src/HOL/Complete_Lattices.thy
author haftmann
Sat Sep 10 10:29:24 2011 +0200 (2011-09-10)
changeset 44860 56101fa00193
parent 44845 src/HOL/Complete_Lattice.thy@5e51075cbd97
child 44918 6a80fbc4e72c
permissions -rw-r--r--
renamed theory Complete_Lattice to Complete_Lattices, in accordance with Lattices, Orderings etc.
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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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proof (rule ext)+
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  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
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    by (fact dual_complete_lattice)
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  fix f :: "'b \<Rightarrow> 'a" and A
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  show "complete_lattice.SUPR Inf A f = (\<Sqinter>a\<in>A. f a)"
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    by (simp only: dual.SUP_def INF_def)
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qed
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lemma SUP_foundation_dual [no_atp]:
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  "complete_lattice.INFI Sup = SUPR"
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proof (rule ext)+
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  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
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    by (fact dual_complete_lattice)
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  fix f :: "'b \<Rightarrow> 'a" and A
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  show "complete_lattice.INFI Sup A f = (\<Squnion>a\<in>A. f a)"
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    by (simp only: dual.INF_def SUP_def)
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qed
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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 (*[simp]*): "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 (*[simp]*): "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 (*[simp]*): "\<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 (*[simp]*): "(\<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 Inf_insert)
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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 Sup_insert)
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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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  by (force intro!: Inf_mono simp: INF_def)
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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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  by (force intro!: Sup_mono simp: SUP_def)
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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)
haftmann@44041
   270
haftmann@43868
   271
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
haftmann@43868
   272
  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
haftmann@43868
   273
haftmann@44041
   274
lemma SUP_union:
haftmann@44041
   275
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@44103
   276
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
haftmann@44041
   277
haftmann@44041
   278
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
   279
  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041
   280
haftmann@44041
   281
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)"
haftmann@44103
   282
  by (rule antisym) (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono,
haftmann@44103
   283
    rule SUP_least, auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
haftmann@44041
   284
haftmann@44085
   285
lemma Inf_top_conv (*[simp]*) [no_atp]:
haftmann@43868
   286
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   287
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   288
proof -
haftmann@43868
   289
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   290
  proof
haftmann@43868
   291
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   292
    then have "A = {} \<or> A = {\<top>}" by auto
haftmann@44083
   293
    then show "\<Sqinter>A = \<top>" by (auto simp add: Inf_insert)
haftmann@43868
   294
  next
haftmann@43868
   295
    assume "\<Sqinter>A = \<top>"
haftmann@43868
   296
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   297
    proof (rule ccontr)
haftmann@43868
   298
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   299
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868
   300
      then obtain B where "A = insert x B" by blast
haftmann@43868
   301
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
haftmann@43868
   302
    qed
haftmann@43868
   303
  qed
haftmann@43868
   304
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868
   305
qed
haftmann@43868
   306
haftmann@44085
   307
lemma INF_top_conv (*[simp]*):
haftmann@44041
   308
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@44041
   309
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@44041
   310
  by (auto simp add: INF_def Inf_top_conv)
haftmann@44041
   311
haftmann@44085
   312
lemma Sup_bot_conv (*[simp]*) [no_atp]:
haftmann@43868
   313
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
haftmann@43868
   314
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
haftmann@43868
   315
proof -
krauss@44845
   316
  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
haftmann@43868
   317
    by (fact dual_complete_lattice)
haftmann@43868
   318
  from dual.Inf_top_conv show ?P and ?Q by simp_all
haftmann@43868
   319
qed
haftmann@43868
   320
haftmann@44085
   321
lemma SUP_bot_conv (*[simp]*):
haftmann@44041
   322
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041
   323
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041
   324
  by (auto simp add: SUP_def Sup_bot_conv)
haftmann@44041
   325
haftmann@43865
   326
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103
   327
  by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077
   328
haftmann@43870
   329
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103
   330
  by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870
   331
haftmann@44085
   332
lemma INF_top (*[simp]*): "(\<Sqinter>x\<in>A. \<top>) = \<top>"
haftmann@43900
   333
  by (cases "A = {}") (simp_all add: INF_empty)
haftmann@43900
   334
haftmann@44085
   335
lemma SUP_bot (*[simp]*): "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
haftmann@43900
   336
  by (cases "A = {}") (simp_all add: SUP_empty)
haftmann@43900
   337
haftmann@43865
   338
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
   339
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865
   340
haftmann@43870
   341
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
   342
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870
   343
haftmann@43871
   344
lemma INF_absorb:
haftmann@43868
   345
  assumes "k \<in> I"
haftmann@43868
   346
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   347
proof -
haftmann@43868
   348
  from assms obtain J where "I = insert k J" by blast
haftmann@43868
   349
  then show ?thesis by (simp add: INF_insert)
haftmann@43868
   350
qed
haftmann@43868
   351
haftmann@43871
   352
lemma SUP_absorb:
haftmann@43871
   353
  assumes "k \<in> I"
haftmann@43871
   354
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   355
proof -
haftmann@43871
   356
  from assms obtain J where "I = insert k J" by blast
haftmann@43871
   357
  then show ?thesis by (simp add: SUP_insert)
haftmann@43871
   358
qed
haftmann@43871
   359
haftmann@43871
   360
lemma INF_constant:
haftmann@43868
   361
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
haftmann@43868
   362
  by (simp add: INF_empty)
haftmann@43868
   363
haftmann@43871
   364
lemma SUP_constant:
haftmann@43871
   365
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
haftmann@43871
   366
  by (simp add: SUP_empty)
haftmann@43871
   367
haftmann@43943
   368
lemma less_INF_D:
haftmann@43943
   369
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   370
proof -
haftmann@43943
   371
  note `y < (\<Sqinter>i\<in>A. f i)`
haftmann@43943
   372
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
haftmann@44103
   373
    by (rule INF_lower)
haftmann@43943
   374
  finally show "y < f i" .
