infinite |
|
2310 |
from `c \<noteq> \<omega>` obtain x where [simp]: "c = Real x" by (cases c) auto |
|
2311 |
from SUP_\<omega>_imp[OF infinite] obtain i where "Real x < f i" by auto |
|
2312 |
have "(INF i. c - f i) \<le> c - f i" |
|
2313 |
by (auto intro!: complete_lattice_class.INF_leI) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2314 |
also have "\<dots> = 0" using `Real x < f i` by (auto simp: minus_pextreal_eq) |
38656 | 2315 |
finally show ?thesis using infinite by auto |
2316 |
next |
|
2317 |
case (preal r) |
|
2318 |
from `c \<noteq> \<omega>` obtain x where c: "c = Real x" by (cases c) auto |
|
2319 |
||
2320 |
show ?thesis unfolding c |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2321 |
proof (rule pextreal_INFI) |
38656 | 2322 |
fix i have "f i \<le> (SUP i. f i)" by (rule le_SUPI) simp |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2323 |
thus "Real x - (SUP i. f i) \<le> Real x - f i" by (rule pextreal_minus_le_cancel) |
38656 | 2324 |
next |
2325 |
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> y \<le> Real x - f i" |
|
2326 |
from this[of 0] obtain p where p: "y = Real p" "0 \<le> p" |
|
2327 |
by (cases "f 0", cases y, auto split: split_if_asm) |
|
2328 |
hence "\<And>i. Real p \<le> Real x - f i" using * by auto |
|
2329 |
hence *: "\<And>i. Real x \<le> f i \<Longrightarrow> Real p = 0" |
|
2330 |
"\<And>i. f i < Real x \<Longrightarrow> Real p + f i \<le> Real x" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2331 |
unfolding pextreal_le_minus_iff by auto |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2332 |
show "y \<le> Real x - (SUP i. f i)" unfolding p pextreal_le_minus_iff |
38656 | 2333 |
proof safe |
2334 |
assume x_less: "Real x \<le> (SUP i. f i)" |
|
2335 |
show "Real p = 0" |
|
2336 |
proof (rule ccontr) |
|
2337 |
assume "Real p \<noteq> 0" |
|
2338 |
hence "0 < Real p" by auto |
|
2339 |
from Sup_close[OF this, of "range f"] |
|
2340 |
obtain i where e: "(SUP i. f i) < f i + Real p" |
|
2341 |
using preal unfolding SUPR_def by auto |
|
2342 |
hence "Real x \<le> f i + Real p" using x_less by auto |
|
2343 |
show False |
|
2344 |
proof cases |
|
2345 |
assume "\<forall>i. f i < Real x" |
|
2346 |
hence "Real p + f i \<le> Real x" using * by auto |
|
2347 |
hence "f i + Real p \<le> (SUP i. f i)" using x_less by (auto simp: field_simps) |
|
2348 |
thus False using e by auto |
|
2349 |
next |
|
2350 |
assume "\<not> (\<forall>i. f i < Real x)" |
|
2351 |
then obtain i where "Real x \<le> f i" by (auto simp: not_less) |
|
2352 |
from *(1)[OF this] show False using `Real p \<noteq> 0` by auto |
|
2353 |
qed |
|
2354 |
qed |
|
2355 |
next |
|
2356 |
have "\<And>i. f i \<le> (SUP i. f i)" by (rule complete_lattice_class.le_SUPI) auto |
|
2357 |
also assume "(SUP i. f i) < Real x" |
|
2358 |
finally have "\<And>i. f i < Real x" by auto |
|
2359 |
hence *: "\<And>i. Real p + f i \<le> Real x" using * by auto |
|
2360 |
have "Real p \<le> Real x" using *[of 0] by (cases "f 0") (auto split: split_if_asm) |
|
2361 |
||
2362 |
have SUP_eq: "(SUP i. f i) \<le> Real x - Real p" |
