src/HOL/Complete_Lattices.thy
author wenzelm
Fri Mar 07 22:30:58 2014 +0100 (2014-03-07)
changeset 55990 41c6b99c5fb7
parent 54414 72949fae4f81
child 56015 57e2cfba9c6e
permissions -rw-r--r--
more antiquotations;
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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)
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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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begin
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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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lemma INF_image [simp]: "INFI (f`A) g = INFI A (\<lambda>x. g (f x))"
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  by (simp add: INF_def image_image)
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lemma INF_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> INFI A C = INFI B D"
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  by (simp add: INF_def image_def)
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end
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class Sup =
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  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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begin
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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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lemma SUP_image [simp]: "SUPR (f`A) g = SUPR A (%x. g (f x))"
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  by (simp add: SUP_def image_image)
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lemma SUP_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPR A C = SUPR B D"
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  by (simp add: SUP_def image_def)
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end
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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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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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subsection {* Abstract complete lattices *}
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text {* A complete lattice always has a bottom and a top,
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so we include them into the following type class,
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along with assumptions that define bottom and top
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in terms of infimum and supremum. *}
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class complete_lattice = lattice + Inf + Sup + bot + top +
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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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  assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
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  assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
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begin
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subclass bounded_lattice
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proof
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  fix a
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  show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
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  show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
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qed
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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_lattice)
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    (unfold_locales, (fact Inf_empty Sup_empty
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        Sup_upper Sup_least Inf_lower Inf_greatest)+)
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end
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context complete_lattice
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begin
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lemma INF_foundation_dual:
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  "Sup.SUPR Inf = INFI"
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  by (simp add: fun_eq_iff INF_def Sup.SUP_def)
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lemma SUP_foundation_dual:
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  "Inf.INFI Sup = SUPR" by (simp add: fun_eq_iff SUP_def Inf.INF_def)
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lemma Sup_eqI:
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  "(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x"
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  by (blast intro: antisym Sup_least Sup_upper)
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lemma Inf_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x"
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  by (blast intro: antisym Inf_greatest Inf_lower)
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lemma SUP_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x"
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  unfolding SUP_def by (rule Sup_eqI) auto
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lemma INF_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x"
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  unfolding INF_def by (rule Inf_eqI) auto
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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_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_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]: "(\<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_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_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 from `v \<in> A` assms(1) 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 from `v \<in> A` assms(1) 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 SUPR_eq:
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   268
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
hoelzl@51328
   269
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
hoelzl@51328
   270
  shows "(SUP i:A. f i) = (SUP j:B. g j)"
hoelzl@51328
   271
  by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
hoelzl@51328
   272
hoelzl@51328
   273
lemma INFI_eq:
hoelzl@51328
   274
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
hoelzl@51328
   275
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
hoelzl@51328
   276
  shows "(INF i:A. f i) = (INF j:B. g j)"
hoelzl@51328
   277
  by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
hoelzl@51328
   278
haftmann@43899
   279
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43868
   280
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43868
   281
haftmann@43899
   282
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
haftmann@43868
   283
  by (auto intro: Sup_least Sup_upper)
haftmann@43868
   284
haftmann@43868
   285
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43868
   286
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
haftmann@43868
   287
haftmann@44041
   288
lemma INF_union:
haftmann@44041
   289
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@44103
   290
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
haftmann@44041
   291
haftmann@43868
   292
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
haftmann@43868
   293
