src/HOL/Complete_Lattices.thy
author wenzelm
Tue Sep 26 20:54:40 2017 +0200 (22 months ago)
changeset 66695 91500c024c7f
parent 64966 d53d7ca3303e
child 67399 eab6ce8368fa
permissions -rw-r--r--
tuned;
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(*  Title:      HOL/Complete_Lattices.thy
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Florian Haftmann
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*)
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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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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 "INFIMUM A f \<equiv> \<Sqinter>(f ` A)"
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lemma INF_image [simp]: "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]: "INFIMUM A (\<lambda>x. x) = \<Sqinter>A"
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  by simp
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lemma INF_id_eq [simp]: "INFIMUM A id = \<Sqinter>A"
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  by simp
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lemma INF_cong: "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 "SUPREMUM A f \<equiv> \<Squnion>(f ` A)"
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lemma SUP_image [simp]: "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]: "SUPREMUM A (\<lambda>x. x) = \<Squnion>A"
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  by simp
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lemma SUP_id_eq [simp]: "SUPREMUM A id = \<Squnion>A"
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  by (simp add: id_def)
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lemma SUP_cong: "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 \<le> x"
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    and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> \<Sqinter>A"
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    and Sup_upper: "x \<in> A \<Longrightarrow> x \<le> \<Squnion>A"
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    and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> \<Squnion>A \<le> z"
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    and Inf_empty [simp]: "\<Sqinter>{} = \<top>"
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    and 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"
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    by (auto intro: Sup_least simp only: Sup_empty [symmetric])
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  show "a \<le> \<top>"
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    by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
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qed
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lemma dual_complete_lattice: "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 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) \<le> 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 \<le> f i) \<Longrightarrow> u \<le> (\<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 \<le> (\<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 \<le> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<le> 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 \<le> v \<Longrightarrow> \<Sqinter>A \<le> 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 \<le> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<le> 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 \<le> u \<Longrightarrow> v \<le> \<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 \<le> f i \<Longrightarrow> u \<le> (\<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 \<le> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<le> a)"
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  by (auto intro: Inf_greatest dest: Inf_lower)
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lemma le_INF_iff: "u \<le> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<le> 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 \<le> b \<longleftrightarrow> (\<forall>a\<in>A. a \<le> 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) \<le> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<le> 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]: "\<Sqinter>UNIV = \<bottom>"
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  by (auto intro!: antisym Inf_lower)
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lemma Sup_UNIV [simp]: "\<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 \<le> 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 \<le> 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 \<le> \<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 \<le> \<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 \<le> b"
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  shows "\<Sqinter>A \<le> \<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 \<le> b" by blast
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  from \<open>a \<in> A\<close> have "\<Sqinter>A \<le> a" by (rule Inf_lower)
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  with \<open>a \<le> b\<close> show "\<Sqinter>A \<le> b" by auto
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qed
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lemma INF_mono: "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<le> (\<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 \<le> b"
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  shows "\<Squnion>A \<le> \<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 \<le> b" by blast
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  from \<open>b \<in> B\<close> have "b \<le> \<Squnion>B" by (rule Sup_upper)
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  with \<open>a \<le> b\<close> show "a \<le> \<Squnion>B" by auto
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qed
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lemma SUP_mono: "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<le> (\<Squnion>n\<in>B. g n)"
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  using Sup_mono [of "f ` A" "g ` B"] by auto
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lemma INF_superset_mono: "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<le> (\<Sqinter>x\<in>B. g x)"
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  \<comment> \<open>The last inclusion is POSITIVE!\<close>
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  by (blast intro: INF_mono dest: subsetD)
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lemma SUP_subset_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<le> (\<Squnion>x\<in>B. g x)"
haftmann@44041
   259
  by (blast intro: SUP_mono dest: subsetD)
haftmann@44041
   260
haftmann@43868
   261
lemma Inf_less_eq:
haftmann@63820
   262
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<le> u"
haftmann@43868
   263
    and "A \<noteq> {}"
haftmann@63820
   264
  shows "\<Sqinter>A \<le> u"
haftmann@43868
   265
proof -
wenzelm@60758
   266
  from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
haftmann@63820
   267
  moreover from \<open>v \<in> A\<close> assms(1) have "v \<le> u" by blast
haftmann@43868
   268
  ultimately show ?thesis by (rule Inf_lower2)
haftmann@43868
   269
qed
haftmann@43868
   270
haftmann@43868
   271
lemma less_eq_Sup:
haftmann@63820
   272
  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<le> v"
haftmann@43868
   273
    and "A \<noteq> {}"
haftmann@63820
   274
  shows "u \<le> \<Squnion>A"
haftmann@43868
   275
proof -
wenzelm@60758
   276
  from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
haftmann@63820
   277
  moreover from \<open>v \<in> A\<close> assms(1) have "u \<le> v" by blast
haftmann@43868
   278
  ultimately show ?thesis by (rule Sup_upper2)
haftmann@43868
   279
qed
haftmann@43868
   280
haftmann@62343
   281
lemma INF_eq:
haftmann@62343
   282
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
wenzelm@63575
   283
    and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
haftmann@62343
   284
  shows "INFIMUM A f = INFIMUM B g"
haftmann@62343
   285
  by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
haftmann@62343
   286
haftmann@56212
   287
lemma SUP_eq:
hoelzl@51328
   288
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
wenzelm@63575
   289
    and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
haftmann@62343
   290
  shows "SUPREMUM A f = SUPREMUM B g"
hoelzl@51328
   291
  by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
hoelzl@51328
   292
haftmann@63820
   293
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<le> \<Sqinter>(A \<inter> B)"
haftmann@43868
   294
