src/HOL/Complete_Lattices.thy
author webertj
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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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class Sup =
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  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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subsection {* Abstract complete lattices *}
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class complete_lattice = bounded_lattice + Inf + Sup +
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  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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begin
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lemma dual_complete_lattice:
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  "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
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  by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
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    (unfold_locales, (fact bot_least top_greatest
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        Sup_upper Sup_least Inf_lower Inf_greatest)+)
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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  INF_def: "INFI A f = \<Sqinter>(f ` A)"
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  SUP_def: "SUPR A f = \<Squnion>(f ` A)"
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text {*
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  Note: must use names @{const INFI} and @{const SUPR} here instead of
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  @{text INF} and @{text SUP} to allow the following syntax coexist
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  with the plain constant names.
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*}
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end
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syntax
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
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translations
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  "INF x y. B"   == "INF x. INF y. B"
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  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
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  "INF x. B"     == "INF x:CONST UNIV. B"
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  "INF x:A. B"   == "CONST INFI A (%x. B)"
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  "SUP x y. B"   == "SUP x. SUP y. B"
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  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
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  "SUP x. B"     == "SUP x:CONST UNIV. B"
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  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
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print_translation {*
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
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*} -- {* to avoid eta-contraction of body *}
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context complete_lattice
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begin
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lemma INF_foundation_dual [no_atp]:
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  "complete_lattice.SUPR Inf = INFI"
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  by (simp add: fun_eq_iff INF_def
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    complete_lattice.SUP_def [OF dual_complete_lattice])
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lemma SUP_foundation_dual [no_atp]:
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  "complete_lattice.INFI Sup = SUPR"
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  by (simp add: fun_eq_iff SUP_def
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    complete_lattice.INF_def [OF dual_complete_lattice])
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lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
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  by (auto simp add: INF_def intro: Inf_lower)
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lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
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  by (auto simp add: INF_def intro: Inf_greatest)
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lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  by (auto simp add: SUP_def intro: Sup_upper)
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lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
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  by (auto simp add: SUP_def intro: Sup_least)
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
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  using Inf_lower [of u A] by auto
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lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
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  using INF_lower [of i A f] by auto
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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
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  using Sup_upper [of u A] by auto
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lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  using SUP_upper [of i A f] by auto
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lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
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  by (auto intro: Inf_greatest dest: Inf_lower)
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lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
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  by (auto simp add: INF_def le_Inf_iff)
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lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
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  by (auto intro: Sup_least dest: Sup_upper)
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lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
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  by (auto simp add: SUP_def Sup_le_iff)
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lemma Inf_empty [simp]:
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  "\<Sqinter>{} = \<top>"
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  by (auto intro: antisym Inf_greatest)
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lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
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  by (simp add: INF_def)
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lemma Sup_empty [simp]:
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  "\<Squnion>{} = \<bottom>"
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  by (auto intro: antisym Sup_least)
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lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
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  by (simp add: SUP_def)
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lemma Inf_UNIV [simp]:
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  "\<Sqinter>UNIV = \<bottom>"
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  by (auto intro!: antisym Inf_lower)
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lemma Sup_UNIV [simp]:
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  "\<Squnion>UNIV = \<top>"
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  by (auto intro!: antisym Sup_upper)
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lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
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  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
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lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
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  by (simp add: INF_def)
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lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
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  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
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lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
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  by (simp add: SUP_def)
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lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
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  by (simp add: INF_def image_image)
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lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
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  by (simp add: SUP_def image_image)
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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  by (auto intro: Inf_greatest Inf_lower)
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lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
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  by (auto intro: Sup_least Sup_upper)
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lemma INF_cong:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
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  by (simp add: INF_def image_def)
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lemma SUP_cong:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
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  by (simp add: SUP_def image_def)
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lemma Inf_mono:
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  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
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  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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proof (rule Inf_greatest)
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  fix b assume "b \<in> B"
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  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
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  from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
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  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
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qed
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lemma INF_mono:
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  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
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  unfolding INF_def by (rule Inf_mono) fast
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lemma Sup_mono:
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  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
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  shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
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proof (rule Sup_least)
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  fix a assume "a \<in> A"
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  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
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  from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
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  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