haftmann@43943
   375
qed
haftmann@43943
   376
haftmann@43943
   377
lemma SUP_lessD:
haftmann@43943
   378
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   379
proof -
haftmann@43943
   380
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
haftmann@44103
   381
    by (rule SUP_upper)
haftmann@43943
   382
  also note `(\<Squnion>i\<in>A. f i) < y`
haftmann@43943
   383
  finally show "f i < y" .
haftmann@43943
   384
qed
haftmann@43943
   385
haftmann@43873
   386
lemma INF_UNIV_bool_expand:
haftmann@43868
   387
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@43868
   388
  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
haftmann@43868
   389
haftmann@43873
   390
lemma SUP_UNIV_bool_expand:
haftmann@43871
   391
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@43871
   392
  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
haftmann@43871
   393
haftmann@32077
   394
end
haftmann@32077
   395
haftmann@44024
   396
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   397
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   398
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   399
begin
haftmann@44024
   400
haftmann@44039
   401
lemma sup_INF:
haftmann@44039
   402
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@44039
   403
  by (simp add: INF_def sup_Inf image_image)
haftmann@44039
   404
haftmann@44039
   405
lemma inf_SUP:
haftmann@44039
   406
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@44039
   407
  by (simp add: SUP_def inf_Sup image_image)
haftmann@44039
   408
haftmann@44032
   409
lemma dual_complete_distrib_lattice:
krauss@44845
   410
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   411
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   412
  apply (fact dual_complete_lattice)
haftmann@44024
   413
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032
   414
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032
   415
  done
haftmann@44024
   416
haftmann@44322
   417
subclass distrib_lattice proof
haftmann@44024
   418
  fix a b c
haftmann@44024
   419
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
haftmann@44082
   420
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def Inf_insert)
haftmann@44024
   421
qed
haftmann@44024
   422
haftmann@44039
   423
lemma Inf_sup:
haftmann@44039
   424
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   425
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   426
haftmann@44039
   427
lemma Sup_inf:
haftmann@44039
   428
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   429
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   430
haftmann@44039
   431
lemma INF_sup: 
haftmann@44039
   432
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   433
  by (simp add: sup_INF sup_commute)
haftmann@44039
   434
haftmann@44039
   435
lemma SUP_inf:
haftmann@44039
   436
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   437
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   438
haftmann@44039
   439
lemma Inf_sup_eq_top_iff:
haftmann@44039
   440
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   441
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   442
haftmann@44039
   443
lemma Sup_inf_eq_bot_iff:
haftmann@44039
   444
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   445
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   446
haftmann@44039
   447
lemma INF_sup_distrib2:
haftmann@44039
   448
  "(\<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
   449
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   450
haftmann@44039
   451
lemma SUP_inf_distrib2:
haftmann@44039
   452
  "(\<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
   453
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   454
haftmann@44024
   455
end
haftmann@44024
   456
haftmann@44032
   457
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   458
begin
haftmann@43873
   459
haftmann@43943
   460
lemma dual_complete_boolean_algebra:
krauss@44845
   461
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   462
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   463
haftmann@43873
   464
lemma uminus_Inf:
haftmann@43873
   465
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   466
proof (rule antisym)
haftmann@43873
   467
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   468
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   469
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   470
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   471
qed
haftmann@43873
   472
haftmann@44041
   473
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@44041
   474
  by (simp add: INF_def SUP_def uminus_Inf image_image)
haftmann@44041
   475
haftmann@43873
   476
lemma uminus_Sup:
haftmann@43873
   477
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   478
proof -
haftmann@43873
   479
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
haftmann@43873
   480
  then show ?thesis by simp
haftmann@43873
   481
qed
haftmann@43873
   482
  
haftmann@43873
   483
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@43873
   484
  by (simp add: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   485
haftmann@43873
   486
end
haftmann@43873
   487
haftmann@43940
   488
class complete_linorder = linorder + complete_lattice
haftmann@43940
   489
begin
haftmann@43940
   490
haftmann@43943
   491
lemma dual_complete_linorder:
krauss@44845
   492
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   493
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   494
haftmann@44085
   495
lemma Inf_less_iff (*[simp]*):
haftmann@43940
   496
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   497
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   498
haftmann@44085
   499
lemma INF_less_iff (*[simp]*):
haftmann@44041
   500
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@44041
   501
  unfolding INF_def Inf_less_iff by auto
haftmann@44041
   502
haftmann@44085
   503
lemma less_Sup_iff (*[simp]*):
haftmann@43940
   504
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   505
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   506
haftmann@44085
   507
lemma less_SUP_iff (*[simp]*):
haftmann@43940
   508
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@43940
   509
  unfolding SUP_def less_Sup_iff by auto
haftmann@43940
   510
haftmann@44085
   511
lemma Sup_eq_top_iff (*[simp]*):
haftmann@43943
   512
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   513
proof
haftmann@43943
   514
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   515
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   516
  proof (intro allI impI)
haftmann@43943
   517
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   518
      unfolding less_Sup_iff by auto
haftmann@43943
   519
  qed
haftmann@43943
   520
next
haftmann@43943
   521
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   522
  show "\<Squnion>A = \<top>"
haftmann@43943
   523
  proof (rule ccontr)
haftmann@43943
   524
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   525
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   526
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   527
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   528
      using * unfolding less_Sup_iff by auto
haftmann@43943
   529
    then show False by auto
haftmann@43943
   530
  qed
haftmann@43943
   531
qed
haftmann@43943
   532
haftmann@44085
   533
lemma SUP_eq_top_iff (*[simp]*):
haftmann@44041
   534
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
haftmann@44041
   535
  unfolding SUP_def Sup_eq_top_iff by auto
haftmann@44041
   536
haftmann@44085
   537
lemma Inf_eq_bot_iff (*[simp]*):
haftmann@43943
   538
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
haftmann@43943
   539
proof -
krauss@44845
   540
  interpret dual: complete_linorder Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
haftmann@43943
   541
    by (fact dual_complete_linorder)
haftmann@43943
   542
  from dual.Sup_eq_top_iff show ?thesis .
haftmann@43943
   543
qed
haftmann@43943
   544
haftmann@44085
   545
lemma INF_eq_bot_iff (*[simp]*):
haftmann@43967
   546
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
haftmann@43967
   547
  unfolding INF_def Inf_eq_bot_iff by auto
haftmann@43967
   548
haftmann@43940
   549
end
haftmann@43940
   550
haftmann@43873
   551
haftmann@32139
   552
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077
   553
haftmann@44024
   554
instantiation bool :: complete_lattice
haftmann@32077
   555
begin
haftmann@32077
   556
haftmann@32077
   557
definition
haftmann@44322
   558
  [simp]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   559
haftmann@32077
   560
definition
haftmann@44322
   561
  [simp]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   562
haftmann@32077
   563
instance proof
haftmann@44322
   564
qed (auto intro: bool_induct)
haftmann@32077
   565
haftmann@32077
   566
end
haftmann@32077
   567
haftmann@43873
   568
lemma INF_bool_eq [simp]:
haftmann@32120
   569
  "INFI = Ball"
haftmann@32120
   570
proof (rule ext)+
haftmann@32120
   571
  fix A :: "'a set"
haftmann@32120
   572
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   573
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
haftmann@44322
   574
    by (auto simp add: INF_def)
haftmann@32120
   575
qed
haftmann@32120
   576
haftmann@43873
   577
lemma SUP_bool_eq [simp]:
haftmann@32120
   578
  "SUPR = Bex"
haftmann@32120
   579
proof (rule ext)+
haftmann@32120
   580
  fix A :: "'a set"
haftmann@32120
   581
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   582
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@44322
   583
    by (auto simp add: SUP_def)
haftmann@32120
   584
qed
haftmann@32120
   585
haftmann@44032
   586
instance bool :: complete_boolean_algebra proof
haftmann@44322
   587
qed (auto intro: bool_induct)
haftmann@44024
   588
haftmann@32077
   589
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   590
begin
haftmann@32077
   591
haftmann@32077
   592
definition
haftmann@44024
   593
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   594
haftmann@41080
   595
lemma Inf_apply:
haftmann@44024
   596
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   597
  by (simp add: Inf_fun_def)
haftmann@32077
   598
haftmann@32077
   599
definition
haftmann@44024
   600
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   601
haftmann@41080
   602
lemma Sup_apply:
haftmann@44024
   603
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   604
  by (simp add: Sup_fun_def)
haftmann@32077
   605
haftmann@32077
   606
instance proof
haftmann@44103
   607
qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   608
haftmann@32077
   609
end
haftmann@32077
   610
haftmann@43873
   611
lemma INF_apply:
haftmann@41080
   612
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@43872
   613
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
hoelzl@38705
   614
haftmann@43873
   615
lemma SUP_apply:
haftmann@41080
   616
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@43872
   617
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
haftmann@32077
   618
haftmann@44024
   619
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
haftmann@44024
   620
qed (auto simp add: inf_apply sup_apply Inf_apply Sup_apply INF_def SUP_def inf_Sup sup_Inf image_image)
haftmann@44024
   621
haftmann@43873
   622
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   623
haftmann@32077
   624
haftmann@41082
   625
subsection {* Inter *}
haftmann@41082
   626
haftmann@41082
   627
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   628
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   629
  
haftmann@41082
   630
notation (xsymbols)
haftmann@41082
   631
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   632
haftmann@41082
   633
lemma Inter_eq:
haftmann@41082
   634
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   635
proof (rule set_eqI)
haftmann@41082
   636
  fix x
haftmann@41082
   637
  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
   638
    by auto
haftmann@41082
   639
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@44322
   640
    by (simp add: Inf_fun_def) (simp add: mem_def)
haftmann@41082
   641
qed
haftmann@41082
   642
haftmann@43741
   643
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   644
  by (unfold Inter_eq) blast
haftmann@41082
   645
haftmann@43741
   646
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   647
  by (simp add: Inter_eq)
haftmann@41082
   648
haftmann@41082
   649
text {*
haftmann@41082
   650
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   651
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   652
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   653
*}
haftmann@41082
   654
haftmann@43741
   655
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   656
  by auto
haftmann@41082
   657
haftmann@43741
   658
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   659
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   660
    @{prop "X \<in> C"}. *}
haftmann@41082
   661
  by (unfold Inter_eq) blast
haftmann@41082
   662
haftmann@43741
   663
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   664
  by (fact Inf_lower)
haftmann@43740
   665
haftmann@41082
   666
lemma Inter_subset:
haftmann@43755
   667
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   668
  by (fact Inf_less_eq)
haftmann@41082
   669
haftmann@43755
   670
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   671
  by (fact Inf_greatest)
haftmann@41082
   672
huffman@44067
   673
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   674
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   675
huffman@44067
   676
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   677
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   678
haftmann@41082
   679
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739
   680
  by (fact Inf_insert)
haftmann@41082
   681
haftmann@41082
   682
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   683
  by (fact less_eq_Inf_inter)
haftmann@41082
   684
haftmann@41082
   685
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   686
  by (fact Inf_union_distrib)
haftmann@43756
   687
haftmann@43868
   688
lemma Inter_UNIV_conv [simp, no_atp]:
haftmann@43741
   689
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   690
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   691
  by (fact Inf_top_conv)+
haftmann@41082
   692
haftmann@43741
   693
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   694
  by (fact Inf_superset_mono)
haftmann@41082
   695
haftmann@41082
   696
haftmann@41082
   697
subsection {* Intersections of families *}
haftmann@41082
   698
haftmann@41082
   699
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   700
  "INTER \<equiv> INFI"
haftmann@41082
   701
haftmann@43872
   702
text {*
haftmann@43872
   703
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   704
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   705
*}
haftmann@43872
   706
haftmann@41082
   707
syntax
haftmann@41082
   708
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   709
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   710
haftmann@41082
   711
syntax (xsymbols)
haftmann@41082
   712
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   713
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   714
haftmann@41082
   715
syntax (latex output)
haftmann@41082
   716
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   717
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   718
haftmann@41082
   719
translations
haftmann@41082
   720
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   721
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   722
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   723
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   724
haftmann@41082
   725
print_translation {*
wenzelm@42284
   726
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   727
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   728
haftmann@44085
   729
lemma INTER_eq:
haftmann@41082
   730
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@44085
   731
  by (auto simp add: INF_def)
haftmann@41082
   732
haftmann@41082
   733
lemma Inter_image_eq [simp]:
haftmann@41082
   734
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43872
   735
  by (rule sym) (fact INF_def)
haftmann@41082
   736
haftmann@43817
   737
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@44085
   738
  by (auto simp add: INF_def image_def)
haftmann@41082
   739
haftmann@43817
   740
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
   741
  by (auto simp add: INF_def image_def)
haftmann@41082
   742
haftmann@43852
   743
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
   744
  by auto
haftmann@41082
   745
haftmann@43852
   746
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
   747
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@44085
   748
  by (auto simp add: INF_def image_def)
haftmann@41082
   749
haftmann@41082
   750
lemma INT_cong [cong]:
haftmann@43854
   751
  "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
   752
  by (fact INF_cong)
haftmann@41082
   753
haftmann@41082
   754
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   755
  by blast
haftmann@41082
   756
haftmann@41082
   757
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   758
  by blast
haftmann@41082
   759
haftmann@43817
   760
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
   761
  by (fact INF_lower)
haftmann@41082
   762
haftmann@43817
   763
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
   764
  by (fact INF_greatest)
haftmann@41082
   765
huffman@44067
   766
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
   767
  by (fact INF_empty)
haftmann@43854
   768
haftmann@43817
   769
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   770
  by (fact INF_absorb)
haftmann@41082
   771
haftmann@43854
   772
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   773
  by (fact le_INF_iff)
haftmann@41082
   774
haftmann@41082
   775
