author  haftmann 
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permissions  rwrr 
32139  1 
(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *) 
11979  2 

32139  3 
header {* Complete lattices, with special focus on sets *} 
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32139  5 
theory Complete_Lattice 
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imports Set 

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begin 

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notation 
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less_eq (infix "\<sqsubseteq>" 50) and 
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less (infix "\<sqsubset>" 50) and 
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inf (infixl "\<sqinter>" 70) and 
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sup (infixl "\<squnion>" 65) 
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subsection {* Abstract complete lattices *} 

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class complete_lattice = lattice + bot + top + 
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fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) 
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and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) 
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assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" 
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and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" 
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assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" 
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and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" 
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begin 
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}" 
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
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lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}" 
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
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lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}" 
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unfolding Sup_Inf by auto 
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lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}" 
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unfolding Inf_Sup by auto 
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lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" 
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by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) 
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lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" 
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by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) 
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lemma Inf_singleton [simp]: 
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"\<Sqinter>{a} = a" 
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by (auto intro: antisym Inf_lower Inf_greatest) 
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lemma Sup_singleton [simp]: 
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"\<Squnion>{a} = a" 
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by (auto intro: antisym Sup_upper Sup_least) 
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lemma Inf_insert_simp: 
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"\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)" 
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by (cases "A = {}") (simp_all, simp add: Inf_insert) 
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lemma Sup_insert_simp: 
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"\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)" 
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by (cases "A = {}") (simp_all, simp add: Sup_insert) 
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lemma Inf_binary: 
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"\<Sqinter>{a, b} = a \<sqinter> b" 
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by (auto simp add: Inf_insert_simp) 
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lemma Sup_binary: 
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"\<Squnion>{a, b} = a \<squnion> b" 
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by (auto simp add: Sup_insert_simp) 
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lemma bot_def: 
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"bot = \<Squnion>{}" 
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by (auto intro: antisym Sup_least) 
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lemma top_def: 
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"top = \<Sqinter>{}" 
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by (auto intro: antisym Inf_greatest) 
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lemma sup_bot [simp]: 
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"x \<squnion> bot = x" 
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using bot_least [of x] by (simp add: sup_commute) 
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lemma inf_top [simp]: 
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"x \<sqinter> top = x" 
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using top_greatest [of x] by (simp add: inf_commute) 
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
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"SUPR A f = \<Squnion> (f ` A)" 
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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
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"INFI A f = \<Sqinter> (f ` A)" 
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end 
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syntax 
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"_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10) 
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"_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10) 
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"_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10) 
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"_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10) 
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translations 
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"SUP x y. B" == "SUP x. SUP y. B" 
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"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" 
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"SUP x. B" == "SUP x:CONST UNIV. B" 
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"SUP x:A. B" == "CONST SUPR A (%x. B)" 
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"INF x y. B" == "INF x. INF y. B" 
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"INF x. B" == "CONST INFI CONST UNIV (%x. B)" 
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"INF x. B" == "INF x:CONST UNIV. B" 
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"INF x:A. B" == "CONST INFI A (%x. B)" 
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print_translation {* [ 
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Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} "_SUP", 
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Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} "_INF" 
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] *}  {* to avoid etacontraction of body *} 
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context complete_lattice 
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begin 
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lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)" 
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by (auto simp add: SUPR_def intro: Sup_upper) 
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lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u" 
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by (auto simp add: SUPR_def intro: Sup_least) 
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lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i" 
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by (auto simp add: INFI_def intro: Inf_lower) 
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lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)" 
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by (auto simp add: INFI_def intro: Inf_greatest) 
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128 

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lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" 
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by (auto intro: antisym SUP_leI le_SUPI) 
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131 

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lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" 
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by (auto intro: antisym INF_leI le_INFI) 
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end 
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32139  138 
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *} 
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instantiation bool :: complete_lattice 
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begin 
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definition 
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Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" 
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definition 
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Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" 
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instance proof 
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qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) 
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end 
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153 