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43868
   295
haftmann@63820
   296
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<le> \<Squnion>A \<sqinter> \<Squnion>B "
haftmann@43868
   297
  by (auto intro: Sup_least Sup_upper)
haftmann@43868
   298
haftmann@43868
   299
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43868
   300
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
haftmann@43868
   301
wenzelm@63575
   302
lemma INF_union: "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@44103
   303
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
haftmann@44041
   304
haftmann@43868
   305
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
haftmann@43868
   306
  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
haftmann@43868
   307
wenzelm@63575
   308
lemma SUP_union: "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@44103
   309
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
haftmann@44041
   310
haftmann@44041
   311
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
   312
  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041
   313
wenzelm@63575
   314
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)"
wenzelm@63575
   315
  (is "?L = ?R")
noschinl@44918
   316
proof (rule antisym)
wenzelm@63575
   317
  show "?L \<le> ?R"
wenzelm@63575
   318
    by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
wenzelm@63575
   319
  show "?R \<le> ?L"
wenzelm@63575
   320
    by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
noschinl@44918
   321
qed
haftmann@44041
   322
blanchet@54147
   323
lemma Inf_top_conv [simp]:
haftmann@43868
   324
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   325
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   326
proof -
haftmann@43868
   327
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   328
  proof
haftmann@43868
   329
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   330
    then have "A = {} \<or> A = {\<top>}" by auto
noschinl@44919
   331
    then show "\<Sqinter>A = \<top>" by auto
haftmann@43868
   332
  next
haftmann@43868
   333
    assume "\<Sqinter>A = \<top>"
haftmann@43868
   334
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   335
    proof (rule ccontr)
haftmann@43868
   336
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   337
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868
   338
      then obtain B where "A = insert x B" by blast
wenzelm@60758
   339
      with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp
haftmann@43868
   340
    qed
haftmann@43868
   341
  qed
haftmann@43868
   342
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868
   343
qed
haftmann@43868
   344
noschinl@44918
   345
lemma INF_top_conv [simp]:
haftmann@56166
   346
  "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   347
  "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   348
  using Inf_top_conv [of "B ` A"] by simp_all
haftmann@44041
   349
blanchet@54147
   350
lemma Sup_bot_conv [simp]:
wenzelm@63575
   351
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)"
wenzelm@63575
   352
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)"
huffman@44920
   353
  using dual_complete_lattice
huffman@44920
   354
  by (rule complete_lattice.Inf_top_conv)+
haftmann@43868
   355
noschinl@44918
   356
lemma SUP_bot_conv [simp]:
wenzelm@63575
   357
  "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
wenzelm@63575
   358
  "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@56166
   359
  using Sup_bot_conv [of "B ` A"] by simp_all
haftmann@44041
   360
haftmann@43865
   361
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103
   362
  by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077
   363
haftmann@43870
   364
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103
   365
  by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870
   366
noschinl@44918
   367
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
huffman@44921
   368
  by (cases "A = {}") simp_all
haftmann@43900
   369
noschinl@44918
   370
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
huffman@44921
   371
  by (cases "A = {}") simp_all
haftmann@43900
   372
haftmann@43865
   373
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
   374
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865
   375
haftmann@43870
   376
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
   377
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870
   378
haftmann@43871
   379
lemma INF_absorb:
haftmann@43868
   380
  assumes "k \<in> I"
haftmann@43868
   381
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   382
proof -
haftmann@43868
   383
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   384
  then show ?thesis by simp
haftmann@43868
   385
qed
haftmann@43868
   386
haftmann@43871
   387
lemma SUP_absorb:
haftmann@43871
   388
  assumes "k \<in> I"
haftmann@43871
   389
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   390
proof -
haftmann@43871
   391
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   392
  then show ?thesis by simp
haftmann@43871
   393
qed
haftmann@43871
   394
wenzelm@63575
   395
lemma INF_inf_const1: "I \<noteq> {} \<Longrightarrow> (INF i:I. inf x (f i)) = inf x (INF i:I. f i)"
hoelzl@57448
   396
  by (intro antisym INF_greatest inf_mono order_refl INF_lower)
hoelzl@57448
   397
     (auto intro: INF_lower2 le_infI2 intro!: INF_mono)
hoelzl@57448
   398
wenzelm@63575
   399
lemma INF_inf_const2: "I \<noteq> {} \<Longrightarrow> (INF i:I. inf (f i) x) = inf (INF i:I. f i) x"
hoelzl@57448
   400
  using INF_inf_const1[of I x f] by (simp add: inf_commute)
hoelzl@57448
   401
wenzelm@63575
   402
lemma INF_constant: "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921
   403
  by simp
haftmann@43868
   404
wenzelm@63575
   405
lemma SUP_constant: "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921
   406
  by simp
haftmann@43871
   407
haftmann@43943
   408
lemma less_INF_D:
wenzelm@63575
   409
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A"
wenzelm@63575
   410
  shows "y < f i"
haftmann@43943
   411
proof -
wenzelm@60758
   412
  note \<open>y < (\<Sqinter>i\<in>A. f i)\<close>
wenzelm@60758
   413
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using \<open>i \<in> A\<close>
haftmann@44103
   414
    by (rule INF_lower)
haftmann@43943
   415
  finally show "y < f i" .
haftmann@43943
   416
qed
haftmann@43943
   417
haftmann@43943
   418
lemma SUP_lessD:
wenzelm@63575
   419
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A"
wenzelm@63575
   420
  shows "f i < y"
haftmann@43943
   421
proof -
wenzelm@63575
   422
  have "f i \<le> (\<Squnion>i\<in>A. f i)"
wenzelm@63575
   423
    using \<open>i \<in> A\<close> by (rule SUP_upper)
wenzelm@60758
   424
  also note \<open>(\<Squnion>i\<in>A. f i) < y\<close>
haftmann@43943
   425
  finally show "f i < y" .
haftmann@43943
   426
qed
haftmann@43943
   427
wenzelm@63575
   428
lemma INF_UNIV_bool_expand: "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@56166
   429
  by (simp add: UNIV_bool inf_commute)
haftmann@43868
   430
wenzelm@63575
   431
lemma SUP_UNIV_bool_expand: "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@56166
   432
  by (simp add: UNIV_bool sup_commute)
haftmann@43871
   433
hoelzl@51328
   434
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@51328
   435
  by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
hoelzl@51328
   436
haftmann@56218
   437
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f"
haftmann@56166
   438
  using Inf_le_Sup [of "f ` A"] by simp
hoelzl@51328
   439
wenzelm@63575
   440
lemma INF_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x"
hoelzl@54414
   441
  by (auto intro: INF_eqI)
hoelzl@54414
   442
wenzelm@63575
   443
lemma SUP_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x"
haftmann@56248
   444
  by (auto intro: SUP_eqI)
hoelzl@54414
   445
wenzelm@63575
   446
lemma INF_eq_iff: "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
   447
  using INF_eq_const [of I f c] INF_lower [of _ I f]
haftmann@56248
   448
  by (auto intro: antisym cong del: strong_INF_cong)
haftmann@56248
   449
wenzelm@63575
   450
lemma SUP_eq_iff: "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
   451
  using SUP_eq_const [of I f c] SUP_upper [of _ I f]
haftmann@56248
   452
  by (auto intro: antisym cong del: strong_SUP_cong)
hoelzl@54414
   453
haftmann@32077
   454
end
haftmann@32077
   455
haftmann@44024
   456
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   457
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
wenzelm@63575
   458
    and inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   459
