src/HOL/Complete_Lattice.thy
author haftmann
Wed Sep 16 13:43:05 2009 +0200 (2009-09-16)
changeset 32587 caa5ada96a00
parent 32436 10cd49e0c067
child 32606 b5c3a8a75772
permissions -rw-r--r--
Inter and Union are mere abbreviations for Inf and Sup
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(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
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header {* Complete lattices, with special focus on sets *}
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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 eta-contraction 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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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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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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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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lemma Inf_empty_bool [simp]:
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  "\<Sqinter>{}"
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  unfolding Inf_bool_def by auto
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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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begin
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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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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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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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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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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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  by blast
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lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
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  by blast
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lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
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  by blast
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lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
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  by blast
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lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
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  by blast
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lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
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  by blast
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lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
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  by blast
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lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
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  by blast
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lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
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  by blast
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lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
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  by blast
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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 eta-contraction of body *}
haftmann@32077
   314
haftmann@32135
   315
lemma UNION_eq_Union_image:
haftmann@32135
   316
  "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
haftmann@32587
   317
  by (simp add: SUPR_def SUPR_set_eq [symmetric])
haftmann@32115
   318
haftmann@32115
   319
lemma Union_def:
haftmann@32117
   320
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   321
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   322
haftmann@32115
   323
lemma UNION_def [noatp]:
haftmann@32135
   324
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   325
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   326
  
haftmann@32115
   327
lemma Union_image_eq [simp]:
haftmann@32115
   328
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   329
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   330
  
wenzelm@11979
   331
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979
   332
  by (unfold UNION_def) blast
wenzelm@11979
   333
wenzelm@11979
   334
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
wenzelm@11979
   335
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   336
    @{term b} may be flexible. *}
wenzelm@11979
   337
  by auto
wenzelm@11979
   338
wenzelm@11979
   339
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979
   340
  by (unfold UNION_def) blast
clasohm@923
   341
wenzelm@11979
   342
lemma UN_cong [cong]:
wenzelm@11979
   343
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979
   344
  by (simp add: UNION_def)
wenzelm@11979
   345
berghofe@29691
   346
lemma strong_UN_cong:
berghofe@29691
   347
    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
berghofe@29691
   348
  by (simp add: UNION_def simp_implies_def)
berghofe@29691
   349
haftmann@32077
   350
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
haftmann@32077
   351
  by blast
haftmann@32077
   352
haftmann@32135
   353
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@32135
   354
  by blast
haftmann@32135
   355
haftmann@32135
   356
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135
   357
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135
   358
haftmann@32135
   359
lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   360
  by blast
haftmann@32135
   361
haftmann@32135
   362
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
   363
  by blast
haftmann@32135
   364
haftmann@32135
   365
lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135
   366
  by blast
haftmann@32135
   367
haftmann@32135
   368
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135
   369
  by blast
haftmann@32135
   370
haftmann@32135
   371
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   372
  by blast
haftmann@32135
   373
haftmann@32135
   374
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135
   375
  by auto
haftmann@32135
   376
haftmann@32135
   377
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135
