src/HOL/Complete_Lattice.thy
author wenzelm
Mon Mar 22 20:58:52 2010 +0100 (2010-03-22)
changeset 35898 c890a3835d15
parent 35828 46cfc4b8112e
child 36364 0e2679025aeb
permissions -rw-r--r--
recovered header;
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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) and
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  top ("\<top>") and
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  bot ("\<bottom>")
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subsection {* Syntactic infimum and supremum operations *}
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class Inf =
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  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
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class Sup =
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  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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subsection {* Abstract complete lattices *}
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class complete_lattice = bounded_lattice + Inf + Sup +
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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 dual_complete_lattice:
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  "complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
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  by (auto intro!: complete_lattice.intro dual_bounded_lattice)
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    (unfold_locales, (fact bot_least top_greatest
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        Sup_upper Sup_least Inf_lower Inf_greatest)+)
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> 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 \<sqsubseteq> b}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Inf_empty:
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  "\<Sqinter>{} = \<top>"
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  by (auto intro: antisym Inf_greatest)
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lemma Sup_empty:
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  "\<Squnion>{} = \<bottom>"
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  by (auto intro: antisym Sup_least)
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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_binary:
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  "\<Sqinter>{a, b} = a \<sqinter> b"
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  by (simp add: Inf_empty Inf_insert)
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lemma Sup_binary:
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  "\<Squnion>{a, b} = a \<squnion> b"
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  by (simp add: Sup_empty Sup_insert)
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lemma Inf_UNIV:
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  "\<Sqinter>UNIV = bot"
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  by (simp add: Sup_Inf Sup_empty [symmetric])
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lemma Sup_UNIV:
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  "\<Squnion>UNIV = top"
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  by (simp add: Inf_Sup Inf_empty [symmetric])
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lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
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  by (auto intro: Sup_least dest: Sup_upper)
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lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
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  by (auto intro: Inf_greatest dest: Inf_lower)
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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} @{syntax_const "_SUP"},
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    Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_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 \<sqsubseteq> (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 \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> 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) \<sqsubseteq> 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 \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"
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  by (auto simp add: INFI_def intro: Inf_greatest)
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lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
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  unfolding SUPR_def by (auto simp add: Sup_le_iff)
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lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
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  unfolding INFI_def by (auto simp add: le_Inf_iff)
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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, no_atp]:
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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,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
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  by blast
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lemma empty_Union_conv [simp,no_atp]: "({} = \<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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abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
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  "UNION \<equiv> SUPR"
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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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haftmann@32077
   314
text {*
haftmann@32077
   315
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   316
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   317
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   318
  former does not make the index expression a subscript of the
haftmann@32077
   319
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   320
  subscripts in Proof General.
haftmann@32077
   321
*}
haftmann@32077
   322
wenzelm@35115
   323
print_translation {*
wenzelm@35115
   324
  [Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   325
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   326
haftmann@32135
   327
lemma UNION_eq_Union_image:
haftmann@32135
   328
  "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
haftmann@32606
   329
  by (fact SUPR_def)
haftmann@32115
   330
haftmann@32115
   331
lemma Union_def:
haftmann@32117
   332
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   333
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   334
blanchet@35828
   335
lemma UNION_def [no_atp]:
haftmann@32135
   336
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   337
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   338
  
haftmann@32115
   339
lemma Union_image_eq [simp]:
haftmann@32115
   340
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   341
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   342
  
wenzelm@11979
   343