|
2363 |
proof (rule SUP_leI) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2364 |
fix i show "f i \<le> Real x - Real p" unfolding pextreal_le_minus_iff |
38656 | 2365 |
proof safe |
2366 |
assume "Real x \<le> Real p" |
|
2367 |
with *[of i] show "f i = 0" |
|
2368 |
by (cases "f i") (auto split: split_if_asm) |
|
2369 |
next |
|
2370 |
assume "Real p < Real x" |
|
2371 |
show "f i + Real p \<le> Real x" using * by (auto simp: field_simps) |
|
2372 |
qed |
|
2373 |
qed |
|
2374 |
||
2375 |
show "Real p + (SUP i. f i) \<le> Real x" |
|
2376 |
proof cases |
|
2377 |
assume "Real x \<le> Real p" |
|
2378 |
with `Real p \<le> Real x` have [simp]: "Real p = Real x" by (rule antisym) |
|
2379 |
{ fix i have "f i = 0" using *[of i] by (cases "f i") (auto split: split_if_asm) } |
|
2380 |
hence "(SUP i. f i) \<le> 0" by (auto intro!: SUP_leI) |
|
2381 |
thus ?thesis by simp |
|
2382 |
next |
|
2383 |
assume "\<not> Real x \<le> Real p" hence "Real p < Real x" unfolding not_le . |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2384 |
with SUP_eq show ?thesis unfolding pextreal_le_minus_iff by (auto simp: field_simps) |
38656 | 2385 |
qed |
2386 |
qed |
|
2387 |
qed |
|
2388 |
qed |
|
2389 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2390 |
lemma pextreal_SUP_minus: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2391 |
fixes f :: "nat \<Rightarrow> pextreal" |
38656 | 2392 |
shows "(SUP i. c - f i) = c - (INF i. f i)" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2393 |
proof (rule pextreal_SUPI) |
38656 | 2394 |
fix i have "(INF i. f i) \<le> f i" by (rule INF_leI) simp |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2395 |
thus "c - f i \<le> c - (INF i. f i)" by (rule pextreal_minus_le_cancel) |
38656 | 2396 |
next |
2397 |
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c - f i \<le> y" |
|
2398 |
show "c - (INF i. f i) \<le> y" |
|
2399 |
proof (cases y) |
|
2400 |
case (preal p) |
|
2401 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2402 |
show ?thesis unfolding pextreal_minus_le_iff preal |
38656 | 2403 |
proof safe |
2404 |
assume INF_le_x: "(INF i. f i) \<le> c" |
|
2405 |
from * have *: "\<And>i. f i \<le> c \<Longrightarrow> c \<le> Real p + f i" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2406 |
unfolding pextreal_minus_le_iff preal by auto |
38656 | 2407 |
|
2408 |
have INF_eq: "c - Real p \<le> (INF i. f i)" |
|
2409 |
proof (rule le_INFI) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2410 |
fix i show "c - Real p \<le> f i" unfolding pextreal_minus_le_iff |
38656 | 2411 |
proof safe |
2412 |
assume "Real p \<le> c" |
|
2413 |
show "c \<le> f i + Real p" |
|
2414 |
proof cases |
|
2415 |
assume "f i \<le> c" from *[OF this] |
|
2416 |
show ?thesis by (simp add: field_simps) |
|
2417 |
next |
|
2418 |
assume "\<not> f i \<le> c" |
|
2419 |
hence "c \<le> f i" by auto |
|
2420 |
also have "\<dots> \<le> f i + Real p" by auto |
|
2421 |
finally show ?thesis . |
|
2422 |
qed |
|
2423 |
qed |
|
2424 |
qed |
|
2425 |
||
2426 |
show "c \<le> Real p + (INF i. f i)" |