begin
haftmann@44024
   460
wenzelm@63575
   461
lemma sup_INF: "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
wenzelm@63172
   462
  by (simp add: sup_Inf)
haftmann@44039
   463
wenzelm@63575
   464
lemma inf_SUP: "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
wenzelm@63172
   465
  by (simp add: inf_Sup)
haftmann@44039
   466
haftmann@44032
   467
lemma dual_complete_distrib_lattice:
krauss@44845
   468
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   469
  apply (rule class.complete_distrib_lattice.intro)
wenzelm@63575
   470
   apply (fact dual_complete_lattice)
haftmann@44024
   471
  apply (rule class.complete_distrib_lattice_axioms.intro)
wenzelm@63575
   472
   apply (simp_all add: inf_Sup sup_Inf)
haftmann@44032
   473
  done
haftmann@44024
   474
wenzelm@63575
   475
subclass distrib_lattice
wenzelm@63575
   476
proof
haftmann@44024
   477
  fix a b c
wenzelm@63575
   478
  have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" by (rule sup_Inf)
haftmann@62343
   479
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by simp
haftmann@44024
   480
qed
haftmann@44024
   481
wenzelm@63575
   482
lemma Inf_sup: "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   483
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   484
wenzelm@63575
   485
lemma Sup_inf: "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   486
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   487
wenzelm@63575
   488
lemma INF_sup: "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   489
  by (simp add: sup_INF sup_commute)
haftmann@44039
   490
wenzelm@63575
   491
lemma SUP_inf: "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   492
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   493
wenzelm@63575
   494
lemma Inf_sup_eq_top_iff: "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   495
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   496
wenzelm@63575
   497
lemma Sup_inf_eq_bot_iff: "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   498
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   499
wenzelm@63575
   500
lemma INF_sup_distrib2: "(\<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
   501
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   502
wenzelm@63575
   503
lemma SUP_inf_distrib2: "(\<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
   504
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   505
haftmann@56074
   506
context
haftmann@56074
   507
  fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
haftmann@56074
   508
  assumes "mono f"
haftmann@56074
   509
begin
haftmann@56074
   510
wenzelm@63575
   511
lemma mono_Inf: "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
wenzelm@60758
   512
  using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
haftmann@56074
   513
wenzelm@63575
   514
lemma mono_Sup: "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
wenzelm@60758
   515
  using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
haftmann@56074
   516
wenzelm@63575
   517
lemma mono_INF: "f (INF i : I. A i) \<le> (INF x : I. f (A x))"
wenzelm@60758
   518
  by (intro complete_lattice_class.INF_greatest monoD[OF \<open>mono f\<close>] INF_lower)
hoelzl@60172
   519
wenzelm@63575
   520
lemma mono_SUP: "(SUP x : I. f (A x)) \<le> f (SUP i : I. A i)"
wenzelm@60758
   521
  by (intro complete_lattice_class.SUP_least monoD[OF \<open>mono f\<close>] SUP_upper)
hoelzl@60172
   522
haftmann@56074
   523
end
haftmann@56074
   524
haftmann@44024
   525
end
haftmann@44024
   526
haftmann@44032
   527
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   528
begin
haftmann@43873
   529
haftmann@43943
   530
lemma dual_complete_boolean_algebra:
krauss@44845
   531
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
wenzelm@63575
   532
  by (rule class.complete_boolean_algebra.intro,
wenzelm@63575
   533
      rule dual_complete_distrib_lattice,
wenzelm@63575
   534
      rule dual_boolean_algebra)
haftmann@43943
   535
wenzelm@63575
   536
lemma uminus_Inf: "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   537
proof (rule antisym)
haftmann@43873
   538
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   539
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   540
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   541
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   542
qed
haftmann@43873
   543
haftmann@44041
   544
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@62343
   545
  by (simp add: uminus_Inf image_image)
haftmann@44041
   546
wenzelm@63575
   547
lemma uminus_Sup: "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   548
proof -
wenzelm@63575
   549
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)"
wenzelm@63575
   550
    by (simp add: image_image uminus_INF)
haftmann@43873
   551
  then show ?thesis by simp
haftmann@43873
   552
qed
wenzelm@63575
   553
haftmann@43873
   554
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@62343
   555
  by (simp add: uminus_Sup image_image)
haftmann@43873
   556
haftmann@43873
   557
end
haftmann@43873
   558
haftmann@43940
   559
class complete_linorder = linorder + complete_lattice
haftmann@43940
   560
begin
haftmann@43940
   561
haftmann@43943
   562
lemma dual_complete_linorder:
krauss@44845
   563
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   564
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   565
haftmann@51386
   566
lemma complete_linorder_inf_min: "inf = min"
haftmann@51540
   567
  by (auto intro: antisym simp add: min_def fun_eq_iff)
haftmann@51386
   568
haftmann@51386
   569
lemma complete_linorder_sup_max: "sup = max"
haftmann@51540
   570
  by (auto intro: antisym simp add: max_def fun_eq_iff)
haftmann@51386
   571
haftmann@63820
   572
lemma Inf_less_iff: "\<Sqinter>S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
wenzelm@63172
   573
  by (simp add: not_le [symmetric] le_Inf_iff)
haftmann@43940
   574
haftmann@63820
   575
lemma INF_less_iff: "(\<Sqinter>i\<in>A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
wenzelm@63172
   576
  by (simp add: Inf_less_iff [of "f ` A"])
haftmann@44041
   577
haftmann@63820
   578
lemma less_Sup_iff: "a < \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"
wenzelm@63172
   579
  by (simp add: not_le [symmetric] Sup_le_iff)
haftmann@43940
   580
haftmann@63820
   581
lemma less_SUP_iff: "a < (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
wenzelm@63172
   582
  by (simp add: less_Sup_iff [of _ "f ` A"])
haftmann@43940
   583
wenzelm@63575
   584
lemma Sup_eq_top_iff [simp]: "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   585
proof
haftmann@43943
   586
  assume *: "\<Squnion>A = \<top>"
wenzelm@63575
   587
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
wenzelm@63575
   588
    unfolding * [symmetric]
haftmann@43943
   589
  proof (intro allI impI)
wenzelm@63575
   590
    fix x
wenzelm@63575
   591
    assume "x < \<Squnion>A"
wenzelm@63575
   592
    then show "\<exists>i\<in>A. x < i"
wenzelm@63172
   593
      by (simp add: less_Sup_iff)
haftmann@43943
   594
  qed
haftmann@43943
   595
next
haftmann@43943
   596
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   597
  show "\<Squnion>A = \<top>"
haftmann@43943
   598
  proof (rule ccontr)
haftmann@43943
   599
    assume "\<Squnion>A \<noteq> \<top>"
wenzelm@63575
   600
    with top_greatest [of "\<Squnion>A"] have "\<Squnion>A < \<top>"
wenzelm@63575
   601
      unfolding le_less by auto
wenzelm@63575
   602
    with * have "\<Squnion>A < \<Squnion>A"
wenzelm@63575
   603
      unfolding less_Sup_iff by auto
haftmann@43943
   604
    then show False by auto
haftmann@43943
   605
  qed
haftmann@43943
   606
qed
haftmann@43943
   607
wenzelm@63575
   608
lemma SUP_eq_top_iff [simp]: "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
haftmann@56166
   609
  using Sup_eq_top_iff [of "f ` A"] by simp
haftmann@44041
   610
wenzelm@63575
   611
lemma Inf_eq_bot_iff [simp]: "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920
   612
  using dual_complete_linorder
huffman@44920
   613
  by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943
   614
wenzelm@63575
   615
lemma INF_eq_bot_iff [simp]: "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
haftmann@56166
   616
  using Inf_eq_bot_iff [of "f ` A"] by simp
hoelzl@51328
   617
hoelzl@51328
   618
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
hoelzl@51328
   619
proof safe
wenzelm@63575
   620
  fix y
wenzelm@63575
   621
  assume "x \<ge> \<Sqinter>A" "y > x"
hoelzl@51328
   622
  then have "y > \<Sqinter>A" by auto
hoelzl@51328
   623
  then show "\<exists>a\<in>A. y > a"
hoelzl@51328
   624
    unfolding Inf_less_iff .
hoelzl@51328
   625
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
hoelzl@51328
   626
wenzelm@63575
   627
lemma INF_le_iff: "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
haftmann@56166
   628
  using Inf_le_iff [of "f ` A"] by simp
haftmann@56166
   629
haftmann@56166
   630
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
haftmann@56166
   631
proof safe
wenzelm@63575
   632
  fix y
wenzelm@63575
   633
  assume "x \<le> \<Squnion>A" "y < x"
haftmann@56166
   634
  then have "y < \<Squnion>A" by auto
haftmann@56166
   635
  then show "\<exists>a\<in>A. y < a"
haftmann@56166
   636
    unfolding less_Sup_iff .
haftmann@56166
   637
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
haftmann@56166
   638
haftmann@56218
   639
lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
haftmann@56166
   640
  using le_Sup_iff [of _ "f ` A"] by simp
hoelzl@51328
   641
haftmann@51386
   642
subclass complete_distrib_lattice
haftmann@51386
   643
proof
haftmann@51386
   644
  fix a and B
haftmann@51386
   645
  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
   646
    by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
haftmann@51386
   647
      (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
haftmann@51386
   648
        le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
haftmann@51386
   649
qed
haftmann@51386
   650
haftmann@43940
   651
end
haftmann@43940
   652
hoelzl@51341
   653
wenzelm@60758
   654
subsection \<open>Complete lattice on @{typ bool}\<close>
haftmann@32077
   655
haftmann@44024
   656
instantiation bool :: complete_lattice
haftmann@32077
   657
begin
haftmann@32077
   658
wenzelm@63575
   659
definition [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   660
wenzelm@63575
   661
definition [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   662
wenzelm@63575
   663
instance
wenzelm@63575
   664
  by standard (auto intro: bool_induct)
haftmann@32077
   665
haftmann@32077
   666
end
haftmann@32077
   667
wenzelm@63575
   668
lemma not_False_in_image_Ball [simp]: "False \<notin> P ` A \<longleftrightarrow> Ball A P"
haftmann@49905
   669
  by auto
haftmann@49905
   670
wenzelm@63575
   671
lemma True_in_image_Bex [simp]: "True \<in> P ` A \<longleftrightarrow> Bex A P"
haftmann@49905
   672
  by auto
haftmann@49905
   673
wenzelm@63575
   674
lemma INF_bool_eq [simp]: "INFIMUM = Ball"
haftmann@62343
   675
  by (simp add: fun_eq_iff)
haftmann@32120
   676
wenzelm@63575
   677
lemma SUP_bool_eq [simp]: "SUPREMUM = Bex"
haftmann@62343
   678
  by (simp add: fun_eq_iff)
haftmann@32120
   679
wenzelm@63575
   680
instance bool :: complete_boolean_algebra
wenzelm@63575
   681
  by standard (auto intro: bool_induct)
haftmann@44024
   682
haftmann@46631
   683
wenzelm@60758
   684
subsection \<open>Complete lattice on @{typ "_ \<Rightarrow> _"}\<close>
haftmann@46631
   685
nipkow@57197
   686
instantiation "fun" :: (type, Inf) Inf
haftmann@32077
   687
begin
haftmann@32077
   688
wenzelm@63575
   689
definition "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   690
wenzelm@63575
   691
lemma Inf_apply [simp, code]: "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   692
  by (simp add: Inf_fun_def)
haftmann@32077
   693
nipkow@57197
   694
instance ..
nipkow@57197
   695
nipkow@57197
   696
end
nipkow@57197
   697
nipkow@57197
   698
instantiation "fun" :: (type, Sup) Sup
nipkow@57197
   699
begin
nipkow@57197
   700
wenzelm@63575
   701
definition "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   702
wenzelm@63575
   703
lemma Sup_apply [simp, code]: "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   704
  by (simp add: Sup_fun_def)
haftmann@32077
   705
nipkow@57197
   706
instance ..
nipkow@57197
   707
nipkow@57197
   708
end
nipkow@57197
   709
nipkow@57197
   710
instantiation "fun" :: (type, complete_lattice) complete_lattice
nipkow@57197
   711
begin
nipkow@57197
   712
wenzelm@63575
   713
instance
wenzelm@63575
   714
  by standard (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   715
haftmann@32077
   716
end
haftmann@32077
   717
wenzelm@63575
   718
lemma INF_apply [simp]: "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@56166
   719
  using Inf_apply [of "f ` A"] by (simp add: comp_def)
hoelzl@38705
   720
wenzelm@63575
   721
lemma SUP_apply [simp]: "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@56166
   722
  using Sup_apply [of "f ` A"] by (simp add: comp_def)
haftmann@32077
   723
wenzelm@63575
   724
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice
wenzelm@63575
   725
  by standard (auto simp add: inf_Sup sup_Inf fun_eq_iff image_image)
haftmann@44024
   726
haftmann@43873
   727
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   728
haftmann@46631
   729
wenzelm@60758
   730
subsection \<open>Complete lattice on unary and binary predicates\<close>
haftmann@46631
   731
wenzelm@63575
   732
lemma Inf1_I: "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a"
noschinl@46884
   733
  by auto
haftmann@46631
   734
wenzelm@63575
   735
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   736
  by simp
haftmann@56742
   737
wenzelm@63575
   738
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   739
  by simp
haftmann@56742
   740
wenzelm@63575
   741
lemma Inf2_I: "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b"
noschinl@46884
   742
  by auto
haftmann@46631
   743
wenzelm@63575
   744
lemma Inf1_D: "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a"
noschinl@46884
   745
  by auto
haftmann@46631
   746
wenzelm@63575
   747
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
haftmann@56742
   748
  by simp
haftmann@56742
   749
wenzelm@63575
   750
lemma Inf2_D: "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b"
noschinl@46884
   751
  by auto
haftmann@46631
   752
wenzelm@63575
   753
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
haftmann@56742
   754
  by simp
haftmann@56742
   755
haftmann@56742
   756
lemma Inf1_E:
haftmann@56742
   757
  assumes "(\<Sqinter>A) a"
haftmann@56742
   758
  obtains "P a" | "P \<notin> A"
haftmann@56742
   759
  using assms by auto
haftmann@46631
   760
haftmann@56742
   761
lemma INF1_E:
haftmann@56742
   762
  assumes "(\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   763
  obtains "B a b" | "a \<notin> A"
haftmann@56742
   764
  using assms by auto
haftmann@56742
   765
haftmann@56742
   766
lemma Inf2_E:
haftmann@56742
   767
  assumes "(\<Sqinter>A) a b"
haftmann@56742
   768
  obtains "r a b" | "r \<notin> A"
haftmann@56742
   769
  using assms by auto
haftmann@56742
   770
haftmann@56742
   771
lemma INF2_E:
haftmann@56742
   772
  assumes "(\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   773
  obtains "B a b c" | "a \<notin> A"
haftmann@56742
   774
  using assms by auto
haftmann@56742
   775
wenzelm@63575
   776
lemma Sup1_I: "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a"
noschinl@46884
   777
  by auto
haftmann@46631
   778
wenzelm@63575
   779
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
haftmann@56742
   780
  by auto
haftmann@56742
   781
wenzelm@63575
   782
lemma Sup2_I: "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b"
haftmann@56742
   783
  by auto
haftmann@56742
   784
wenzelm@63575
   785
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
noschinl@46884
   786
  by auto
haftmann@46631
   787
haftmann@56742
   788
lemma Sup1_E:
haftmann@56742
   789
  assumes "(\<Squnion>A) a"
haftmann@56742
   790
  obtains P where "P \<in> A" and "P a"
haftmann@56742
   791
  using assms by auto
haftmann@56742
   792
haftmann@56742
   793
lemma SUP1_E:
haftmann@56742
   794
  assumes "(\<Squnion>x\<in>A. B x) b"
haftmann@56742
   795
  obtains x where "x \<in> A" and "B x b"
haftmann@56742
   796
  using assms by auto
haftmann@46631
   797
haftmann@56742
   798
lemma Sup2_E:
haftmann@56742
   799
  assumes "(\<Squnion>A) a b"
haftmann@56742
   800
  obtains r where "r \<in> A" "r a b"
haftmann@56742
   801
  using assms by auto
haftmann@56742
   802
haftmann@56742
   803
lemma SUP2_E:
haftmann@56742
   804
  assumes "(\<Squnion>x\<in>A. B x) b c"
haftmann@56742
   805
  obtains x where "x \<in> A" "B x b c"
haftmann@56742
   806
  using assms by auto
haftmann@46631
   807
haftmann@46631
   808
wenzelm@60758
   809
subsection \<open>Complete lattice on @{typ "_ set"}\<close>
haftmann@46631
   810
haftmann@45960
   811
instantiation "set" :: (type) complete_lattice
haftmann@45960
   812
begin
haftmann@45960
   813
wenzelm@63575
   814
definition "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   815
wenzelm@63575
   816
definition "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   817
wenzelm@63575
   818
instance
wenzelm@63575
   819
  by standard (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
haftmann@45960
   820
haftmann@45960
   821
end
haftmann@45960
   822
haftmann@45960
   823
instance "set" :: (type) complete_boolean_algebra
wenzelm@63575
   824
  by standard (auto simp add: Inf_set_def Sup_set_def image_def)
wenzelm@63575
   825
haftmann@32077
   826
wenzelm@60758
   827
subsubsection \<open>Inter\<close>
haftmann@41082
   828
wenzelm@61952
   829
abbreviation Inter :: "'a set set \<Rightarrow> 'a set"  ("\<Inter>_" [900] 900)
wenzelm@61952
   830
  where "\<Inter>S \<equiv> \<Sqinter>S"
wenzelm@63575
   831
wenzelm@63575
   832
lemma Inter_eq: "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   833
proof (rule set_eqI)
haftmann@41082
   834
  fix x
haftmann@41082
   835
  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
   836
    by auto
haftmann@41082
   837
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960
   838
    by (simp add: Inf_set_def image_def)
haftmann@41082
   839
qed
haftmann@41082
   840
blanchet@54147
   841
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   842
  by (unfold Inter_eq) blast
haftmann@41082
   843
haftmann@43741
   844
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   845
  by (simp add: Inter_eq)
haftmann@41082
   846
wenzelm@60758
   847
text \<open>
wenzelm@63575
   848
  \<^medskip> A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   849
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
wenzelm@61799
   850
  @{prop "X \<in> C"} does not!  This rule is analogous to \<open>spec\<close>.
wenzelm@60758
   851
\<close>
haftmann@41082
   852
haftmann@43741
   853
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   854
  by auto
haftmann@41082
   855
haftmann@43741
   856
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@61799
   857
  \<comment> \<open>``Classical'' elimination rule -- does not require proving
wenzelm@60758
   858
    @{prop "X \<in> C"}.\<close>
wenzelm@63575
   859
  unfolding Inter_eq by blast
haftmann@41082
   860
haftmann@43741
   861
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   862
  by (fact Inf_lower)
haftmann@43740
   863
wenzelm@63575
   864
lemma Inter_subset: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   865
  by (fact Inf_less_eq)
haftmann@41082
   866
wenzelm@61952
   867
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A"
haftmann@43740
   868
  by (fact Inf_greatest)
haftmann@41082
   869
huffman@44067
   870
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   871
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   872
huffman@44067
   873
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   874
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   875
huffman@44920
   876
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920
   877
  by (fact Inf_insert) (* already simp *)
haftmann@41082
   878
haftmann@41082
   879
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   880
  by (fact less_eq_Inf_inter)
haftmann@41082
   881
haftmann@41082
   882
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   883
  by (fact Inf_union_distrib)
haftmann@43756
   884
blanchet@54147
   885
lemma Inter_UNIV_conv [simp]:
haftmann@43741
   886
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   887
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   888
  by (fact Inf_top_conv)+
haftmann@41082
   889
haftmann@43741
   890
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   891
  by (fact Inf_superset_mono)
haftmann@41082
   892
haftmann@41082
   893
wenzelm@60758
   894
subsubsection \<open>Intersections of families\<close>
haftmann@41082
   895
wenzelm@61955
   896
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
wenzelm@61955
   897
  where "INTER \<equiv> INFIMUM"
haftmann@41082
   898
wenzelm@60758
   899
text \<open>
wenzelm@61799
   900
  Note: must use name @{const INTER} here instead of \<open>INT\<close>
haftmann@43872
   901
  to allow the following syntax coexist with the plain constant name.
wenzelm@60758
   902
\<close>
haftmann@43872
   903
wenzelm@61955
   904
syntax (ASCII)
wenzelm@61955
   905
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3INT _./ _)" [0, 10] 10)
wenzelm@61955
   906
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   907
haftmann@41082
   908
syntax (latex output)
wenzelm@62789
   909
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
wenzelm@62789
   910
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
wenzelm@61955
   911
wenzelm@61955
   912
syntax
wenzelm@61955
   913
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
wenzelm@61955
   914
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   915
haftmann@41082
   916
translations
wenzelm@61955
   917
  "\<Inter>x y. B"  \<rightleftharpoons> "\<Inter>x. \<Inter>y. B"
wenzelm@61955
   918
  "\<Inter>x. B"    \<rightleftharpoons> "CONST INTER CONST UNIV (\<lambda>x. B)"
wenzelm@61955
   919
  "\<Inter>x. B"    \<rightleftharpoons> "\<Inter>x \<in> CONST UNIV. B"
wenzelm@61955
   920
  "\<Inter>x\<in>A. B"  \<rightleftharpoons> "CONST INTER A (\<lambda>x. B)"
haftmann@41082
   921
wenzelm@60758
   922
print_translation \<open>
wenzelm@42284
   923
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
wenzelm@61799
   924
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
haftmann@41082
   925
wenzelm@63575
   926
lemma INTER_eq: "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@56166
   927
  by (auto intro!: INF_eqI)
haftmann@41082
   928
haftmann@43817
   929
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@56166
   930
  using Inter_iff [of _ "B ` A"] by simp
haftmann@41082
   931
haftmann@43817
   932
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
   933
  by auto
haftmann@41082
   934
haftmann@43852
   935
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
   936
  by auto
haftmann@41082
   937
haftmann@43852
   938
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
   939
  \<comment> \<open>"Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}.\<close>
haftmann@62343
   940
  by auto
haftmann@41082
   941
haftmann@41082
   942
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   943
  by blast
haftmann@41082
   944
haftmann@41082
   945
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   946
  by blast
haftmann@41082
   947
haftmann@43817
   948
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
   949
  by (fact INF_lower)
haftmann@41082
   950
haftmann@43817
   951
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
   952
  by (fact INF_greatest)
haftmann@41082
   953
huffman@44067
   954
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
   955
  by (fact INF_empty)
haftmann@43854
   956
haftmann@43817
   957
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   958
  by (fact INF_absorb)
haftmann@41082
   959
haftmann@43854
   960
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   961
  by (fact le_INF_iff)
haftmann@41082
   962
haftmann@41082
   963
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   964
  by (fact INF_insert)
haftmann@43865
   965
haftmann@43865
   966
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
   967
  by (fact INF_union)
haftmann@43865
   968
wenzelm@63575
   969
lemma INT_insert_distrib: "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   970
  by blast
haftmann@43854
   971
haftmann@41082
   972
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   973
  by (fact INF_constant)
haftmann@43865
   974
huffman@44920
   975
lemma INTER_UNIV_conv:
wenzelm@63575
   976
  "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
wenzelm@63575
   977
  "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920
   978
  by (fact INF_top_conv)+ (* already simp *)
haftmann@43865
   979
haftmann@43865
   980
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   981
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   982
wenzelm@63575
   983
lemma INT_anti_mono: "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
   984
  \<comment> \<open>The last inclusion is POSITIVE!\<close>
haftmann@43940
   985
  by (fact INF_superset_mono)
haftmann@41082
   986
haftmann@41082
   987
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   988
  by blast
haftmann@41082
   989
haftmann@43817
   990
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   991
  by blast
haftmann@41082
   992
haftmann@41082
   993
wenzelm@60758
   994
subsubsection \<open>Union\<close>
haftmann@32115
   995
wenzelm@61952
   996
abbreviation Union :: "'a set set \<Rightarrow> 'a set"  ("\<Union>_" [900] 900)
wenzelm@61952
   997
  where "\<Union>S \<equiv> \<Squnion>S"
haftmann@32115
   998
wenzelm@63575
   999
lemma Union_eq: "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
  1000
proof (rule set_eqI)
haftmann@32115
  1001
  fix x
haftmann@32135
  1002
  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
  1003
    by auto
haftmann@32135
  1004
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960
  1005
    by (simp add: Sup_set_def image_def)
haftmann@32115
  1006
qed
haftmann@32115
  1007
wenzelm@63575
  1008
lemma Union_iff [simp]: "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
  1009
  by (unfold Union_eq) blast
haftmann@32115
  1010
wenzelm@63575
  1011
lemma UnionI [intro]: "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
wenzelm@61799
  1012
  \<comment> \<open>The order of the premises presupposes that @{term C} is rigid;
wenzelm@60758
  1013
    @{term A} may be flexible.\<close>
haftmann@32115
  1014
  by auto
haftmann@32115
  1015
wenzelm@63575
  1016
lemma UnionE [elim!]: "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
  1017
  by auto
haftmann@32115
  1018
haftmann@43817
  1019
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
  1020
  by (fact Sup_upper)
haftmann@32135
  1021
haftmann@43817
  1022
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
  1023
  by (fact Sup_least)
haftmann@32135
  1024
huffman@44920
  1025
lemma Union_empty: "\<Union>{} = {}"
huffman@44920
  1026
  by (fact Sup_empty) (* already simp *)
haftmann@32135
  1027
huffman@44920
  1028
lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920
  1029
  by (fact Sup_UNIV) (* already simp *)
haftmann@32135
  1030
huffman@44920
  1031
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920
  1032
  by (fact Sup_insert) (* already simp *)
haftmann@32135
  1033
haftmann@43817
  1034
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
  1035
  by (fact Sup_union_distrib)
haftmann@32135
  1036
haftmann@32135
  1037
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
  1038
  by (fact Sup_inter_less_eq)
haftmann@32135
  1039
blanchet@54147
  1040
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1041
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1042
blanchet@54147
  1043
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1044
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1045
haftmann@32135
  1046
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
  1047
  by blast
haftmann@32135
  1048
haftmann@32135
  1049
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
  1050
  by blast
haftmann@32135
  1051
haftmann@43817
  1052
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
  1053
  by (fact Sup_subset_mono)
haftmann@32135
  1054
lp15@63469
  1055
lemma Union_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> x \<subseteq> y) \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
lp15@63469
  1056
  by blast
haftmann@32115
  1057
lp15@63879
  1058
lemma disjnt_inj_on_iff:
lp15@63879
  1059
     "\<lbrakk>inj_on f (\<Union>\<A>); X \<in> \<A>; Y \<in> \<A>\<rbrakk> \<Longrightarrow> disjnt (f ` X) (f ` Y) \<longleftrightarrow> disjnt X Y"
lp15@63879
  1060
  apply (auto simp: disjnt_def)
lp15@63879
  1061
  using inj_on_eq_iff by fastforce
lp15@63879
  1062
wenzelm@63575
  1063
wenzelm@60758
  1064
subsubsection \<open>Unions of families\<close>
haftmann@32077
  1065
wenzelm@61955
  1066
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
wenzelm@61955
  1067
  where "UNION \<equiv> SUPREMUM"
haftmann@32077
  1068
wenzelm@60758
  1069
text \<open>
wenzelm@61799
  1070
  Note: must use name @{const UNION} here instead of \<open>UN\<close>
haftmann@43872
  1071
  to allow the following syntax coexist with the plain constant name.
wenzelm@60758
  1072
\<close>
haftmann@43872
  1073
wenzelm@61955
  1074
syntax (ASCII)
wenzelm@35115
  1075
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
  1076
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
  1077
haftmann@32077
  1078
syntax (latex output)
wenzelm@62789
  1079
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
wenzelm@62789
  1080
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
  1081
wenzelm@61955
  1082
syntax
wenzelm@61955
  1083
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
wenzelm@61955
  1084
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
wenzelm@61955
  1085
haftmann@32077
  1086
translations
wenzelm@61955
  1087
  "\<Union>x y. B"   \<rightleftharpoons> "\<Union>x. \<Union>y. B"
wenzelm@61955
  1088
  "\<Union>x. B"     \<rightleftharpoons> "CONST UNION CONST UNIV (\<lambda>x. B)"
wenzelm@61955
  1089
  "\<Union>x. B"     \<rightleftharpoons> "\<Union>x \<in> CONST UNIV. B"
wenzelm@61955
  1090
  "\<Union>x\<in>A. B"   \<rightleftharpoons> "CONST UNION A (\<lambda>x. B)"
haftmann@32077
  1091
wenzelm@60758
  1092
text \<open>
wenzelm@61955
  1093
  Note the difference between ordinary syntax of indexed
wenzelm@61799
  1094
  unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>)
wenzelm@61955
  1095
  and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}.
wenzelm@60758
  1096
\<close>
haftmann@32077
  1097
wenzelm@60758
  1098
print_translation \<open>
wenzelm@42284
  1099
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@61799
  1100
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
haftmann@32077
  1101
wenzelm@63575
  1102
lemma UNION_eq: "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@56166
  1103
  by (auto intro!: SUP_eqI)
huffman@44920
  1104
wenzelm@63575
  1105
lemma bind_UNION [code]: "Set.bind A f = UNION A f"
haftmann@45960
  1106
  by (simp add: bind_def UNION_eq)
haftmann@45960
  1107
wenzelm@63575
  1108
lemma member_bind [simp]: "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036
  1109
  by (simp add: bind_UNION)
haftmann@46036
  1110
wenzelm@60585
  1111
lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
lp15@60307
  1112
  by blast
lp15@60307
  1113
haftmann@46036
  1114
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@56166
  1115
  using Union_iff [of _ "B ` A"] by simp
wenzelm@11979
  1116
haftmann@43852
  1117
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@61799
  1118
  \<comment> \<open>The order of the premises presupposes that @{term A} is rigid;
wenzelm@60758
  1119
    @{term b} may be flexible.\<close>
wenzelm@11979
  1120
  by auto
wenzelm@11979
  1121
haftmann@43852
  1122
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
  1123
  by auto
haftmann@32077
  1124
haftmann@43817
  1125
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
  1126
  by (fact SUP_upper)
haftmann@32135
  1127
haftmann@43817
  1128
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
  1129
  by (fact SUP_least)
haftmann@32135
  1130
blanchet@54147
  1131
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
  1132
  by blast
haftmann@32135
  1133
haftmann@43817
  1134
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
  1135
  by blast
haftmann@32135
  1136
blanchet@54147
  1137
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
  1138
  by (fact SUP_empty)
haftmann@32135
  1139
huffman@44920
  1140
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920
  1141
  by (fact SUP_bot) (* already simp *)
haftmann@32135
  1142
haftmann@43817
  1143
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
  1144
  by (fact SUP_absorb)
haftmann@32135
  1145
haftmann@32135
  1146
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
  1147
  by (fact SUP_insert)
haftmann@32135
  1148
haftmann@44085
  1149
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
  1150
  by (fact SUP_union)
haftmann@32135
  1151
haftmann@43967
  1152
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
  1153
  by blast
haftmann@32135
  1154
haftmann@32135
  1155
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
  1156
  by (fact SUP_le_iff)
haftmann@32135
  1157
haftmann@32135
  1158
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
  1159
  by (fact SUP_constant)
haftmann@32135
  1160
haftmann@43944
  1161
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
  1162
  by blast
haftmann@32135
  1163
huffman@44920
  1164
lemma UNION_empty_conv:
haftmann@43817
  1165
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1166
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920
  1167
  by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135
  1168
blanchet@54147
  1169
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1170
  by blast
haftmann@32135
  1171
haftmann@43900
  1172
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
  1173
  by blast
haftmann@32135
  1174
haftmann@43900
  1175
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
  1176
  by blast
haftmann@32135
  1177
haftmann@32135
  1178
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
nipkow@62390
  1179
  by safe (auto simp add: if_split_mem2)
haftmann@32135
  1180
haftmann@43817
  1181
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1182
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1183
haftmann@32135
  1184
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1185
  by blast
haftmann@32135
  1186
haftmann@32135
  1187
lemma UN_mono:
haftmann@43817
  1188
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1189
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1190
  by (fact SUP_subset_mono)
haftmann@32135
  1191
haftmann@43817
  1192
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1193
  by blast
haftmann@32135
  1194
haftmann@43817
  1195
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1196
  by blast
haftmann@32135
  1197
haftmann@43817
  1198
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
wenzelm@61799
  1199
  \<comment> \<open>NOT suitable for rewriting\<close>
haftmann@32135
  1200
  by blast
haftmann@32135
  1201
haftmann@43817
  1202
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1203
  by blast
haftmann@32135
  1204
haftmann@45013
  1205
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013
  1206
  by blast
haftmann@45013
  1207
eberlm@63099
  1208
lemma inj_on_image: "inj_on f (\<Union>A) \<Longrightarrow> inj_on (op ` f) A"
eberlm@63099
  1209
  unfolding inj_on_def by blast
wenzelm@11979
  1210
wenzelm@63575
  1211
wenzelm@60758
  1212
subsubsection \<open>Distributive laws\<close>
wenzelm@12897
  1213
wenzelm@12897
  1214
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1215
  by (fact inf_Sup)
wenzelm@12897
  1216
haftmann@44039
  1217
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1218
  by (fact sup_Inf)
haftmann@44039
  1219
wenzelm@12897
  1220
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1221
  by (fact Sup_inf)
haftmann@44039
  1222
haftmann@44039
  1223
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
  1224
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1225
haftmann@44039
  1226
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
  1227
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1228
wenzelm@63575
  1229
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"  (* FIXME drop *)
haftmann@56166
  1230
  by (simp add: INT_Int_distrib)
wenzelm@12897
  1231
wenzelm@63575
  1232
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"  (* FIXME drop *)
wenzelm@61799
  1233
  \<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close>
wenzelm@61799
  1234
  \<comment> \<open>Union of a family of unions\<close>
haftmann@56166
  1235
  by (simp add: UN_Un_distrib)
wenzelm@12897
  1236
haftmann@44039
  1237
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1238
  by (fact sup_INF)
wenzelm@12897
  1239
wenzelm@12897
  1240
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@61799
  1241
  \<comment> \<open>Halmos, Naive Set Theory, page 35.\<close>
haftmann@44039
  1242
  by (fact inf_SUP)
wenzelm@12897
  1243
wenzelm@12897
  1244
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
  1245
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1246
wenzelm@12897
  1247
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
  1248
  by (fact INF_sup_distrib2)
haftmann@44039
  1249
haftmann@44039
  1250
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1251
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1252
Andreas@61630
  1253
lemma SUP_UNION: "(SUP x:(UN y:A. g y). f x) = (SUP y:A. SUP x:g y. f x :: _ :: complete_lattice)"
wenzelm@63575
  1254
  by (rule order_antisym) (blast intro: SUP_least SUP_upper2)+
wenzelm@63575
  1255
wenzelm@12897
  1256
wenzelm@60758
  1257
subsection \<open>Injections and bijections\<close>
haftmann@56015
  1258
wenzelm@63575
  1259
lemma inj_on_Inter: "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
haftmann@56015
  1260
  unfolding inj_on_def by blast
haftmann@56015
  1261
wenzelm@63575
  1262
lemma inj_on_INTER: "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
  1263
  unfolding inj_on_def by safe simp
haftmann@56015
  1264
haftmann@56015
  1265
lemma inj_on_UNION_chain:
wenzelm@63575
  1266
  assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i"
wenzelm@63575
  1267
    and inj: "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)"
wenzelm@60585
  1268
  shows "inj_on f (\<Union>i \<in> I. A i)"
haftmann@56015
  1269
proof -
wenzelm@63575
  1270
  have "x = y"
wenzelm@63575
  1271
    if *: "i \<in> I" "j \<in> I"
wenzelm@63575
  1272
    and **: "x \<in> A i" "y \<in> A j"
wenzelm@63575
  1273
    and ***: "f x = f y"
wenzelm@63575
  1274
    for i j x y
wenzelm@63575
  1275
    using chain [OF *]
wenzelm@63575
  1276
  proof
wenzelm@63575
  1277
    assume "A i \<le> A j"
wenzelm@63575
  1278
    with ** have "x \<in> A j" by auto
wenzelm@63575
  1279
    with inj * ** *** show ?thesis
wenzelm@63575
  1280
      by (auto simp add: inj_on_def)
wenzelm@63575
  1281
  next
wenzelm@63575
  1282
    assume "A j \<le> A i"
wenzelm@63575
  1283
    with ** have "y \<in> A i" by auto
wenzelm@63575
  1284
    with inj * ** *** show ?thesis
wenzelm@63575
  1285
      by (auto simp add: inj_on_def)
wenzelm@63575
  1286
  qed
wenzelm@63575
  1287
  then show ?thesis
wenzelm@63575
  1288
    by (unfold inj_on_def UNION_eq) auto
haftmann@56015
  1289
qed
haftmann@56015
  1290
haftmann@56015
  1291
lemma bij_betw_UNION_chain:
wenzelm@63575
  1292
  assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i"
wenzelm@63575
  1293
    and bij: "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
wenzelm@60585
  1294
  shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)"
wenzelm@63575
  1295
  unfolding bij_betw_def
wenzelm@63576
  1296
proof safe
wenzelm@63575
  1297
  have "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)"
wenzelm@63575
  1298
    using bij bij_betw_def[of f] by auto
wenzelm@63576
  1299
  then show "inj_on f (UNION I A)"
wenzelm@63575
  1300
    using chain inj_on_UNION_chain[of I A f] by auto
haftmann@56015
  1301
next
haftmann@56015
  1302
  fix i x
haftmann@56015
  1303
  assume *: "i \<in> I" "x \<in> A i"
wenzelm@63576
  1304
  with bij have "f x \<in> A' i"
wenzelm@63576
  1305
    by (auto simp: bij_betw_def)
wenzelm@63576
  1306
  with * show "f x \<in> UNION I A'" by blast
haftmann@56015
  1307
next
haftmann@56015
  1308
  fix i x'
haftmann@56015
  1309
  assume *: "i \<in> I" "x' \<in> A' i"
wenzelm@63576
  1310
  with bij have "\<exists>x \<in> A i. x' = f x"
wenzelm@63576
  1311
    unfolding bij_betw_def by blast
wenzelm@63575
  1312
  with * have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
wenzelm@63575
  1313
    by blast
wenzelm@63576
  1314
  then show "x' \<in> f ` UNION I A"
wenzelm@63575
  1315
    by blast
haftmann@56015
  1316
qed
haftmann@56015
  1317
haftmann@56015
  1318
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
wenzelm@63575
  1319
lemma image_INT: "inj_on f C \<Longrightarrow> \<forall>x\<in>A. B x \<subseteq> C \<Longrightarrow> j \<in> A \<Longrightarrow> f ` (INTER A B) = (INT x:A. f ` B x)"
wenzelm@63575
  1320
  by (auto simp add: inj_on_def) blast
haftmann@56015
  1321
wenzelm@63575
  1322
lemma bij_image_INT: "bij f \<Longrightarrow> f ` (INTER A B) = (INT x:A. f ` B x)"
wenzelm@64966
  1323
  by (auto simp: bij_def inj_def surj_def) blast
haftmann@56015
  1324
wenzelm@63575
  1325
lemma UNION_fun_upd: "UNION J (A(i := B)) = UNION (J - {i}) A \<union> (if i \<in> J then B else {})"
haftmann@62343
  1326
  by (auto simp add: set_eq_iff)
haftmann@63365
  1327
haftmann@63365
  1328
lemma bij_betw_Pow:
haftmann@63365
  1329
  assumes "bij_betw f A B"
haftmann@63365
  1330
  shows "bij_betw (image f) (Pow A) (Pow B)"
haftmann@63365
  1331
proof -
haftmann@63365
  1332
  from assms have "inj_on f A"
haftmann@63365
  1333
    by (rule bij_betw_imp_inj_on)
haftmann@63365
  1334
  then have "inj_on f (\<Union>Pow A)"
haftmann@63365
  1335
    by simp
haftmann@63365
  1336
  then have "inj_on (image f) (Pow A)"
haftmann@63365
  1337
    by (rule inj_on_image)
haftmann@63365
  1338
  then have "bij_betw (image f) (Pow A) (image f ` Pow A)"
haftmann@63365
  1339
    by (rule inj_on_imp_bij_betw)
haftmann@63365
  1340
  moreover from assms have "f ` A = B"
haftmann@63365
  1341
    by (rule bij_betw_imp_surj_on)
haftmann@63365
  1342
  then have "image f ` Pow A = Pow B"
haftmann@63365
  1343
    by (rule image_Pow_surj)
haftmann@63365
  1344
  ultimately show ?thesis by simp
haftmann@63365
  1345
qed
haftmann@63365
  1346
haftmann@56015
  1347
wenzelm@60758
  1348
subsubsection \<open>Complement\<close>
haftmann@32135
  1349
haftmann@43873
  1350
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1351
  by (fact uminus_INF)
wenzelm@12897
  1352
haftmann@43873
  1353
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1354
  by (fact uminus_SUP)
wenzelm@12897
  1355
wenzelm@12897
  1356
wenzelm@60758
  1357
subsubsection \<open>Miniscoping and maxiscoping\<close>
wenzelm@12897
  1358
wenzelm@63575
  1359
text \<open>\<^medskip> Miniscoping: pushing in quantifiers and big Unions and Intersections.\<close>
wenzelm@12897
  1360
wenzelm@12897
  1361
lemma UN_simps [simp]:
haftmann@43817
  1362
  "\<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
  1363
  "\<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
  1364
  "\<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
  1365
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1366
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1367
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1368
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1369
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1370
  "\<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
  1371
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1372
  by auto
wenzelm@12897
  1373
wenzelm@12897
  1374
lemma INT_simps [simp]:
haftmann@44032
  1375
  "\<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
  1376
  "\<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
  1377
  "\<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
  1378
  "\<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
  1379
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1380
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1381
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1382
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1383
  "\<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
  1384
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1385
  by auto
wenzelm@12897
  1386
blanchet@54147
  1387
lemma UN_ball_bex_simps [simp]:
haftmann@43852
  1388
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1389
  "\<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
  1390
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1391
  "\<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
  1392
  by auto
wenzelm@12897
  1393
haftmann@43943
  1394
wenzelm@63575
  1395
text \<open>\<^medskip> Maxiscoping: pulling out big Unions and Intersections.\<close>
paulson@13860
  1396
paulson@13860
  1397
lemma UN_extend_simps:
haftmann@43817
  1398
  "\<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
  1399
  "\<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
  1400
  "\<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
  1401
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1402
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1403
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1404
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1405
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1406
  "\<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
  1407
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1408
  by auto
paulson@13860
  1409
paulson@13860
  1410
lemma INT_extend_simps:
haftmann@43852
  1411
  "\<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
  1412
  "\<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
  1413
  "\<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
  1414
  "\<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
  1415
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1416
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1417
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1418
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1419
  "\<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
  1420
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1421
  by auto
paulson@13860
  1422
wenzelm@60758
  1423
text \<open>Finally\<close>
haftmann@43872
  1424
haftmann@30596
  1425
lemmas mem_simps =
haftmann@30596
  1426
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1427
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
wenzelm@61799
  1428
  \<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close>
wenzelm@21669
  1429
wenzelm@11979
  1430
end