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qed
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lemma SUP_mono:
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  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
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  unfolding SUP_def by (rule Sup_mono) fast
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lemma INF_superset_mono:
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  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
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  -- {* The last inclusion is POSITIVE! *}
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  by (blast intro: INF_mono dest: subsetD)
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lemma SUP_subset_mono:
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  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
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  by (blast intro: SUP_mono dest: subsetD)
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lemma Inf_less_eq:
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  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
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    and "A \<noteq> {}"
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  shows "\<Sqinter>A \<sqsubseteq> u"
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proof -
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  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
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  moreover with assms have "v \<sqsubseteq> u" by blast
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  ultimately show ?thesis by (rule Inf_lower2)
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qed
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lemma less_eq_Sup:
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  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
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    and "A \<noteq> {}"
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  shows "u \<sqsubseteq> \<Squnion>A"
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proof -
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  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
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  moreover with assms have "u \<sqsubseteq> v" by blast
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  ultimately show ?thesis by (rule Sup_upper2)
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qed
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lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
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  by (auto intro: Inf_greatest Inf_lower)
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lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
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  by (auto intro: Sup_least Sup_upper)
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lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
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  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
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lemma INF_union:
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  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
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  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
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lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
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  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
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lemma SUP_union:
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  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
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  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
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lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)"
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  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
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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")
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proof (rule antisym)
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   270
  show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   271
next
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   272
  show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   273
qed
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   274
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   275
lemma Inf_top_conv [simp, no_atp]:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   276
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   277
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   278
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   279
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   280
  proof
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   281
    assume "\<forall>x\<in>A. x = \<top>"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   282
    then have "A = {} \<or> A = {\<top>}" by auto
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   283
    then show "\<Sqinter>A = \<top>" by auto
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   284
  next
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   285
    assume "\<Sqinter>A = \<top>"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   286
    show "\<forall>x\<in>A. x = \<top>"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   287
    proof (rule ccontr)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   288
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   289
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   290
      then obtain B where "A = insert x B" by blast
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   291
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   292
    qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   293
  qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   294
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   295
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   296
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   297
lemma INF_top_conv [simp]:
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   298
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   299
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   300
  by (auto simp add: INF_def)
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   301
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   302
lemma Sup_bot_conv [simp, no_atp]:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   303
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   304
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   305
  using dual_complete_lattice
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   306
  by (rule complete_lattice.Inf_top_conv)+
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   307
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   308
lemma SUP_bot_conv [simp]:
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   309
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   310
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   311
  by (auto simp add: SUP_def)
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   312
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   313
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   314
  by (auto intro: antisym INF_lower INF_greatest)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   315
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   316
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   317
  by (auto intro: antisym SUP_upper SUP_least)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   318
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   319
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
44921
58eef4843641 tuned proofs
huffman
parents: 44920
diff changeset
   320
  by (cases "A = {}") simp_all
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   321
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   322
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
44921
58eef4843641 tuned proofs
huffman
parents: 44920
diff changeset
   323
  by (cases "A = {}") simp_all
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   324
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   325
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)"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   326
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   327
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   328
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)"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   329
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   330
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   331
lemma INF_absorb:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   332
  assumes "k \<in> I"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   333
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   334
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   335
  from assms obtain J where "I = insert k J" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   336
  then show ?thesis by (simp add: INF_insert)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   337
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   338
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   339
lemma SUP_absorb:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   340
  assumes "k \<in> I"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   341
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   342
proof -
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   343
  from assms obtain J where "I = insert k J" by blast
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   344
  then show ?thesis by (simp add: SUP_insert)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   345
qed
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   346
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   347
lemma INF_constant:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   348
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
44921
58eef4843641 tuned proofs
huffman
parents: 44920
diff changeset
   349
  by simp
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   350
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   351
lemma SUP_constant:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   352
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
44921
58eef4843641 tuned proofs
huffman
parents: 44920
diff changeset
   353
  by simp
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   354
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   355
lemma less_INF_D:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   356
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   357
proof -
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   358
  note `y < (\<Sqinter>i\<in>A. f i)`
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   359
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   360
    by (rule INF_lower)
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   361
  finally show "y < f i" .