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lemma Inf_empty_bool [simp]: 
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"\<Sqinter>{}" 
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unfolding Inf_bool_def by auto 
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157 

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lemma not_Sup_empty_bool [simp]: 
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"\<not> \<Squnion>{}" 
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unfolding Sup_bool_def by auto 
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lemma INFI_bool_eq: 
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"INFI = Ball" 
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proof (rule ext)+ 
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fix A :: "'a set" 
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fix P :: "'a \<Rightarrow> bool" 
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show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)" 
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by (auto simp add: Ball_def INFI_def Inf_bool_def) 
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qed 
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lemma SUPR_bool_eq: 
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"SUPR = Bex" 
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proof (rule ext)+ 
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fix A :: "'a set" 
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fix P :: "'a \<Rightarrow> bool" 
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show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)" 
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by (auto simp add: Bex_def SUPR_def Sup_bool_def) 
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qed 
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instantiation "fun" :: (type, complete_lattice) complete_lattice 
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181 
begin 
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182 

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definition 
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Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" 
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185 

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definition 
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Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" 
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instance proof 
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qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def 
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intro: Inf_lower Sup_upper Inf_greatest Sup_least) 
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end 
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194 

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lemma Inf_empty_fun: 
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"\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})" 
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by (simp add: Inf_fun_def) 
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lemma Sup_empty_fun: 
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"\<Squnion>{} = (\<lambda>_. \<Squnion>{})" 
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by (simp add: Sup_fun_def) 
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202 

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32139  204 
subsection {* Union *} 
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abbreviation Union :: "'a set set \<Rightarrow> 'a set" where 
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"Union S \<equiv> \<Squnion>S" 
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notation (xsymbols) 
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Union ("\<Union>_" [90] 90) 
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lemma Union_eq: 
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"\<Union>A = {x. \<exists>B \<in> A. x \<in> B}" 
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proof (rule set_ext) 
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fix x 
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have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)" 
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by auto 
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then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" 
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by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def) 
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qed 
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lemma Union_iff [simp, noatp]: 
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"A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)" 
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by (unfold Union_eq) blast 
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lemma UnionI [intro]: 
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"X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C" 
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 {* The order of the premises presupposes that @{term C} is rigid; 
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@{term A} may be flexible. *} 
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by auto 
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lemma UnionE [elim!]: 
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"A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R" 
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by auto 
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lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A" 
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by (iprover intro: subsetI UnionI) 
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lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" 
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by (iprover intro: subsetI elim: UnionE dest: subsetD) 
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lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}" 
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by blast 
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244 

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lemma Union_empty [simp]: "Union({}) = {}" 
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by blast 
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lemma Union_UNIV [simp]: "Union UNIV = UNIV" 
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lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B" 
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lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B" 
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lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" 
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lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})" 
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lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})" 
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lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})" 
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lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)" 
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lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A" 
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lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B" 
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32139  279 
subsection {* Unions of families *} 
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definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 
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SUPR_set_eq [symmetric]: "UNION S f = (SUP x:S. f x)" 
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syntax 
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"@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) 
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"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10) 
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syntax (xsymbols) 
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"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) 
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"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10) 
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syntax (latex output) 
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"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 
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"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 
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translations 
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"UN x y. B" == "UN x. UN y. B" 
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"UN x. B" == "CONST UNION CONST UNIV (%x. B)" 
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"UN x. B" == "UN x:CONST UNIV. B" 
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"UN x:A. B" == "CONST UNION A (%x. B)" 
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text {* 
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Note the difference between ordinary xsymbol syntax of indexed 
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unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) 
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and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The 
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former does not make the index expression a subscript of the 
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union/intersection symbol because this leads to problems with nested 
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subscripts in Proof General. 
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*} 
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print_translation {* [ 
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Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} "@UNION" 
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] *}  {* to avoid etacontraction of body *} 
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lemma UNION_eq_Union_image: 
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"(\<Union>x\<in>A. B x) = \<Union>(B`A)" 
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by (simp add: SUPR_def SUPR_set_eq [symmetric]) 
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lemma Union_def: 
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"\<Union>S = (\<Union>x\<in>S. x)" 
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by (simp add: UNION_eq_Union_image image_def) 
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lemma UNION_def [noatp]: 
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"(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" 
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by (auto simp add: UNION_eq_Union_image Union_eq) 
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lemma Union_image_eq [simp]: 
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"\<Union>(B`A) = (\<Union>x\<in>A. B x)" 
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by (rule sym) (fact UNION_eq_Union_image) 
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11979  331 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 
332 
by (unfold UNION_def) blast 