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   776
  by (fact INF_insert)
haftmann@43865
   777
haftmann@43865
   778
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
   779
  by (fact INF_union)
haftmann@43865
   780
haftmann@43865
   781
lemma INT_insert_distrib:
haftmann@43865
   782
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   783
  by blast
haftmann@43854
   784
haftmann@41082
   785
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   786
  by (fact INF_constant)
haftmann@43865
   787
haftmann@43854
   788
lemma INTER_UNIV_conv [simp]:
haftmann@43817
   789
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   790
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43865
   791
  by (fact INF_top_conv)+
haftmann@43865
   792
haftmann@43865
   793
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   794
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   795
haftmann@43865
   796
lemma INT_anti_mono:
haftmann@43900
   797
  "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
   798
  -- {* The last inclusion is POSITIVE! *}
haftmann@43940
   799
  by (fact INF_superset_mono)
haftmann@41082
   800
haftmann@41082
   801
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   802
  by blast
haftmann@41082
   803
haftmann@43817
   804
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   805
  by blast
haftmann@41082
   806
haftmann@41082
   807
haftmann@32139
   808
subsection {* Union *}
haftmann@32115
   809
haftmann@32587
   810
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   811
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   812
haftmann@32115
   813
notation (xsymbols)
haftmann@32115
   814
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   815
haftmann@32135
   816
lemma Union_eq:
haftmann@32135
   817
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   818
proof (rule set_eqI)
haftmann@32115
   819
  fix x
haftmann@32135
   820
  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
   821
    by auto
haftmann@32135
   822
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@44322
   823
    by (simp add: Sup_fun_def) (simp add: mem_def)
haftmann@32115
   824
qed
haftmann@32115
   825
blanchet@35828
   826
lemma Union_iff [simp, no_atp]:
haftmann@32115
   827
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   828
  by (unfold Union_eq) blast
haftmann@32115
   829
haftmann@32115
   830
lemma UnionI [intro]:
haftmann@32115
   831
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   832
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   833
    @{term A} may be flexible. *}
haftmann@32115
   834
  by auto
haftmann@32115
   835
haftmann@32115
   836
lemma UnionE [elim!]:
haftmann@43817
   837
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   838
  by auto
haftmann@32115
   839
haftmann@43817
   840
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
   841
  by (fact Sup_upper)
haftmann@32135
   842
haftmann@43817
   843
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
   844
  by (fact Sup_least)
haftmann@32135
   845
haftmann@43817
   846
lemma Union_empty [simp]: "\<Union>{} = {}"
haftmann@43901
   847
  by (fact Sup_empty)
haftmann@32135
   848
haftmann@43817
   849
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
haftmann@43901
   850
  by (fact Sup_UNIV)
haftmann@32135
   851
haftmann@43817
   852
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
haftmann@43901
   853
  by (fact Sup_insert)
haftmann@32135
   854
haftmann@43817
   855
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
   856
  by (fact Sup_union_distrib)
haftmann@32135
   857
haftmann@32135
   858
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
   859
  by (fact Sup_inter_less_eq)
haftmann@32135
   860
haftmann@43817
   861
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@43901
   862
  by (fact Sup_bot_conv)
haftmann@32135
   863
haftmann@43817
   864
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@43901
   865
  by (fact Sup_bot_conv)
haftmann@32135
   866
haftmann@32135
   867
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   868
  by blast
haftmann@32135
   869
haftmann@32135
   870
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   871
  by blast
haftmann@32135
   872
haftmann@43817
   873
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
   874
  by (fact Sup_subset_mono)
haftmann@32135
   875
haftmann@32115
   876
haftmann@32139
   877
subsection {* Unions of families *}
haftmann@32077
   878
haftmann@32606
   879
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   880
  "UNION \<equiv> SUPR"
haftmann@32077
   881
haftmann@43872
   882
text {*
haftmann@43872
   883
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
   884
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   885
*}
haftmann@43872
   886
haftmann@32077
   887
syntax
wenzelm@35115
   888
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   889
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   890
haftmann@32077
   891
syntax (xsymbols)
wenzelm@35115
   892
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   893
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   894
haftmann@32077
   895
syntax (latex output)
wenzelm@35115
   896
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   897
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   898
haftmann@32077
   899
translations
haftmann@32077
   900
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   901
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   902
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   903
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   904
haftmann@32077
   905
text {*
haftmann@32077
   906
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   907
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   908
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   909
  former does not make the index expression a subscript of the
haftmann@32077
   910
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   911
  subscripts in Proof General.
haftmann@32077
   912
*}
haftmann@32077
   913
wenzelm@35115
   914
print_translation {*
wenzelm@42284
   915
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   916
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   917
haftmann@44085
   918
lemma UNION_eq [no_atp]:
haftmann@32135
   919
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@44085
   920
  by (auto simp add: SUP_def)
haftmann@32115
   921
  
haftmann@32115
   922
lemma Union_image_eq [simp]:
haftmann@43817
   923
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@44085
   924
  by (auto simp add: UNION_eq)
haftmann@32115
   925
  
haftmann@43852
   926
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
haftmann@44085
   927
  by (auto simp add: SUP_def image_def)
wenzelm@11979
   928
haftmann@43852
   929
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
   930
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   931
    @{term b} may be flexible. *}
wenzelm@11979
   932
  by auto
wenzelm@11979
   933
haftmann@43852
   934
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
   935
  by (auto simp add: SUP_def image_def)
clasohm@923
   936
wenzelm@11979
   937
lemma UN_cong [cong]:
haftmann@43900
   938
  "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
   939
  by (fact SUP_cong)
wenzelm@11979
   940
berghofe@29691
   941
lemma strong_UN_cong:
haftmann@43900
   942
  "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
   943
  by (unfold simp_implies_def) (fact UN_cong)
berghofe@29691
   944
haftmann@43817
   945
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
   946
  by blast
haftmann@32077
   947
haftmann@43817
   948
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
   949
  by (fact SUP_upper)
haftmann@32135
   950
haftmann@43817
   951
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
   952
  by (fact SUP_least)
haftmann@32135
   953
blanchet@35828
   954
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   955
  by blast
haftmann@32135
   956
haftmann@43817
   957
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
   958
  by blast
haftmann@32135
   959
huffman@44067
   960
lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
   961
  by (fact SUP_empty)
haftmann@32135
   962
haftmann@32135
   963
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@43900
   964
  by (fact SUP_bot)
haftmann@32135
   965
haftmann@43817
   966
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
   967
  by (fact SUP_absorb)
haftmann@32135
   968
haftmann@32135
   969
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
   970
  by (fact SUP_insert)
haftmann@32135
   971
haftmann@44085
   972
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
   973
  by (fact SUP_union)
haftmann@32135
   974
haftmann@43967
   975
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
   976
  by blast
haftmann@32135
   977
haftmann@32135
   978
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   979
  by (fact SUP_le_iff)
haftmann@32135
   980
haftmann@32135
   981
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
   982
  by (fact SUP_constant)
haftmann@32135
   983
haftmann@43944
   984
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   985
  by blast
haftmann@32135
   986
haftmann@32135
   987
lemma UNION_empty_conv[simp]:
haftmann@43817
   988
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
   989
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43900
   990
  by (fact SUP_bot_conv)+
haftmann@32135
   991
blanchet@35828
   992
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   993
  by blast
haftmann@32135
   994
haftmann@43900
   995
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
   996
  by blast
haftmann@32135
   997
haftmann@43900
   998
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
   999
  by blast
haftmann@32135
  1000
haftmann@32135
  1001
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
  1002
  by (auto simp add: split_if_mem2)
haftmann@32135
  1003
haftmann@43817
  1004
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1005
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1006
haftmann@32135
  1007
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1008
  by blast
haftmann@32135
  1009
haftmann@32135
  1010
lemma UN_mono:
haftmann@43817
  1011
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1012
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1013
  by (fact SUP_subset_mono)
haftmann@32135
  1014
haftmann@43817
  1015
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1016
  by blast
haftmann@32135
  1017
haftmann@43817
  1018
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1019
  by blast
haftmann@32135
  1020
haftmann@43817
  1021
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
  1022
  -- {* NOT suitable for rewriting *}
haftmann@32135
  1023
  by blast
haftmann@32135
  1024
haftmann@43817
  1025
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1026
  by blast
haftmann@32135
  1027
wenzelm@11979
  1028
haftmann@32139
  1029
subsection {* Distributive laws *}
wenzelm@12897
  1030
wenzelm@12897
  1031
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1032
  by (fact inf_Sup)
wenzelm@12897
  1033
haftmann@44039
  1034
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1035
  by (fact sup_Inf)
haftmann@44039
  1036
wenzelm@12897
  1037
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1038
  by (fact Sup_inf)
haftmann@44039
  1039
haftmann@44039
  1040
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
  1041
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1042
haftmann@44039
  1043
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
  1044
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1045
haftmann@44039
  1046
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
haftmann@44039
  1047
  by (simp only: INT_Int_distrib INF_def)
wenzelm@12897
  1048
haftmann@43817
  1049
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897
  1050
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1051
  -- {* Union of a family of unions *}
haftmann@44039
  1052
  by (simp only: UN_Un_distrib SUP_def)
wenzelm@12897
  1053
haftmann@44039
  1054
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1055
  by (fact sup_INF)
wenzelm@12897
  1056
wenzelm@12897
  1057
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1058
  -- {* Halmos, Naive Set Theory, page 35. *}
haftmann@44039
  1059
  by (fact inf_SUP)
wenzelm@12897
  1060
wenzelm@12897
  1061
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
  1062
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1063
wenzelm@12897
  1064
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
  1065
  by (fact INF_sup_distrib2)
haftmann@44039
  1066
haftmann@44039
  1067
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1068
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1069
wenzelm@12897
  1070
haftmann@32139
  1071
subsection {* Complement *}
haftmann@32135
  1072
haftmann@43873
  1073
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1074
  by (fact uminus_INF)
wenzelm@12897
  1075
haftmann@43873
  1076
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1077
  by (fact uminus_SUP)
wenzelm@12897
  1078
wenzelm@12897
  1079
haftmann@32139
  1080
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
  1081
paulson@13860
  1082
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1083
           and Intersections. *}
wenzelm@12897
  1084
wenzelm@12897
  1085
lemma UN_simps [simp]:
haftmann@43817
  1086
  "\<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
  1087
  "\<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
  1088
  "\<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
  1089
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1090
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1091
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1092
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1093
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1094
  "\<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
  1095
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1096
  by auto
wenzelm@12897
  1097
wenzelm@12897
  1098
lemma INT_simps [simp]:
haftmann@44032
  1099
  "\<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
  1100
  "\<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
  1101
  "\<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
  1102
  "\<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
  1103
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1104
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1105
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1106
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1107
  "\<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
  1108
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1109
  by auto
wenzelm@12897
  1110
haftmann@43967
  1111
lemma UN_ball_bex_simps [simp, no_atp]:
haftmann@43852
  1112
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1113
  "\<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
  1114
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1115
  "\<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
  1116
  by auto
wenzelm@12897
  1117
haftmann@43943
  1118
paulson@13860
  1119
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1120
paulson@13860
  1121
lemma UN_extend_simps:
haftmann@43817
  1122
  "\<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
  1123
  "\<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
  1124
  "\<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
  1125
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1126
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1127
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1128
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1129
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1130
  "\<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
  1131
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1132
  by auto
paulson@13860
  1133
paulson@13860
  1134
lemma INT_extend_simps:
haftmann@43852
  1135
  "\<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
  1136
  "\<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
  1137
  "\<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
  1138
  "\<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
  1139
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1140
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1141
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1142
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1143
  "\<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
  1144
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1145
  by auto
paulson@13860
  1146
paulson@13860
  1147
haftmann@43872
  1148
text {* Legacy names *}
haftmann@43872
  1149
haftmann@44083
  1150
lemma (in complete_lattice) Inf_singleton [simp]:
haftmann@44082
  1151
  "\<Sqinter>{a} = a"
haftmann@44083
  1152
  by (simp add: Inf_insert)
haftmann@44082
  1153
haftmann@44083
  1154
lemma (in complete_lattice) Sup_singleton [simp]:
haftmann@44082
  1155
  "\<Squnion>{a} = a"
haftmann@44083
  1156
  by (simp add: Sup_insert)
haftmann@44082
  1157
haftmann@44083
  1158
lemma (in complete_lattice) Inf_binary:
haftmann@44082
  1159
  "\<Sqinter>{a, b} = a \<sqinter> b"
haftmann@44082
  1160
  by (simp add: Inf_insert)
haftmann@44082
  1161
haftmann@44083
  1162
lemma (in complete_lattice) Sup_binary:
haftmann@44082
  1163
  "\<Squnion>{a, b} = a \<squnion> b"
haftmann@44082
  1164
  by (simp add: Sup_insert)
haftmann@44082
  1165
haftmann@43872
  1166
lemmas (in complete_lattice) INFI_def = INF_def
haftmann@43872
  1167
lemmas (in complete_lattice) SUPR_def = SUP_def
haftmann@44103
  1168
lemmas (in complete_lattice) INF_leI = INF_lower
haftmann@44103
  1169
lemmas (in complete_lattice) INF_leI2 = INF_lower2
haftmann@44103
  1170
lemmas (in complete_lattice) le_INFI = INF_greatest
haftmann@44103
  1171
lemmas (in complete_lattice) le_SUPI = SUP_upper
haftmann@44103
  1172
lemmas (in complete_lattice) le_SUPI2 = SUP_upper2
haftmann@44103
  1173
lemmas (in complete_lattice) SUP_leI = SUP_least
haftmann@43943
  1174
lemmas (in complete_lattice) less_INFD = less_INF_D
haftmann@43940
  1175
haftmann@44085
  1176
lemmas INFI_apply = INF_apply
haftmann@44085
  1177
lemmas SUPR_apply = SUP_apply
haftmann@44085
  1178
haftmann@44085
  1179
text {* Grep and put to news from here *}
haftmann@44085
  1180
haftmann@44085
  1181
lemma (in complete_lattice) INF_eq:
haftmann@44085
  1182
  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@44085
  1183
  by (simp add: INF_def image_def)
haftmann@44085
  1184
haftmann@44085
  1185
lemma (in complete_lattice) SUP_eq:
haftmann@44085
  1186
  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@44085
  1187
  by (simp add: SUP_def image_def)
haftmann@44085
  1188
haftmann@43940
  1189
lemma (in complete_lattice) INF_subset:
haftmann@43940
  1190
  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
haftmann@43940
  1191
  by (rule INF_superset_mono) auto
haftmann@43940
  1192
haftmann@44085
  1193
lemma (in complete_lattice) INF_UNIV_range:
haftmann@44085
  1194
  "(\<Sqinter>x. f x) = \<Sqinter>range f"
haftmann@44085
  1195
  by (fact INF_def)
haftmann@44085
  1196
haftmann@44085
  1197
lemma (in complete_lattice) SUP_UNIV_range:
haftmann@44085
  1198
  "(\<Squnion>x. f x) = \<Squnion>range f"
haftmann@44085
  1199
  by (fact SUP_def)
haftmann@44085
  1200
haftmann@44085
  1201
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@44085
  1202
  by (simp add: Inf_insert)
haftmann@44085
  1203
haftmann@44085
  1204
lemma INTER_eq_Inter_image:
haftmann@44085
  1205
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@44085
  1206
  by (fact INF_def)
haftmann@44085
  1207
  
haftmann@44085
  1208
lemma Inter_def:
haftmann@44085
  1209
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@44085
  1210
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@44085
  1211
haftmann@44085
  1212
lemmas INTER_def = INTER_eq
haftmann@44085
  1213
lemmas UNION_def = UNION_eq
haftmann@44085
  1214
haftmann@44085
  1215
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@44085
  1216
  by (fact INF_eq)
haftmann@44085
  1217
haftmann@44085
  1218
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@44085
  1219
  by blast
haftmann@44085
  1220
haftmann@44085
  1221
lemma UNION_eq_Union_image:
haftmann@44085
  1222
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
haftmann@44085
  1223
  by (fact SUP_def)
haftmann@44085
  1224
haftmann@44085
  1225
lemma Union_def:
haftmann@44085
  1226
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@44085
  1227
  by (simp add: UNION_eq_Union_image image_def)
haftmann@44085
  1228
haftmann@44085
  1229
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@44085
  1230
  by blast
haftmann@44085
  1231
haftmann@44085
  1232
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@44085
  1233
  by (fact SUP_eq)
haftmann@44085
  1234
haftmann@43872
  1235
haftmann@43872
  1236
text {* Finally *}
haftmann@43872
  1237
haftmann@32135
  1238
no_notation
haftmann@32135
  1239
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
  1240
  less (infix "\<sqsubset>" 50) and
haftmann@41082
  1241
  bot ("\<bottom>") and
haftmann@41082
  1242
  top ("\<top>") and
haftmann@32135
  1243
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
  1244
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
  1245
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
  1246
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
  1247
haftmann@41080
  1248
no_syntax (xsymbols)
haftmann@41082
  1249
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
  1250
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1251
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
  1252
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1253
haftmann@30596
  1254
lemmas mem_simps =
haftmann@30596
  1255
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1256
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1257
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1258
wenzelm@11979
  1259
end