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   362
qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   363
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   364
lemma SUP_lessD:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   365
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   366
proof -
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   367
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   368
    by (rule SUP_upper)
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   369
  also note `(\<Squnion>i\<in>A. f i) < y`
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   370
  finally show "f i < y" .
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   371
qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   372
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   373
lemma INF_UNIV_bool_expand:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   374
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
44921
58eef4843641 tuned proofs
huffman
parents: 44920
diff changeset
   375
  by (simp add: UNIV_bool INF_insert inf_commute)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   376
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   377
lemma SUP_UNIV_bool_expand:
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   378
  "(\<Squnion>b. A b) = A True \<squnion> A False"
44921
58eef4843641 tuned proofs
huffman
parents: 44920
diff changeset
   379
  by (simp add: UNIV_bool SUP_insert sup_commute)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   380
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   381
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   382
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   383
class complete_distrib_lattice = complete_lattice +
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   384
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   385
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   386
begin
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   387
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   388
lemma sup_INF:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   389
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   390
  by (simp add: INF_def sup_Inf image_image)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   391
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   392
lemma inf_SUP:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   393
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   394
  by (simp add: SUP_def inf_Sup image_image)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   395
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
   396
lemma dual_complete_distrib_lattice:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44322
diff changeset
   397
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   398
  apply (rule class.complete_distrib_lattice.intro)
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   399
  apply (fact dual_complete_lattice)
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   400
  apply (rule class.complete_distrib_lattice_axioms.intro)
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
   401
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
   402
  done
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   403
44322
43b465f4c480 more concise definition for Inf, Sup on bool
haftmann
parents: 44104
diff changeset
   404
subclass distrib_lattice proof
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   405
  fix a b c
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   406
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   407
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   408
qed
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   409
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   410
lemma Inf_sup:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   411
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   412
  by (simp add: sup_Inf sup_commute)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   413
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   414
lemma Sup_inf:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   415
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   416
  by (simp add: inf_Sup inf_commute)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   417
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   418
lemma INF_sup: 
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   419
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   420
  by (simp add: sup_INF sup_commute)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   421
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   422
lemma SUP_inf:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   423
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   424
  by (simp add: inf_SUP inf_commute)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   425
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   426
lemma Inf_sup_eq_top_iff:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   427
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   428
  by (simp only: Inf_sup INF_top_conv)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   429
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   430
lemma Sup_inf_eq_bot_iff:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   431
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   432
  by (simp only: Sup_inf SUP_bot_conv)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   433
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   434
lemma INF_sup_distrib2:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   435
  "(\<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)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   436
  by (subst INF_commute) (simp add: sup_INF INF_sup)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   437
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   438
lemma SUP_inf_distrib2:
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   439
  "(\<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)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   440
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
   441
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   442
end
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   443
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
   444
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   445
begin
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   446
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   447
lemma dual_complete_boolean_algebra:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44322
diff changeset
   448
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
   449
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   450
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   451
lemma uminus_Inf:
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   452
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   453
proof (rule antisym)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   454
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   455
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   456
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   457
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   458
qed
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   459
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   460
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   461
  by (simp add: INF_def SUP_def uminus_Inf image_image)
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   462
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   463
lemma uminus_Sup:
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   464
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   465
proof -
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   466
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   467
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   468
qed
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   469
  
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   470
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   471
  by (simp add: INF_def SUP_def uminus_Sup image_image)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   472
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   473
end
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   474
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   475
class complete_linorder = linorder + complete_lattice
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   476
begin
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   477
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   478
lemma dual_complete_linorder:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44322
diff changeset
   479
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   480
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   481
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   482
lemma Inf_less_iff:
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   483
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   484
  unfolding not_le [symmetric] le_Inf_iff by auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   485
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   486
lemma INF_less_iff:
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   487
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   488
  unfolding INF_def Inf_less_iff by auto
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   489
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   490
lemma less_Sup_iff:
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   491
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   492
  unfolding not_le [symmetric] Sup_le_iff by auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   493