  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
haftmann@43868
   294
haftmann@44041
   295
lemma SUP_union:
haftmann@44041
   296
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@44103
   297
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
haftmann@44041
   298
haftmann@44041
   299
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
   300
  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041
   301
noschinl@44918
   302
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
   303
proof (rule antisym)
noschinl@44918
   304
  show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
noschinl@44918
   305
next
noschinl@44918
   306
  show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
noschinl@44918
   307
qed
haftmann@44041
   308
blanchet@54147
   309
lemma Inf_top_conv [simp]:
haftmann@43868
   310
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   311
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   312
proof -
haftmann@43868
   313
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   314
  proof
haftmann@43868
   315
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   316
    then have "A = {} \<or> A = {\<top>}" by auto
noschinl@44919
   317
    then show "\<Sqinter>A = \<top>" by auto
haftmann@43868
   318
  next
haftmann@43868
   319
    assume "\<Sqinter>A = \<top>"
haftmann@43868
   320
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   321
    proof (rule ccontr)
haftmann@43868
   322
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   323
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868
   324
      then obtain B where "A = insert x B" by blast
noschinl@44919
   325
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
haftmann@43868
   326
    qed
haftmann@43868
   327
  qed
haftmann@43868
   328
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868
   329
qed
haftmann@43868
   330
noschinl@44918
   331
lemma INF_top_conv [simp]:
haftmann@44041
   332
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@44041
   333
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
noschinl@44919
   334
  by (auto simp add: INF_def)
haftmann@44041
   335
blanchet@54147
   336
lemma Sup_bot_conv [simp]:
haftmann@43868
   337
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
haftmann@43868
   338
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
huffman@44920
   339
  using dual_complete_lattice
huffman@44920
   340
  by (rule complete_lattice.Inf_top_conv)+
haftmann@43868
   341
noschinl@44918
   342
lemma SUP_bot_conv [simp]:
haftmann@44041
   343
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041
   344
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
noschinl@44919
   345
  by (auto simp add: SUP_def)
haftmann@44041
   346
haftmann@43865
   347
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103
   348
  by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077
   349
haftmann@43870
   350
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103
   351
  by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870
   352
noschinl@44918
   353
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
huffman@44921
   354
  by (cases "A = {}") simp_all
haftmann@43900
   355
noschinl@44918
   356
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
huffman@44921
   357
  by (cases "A = {}") simp_all
haftmann@43900
   358
haftmann@43865
   359
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
   360
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865
   361
haftmann@43870
   362
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
   363
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870
   364
haftmann@43871
   365
lemma INF_absorb:
haftmann@43868
   366
  assumes "k \<in> I"
haftmann@43868
   367
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   368
proof -
haftmann@43868
   369
  from assms obtain J where "I = insert k J" by blast
haftmann@43868
   370
  then show ?thesis by (simp add: INF_insert)
haftmann@43868
   371
qed
haftmann@43868
   372
haftmann@43871
   373
lemma SUP_absorb:
haftmann@43871
   374
  assumes "k \<in> I"
haftmann@43871
   375
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   376
proof -
haftmann@43871
   377
  from assms obtain J where "I = insert k J" by blast
haftmann@43871
   378
  then show ?thesis by (simp add: SUP_insert)
haftmann@43871
   379
qed
haftmann@43871
   380
haftmann@43871
   381
lemma INF_constant:
haftmann@43868
   382
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921
   383
  by simp
haftmann@43868
   384
haftmann@43871
   385
lemma SUP_constant:
haftmann@43871
   386
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921
   387
  by simp
haftmann@43871
   388
haftmann@43943
   389
lemma less_INF_D:
haftmann@43943
   390
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   391
proof -
haftmann@43943
   392
  note `y < (\<Sqinter>i\<in>A. f i)`
haftmann@43943
   393
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
haftmann@44103
   394
    by (rule INF_lower)
haftmann@43943
   395
  finally show "y < f i" .
haftmann@43943
   396
qed
haftmann@43943
   397
haftmann@43943
   398
lemma SUP_lessD:
haftmann@43943
   399
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   400
proof -
haftmann@43943
   401
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
haftmann@44103
   402
    by (rule SUP_upper)
haftmann@43943
   403
  also note `(\<Squnion>i\<in>A. f i) < y`
haftmann@43943
   404
  finally show "f i < y" .
haftmann@43943
   405
qed
haftmann@43943
   406
haftmann@43873
   407
lemma INF_UNIV_bool_expand:
haftmann@43868
   408
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
huffman@44921
   409
  by (simp add: UNIV_bool INF_insert inf_commute)
haftmann@43868
   410
haftmann@43873
   411
lemma SUP_UNIV_bool_expand:
haftmann@43871
   412
  "(\<Squnion>b. A b) = A True \<squnion> A False"
huffman@44921
   413
  by (simp add: UNIV_bool SUP_insert sup_commute)
haftmann@43871
   414
hoelzl@51328
   415
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@51328
   416
  by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
hoelzl@51328
   417
hoelzl@51328
   418
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFI A f \<le> SUPR A f"
hoelzl@51328
   419
  unfolding INF_def SUP_def by (rule Inf_le_Sup) auto
hoelzl@51328
   420
hoelzl@54414
   421
lemma SUP_eq_const:
hoelzl@54414
   422
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPR I f = x"
hoelzl@54414
   423
  by (auto intro: SUP_eqI)
hoelzl@54414
   424
hoelzl@54414
   425
lemma INF_eq_const:
hoelzl@54414
   426
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFI I f = x"
hoelzl@54414
   427
  by (auto intro: INF_eqI)
hoelzl@54414
   428
hoelzl@54414
   429
lemma SUP_eq_iff:
hoelzl@54414
   430
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> (SUPR I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
hoelzl@54414
   431
  using SUP_eq_const[of I f c] SUP_upper[of _ I f] by (auto intro: antisym)
hoelzl@54414
   432
hoelzl@54414
   433
lemma INF_eq_iff:
hoelzl@54414
   434
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> (INFI I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
hoelzl@54414
   435
  using INF_eq_const[of I f c] INF_lower[of _ I f] by (auto intro: antisym)
hoelzl@54414
   436
haftmann@32077
   437
end
haftmann@32077
   438
haftmann@44024
   439
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   440
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   441
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   442