|
2427 |
proof cases |
|
2428 |
assume "Real p \<le> c" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2429 |
with INF_eq show ?thesis unfolding pextreal_minus_le_iff by (auto simp: field_simps) |
38656 | 2430 |
next |
2431 |
assume "\<not> Real p \<le> c" |
|
2432 |
hence "c \<le> Real p" by auto |
|
2433 |
also have "Real p \<le> Real p + (INF i. f i)" by auto |
|
2434 |
finally show ?thesis . |
|
2435 |
qed |
|
2436 |
qed |
|
2437 |
qed simp |
|
2438 |
qed |
|
2439 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2440 |
lemma pextreal_le_minus_imp_0: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2441 |
fixes a b :: pextreal |
38656 | 2442 |
shows "a \<le> a - b \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a \<noteq> \<omega> \<Longrightarrow> b = 0" |
2443 |
by (cases a, cases b, auto split: split_if_asm) |
|
2444 |
||
2445 |
lemma lim_INF_eq_lim_SUP: |
|
2446 |
fixes X :: "nat \<Rightarrow> real" |
|
2447 |
assumes "\<And>i. 0 \<le> X i" and "0 \<le> x" |
|
2448 |
and lim_INF: "(SUP n. INF m. Real (X (n + m))) = Real x" (is "(SUP n. ?INF n) = _") |
|
2449 |
and lim_SUP: "(INF n. SUP m. Real (X (n + m))) = Real x" (is "(INF n. ?SUP n) = _") |
|
2450 |
shows "X ----> x" |
|
2451 |
proof (rule LIMSEQ_I) |
|
2452 |
fix r :: real assume "0 < r" |
|
2453 |
hence "0 \<le> r" by auto |
|
2454 |
from Sup_close[of "Real r" "range ?INF"] |
|
2455 |
obtain n where inf: "Real x < ?INF n + Real r" |
|
2456 |
unfolding SUPR_def lim_INF[unfolded SUPR_def] using `0 < r` by auto |
|
2457 |
||
2458 |
from Inf_close[of "range ?SUP" "Real r"] |
|
2459 |
obtain n' where sup: "?SUP n' < Real x + Real r" |
|
2460 |
unfolding INFI_def lim_SUP[unfolded INFI_def] using `0 < r` by auto |
|
2461 |
||
2462 |
show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r" |
|
2463 |
proof (safe intro!: exI[of _ "max n n'"]) |
|
2464 |
fix m assume "max n n' \<le> m" hence "n \<le> m" "n' \<le> m" by auto |
|
2465 |
||
2466 |
note inf |
|
2467 |
also have "?INF n + Real r \<le> Real (X (n + (m - n))) + Real r" |
|
2468 |
by (rule le_add_Real, auto simp: `0 \<le> r` intro: INF_leI) |
|
2469 |
finally have up: "x < X m + r" |
|
2470 |
using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n \<le> m` by auto |
|
2471 |
||
2472 |
have "Real (X (n' + (m - n'))) \<le> ?SUP n'" |
|
2473 |
by (auto simp: `0 \<le> r` intro: le_SUPI) |
|
2474 |
also note sup |
|
2475 |
finally have down: "X m < x + r" |
|
2476 |
using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n' \<le> m` by auto |
|
2477 |
||
2478 |
show "norm (X m - x) < r" using up down by auto |
|
2479 |
qed |
|
2480 |
qed |
|
2481 |
||
2482 |
lemma Sup_countable_SUPR: |
|
2483 |
assumes "Sup A \<noteq> \<omega>" "A \<noteq> {}" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2484 |
shows "\<exists> f::nat \<Rightarrow> pextreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f" |
38656 | 2485 |
proof - |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2486 |
have "\<And>n. 0 < 1 / (of_nat n :: pextreal)" by auto |
38656 | 2487 |
from Sup_close[OF this assms] |