333 

334 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

335 
 {* The order of the premises presupposes that @{term A} is rigid; 

336 
@{term b} may be flexible. *} 

337 
by auto 

338 

339 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

340 
by (unfold UNION_def) blast 

923  341 

11979  342 
lemma UN_cong [cong]: 
343 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

344 
by (simp add: UNION_def) 

345 

29691  346 
lemma strong_UN_cong: 
347 
"A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

348 
by (simp add: UNION_def simp_implies_def) 

349 

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lemma image_eq_UN: "f`A = (UN x:A. {f x})" 
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lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)" 
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lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" 
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by (iprover intro: subsetI elim: UN_E dest: subsetD) 
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lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" 
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lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" 
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lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}" 
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lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}" 
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lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" 
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lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)" 
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by auto 
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376 

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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

377 
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

378 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

379 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

380 
lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

381 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

382 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

383 
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

384 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

385 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

386 
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

387 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

388 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

389 
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

390 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

391 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

392 
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

393 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

394 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

395 
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

396 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

397 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

398 
lemma UNION_empty_conv[simp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

399 
"({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

400 
"((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

401 
by blast+ 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

402 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

403 
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

404 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

405 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

406 
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

407 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

408 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

409 
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

410 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

411 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

412 
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

413 
by (auto simp add: split_if_mem2) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

414 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

415 
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

416 
by (auto intro: bool_contrapos) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

417 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

418 
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

419 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

420 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

421 
lemma UN_mono: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

422 
"A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==> 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

423 
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

424 
by (blast dest: subsetD) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

425 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

426 
lemma vimage_Union: "f ` (Union A) = (UN X:A. f ` X)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

427 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

428 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

429 
lemma vimage_UN: "f`(UN x:A. B x) = (UN x:A. f ` B x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

430 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

431 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

432 
lemma vimage_eq_UN: "f`B = (UN y: B. f`{y})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

433 
 {* NOT suitable for rewriting *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

434 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

435 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

436 
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

437 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

438 

11979  439 

32139  440 
subsection {* Inter *} 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

441 

32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

442 
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where 
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

443 
"Inter S \<equiv> \<Sqinter>S" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

444 

32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

445 
notation (xsymbols) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

446 
Inter ("\<Inter>_" [90] 90) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

447 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

448 
lemma Inter_eq [code del]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

449 
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

450 
proof (rule set_ext) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

451 
fix x 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

452 
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

453 
by auto 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

454 
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" 
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

455 
by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

456 
qed 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

457 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

458 
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

459 
by (unfold Inter_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

460 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

461 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

462 
by (simp add: Inter_eq) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

463 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

464 
text {* 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

465 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

466 
contains @{term A} as an element, but @{prop "A:X"} can hold when 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

467 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

468 
*} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

469 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

470 
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

471 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

472 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

473 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

474 
 {* ``Classical'' elimination rule  does not require proving 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

475 
@{prop "X:C"}. *} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

476 
by (unfold Inter_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

477 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

478 
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

479 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

480 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

481 
lemma Inter_subset: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

482 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

483 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

484 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

485 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

486 
by (iprover intro: InterI subsetI dest: subsetD) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

487 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

488 
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

489 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

490 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

491 
lemma Inter_empty [simp]: "\<Inter>{} = UNIV" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

492 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

493 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

494 
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

495 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

496 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

497 
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

498 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

499 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

500 
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

501 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

502 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

503 
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

504 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

505 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

506 
lemma Inter_UNIV_conv [simp,noatp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

507 
"(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

508 
"(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

509 
by blast+ 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

510 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

511 
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

512 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

513 

32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

514 

32139  515 
subsection {* Intersections of families *} 
11979  516 

32081  517 
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

518 
INFI_set_eq [symmetric]: "INTER S f = (INF x:S. f x)" 
32081  519 

520 
syntax 

521 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 

522 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10) 

523 

524 
syntax (xsymbols) 

525 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 

526 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10) 

527 

528 
syntax (latex output) 

529 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

530 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 

531 

532 
translations 

533 
"INT x y. B" == "INT x. INT y. B" 

534 
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" 

535 
"INT x. B" == "INT x:CONST UNIV. B" 

536 
"INT x:A. B" == "CONST INTER A (%x. B)" 

537 

32120
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

538 
print_translation {* [ 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

539 
Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} "@INTER" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

540 
] *}  {* to avoid etacontraction of body *} 
32081  541 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

542 
lemma INTER_eq_Inter_image: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

543 
"(\<Inter>x\<in>A. B x) = \<Inter>(B`A)" 
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

544 
by (simp add: INFI_def INFI_set_eq [symmetric]) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

545 

32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

546 
lemma Inter_def: 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

547 
"\<Inter>S = (\<Inter>x\<in>S. x)" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

548 
by (simp add: INTER_eq_Inter_image image_def) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

549 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

550 
lemma INTER_def: 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

551 
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

552 
by (auto simp add: INTER_eq_Inter_image Inter_eq) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

553 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

554 
lemma Inter_image_eq [simp]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

555 
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

556 
by (rule sym) (fact INTER_eq_Inter_image) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

557 

11979  558 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 
559 
by (unfold INTER_def) blast 

923  560 

11979  561 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
562 
by (unfold INTER_def) blast 

563 

564 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

565 
by auto 

566 

567 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

568 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

569 
by (unfold INTER_def) blast 

570 

571 
lemma INT_cong [cong]: 

572 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

573 
by (simp add: INTER_def) 

7238
36e58620ffc8
replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents:
5931
diff
changeset

574 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

575 
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

576 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

577 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

578 
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

579 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

580 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

581 
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

582 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

583 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

584 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 
17589  585 
by (iprover intro: INT_I subsetI dest: subsetD) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

586 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

587 
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

588 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

589 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

590 
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

591 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

592 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

593 
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

594 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

595 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

596 
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

597 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

598 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

599 
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

600 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

601 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

602 
lemma INT_insert_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

603 
"u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

604 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

605 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

606 
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

607 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

608 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

609 
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

610 
 {* Look: it has an \emph{existential} quantifier *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

611 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

612 

18447  613 
lemma INTER_UNIV_conv[simp]: 
13653  614 
"(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)" 
615 
"((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" 

616 
by blast+ 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

617 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

618 
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

619 
by (auto intro: bool_induct) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

620 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

621 
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

622 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

623 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

624 
lemma INT_anti_mono: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

625 
"B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==> 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

626 
(\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

627 
 {* The last inclusion is POSITIVE! *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

628 
by (blast dest: subsetD) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

629 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

630 
lemma vimage_INT: "f`(INT x:A. B x) = (INT x:A. f ` B x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

631 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

632 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

633 

32139  634 
subsection {* Distributive laws *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

635 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

636 
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

637 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

638 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

639 
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

640 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

641 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

642 
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

643 
 {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

644 
 {* Union of a family of unions *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

645 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

646 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

647 
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

648 
 {* Equivalent version *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

649 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

650 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

651 
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

652 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

653 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

654 
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

655 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

656 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

657 
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

658 
 {* Equivalent version *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

659 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

660 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

661 
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

662 
 {* Halmos, Naive Set Theory, page 35. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

663 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

664 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

665 
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

666 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

667 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

668 
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

669 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

670 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

671 
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

672 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

673 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

674 

32139  675 
subsection {* Complement *} 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

676 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

677 
lemma Compl_UN [simp]: "(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. B x)" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

678 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

679 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

680 
lemma Compl_INT [simp]: "(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. B x)" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

681 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

682 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

683 

32139  684 
subsection {* Miniscoping and maxiscoping *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

685 

13860  686 
text {* \medskip Miniscoping: pushing in quantifiers and big Unions 
687 
and Intersections. *} 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

688 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

689 
lemma UN_simps [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

690 
"!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

691 
"!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

692 
"!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

693 
"!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

694 
"!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

695 
"!!A B C. (UN x:C. A x  B) = ((UN x:C. A x)  B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

696 
"!!A B C. (UN x:C. A  B x) = (A  (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

697 
"!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

698 
"!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

699 
"!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

700 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

701 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

702 
lemma INT_simps [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

703 
"!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

704 
"!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

705 
"!!A B C. (INT x:C. A x  B) = (if C={} then UNIV else (INT x:C. A x)  B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

706 
"!!A B C. (INT x:C. A  B x) = (if C={} then UNIV else A  (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

707 
"!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

708 
"!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

709 
"!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

710 
"!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

711 
"!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

712 
"!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

713 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

714 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

715 
lemma ball_simps [simp,noatp]: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

716 
"!!A P Q. (ALL x:A. P x  Q) = ((ALL x:A. P x)  Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

717 
"!!A P Q. (ALL x:A. P  Q x) = (P  (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

718 
"!!A P Q. (ALL x:A. P > Q x) = (P > (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

719 
"!!A P Q. (ALL x:A. P x > Q) = ((EX x:A. P x) > Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

720 
"!!P. (ALL x:{}. P x) = True" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

721 
"!!P. (ALL x:UNIV. P x) = (ALL x. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

722 
"!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

723 
"!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

724 
"!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

725 
"!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x > P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

726 
"!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

727 
"!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

728 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

729 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

730 
lemma bex_simps [simp,noatp]: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

731 
"!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

732 
"!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

733 
"!!P. (EX x:{}. P x) = False" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

734 
"!!P. (EX x:UNIV. P x) = (EX x. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

735 
"!!a B P. (EX x:insert a B. P x) = (P(a)  (EX x:B. P x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

736 
"!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

737 
"!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

738 
"!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

739 
"!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

740 
"!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

741 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

742 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

743 
lemma ball_conj_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

744 
"(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

745 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

746 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

747 
lemma bex_disj_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

748 
"(EX x:A. P x  Q x) = ((EX x:A. P x)  (EX x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

749 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

750 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

751 

13860  752 
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} 
753 

754 
lemma UN_extend_simps: 

755 
"!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))" 

756 
"!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))" 

757 
"!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))" 

758 
"!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)" 

759 
"!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)" 

760 
"!!A B C. ((UN x:C. A x)  B) = (UN x:C. A x  B)" 

761 
"!!A B C. (A  (INT x:C. B x)) = (UN x:C. A  B x)" 

762 
"!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)" 

763 
"!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)" 

764 
"!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)" 

765 
by auto 

766 

767 
lemma INT_extend_simps: 

768 
"!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))" 

769 
"!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))" 

770 
"!!A B C. (INT x:C. A x)  B = (if C={} then UNIVB else (INT x:C. A x  B))" 

771 
"!!A B C. A  (UN x:C. B x) = (if C={} then A else (INT x:C. A  B x))" 

772 
"!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))" 

773 
"!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)" 

774 
"!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)" 

775 
"!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)" 

776 
"!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)" 

777 
"!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)" 

778 
by auto 

779 

780 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

781 
no_notation 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

782 
less_eq (infix "\<sqsubseteq>" 50) and 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

783 
less (infix "\<sqsubset>" 50) and 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

784 
inf (infixl "\<sqinter>" 70) and 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

785 
sup (infixl "\<squnion>" 65) and 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

786 
Inf ("\<Sqinter>_" [900] 900) and 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

787 
Sup ("\<Squnion>_" [900] 900) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

788 

30596  789 
lemmas mem_simps = 
790 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

791 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

792 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

21669  793 

11979  794 
end 