begin
haftmann@44024
   443
haftmann@44039
   444
lemma sup_INF:
haftmann@44039
   445
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@44039
   446
  by (simp add: INF_def sup_Inf image_image)
haftmann@44039
   447
haftmann@44039
   448
lemma inf_SUP:
haftmann@44039
   449
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@44039
   450
  by (simp add: SUP_def inf_Sup image_image)
haftmann@44039
   451
haftmann@44032
   452
lemma dual_complete_distrib_lattice:
krauss@44845
   453
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   454
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   455
  apply (fact dual_complete_lattice)
haftmann@44024
   456
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032
   457
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032
   458
  done
haftmann@44024
   459
haftmann@44322
   460
subclass distrib_lattice proof
haftmann@44024
   461
  fix a b c
haftmann@44024
   462
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
noschinl@44919
   463
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
haftmann@44024
   464
qed
haftmann@44024
   465
haftmann@44039
   466
lemma Inf_sup:
haftmann@44039
   467
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   468
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   469
haftmann@44039
   470
lemma Sup_inf:
haftmann@44039
   471
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   472
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   473
haftmann@44039
   474
lemma INF_sup: 
haftmann@44039
   475
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   476
  by (simp add: sup_INF sup_commute)
haftmann@44039
   477
haftmann@44039
   478
lemma SUP_inf:
haftmann@44039
   479
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   480
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   481
haftmann@44039
   482
lemma Inf_sup_eq_top_iff:
haftmann@44039
   483
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   484
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   485
haftmann@44039
   486
lemma Sup_inf_eq_bot_iff:
haftmann@44039
   487
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   488
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   489
haftmann@44039
   490
lemma INF_sup_distrib2:
haftmann@44039
   491
  "(\<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
   492
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   493
haftmann@44039
   494
lemma SUP_inf_distrib2:
haftmann@44039
   495
  "(\<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
   496
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   497
haftmann@44024
   498
end
haftmann@44024
   499
haftmann@44032
   500
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   501
begin
haftmann@43873
   502
haftmann@43943
   503
lemma dual_complete_boolean_algebra:
krauss@44845
   504
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   505
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   506
haftmann@43873
   507
lemma uminus_Inf:
haftmann@43873
   508
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   509
proof (rule antisym)
haftmann@43873
   510
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   511
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   512
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   513
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   514
qed
haftmann@43873
   515
haftmann@44041
   516
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@44041
   517
  by (simp add: INF_def SUP_def uminus_Inf image_image)
haftmann@44041
   518
haftmann@43873
   519
lemma uminus_Sup:
haftmann@43873
   520
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   521
proof -
haftmann@43873
   522
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
haftmann@43873
   523
  then show ?thesis by simp
haftmann@43873
   524
qed
haftmann@43873
   525
  
haftmann@43873
   526
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@43873
   527
  by (simp add: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   528
haftmann@43873
   529
end
haftmann@43873
   530
haftmann@43940
   531
class complete_linorder = linorder + complete_lattice
haftmann@43940
   532
begin
haftmann@43940
   533
haftmann@43943
   534
lemma dual_complete_linorder:
krauss@44845
   535
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   536
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   537
haftmann@51386
   538
lemma complete_linorder_inf_min: "inf = min"
haftmann@51540
   539
  by (auto intro: antisym simp add: min_def fun_eq_iff)
haftmann@51386
   540
haftmann@51386
   541
lemma complete_linorder_sup_max: "sup = max"
haftmann@51540
   542
  by (auto intro: antisym simp add: max_def fun_eq_iff)
haftmann@51386
   543
noschinl@44918
   544
lemma Inf_less_iff:
haftmann@43940
   545
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   546
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   547
noschinl@44918
   548
lemma INF_less_iff:
haftmann@44041
   549
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@44041
   550
  unfolding INF_def Inf_less_iff by auto
haftmann@44041
   551
noschinl@44918
   552
lemma less_Sup_iff:
haftmann@43940
   553
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   554
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   555
noschinl@44918
   556
lemma less_SUP_iff:
haftmann@43940
   557
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@43940
   558
  unfolding SUP_def less_Sup_iff by auto
haftmann@43940
   559
noschinl@44918
   560
lemma Sup_eq_top_iff [simp]:
haftmann@43943
   561
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   562
proof
haftmann@43943
   563
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   564
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   565
  proof (intro allI impI)
haftmann@43943
   566
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   567
      unfolding less_Sup_iff by auto
haftmann@43943
   568
  qed
haftmann@43943
   569
next
haftmann@43943
   570
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   571
  show "\<Squnion>A = \<top>"
haftmann@43943
   572
  proof (rule ccontr)
haftmann@43943
   573
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   574
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   575
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   576
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   577
      using * unfolding less_Sup_iff by auto
haftmann@43943
   578
    then show False by auto
haftmann@43943
   579
  qed
haftmann@43943
   580
qed
haftmann@43943
   581
noschinl@44918
   582
lemma SUP_eq_top_iff [simp]:
haftmann@44041
   583
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
noschinl@44919
   584
  unfolding SUP_def by auto
haftmann@44041
   585
noschinl@44918
   586
lemma Inf_eq_bot_iff [simp]:
haftmann@43943
   587
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920
   588
  using dual_complete_linorder
huffman@44920
   589
  by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943
   590
noschinl@44918
   591
lemma INF_eq_bot_iff [simp]:
haftmann@43967
   592
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
noschinl@44919
   593
  unfolding INF_def by auto
haftmann@43967
   594
hoelzl@51328
   595
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
hoelzl@51328
   596
proof safe
hoelzl@51328
   597
  fix y assume "x \<le> \<Squnion>A" "y < x"
hoelzl@51328
   598
  then have "y < \<Squnion>A" by auto
hoelzl@51328
   599
  then show "\<exists>a\<in>A. y < a"
hoelzl@51328
   600
    unfolding less_Sup_iff .