2488 |
have "\<forall>n. \<exists>x. x \<in> A \<and> Sup A < x + 1 / of_nat n" by blast |
|
2489 |
from choice[OF this] obtain f where "range f \<subseteq> A" and |
|
2490 |
epsilon: "\<And>n. Sup A < f n + 1 / of_nat n" by blast |
|
2491 |
have "SUPR UNIV f = Sup A" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2492 |
proof (rule pextreal_SUPI) |
38656 | 2493 |
fix i show "f i \<le> Sup A" using `range f \<subseteq> A` |
2494 |
by (auto intro!: complete_lattice_class.Sup_upper) |
|
2495 |
next |
|
2496 |
fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y" |
|
2497 |
show "Sup A \<le> y" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2498 |
proof (rule pextreal_le_epsilon) |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2499 |
fix e :: pextreal assume "0 < e" |
38656 | 2500 |
show "Sup A \<le> y + e" |
2501 |
proof (cases e) |
|
2502 |
case (preal r) |
|
2503 |
hence "0 < r" using `0 < e` by auto |
|
2504 |
then obtain n where *: "inverse (of_nat n) < r" "0 < n" |
|
2505 |
using ex_inverse_of_nat_less by auto |
|
2506 |
have "Sup A \<le> f n + 1 / of_nat n" using epsilon[of n] by auto |
|
2507 |
also have "1 / of_nat n \<le> e" using preal * by (auto simp: real_eq_of_nat) |
|
2508 |
with bound have "f n + 1 / of_nat n \<le> y + e" by (rule add_mono) simp |
|
2509 |
finally show "Sup A \<le> y + e" . |
|
2510 |
qed simp |
|
2511 |
qed |
|
2512 |
qed |
|
2513 |
with `range f \<subseteq> A` show ?thesis by (auto intro!: exI[of _ f]) |
|
2514 |
qed |
|
2515 |
||
2516 |
lemma SUPR_countable_SUPR: |
|
2517 |
assumes "SUPR A g \<noteq> \<omega>" "A \<noteq> {}" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2518 |
shows "\<exists> f::nat \<Rightarrow> pextreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f" |
38656 | 2519 |
proof - |
2520 |
have "Sup (g`A) \<noteq> \<omega>" "g`A \<noteq> {}" using assms unfolding SUPR_def by auto |
|
2521 |
from Sup_countable_SUPR[OF this] |
|
2522 |
show ?thesis unfolding SUPR_def . |
|
2523 |
qed |
|
2524 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2525 |
lemma pextreal_setsum_subtractf: |
38656 | 2526 |
assumes "\<And>i. i\<in>A \<Longrightarrow> g i \<le> f i" and "\<And>i. i\<in>A \<Longrightarrow> f i \<noteq> \<omega>" |
2527 |
shows "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)" |
|
2528 |
proof cases |
|
2529 |
assume "finite A" from this assms show ?thesis |
|
2530 |
proof induct |
|
2531 |
case (insert x A) |
|
2532 |
hence hyp: "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)" |
|
2533 |
by auto |
|
2534 |
{ fix i assume *: "i \<in> insert x A" |
|
2535 |
hence "g i \<le> f i" using insert by simp |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2536 |
also have "f i < \<omega>" using * insert by (simp add: pextreal_less_\<omega>) |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2537 |
finally have "g i \<noteq> \<omega>" by (simp add: pextreal_less_\<omega>) } |
38656 | 2538 |
hence "setsum g A \<noteq> \<omega>" "g x \<noteq> \<omega>" by (auto simp: setsum_\<omega>) |
2539 |
