src/HOL/Complete_Lattices.thy
author haftmann
Wed Feb 17 21:51:56 2016 +0100 (2016-02-17)
changeset 62343 24106dc44def
parent 62048 fefd79f6b232
child 62390 842917225d56
permissions -rw-r--r--
prefer abbreviations for compound operators INFIMUM and SUPREMUM
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(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
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section \<open>Complete lattices\<close>
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theory Complete_Lattices
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imports Fun
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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 \<open>Syntactic infimum and supremum operations\<close>
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class Inf =
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  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
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begin
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abbreviation INFIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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where
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  "INFIMUM A f \<equiv> \<Sqinter>(f ` A)"
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lemma INF_image [simp]:
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  "INFIMUM (f ` A) g = INFIMUM A (g \<circ> f)"
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  by (simp add: image_comp)
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lemma INF_identity_eq [simp]:
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  "INFIMUM A (\<lambda>x. x) = \<Sqinter>A"
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  by simp
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lemma INF_id_eq [simp]:
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  "INFIMUM A id = \<Sqinter>A"
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  by simp
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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> INFIMUM A C = INFIMUM B D"
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  by (simp add: image_def)
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lemma strong_INF_cong [cong]:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D"
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  unfolding simp_implies_def by (fact INF_cong)
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end
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class Sup =
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  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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begin
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abbreviation SUPREMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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where
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  "SUPREMUM A f \<equiv> \<Squnion>(f ` A)"
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lemma SUP_image [simp]:
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  "SUPREMUM (f ` A) g = SUPREMUM A (g \<circ> f)"
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  by (simp add: image_comp)
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lemma SUP_identity_eq [simp]:
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  "SUPREMUM A (\<lambda>x. x) = \<Squnion>A"
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  by simp
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lemma SUP_id_eq [simp]:
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  "SUPREMUM A id = \<Squnion>A"
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  by (simp add: id_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> SUPREMUM A C = SUPREMUM B D"
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  by (simp add: image_def)
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lemma strong_SUP_cong [cong]:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D"
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  unfolding simp_implies_def by (fact SUP_cong)
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end
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text \<open>
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  Note: must use names @{const INFIMUM} and @{const SUPREMUM} here instead of
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  \<open>INF\<close> and \<open>SUP\<close> to allow the following syntax coexist
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  with the plain constant names.
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\<close>
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syntax (ASCII)
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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 (output)
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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
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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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  "\<Sqinter>x y. B"   \<rightleftharpoons> "\<Sqinter>x. \<Sqinter>y. B"
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  "\<Sqinter>x. B"     \<rightleftharpoons> "CONST INFIMUM CONST UNIV (\<lambda>x. B)"
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  "\<Sqinter>x. B"     \<rightleftharpoons> "\<Sqinter>x \<in> CONST UNIV. B"
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  "\<Sqinter>x\<in>A. B"   \<rightleftharpoons> "CONST INFIMUM A (\<lambda>x. B)"
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  "\<Squnion>x y. B"   \<rightleftharpoons> "\<Squnion>x. \<Squnion>y. B"
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  "\<Squnion>x. B"     \<rightleftharpoons> "CONST SUPREMUM CONST UNIV (\<lambda>x. B)"
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  "\<Squnion>x. B"     \<rightleftharpoons> "\<Squnion>x \<in> CONST UNIV. B"
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  "\<Squnion>x\<in>A. B"   \<rightleftharpoons> "CONST SUPREMUM A (\<lambda>x. B)"
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print_translation \<open>
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFIMUM} @{syntax_const "_INF"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPREMUM} @{syntax_const "_SUP"}]
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\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
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subsection \<open>Abstract complete lattices\<close>
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text \<open>A complete lattice always has a bottom and a top,
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so we include them into the following type class,
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along with assumptions that define bottom and top
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in terms of infimum and supremum.\<close>
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class complete_lattice = lattice + Inf + Sup + bot + top +
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  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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  assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
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  assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
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begin
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subclass bounded_lattice
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proof
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  fix a
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  show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
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  show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
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qed
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lemma dual_complete_lattice:
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  "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
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  by (auto intro!: class.complete_lattice.intro dual_lattice)
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    (unfold_locales, (fact Inf_empty Sup_empty
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        Sup_upper Sup_least Inf_lower Inf_greatest)+)
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end
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context complete_lattice
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begin
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lemma Sup_eqI:
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  "(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x"
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  by (blast intro: antisym Sup_least Sup_upper)
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lemma Inf_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x"
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  by (blast intro: antisym Inf_greatest Inf_lower)
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lemma SUP_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x"
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  using Sup_eqI [of "f ` A" x] by auto
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lemma INF_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x"
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  using Inf_eqI [of "f ` A" x] by auto
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lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
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  using Inf_lower [of _ "f ` A"] by simp
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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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  using Inf_greatest [of "f ` A"] by auto
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lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  using Sup_upper [of _ "f ` A"] by simp
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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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  using Sup_least [of "f ` A"] by auto
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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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  using le_Inf_iff [of _ "f ` A"] by simp
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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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  using Sup_le_iff [of "f ` A"] by simp
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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 [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFIMUM A f"
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  by (simp cong del: strong_INF_cong)
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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 [simp]: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPREMUM A f"
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  by (simp cong del: strong_SUP_cong)
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lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
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  by (simp cong del: strong_INF_cong)
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lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