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   494
lemma less_SUP_iff:
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   495
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   496
  unfolding SUP_def less_Sup_iff by auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   497
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   498
lemma Sup_eq_top_iff [simp]:
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   499
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   500
proof
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   501
  assume *: "\<Squnion>A = \<top>"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   502
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   503
  proof (intro allI impI)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   504
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   505
      unfolding less_Sup_iff by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   506
  qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   507
next
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   508
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   509
  show "\<Squnion>A = \<top>"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   510
  proof (rule ccontr)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   511
    assume "\<Squnion>A \<noteq> \<top>"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   512
    with top_greatest [of "\<Squnion>A"]
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   513
    have "\<Squnion>A < \<top>" unfolding le_less by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   514
    then have "\<Squnion>A < \<Squnion>A"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   515
      using * unfolding less_Sup_iff by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   516
    then show False by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   517
  qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   518
qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   519
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   520
lemma SUP_eq_top_iff [simp]:
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   521
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   522
  unfolding SUP_def by auto
44041
01d6ab227069 tuned order: pushing INF and SUP to Inf and Sup
haftmann
parents: 44040
diff changeset
   523
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   524
lemma Inf_eq_bot_iff [simp]:
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   525
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   526
  using dual_complete_linorder
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   527
  by (rule complete_linorder.Sup_eq_top_iff)
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   528
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44860
diff changeset
   529
lemma INF_eq_bot_iff [simp]:
43967
610efb6bda1f more coherent structure in and across theories
haftmann
parents: 43944
diff changeset
   530
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   531
  unfolding INF_def by auto
43967
610efb6bda1f more coherent structure in and across theories
haftmann
parents: 43944
diff changeset
   532
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   533
end
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   534
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   535
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   536
subsection {* Complete lattice on @{typ bool} *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   537
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   538
instantiation bool :: complete_lattice
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   539
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   540
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   541
definition
46154
5115e47a7752 use Inf/Sup_bool_def/apply as code equations
haftmann
parents: 46036
diff changeset
   542
  [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   543
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   544
definition
46154
5115e47a7752 use Inf/Sup_bool_def/apply as code equations
haftmann
parents: 46036
diff changeset
   545
  [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   546
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   547
instance proof
44322
43b465f4c480 more concise definition for Inf, Sup on bool
haftmann
parents: 44104
diff changeset
   548
qed (auto intro: bool_induct)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   549
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   550
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   551
49905
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   552
lemma not_False_in_image_Ball [simp]:
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   553
  "False \<notin> P ` A \<longleftrightarrow> Ball A P"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   554
  by auto
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   555
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   556
lemma True_in_image_Bex [simp]:
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   557
  "True \<in> P ` A \<longleftrightarrow> Bex A P"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   558
  by auto
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   559
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   560
lemma INF_bool_eq [simp]:
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   561
  "INFI = Ball"
49905
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   562
  by (simp add: fun_eq_iff INF_def)
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   563
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   564
lemma SUP_bool_eq [simp]:
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   565
  "SUPR = Bex"
49905
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 46884
diff changeset
   566
  by (simp add: fun_eq_iff SUP_def)
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   567
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
   568
instance bool :: complete_boolean_algebra proof
44322
43b465f4c480 more concise definition for Inf, Sup on bool
haftmann
parents: 44104
diff changeset
   569
qed (auto intro: bool_induct)
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   570
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   571
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   572
subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   573
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   574
instantiation "fun" :: (type, complete_lattice) complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   575
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   576
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   577
definition
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   578
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   579
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46693
diff changeset
   580
lemma Inf_apply [simp, code]:
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   581
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   582
  by (simp add: Inf_fun_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   583
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   584
definition
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   585
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   586
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46693
diff changeset
   587
lemma Sup_apply [simp, code]:
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   588
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   589
  by (simp add: Sup_fun_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   590
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   591
instance proof
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   592
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   593
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   594
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   595
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46693
diff changeset
   596
lemma INF_apply [simp]:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   597
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   598
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   599
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46693
diff changeset
   600
lemma SUP_apply [simp]:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   601
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   602
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   603
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   604
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   605
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image)
44024
de7642fcbe1e class complete_distrib_lattice
haftmann
parents: 43967
diff changeset
   606
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   607
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   608
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   609
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   610
subsection {* Complete lattice on unary and binary predicates *}
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   611
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   612
lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   613
  by simp
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   614
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   615
lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   616
  by simp
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   617
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   618
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   619
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   620
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   621