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979
   344
  by (unfold UNION_def) blast
wenzelm@11979
   345
wenzelm@11979
   346
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
wenzelm@11979
   347
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   348
    @{term b} may be flexible. *}
wenzelm@11979
   349
  by auto
wenzelm@11979
   350
wenzelm@11979
   351
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979
   352
  by (unfold UNION_def) blast
clasohm@923
   353
wenzelm@11979
   354
lemma UN_cong [cong]:
wenzelm@11979
   355
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979
   356
  by (simp add: UNION_def)
wenzelm@11979
   357
berghofe@29691
   358
lemma strong_UN_cong:
berghofe@29691
   359
    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
berghofe@29691
   360
  by (simp add: UNION_def simp_implies_def)
berghofe@29691
   361
haftmann@32077
   362
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
haftmann@32077
   363
  by blast
haftmann@32077
   364
haftmann@32135
   365
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@32606
   366
  by (fact le_SUPI)
haftmann@32135
   367
haftmann@32135
   368
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135
   369
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135
   370
blanchet@35828
   371
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   372
  by blast
haftmann@32135
   373
haftmann@32135
   374
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
   375
  by blast
haftmann@32135
   376
blanchet@35828
   377
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135
   378
  by blast
haftmann@32135
   379
haftmann@32135
   380
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135
   381
  by blast
haftmann@32135
   382
haftmann@32135
   383
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   384
  by blast
haftmann@32135
   385
haftmann@32135
   386
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135
   387
  by auto
haftmann@32135
   388
haftmann@32135
   389
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135
   390
  by blast
haftmann@32135
   391
haftmann@32135
   392
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
   393
  by blast
haftmann@32135
   394
haftmann@32135
   395
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
   396
  by blast
haftmann@32135
   397
haftmann@32135
   398
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   399
  by (fact SUP_le_iff)
haftmann@32135
   400
haftmann@32135
   401
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   402
  by blast
haftmann@32135
   403
haftmann@32135
   404
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135
   405
  by auto
haftmann@32135
   406
haftmann@32135
   407
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135
   408
  by blast
haftmann@32135
   409
haftmann@32135
   410
lemma UNION_empty_conv[simp]:
haftmann@32135
   411
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   412
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   413
by blast+
haftmann@32135
   414
blanchet@35828
   415
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   416
  by blast
haftmann@32135
   417
haftmann@32135
   418
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
   419
  by blast
haftmann@32135
   420
haftmann@32135
   421
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
   422
  by blast
haftmann@32135
   423
haftmann@32135
   424
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   425
  by (auto simp add: split_if_mem2)
haftmann@32135
   426
haftmann@32135
   427
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
haftmann@32135
   428
  by (auto intro: bool_contrapos)
haftmann@32135
   429
haftmann@32135
   430
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   431
  by blast
haftmann@32135
   432
haftmann@32135
   433
lemma UN_mono:
haftmann@32135
   434
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135
   435
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135
   436
  by (blast dest: subsetD)
haftmann@32135
   437
haftmann@32135
   438
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
haftmann@32135
   439
  by blast
haftmann@32135
   440
haftmann@32135
   441
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
haftmann@32135
   442
  by blast
haftmann@32135
   443
haftmann@32135
   444
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
haftmann@32135
   445
  -- {* NOT suitable for rewriting *}
haftmann@32135
   446
  by blast
haftmann@32135
   447
haftmann@32135
   448
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
haftmann@32135
   449
by blast
haftmann@32135
   450
wenzelm@11979
   451
haftmann@32139
   452
subsection {* Inter *}
haftmann@32115
   453
haftmann@32587
   454
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   455
  "Inter S \<equiv> \<Sqinter>S"
haftmann@32135
   456
  
haftmann@32115
   457
notation (xsymbols)
haftmann@32115
   458
  Inter  ("\<Inter>_" [90] 90)
haftmann@32115
   459
haftmann@32135
   460
lemma Inter_eq [code del]:
haftmann@32135
   461
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@32115
   462
proof (rule set_ext)
haftmann@32115
   463
  fix x
haftmann@32135
   464
  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
   465
    by auto
haftmann@32135
   466
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@32587
   467
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@32115
   468
qed
haftmann@32115
   469
blanchet@35828
   470
lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"
haftmann@32115
   471
  by (unfold Inter_eq) blast
haftmann@32115
   472
haftmann@32115
   473
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
haftmann@32115
   474
  by (simp add: Inter_eq)
haftmann@32115
   475
haftmann@32115
   476
text {*
haftmann@32115
   477
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@32115
   478
  contains @{term A} as an element, but @{prop "A:X"} can hold when
haftmann@32115
   479
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
haftmann@32115
   480
*}
haftmann@32115
   481
haftmann@32115
   482
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
haftmann@32115
   483
  by auto
haftmann@32115
   484
haftmann@32115
   485
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
haftmann@32115
   486
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@32115
   487
    @{prop "X:C"}. *}
haftmann@32115
   488
  by (unfold Inter_eq) blast
haftmann@32115
   489
haftmann@32135
   490
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
haftmann@32135
   491
  by blast
haftmann@32135
   492
haftmann@32135
   493
lemma Inter_subset:
haftmann@32135
   494
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
haftmann@32135
   495
  by blast
haftmann@32135
   496
haftmann@32135
   497
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
haftmann@32135
   498
  by (iprover intro: InterI subsetI dest: subsetD)
haftmann@32135
   499
haftmann@32135
   500
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@32135
   501
  by blast
haftmann@32135
   502
haftmann@32135
   503
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@32135
   504
  by blast
haftmann@32135
   505
haftmann@32135
   506
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@32135
   507
  by blast
haftmann@32135
   508
haftmann@32135
   509
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@32135
   510
  by blast
haftmann@32135
   511
haftmann@32135
   512
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@32135
   513
  by blast
haftmann@32135
   514
haftmann@32135
   515
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@32135
   516
  by blast
haftmann@32135
   517
blanchet@35828
   518
lemma Inter_UNIV_conv [simp,no_atp]:
haftmann@32135
   519
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
haftmann@32135
   520
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
haftmann@32135
   521
  by blast+
haftmann@32135
   522
haftmann@32135
   523
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
haftmann@32135
   524
  by blast
haftmann@32135
   525
haftmann@32115
   526
haftmann@32139
   527
subsection {* Intersections of families *}
wenzelm@11979
   528
haftmann@32606
   529
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   530
  "INTER \<equiv> INFI"
haftmann@32081
   531
haftmann@32081
   532
syntax
wenzelm@35115
   533
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
wenzelm@35115
   534
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
haftmann@32081
   535
haftmann@32081
   536
syntax (xsymbols)
wenzelm@35115
   537
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
wenzelm@35115
   538
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
haftmann@32081
   539
haftmann@32081
   540
syntax (latex output)
wenzelm@35115
   541
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
wenzelm@35115
   542
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
haftmann@32081
   543
haftmann@32081
   544
translations
haftmann@32081
   545
  "INT x y. B"  == "INT x. INT y. B"
haftmann@32081
   546
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@32081
   547
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@32081
   548
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@32081
   549
wenzelm@35115
   550
print_translation {*
wenzelm@35115
   551
  [Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
wenzelm@35115
   552
*} -- {* to avoid eta-contraction of body *}
haftmann@32081
   553
haftmann@32135
   554
lemma INTER_eq_Inter_image:
haftmann@32135
   555
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@32606
   556
  by (fact INFI_def)
haftmann@32135
   557
  
haftmann@32115
   558
lemma Inter_def:
haftmann@32135
   559
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@32115
   560
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@32115
   561
haftmann@32115
   562
lemma INTER_def:
haftmann@32135
   563
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@32117
   564
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@32115
   565
haftmann@32115
   566
lemma Inter_image_eq [simp]:
haftmann@32115
   567
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@32115
   568
  by (rule sym) (fact INTER_eq_Inter_image)
haftmann@32115
   569
wenzelm@11979
   570
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
wenzelm@11979
   571
  by (unfold INTER_def) blast
clasohm@923
   572
wenzelm@11979
   573
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
wenzelm@11979
   574
  by (unfold INTER_def) blast
wenzelm@11979
   575
wenzelm@11979
   576
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
wenzelm@11979
   577
  by auto
wenzelm@11979
   578
wenzelm@11979
   579
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
wenzelm@11979
   580
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
wenzelm@11979
   581
  by (unfold INTER_def) blast
wenzelm@11979
   582
wenzelm@11979
   583
lemma INT_cong [cong]:
wenzelm@11979
   584
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
wenzelm@11979
   585
  by (simp add: INTER_def)
wenzelm@7238
   586
haftmann@32135
   587
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@30531
   588
  by blast
haftmann@30531
   589
haftmann@32135
   590
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
wenzelm@12897
   591
  by blast
wenzelm@12897
   592
wenzelm@12897
   593
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@32606
   594
  by (fact INF_leI)
wenzelm@12897
   595
wenzelm@12897
   596
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@32606
   597
  by (fact le_INFI)
wenzelm@12897
   598
wenzelm@12897
   599
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
wenzelm@12897
   600
  by blast
wenzelm@12897
   601
wenzelm@12897
   602
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
wenzelm@12897
   603
  by blast
wenzelm@12897
   604
wenzelm@12897
   605
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
huffman@35629
   606
  by (fact le_INF_iff)
wenzelm@12897
   607
wenzelm@12897
   608
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
wenzelm@12897
   609
  by blast
wenzelm@12897
   610
wenzelm@12897
   611
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
   612
  by blast
wenzelm@12897
   613
wenzelm@12897
   614
lemma INT_insert_distrib:
wenzelm@12897
   615
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
wenzelm@12897
   616
  by blast
wenzelm@12897
   617
wenzelm@12897
   618
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
wenzelm@12897
   619
  by auto
wenzelm@12897
   620
wenzelm@12897
   621
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
wenzelm@12897
   622
  -- {* Look: it has an \emph{existential} quantifier *}
wenzelm@12897
   623
  by blast
wenzelm@12897
   624
paulson@18447
   625
lemma INTER_UNIV_conv[simp]:
nipkow@13653
   626
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
   627
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
   628
by blast+
wenzelm@12897
   629
haftmann@32135
   630
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
haftmann@32135
   631
  by (auto intro: bool_induct)
haftmann@32135
   632
haftmann@32135
   633
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@32135
   634
  by blast
haftmann@32135
   635
haftmann@32135
   636
lemma INT_anti_mono:
haftmann@32135
   637
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135
   638
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@32135
   639
  -- {* The last inclusion is POSITIVE! *}
haftmann@32135
   640
  by (blast dest: subsetD)
haftmann@32135
   641
haftmann@32135
   642
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
haftmann@32135
   643
  by blast
haftmann@32135
   644
haftmann@32135
   645
haftmann@32139
   646
subsection {* Distributive laws *}
wenzelm@12897
   647
wenzelm@12897
   648
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   649
  by blast
wenzelm@12897
   650
wenzelm@12897
   651
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   652
  by blast
wenzelm@12897
   653
wenzelm@12897
   654
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
wenzelm@12897
   655
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   656
  -- {* Union of a family of unions *}
wenzelm@12897
   657
  by blast
wenzelm@12897
   658
wenzelm@12897
   659
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
   660
  -- {* Equivalent version *}
wenzelm@12897
   661
  by blast
wenzelm@12897
   662
wenzelm@12897
   663
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   664
  by blast
wenzelm@12897
   665
wenzelm@12897
   666
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897
   667
  by blast
wenzelm@12897
   668
wenzelm@12897
   669
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
   670
  -- {* Equivalent version *}
wenzelm@12897
   671
  by blast
wenzelm@12897
   672
wenzelm@12897
   673
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   674
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   675
  by blast
wenzelm@12897
   676
wenzelm@12897
   677
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   678
  by blast
wenzelm@12897
   679
wenzelm@12897
   680
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
   681
  by blast
wenzelm@12897
   682
wenzelm@12897
   683
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
   684
  by blast
wenzelm@12897
   685
wenzelm@12897
   686
haftmann@32139
   687
subsection {* Complement *}
haftmann@32135
   688
haftmann@32135
   689
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
   690
  by blast
wenzelm@12897
   691
haftmann@32135
   692
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
   693
  by blast
wenzelm@12897
   694
wenzelm@12897
   695
haftmann@32139
   696
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   697
paulson@13860
   698
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   699
           and Intersections. *}
wenzelm@12897
   700
wenzelm@12897
   701
lemma UN_simps [simp]:
wenzelm@12897
   702
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897
   703
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897
   704
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897
   705
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897
   706
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897
   707
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897
   708
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897
   709
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897
   710
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897
   711
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
   712
  by auto
wenzelm@12897
   713
wenzelm@12897
   714
lemma INT_simps [simp]:
wenzelm@12897
   715
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897
   716
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897
   717
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897
   718
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897
   719
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897
   720
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897
   721
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897
   722
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897
   723
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897
   724
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
   725
  by auto
wenzelm@12897
   726
blanchet@35828
   727
lemma ball_simps [simp,no_atp]:
wenzelm@12897
   728
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897
   729
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897
   730
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897
   731
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897
   732
  "!!P. (ALL x:{}. P x) = True"
wenzelm@12897
   733
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897
   734
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897
   735
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897
   736
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897
   737
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897
   738
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897
   739
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897
   740
  by auto
wenzelm@12897
   741
blanchet@35828
   742
lemma bex_simps [simp,no_atp]:
wenzelm@12897
   743
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897
   744
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897
   745
  "!!P. (EX x:{}. P x) = False"
wenzelm@12897
   746
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897
   747
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897
   748
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897
   749
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897
   750
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897
   751
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897
   752
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897
   753
  by auto
wenzelm@12897
   754
wenzelm@12897
   755
lemma ball_conj_distrib:
wenzelm@12897
   756
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897
   757
  by blast
wenzelm@12897
   758
wenzelm@12897
   759
lemma bex_disj_distrib:
wenzelm@12897
   760
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897
   761
  by blast
wenzelm@12897
   762
wenzelm@12897
   763
paulson@13860
   764
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
   765
paulson@13860
   766
lemma UN_extend_simps:
paulson@13860
   767
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860
   768
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860
   769
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860
   770
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860
   771
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860
   772
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860
   773
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860
   774
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860
   775
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860
   776
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
   777
  by auto
paulson@13860
   778
paulson@13860
   779
lemma INT_extend_simps:
paulson@13860
   780
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860
   781
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860
   782
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860
   783
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860
   784
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860
   785
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860
   786
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860
   787
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860
   788
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860
   789
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
   790
  by auto
paulson@13860
   791
paulson@13860
   792
haftmann@32135
   793
no_notation
haftmann@32135
   794
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
   795
  less (infix "\<sqsubset>" 50) and
haftmann@32135
   796
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
   797
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
   798
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@32678
   799
  Sup  ("\<Squnion>_" [900] 900) and
haftmann@32678
   800
  top ("\<top>") and
haftmann@32678
   801
  bot ("\<bottom>")
haftmann@32135
   802
haftmann@30596
   803
lemmas mem_simps =
haftmann@30596
   804
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
   805
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
   806
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
   807
wenzelm@11979
   808
end