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  by (simp cong del: strong_SUP_cong)
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lemma Inf_UNIV [simp]:
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  "\<Sqinter>UNIV = \<bottom>"
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  by (auto intro!: antisym Inf_lower)
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lemma Sup_UNIV [simp]:
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  "\<Squnion>UNIV = \<top>"
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  by (auto intro!: antisym Sup_upper)
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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  by (auto intro: Inf_greatest Inf_lower)
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lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
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  by (auto intro: Sup_least Sup_upper)
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lemma Inf_mono:
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  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
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  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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proof (rule Inf_greatest)
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  fix b assume "b \<in> B"
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  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
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  from \<open>a \<in> A\<close> have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
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  with \<open>a \<sqsubseteq> b\<close> 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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  using Inf_mono [of "g ` B" "f ` A"] by auto
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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 \<open>b \<in> B\<close> have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
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  with \<open>a \<sqsubseteq> b\<close> 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)"
haftmann@56166
   266
  using Sup_mono [of "f ` A" "g ` B"] by auto
haftmann@44041
   267
haftmann@44041
   268
lemma INF_superset_mono:
haftmann@44041
   269
  "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)"
wenzelm@61799
   270
  \<comment> \<open>The last inclusion is POSITIVE!\<close>
haftmann@44041
   271
  by (blast intro: INF_mono dest: subsetD)
haftmann@44041
   272
haftmann@44041
   273
lemma SUP_subset_mono:
haftmann@44041
   274
  "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)"
haftmann@44041
   275
  by (blast intro: SUP_mono dest: subsetD)
haftmann@44041
   276
haftmann@43868
   277
lemma Inf_less_eq:
haftmann@43868
   278
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43868
   279
    and "A \<noteq> {}"
haftmann@43868
   280
  shows "\<Sqinter>A \<sqsubseteq> u"
haftmann@43868
   281
proof -
wenzelm@60758
   282
  from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
wenzelm@60758
   283
  moreover from \<open>v \<in> A\<close> assms(1) have "v \<sqsubseteq> u" by blast
haftmann@43868
   284
  ultimately show ?thesis by (rule Inf_lower2)
haftmann@43868
   285
qed
haftmann@43868
   286
haftmann@43868
   287
lemma less_eq_Sup:
haftmann@43868
   288
  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
haftmann@43868
   289
    and "A \<noteq> {}"
haftmann@43868
   290
  shows "u \<sqsubseteq> \<Squnion>A"
haftmann@43868
   291
proof -
wenzelm@60758
   292
  from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
wenzelm@60758
   293
  moreover from \<open>v \<in> A\<close> assms(1) have "u \<sqsubseteq> v" by blast
haftmann@43868
   294
  ultimately show ?thesis by (rule Sup_upper2)
haftmann@43868
   295
qed
haftmann@43868
   296
haftmann@62343
   297
lemma INF_eq:
haftmann@62343
   298
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
haftmann@62343
   299
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
haftmann@62343
   300
  shows "INFIMUM A f = INFIMUM B g"
haftmann@62343
   301
  by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
haftmann@62343
   302
haftmann@56212
   303
lemma SUP_eq:
hoelzl@51328
   304
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
hoelzl@51328
   305
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
haftmann@62343
   306
  shows "SUPREMUM A f = SUPREMUM B g"
hoelzl@51328
   307
  by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
hoelzl@51328
   308
haftmann@43899
   309
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43868
   310
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43868
   311
haftmann@43899
   312
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
haftmann@43868
   313
  by (auto intro: Sup_least Sup_upper)
haftmann@43868
   314
haftmann@43868
   315
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43868
   316
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
haftmann@43868
   317
haftmann@44041
   318
lemma INF_union:
haftmann@44041
   319
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@44103
   320
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
haftmann@44041
   321
haftmann@43868
   322
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
haftmann@43868
   323
  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
haftmann@43868
   324
haftmann@44041
   325
lemma SUP_union:
haftmann@44041
   326
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@44103
   327
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
haftmann@44041
   328
haftmann@44041
   329
lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)"
haftmann@44103
   330
  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041
   331
noschinl@44918
   332
lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" (is "?L = ?R")
noschinl@44918
   333
proof (rule antisym)
noschinl@44918
   334
  show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
noschinl@44918
   335
next
noschinl@44918
   336
  show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
noschinl@44918
   337
qed
haftmann@44041
   338
blanchet@54147
   339
lemma Inf_top_conv [simp]:
haftmann@43868
   340
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   341
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   342
proof -
haftmann@43868
   343
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   344
  proof
haftmann@43868
   345
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   346
    then have "A = {} \<or> A = {\<top>}" by auto
noschinl@44919
   347
    then show "\<Sqinter>A = \<top>" by auto
haftmann@43868
   348
  next
haftmann@43868
   349
    assume "\<Sqinter>A = \<top>"
haftmann@43868
   350
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   351
    proof (rule ccontr)
haftmann@43868
   352
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   353
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868
   354
      then obtain B where "A = insert x B" by blast
wenzelm@60758
   355
      with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp
haftmann@43868
   356
    qed
haftmann@43868
   357
  qed
haftmann@43868
   358
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868
   359
qed
haftmann@43868
   360
noschinl@44918
   361
lemma INF_top_conv [simp]:
haftmann@56166
   362
  "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   363
  "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   364
  using Inf_top_conv [of "B ` A"] by simp_all
haftmann@44041
   365
blanchet@54147
   366
lemma Sup_bot_conv [simp]:
haftmann@43868
   367
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
haftmann@43868
   368
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
huffman@44920
   369
  using dual_complete_lattice
huffman@44920
   370
  by (rule complete_lattice.Inf_top_conv)+
haftmann@43868
   371
noschinl@44918
   372
lemma SUP_bot_conv [simp]:
haftmann@44041
   373
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041
   374
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@56166
   375
  using Sup_bot_conv [of "B ` A"] by simp_all
haftmann@44041
   376
haftmann@43865
   377
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103
   378
  by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077
   379
haftmann@43870
   380
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103
   381
  by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870
   382
noschinl@44918
   383
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
huffman@44921
   384
  by (cases "A = {}") simp_all
haftmann@43900
   385
noschinl@44918
   386
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
huffman@44921
   387
  by (cases "A = {}") simp_all
haftmann@43900
   388
haftmann@43865
   389
lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
haftmann@44103
   390
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865
   391
haftmann@43870
   392
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)"
haftmann@44103
   393
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870
   394
haftmann@43871
   395
lemma INF_absorb:
haftmann@43868
   396
  assumes "k \<in> I"
haftmann@43868
   397
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   398
proof -
haftmann@43868
   399
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   400
  then show ?thesis by simp
haftmann@43868
   401
qed
haftmann@43868
   402
haftmann@43871
   403
lemma SUP_absorb:
haftmann@43871
   404
  assumes "k \<in> I"
haftmann@43871
   405
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   406
proof -
haftmann@43871
   407
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   408
  then show ?thesis by simp
haftmann@43871
   409
qed
haftmann@43871
   410
hoelzl@57448
   411
lemma INF_inf_const1:
hoelzl@57448
   412
  "I \<noteq> {} \<Longrightarrow> (INF i:I. inf x (f i)) = inf x (INF i:I. f i)"
hoelzl@57448
   413
  by (intro antisym INF_greatest inf_mono order_refl INF_lower)
hoelzl@57448
   414
     (auto intro: INF_lower2 le_infI2 intro!: INF_mono)
hoelzl@57448
   415
hoelzl@57448
   416
lemma INF_inf_const2:
hoelzl@57448
   417
  "I \<noteq> {} \<Longrightarrow> (INF i:I. inf (f i) x) = inf (INF i:I. f i) x"
hoelzl@57448
   418
  using INF_inf_const1[of I x f] by (simp add: inf_commute)
hoelzl@57448
   419
haftmann@43871
   420
lemma INF_constant:
haftmann@43868
   421
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921
   422
  by simp
haftmann@43868
   423
haftmann@43871
   424
lemma SUP_constant:
haftmann@43871
   425
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921
   426
  by simp
haftmann@43871
   427
haftmann@43943
   428
lemma less_INF_D:
haftmann@43943
   429
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   430
proof -
wenzelm@60758
   431
  note \<open>y < (\<Sqinter>i\<in>A. f i)\<close>
wenzelm@60758
   432
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using \<open>i \<in> A\<close>
haftmann@44103
   433
    by (rule INF_lower)
haftmann@43943
   434
  finally show "y < f i" .
haftmann@43943
   435
qed
haftmann@43943
   436
haftmann@43943
   437
lemma SUP_lessD:
haftmann@43943
   438
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   439
proof -
wenzelm@60758
   440
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using \<open>i \<in> A\<close>
haftmann@44103
   441
    by (rule SUP_upper)
wenzelm@60758
   442
  also note \<open>(\<Squnion>i\<in>A. f i) < y\<close>
haftmann@43943
   443
  finally show "f i < y" .