   378
  by blast
haftmann@32135
   379
haftmann@32135
   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)"
haftmann@32135
   381
  by blast
haftmann@32135
   382
haftmann@32135
   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)"
haftmann@32135
   384
  by blast
haftmann@32135
   385
haftmann@32135
   386
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
haftmann@32135
   387
  by blast
haftmann@32135
   388
haftmann@32135
   389
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   390
  by blast
haftmann@32135
   391
haftmann@32135
   392
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135
   393
  by auto
haftmann@32135
   394
haftmann@32135
   395
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135
   396
  by blast
haftmann@32135
   397
haftmann@32135
   398
lemma UNION_empty_conv[simp]:
haftmann@32135
   399
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   400
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   401
by blast+
haftmann@32135
   402
haftmann@32135
   403
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   404
  by blast
haftmann@32135
   405
haftmann@32135
   406
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
   407
  by blast
haftmann@32135
   408
haftmann@32135
   409
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
   410
  by blast
haftmann@32135
   411
haftmann@32135
   412
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   413
  by (auto simp add: split_if_mem2)
haftmann@32135
   414
haftmann@32135
   415
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
haftmann@32135
   416
  by (auto intro: bool_contrapos)
haftmann@32135
   417
haftmann@32135
   418
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   419
  by blast
haftmann@32135
   420
haftmann@32135
   421
lemma UN_mono:
haftmann@32135
   422
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135
   423
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135
   424
  by (blast dest: subsetD)
haftmann@32135
   425
haftmann@32135
   426
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
haftmann@32135
   427
  by blast
haftmann@32135
   428
haftmann@32135
   429
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
haftmann@32135
   430
  by blast
haftmann@32135
   431
haftmann@32135
   432
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
haftmann@32135
   433
  -- {* NOT suitable for rewriting *}
haftmann@32135
   434
  by blast
haftmann@32135
   435
haftmann@32135
   436
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
haftmann@32135
   437
by blast
haftmann@32135
   438
wenzelm@11979
   439
haftmann@32139
   440
subsection {* Inter *}
haftmann@32115
   441
haftmann@32587
   442
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   443
  "Inter S \<equiv> \<Sqinter>S"
haftmann@32135
   444
  
haftmann@32115
   445
notation (xsymbols)
haftmann@32115
   446
  Inter  ("\<Inter>_" [90] 90)
haftmann@32115
   447
haftmann@32135
   448
lemma Inter_eq [code del]:
haftmann@32135
   449
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@32115
   450
proof (rule set_ext)
haftmann@32115
   451
  fix x
haftmann@32135
   452
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
haftmann@32115
   453
    by auto
haftmann@32135
   454
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@32587
   455
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@32115
   456
qed
haftmann@32115
   457
haftmann@32115
   458
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
haftmann@32115
   459
  by (unfold Inter_eq) blast
haftmann@32115
   460
haftmann@32115
   461
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
haftmann@32115
   462
  by (simp add: Inter_eq)
haftmann@32115
   463
haftmann@32115
   464
text {*
haftmann@32115
   465
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@32115
   466
  contains @{term A} as an element, but @{prop "A:X"} can hold when
haftmann@32115
   467
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
haftmann@32115
   468
*}
haftmann@32115
   469
haftmann@32115
   470
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
haftmann@32115
   471
  by auto
haftmann@32115
   472
haftmann@32115
   473
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
haftmann@32115
   474
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@32115
   475
    @{prop "X:C"}. *}
haftmann@32115
   476
  by (unfold Inter_eq) blast
haftmann@32115
   477
haftmann@32135
   478
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
haftmann@32135
   479
  by blast
haftmann@32135
   480
haftmann@32135
   481
lemma Inter_subset:
haftmann@32135
   482
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
haftmann@32135
   483
  by blast
haftmann@32135
   484
haftmann@32135
   485
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
haftmann@32135
   486
  by (iprover intro: InterI subsetI dest: subsetD)
haftmann@32135
   487
haftmann@32135
   488
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@32135
   489
  by blast
haftmann@32135
   490
haftmann@32135
   491
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@32135
   492
  by blast
haftmann@32135
   493
haftmann@32135
   494
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@32135
   495
  by blast
haftmann@32135
   496
haftmann@32135
   497
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@32135
   498
  by blast
haftmann@32135
   499
haftmann@32135
   500
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@32135
   501
  by blast
haftmann@32135
   502
haftmann@32135
   503
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@32135
   504
  by blast
haftmann@32135
   505
haftmann@32135
   506
lemma Inter_UNIV_conv [simp,noatp]:
haftmann@32135
   507
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
haftmann@32135
   508
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
haftmann@32135
   509
  by blast+
haftmann@32135
   510
haftmann@32135
   511
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
haftmann@32135
   512
  by blast
haftmann@32135
   513
haftmann@32115
   514
haftmann@32139
   515
subsection {* Intersections of families *}
wenzelm@11979
   516
haftmann@32081
   517
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32135
   518
  INFI_set_eq [symmetric]: "INTER S f = (INF x:S. f x)"
haftmann@32081
   519
haftmann@32081
   520
syntax
haftmann@32081
   521
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@32081
   522
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
haftmann@32081
   523
haftmann@32081
   524
syntax (xsymbols)
haftmann@32081
   525