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   622
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   623
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   624
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   625
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   626
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   627
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   628
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   629
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   630
lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   631
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   632
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   633
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"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   634
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   635
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   636
lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   637
  by simp
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   638
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   639
lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   640
  by simp
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   641
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   642
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   643
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   644
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   645
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   646
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   647
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   648
lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   649
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   650
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   651
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"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   652
  by auto
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   653
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   654
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   655
subsection {* Complete lattice on @{typ "_ set"} *}
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   656
45960
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   657
instantiation "set" :: (type) complete_lattice
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   658
begin
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   659
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   660
definition
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   661
  "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   662
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   663
definition
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   664
  "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   665
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   666
instance proof
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   667
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def Inf_bool_def Sup_bool_def le_fun_def)
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   668
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   669
end
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   670
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   671
instance "set" :: (type) complete_boolean_algebra
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   672
proof
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   673
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   674
  
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   675
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   676
subsubsection {* Inter *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   677
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   678
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   679
  "Inter S \<equiv> \<Sqinter>S"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   680
  
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   681
notation (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   682
  Inter  ("\<Inter>_" [90] 90)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   683
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   684
lemma Inter_eq:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   685
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   686
proof (rule set_eqI)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   687
  fix x
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   688
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   689
    by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   690
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
45960
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   691
    by (simp add: Inf_set_def image_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   692
qed
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   693
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   694
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   695
  by (unfold Inter_eq) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   696
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   697
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   698
  by (simp add: Inter_eq)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   699
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   700
text {*
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   701
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   702
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   703
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   704
*}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   705
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   706
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   707
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   708
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   709
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   710
  -- {* ``Classical'' elimination rule -- does not require proving
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   711
    @{prop "X \<in> C"}. *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   712
  by (unfold Inter_eq) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   713
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   714
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   715
  by (fact Inf_lower)
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   716
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   717
lemma Inter_subset:
43755
5e4a595e63fc tuned notation
haftmann
parents: 43754
diff changeset
   718
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   719
  by (fact Inf_less_eq)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   720
43755
5e4a595e63fc tuned notation
haftmann
parents: 43754
diff changeset
   721
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   722
  by (fact Inf_greatest)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   723
44067
5feac96f0e78 declare {INF,SUP}_empty [simp]
huffman
parents: 44041
diff changeset
   724
lemma Inter_empty: "\<Inter>{} = UNIV"
5feac96f0e78 declare {INF,SUP}_empty [simp]
huffman
parents: 44041
diff changeset
   725
  by (fact Inf_empty) (* already simp *)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   726
44067
5feac96f0e78 declare {INF,SUP}_empty [simp]
huffman
parents: 44041
diff changeset
   727
lemma Inter_UNIV: "\<Inter>UNIV = {}"
5feac96f0e78 declare {INF,SUP}_empty [simp]
huffman
parents: 44041
diff changeset
   728
  by (fact Inf_UNIV) (* already simp *)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   729
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   730
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   731
  by (fact Inf_insert) (* already simp *)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   732
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   733
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
   734
  by (fact less_eq_Inf_inter)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   735
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   736
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
43756
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   737
  by (fact Inf_union_distrib)
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   738
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   739
lemma Inter_UNIV_conv [simp, no_atp]:
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   740
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   741
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
43801
097732301fc0 more generalization towards complete lattices
haftmann
parents: 43756
diff changeset
   742
  by (fact Inf_top_conv)+
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   743
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   744
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
   745
  by (fact Inf_superset_mono)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   746
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   747
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   748
subsubsection {* Intersections of families *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   749
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   750
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   751
  "INTER \<equiv> INFI"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   752
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   753
text {*
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   754
  Note: must use name @{const INTER} here instead of @{text INT}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   755
  to allow the following syntax coexist with the plain constant name.