haftmann@43943
   444
qed
haftmann@43943
   445
haftmann@43873
   446
lemma INF_UNIV_bool_expand:
haftmann@43868
   447
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@56166
   448
  by (simp add: UNIV_bool inf_commute)
haftmann@43868
   449
haftmann@43873
   450
lemma SUP_UNIV_bool_expand:
haftmann@43871
   451
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@56166
   452
  by (simp add: UNIV_bool sup_commute)
haftmann@43871
   453
hoelzl@51328
   454
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@51328
   455
  by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
hoelzl@51328
   456
haftmann@56218
   457
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f"
haftmann@56166
   458
  using Inf_le_Sup [of "f ` A"] by simp
hoelzl@51328
   459
hoelzl@54414
   460
lemma INF_eq_const:
haftmann@56218
   461
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x"
hoelzl@54414
   462
  by (auto intro: INF_eqI)
hoelzl@54414
   463
haftmann@56248
   464
lemma SUP_eq_const:
haftmann@56248
   465
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x"
haftmann@56248
   466
  by (auto intro: SUP_eqI)
hoelzl@54414
   467
hoelzl@54414
   468
lemma INF_eq_iff:
haftmann@56218
   469
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> (INFIMUM I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
haftmann@56248
   470
  using INF_eq_const [of I f c] INF_lower [of _ I f]
haftmann@56248
   471
  by (auto intro: antisym cong del: strong_INF_cong)
haftmann@56248
   472
haftmann@56248
   473
lemma SUP_eq_iff:
haftmann@56248
   474
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> (SUPREMUM I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
haftmann@56248
   475
  using SUP_eq_const [of I f c] SUP_upper [of _ I f]
haftmann@56248
   476
  by (auto intro: antisym cong del: strong_SUP_cong)
hoelzl@54414
   477
haftmann@32077
   478
end
haftmann@32077
   479
haftmann@44024
   480
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   481
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   482
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   483
begin
haftmann@44024
   484
haftmann@44039
   485
lemma sup_INF:
haftmann@44039
   486
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@62343
   487
  unfolding sup_Inf by simp
haftmann@44039
   488
haftmann@44039
   489
lemma inf_SUP:
haftmann@44039
   490
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@62343
   491
  unfolding inf_Sup by simp
haftmann@44039
   492
haftmann@44032
   493
lemma dual_complete_distrib_lattice:
krauss@44845
   494
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   495
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   496
  apply (fact dual_complete_lattice)
haftmann@44024
   497
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@62343
   498
  apply (simp_all add: inf_Sup sup_Inf)
haftmann@44032
   499
  done
haftmann@44024
   500
haftmann@44322
   501
subclass distrib_lattice proof
haftmann@44024
   502
  fix a b c
haftmann@44024
   503
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
haftmann@62343
   504
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by simp
haftmann@44024
   505
qed
haftmann@44024
   506
haftmann@44039
   507
lemma Inf_sup:
haftmann@44039
   508
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   509
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   510
haftmann@44039
   511
lemma Sup_inf:
haftmann@44039
   512
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   513
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   514
haftmann@44039
   515
lemma INF_sup: 
haftmann@44039
   516
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   517
  by (simp add: sup_INF sup_commute)
haftmann@44039
   518
haftmann@44039
   519
lemma SUP_inf:
haftmann@44039
   520
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   521
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   522
haftmann@44039
   523
lemma Inf_sup_eq_top_iff:
haftmann@44039
   524
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   525
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   526
haftmann@44039
   527
lemma Sup_inf_eq_bot_iff:
haftmann@44039
   528
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   529
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   530
haftmann@44039
   531
lemma INF_sup_distrib2:
haftmann@44039
   532
  "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)"
haftmann@44039
   533
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   534
haftmann@44039
   535
lemma SUP_inf_distrib2:
haftmann@44039
   536
  "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)"
haftmann@44039
   537
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   538
haftmann@56074
   539
context
haftmann@56074
   540
  fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
haftmann@56074
   541
  assumes "mono f"
haftmann@56074
   542
begin
haftmann@56074
   543
haftmann@56074
   544
lemma mono_Inf:
haftmann@56074
   545
  shows "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
wenzelm@60758
   546
  using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
haftmann@56074
   547
haftmann@56074
   548
lemma mono_Sup:
haftmann@56074
   549
  shows "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
wenzelm@60758
   550
  using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
haftmann@56074
   551
hoelzl@60172
   552
lemma mono_INF:
hoelzl@60172
   553
  "f (INF i : I. A i) \<le> (INF x : I. f (A x))"
wenzelm@60758
   554
  by (intro complete_lattice_class.INF_greatest monoD[OF \<open>mono f\<close>] INF_lower)
hoelzl@60172
   555
hoelzl@60172
   556
lemma mono_SUP:
hoelzl@60172
   557
  "(SUP x : I. f (A x)) \<le> f (SUP i : I. A i)"
wenzelm@60758
   558
  by (intro complete_lattice_class.SUP_least monoD[OF \<open>mono f\<close>] SUP_upper)
hoelzl@60172
   559
haftmann@56074
   560
end
haftmann@56074
   561
haftmann@44024
   562
end
haftmann@44024
   563
haftmann@44032
   564
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   565
begin
haftmann@43873
   566
haftmann@43943
   567
lemma dual_complete_boolean_algebra:
krauss@44845
   568
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   569
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   570
haftmann@43873
   571
lemma uminus_Inf:
haftmann@43873
   572
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   573
proof (rule antisym)
haftmann@43873
   574
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   575
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   576
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   577
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   578
qed
haftmann@43873
   579
haftmann@44041
   580
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@62343
   581
  by (simp add: uminus_Inf image_image)
haftmann@44041
   582
haftmann@43873
   583
lemma uminus_Sup:
haftmann@43873
   584
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   585
proof -
haftmann@56166
   586
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_INF)
haftmann@43873
   587
  then show ?thesis by simp
haftmann@43873
   588
qed
haftmann@43873
   589
  
haftmann@43873
   590
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@62343
   591
  by (simp add: uminus_Sup image_image)
haftmann@43873
   592
haftmann@43873
   593
end
haftmann@43873
   594
haftmann@43940
   595
class complete_linorder = linorder + complete_lattice
haftmann@43940
   596
begin
haftmann@43940
   597
haftmann@43943
   598
lemma dual_complete_linorder:
krauss@44845
   599
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   600
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   601
haftmann@51386
   602
lemma complete_linorder_inf_min: "inf = min"
haftmann@51540
   603
  by (auto intro: antisym simp add: min_def fun_eq_iff)
haftmann@51386
   604
haftmann@51386
   605
lemma complete_linorder_sup_max: "sup = max"
haftmann@51540
   606
  by (auto intro: antisym simp add: max_def fun_eq_iff)
haftmann@51386
   607
noschinl@44918
   608
lemma Inf_less_iff:
haftmann@43940
   609
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   610
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   611
noschinl@44918
   612
lemma INF_less_iff:
haftmann@44041
   613
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@56166
   614
  using Inf_less_iff [of "f ` A"] by simp
haftmann@44041
   615
noschinl@44918
   616
lemma less_Sup_iff:
haftmann@43940
   617
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   618
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   619
noschinl@44918
   620
lemma less_SUP_iff:
haftmann@43940
   621
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@56166
   622
  using less_Sup_iff [of _ "f ` A"] by simp
haftmann@43940
   623
noschinl@44918
   624
lemma Sup_eq_top_iff [simp]:
haftmann@43943
   625
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   626
proof
haftmann@43943
   627
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   628
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   629
  proof (intro allI impI)
haftmann@43943
   630
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   631
      unfolding less_Sup_iff by auto
haftmann@43943
   632
  qed
haftmann@43943
   633
next
haftmann@43943
   634
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   635
  show "\<Squnion>A = \<top>"
haftmann@43943
   636
  proof (rule ccontr)
haftmann@43943
   637
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   638
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   639
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   640
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   641
      using * unfolding less_Sup_iff by auto
haftmann@43943
   642
    then show False by auto
haftmann@43943
   643
  qed
haftmann@43943
   644
qed
haftmann@43943
   645
noschinl@44918
   646
lemma SUP_eq_top_iff [simp]:
haftmann@44041
   647
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
haftmann@56166
   648
  using Sup_eq_top_iff [of "f ` A"] by simp
haftmann@44041
   649
noschinl@44918
   650
lemma Inf_eq_bot_iff [simp]:
haftmann@43943
   651
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920
   652
  using dual_complete_linorder
huffman@44920
   653
  by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943
   654
noschinl@44918
   655
lemma INF_eq_bot_iff [simp]:
haftmann@43967
   656
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
haftmann@56166
   657
  using Inf_eq_bot_iff [of "f ` A"] by simp
hoelzl@51328
   658
hoelzl@51328
   659
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
hoelzl@51328
   660
proof safe
hoelzl@51328
   661
  fix y assume "x \<ge> \<Sqinter>A" "y > x"
hoelzl@51328
   662
  then have "y > \<Sqinter>A" by auto
hoelzl@51328
   663
  then show "\<exists>a\<in>A. y > a"
hoelzl@51328
   664
    unfolding Inf_less_iff .
hoelzl@51328
   665
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
hoelzl@51328
   666
hoelzl@51328
   667
lemma INF_le_iff:
haftmann@56218
   668
  "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
haftmann@56166
   669
  using Inf_le_iff [of "f ` A"] by simp
haftmann@56166
   670
haftmann@56166
   671
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
haftmann@56166
   672
proof safe
haftmann@56166
   673
  fix y assume "x \<le> \<Squnion>A" "y < x"
haftmann@56166
   674
  then have "y < \<Squnion>A" by auto
haftmann@56166
   675
  then show "\<exists>a\<in>A. y < a"
haftmann@56166
   676
    unfolding less_Sup_iff .
haftmann@56166
   677
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
haftmann@56166
   678
haftmann@56218
   679
lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
haftmann@56166
   680
  using le_Sup_iff [of _ "f ` A"] by simp
hoelzl@51328
   681
haftmann@51386
   682
subclass complete_distrib_lattice
haftmann@51386
   683
proof
haftmann@51386
   684
  fix a and B
haftmann@51386
   685
  show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@51386
   686
    by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
haftmann@51386
   687
      (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
haftmann@51386
   688
        le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
haftmann@51386
   689
qed
haftmann@51386
   690
haftmann@43940
   691
end
haftmann@43940
   692
hoelzl@51341
   693
wenzelm@60758
   694
subsection \<open>Complete lattice on @{typ bool}\<close>
haftmann@32077
   695
haftmann@44024
   696
instantiation bool :: complete_lattice
haftmann@32077
   697
begin
haftmann@32077
   698
haftmann@32077
   699
definition
haftmann@46154
   700
  [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   701
haftmann@32077
   702
definition
haftmann@46154
   703
  [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   704
haftmann@32077
   705
instance proof
haftmann@44322
   706
qed (auto intro: bool_induct)
haftmann@32077
   707
haftmann@32077
   708
end
haftmann@32077
   709
haftmann@49905
   710
lemma not_False_in_image_Ball [simp]:
haftmann@49905
   711
  "False \<notin> P ` A \<longleftrightarrow> Ball A P"
haftmann@49905
   712
  by auto
haftmann@49905
   713
haftmann@49905
   714
lemma True_in_image_Bex [simp]:
haftmann@49905
   715
  "True \<in> P ` A \<longleftrightarrow> Bex A P"
haftmann@49905
   716
  by auto
haftmann@49905
   717
haftmann@43873
   718
lemma INF_bool_eq [simp]:
haftmann@56218
   719
  "INFIMUM = Ball"
haftmann@62343
   720
  by (simp add: fun_eq_iff)
haftmann@32120
   721
haftmann@43873
   722
lemma SUP_bool_eq [simp]:
haftmann@56218
   723
  "SUPREMUM = Bex"
haftmann@62343
   724
  by (simp add: fun_eq_iff)
haftmann@32120
   725
haftmann@44032
   726
instance bool :: complete_boolean_algebra proof
haftmann@44322
   727
qed (auto intro: bool_induct)
haftmann@44024
   728
haftmann@46631
   729
wenzelm@60758
   730
subsection \<open>Complete lattice on @{typ "_ \<Rightarrow> _"}\<close>
haftmann@46631
   731
nipkow@57197
   732
instantiation "fun" :: (type, Inf) Inf
haftmann@32077
   733
begin
haftmann@32077
   734
haftmann@32077
   735
definition
haftmann@44024
   736
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   737
noschinl@46882
   738
lemma Inf_apply [simp, code]:
haftmann@44024
   739
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   740
  by (simp add: Inf_fun_def)
haftmann@32077
   741
nipkow@57197
   742
instance ..
nipkow@57197
   743
nipkow@57197
   744
end
nipkow@57197
   745
nipkow@57197
   746
instantiation "fun" :: (type, Sup) Sup
nipkow@57197
   747
begin
nipkow@57197
   748
haftmann@32077
   749
definition
haftmann@44024
   750
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   751
noschinl@46882
   752
lemma Sup_apply [simp, code]:
haftmann@44024
   753
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   754
  by (simp add: Sup_fun_def)
haftmann@32077
   755
nipkow@57197
   756
instance ..