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@32081
   526
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
haftmann@32081
   527
haftmann@32081
   528
syntax (latex output)
haftmann@32081
   529
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@32081
   530
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
haftmann@32081
   531
haftmann@32081
   532
translations
haftmann@32081
   533
  "INT x y. B"  == "INT x. INT y. B"
haftmann@32081
   534
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@32081
   535
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@32081
   536
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@32081
   537
haftmann@32120
   538
print_translation {* [
haftmann@32120
   539
Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} "@INTER"
haftmann@32120
   540
] *} -- {* to avoid eta-contraction of body *}
haftmann@32081
   541
haftmann@32135
   542
lemma INTER_eq_Inter_image:
haftmann@32135
   543
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@32587
   544
  by (simp add: INFI_def INFI_set_eq [symmetric])
haftmann@32135
   545
  
haftmann@32115
   546
lemma Inter_def:
haftmann@32135
   547
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@32115
   548
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@32115
   549
haftmann@32115
   550
lemma INTER_def:
haftmann@32135
   551
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@32117
   552
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@32115
   553
haftmann@32115
   554
lemma Inter_image_eq [simp]:
haftmann@32115
   555
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@32115
   556
  by (rule sym) (fact INTER_eq_Inter_image)
haftmann@32115
   557
wenzelm@11979
   558
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
wenzelm@11979
   559
  by (unfold INTER_def) blast
clasohm@923
   560
wenzelm@11979
   561
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
wenzelm@11979
   562
  by (unfold INTER_def) blast
wenzelm@11979
   563
wenzelm@11979
   564
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
wenzelm@11979
   565
  by auto
wenzelm@11979
   566
wenzelm@11979
   567
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
wenzelm@11979
   568
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
wenzelm@11979
   569
  by (unfold INTER_def) blast
wenzelm@11979
   570
wenzelm@11979
   571
lemma INT_cong [cong]:
wenzelm@11979
   572
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
wenzelm@11979
   573
  by (simp add: INTER_def)
wenzelm@7238
   574
haftmann@32135
   575
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@30531
   576
  by blast
haftmann@30531
   577
haftmann@32135
   578
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
wenzelm@12897
   579
  by blast
wenzelm@12897
   580
wenzelm@12897
   581
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
wenzelm@12897
   582
  by blast
wenzelm@12897
   583
wenzelm@12897
   584
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
nipkow@17589
   585
  by (iprover intro: INT_I subsetI dest: subsetD)
wenzelm@12897
   586
wenzelm@12897
   587
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
wenzelm@12897
   588
  by blast
wenzelm@12897
   589
wenzelm@12897
   590
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
wenzelm@12897
   591
  by blast
wenzelm@12897
   592
wenzelm@12897
   593
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
wenzelm@12897
   594
  by blast
wenzelm@12897
   595
wenzelm@12897
   596
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
wenzelm@12897
   597
  by blast
wenzelm@12897
   598
wenzelm@12897
   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)"
wenzelm@12897
   600
  by blast
wenzelm@12897
   601
wenzelm@12897
   602
lemma INT_insert_distrib:
wenzelm@12897
   603
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
wenzelm@12897
   604
  by blast
wenzelm@12897
   605
wenzelm@12897
   606
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
wenzelm@12897
   607
  by auto
wenzelm@12897
   608
wenzelm@12897
   609
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
wenzelm@12897
   610
  -- {* Look: it has an \emph{existential} quantifier *}
wenzelm@12897
   611
  by blast
wenzelm@12897
   612
paulson@18447
   613
lemma INTER_UNIV_conv[simp]:
nipkow@13653
   614
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
   615
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
   616
by blast+
wenzelm@12897
   617
haftmann@32135
   618
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
haftmann@32135
   619
  by (auto intro: bool_induct)
haftmann@32135
   620
haftmann@32135
   621
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@32135
   622
  by blast
haftmann@32135
   623
haftmann@32135
   624
lemma INT_anti_mono:
haftmann@32135
   625
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135
   626
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@32135
   627
  -- {* The last inclusion is POSITIVE! *}
haftmann@32135
   628
  by (blast dest: subsetD)
haftmann@32135
   629
haftmann@32135
   630
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
haftmann@32135
   631
  by blast
haftmann@32135
   632
haftmann@32135
   633
haftmann@32139
   634
subsection {* Distributive laws *}
wenzelm@12897
   635
wenzelm@12897
   636
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   637
  by blast
wenzelm@12897
   638
wenzelm@12897
   639
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   640
  by blast
wenzelm@12897
   641
wenzelm@12897
   642
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
wenzelm@12897
   643
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   644
  -- {* Union of a family of unions *}
wenzelm@12897
   645
  by blast
wenzelm@12897
   646
wenzelm@12897
   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)"
wenzelm@12897
   648
  -- {* Equivalent version *}
wenzelm@12897
   649
  by blast
wenzelm@12897
   650
wenzelm@12897
   651
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   652
  by blast
wenzelm@12897
   653
wenzelm@12897
   654
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897
   655
  by blast
wenzelm@12897
   656
wenzelm@12897
   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)"
wenzelm@12897
   658
  -- {* Equivalent version *}
wenzelm@12897
   659
  by blast
wenzelm@12897
   660
wenzelm@12897
   661
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   662
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   663
  by blast
wenzelm@12897
   664
wenzelm@12897
   665
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   666