hoelzl@51328
   601
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
hoelzl@51328
   602
hoelzl@51328
   603
lemma le_SUP_iff: "x \<le> SUPR A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
hoelzl@51328
   604
  unfolding le_Sup_iff SUP_def by simp
hoelzl@51328
   605
hoelzl@51328
   606
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
hoelzl@51328
   607
proof safe
hoelzl@51328
   608
  fix y assume "x \<ge> \<Sqinter>A" "y > x"
hoelzl@51328
   609
  then have "y > \<Sqinter>A" by auto
hoelzl@51328
   610
  then show "\<exists>a\<in>A. y > a"
hoelzl@51328
   611
    unfolding Inf_less_iff .
hoelzl@51328
   612
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
hoelzl@51328
   613
hoelzl@51328
   614
lemma INF_le_iff:
hoelzl@51328
   615
  "INFI A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
hoelzl@51328
   616
  unfolding Inf_le_iff INF_def by simp
hoelzl@51328
   617
haftmann@51386
   618
subclass complete_distrib_lattice
haftmann@51386
   619
proof
haftmann@51386
   620
  fix a and B
haftmann@51386
   621
  show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@51386
   622
    by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
haftmann@51386
   623
      (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
haftmann@51386
   624
        le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
haftmann@51386
   625
qed
haftmann@51386
   626
haftmann@43940
   627
end
haftmann@43940
   628
hoelzl@51341
   629
haftmann@46631
   630
subsection {* Complete lattice on @{typ bool} *}
haftmann@32077
   631
haftmann@44024
   632
instantiation bool :: complete_lattice
haftmann@32077
   633
begin
haftmann@32077
   634
haftmann@32077
   635
definition
haftmann@46154
   636
  [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   637
haftmann@32077
   638
definition
haftmann@46154
   639
  [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   640
haftmann@32077
   641
instance proof
haftmann@44322
   642
qed (auto intro: bool_induct)
haftmann@32077
   643
haftmann@32077
   644
end
haftmann@32077
   645
haftmann@49905
   646
lemma not_False_in_image_Ball [simp]:
haftmann@49905
   647
  "False \<notin> P ` A \<longleftrightarrow> Ball A P"
haftmann@49905
   648
  by auto
haftmann@49905
   649
haftmann@49905
   650
lemma True_in_image_Bex [simp]:
haftmann@49905
   651
  "True \<in> P ` A \<longleftrightarrow> Bex A P"
haftmann@49905
   652
  by auto
haftmann@49905
   653
haftmann@43873
   654
lemma INF_bool_eq [simp]:
haftmann@32120
   655
  "INFI = Ball"
haftmann@49905
   656
  by (simp add: fun_eq_iff INF_def)
haftmann@32120
   657
haftmann@43873
   658
lemma SUP_bool_eq [simp]:
haftmann@32120
   659
  "SUPR = Bex"
haftmann@49905
   660
  by (simp add: fun_eq_iff SUP_def)
haftmann@32120
   661
haftmann@44032
   662
instance bool :: complete_boolean_algebra proof
haftmann@44322
   663
qed (auto intro: bool_induct)
haftmann@44024
   664
haftmann@46631
   665
haftmann@46631
   666
subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
haftmann@46631
   667
haftmann@32077
   668
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   669
begin
haftmann@32077
   670
haftmann@32077
   671
definition
haftmann@44024
   672
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   673
noschinl@46882
   674
lemma Inf_apply [simp, code]:
haftmann@44024
   675
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   676
  by (simp add: Inf_fun_def)
haftmann@32077
   677
haftmann@32077
   678
definition
haftmann@44024
   679
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   680
noschinl@46882
   681
lemma Sup_apply [simp, code]:
haftmann@44024
   682
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   683
  by (simp add: Sup_fun_def)
haftmann@32077
   684
haftmann@32077
   685
instance proof
noschinl@46884
   686
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   687
haftmann@32077
   688
end
haftmann@32077
   689
noschinl@46882
   690
lemma INF_apply [simp]:
haftmann@41080
   691
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
noschinl@46884
   692
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def)
hoelzl@38705
   693
noschinl@46882
   694
lemma SUP_apply [simp]:
haftmann@41080
   695
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
noschinl@46884
   696
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def)
haftmann@32077
   697
haftmann@44024
   698
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
noschinl@46884
   699
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image)
haftmann@44024
   700
haftmann@43873
   701
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   702
haftmann@46631
   703
haftmann@46631
   704
subsection {* Complete lattice on unary and binary predicates *}
haftmann@46631
   705
haftmann@46631
   706
lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
noschinl@46884
   707
  by simp
haftmann@46631
   708
haftmann@46631
   709
lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
noschinl@46884
   710
  by simp
haftmann@46631
   711
haftmann@46631
   712
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
noschinl@46884
   713
  by auto
haftmann@46631
   714
haftmann@46631
   715
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
noschinl@46884
   716
  by auto
haftmann@46631
   717
haftmann@46631
   718
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
noschinl@46884
   719
  by auto
haftmann@46631
   720
haftmann@46631
   721
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
noschinl@46884
   722
  by auto
haftmann@46631
   723
haftmann@46631
   724
lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   725
  by auto
haftmann@46631
   726
haftmann@46631
   727
lemma INF2_E: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   728
  by auto
haftmann@46631
   729
haftmann@46631
   730
lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
noschinl@46884
   731
  by simp
haftmann@46631
   732
haftmann@46631
   733
lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
noschinl@46884
   734
  by simp
haftmann@46631
   735
haftmann@46631
   736
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
noschinl@46884
   737
  by auto
haftmann@46631
   738
haftmann@46631
   739
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
noschinl@46884
   740
  by auto
haftmann@46631
   741
haftmann@46631
   742
lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   743
  by auto
haftmann@46631
   744
haftmann@46631
   745
lemma SUP2_E: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   746
  by auto
haftmann@46631
   747
haftmann@46631
   748
haftmann@46631
   749
subsection {* Complete lattice on @{typ "_ set"} *}
haftmann@46631
   750
haftmann@45960
   751
instantiation "set" :: (type) complete_lattice
haftmann@45960
   752
begin
haftmann@45960