nipkow@57197
   757
nipkow@57197
   758
end
nipkow@57197
   759
nipkow@57197
   760
instantiation "fun" :: (type, complete_lattice) complete_lattice
nipkow@57197
   761
begin
nipkow@57197
   762
haftmann@32077
   763
instance proof
noschinl@46884
   764
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   765
haftmann@32077
   766
end
haftmann@32077
   767
noschinl@46882
   768
lemma INF_apply [simp]:
haftmann@41080
   769
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@56166
   770
  using Inf_apply [of "f ` A"] by (simp add: comp_def)
hoelzl@38705
   771
noschinl@46882
   772
lemma SUP_apply [simp]:
haftmann@41080
   773
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@56166
   774
  using Sup_apply [of "f ` A"] by (simp add: comp_def)
haftmann@32077
   775
haftmann@44024
   776
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
haftmann@62343
   777
qed (auto simp add: inf_Sup sup_Inf fun_eq_iff image_image)
haftmann@44024
   778
haftmann@43873
   779
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   780
haftmann@46631
   781
wenzelm@60758
   782
subsection \<open>Complete lattice on unary and binary predicates\<close>
haftmann@46631
   783
haftmann@56742
   784
lemma Inf1_I: 
haftmann@56742
   785
  "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a"
noschinl@46884
   786
  by auto
haftmann@46631
   787
haftmann@56742
   788
lemma INF1_I:
haftmann@56742
   789
  "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   790
  by simp
haftmann@56742
   791
haftmann@56742
   792
lemma INF2_I:
haftmann@56742
   793
  "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   794
  by simp
haftmann@56742
   795
haftmann@56742
   796
lemma Inf2_I: 
haftmann@56742
   797
  "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b"
noschinl@46884
   798
  by auto
haftmann@46631
   799
haftmann@56742
   800
lemma Inf1_D:
haftmann@56742
   801
  "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a"
noschinl@46884
   802
  by auto
haftmann@46631
   803
haftmann@56742
   804
lemma INF1_D:
haftmann@56742
   805
  "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
haftmann@56742
   806
  by simp
haftmann@56742
   807
haftmann@56742
   808
lemma Inf2_D:
haftmann@56742
   809
  "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b"
noschinl@46884
   810
  by auto
haftmann@46631
   811
haftmann@56742
   812
lemma INF2_D:
haftmann@56742
   813
  "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
haftmann@56742
   814
  by simp
haftmann@56742
   815
haftmann@56742
   816
lemma Inf1_E:
haftmann@56742
   817
  assumes "(\<Sqinter>A) a"
haftmann@56742
   818
  obtains "P a" | "P \<notin> A"
haftmann@56742
   819
  using assms by auto
haftmann@46631
   820
haftmann@56742
   821
lemma INF1_E:
haftmann@56742
   822
  assumes "(\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   823
  obtains "B a b" | "a \<notin> A"
haftmann@56742
   824
  using assms by auto
haftmann@56742
   825
haftmann@56742
   826
lemma Inf2_E:
haftmann@56742
   827
  assumes "(\<Sqinter>A) a b"
haftmann@56742
   828
  obtains "r a b" | "r \<notin> A"
haftmann@56742
   829
  using assms by auto
haftmann@56742
   830
haftmann@56742
   831
lemma INF2_E:
haftmann@56742
   832
  assumes "(\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   833
  obtains "B a b c" | "a \<notin> A"
haftmann@56742
   834
  using assms by auto
haftmann@56742
   835
haftmann@56742
   836
lemma Sup1_I:
haftmann@56742
   837
  "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a"
noschinl@46884
   838
  by auto
haftmann@46631
   839
haftmann@56742
   840
lemma SUP1_I:
haftmann@56742
   841
  "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
haftmann@56742
   842
  by auto
haftmann@56742
   843
haftmann@56742
   844
lemma Sup2_I:
haftmann@56742
   845
  "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b"
haftmann@56742
   846
  by auto
haftmann@56742
   847
haftmann@56742
   848
lemma SUP2_I:
haftmann@56742
   849
  "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
noschinl@46884
   850
  by auto
haftmann@46631
   851
haftmann@56742
   852
lemma Sup1_E:
haftmann@56742
   853
  assumes "(\<Squnion>A) a"
haftmann@56742
   854
  obtains P where "P \<in> A" and "P a"
haftmann@56742
   855
  using assms by auto
haftmann@56742
   856
haftmann@56742
   857
lemma SUP1_E:
haftmann@56742
   858
  assumes "(\<Squnion>x\<in>A. B x) b"
haftmann@56742
   859
  obtains x where "x \<in> A" and "B x b"
haftmann@56742
   860
  using assms by auto
haftmann@46631
   861
haftmann@56742
   862
lemma Sup2_E:
haftmann@56742
   863
  assumes "(\<Squnion>A) a b"
haftmann@56742
   864
  obtains r where "r \<in> A" "r a b"
haftmann@56742
   865
  using assms by auto
haftmann@56742
   866
haftmann@56742
   867
lemma SUP2_E:
haftmann@56742
   868
  assumes "(\<Squnion>x\<in>A. B x) b c"
haftmann@56742
   869
  obtains x where "x \<in> A" "B x b c"
haftmann@56742
   870
  using assms by auto
haftmann@46631
   871
haftmann@46631
   872
wenzelm@60758
   873
subsection \<open>Complete lattice on @{typ "_ set"}\<close>
haftmann@46631
   874
haftmann@45960
   875
instantiation "set" :: (type) complete_lattice
haftmann@45960
   876
begin
haftmann@45960
   877
haftmann@45960
   878
definition
haftmann@45960
   879
  "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   880
haftmann@45960
   881
definition
haftmann@45960
   882
  "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   883
haftmann@45960
   884
instance proof
haftmann@51386
   885
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
haftmann@45960
   886
haftmann@45960
   887
end
haftmann@45960
   888
haftmann@45960
   889
instance "set" :: (type) complete_boolean_algebra
haftmann@45960
   890
proof
haftmann@62343
   891
qed (auto simp add: Inf_set_def Sup_set_def image_def)
haftmann@45960
   892
  
haftmann@32077
   893
wenzelm@60758
   894
subsubsection \<open>Inter\<close>
haftmann@41082
   895
wenzelm@61952
   896
abbreviation Inter :: "'a set set \<Rightarrow> 'a set"  ("\<Inter>_" [900] 900)
wenzelm@61952
   897
  where "\<Inter>S \<equiv> \<Sqinter>S"
haftmann@41082
   898
  
haftmann@41082
   899
lemma Inter_eq:
haftmann@41082
   900
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   901
proof (rule set_eqI)
haftmann@41082
   902
  fix x
haftmann@41082
   903
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
haftmann@41082
   904
    by auto
haftmann@41082
   905
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960
   906
    by (simp add: Inf_set_def image_def)
haftmann@41082
   907
qed
haftmann@41082
   908
blanchet@54147
   909
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   910
  by (unfold Inter_eq) blast
haftmann@41082
   911
haftmann@43741
   912
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   913
  by (simp add: Inter_eq)
haftmann@41082
   914
wenzelm@60758
   915
text \<open>
haftmann@41082
   916
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   917
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
wenzelm@61799
   918
  @{prop "X \<in> C"} does not!  This rule is analogous to \<open>spec\<close>.
wenzelm@60758
   919
\<close>
haftmann@41082
   920
haftmann@43741
   921
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   922
  by auto
haftmann@41082
   923
haftmann@43741
   924
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@61799
   925
  \<comment> \<open>``Classical'' elimination rule -- does not require proving
wenzelm@60758
   926
    @{prop "X \<in> C"}.\<close>
haftmann@41082
   927
  by (unfold Inter_eq) blast
haftmann@41082
   928
haftmann@43741
   929
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   930
  by (fact Inf_lower)
haftmann@43740
   931
haftmann@41082
   932
lemma Inter_subset:
haftmann@43755
   933
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   934
  by (fact Inf_less_eq)
haftmann@41082
   935
wenzelm@61952
   936
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A"
haftmann@43740
   937
  by (fact Inf_greatest)
haftmann@41082
   938
huffman@44067
   939
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   940
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   941
huffman@44067
   942
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   943
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   944
huffman@44920
   945
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920
   946
  by (fact Inf_insert) (* already simp *)
haftmann@41082
   947
haftmann@41082
   948
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   949
  by (fact less_eq_Inf_inter)
haftmann@41082
   950
haftmann@41082
   951
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   952
  by (fact Inf_union_distrib)
haftmann@43756
   953
blanchet@54147
   954
lemma Inter_UNIV_conv [simp]:
haftmann@43741
   955
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   956
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   957
  by (fact Inf_top_conv)+
haftmann@41082
   958
haftmann@43741
   959
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   960
  by (fact Inf_superset_mono)
haftmann@41082
   961
haftmann@41082
   962
wenzelm@60758
   963
subsubsection \<open>Intersections of families\<close>
haftmann@41082
   964
wenzelm@61955
   965
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
wenzelm@61955
   966
  where "INTER \<equiv> INFIMUM"
haftmann@41082
   967
wenzelm@60758
   968
text \<open>
wenzelm@61799
   969
  Note: must use name @{const INTER} here instead of \<open>INT\<close>
haftmann@43872
   970
  to allow the following syntax coexist with the plain constant name.
wenzelm@60758
   971
\<close>
haftmann@43872
   972
wenzelm@61955
   973
syntax (ASCII)
wenzelm@61955
   974
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3INT _./ _)" [0, 10] 10)
wenzelm@61955
   975
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   976
haftmann@41082
   977
syntax (latex output)
wenzelm@61955
   978
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
wenzelm@61955
   979
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
wenzelm@61955
   980
wenzelm@61955
   981
syntax
wenzelm@61955
   982
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
wenzelm@61955
   983
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   984
haftmann@41082
   985
translations
wenzelm@61955
   986
  "\<Inter>x y. B"  \<rightleftharpoons> "\<Inter>x. \<Inter>y. B"
wenzelm@61955
   987
  "\<Inter>x. B"    \<rightleftharpoons> "CONST INTER CONST UNIV (\<lambda>x. B)"
wenzelm@61955
   988
  "\<Inter>x. B"    \<rightleftharpoons> "\<Inter>x \<in> CONST UNIV. B"
wenzelm@61955
   989
  "\<Inter>x\<in>A. B"  \<rightleftharpoons> "CONST INTER A (\<lambda>x. B)"
haftmann@41082
   990
wenzelm@60758
   991
print_translation \<open>
wenzelm@42284
   992
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
wenzelm@61799
   993
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
haftmann@41082
   994
haftmann@44085
   995
lemma INTER_eq:
haftmann@41082
   996
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@56166
   997
  by (auto intro!: INF_eqI)
haftmann@41082
   998
haftmann@43817
   999
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@56166
  1000
  using Inter_iff [of _ "B ` A"] by simp
haftmann@41082
  1001
haftmann@43817
  1002
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
haftmann@62343
  1003
  by auto
haftmann@41082
  1004
haftmann@43852
  1005
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
haftmann@41082
  1006
  by auto
haftmann@41082
  1007
haftmann@43852
  1008
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"
wenzelm@61799
  1009
  \<comment> \<open>"Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}.\<close>
haftmann@62343
  1010
  by auto
haftmann@41082
  1011
haftmann@41082
  1012
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
  1013
  by blast
haftmann@41082
  1014
haftmann@41082
  1015
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
  1016
  by blast
haftmann@41082
  1017
haftmann@43817
  1018
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
  1019
  by (fact INF_lower)
haftmann@41082
  1020
haftmann@43817
  1021
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
  1022
  by (fact INF_greatest)
haftmann@41082
  1023
huffman@44067
  1024
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
  1025
  by (fact INF_empty)
haftmann@43854
  1026
haftmann@43817
  1027
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
  1028
  by (fact INF_absorb)
haftmann@41082
  1029
haftmann@43854
  1030
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
  1031
  by (fact le_INF_iff)
haftmann@41082
  1032
haftmann@41082
  1033
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
  1034
  by (fact INF_insert)
haftmann@43865
  1035
haftmann@43865
  1036
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
haftmann@43865
  1037
  by (fact INF_union)
haftmann@43865
  1038
haftmann@43865
  1039
lemma INT_insert_distrib:
haftmann@43865
  1040
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
  1041
  by blast
haftmann@43854
  1042
haftmann@41082
  1043
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
  1044
  by (fact INF_constant)
haftmann@43865
  1045
huffman@44920
  1046
lemma INTER_UNIV_conv:
haftmann@43817
  1047
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
  1048
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920
  1049
  by (fact INF_top_conv)+ (* already simp *)
haftmann@43865
  1050
haftmann@43865
  1051
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
  1052
  by (fact INF_UNIV_bool_expand)
haftmann@43865
  1053
haftmann@43865
  1054
lemma INT_anti_mono:
haftmann@43900
  1055
  "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)"
wenzelm@61799
  1056
  \<comment> \<open>The last inclusion is POSITIVE!\<close>
haftmann@43940
  1057
  by (fact INF_superset_mono)
haftmann@41082
  1058
haftmann@41082
  1059
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
  1060
  by blast
haftmann@41082
  1061
haftmann@43817
  1062
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
  1063
  by blast
haftmann@41082
  1064
haftmann@41082
  1065
wenzelm@60758
  1066
subsubsection \<open>Union\<close>
haftmann@32115
  1067
wenzelm@61952
  1068
abbreviation Union :: "'a set set \<Rightarrow> 'a set"  ("\<Union>_" [900] 900)
wenzelm@61952
  1069
  where "\<Union>S \<equiv> \<Squnion>S"
haftmann@32115
  1070
haftmann@32135
  1071
lemma Union_eq:
haftmann@32135
  1072
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
  1073
proof (rule set_eqI)
haftmann@32115
  1074
  fix x
haftmann@32135
  1075
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
haftmann@32115
  1076
    by auto
haftmann@32135
  1077
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960
  1078
    by (simp add: Sup_set_def image_def)
haftmann@32115
  1079
qed
haftmann@32115
  1080
blanchet@54147
  1081
lemma Union_iff [simp]:
haftmann@32115
  1082
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
  1083
  by (unfold Union_eq) blast
haftmann@32115
  1084
haftmann@32115
  1085
lemma UnionI [intro]:
haftmann@32115
  1086
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
wenzelm@61799
  1087
  \<comment> \<open>The order of the premises presupposes that @{term C} is rigid;
wenzelm@60758
  1088
    @{term A} may be flexible.\<close>
haftmann@32115
  1089
  by auto
haftmann@32115
  1090
haftmann@32115
  1091
lemma UnionE [elim!]:
haftmann@43817
  1092
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
  1093
  by auto
haftmann@32115
  1094
haftmann@43817
  1095
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
  1096
  by (fact Sup_upper)
haftmann@32135
  1097
haftmann@43817
  1098
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
  1099
  by (fact Sup_least)
haftmann@32135
  1100
huffman@44920
  1101
lemma Union_empty: "\<Union>{} = {}"
huffman@44920
  1102
  by (fact Sup_empty) (* already simp *)
haftmann@32135
  1103
huffman@44920
  1104
lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920
  1105
  by (fact Sup_UNIV) (* already simp *)
haftmann@32135
  1106
huffman@44920
  1107
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920
  1108
  by (fact Sup_insert) (* already simp *)
haftmann@32135
  1109
haftmann@43817
  1110
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
  1111
  by (fact Sup_union_distrib)
haftmann@32135
  1112
haftmann@32135
  1113
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
  1114
  by (fact Sup_inter_less_eq)
haftmann@32135
  1115
blanchet@54147
  1116
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1117
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1118
blanchet@54147
  1119
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1120
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1121
haftmann@32135
  1122
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
  1123
  by blast
haftmann@32135
  1124
haftmann@32135
  1125
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
  1126
  by blast
haftmann@32135
  1127
haftmann@43817
  1128
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
  1129
  by (fact Sup_subset_mono)
haftmann@32135
  1130
haftmann@32115
  1131
wenzelm@60758
  1132
subsubsection \<open>Unions of families\<close>
haftmann@32077
  1133
wenzelm@61955
  1134
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
wenzelm@61955
  1135
  where "UNION \<equiv> SUPREMUM"
haftmann@32077
  1136
wenzelm@60758
  1137
text \<open>
wenzelm@61799
  1138
  Note: must use name @{const UNION} here instead of \<open>UN\<close>
haftmann@43872
  1139
  to allow the following syntax coexist with the plain constant name.