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   756
*}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   757
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   758
syntax
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   759
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   760
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   761
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   762
syntax (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   763
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   764
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   765
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   766
syntax (latex output)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   767
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   768
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   769
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   770
translations
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   771
  "INT x y. B"  == "INT x. INT y. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   772
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   773
  "INT x. B"    == "INT x:CONST UNIV. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   774
  "INT x:A. B"  == "CONST INTER A (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   775
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   776
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   777
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   778
*} -- {* to avoid eta-contraction of body *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   779
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   780
lemma INTER_eq:
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   781
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   782
  by (auto simp add: INF_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   783
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   784
lemma Inter_image_eq [simp]:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   785
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   786
  by (rule sym) (fact INF_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   787
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   788
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   789
  by (auto simp add: INF_def image_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   790
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   791
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   792
  by (auto simp add: INF_def image_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   793
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   794
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   795
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   796
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   797
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"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   798
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   799
  by (auto simp add: INF_def image_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   800
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   801
lemma INT_cong [cong]:
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   802
  "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)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   803
  by (fact INF_cong)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   804
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   805
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   806
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   807
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   808
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   809
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   810
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   811
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   812
  by (fact INF_lower)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   813
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   814
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
   815
  by (fact INF_greatest)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   816
44067
5feac96f0e78 declare {INF,SUP}_empty [simp]
huffman
parents: 44041
diff changeset
   817
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   818
  by (fact INF_empty)
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   819
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   820
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   821
  by (fact INF_absorb)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   822
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   823
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   824
  by (fact le_INF_iff)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   825
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   826
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   827
  by (fact INF_insert)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   828
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   829
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   830
  by (fact INF_union)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   831
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   832
lemma INT_insert_distrib:
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   833
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   834
  by blast
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   835
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   836
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   837
  by (fact INF_constant)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   838
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   839
lemma INTER_UNIV_conv:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   840
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   841
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   842
  by (fact INF_top_conv)+ (* already simp *)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   843
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   844
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   845
  by (fact INF_UNIV_bool_expand)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   846
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   847
lemma INT_anti_mono:
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   848
  "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)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   849
  -- {* The last inclusion is POSITIVE! *}
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   850
  by (fact INF_superset_mono)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   851
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   852
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   853
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   854
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   855
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   856
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   857
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   858
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   859
subsubsection {* Union *}
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   860
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   861
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   862
  "Union S \<equiv> \<Squnion>S"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   863
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   864
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   865
  Union  ("\<Union>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   866
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   867
lemma Union_eq:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   868
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 38705
diff changeset
   869
proof (rule set_eqI)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   870
  fix x
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   871
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   872
    by auto
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   873
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
45960
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   874
    by (simp add: Sup_set_def image_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   875
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   876
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   877
lemma Union_iff [simp, no_atp]:
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   878
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   879
  by (unfold Union_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   880
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   881
lemma UnionI [intro]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   882
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   883
  -- {* The order of the premises presupposes that @{term C} is rigid;
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   884
    @{term A} may be flexible. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   885
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   886
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   887
lemma UnionE [elim!]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   888
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   889
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   890
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   891
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   892
  by (fact Sup_upper)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   893
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   894
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   895
  by (fact Sup_least)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   896
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   897
lemma Union_empty: "\<Union>{} = {}"
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   898
  by (fact Sup_empty) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   899
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   900
lemma Union_UNIV: "\<Union>UNIV = UNIV"
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   901
  by (fact Sup_UNIV) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   902
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   903
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   904
  by (fact Sup_insert) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   905
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   906
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   907
  by (fact Sup_union_distrib)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   908
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   909
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   910
  by (fact Sup_inter_less_eq)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   911
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   912
lemma Union_empty_conv [no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   913
  by (fact Sup_bot_conv) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   914
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   915
lemma empty_Union_conv [no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   916
  by (fact Sup_bot_conv) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   917
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   918
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   919
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   920
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   921
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   922
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   923
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   924
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   925
  by (fact Sup_subset_mono)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   926
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   927
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   928
subsubsection {* Unions of families *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   929
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   930
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   931
  "UNION \<equiv> SUPR"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   932
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   933
text {*
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   934
  Note: must use name @{const UNION} here instead of @{text UN}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   935
  to allow the following syntax coexist with the plain constant name.