   753
haftmann@45960
   754
definition
haftmann@45960
   755
  "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   756
haftmann@45960
   757
definition
haftmann@45960
   758
  "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   759
haftmann@45960
   760
instance proof
haftmann@51386
   761
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
haftmann@45960
   762
haftmann@45960
   763
end
haftmann@45960
   764
haftmann@45960
   765
instance "set" :: (type) complete_boolean_algebra
haftmann@45960
   766
proof
haftmann@45960
   767
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
haftmann@45960
   768
  
haftmann@32077
   769
haftmann@46631
   770
subsubsection {* Inter *}
haftmann@41082
   771
haftmann@41082
   772
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   773
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   774
  
haftmann@41082
   775
notation (xsymbols)
haftmann@52141
   776
  Inter  ("\<Inter>_" [900] 900)
haftmann@41082
   777
haftmann@41082
   778
lemma Inter_eq:
haftmann@41082
   779
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   780
proof (rule set_eqI)
haftmann@41082
   781
  fix x
haftmann@41082
   782
  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
   783
    by auto
haftmann@41082
   784
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960
   785
    by (simp add: Inf_set_def image_def)
haftmann@41082
   786
qed
haftmann@41082
   787
blanchet@54147
   788
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   789
  by (unfold Inter_eq) blast
haftmann@41082
   790
haftmann@43741
   791
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   792
  by (simp add: Inter_eq)
haftmann@41082
   793
haftmann@41082
   794
text {*
haftmann@41082
   795
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   796
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   797
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   798
*}
haftmann@41082
   799
haftmann@43741
   800
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   801
  by auto
haftmann@41082
   802
haftmann@43741
   803
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   804
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   805
    @{prop "X \<in> C"}. *}
haftmann@41082
   806
  by (unfold Inter_eq) blast
haftmann@41082
   807
haftmann@43741
   808
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   809
  by (fact Inf_lower)
haftmann@43740
   810
haftmann@41082
   811
lemma Inter_subset:
haftmann@43755
   812
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   813
  by (fact Inf_less_eq)
haftmann@41082
   814
haftmann@43755
   815
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   816
  by (fact Inf_greatest)
haftmann@41082
   817
huffman@44067
   818
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   819
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   820
huffman@44067
   821
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   822
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   823
huffman@44920
   824
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920
   825
  by (fact Inf_insert) (* already simp *)
haftmann@41082
   826
haftmann@41082
   827
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   828
  by (fact less_eq_Inf_inter)
haftmann@41082
   829
haftmann@41082
   830
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   831
  by (fact Inf_union_distrib)
haftmann@43756
   832
blanchet@54147
   833
lemma Inter_UNIV_conv [simp]:
haftmann@43741
   834
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   835
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   836
  by (fact Inf_top_conv)+
haftmann@41082
   837
haftmann@43741
   838
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   839
  by (fact Inf_superset_mono)
haftmann@41082
   840
haftmann@41082
   841
haftmann@46631
   842
subsubsection {* Intersections of families *}
haftmann@41082
   843
haftmann@41082
   844
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   845
  "INTER \<equiv> INFI"
haftmann@41082
   846
haftmann@43872
   847
text {*
haftmann@43872
   848
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   849
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   850
*}
haftmann@43872
   851
haftmann@41082
   852
syntax
haftmann@41082
   853
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   854
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   855
haftmann@41082
   856
syntax (xsymbols)
haftmann@41082
   857
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   858
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   859
haftmann@41082
   860
syntax (latex output)
haftmann@41082
   861
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   862
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   863
haftmann@41082
   864
translations
haftmann@41082
   865
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   866
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   867
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   868
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   869
haftmann@41082
   870
print_translation {*
wenzelm@42284
   871
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   872
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   873
haftmann@44085
   874
lemma INTER_eq:
haftmann@41082
   875
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@44085
   876
  by (auto simp add: INF_def)
haftmann@41082
   877
haftmann@41082
   878
lemma Inter_image_eq [simp]:
haftmann@41082
   879
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43872
   880
  by (rule sym) (fact INF_def)
haftmann@41082
   881
haftmann@43817
   882
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@44085
   883
  by (auto simp add: INF_def image_def)
haftmann@41082
   884
haftmann@43817
   885
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
   886
  by (auto simp add: INF_def image_def)
haftmann@41082
   887
haftmann@43852
   888
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
   889
  by auto
haftmann@41082
   890
haftmann@43852
   891
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
   892
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@44085
   893
  by (auto simp add: INF_def image_def)
haftmann@41082
   894
haftmann@41082
   895
lemma INT_cong [cong]:
haftmann@43854
   896
  "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
   897
  by (fact INF_cong)
haftmann@41082
   898
haftmann@41082
   899
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   900
  by blast
haftmann@41082
   901
haftmann@41082
   902
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   903
  by blast
haftmann@41082
   904
haftmann@43817
   905
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
   906
  by (fact INF_lower)