wenzelm@60758
  1140
\<close>
haftmann@43872
  1141
wenzelm@61955
  1142
syntax (ASCII)
wenzelm@35115
  1143
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
  1144
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
  1145
haftmann@32077
  1146
syntax (latex output)
wenzelm@35115
  1147
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
  1148
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
  1149
wenzelm@61955
  1150
syntax
wenzelm@61955
  1151
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
wenzelm@61955
  1152
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
wenzelm@61955
  1153
haftmann@32077
  1154
translations
wenzelm@61955
  1155
  "\<Union>x y. B"   \<rightleftharpoons> "\<Union>x. \<Union>y. B"
wenzelm@61955
  1156
  "\<Union>x. B"     \<rightleftharpoons> "CONST UNION CONST UNIV (\<lambda>x. B)"
wenzelm@61955
  1157
  "\<Union>x. B"     \<rightleftharpoons> "\<Union>x \<in> CONST UNIV. B"
wenzelm@61955
  1158
  "\<Union>x\<in>A. B"   \<rightleftharpoons> "CONST UNION A (\<lambda>x. B)"
haftmann@32077
  1159
wenzelm@60758
  1160
text \<open>
wenzelm@61955
  1161
  Note the difference between ordinary syntax of indexed
wenzelm@61799
  1162
  unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>)
wenzelm@61955
  1163
  and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}.
wenzelm@60758
  1164
\<close>
haftmann@32077
  1165
wenzelm@60758
  1166
print_translation \<open>
wenzelm@42284
  1167
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@61799
  1168
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
haftmann@32077
  1169
blanchet@54147
  1170
lemma UNION_eq:
haftmann@32135
  1171
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@56166
  1172
  by (auto intro!: SUP_eqI)
huffman@44920
  1173
haftmann@45960
  1174
lemma bind_UNION [code]:
haftmann@45960
  1175
  "Set.bind A f = UNION A f"
haftmann@45960
  1176
  by (simp add: bind_def UNION_eq)
haftmann@45960
  1177
haftmann@46036
  1178
lemma member_bind [simp]:
haftmann@46036
  1179
  "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036
  1180
  by (simp add: bind_UNION)
haftmann@46036
  1181
wenzelm@60585
  1182
lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
lp15@60307
  1183
  by blast
lp15@60307
  1184
haftmann@46036
  1185
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@56166
  1186
  using Union_iff [of _ "B ` A"] by simp
wenzelm@11979
  1187
haftmann@43852
  1188
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@61799
  1189
  \<comment> \<open>The order of the premises presupposes that @{term A} is rigid;
wenzelm@60758
  1190
    @{term b} may be flexible.\<close>
wenzelm@11979
  1191
  by auto
wenzelm@11979
  1192
haftmann@43852
  1193
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@62343
  1194
  by auto
haftmann@32077
  1195
haftmann@43817
  1196
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
  1197
  by (fact SUP_upper)
haftmann@32135
  1198
haftmann@43817
  1199
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
  1200
  by (fact SUP_least)
haftmann@32135
  1201
blanchet@54147
  1202
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
  1203
  by blast
haftmann@32135
  1204
haftmann@43817
  1205
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
  1206
  by blast
haftmann@32135
  1207
blanchet@54147
  1208
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
  1209
  by (fact SUP_empty)
haftmann@32135
  1210
huffman@44920
  1211
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920
  1212
  by (fact SUP_bot) (* already simp *)
haftmann@32135
  1213
haftmann@43817
  1214
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
  1215
  by (fact SUP_absorb)
haftmann@32135
  1216
haftmann@32135
  1217
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
  1218
  by (fact SUP_insert)
haftmann@32135
  1219
haftmann@44085
  1220
lemma UN_Un [simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
haftmann@43900
  1221
  by (fact SUP_union)
haftmann@32135
  1222
haftmann@43967
  1223
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
haftmann@32135
  1224
  by blast
haftmann@32135
  1225
haftmann@32135
  1226
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
  1227
  by (fact SUP_le_iff)
haftmann@32135
  1228
haftmann@32135
  1229
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
  1230
  by (fact SUP_constant)
haftmann@32135
  1231
haftmann@43944
  1232
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
  1233
  by blast
haftmann@32135
  1234
huffman@44920
  1235
lemma UNION_empty_conv:
haftmann@43817
  1236
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1237
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920
  1238
  by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135
  1239
blanchet@54147
  1240
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1241
  by blast
haftmann@32135
  1242
haftmann@43900
  1243
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
  1244
  by blast
haftmann@32135
  1245
haftmann@43900
  1246
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
  1247
  by blast
haftmann@32135
  1248
haftmann@32135
  1249
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@62343
  1250
  by safe (auto simp add: split_if_mem2)
haftmann@32135
  1251
haftmann@43817
  1252
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1253
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1254
haftmann@32135
  1255
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1256
  by blast
haftmann@32135
  1257
haftmann@32135
  1258
lemma UN_mono:
haftmann@43817
  1259
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1260
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1261
  by (fact SUP_subset_mono)
haftmann@32135
  1262
haftmann@43817
  1263
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1264
  by blast
haftmann@32135
  1265
haftmann@43817
  1266
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1267
  by blast
haftmann@32135
  1268
haftmann@43817
  1269
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
wenzelm@61799
  1270
  \<comment> \<open>NOT suitable for rewriting\<close>
haftmann@32135
  1271
  by blast
haftmann@32135
  1272
haftmann@43817
  1273
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1274
  by blast
haftmann@32135
  1275
haftmann@45013
  1276
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013
  1277
  by blast
haftmann@45013
  1278
wenzelm@11979
  1279
wenzelm@60758
  1280
subsubsection \<open>Distributive laws\<close>
wenzelm@12897
  1281
wenzelm@12897
  1282
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1283
  by (fact inf_Sup)
wenzelm@12897
  1284
haftmann@44039
  1285
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1286
  by (fact sup_Inf)
haftmann@44039
  1287
wenzelm@12897
  1288
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1289
  by (fact Sup_inf)
haftmann@44039
  1290
haftmann@44039
  1291
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
haftmann@44039
  1292
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1293
haftmann@44039
  1294
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
haftmann@44039
  1295
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1296
wenzelm@61799
  1297
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" \<comment> \<open>FIXME drop\<close>
haftmann@56166
  1298
  by (simp add: INT_Int_distrib)
wenzelm@12897
  1299
wenzelm@61799
  1300
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" \<comment> \<open>FIXME drop\<close>
wenzelm@61799
  1301
  \<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close>
wenzelm@61799
  1302
  \<comment> \<open>Union of a family of unions\<close>
haftmann@56166
  1303
  by (simp add: UN_Un_distrib)
wenzelm@12897
  1304
haftmann@44039
  1305
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1306
  by (fact sup_INF)
wenzelm@12897
  1307
wenzelm@12897
  1308
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@61799
  1309
  \<comment> \<open>Halmos, Naive Set Theory, page 35.\<close>
haftmann@44039
  1310
  by (fact inf_SUP)
wenzelm@12897
  1311
wenzelm@12897
  1312
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
haftmann@44039
  1313
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1314
wenzelm@12897
  1315
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
haftmann@44039
  1316
  by (fact INF_sup_distrib2)
haftmann@44039
  1317
haftmann@44039
  1318
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1319
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1320
Andreas@61630
  1321
lemma SUP_UNION: "(SUP x:(UN y:A. g y). f x) = (SUP y:A. SUP x:g y. f x :: _ :: complete_lattice)"
Andreas@61630
  1322
by(rule order_antisym)(blast intro: SUP_least SUP_upper2)+
wenzelm@12897
  1323
wenzelm@60758
  1324
subsection \<open>Injections and bijections\<close>
haftmann@56015
  1325
haftmann@56015
  1326
lemma inj_on_Inter:
haftmann@56015
  1327
  "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
haftmann@56015
