src/HOL/Complete_Lattices.thy
author haftmann
Thu Dec 29 14:23:40 2011 +0100 (2011-12-29)
changeset 46036 6a86cc88b02f
parent 45960 e1b09bfb52f1
child 46154 5115e47a7752
permissions -rw-r--r--
fundamental theorems on Set.bind
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 (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
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header {* Complete lattices *}
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theory Complete_Lattices
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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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  "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
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  by (auto intro!: class.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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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  INF_def: "INFI A f = \<Sqinter>(f ` A)"
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  SUP_def: "SUPR A f = \<Squnion>(f ` A)"
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text {*
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  Note: must use names @{const INFI} and @{const SUPR} here instead of
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  @{text INF} and @{text SUP} to allow the following syntax coexist
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  with the plain constant names.
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*}
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end
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syntax
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
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translations
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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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  "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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print_translation {*
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
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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 INF_foundation_dual [no_atp]:
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  "complete_lattice.SUPR Inf = INFI"
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  by (simp add: fun_eq_iff INF_def
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    complete_lattice.SUP_def [OF dual_complete_lattice])
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lemma SUP_foundation_dual [no_atp]:
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  "complete_lattice.INFI Sup = SUPR"
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  by (simp add: fun_eq_iff SUP_def
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    complete_lattice.INF_def [OF dual_complete_lattice])
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lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
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  by (auto simp add: INF_def intro: Inf_lower)
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lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
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  by (auto simp add: INF_def intro: Inf_greatest)
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lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  by (auto simp add: SUP_def intro: Sup_upper)
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lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
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  by (auto simp add: SUP_def intro: Sup_least)
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
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  using Inf_lower [of u A] by auto
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lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
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  using INF_lower [of i A f] by auto
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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
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  using Sup_upper [of u A] by auto
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lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  using SUP_upper [of i A f] by auto
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lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>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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lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
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  by (auto simp add: INF_def le_Inf_iff)
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lemma Sup_le_iff: "\<Squnion>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 SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
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  by (auto simp add: SUP_def Sup_le_iff)
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lemma Inf_empty [simp]:
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  "\<Sqinter>{} = \<top>"
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  by (auto intro: antisym Inf_greatest)
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lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
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  by (simp add: INF_def)
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lemma Sup_empty [simp]:
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  "\<Squnion>{} = \<bottom>"
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  by (auto intro: antisym Sup_least)
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lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
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  by (simp add: SUP_def)
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lemma Inf_UNIV [simp]:
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  "\<Sqinter>UNIV = \<bottom>"
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  by (auto intro!: antisym Inf_lower)
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lemma Sup_UNIV [simp]:
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  "\<Squnion>UNIV = \<top>"
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  by (auto intro!: antisym Sup_upper)
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lemma Inf_insert [simp]: "\<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 INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
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  by (simp add: INF_def)
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lemma Sup_insert [simp]: "\<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 SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
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  by (simp add: SUP_def)
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lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
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  by (simp add: INF_def image_image)
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lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
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  by (simp add: SUP_def image_image)
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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_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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  by (auto intro: Inf_greatest Inf_lower)
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lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
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  by (auto intro: Sup_least Sup_upper)
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lemma INF_cong:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
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  by (simp add: INF_def image_def)
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lemma SUP_cong:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
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  by (simp add: SUP_def image_def)
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lemma Inf_mono:
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  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
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  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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proof (rule Inf_greatest)
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  fix b assume "b \<in> B"
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  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
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  from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
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  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
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qed
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lemma INF_mono:
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  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
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  unfolding INF_def by (rule Inf_mono) fast
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lemma Sup_mono:
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  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
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  shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
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proof (rule Sup_least)