haftmann@41082
   907
haftmann@43817
   908
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
   909
  by (fact INF_greatest)
haftmann@41082
   910
huffman@44067
   911
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
   912
  by (fact INF_empty)
haftmann@43854
   913
haftmann@43817
   914
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   915
  by (fact INF_absorb)
haftmann@41082
   916
haftmann@43854
   917
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   918
  by (fact le_INF_iff)
haftmann@41082
   919
haftmann@41082
   920
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   921
  by (fact INF_insert)
haftmann@43865
   922
haftmann@43865
   923
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
   924
  by (fact INF_union)
haftmann@43865
   925
haftmann@43865
   926
lemma INT_insert_distrib:
haftmann@43865
   927
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   928
  by blast
haftmann@43854
   929
haftmann@41082
   930
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   931
  by (fact INF_constant)
haftmann@43865
   932
huffman@44920
   933
lemma INTER_UNIV_conv:
haftmann@43817
   934
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   935
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920
   936
  by (fact INF_top_conv)+ (* already simp *)
haftmann@43865
   937
haftmann@43865
   938
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   939
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   940
haftmann@43865
   941
lemma INT_anti_mono:
haftmann@43900
   942
  "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
   943
  -- {* The last inclusion is POSITIVE! *}
haftmann@43940
   944
  by (fact INF_superset_mono)
haftmann@41082
   945
haftmann@41082
   946
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   947
  by blast
haftmann@41082
   948
haftmann@43817
   949
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   950
  by blast
haftmann@41082
   951
haftmann@41082
   952
haftmann@46631
   953
subsubsection {* Union *}
haftmann@32115
   954
haftmann@32587
   955
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   956
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   957
haftmann@32115
   958
notation (xsymbols)
haftmann@52141
   959
  Union  ("\<Union>_" [900] 900)
haftmann@32115
   960
haftmann@32135
   961
lemma Union_eq:
haftmann@32135
   962
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   963
proof (rule set_eqI)
haftmann@32115
   964
  fix x
haftmann@32135
   965
  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
   966
    by auto
haftmann@32135
   967
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960
   968
    by (simp add: Sup_set_def image_def)
haftmann@32115
   969
qed
haftmann@32115
   970
blanchet@54147
   971
lemma Union_iff [simp]:
haftmann@32115
   972
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   973
  by (unfold Union_eq) blast
haftmann@32115
   974
haftmann@32115
   975
lemma UnionI [intro]:
haftmann@32115
   976
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   977
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   978
    @{term A} may be flexible. *}
haftmann@32115
   979
  by auto
haftmann@32115
   980
haftmann@32115
   981
lemma UnionE [elim!]:
haftmann@43817
   982
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   983
  by auto
haftmann@32115
   984
haftmann@43817
   985
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
   986
  by (fact Sup_upper)
haftmann@32135
   987
haftmann@43817
   988
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
   989
  by (fact Sup_least)
haftmann@32135
   990
huffman@44920
   991
lemma Union_empty: "\<Union>{} = {}"
huffman@44920
   992
  by (fact Sup_empty) (* already simp *)
haftmann@32135
   993
huffman@44920
   994
lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920
   995
  by (fact Sup_UNIV) (* already simp *)
haftmann@32135
   996
huffman@44920
   997
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920
   998
  by (fact Sup_insert) (* already simp *)
haftmann@32135
   999
haftmann@43817
  1000
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
  1001
  by (fact Sup_union_distrib)
haftmann@32135
  1002
haftmann@32135
  1003
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
  1004
  by (fact Sup_inter_less_eq)
haftmann@32135
  1005
blanchet@54147
  1006
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1007
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1008
blanchet@54147
  1009
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1010
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1011
haftmann@32135
  1012
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
  1013
  by blast
haftmann@32135
  1014
haftmann@32135
  1015
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
  1016
  by blast
haftmann@32135
  1017
haftmann@43817
  1018
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
  1019
  by (fact Sup_subset_mono)
haftmann@32135
  1020
haftmann@32115
  1021
haftmann@46631
  1022
subsubsection {* Unions of families *}
haftmann@32077
  1023
haftmann@32606
  1024
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
  1025
  "UNION \<equiv> SUPR"
haftmann@32077
  1026
haftmann@43872
  1027
text {*
haftmann@43872
  1028
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
  1029
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
  1030
*}
haftmann@43872
  1031
haftmann@32077
  1032
syntax
wenzelm@35115
  1033
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
  1034
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
  1035
haftmann@32077
  1036
syntax (xsymbols)
wenzelm@35115
  1037
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
  1038
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
  1039
haftmann@32077
  1040
syntax (latex output)
wenzelm@35115
  1041
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
  1042
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
  1043
haftmann@32077
  1044
translations
haftmann@32077
  1045
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
  1046
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
  1047
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
  1048
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
  1049
haftmann@32077
  1050
text {*
haftmann@32077
  1051
  Note the difference between ordinary xsymbol syntax of indexed
wenzelm@53015
  1052
  unions and intersections (e.g.\ @{text"\<Union>a\<^sub>1\<in>A\<^sub>1. B"})
wenzelm@53015
  1053
  and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. The
haftmann@32077
  1054
  former does not make the index expression a subscript of the
haftmann@32077
  1055
  union/intersection symbol because this leads to problems with nested
haftmann@32077
  1056
  subscripts in Proof General.
haftmann@32077
  1057
*}
haftmann@32077
  1058
wenzelm@35115
  1059
print_translation {*
wenzelm@42284
  1060
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
  1061
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
  1062
blanchet@54147
  1063
lemma UNION_eq:
haftmann@32135
  1064
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@44085
  1065
  by (auto simp add: SUP_def)
huffman@44920
  1066
haftmann@45960
  1067
lemma bind_UNION [code]:
haftmann@45960
  1068
  "Set.bind A f = UNION A f"
haftmann@45960
  1069
  by (simp add: bind_def UNION_eq)
haftmann@45960
  1070
haftmann@46036
  1071
lemma member_bind [simp]:
haftmann@46036
  1072
  "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036
  1073
  by (simp add: bind_UNION)
haftmann@46036
  1074
haftmann@32115
  1075
lemma Union_image_eq [simp]:
haftmann@43817
  1076
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
huffman@44920
  1077
  by (rule sym) (fact SUP_def)
huffman@44920
  1078
haftmann@46036
  1079
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@44085
  1080
  by (auto simp add: SUP_def image_def)
wenzelm@11979
  1081
haftmann@43852
  1082
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
  1083
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
  1084
    @{term b} may be flexible. *}
wenzelm@11979
  1085
  by auto
wenzelm@11979
  1086
haftmann@43852
  1087
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
  1088
  by (auto simp add: SUP_def image_def)
clasohm@923
  1089
wenzelm@11979
  1090
lemma UN_cong [cong]:
haftmann@43900
  1091
  "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
  1092
  by (fact SUP_cong)
wenzelm@11979
  1093
berghofe@29691
  1094
lemma strong_UN_cong:
haftmann@43900
  1095
  "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
  1096
  by (unfold simp_implies_def) (fact UN_cong)
berghofe@29691
  1097
haftmann@43817
  1098
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
  1099
  by blast
haftmann@32077
  1100
haftmann@43817
  1101
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
  1102
  by (fact SUP_upper)
haftmann@32135
  1103
haftmann@43817
  1104
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
  1105
  by (fact SUP_least)
haftmann@32135
  1106
blanchet@54147
  1107
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
  1108
  by blast
haftmann@32135
  1109
haftmann@43817
  1110
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
  1111
  by blast
haftmann@32135
  1112
blanchet@54147
  1113
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
  1114
  by (fact SUP_empty)
haftmann@32135
  1115
huffman@44920
  1116
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920
  1117
  by (fact SUP_bot) (* already simp *)
haftmann@32135
  1118
haftmann@43817
  1119
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
  1120
  by (fact SUP_absorb)
haftmann@32135
  1121
haftmann@32135
  1122
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
  1123
  by (fact SUP_insert)
haftmann@32135
  1124
haftmann@44085
  1125
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
  1126
  by (fact SUP_union)
haftmann@32135
  1127
haftmann@43967
  1128
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
  1129
  by blast
haftmann@32135
  1130
haftmann@32135
  1131
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
  1132
  by (fact SUP_le_iff)
haftmann@32135
  1133
haftmann@32135
  1134
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
  1135
  by (fact SUP_constant)
haftmann@32135
  1136
haftmann@43944
  1137
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
  1138
  by blast
haftmann@32135
  1139
huffman@44920
  1140
lemma UNION_empty_conv:
haftmann@43817
  1141
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1142
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920
  1143
  by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135
  1144
blanchet@54147
  1145
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1146
  by blast
haftmann@32135
  1147
haftmann@43900
  1148
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
  1149
  by blast
haftmann@32135
  1150
haftmann@43900
  1151
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
  1152
  by blast
haftmann@32135
  1153
haftmann@32135
  1154
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
  1155
  by (auto simp add: split_if_mem2)
haftmann@32135
  1156
haftmann@43817
  1157
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1158
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1159
haftmann@32135
  1160
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1161
  by blast
haftmann@32135
  1162
haftmann@32135
  1163
lemma UN_mono:
haftmann@43817
  1164
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1165
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1166
  by (fact SUP_subset_mono)
haftmann@32135
  1167
haftmann@43817
  1168
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1169
  by blast
haftmann@32135
  1170
haftmann@43817
  1171
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1172
  by blast
haftmann@32135
  1173
haftmann@43817
  1174
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
  1175
  -- {* NOT suitable for rewriting *}
haftmann@32135
  1176
  by blast
haftmann@32135
  1177
haftmann@43817
  1178
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1179
  by blast
haftmann@32135
  1180
haftmann@45013
  1181
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013
  1182
  by blast
haftmann@45013
  1183
wenzelm@11979
  1184
haftmann@46631
  1185
subsubsection {* Distributive laws *}
wenzelm@12897
  1186
wenzelm@12897
  1187
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1188
  by (fact inf_Sup)
wenzelm@12897
  1189
haftmann@44039
  1190
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1191
  by (fact sup_Inf)
haftmann@44039
  1192
wenzelm@12897
  1193
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1194