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   936
*}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   937
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   938
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   939
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   940
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   941
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   942
syntax (xsymbols)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   943
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   944
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   945
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   946
syntax (latex output)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   947
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   948
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   949
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   950
translations
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   951
  "UN x y. B"   == "UN x. UN y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   952
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   953
  "UN x. B"     == "UN x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   954
  "UN x:A. B"   == "CONST UNION A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   955
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   956
text {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   957
  Note the difference between ordinary xsymbol syntax of indexed
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   958
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   959
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   960
  former does not make the index expression a subscript of the
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   961
  union/intersection symbol because this leads to problems with nested
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   962
  subscripts in Proof General.
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   963
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   964
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   965
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   966
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   967
*} -- {* to avoid eta-contraction of body *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   968
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   969
lemma UNION_eq [no_atp]:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   970
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   971
  by (auto simp add: SUP_def)
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   972
45960
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   973
lemma bind_UNION [code]:
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   974
  "Set.bind A f = UNION A f"
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   975
  by (simp add: bind_def UNION_eq)
e1b09bfb52f1 lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents: 45013
diff changeset
   976
46036
6a86cc88b02f fundamental theorems on Set.bind
haftmann
parents: 45960
diff changeset
   977
lemma member_bind [simp]:
6a86cc88b02f fundamental theorems on Set.bind
haftmann
parents: 45960
diff changeset
   978
  "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
6a86cc88b02f fundamental theorems on Set.bind
haftmann
parents: 45960
diff changeset
   979
  by (simp add: bind_UNION)
6a86cc88b02f fundamental theorems on Set.bind
haftmann
parents: 45960
diff changeset
   980
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   981
lemma Union_image_eq [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   982
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   983
  by (rule sym) (fact SUP_def)
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
   984
46036
6a86cc88b02f fundamental theorems on Set.bind
haftmann
parents: 45960
diff changeset
   985
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   986
  by (auto simp add: SUP_def image_def)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   987
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   988
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   989
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   990
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   991
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   992
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   993
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"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
   994
  by (auto simp add: SUP_def image_def)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   995
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   996
lemma UN_cong [cong]:
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   997
  "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)"
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   998
  by (fact SUP_cong)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   999
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1000
lemma strong_UN_cong:
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1001
  "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)"
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1002
  by (unfold simp_implies_def) (fact UN_cong)
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1003
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1004
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1005
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1006
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1007
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
  1008
  by (fact SUP_upper)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1009
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1010
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
44103
cedaca00789f more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents: 44085
diff changeset
  1011
  by (fact SUP_least)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1012
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
  1013
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1014
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1015
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1016
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1017
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1018
44067
5feac96f0e78 declare {INF,SUP}_empty [simp]
huffman
parents: 44041
diff changeset
  1019
lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}"
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
  1020
  by (fact SUP_empty)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1021
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
  1022
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
  1023
  by (fact SUP_bot) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1024
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1025
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1026
  by (fact SUP_absorb)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1027
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1028
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1029
  by (fact SUP_insert)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1030
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 44084
diff changeset
  1031
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)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1032
  by (fact SUP_union)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1033
43967
610efb6bda1f more coherent structure in and across theories
haftmann
parents: 43944
diff changeset
  1034
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)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1035
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1036
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1037
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
  1038
  by (fact SUP_le_iff)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1039
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1040
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1041
  by (fact SUP_constant)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1042
43944
b1b436f75070 dropped errorneous hint
haftmann
parents: 43943
diff changeset
  1043
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1044
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1045
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
  1046
lemma UNION_empty_conv:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1047
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1048
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
44920
4657b4c11138 remove some redundant [simp] declarations;
huffman
parents: 44919
diff changeset
  1049
  by (fact SUP_bot_conv)+ (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1050
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
  1051
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1052
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1053
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1054
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1055
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1056
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1057
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1058
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1059
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1060
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1061
  by (auto simp add: split_if_mem2)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1062
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1063
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
  1064
  by (fact SUP_UNIV_bool_expand)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1065
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1066
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1067
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1068
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1069
lemma UN_mono:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1070
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1071
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
  1072
  by (fact SUP_subset_mono)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1073