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  fix a assume "a \<in> A"
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  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
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  from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
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  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
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qed
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lemma SUP_mono:
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  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
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  unfolding SUP_def by (rule Sup_mono) fast
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lemma INF_superset_mono:
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  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
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  -- {* The last inclusion is POSITIVE! *}
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  by (blast intro: INF_mono dest: subsetD)
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lemma SUP_subset_mono:
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  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
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  by (blast intro: SUP_mono dest: subsetD)
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lemma Inf_less_eq:
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  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
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    and "A \<noteq> {}"
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  shows "\<Sqinter>A \<sqsubseteq> u"
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proof -
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  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
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  moreover with assms have "v \<sqsubseteq> u" by blast
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  ultimately show ?thesis by (rule Inf_lower2)
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qed
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lemma less_eq_Sup:
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  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
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    and "A \<noteq> {}"
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  shows "u \<sqsubseteq> \<Squnion>A"
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proof -
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  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
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  moreover with assms have "u \<sqsubseteq> v" by blast
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  ultimately show ?thesis by (rule Sup_upper2)
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qed
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lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
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  by (auto intro: Inf_greatest Inf_lower)
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lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
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  by (auto intro: Sup_least Sup_upper)
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lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
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  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
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lemma INF_union:
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  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
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  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
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lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
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  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
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lemma SUP_union:
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  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
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  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
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lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)"
haftmann@44103
   269
  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041
   270
noschinl@44918
   271
lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" (is "?L = ?R")
noschinl@44918
   272
proof (rule antisym)
noschinl@44918
   273
  show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
noschinl@44918
   274
next
noschinl@44918
   275
  show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
noschinl@44918
   276
qed
haftmann@44041
   277
noschinl@44918
   278
lemma Inf_top_conv [simp, no_atp]:
haftmann@43868
   279
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   280
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   281
proof -
haftmann@43868
   282
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   283
  proof
haftmann@43868
   284
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   285
    then have "A = {} \<or> A = {\<top>}" by auto
noschinl@44919
   286
    then show "\<Sqinter>A = \<top>" by auto
haftmann@43868
   287
  next
haftmann@43868
   288
    assume "\<Sqinter>A = \<top>"
haftmann@43868
   289
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   290
    proof (rule ccontr)
haftmann@43868
   291
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   292
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868
   293
      then obtain B where "A = insert x B" by blast
noschinl@44919
   294
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
haftmann@43868
   295
    qed
haftmann@43868
   296
  qed
haftmann@43868
   297
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868
   298
qed
haftmann@43868
   299
noschinl@44918
   300
lemma INF_top_conv [simp]:
haftmann@44041
   301
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@44041
   302
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
noschinl@44919
   303
  by (auto simp add: INF_def)
haftmann@44041
   304
noschinl@44918
   305
lemma Sup_bot_conv [simp, no_atp]:
haftmann@43868
   306
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
haftmann@43868
   307
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
huffman@44920
   308
  using dual_complete_lattice
huffman@44920
   309
  by (rule complete_lattice.Inf_top_conv)+
haftmann@43868
   310
noschinl@44918
   311
lemma SUP_bot_conv [simp]:
haftmann@44041
   312
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041
   313
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
noschinl@44919
   314
  by (auto simp add: SUP_def)
haftmann@44041
   315
haftmann@43865
   316
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103
   317
  by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077
   318
haftmann@43870
   319
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103
   320
  by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870
   321
noschinl@44918
   322
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
huffman@44921
   323
  by (cases "A = {}") simp_all
haftmann@43900
   324
noschinl@44918
   325
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
huffman@44921
   326
  by (cases "A = {}") simp_all
haftmann@43900
   327
haftmann@43865
   328
lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
haftmann@44103
   329
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865
   330
haftmann@43870
   331
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)"
haftmann@44103
   332
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870
   333
haftmann@43871
   334
lemma INF_absorb:
haftmann@43868
   335
  assumes "k \<in> I"
haftmann@43868
   336
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   337
proof -
haftmann@43868
   338
  from assms obtain J where "I = insert k J" by blast
haftmann@43868
   339
  then show ?thesis by (simp add: INF_insert)
haftmann@43868
   340
qed
haftmann@43868
   341
haftmann@43871
   342
lemma SUP_absorb:
haftmann@43871
   343
  assumes "k \<in> I"
haftmann@43871
   344
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   345
proof -
haftmann@43871
   346
  from assms obtain J where "I = insert k J" by blast
haftmann@43871
   347
  then show ?thesis by (simp add: SUP_insert)
haftmann@43871
   348
qed
haftmann@43871
   349
haftmann@43871
   350
lemma INF_constant:
haftmann@43868
   351
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921
   352
  by simp
haftmann@43868
   353
haftmann@43871
   354
lemma SUP_constant:
haftmann@43871
   355
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921
   356
  by simp
haftmann@43871
   357
haftmann@43943
   358
lemma less_INF_D:
haftmann@43943
   359
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   360
proof -
haftmann@43943
   361
  note `y < (\<Sqinter>i\<in>A. f i)`
haftmann@43943
   362
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
haftmann@44103
   363
    by (rule INF_lower)
haftmann@43943
   364
  finally show "y < f i" .
haftmann@43943
   365
qed
haftmann@43943
   366
haftmann@43943
   367
lemma SUP_lessD:
haftmann@43943
   368
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   369
proof -
haftmann@43943
   370
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
haftmann@44103
   371
    by (rule SUP_upper)
haftmann@43943
   372
  also note `(\<Squnion>i\<in>A. f i) < y`
haftmann@43943
   373
  finally show "f i < y" .
haftmann@43943
   374
qed
haftmann@43943
   375
haftmann@43873
   376
lemma INF_UNIV_bool_expand:
haftmann@43868
   377
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
huffman@44921
   378
  by (simp add: UNIV_bool INF_insert inf_commute)
haftmann@43868
   379
haftmann@43873
   380
lemma SUP_UNIV_bool_expand:
haftmann@43871
   381
  "(\<Squnion>b. A b) = A True \<squnion> A False"
huffman@44921
   382
  by (simp add: UNIV_bool SUP_insert sup_commute)
haftmann@43871
   383
haftmann@32077
   384
end
haftmann@32077
   385
haftmann@44024
   386
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   387
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   388
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   389
begin
haftmann@44024
   390
haftmann@44039
   391
lemma sup_INF:
haftmann@44039
   392
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@44039
   393
  by (simp add: INF_def sup_Inf image_image)
haftmann@44039
   394
haftmann@44039
   395
lemma inf_SUP:
haftmann@44039
   396
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@44039
   397
  by (simp add: SUP_def inf_Sup image_image)
haftmann@44039
   398
haftmann@44032
   399
lemma dual_complete_distrib_lattice:
krauss@44845
   400
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   401
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   402
  apply (fact dual_complete_lattice)
haftmann@44024
   403
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032
   404
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032
   405
  done
haftmann@44024
   406
haftmann@44322
   407
subclass distrib_lattice proof
haftmann@44024
   408
  fix a b c
haftmann@44024
   409
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
noschinl@44919
   410
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
haftmann@44024
   411
qed
haftmann@44024
   412
haftmann@44039
   413
lemma Inf_sup:
haftmann@44039
   414
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   415
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   416
haftmann@44039
   417
lemma Sup_inf:
haftmann@44039
   418
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   419
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   420
haftmann@44039
   421
lemma INF_sup: 
haftmann@44039
   422
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   423
  by (simp add: sup_INF sup_commute)
haftmann@44039
   424
haftmann@44039
   425
lemma SUP_inf:
haftmann@44039
   426
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   427
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   428
haftmann@44039
   429
lemma Inf_sup_eq_top_iff:
haftmann@44039
   430
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   431
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   432
haftmann@44039
   433
lemma Sup_inf_eq_bot_iff:
haftmann@44039
   434
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   435
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   436
haftmann@44039
   437
lemma INF_sup_distrib2:
haftmann@44039
   438
  "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)"
haftmann@44039
   439
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   440
haftmann@44039
   441
lemma SUP_inf_distrib2:
haftmann@44039
   442
  "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)"
haftmann@44039
   443
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   444
haftmann@44024
   445
end
haftmann@44024
   446
haftmann@44032
   447
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   448
begin
haftmann@43873
   449
haftmann@43943
   450
lemma dual_complete_boolean_algebra:
krauss@44845
   451
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   452
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   453
haftmann@43873
   454
lemma uminus_Inf:
haftmann@43873
   455
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   456
proof (rule antisym)
haftmann@43873
   457
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   458
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   459
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   460
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   461
qed
haftmann@43873
   462
haftmann@44041
   463
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@44041
   464
  by (simp add: INF_def SUP_def uminus_Inf image_image)
haftmann@44041
   465
haftmann@43873
   466
lemma uminus_Sup:
haftmann@43873
   467
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   468
proof -
haftmann@43873
   469
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
haftmann@43873
   470
  then show ?thesis by simp
haftmann@43873
   471
qed
haftmann@43873
   472
  
haftmann@43873
   473
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@43873
   474
  by (simp add: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   475
haftmann@43873
   476
end
haftmann@43873
   477
haftmann@43940
   478
class complete_linorder = linorder + complete_lattice
haftmann@43940
   479
begin
haftmann@43940
   480
haftmann@43943
   481
lemma dual_complete_linorder:
krauss@44845
   482
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   483
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   484
noschinl@44918
   485
lemma Inf_less_iff:
haftmann@43940
   486
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   487
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   488
noschinl@44918
   489
lemma INF_less_iff:
haftmann@44041
   490
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@44041
   491
  unfolding INF_def Inf_less_iff by auto
haftmann@44041
   492
noschinl@44918
   493
lemma less_Sup_iff:
haftmann@43940
   494
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   495
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   496
noschinl@44918
   497
lemma less_SUP_iff:
haftmann@43940
   498
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@43940
   499
  unfolding SUP_def less_Sup_iff by auto
haftmann@43940
   500
noschinl@44918
   501
lemma Sup_eq_top_iff [simp]:
haftmann@43943
   502
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   503
proof
haftmann@43943
   504
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   505
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   506
  proof (intro allI impI)
haftmann@43943
   507
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   508
      unfolding less_Sup_iff by auto
haftmann@43943
   509
  qed
haftmann@43943
   510
next
haftmann@43943
   511
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   512
  show "\<Squnion>A = \<top>"
haftmann@43943
   513
  proof (rule ccontr)
haftmann@43943
   514
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   515
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   516
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   517
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   518
      using * unfolding less_Sup_iff by auto
haftmann@43943
   519
    then show False by auto
haftmann@43943
   520
  qed
haftmann@43943
   521
qed
haftmann@43943
   522
noschinl@44918
   523
lemma SUP_eq_top_iff [simp]:
haftmann@44041
   524
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
noschinl@44919
   525
  unfolding SUP_def by auto
haftmann@44041
   526
noschinl@44918
   527
lemma Inf_eq_bot_iff [simp]:
haftmann@43943
   528
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920
   529
  using dual_complete_linorder
huffman@44920
   530
  by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943
   531
noschinl@44918
   532
lemma INF_eq_bot_iff [simp]:
haftmann@43967
   533
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
noschinl@44919
   534
  unfolding INF_def by auto
haftmann@43967
   535
haftmann@43940
   536
end
haftmann@43940
   537
haftmann@43873
   538
haftmann@32139
   539
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077
   540
haftmann@44024
   541
instantiation bool :: complete_lattice
haftmann@32077
   542
begin
haftmann@32077
   543
haftmann@32077
   544
definition
haftmann@44322
   545
  [simp]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   546
haftmann@32077
   547
definition
haftmann@44322
   548
  [simp]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   549
haftmann@32077
   550
instance proof
haftmann@44322
   551
qed (auto intro: bool_induct)
haftmann@32077
   552
haftmann@32077
   553
end
haftmann@32077
   554
haftmann@43873
   555
lemma INF_bool_eq [simp]:
haftmann@32120
   556
  "INFI = Ball"
haftmann@32120
   557
proof (rule ext)+
haftmann@32120
   558
  fix A :: "'a set"
haftmann@32120
   559
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   560
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
haftmann@44322
   561
    by (auto simp add: INF_def)
haftmann@32120
   562
qed
haftmann@32120
   563
haftmann@43873
   564
lemma SUP_bool_eq [simp]:
haftmann@32120
   565
  "SUPR = Bex"
haftmann@32120
   566
proof (rule ext)+
haftmann@32120
   567
  fix A :: "'a set"
haftmann@32120
   568
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   569
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@44322
   570
    by (auto simp add: SUP_def)
haftmann@32120
   571
qed
haftmann@32120
   572
haftmann@44032
   573
instance bool :: complete_boolean_algebra proof
haftmann@44322
   574
qed (auto intro: bool_induct)
haftmann@44024
   575
haftmann@32077
   576
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   577
begin
haftmann@32077
   578
haftmann@32077
   579
definition
haftmann@44024
   580
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   581
haftmann@41080
   582
lemma Inf_apply:
haftmann@44024
   583
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   584
  by (simp add: Inf_fun_def)
haftmann@32077
   585
haftmann@32077
   586
definition
haftmann@44024
   587
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   588
haftmann@41080
   589
lemma Sup_apply:
haftmann@44024
   590
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   591
  by (simp add: Sup_fun_def)
haftmann@32077
   592
haftmann@32077
   593
instance proof
haftmann@44103
   594
qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   595
haftmann@32077
   596
end
haftmann@32077
   597
haftmann@43873
   598
lemma INF_apply:
haftmann@41080
   599
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@43872
   600
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
hoelzl@38705
   601
haftmann@43873
   602
lemma SUP_apply:
haftmann@41080
   603
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@43872
   604
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
haftmann@32077
   605
haftmann@44024
   606
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
haftmann@44024
   607
qed (auto simp add: inf_apply sup_apply Inf_apply Sup_apply INF_def SUP_def inf_Sup sup_Inf image_image)
haftmann@44024
   608
haftmann@43873
   609
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   610
haftmann@45960
   611
instantiation "set" :: (type) complete_lattice
haftmann@45960
   612
begin
haftmann@45960
   613
haftmann@45960
   614
definition
haftmann@45960
   615
  "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   616
haftmann@45960
   617
definition
haftmann@45960
   618
  "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   619
haftmann@45960
   620
instance proof
haftmann@45960
   621
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def Inf_bool_def Sup_bool_def le_fun_def)
haftmann@45960
   622
haftmann@45960
   623
end
haftmann@45960
   624
haftmann@45960
   625
instance "set" :: (type) complete_boolean_algebra
haftmann@45960
   626
proof
haftmann@45960
   627
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
haftmann@45960
   628
  
haftmann@32077
   629
haftmann@41082
   630
subsection {* Inter *}
haftmann@41082
   631
haftmann@41082
   632
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   633
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   634
  
haftmann@41082
   635
notation (xsymbols)
haftmann@41082
   636
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   637
haftmann@41082
   638
lemma Inter_eq:
haftmann@41082
   639
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   640
proof (rule set_eqI)
haftmann@41082
   641
  fix x
haftmann@41082
   642
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
haftmann@41082
   643
    by auto
haftmann@41082
   644
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960
   645
    by (simp add: Inf_set_def image_def)
haftmann@41082
   646
qed
haftmann@41082
   647
haftmann@43741
   648
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   649
  by (unfold Inter_eq) blast
haftmann@41082
   650
haftmann@43741
   651
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   652
  by (simp add: Inter_eq)
haftmann@41082
   653
haftmann@41082
   654
text {*
haftmann@41082
   655
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   656
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   657
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   658
*}
haftmann@41082
   659
haftmann@43741
   660
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   661
  by auto
haftmann@41082
   662
haftmann@43741
   663
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   664
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   665
    @{prop "X \<in> C"}. *}
haftmann@41082
   666
  by (unfold Inter_eq) blast
haftmann@41082
   667
haftmann@43741
   668
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   669
  by (fact Inf_lower)
haftmann@43740
   670
haftmann@41082
   671
lemma Inter_subset:
haftmann@43755
   672
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   673
  by (fact Inf_less_eq)
haftmann@41082
   674
haftmann@43755
   675
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   676
  by (fact Inf_greatest)
haftmann@41082
   677
huffman@44067
   678
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   679
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   680
huffman@44067
   681
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   682
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   683
huffman@44920
   684
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920
   685
  by (fact Inf_insert) (* already simp *)
haftmann@41082
   686
haftmann@41082
   687
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   688
  by (fact less_eq_Inf_inter)
haftmann@41082
   689
haftmann@41082
   690
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   691
  by (fact Inf_union_distrib)
haftmann@43756
   692
haftmann@43868
   693
lemma Inter_UNIV_conv [simp, no_atp]:
haftmann@43741
   694
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   695
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   696
  by (fact Inf_top_conv)+
haftmann@41082
   697
haftmann@43741
   698
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   699
  by (fact Inf_superset_mono)
haftmann@41082
   700
haftmann@41082
   701
haftmann@41082
   702
subsection {* Intersections of families *}
haftmann@41082
   703
haftmann@41082
   704
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   705
  "INTER \<equiv> INFI"
haftmann@41082
   706
haftmann@43872
   707
text {*
haftmann@43872
   708
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   709
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   710
*}
haftmann@43872
   711
haftmann@41082
   712
syntax
haftmann@41082
   713
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   714
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   715
haftmann@41082
   716
syntax (xsymbols)
haftmann@41082
   717
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   718
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   719
haftmann@41082
   720
syntax (latex output)
haftmann@41082
   721
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   722
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   723
haftmann@41082
   724
translations
haftmann@41082
   725
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   726
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   727
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   728
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   729
haftmann@41082
   730
print_translation {*
wenzelm@42284
   731
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   732
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   733
haftmann@44085
   734
lemma INTER_eq:
haftmann@41082
   735
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@44085
   736
  by (auto simp add: INF_def)
haftmann@41082
   737
haftmann@41082
   738
lemma Inter_image_eq [simp]:
haftmann@41082
   739
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43872
   740
  by (rule sym) (fact INF_def)
haftmann@41082
   741
haftmann@43817
   742
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@44085
   743
  by (auto simp add: INF_def image_def)
haftmann@41082
   744
haftmann@43817
   745
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
haftmann@44085
   746
  by (auto simp add: INF_def image_def)
haftmann@41082
   747
haftmann@43852
   748
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
haftmann@41082
   749
  by auto
haftmann@41082
   750
haftmann@43852
   751
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@43852
   752
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@44085
   753
  by (auto simp add: INF_def image_def)
haftmann@41082
   754
haftmann@41082
   755
lemma INT_cong [cong]:
haftmann@43854
   756
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
haftmann@43865
   757
  by (fact INF_cong)
haftmann@41082
   758
haftmann@41082
   759
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   760
  by blast
haftmann@41082
   761
haftmann@41082
   762
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   763
  by blast
haftmann@41082
   764
haftmann@43817
   765
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
   766
  by (fact INF_lower)
haftmann@41082
   767
haftmann@43817
   768
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
   769
  by (fact INF_greatest)
haftmann@41082
   770
huffman@44067
   771
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
   772
  by (fact INF_empty)
haftmann@43854
   773
haftmann@43817
   774
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   775
  by (fact INF_absorb)
haftmann@41082
   776
haftmann@43854
   777
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   778
  by (fact le_INF_iff)
haftmann@41082
   779
haftmann@41082
   780
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   781
  by (fact INF_insert)
haftmann@43865
   782
haftmann@43865
   783
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
haftmann@43865
   784
  by (fact INF_union)
haftmann@43865
   785
haftmann@43865
   786
lemma INT_insert_distrib:
haftmann@43865
   787
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   788
  by blast
haftmann@43854
   789
haftmann@41082
   790
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   791
  by (fact INF_constant)
haftmann@43865
   792
huffman@44920
   793
lemma INTER_UNIV_conv:
haftmann@43817
   794
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   795
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920
   796
  by (fact INF_top_conv)+ (* already simp *)
haftmann@43865
   797
haftmann@43865
   798
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   799
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   800
haftmann@43865
   801
lemma INT_anti_mono:
haftmann@43900
   802
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@43865
   803
  -- {* The last inclusion is POSITIVE! *}
haftmann@43940
   804
  by (fact INF_superset_mono)
haftmann@41082
   805
haftmann@41082
   806
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   807
  by blast
haftmann@41082
   808
haftmann@43817
   809
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   810
  by blast
haftmann@41082
   811
haftmann@41082
   812
haftmann@32139
   813
subsection {* Union *}
haftmann@32115
   814
haftmann@32587
   815
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   816
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   817
haftmann@32115
   818
notation (xsymbols)
haftmann@32115
   819
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   820
haftmann@32135
   821
lemma Union_eq:
haftmann@32135
   822
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   823
proof (rule set_eqI)
haftmann@32115
   824
  fix x
haftmann@32135
   825
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
haftmann@32115
   826
    by auto
haftmann@32135
   827
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960
   828
    by (simp add: Sup_set_def image_def)
haftmann@32115
   829
qed
haftmann@32115
   830
blanchet@35828
   831
lemma Union_iff [simp, no_atp]:
haftmann@32115
   832
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   833
  by (unfold Union_eq) blast
haftmann@32115
   834
haftmann@32115
   835
lemma UnionI [intro]:
haftmann@32115
   836
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   837
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   838
    @{term A} may be flexible. *}
haftmann@32115
   839
  by auto
haftmann@32115
   840
haftmann@32115
   841
lemma UnionE [elim!]:
haftmann@43817
   842
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   843
  by auto
haftmann@32115
   844
haftmann@43817
   845
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
   846
  by (fact Sup_upper)
haftmann@32135
   847
haftmann@43817
   848
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
   849
  by (fact Sup_least)
haftmann@32135
   850
huffman@44920
   851
lemma Union_empty: "\<Union>{} = {}"
huffman@44920
   852
  by (fact Sup_empty) (* already simp *)
haftmann@32135
   853
huffman@44920
   854
lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920
   855
  by (fact Sup_UNIV) (* already simp *)
haftmann@32135
   856
huffman@44920
   857
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920
   858
  by (fact Sup_insert) (* already simp *)
haftmann@32135
   859
haftmann@43817
   860
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
   861
  by (fact Sup_union_distrib)
haftmann@32135
   862
haftmann@32135
   863
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
   864
  by (fact Sup_inter_less_eq)
haftmann@32135
   865
huffman@44920
   866
lemma Union_empty_conv [no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
   867
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
   868
huffman@44920
   869
lemma empty_Union_conv [no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
   870
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
   871
haftmann@32135
   872
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   873
  by blast
haftmann@32135
   874
haftmann@32135
   875
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   876
  by blast
haftmann@32135
   877
haftmann@43817
   878
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
   879
  by (fact Sup_subset_mono)
haftmann@32135
   880
haftmann@32115
   881
haftmann@32139
   882
subsection {* Unions of families *}
haftmann@32077
   883
haftmann@32606
   884
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   885
  "UNION \<equiv> SUPR"
haftmann@32077
   886
haftmann@43872
   887
text {*
haftmann@43872
   888
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
   889
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   890
*}
haftmann@43872
   891
haftmann@32077
   892
syntax
wenzelm@35115
   893
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   894
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   895
haftmann@32077
   896
syntax (xsymbols)
wenzelm@35115
   897
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   898
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   899
haftmann@32077
   900
syntax (latex output)
wenzelm@35115
   901
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   902
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   903
haftmann@32077
   904
translations
haftmann@32077
   905
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   906
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   907
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   908
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   909
haftmann@32077
   910
text {*
haftmann@32077
   911
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   912
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   913
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   914
  former does not make the index expression a subscript of the
haftmann@32077
   915
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   916
  subscripts in Proof General.
haftmann@32077
   917
*}
haftmann@32077
   918
wenzelm@35115
   919
print_translation {*
wenzelm@42284
   920
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   921
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   922
haftmann@44085
   923
lemma UNION_eq [no_atp]:
haftmann@32135
   924
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@44085
   925
  by (auto simp add: SUP_def)
huffman@44920
   926
haftmann@45960
   927
lemma bind_UNION [code]:
haftmann@45960
   928
  "Set.bind A f = UNION A f"
haftmann@45960
   929
  by (simp add: bind_def UNION_eq)
haftmann@45960
   930
haftmann@46036
   931
lemma member_bind [simp]:
haftmann@46036
   932
  "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036
   933
  by (simp add: bind_UNION)
haftmann@46036
   934
haftmann@32115
   935
lemma Union_image_eq [simp]:
haftmann@43817
   936
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
huffman@44920
   937
  by (rule sym) (fact SUP_def)
huffman@44920
   938
haftmann@46036
   939
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@44085
   940
  by (auto simp add: SUP_def image_def)
wenzelm@11979
   941
haftmann@43852
   942
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
   943
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   944
    @{term b} may be flexible. *}
wenzelm@11979
   945
  by auto
wenzelm@11979
   946
haftmann@43852
   947
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@44085
   948
  by (auto simp add: SUP_def image_def)
clasohm@923
   949
wenzelm@11979
   950
lemma UN_cong [cong]:
haftmann@43900
   951
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
haftmann@43900
   952
  by (fact SUP_cong)
wenzelm@11979
   953
berghofe@29691
   954
lemma strong_UN_cong:
haftmann@43900
   955
  "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
haftmann@43900
   956
  by (unfold simp_implies_def) (fact UN_cong)
berghofe@29691
   957
haftmann@43817
   958
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
   959
  by blast
haftmann@32077
   960
haftmann@43817
   961
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
   962
  by (fact SUP_upper)
haftmann@32135
   963
haftmann@43817
   964
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
   965
  by (fact SUP_least)
haftmann@32135
   966
blanchet@35828
   967
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   968
  by blast
haftmann@32135
   969
haftmann@43817
   970
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
   971
  by blast
haftmann@32135
   972
huffman@44067
   973
lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
   974
  by (fact SUP_empty)
haftmann@32135
   975
huffman@44920
   976
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920
   977
  by (fact SUP_bot) (* already simp *)
haftmann@32135
   978
haftmann@43817
   979
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
   980
  by (fact SUP_absorb)
haftmann@32135
   981
haftmann@32135
   982
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
   983
  by (fact SUP_insert)
haftmann@32135
   984
haftmann@44085
   985
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@43900
   986
  by (fact SUP_union)
haftmann@32135
   987
haftmann@43967
   988
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
   989
  by blast
haftmann@32135
   990
haftmann@32135
   991
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   992
  by (fact SUP_le_iff)
haftmann@32135
   993
haftmann@32135
   994
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
   995
  by (fact SUP_constant)
haftmann@32135
   996
haftmann@43944
   997
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   998
  by blast
haftmann@32135
   999
huffman@44920
  1000
lemma UNION_empty_conv:
haftmann@43817
  1001
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1002
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920
  1003
  by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135
  1004
blanchet@35828
  1005
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1006
  by blast
haftmann@32135
  1007
haftmann@43900
  1008
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
  1009
  by blast
haftmann@32135
  1010
haftmann@43900
  1011
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
  1012
  by blast
haftmann@32135
  1013
haftmann@32135
  1014
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
  1015
  by (auto simp add: split_if_mem2)
haftmann@32135
  1016
haftmann@43817
  1017
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1018
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1019
haftmann@32135
  1020
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1021
  by blast
haftmann@32135
  1022
haftmann@32135
  1023
lemma UN_mono:
haftmann@43817
  1024
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1025
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1026
  by (fact SUP_subset_mono)
haftmann@32135
  1027
haftmann@43817
  1028
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1029
  by blast
haftmann@32135
  1030
haftmann@43817
  1031
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1032
  by blast
haftmann@32135
  1033
haftmann@43817
  1034
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
  1035
  -- {* NOT suitable for rewriting *}
haftmann@32135
  1036
  by blast
haftmann@32135
  1037
haftmann@43817
  1038
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1039
  by blast
haftmann@32135
  1040
haftmann@45013
  1041
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013
  1042
  by blast
haftmann@45013
  1043
wenzelm@11979
  1044
haftmann@32139
  1045
subsection {* Distributive laws *}
wenzelm@12897
  1046
wenzelm@12897
  1047
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1048
  by (fact inf_Sup)
wenzelm@12897
  1049
haftmann@44039
  1050
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1051
  by (fact sup_Inf)
haftmann@44039
  1052
wenzelm@12897
  1053
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1054
  by (fact Sup_inf)
haftmann@44039
  1055
haftmann@44039
  1056
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)"
haftmann@44039
  1057
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1058
haftmann@44039
  1059
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)"
haftmann@44039
  1060
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1061
haftmann@44039
  1062
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
haftmann@44039
  1063
  by (simp only: INT_Int_distrib INF_def)
wenzelm@12897
  1064
haftmann@43817
  1065
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897
  1066
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1067
  -- {* Union of a family of unions *}
haftmann@44039
  1068
  by (simp only: UN_Un_distrib SUP_def)
wenzelm@12897
  1069
haftmann@44039
  1070
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1071
  by (fact sup_INF)
wenzelm@12897
  1072
wenzelm@12897
  1073
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1074
  -- {* Halmos, Naive Set Theory, page 35. *}
haftmann@44039
  1075
  by (fact inf_SUP)
wenzelm@12897
  1076
wenzelm@12897
  1077
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)"
haftmann@44039
  1078
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1079
wenzelm@12897
  1080
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)"
haftmann@44039
  1081
  by (fact INF_sup_distrib2)
haftmann@44039
  1082
haftmann@44039
  1083
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1084
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1085
wenzelm@12897
  1086
haftmann@32139
  1087
subsection {* Complement *}
haftmann@32135
  1088
haftmann@43873
  1089
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1090
  by (fact uminus_INF)
wenzelm@12897
  1091
haftmann@43873
  1092
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1093
  by (fact uminus_SUP)
wenzelm@12897
  1094
wenzelm@12897
  1095
haftmann@32139
  1096
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
  1097
paulson@13860
  1098
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1099
           and Intersections. *}
wenzelm@12897
  1100
wenzelm@12897
  1101
lemma UN_simps [simp]:
haftmann@43817
  1102
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
haftmann@44032
  1103
  "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
haftmann@43852
  1104
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
haftmann@44032
  1105
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1106
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1107
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1108
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1109
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1110
  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
haftmann@43831
  1111
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1112
  by auto
wenzelm@12897
  1113
wenzelm@12897
  1114
lemma INT_simps [simp]:
haftmann@44032
  1115
  "\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter> B)"
haftmann@43831
  1116
  "\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))"
haftmann@43852
  1117
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
haftmann@43852
  1118
  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
haftmann@43817
  1119
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1120
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1121
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1122
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1123
  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
haftmann@43852
  1124
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1125
  by auto
wenzelm@12897
  1126
haftmann@43967
  1127
lemma UN_ball_bex_simps [simp, no_atp]:
haftmann@43852
  1128
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1129
  "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
haftmann@43852
  1130
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1131
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
wenzelm@12897
  1132
  by auto
wenzelm@12897
  1133
haftmann@43943
  1134
paulson@13860
  1135
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1136
paulson@13860
  1137
lemma UN_extend_simps:
haftmann@43817
  1138
  "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"
haftmann@44032
  1139
  "\<And>A B C. (\<Union>x\<in>C. A x) \<union> B = (if C={} then B else (\<Union>x\<in>C. A x \<union> B))"
haftmann@43852
  1140
  "\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))"
haftmann@43852
  1141
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1142
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1143
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1144
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1145
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1146
  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
haftmann@43831
  1147
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1148
  by auto
paulson@13860
  1149
paulson@13860
  1150
lemma INT_extend_simps:
haftmann@43852
  1151
  "\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))"
haftmann@43852
  1152
  "\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))"
haftmann@43852
  1153
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
haftmann@43852
  1154
  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
haftmann@43817
  1155
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1156
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1157
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1158
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1159
  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
haftmann@43852
  1160
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1161
  by auto
paulson@13860
  1162
haftmann@43872
  1163
text {* Finally *}
haftmann@43872
  1164
haftmann@32135
  1165
no_notation
haftmann@32135
  1166
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
  1167
  less (infix "\<sqsubset>" 50) and
haftmann@41082
  1168
  bot ("\<bottom>") and
haftmann@41082
  1169
  top ("\<top>") and
haftmann@32135
  1170
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
  1171
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
  1172
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
  1173
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
  1174
haftmann@41080
  1175
no_syntax (xsymbols)
haftmann@41082
  1176
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
  1177
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1178
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
  1179
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1180
haftmann@30596
  1181
lemmas mem_simps =
haftmann@30596
  1182
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1183
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1184
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1185
wenzelm@11979
  1186
end