  by blast
wenzelm@12897
   667
wenzelm@12897
   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)"
wenzelm@12897
   669
  by blast
wenzelm@12897
   670
wenzelm@12897
   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)"
wenzelm@12897
   672
  by blast
wenzelm@12897
   673
wenzelm@12897
   674
haftmann@32139
   675
subsection {* Complement *}
haftmann@32135
   676
haftmann@32135
   677
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
   678
  by blast
wenzelm@12897
   679
haftmann@32135
   680
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
   681
  by blast
wenzelm@12897
   682
wenzelm@12897
   683
haftmann@32139
   684
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   685
paulson@13860
   686
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   687
           and Intersections. *}
wenzelm@12897
   688
wenzelm@12897
   689
lemma UN_simps [simp]:
wenzelm@12897
   690
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897
   691
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897
   692
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897
   693
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897
   694
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897
   695
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897
   696
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897
   697
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897
   698
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897
   699
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
   700
  by auto
wenzelm@12897
   701
wenzelm@12897
   702
lemma INT_simps [simp]:
wenzelm@12897
   703
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897
   704
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897
   705
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897
   706
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897
   707
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897
   708
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897
   709
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897
   710
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897
   711
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897
   712
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
   713
  by auto
wenzelm@12897
   714
paulson@24286
   715
lemma ball_simps [simp,noatp]:
wenzelm@12897
   716
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897
   717
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897
   718
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897
   719
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897
   720
  "!!P. (ALL x:{}. P x) = True"
wenzelm@12897
   721
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897
   722
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897
   723
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897
   724
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897
   725
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897
   726
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897
   727
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897
   728
  by auto
wenzelm@12897
   729
paulson@24286
   730
lemma bex_simps [simp,noatp]:
wenzelm@12897
   731
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897
   732
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897
   733
  "!!P. (EX x:{}. P x) = False"
wenzelm@12897
   734
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897
   735
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897
   736
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897
   737
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897
   738
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897
   739
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897
   740
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897
   741
  by auto
wenzelm@12897
   742
wenzelm@12897
   743
lemma ball_conj_distrib:
wenzelm@12897
   744
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897
   745
  by blast
wenzelm@12897
   746
wenzelm@12897
   747
lemma bex_disj_distrib:
wenzelm@12897
   748
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897
   749
  by blast
wenzelm@12897
   750
wenzelm@12897
   751
paulson@13860
   752
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
   753
paulson@13860
   754
lemma UN_extend_simps:
paulson@13860
   755
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860
   756
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860
   757
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860
   758
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860
   759
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860
   760
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860
   761
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860
   762
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860
   763
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860
   764
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
   765
  by auto
paulson@13860
   766
paulson@13860
   767
lemma INT_extend_simps:
paulson@13860
   768
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860
   769
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860
   770
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860
   771
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860
   772
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860
   773
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860
   774
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860
   775
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860
   776
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860
   777
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
   778
  by auto
paulson@13860
   779
paulson@13860
   780
haftmann@32135
   781
no_notation
haftmann@32135
   782
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
   783
  less (infix "\<sqsubset>" 50) and
haftmann@32135
   784
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
   785
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
   786
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@32135
   787
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
   788
haftmann@30596
   789
lemmas mem_simps =
haftmann@30596
   790
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
   791
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
   792
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
   793
wenzelm@11979
   794
end