moreover have "setsum f A \<noteq> \<omega>" "f x \<noteq> \<omega>" using insert by (auto simp: setsum_\<omega>) |
|
2540 |
moreover have "g x \<le> f x" using insert by auto |
|
2541 |
moreover have "(\<Sum>i\<in>A. g i) \<le> (\<Sum>i\<in>A. f i)" using insert by (auto intro!: setsum_mono) |
|
2542 |
ultimately show ?case using `finite A` `x \<notin> A` hyp |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2543 |
by (auto simp: pextreal_noteq_omega_Ex) |
38656 | 2544 |
qed simp |
2545 |
qed simp |
|
2546 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2547 |
lemma real_of_pextreal_diff: |
38656 | 2548 |
"y \<le> x \<Longrightarrow> x \<noteq> \<omega> \<Longrightarrow> real x - real y = real (x - y)" |
2549 |
by (cases x, cases y) auto |
|
2550 |
||
2551 |
lemma psuminf_minus: |
|
2552 |
assumes ord: "\<And>i. g i \<le> f i" and fin: "psuminf g \<noteq> \<omega>" "psuminf f \<noteq> \<omega>" |
|
2553 |
shows "(\<Sum>\<^isub>\<infinity> i. f i - g i) = psuminf f - psuminf g" |
|
2554 |
proof - |
|
2555 |
have [simp]: "\<And>i. f i \<noteq> \<omega>" using fin by (auto intro: psuminf_\<omega>) |
|
2556 |
from fin have "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity>x. f x)" |
|
2557 |
and "(\<lambda>x. real (g x)) sums real (\<Sum>\<^isub>\<infinity>x. g x)" |
|
2558 |
by (auto intro: psuminf_imp_suminf) |
|
2559 |
from sums_diff[OF this] |
|
2560 |
have "(\<lambda>n. real (f n - g n)) sums (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))" using fin ord |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2561 |
by (subst (asm) (1 2) real_of_pextreal_diff) (auto simp: psuminf_\<omega> psuminf_le) |
38656 | 2562 |
hence "(\<Sum>\<^isub>\<infinity> i. Real (real (f i - g i))) = Real (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))" |
2563 |
by (rule suminf_imp_psuminf) simp |
|
2564 |
thus ?thesis using fin by (simp add: Real_real psuminf_\<omega>) |
|
2565 |
qed |
|
2566 |
||
2567 |
lemma INF_eq_LIMSEQ: |
|
2568 |
assumes "mono (\<lambda>i. - f i)" and "\<And>n. 0 \<le> f n" and "0 \<le> x" |
|
2569 |
shows "(INF n. Real (f n)) = Real x \<longleftrightarrow> f ----> x" |
|
2570 |
proof |
|
2571 |
assume x: "(INF n. Real (f n)) = Real x" |
|
2572 |
{ fix n |
|
2573 |
have "Real x \<le> Real (f n)" using x[symmetric] by (auto intro: INF_leI) |
|
2574 |
hence "x \<le> f n" using assms by simp |
|
2575 |
hence "\<bar>f n - x\<bar> = f n - x" by auto } |
|
2576 |
note abs_eq = this |
|
2577 |
show "f ----> x" |
|
2578 |
proof (rule LIMSEQ_I) |
|
2579 |
fix r :: real assume "0 < r" |
|
2580 |
show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r" |
|
2581 |
proof (rule ccontr) |
|
2582 |
assume *: "\<not> ?thesis" |
|
2583 |
{ fix N |
|
2584 |
from * obtain n where *: "N \<le> n" "r \<le> f n - x" |
|
2585 |
using abs_eq by (auto simp: not_less) |
|
2586 |
hence "x + r \<le> f n" by auto |
|
2587 |
also have "f n \<le> f N" using `mono (\<lambda>i. - f i)` * by (auto dest: monoD) |
|
2588 |
finally have "Real (x + r) \<le> Real (f N)" using `0 \<le> f N` by auto } |
|
2589 |
hence "Real x < Real (x + r)" |
|
2590 |