  1328
  unfolding inj_on_def by blast
haftmann@56015
  1329
haftmann@56015
  1330
lemma inj_on_INTER:
haftmann@56015
  1331
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)) \<Longrightarrow> inj_on f (\<Inter>i \<in> I. A i)"
haftmann@62343
  1332
  unfolding inj_on_def by safe simp
haftmann@56015
  1333
haftmann@56015
  1334
lemma inj_on_UNION_chain:
haftmann@56015
  1335
  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
haftmann@56015
  1336
         INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
wenzelm@60585
  1337
  shows "inj_on f (\<Union>i \<in> I. A i)"
haftmann@56015
  1338
proof -
haftmann@56015
  1339
  {
haftmann@56015
  1340
    fix i j x y
haftmann@56015
  1341
    assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
haftmann@56015
  1342
      and ***: "f x = f y"
haftmann@56015
  1343
    have "x = y"
haftmann@56015
  1344
    proof -
haftmann@56015
  1345
      {
haftmann@56015
  1346
        assume "A i \<le> A j"
haftmann@56015
  1347
        with ** have "x \<in> A j" by auto
haftmann@56015
  1348
        with INJ * ** *** have ?thesis
haftmann@56015
  1349
        by(auto simp add: inj_on_def)
haftmann@56015
  1350
      }
haftmann@56015
  1351
      moreover
haftmann@56015
  1352
      {
haftmann@56015
  1353
        assume "A j \<le> A i"
haftmann@56015
  1354
        with ** have "y \<in> A i" by auto
haftmann@56015
  1355
        with INJ * ** *** have ?thesis
haftmann@56015
  1356
        by(auto simp add: inj_on_def)
haftmann@56015
  1357
      }
haftmann@56015
  1358
      ultimately show ?thesis using CH * by blast
haftmann@56015
  1359
    qed
haftmann@56015
  1360
  }
haftmann@56015
  1361
  then show ?thesis by (unfold inj_on_def UNION_eq) auto
haftmann@56015
  1362
qed
haftmann@56015
  1363
haftmann@56015
  1364
lemma bij_betw_UNION_chain:
haftmann@56015
  1365
  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
haftmann@56015
  1366
         BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
wenzelm@60585
  1367
  shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)"
haftmann@56015
  1368
proof (unfold bij_betw_def, auto)
haftmann@56015
  1369
  have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
haftmann@56015
  1370
  using BIJ bij_betw_def[of f] by auto
wenzelm@60585
  1371
  thus "inj_on f (\<Union>i \<in> I. A i)"
haftmann@56015
  1372
  using CH inj_on_UNION_chain[of I A f] by auto
haftmann@56015
  1373
next
haftmann@56015
  1374
  fix i x
haftmann@56015
  1375
  assume *: "i \<in> I" "x \<in> A i"
haftmann@56015
  1376
  hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
haftmann@56015
  1377
  thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
haftmann@56015
  1378
next
haftmann@56015
  1379
  fix i x'
haftmann@56015
  1380
  assume *: "i \<in> I" "x' \<in> A' i"
haftmann@56015
  1381
  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
haftmann@56015
  1382
  then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
haftmann@56015
  1383
    using * by blast
haftmann@56015
  1384
  then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by blast
haftmann@56015
  1385
qed
haftmann@56015
  1386
haftmann@56015
  1387
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
haftmann@56015
  1388
lemma image_INT:
haftmann@56015
  1389
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
haftmann@56015
  1390
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
haftmann@62343
  1391
  by (simp add: inj_on_def, auto) blast
haftmann@56015
  1392
haftmann@56015
  1393
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
haftmann@62343
  1394
  apply (simp add: bij_def)
haftmann@62343
  1395
  apply (simp add: inj_on_def surj_def)
haftmann@62343
  1396
  apply auto
haftmann@62343
  1397
  apply blast
haftmann@62343
  1398
  done
haftmann@56015
  1399
haftmann@56015
  1400
lemma UNION_fun_upd:
haftmann@62343
  1401
  "UNION J (A(i := B)) = UNION (J - {i}) A \<union> (if i \<in> J then B else {})"
haftmann@62343
  1402
  by (auto simp add: set_eq_iff)
haftmann@62343
  1403
  
haftmann@56015
  1404
wenzelm@60758
  1405
subsubsection \<open>Complement\<close>
haftmann@32135
  1406
haftmann@43873
  1407
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1408
  by (fact uminus_INF)
wenzelm@12897
  1409
haftmann@43873
  1410
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1411
  by (fact uminus_SUP)
wenzelm@12897
  1412
wenzelm@12897
  1413
wenzelm@60758
  1414
subsubsection \<open>Miniscoping and maxiscoping\<close>
wenzelm@12897
  1415
wenzelm@60758
  1416
text \<open>\medskip Miniscoping: pushing in quantifiers and big Unions
wenzelm@60758
  1417
           and Intersections.\<close>
wenzelm@12897
  1418
wenzelm@12897
  1419
lemma UN_simps [simp]:
haftmann@43817
  1420
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
haftmann@44032
  1421
  "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
haftmann@43852
  1422
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
haftmann@44032
  1423
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1424
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1425
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1426
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1427
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1428
  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
haftmann@43831
  1429
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1430
  by auto
wenzelm@12897
  1431
wenzelm@12897
  1432
lemma INT_simps [simp]:
haftmann@44032
  1433
  "\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter> B)"
haftmann@43831
  1434
  "\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))"
haftmann@43852
  1435
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
haftmann@43852
  1436
  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
haftmann@43817
  1437
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1438
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1439
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1440
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1441
  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
haftmann@43852
  1442
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1443
  by auto
wenzelm@12897
  1444
blanchet@54147
  1445
lemma UN_ball_bex_simps [simp]:
haftmann@43852
  1446
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1447
  "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
haftmann@43852
  1448
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1449
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
wenzelm@12897
  1450
  by auto
wenzelm@12897
  1451
haftmann@43943
  1452
wenzelm@60758
  1453
text \<open>\medskip Maxiscoping: pulling out big Unions and Intersections.\<close>
paulson@13860
  1454
paulson@13860
  1455
lemma UN_extend_simps:
haftmann@43817
  1456
  "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"
haftmann@44032
  1457
  "\<And>A B C. (\<Union>x\<in>C. A x) \<union> B = (if C={} then B else (\<Union>x\<in>C. A x \<union> B))"
haftmann@43852
  1458
  "\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))"
haftmann@43852
  1459
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1460
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1461
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1462
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1463
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1464
  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
haftmann@43831
  1465
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1466
  by auto
paulson@13860
  1467
paulson@13860
  1468
lemma INT_extend_simps:
haftmann@43852
  1469
  "\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))"
haftmann@43852
  1470
  "\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))"
haftmann@43852
  1471
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
haftmann@43852
  1472
  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
haftmann@43817
  1473
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1474
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1475
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1476
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1477
  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
haftmann@43852
  1478
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1479
  by auto
paulson@13860
  1480
wenzelm@60758
  1481
text \<open>Finally\<close>
haftmann@43872
  1482
haftmann@32135
  1483
no_notation
haftmann@46691
  1484
  less_eq (infix "\<sqsubseteq>" 50) and
haftmann@46691
  1485
  less (infix "\<sqsubset>" 50)
haftmann@32135
  1486
haftmann@30596
  1487
lemmas mem_simps =
haftmann@30596
  1488
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1489
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
wenzelm@61799
  1490
  \<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close>
wenzelm@21669
  1491
wenzelm@11979
  1492
end
haftmann@49905
  1493