  by (fact Sup_inf)
haftmann@44039
  1195
haftmann@44039
  1196
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
  1197
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1198
haftmann@44039
  1199
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
  1200
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1201
haftmann@44039
  1202
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
haftmann@44039
  1203
  by (simp only: INT_Int_distrib INF_def)
wenzelm@12897
  1204
haftmann@43817
  1205
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897
  1206
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1207
  -- {* Union of a family of unions *}
haftmann@44039
  1208
  by (simp only: UN_Un_distrib SUP_def)
wenzelm@12897
  1209
haftmann@44039
  1210
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1211
  by (fact sup_INF)
wenzelm@12897
  1212
wenzelm@12897
  1213
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1214
  -- {* Halmos, Naive Set Theory, page 35. *}
haftmann@44039
  1215
  by (fact inf_SUP)
wenzelm@12897
  1216
wenzelm@12897
  1217
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
  1218
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1219
wenzelm@12897
  1220
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
  1221
  by (fact INF_sup_distrib2)
haftmann@44039
  1222
haftmann@44039
  1223
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1224
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1225
wenzelm@12897
  1226
haftmann@46631
  1227
subsubsection {* Complement *}
haftmann@32135
  1228
haftmann@43873
  1229
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1230
  by (fact uminus_INF)
wenzelm@12897
  1231
haftmann@43873
  1232
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1233
  by (fact uminus_SUP)
wenzelm@12897
  1234
wenzelm@12897
  1235
haftmann@46631
  1236
subsubsection {* Miniscoping and maxiscoping *}
wenzelm@12897
  1237
paulson@13860
  1238
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1239
           and Intersections. *}
wenzelm@12897
  1240
wenzelm@12897
  1241
lemma UN_simps [simp]:
haftmann@43817
  1242
  "\<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
  1243
  "\<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
  1244
  "\<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
  1245
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1246
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1247
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1248
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1249
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1250
  "\<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
  1251
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1252
  by auto
wenzelm@12897
  1253
wenzelm@12897
  1254
lemma INT_simps [simp]:
haftmann@44032
  1255
  "\<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
  1256
  "\<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
  1257
  "\<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
  1258
  "\<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
  1259
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1260
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1261
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1262
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1263
  "\<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
  1264
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1265
  by auto
wenzelm@12897
  1266
blanchet@54147
  1267
lemma UN_ball_bex_simps [simp]:
haftmann@43852
  1268
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1269
  "\<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
  1270
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1271
  "\<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
  1272
  by auto
wenzelm@12897
  1273
haftmann@43943
  1274
paulson@13860
  1275
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1276
paulson@13860
  1277
lemma UN_extend_simps:
haftmann@43817
  1278
  "\<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
  1279
  "\<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
  1280
  "\<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
  1281
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1282
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1283
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1284
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1285
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1286
  "\<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
  1287
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1288
  by auto
paulson@13860
  1289
paulson@13860
  1290
lemma INT_extend_simps:
haftmann@43852
  1291
  "\<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
  1292
  "\<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
  1293
  "\<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
  1294
  "\<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
  1295
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1296
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1297
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1298
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1299
  "\<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
  1300
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1301
  by auto
paulson@13860
  1302
haftmann@43872
  1303
text {* Finally *}
haftmann@43872
  1304
haftmann@32135
  1305
no_notation
haftmann@46691
  1306
  less_eq (infix "\<sqsubseteq>" 50) and
haftmann@46691
  1307
  less (infix "\<sqsubset>" 50)
haftmann@32135
  1308
haftmann@30596
  1309
lemmas mem_simps =
haftmann@30596
  1310
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1311
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1312
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1313
wenzelm@11979
  1314
end
haftmann@49905
  1315