and "Real (x + r) \<le> (INF n. Real (f n))" using `0 < r` `0 \<le> x` by (auto intro: le_INFI) |
|
2591 |
hence "Real x < (INF n. Real (f n))" by (rule less_le_trans) |
|
2592 |
thus False using x by auto |
|
2593 |
qed |
|
2594 |
qed |
|
2595 |
next |
|
2596 |
assume "f ----> x" |
|
2597 |
show "(INF n. Real (f n)) = Real x" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2598 |
proof (rule pextreal_INFI) |
38656 | 2599 |
fix n |
2600 |
from decseq_le[OF _ `f ----> x`] assms |
|
2601 |
show "Real x \<le> Real (f n)" unfolding decseq_eq_incseq incseq_mono by auto |
|
2602 |
next |
|
2603 |
fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> y \<le> Real (f n)" |
|
2604 |
thus "y \<le> Real x" |
|
2605 |
proof (cases y) |
|
2606 |
case (preal r) |
|
2607 |
with * have "\<exists>N. \<forall>n\<ge>N. r \<le> f n" using assms by fastsimp |
|
2608 |
from LIMSEQ_le_const[OF `f ----> x` this] |
|
2609 |
show "y \<le> Real x" using `0 \<le> x` preal by auto |
|
2610 |
qed simp |
|
2611 |
qed |
|
2612 |
qed |
|
2613 |
||
2614 |
lemma INFI_bound: |
|
2615 |
assumes "\<forall>N. x \<le> f N" |
|
2616 |
shows "x \<le> (INF n. f n)" |
|
2617 |
using assms by (simp add: INFI_def le_Inf_iff) |
|
2618 |
||
2619 |
lemma LIMSEQ_imp_lim_INF: |
|
2620 |
assumes pos: "\<And>i. 0 \<le> X i" and lim: "X ----> x" |
|
2621 |
shows "(SUP n. INF m. Real (X (n + m))) = Real x" |
|
2622 |
proof - |
|
2623 |
have "0 \<le> x" using assms by (auto intro!: LIMSEQ_le_const) |
|
2624 |
||
2625 |
have "\<And>n. (INF m. Real (X (n + m))) \<le> Real (X (n + 0))" by (rule INF_leI) simp |
|
2626 |
also have "\<And>n. Real (X (n + 0)) < \<omega>" by simp |
|
2627 |
finally have "\<forall>n. \<exists>r\<ge>0. (INF m. Real (X (n + m))) = Real r" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2628 |
by (auto simp: pextreal_less_\<omega> pextreal_noteq_omega_Ex) |
38656 | 2629 |
from choice[OF this] obtain r where r: "\<And>n. (INF m. Real (X (n + m))) = Real (r n)" "\<And>n. 0 \<le> r n" |
2630 |
by auto |
|
2631 |
||
2632 |
show ?thesis unfolding r |
|
2633 |
proof (subst SUP_eq_LIMSEQ) |
|
2634 |
show "mono r" unfolding mono_def |
|
2635 |
proof safe |
|
2636 |
fix x y :: nat assume "x \<le> y" |
|
2637 |
have "Real (r x) \<le> Real (r y)" unfolding r(1)[symmetric] using pos |
|
38705 | 2638 |
proof (safe intro!: INF_mono bexI) |
38656 | 2639 |
fix m have "x + (m + y - x) = y + m" |
2640 |
using `x \<le> y` by auto |
|
2641 |
thus "Real (X (x + (m + y - x))) \<le> Real (X (y + m))" by simp |
|
38705 | 2642 |
qed simp |
38656 | 2643 |
thus "r x \<le> r y" using r by auto |
2644 |
qed |
|
2645 |
show "\<And>n. 0 \<le> r n" by fact |
|
2646 |
show "0 \<le> x" by fact |
|
2647 |
show "r ----> x" |
|
2648 |
proof (rule LIMSEQ_I) |
|
2649 |
fix e :: real assume "0 < e" |
|
2650 |
hence "0 < e/2" by auto |
|
2651 |
from LIMSEQ_D[OF lim this] obtain N where *: "\<And>n. N \<le> n \<Longrightarrow> \<bar>X n - x\<bar> < e/2" |
|
2652 |
by auto |
|
2653 |
show "\<exists>N. \<forall>n\<ge>N. norm (r n - x) < e" |
|
2654 |
proof (safe intro!: exI[of _ N]) |
|
2655 |
fix n assume "N \<le> n" |
|
2656 |
show "norm (r n - x) < e" |
|
2657 |
proof cases |
|
2658 |
assume "r n < x" |
|
2659 |
have "x - r n \<le> e/2" |
|
2660 |
proof cases |
|
2661 |
assume e: "e/2 \<le> x" |
|
2662 |
have "Real (x - e/2) \<le> Real (r n)" unfolding r(1)[symmetric] |
|
2663 |
proof (rule le_INFI) |
|
2664 |
fix m show "Real (x - e / 2) \<le> Real (X (n + m))" |
|
2665 |
using *[of "n + m"] `N \<le> n` |
|
2666 |
using pos by (auto simp: field_simps abs_real_def split: split_if_asm) |
|
2667 |
qed |
|
2668 |
with e show ?thesis using pos `0 \<le> x` r(2) by auto |
|
2669 |
next |
|
2670 |
assume "\<not> e/2 \<le> x" hence "x - e/2 < 0" by auto |
|
2671 |
with `0 \<le> r n` show ?thesis by auto |
|
2672 |
qed |
|
2673 |
with `r n < x` show ?thesis by simp |
|
2674 |
next |
|
2675 |
assume e: "\<not> r n < x" |
|
2676 |
have "Real (r n) \<le> Real (X (n + 0))" unfolding r(1)[symmetric] |
|
2677 |
by (rule INF_leI) simp |
|
2678 |
hence "r n - x \<le> X n - x" using r pos by auto |
|
2679 |
also have "\<dots> < e/2" using *[OF `N \<le> n`] by (auto simp: field_simps abs_real_def split: split_if_asm) |
|
2680 |
finally have "r n - x < e" using `0 < e` by auto |
|
2681 |
with e show ?thesis by auto |
|
2682 |
qed |
|
2683 |
qed |
|
2684 |
qed |
|
2685 |
qed |
|
2686 |
qed |
|
2687 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2688 |
lemma real_of_pextreal_strict_mono_iff: |
38656 | 2689 |
"real a < real b \<longleftrightarrow> (b \<noteq> \<omega> \<and> ((a = \<omega> \<and> 0 < b) \<or> (a < b)))" |
2690 |
proof (cases a) |
|
2691 |
case infinite thus ?thesis by (cases b) auto |
|
2692 |
next |
|
2693 |
case preal thus ?thesis by (cases b) auto |
|
2694 |
qed |
|
2695 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2696 |
lemma real_of_pextreal_mono_iff: |
38656 | 2697 |
"real a \<le> real b \<longleftrightarrow> (a = \<omega> \<or> (b \<noteq> \<omega> \<and> a \<le> b) \<or> (b = \<omega> \<and> a = 0))" |
2698 |
proof (cases a) |
|
2699 |
case infinite thus ?thesis by (cases b) auto |
|
2700 |
next |
|
2701 |
case preal thus ?thesis by (cases b) auto |
|
2702 |
qed |
|
2703 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2704 |
lemma ex_pextreal_inverse_of_nat_Suc_less: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2705 |
fixes e :: pextreal assumes "0 < e" shows "\<exists>n. inverse (of_nat (Suc n)) < e" |
38656 | 2706 |
proof (cases e) |
2707 |
case (preal r) |
|
2708 |
with `0 < e` ex_inverse_of_nat_Suc_less[of r] |
|
2709 |
obtain n where "inverse (of_nat (Suc n)) < r" by auto |
|
2710 |
with preal show ?thesis |
|
2711 |
by (auto simp: real_eq_of_nat[symmetric]) |
|
2712 |
qed auto |
|
2713 |
||
2714 |
lemma Lim_eq_Sup_mono: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2715 |
fixes u :: "nat \<Rightarrow> pextreal" assumes "mono u" |
38656 | 2716 |
shows "u ----> (SUP i. u i)" |
2717 |
proof - |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2718 |
from lim_pextreal_increasing[of u] `mono u` |
38656 | 2719 |
obtain l where l: "u ----> l" unfolding mono_def by auto |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2720 |
from SUP_Lim_pextreal[OF _ this] `mono u` |
38656 | 2721 |
have "(SUP i. u i) = l" unfolding mono_def by auto |
2722 |
with l show ?thesis by simp |
|
2723 |
qed |
|
2724 |
||
2725 |
lemma isotone_Lim: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2726 |
fixes x :: pextreal assumes "u \<up> x" |
38656 | 2727 |
shows "u ----> x" (is ?lim) and "mono u" (is ?mono) |
2728 |
proof - |
|
2729 |
show ?mono using assms unfolding mono_iff_le_Suc isoton_def by auto |
|
2730 |
from Lim_eq_Sup_mono[OF this] `u \<up> x` |
|
2731 |
show ?lim unfolding isoton_def by simp |
|
2732 |
qed |
|
2733 |
||
2734 |
lemma isoton_iff_Lim_mono: |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2735 |
fixes u :: "nat \<Rightarrow> pextreal" |
38656 | 2736 |
shows "u \<up> x \<longleftrightarrow> (mono u \<and> u ----> x)" |
2737 |
proof safe |
|
2738 |
assume "mono u" and x: "u ----> x" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2739 |
with SUP_Lim_pextreal[OF _ x] |
38656 | 2740 |
show "u \<up> x" unfolding isoton_def |
2741 |
using `mono u`[unfolded mono_def] |
|
2742 |
using `mono u`[unfolded mono_iff_le_Suc] |
|
2743 |
by auto |
|
2744 |
qed (auto dest: isotone_Lim) |
|
2745 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2746 |
lemma pextreal_inverse_inverse[simp]: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2747 |
fixes x :: pextreal |
38656 | 2748 |
shows "inverse (inverse x) = x" |
2749 |
by (cases x) auto |
|
2750 |
||
2751 |
lemma atLeastAtMost_omega_eq_atLeast: |
|
2752 |
shows "{a .. \<omega>} = {a ..}" |
|
2753 |
by auto |
|
2754 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2755 |
lemma atLeast0AtMost_eq_atMost: "{0 :: pextreal .. a} = {.. a}" by auto |
38656 | 2756 |
|
2757 |
lemma greaterThan_omega_Empty: "{\<omega> <..} = {}" by auto |
|
2758 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2759 |
lemma lessThan_0_Empty: "{..< 0 :: pextreal} = {}" by auto |
38656 | 2760 |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2761 |
lemma real_of_pextreal_inverse[simp]: |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2762 |
fixes X :: pextreal |
40871 | 2763 |
shows "real (inverse X) = 1 / real X" |
2764 |
by (cases X) (auto simp: inverse_eq_divide) |
|
2765 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2766 |
lemma real_of_pextreal_le_0[simp]: "real (X :: pextreal) \<le> 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)" |
40871 | 2767 |
by (cases X) auto |
2768 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2769 |
lemma real_of_pextreal_less_0[simp]: "\<not> (real (X :: pextreal) < 0)" |
40871 | 2770 |
by (cases X) auto |
2771 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2772 |
lemma abs_real_of_pextreal[simp]: "\<bar>real (X :: pextreal)\<bar> = real X" |
40871 | 2773 |
by simp |
2774 |
||
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset
|
2775 |
lemma zero_less_real_of_pextreal: "0 < real (X :: pextreal) \<longleftrightarrow> X \<noteq> 0 \<and> X \<noteq> \<omega>" |
40871 | 2776 |
by (cases X) auto |
2777 |
||
38656 | 2778 |
end |