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1074
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1075
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1076
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1077
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1078
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1079
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1080
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1081
  -- {* NOT suitable for rewriting *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1082
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1083
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1084
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1085
  by blast
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1086
45013
05031b71a89a official status for UN_singleton
haftmann
parents: 44995
diff changeset
  1087
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
05031b71a89a official status for UN_singleton
haftmann
parents: 44995
diff changeset
  1088
  by blast
05031b71a89a official status for UN_singleton
haftmann
parents: 44995
diff changeset
  1089
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1090
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
  1091
subsubsection {* Distributive laws *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1092
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1093
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
  1094
  by (fact inf_Sup)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1095
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1096
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1097
  by (fact sup_Inf)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1098
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1099
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1100
  by (fact Sup_inf)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1101
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1102
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)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1103
  by (rule sym) (rule INF_inf_distrib)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1104
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1105
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)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1106
  by (rule sym) (rule SUP_sup_distrib)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1107
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1108
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1109
  by (simp only: INT_Int_distrib INF_def)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1110
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1111
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1112
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1113
  -- {* Union of a family of unions *}
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1114
  by (simp only: UN_Un_distrib SUP_def)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1115
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1116
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1117
  by (fact sup_INF)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1118
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1119
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1120
  -- {* Halmos, Naive Set Theory, page 35. *}
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1121
  by (fact inf_SUP)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1122
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1123
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)"
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1124
  by (fact SUP_inf_distrib2)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1125
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1126
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)"
44039
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1127
  by (fact INF_sup_distrib2)
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1128
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1129
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
c3d0dac940fc generalized lemmas to complete lattices
haftmann
parents: 44032
diff changeset
  1130
  by (fact Sup_inf_eq_bot_iff)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1131
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1132
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
  1133
subsubsection {* Complement *}
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1134
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1135
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1136
  by (fact uminus_INF)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1137
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1138
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1139
  by (fact uminus_SUP)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1140
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1141
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
  1142
subsubsection {* Miniscoping and maxiscoping *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1143
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1144
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1145
           and Intersections. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1146
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1147
lemma UN_simps [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1148
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
  1149
  "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1150
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
  1151
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1152
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1153
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1154
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1155
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1156
  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1157
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1158
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1159
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1160
lemma INT_simps [simp]:
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
  1161
  "\<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)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1162
  "\<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))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1163
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1164
  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1165
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1166
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1167
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1168
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1169
  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1170
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1172
43967
610efb6bda1f more coherent structure in and across theories
haftmann
parents: 43944
diff changeset
  1173
lemma UN_ball_bex_simps [simp, no_atp]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1174
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
43967
610efb6bda1f more coherent structure in and across theories
haftmann
parents: 43944
diff changeset
  1175
  "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1176
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1177
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1178
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1179
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1180
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1181
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1182
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1183
lemma UN_extend_simps:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1184
  "\<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)))"
44032
cb768f4ceaf9 solving duality problem for complete_distrib_lattice; tuned
haftmann
parents: 44029
diff changeset
  1185
  "\<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))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1186
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1187
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1188
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1189
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1190
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1191
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1192
  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1193
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1194
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1195
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1196
lemma INT_extend_simps:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1197
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1198
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1199
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1200
  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1201
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1202
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1203
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1204
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1205
  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1206
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1207
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1208
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1209
text {* Finally *}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1210
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1211
no_notation
46691
72d81e789106 tuned syntax declarations; tuned structure
haftmann
parents: 46689
diff changeset
  1212
  less_eq (infix "\<sqsubseteq>" 50) and
72d81e789106 tuned syntax declarations; tuned structure
haftmann
parents: 46689
diff changeset
  1213
  less (infix "\<sqsubset>" 50)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1214
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1215
lemmas mem_simps =
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1216
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff