src/HOL/Complete_Lattices.thy
author haftmann
Sat Mar 22 08:37:43 2014 +0100 (2014-03-22)
changeset 56248 67dc9549fa15
parent 56218 1c3f1f2431f9
child 56741 2b3710a4fa94
permissions -rw-r--r--
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
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(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
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header {* Complete lattices *}
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theory Complete_Lattices
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imports Fun
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begin
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notation
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  less_eq (infix "\<sqsubseteq>" 50) and
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  less (infix "\<sqsubset>" 50)
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subsection {* Syntactic infimum and supremum operations *}
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class Inf =
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  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
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begin
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definition INFIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  INF_def: "INFIMUM A f = \<Sqinter>(f ` A)"
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lemma Inf_image_eq [simp]:
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  "\<Sqinter>(f ` A) = INFIMUM A f"
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  by (simp add: INF_def)
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lemma INF_image [simp]:
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  "INFIMUM (f ` A) g = INFIMUM A (g \<circ> f)"
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  by (simp only: INF_def image_comp)
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lemma INF_identity_eq [simp]:
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  "INFIMUM A (\<lambda>x. x) = \<Sqinter>A"
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  by (simp add: INF_def)
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lemma INF_id_eq [simp]:
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  "INFIMUM A id = \<Sqinter>A"
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  by (simp add: id_def)
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lemma INF_cong:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D"
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  by (simp add: INF_def 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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definition SUPREMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  SUP_def: "SUPREMUM A f = \<Squnion>(f ` A)"
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lemma Sup_image_eq [simp]:
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  "\<Squnion>(f ` A) = SUPREMUM A f"
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  by (simp add: SUP_def)
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lemma SUP_image [simp]:
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  "SUPREMUM (f ` A) g = SUPREMUM A (g \<circ> f)"
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  by (simp only: SUP_def image_comp)
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lemma SUP_identity_eq [simp]:
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  "SUPREMUM A (\<lambda>x. x) = \<Squnion>A"
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  by (simp add: SUP_def)
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lemma SUP_id_eq [simp]:
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  "SUPREMUM A id = \<Squnion>A"
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  by (simp add: id_def)
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lemma SUP_cong:
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  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D"
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  by (simp add: SUP_def 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 {*
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  Note: must use names @{const INFIMUM} and @{const SUPREMUM} here instead of
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  @{text INF} and @{text SUP} to allow the following syntax coexist
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  with the plain constant names.
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*}
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syntax
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
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translations
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  "INF x y. B"   == "INF x. INF y. B"
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  "INF x. B"     == "CONST INFIMUM CONST UNIV (%x. B)"
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  "INF x. B"     == "INF x:CONST UNIV. B"
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  "INF x:A. B"   == "CONST INFIMUM A (%x. B)"
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  "SUP x y. B"   == "SUP x. SUP y. B"
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  "SUP x. B"     == "CONST SUPREMUM CONST UNIV (%x. B)"
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  "SUP x. B"     == "SUP x:CONST UNIV. B"
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  "SUP x:A. B"   == "CONST SUPREMUM A (%x. B)"
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print_translation {*
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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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*} -- {* to avoid eta-contraction of body *}
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subsection {* Abstract complete lattices *}
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text {* A complete lattice always has a bottom and a top,
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so we include them into the following type class,
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along with assumptions that define bottom and top
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in terms of infimum and supremum. *}
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class complete_lattice = lattice + Inf + Sup + bot + top +
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  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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  assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
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  assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
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begin
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subclass bounded_lattice
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proof
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  fix a
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  show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
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  show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
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qed
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lemma dual_complete_lattice:
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  "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
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  by (auto intro!: class.complete_lattice.intro dual_lattice)
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    (unfold_locales, (fact Inf_empty Sup_empty
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        Sup_upper Sup_least Inf_lower Inf_greatest)+)
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end
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context complete_lattice
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begin
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lemma INF_foundation_dual:
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  "Sup.SUPREMUM Inf = INFIMUM"
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  by (simp add: fun_eq_iff Sup.SUP_def)
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lemma SUP_foundation_dual:
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  "Inf.INFIMUM Sup = SUPREMUM"
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  by (simp add: fun_eq_iff Inf.INF_def)
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lemma Sup_eqI:
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  "(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x"
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  by (blast intro: antisym Sup_least Sup_upper)
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lemma Inf_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x"
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  by (blast intro: antisym Inf_greatest Inf_lower)
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lemma SUP_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x"
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  using Sup_eqI [of "f ` A" x] by auto
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lemma INF_eqI:
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  "(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x"
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  using Inf_eqI [of "f ` A" x] by auto
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lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
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  using Inf_lower [of _ "f ` A"] by simp
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lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
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  using Inf_greatest [of "f ` A"] by auto
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lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  using Sup_upper [of _ "f ` A"] by simp
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lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
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  using Sup_least [of "f ` A"] by auto
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
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  using Inf_lower [of u A] by auto
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lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
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  using INF_lower [of i A f] by auto
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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
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  using Sup_upper [of u A] by auto
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lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  using SUP_upper [of i A f] by auto
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lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
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  by (auto intro: Inf_greatest dest: Inf_lower)
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lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
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  using le_Inf_iff [of _ "f ` A"] by simp
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lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
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  by (auto intro: Sup_least dest: Sup_upper)
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lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
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  using Sup_le_iff [of "f ` A"] by simp
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lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
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  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
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lemma INF_insert [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFIMUM A f"
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  unfolding INF_def Inf_insert by simp
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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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  unfolding SUP_def Sup_insert by simp
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lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
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  by (simp add: INF_def)
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lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
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  by (simp add: SUP_def)
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lemma Inf_UNIV [simp]:
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  "\<Sqinter>UNIV = \<bottom>"
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  by (auto intro!: antisym Inf_lower)
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lemma Sup_UNIV [simp]:
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  "\<Squnion>UNIV = \<top>"
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  by (auto intro!: antisym Sup_upper)
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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  by (auto intro: Inf_greatest Inf_lower)
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lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
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  by (auto intro: Sup_least Sup_upper)
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lemma Inf_mono:
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  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
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  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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proof (rule Inf_greatest)
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  fix b assume "b \<in> B"
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  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
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  from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
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  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
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qed
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lemma INF_mono:
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  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
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  using Inf_mono [of "g ` B" "f ` A"] by auto
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lemma Sup_mono:
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  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
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  shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
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proof (rule Sup_least)
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  fix a assume "a \<in> A"
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  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
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  from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
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  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
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qed
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lemma SUP_mono:
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  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
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  using Sup_mono [of "f ` A" "g ` B"] by auto
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   274
haftmann@44041
   275
lemma INF_superset_mono:
haftmann@44041
   276
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
haftmann@44041
   277
  -- {* The last inclusion is POSITIVE! *}
haftmann@44041
   278
  by (blast intro: INF_mono dest: subsetD)
haftmann@44041
   279
haftmann@44041
   280
lemma SUP_subset_mono:
haftmann@44041
   281
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
haftmann@44041
   282
  by (blast intro: SUP_mono dest: subsetD)
haftmann@44041
   283
haftmann@43868
   284
lemma Inf_less_eq:
haftmann@43868
   285
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43868
   286
    and "A \<noteq> {}"
haftmann@43868
   287
  shows "\<Sqinter>A \<sqsubseteq> u"
haftmann@43868
   288
proof -
haftmann@43868
   289
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
wenzelm@53374
   290
  moreover from `v \<in> A` assms(1) have "v \<sqsubseteq> u" by blast
haftmann@43868
   291
  ultimately show ?thesis by (rule Inf_lower2)
haftmann@43868
   292
qed
haftmann@43868
   293
haftmann@43868
   294
lemma less_eq_Sup:
haftmann@43868
   295
  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
haftmann@43868
   296
    and "A \<noteq> {}"
haftmann@43868
   297
  shows "u \<sqsubseteq> \<Squnion>A"
haftmann@43868
   298
proof -
haftmann@43868
   299
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
wenzelm@53374
   300
  moreover from `v \<in> A` assms(1) have "u \<sqsubseteq> v" by blast
haftmann@43868
   301
  ultimately show ?thesis by (rule Sup_upper2)
haftmann@43868
   302
qed
haftmann@43868
   303
haftmann@56212
   304
lemma SUP_eq:
hoelzl@51328
   305
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
hoelzl@51328
   306
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
haftmann@56166
   307
  shows "(\<Squnion>i\<in>A. f i) = (\<Squnion>j\<in>B. g j)"
hoelzl@51328
   308
  by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
hoelzl@51328
   309
haftmann@56212
   310
lemma INF_eq:
hoelzl@51328
   311
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
hoelzl@51328
   312
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
haftmann@56166
   313
  shows "(\<Sqinter>i\<in>A. f i) = (\<Sqinter>j\<in>B. g j)"
hoelzl@51328
   314
  by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
hoelzl@51328
   315
haftmann@43899
   316
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43868
   317
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43868
   318
haftmann@43899
   319
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
haftmann@43868
   320
  by (auto intro: Sup_least Sup_upper)
haftmann@43868
   321
haftmann@43868
   322
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43868
   323
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
haftmann@43868
   324
haftmann@44041
   325
lemma INF_union:
haftmann@44041
   326
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@44103
   327
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
haftmann@44041
   328
haftmann@43868
   329
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
haftmann@43868
   330
  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
haftmann@43868
   331
haftmann@44041
   332
lemma SUP_union:
haftmann@44041
   333
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@44103
   334
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
haftmann@44041
   335
haftmann@44041
   336
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
   337
  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041
   338
noschinl@44918
   339
lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" (is "?L = ?R")
noschinl@44918
   340
proof (rule antisym)
noschinl@44918
   341
  show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
noschinl@44918
   342
next
noschinl@44918
   343
  show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
noschinl@44918
   344
qed
haftmann@44041
   345
blanchet@54147
   346
lemma Inf_top_conv [simp]:
haftmann@43868
   347
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   348
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   349
proof -
haftmann@43868
   350
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   351
  proof
haftmann@43868
   352
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   353
    then have "A = {} \<or> A = {\<top>}" by auto
noschinl@44919
   354
    then show "\<Sqinter>A = \<top>" by auto
haftmann@43868
   355
  next
haftmann@43868
   356
    assume "\<Sqinter>A = \<top>"
haftmann@43868
   357
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   358
    proof (rule ccontr)
haftmann@43868
   359
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   360
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868
   361
      then obtain B where "A = insert x B" by blast
noschinl@44919
   362
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
haftmann@43868
   363
    qed
haftmann@43868
   364
  qed
haftmann@43868
   365
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868
   366
qed
haftmann@43868
   367
noschinl@44918
   368
lemma INF_top_conv [simp]:
haftmann@56166
   369
  "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   370
  "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   371
  using Inf_top_conv [of "B ` A"] by simp_all
haftmann@44041
   372
blanchet@54147
   373
lemma Sup_bot_conv [simp]:
haftmann@43868
   374
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
haftmann@43868
   375
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
huffman@44920
   376
  using dual_complete_lattice
huffman@44920
   377
  by (rule complete_lattice.Inf_top_conv)+
haftmann@43868
   378
noschinl@44918
   379
lemma SUP_bot_conv [simp]:
haftmann@44041
   380
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041
   381
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@56166
   382
  using Sup_bot_conv [of "B ` A"] by simp_all
haftmann@44041
   383
haftmann@43865
   384
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103
   385
  by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077
   386
haftmann@43870
   387
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103
   388
  by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870
   389
noschinl@44918
   390
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
huffman@44921
   391
  by (cases "A = {}") simp_all
haftmann@43900
   392
noschinl@44918
   393
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
huffman@44921
   394
  by (cases "A = {}") simp_all
haftmann@43900
   395
haftmann@43865
   396
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
   397
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865
   398
haftmann@43870
   399
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
   400
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870
   401
haftmann@43871
   402
lemma INF_absorb:
haftmann@43868
   403
  assumes "k \<in> I"
haftmann@43868
   404
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   405
proof -
haftmann@43868
   406
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   407
  then show ?thesis by simp
haftmann@43868
   408
qed
haftmann@43868
   409
haftmann@43871
   410
lemma SUP_absorb:
haftmann@43871
   411
  assumes "k \<in> I"
haftmann@43871
   412
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   413
proof -
haftmann@43871
   414
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   415
  then show ?thesis by simp
haftmann@43871
   416
qed
haftmann@43871
   417
haftmann@43871
   418
lemma INF_constant:
haftmann@43868
   419
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921
   420
  by simp
haftmann@43868
   421
haftmann@43871
   422
lemma SUP_constant:
haftmann@43871
   423
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921
   424
  by simp
haftmann@43871
   425
haftmann@43943
   426
lemma less_INF_D:
haftmann@43943
   427
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   428
proof -
haftmann@43943
   429
  note `y < (\<Sqinter>i\<in>A. f i)`
haftmann@43943
   430
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
haftmann@44103
   431
    by (rule INF_lower)
haftmann@43943
   432
  finally show "y < f i" .
haftmann@43943
   433
qed
haftmann@43943
   434
haftmann@43943
   435
lemma SUP_lessD:
haftmann@43943
   436
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   437
proof -
haftmann@43943
   438
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
haftmann@44103
   439
    by (rule SUP_upper)
haftmann@43943
   440
  also note `(\<Squnion>i\<in>A. f i) < y`
haftmann@43943
   441
  finally show "f i < y" .
haftmann@43943
   442
qed
haftmann@43943
   443
haftmann@43873
   444
lemma INF_UNIV_bool_expand:
haftmann@43868
   445
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@56166
   446
  by (simp add: UNIV_bool inf_commute)
haftmann@43868
   447
haftmann@43873
   448
lemma SUP_UNIV_bool_expand:
haftmann@43871
   449
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@56166
   450
  by (simp add: UNIV_bool sup_commute)
haftmann@43871
   451
hoelzl@51328
   452
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@51328
   453
  by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
hoelzl@51328
   454
haftmann@56218
   455
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f"
haftmann@56166
   456
  using Inf_le_Sup [of "f ` A"] by simp
hoelzl@51328
   457
hoelzl@54414
   458
lemma INF_eq_const:
haftmann@56218
   459
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x"
hoelzl@54414
   460
  by (auto intro: INF_eqI)
hoelzl@54414
   461
haftmann@56248
   462
lemma SUP_eq_const:
haftmann@56248
   463
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x"
haftmann@56248
   464
  by (auto intro: SUP_eqI)
hoelzl@54414
   465
hoelzl@54414
   466
lemma INF_eq_iff:
haftmann@56218
   467
  "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
   468
  using INF_eq_const [of I f c] INF_lower [of _ I f]
haftmann@56248
   469
  by (auto intro: antisym cong del: strong_INF_cong)
haftmann@56248
   470
haftmann@56248
   471
lemma SUP_eq_iff:
haftmann@56248
   472
  "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
   473
  using SUP_eq_const [of I f c] SUP_upper [of _ I f]
haftmann@56248
   474
  by (auto intro: antisym cong del: strong_SUP_cong)
hoelzl@54414
   475
haftmann@32077
   476
end
haftmann@32077
   477
haftmann@44024
   478
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   479
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   480
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   481
begin
haftmann@44024
   482
haftmann@44039
   483
lemma sup_INF:
haftmann@44039
   484
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@56166
   485
  by (simp only: INF_def sup_Inf image_image)
haftmann@44039
   486
haftmann@44039
   487
lemma inf_SUP:
haftmann@44039
   488
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@56166
   489
  by (simp only: SUP_def inf_Sup image_image)
haftmann@44039
   490
haftmann@44032
   491
lemma dual_complete_distrib_lattice:
krauss@44845
   492
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   493
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   494
  apply (fact dual_complete_lattice)
haftmann@44024
   495
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032
   496
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032
   497
  done
haftmann@44024
   498
haftmann@44322
   499
subclass distrib_lattice proof
haftmann@44024
   500
  fix a b c
haftmann@44024
   501
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
noschinl@44919
   502
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
haftmann@44024
   503
qed
haftmann@44024
   504
haftmann@44039
   505
lemma Inf_sup:
haftmann@44039
   506
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   507
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   508
haftmann@44039
   509
lemma Sup_inf:
haftmann@44039
   510
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   511
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   512
haftmann@44039
   513
lemma INF_sup: 
haftmann@44039
   514
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   515
  by (simp add: sup_INF sup_commute)
haftmann@44039
   516
haftmann@44039
   517
lemma SUP_inf:
haftmann@44039
   518
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   519
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   520
haftmann@44039
   521
lemma Inf_sup_eq_top_iff:
haftmann@44039
   522
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   523
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   524
haftmann@44039
   525
lemma Sup_inf_eq_bot_iff:
haftmann@44039
   526
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   527
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   528
haftmann@44039
   529
lemma INF_sup_distrib2:
haftmann@44039
   530
  "(\<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
   531
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   532
haftmann@44039
   533
lemma SUP_inf_distrib2:
haftmann@44039
   534
  "(\<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
   535
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   536
haftmann@56074
   537
context
haftmann@56074
   538
  fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
haftmann@56074
   539
  assumes "mono f"
haftmann@56074
   540
begin
haftmann@56074
   541
haftmann@56074
   542
lemma mono_Inf:
haftmann@56074
   543
  shows "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
haftmann@56074
   544
  using `mono f` by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
haftmann@56074
   545
haftmann@56074
   546
lemma mono_Sup:
haftmann@56074
   547
  shows "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
haftmann@56074
   548
  using `mono f` by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
haftmann@56074
   549
haftmann@56074
   550
end
haftmann@56074
   551
haftmann@44024
   552
end
haftmann@44024
   553
haftmann@44032
   554
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   555
begin
haftmann@43873
   556
haftmann@43943
   557
lemma dual_complete_boolean_algebra:
krauss@44845
   558
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   559
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   560
haftmann@43873
   561
lemma uminus_Inf:
haftmann@43873
   562
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   563
proof (rule antisym)
haftmann@43873
   564
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   565
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   566
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   567
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   568
qed
haftmann@43873
   569
haftmann@44041
   570
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@56166
   571
  by (simp only: INF_def SUP_def uminus_Inf image_image)
haftmann@44041
   572
haftmann@43873
   573
lemma uminus_Sup:
haftmann@43873
   574
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   575
proof -
haftmann@56166
   576
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_INF)
haftmann@43873
   577
  then show ?thesis by simp
haftmann@43873
   578
qed
haftmann@43873
   579
  
haftmann@43873
   580
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@56166
   581
  by (simp only: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   582
haftmann@43873
   583
end
haftmann@43873
   584
haftmann@43940
   585
class complete_linorder = linorder + complete_lattice
haftmann@43940
   586
begin
haftmann@43940
   587
haftmann@43943
   588
lemma dual_complete_linorder:
krauss@44845
   589
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   590
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   591
haftmann@51386
   592
lemma complete_linorder_inf_min: "inf = min"
haftmann@51540
   593
  by (auto intro: antisym simp add: min_def fun_eq_iff)
haftmann@51386
   594
haftmann@51386
   595
lemma complete_linorder_sup_max: "sup = max"
haftmann@51540
   596
  by (auto intro: antisym simp add: max_def fun_eq_iff)
haftmann@51386
   597
noschinl@44918
   598
lemma Inf_less_iff:
haftmann@43940
   599
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   600
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   601
noschinl@44918
   602
lemma INF_less_iff:
haftmann@44041
   603
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@56166
   604
  using Inf_less_iff [of "f ` A"] by simp
haftmann@44041
   605
noschinl@44918
   606
lemma less_Sup_iff:
haftmann@43940
   607
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   608
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   609
noschinl@44918
   610
lemma less_SUP_iff:
haftmann@43940
   611
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@56166
   612
  using less_Sup_iff [of _ "f ` A"] by simp
haftmann@43940
   613
noschinl@44918
   614
lemma Sup_eq_top_iff [simp]:
haftmann@43943
   615
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   616
proof
haftmann@43943
   617
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   618
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   619
  proof (intro allI impI)
haftmann@43943
   620
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   621
      unfolding less_Sup_iff by auto
haftmann@43943
   622
  qed
haftmann@43943
   623
next
haftmann@43943
   624
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   625
  show "\<Squnion>A = \<top>"
haftmann@43943
   626
  proof (rule ccontr)
haftmann@43943
   627
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   628
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   629
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   630
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   631
      using * unfolding less_Sup_iff by auto
haftmann@43943
   632
    then show False by auto
haftmann@43943
   633
  qed
haftmann@43943
   634
qed
haftmann@43943
   635
noschinl@44918
   636
lemma SUP_eq_top_iff [simp]:
haftmann@44041
   637
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
haftmann@56166
   638
  using Sup_eq_top_iff [of "f ` A"] by simp
haftmann@44041
   639
noschinl@44918
   640
lemma Inf_eq_bot_iff [simp]:
haftmann@43943
   641
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920
   642
  using dual_complete_linorder
huffman@44920
   643
  by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943
   644
noschinl@44918
   645
lemma INF_eq_bot_iff [simp]:
haftmann@43967
   646
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
haftmann@56166
   647
  using Inf_eq_bot_iff [of "f ` A"] by simp
hoelzl@51328
   648
hoelzl@51328
   649
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
hoelzl@51328
   650
proof safe
hoelzl@51328
   651
  fix y assume "x \<ge> \<Sqinter>A" "y > x"
hoelzl@51328
   652
  then have "y > \<Sqinter>A" by auto
hoelzl@51328
   653
  then show "\<exists>a\<in>A. y > a"
hoelzl@51328
   654
    unfolding Inf_less_iff .
hoelzl@51328
   655
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
hoelzl@51328
   656
hoelzl@51328
   657
lemma INF_le_iff:
haftmann@56218
   658
  "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
haftmann@56166
   659
  using Inf_le_iff [of "f ` A"] by simp
haftmann@56166
   660
haftmann@56166
   661
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
haftmann@56166
   662
proof safe
haftmann@56166
   663
  fix y assume "x \<le> \<Squnion>A" "y < x"
haftmann@56166
   664
  then have "y < \<Squnion>A" by auto
haftmann@56166
   665
  then show "\<exists>a\<in>A. y < a"
haftmann@56166
   666
    unfolding less_Sup_iff .
haftmann@56166
   667
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
haftmann@56166
   668
haftmann@56218
   669
lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
haftmann@56166
   670
  using le_Sup_iff [of _ "f ` A"] by simp
hoelzl@51328
   671
haftmann@51386
   672
subclass complete_distrib_lattice
haftmann@51386
   673
proof
haftmann@51386
   674
  fix a and B
haftmann@51386
   675
  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
   676
    by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
haftmann@51386
   677
      (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
haftmann@51386
   678
        le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
haftmann@51386
   679
qed
haftmann@51386
   680
haftmann@43940
   681
end
haftmann@43940
   682
hoelzl@51341
   683
haftmann@46631
   684
subsection {* Complete lattice on @{typ bool} *}
haftmann@32077
   685
haftmann@44024
   686
instantiation bool :: complete_lattice
haftmann@32077
   687
begin
haftmann@32077
   688
haftmann@32077
   689
definition
haftmann@46154
   690
  [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   691
haftmann@32077
   692
definition
haftmann@46154
   693
  [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   694
haftmann@32077
   695
instance proof
haftmann@44322
   696
qed (auto intro: bool_induct)
haftmann@32077
   697
haftmann@32077
   698
end
haftmann@32077
   699
haftmann@49905
   700
lemma not_False_in_image_Ball [simp]:
haftmann@49905
   701
  "False \<notin> P ` A \<longleftrightarrow> Ball A P"
haftmann@49905
   702
  by auto
haftmann@49905
   703
haftmann@49905
   704
lemma True_in_image_Bex [simp]:
haftmann@49905
   705
  "True \<in> P ` A \<longleftrightarrow> Bex A P"
haftmann@49905
   706
  by auto
haftmann@49905
   707
haftmann@43873
   708
lemma INF_bool_eq [simp]:
haftmann@56218
   709
  "INFIMUM = Ball"
haftmann@49905
   710
  by (simp add: fun_eq_iff INF_def)
haftmann@32120
   711
haftmann@43873
   712
lemma SUP_bool_eq [simp]:
haftmann@56218
   713
  "SUPREMUM = Bex"
haftmann@49905
   714
  by (simp add: fun_eq_iff SUP_def)
haftmann@32120
   715
haftmann@44032
   716
instance bool :: complete_boolean_algebra proof
haftmann@44322
   717
qed (auto intro: bool_induct)
haftmann@44024
   718
haftmann@46631
   719
haftmann@46631
   720
subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
haftmann@46631
   721
haftmann@32077
   722
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   723
begin
haftmann@32077
   724
haftmann@32077
   725
definition
haftmann@44024
   726
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   727
noschinl@46882
   728
lemma Inf_apply [simp, code]:
haftmann@44024
   729
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   730
  by (simp add: Inf_fun_def)
haftmann@32077
   731
haftmann@32077
   732
definition
haftmann@44024
   733
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   734
noschinl@46882
   735
lemma Sup_apply [simp, code]:
haftmann@44024
   736
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   737
  by (simp add: Sup_fun_def)
haftmann@32077
   738
haftmann@32077
   739
instance proof
noschinl@46884
   740
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   741
haftmann@32077
   742
end
haftmann@32077
   743
noschinl@46882
   744
lemma INF_apply [simp]:
haftmann@41080
   745
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@56166
   746
  using Inf_apply [of "f ` A"] by (simp add: comp_def)
hoelzl@38705
   747
noschinl@46882
   748
lemma SUP_apply [simp]:
haftmann@41080
   749
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@56166
   750
  using Sup_apply [of "f ` A"] by (simp add: comp_def)
haftmann@32077
   751
haftmann@44024
   752
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
haftmann@56166
   753
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf fun_eq_iff image_image
haftmann@56166
   754
  simp del: Inf_image_eq Sup_image_eq)
haftmann@44024
   755
haftmann@43873
   756
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   757
haftmann@46631
   758
haftmann@46631
   759
subsection {* Complete lattice on unary and binary predicates *}
haftmann@46631
   760
haftmann@46631
   761
lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
noschinl@46884
   762
  by simp
haftmann@46631
   763
haftmann@46631
   764
lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
noschinl@46884
   765
  by simp
haftmann@46631
   766
haftmann@46631
   767
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
noschinl@46884
   768
  by auto
haftmann@46631
   769
haftmann@46631
   770
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
noschinl@46884
   771
  by auto
haftmann@46631
   772
haftmann@46631
   773
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
noschinl@46884
   774
  by auto
haftmann@46631
   775
haftmann@46631
   776
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
noschinl@46884
   777
  by auto
haftmann@46631
   778
haftmann@46631
   779
lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   780
  by auto
haftmann@46631
   781
haftmann@46631
   782
lemma INF2_E: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   783
  by auto
haftmann@46631
   784
haftmann@46631
   785
lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
noschinl@46884
   786
  by simp
haftmann@46631
   787
haftmann@46631
   788
lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
noschinl@46884
   789
  by simp
haftmann@46631
   790
haftmann@46631
   791
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
noschinl@46884
   792
  by auto
haftmann@46631
   793
haftmann@46631
   794
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
noschinl@46884
   795
  by auto
haftmann@46631
   796
haftmann@46631
   797
lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   798
  by auto
haftmann@46631
   799
haftmann@46631
   800
lemma SUP2_E: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884
   801
  by auto
haftmann@46631
   802
haftmann@46631
   803
haftmann@46631
   804
subsection {* Complete lattice on @{typ "_ set"} *}
haftmann@46631
   805
haftmann@45960
   806
instantiation "set" :: (type) complete_lattice
haftmann@45960
   807
begin
haftmann@45960
   808
haftmann@45960
   809
definition
haftmann@45960
   810
  "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   811
haftmann@45960
   812
definition
haftmann@45960
   813
  "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   814
haftmann@45960
   815
instance proof
haftmann@51386
   816
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
haftmann@45960
   817
haftmann@45960
   818
end
haftmann@45960
   819
haftmann@45960
   820
instance "set" :: (type) complete_boolean_algebra
haftmann@45960
   821
proof
haftmann@45960
   822
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
haftmann@45960
   823
  
haftmann@32077
   824
haftmann@46631
   825
subsubsection {* Inter *}
haftmann@41082
   826
haftmann@41082
   827
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   828
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   829
  
haftmann@41082
   830
notation (xsymbols)
haftmann@52141
   831
  Inter  ("\<Inter>_" [900] 900)
haftmann@41082
   832
haftmann@41082
   833
lemma Inter_eq:
haftmann@41082
   834
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   835
proof (rule set_eqI)
haftmann@41082
   836
  fix x
haftmann@41082
   837
  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
   838
    by auto
haftmann@41082
   839
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960
   840
    by (simp add: Inf_set_def image_def)
haftmann@41082
   841
qed
haftmann@41082
   842
blanchet@54147
   843
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   844
  by (unfold Inter_eq) blast
haftmann@41082
   845
haftmann@43741
   846
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   847
  by (simp add: Inter_eq)
haftmann@41082
   848
haftmann@41082
   849
text {*
haftmann@41082
   850
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   851
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   852
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   853
*}
haftmann@41082
   854
haftmann@43741
   855
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   856
  by auto
haftmann@41082
   857
haftmann@43741
   858
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   859
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   860
    @{prop "X \<in> C"}. *}
haftmann@41082
   861
  by (unfold Inter_eq) blast
haftmann@41082
   862
haftmann@43741
   863
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   864
  by (fact Inf_lower)
haftmann@43740
   865
haftmann@41082
   866
lemma Inter_subset:
haftmann@43755
   867
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   868
  by (fact Inf_less_eq)
haftmann@41082
   869
haftmann@43755
   870
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   871
  by (fact Inf_greatest)
haftmann@41082
   872
huffman@44067
   873
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   874
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   875
huffman@44067
   876
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   877
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   878
huffman@44920
   879
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920
   880
  by (fact Inf_insert) (* already simp *)
haftmann@41082
   881
haftmann@41082
   882
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   883
  by (fact less_eq_Inf_inter)
haftmann@41082
   884
haftmann@41082
   885
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   886
  by (fact Inf_union_distrib)
haftmann@43756
   887
blanchet@54147
   888
lemma Inter_UNIV_conv [simp]:
haftmann@43741
   889
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   890
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   891
  by (fact Inf_top_conv)+
haftmann@41082
   892
haftmann@43741
   893
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   894
  by (fact Inf_superset_mono)
haftmann@41082
   895
haftmann@41082
   896
haftmann@46631
   897
subsubsection {* Intersections of families *}
haftmann@41082
   898
haftmann@41082
   899
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@56218
   900
  "INTER \<equiv> INFIMUM"
haftmann@41082
   901
haftmann@43872
   902
text {*
haftmann@43872
   903
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   904
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   905
*}
haftmann@43872
   906
haftmann@41082
   907
syntax
haftmann@41082
   908
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   909
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   910
haftmann@41082
   911
syntax (xsymbols)
haftmann@41082
   912
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   913
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   914
haftmann@41082
   915
syntax (latex output)
haftmann@41082
   916
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   917
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   918
haftmann@41082
   919
translations
haftmann@41082
   920
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   921
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   922
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   923
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   924
haftmann@41082
   925
print_translation {*
wenzelm@42284
   926
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   927
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   928
haftmann@44085
   929
lemma INTER_eq:
haftmann@41082
   930
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@56166
   931
  by (auto intro!: INF_eqI)
haftmann@41082
   932
haftmann@56166
   933
lemma Inter_image_eq:
haftmann@56166
   934
  "\<Inter>(B ` A) = (\<Inter>x\<in>A. B x)"
haftmann@56166
   935
  by (fact Inf_image_eq)
haftmann@41082
   936
haftmann@43817
   937
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@56166
   938
  using Inter_iff [of _ "B ` A"] by simp
haftmann@41082
   939
haftmann@43817
   940
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
haftmann@44085
   941
  by (auto simp add: INF_def image_def)
haftmann@41082
   942
haftmann@43852
   943
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
   944
  by auto
haftmann@41082
   945
haftmann@43852
   946
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@43852
   947
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@44085
   948
  by (auto simp add: INF_def image_def)
haftmann@41082
   949
haftmann@41082
   950
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   951
  by blast
haftmann@41082
   952
haftmann@41082
   953
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   954
  by blast
haftmann@41082
   955
haftmann@43817
   956
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
   957
  by (fact INF_lower)
haftmann@41082
   958
haftmann@43817
   959
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
   960
  by (fact INF_greatest)
haftmann@41082
   961
huffman@44067
   962
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
   963
  by (fact INF_empty)
haftmann@43854
   964
haftmann@43817
   965
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   966
  by (fact INF_absorb)
haftmann@41082
   967
haftmann@43854
   968
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   969
  by (fact le_INF_iff)
haftmann@41082
   970
haftmann@41082
   971
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   972
  by (fact INF_insert)
haftmann@43865
   973
haftmann@43865
   974
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
   975
  by (fact INF_union)
haftmann@43865
   976
haftmann@43865
   977
lemma INT_insert_distrib:
haftmann@43865
   978
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   979
  by blast
haftmann@43854
   980
haftmann@41082
   981
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   982
  by (fact INF_constant)
haftmann@43865
   983
huffman@44920
   984
lemma INTER_UNIV_conv:
haftmann@43817
   985
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   986
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920
   987
  by (fact INF_top_conv)+ (* already simp *)
haftmann@43865
   988
haftmann@43865
   989
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   990
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   991
haftmann@43865
   992
lemma INT_anti_mono:
haftmann@43900
   993
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@43865
   994
  -- {* The last inclusion is POSITIVE! *}
haftmann@43940
   995
  by (fact INF_superset_mono)
haftmann@41082
   996
haftmann@41082
   997
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   998
  by blast
haftmann@41082
   999
haftmann@43817
  1000
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
  1001
  by blast
haftmann@41082
  1002
haftmann@41082
  1003
haftmann@46631
  1004
subsubsection {* Union *}
haftmann@32115
  1005
haftmann@32587
  1006
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
  1007
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
  1008
haftmann@32115
  1009
notation (xsymbols)
haftmann@52141
  1010
  Union  ("\<Union>_" [900] 900)
haftmann@32115
  1011
haftmann@32135
  1012
lemma Union_eq:
haftmann@32135
  1013
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
  1014
proof (rule set_eqI)
haftmann@32115
  1015
  fix x
haftmann@32135
  1016
  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
  1017
    by auto
haftmann@32135
  1018
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960
  1019
    by (simp add: Sup_set_def image_def)
haftmann@32115
  1020
qed
haftmann@32115
  1021
blanchet@54147
  1022
lemma Union_iff [simp]:
haftmann@32115
  1023
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
  1024
  by (unfold Union_eq) blast
haftmann@32115
  1025
haftmann@32115
  1026
lemma UnionI [intro]:
haftmann@32115
  1027
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
  1028
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
  1029
    @{term A} may be flexible. *}
haftmann@32115
  1030
  by auto
haftmann@32115
  1031
haftmann@32115
  1032
lemma UnionE [elim!]:
haftmann@43817
  1033
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
  1034
  by auto
haftmann@32115
  1035
haftmann@43817
  1036
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
  1037
  by (fact Sup_upper)
haftmann@32135
  1038
haftmann@43817
  1039
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
  1040
  by (fact Sup_least)
haftmann@32135
  1041
huffman@44920
  1042
lemma Union_empty: "\<Union>{} = {}"
huffman@44920
  1043
  by (fact Sup_empty) (* already simp *)
haftmann@32135
  1044
huffman@44920
  1045
lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920
  1046
  by (fact Sup_UNIV) (* already simp *)
haftmann@32135
  1047
huffman@44920
  1048
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920
  1049
  by (fact Sup_insert) (* already simp *)
haftmann@32135
  1050
haftmann@43817
  1051
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
  1052
  by (fact Sup_union_distrib)
haftmann@32135
  1053
haftmann@32135
  1054
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
  1055
  by (fact Sup_inter_less_eq)
haftmann@32135
  1056
blanchet@54147
  1057
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1058
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1059
blanchet@54147
  1060
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1061
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1062
haftmann@32135
  1063
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
  1064
  by blast
haftmann@32135
  1065
haftmann@32135
  1066
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
  1067
  by blast
haftmann@32135
  1068
haftmann@43817
  1069
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
  1070
  by (fact Sup_subset_mono)
haftmann@32135
  1071
haftmann@32115
  1072
haftmann@46631
  1073
subsubsection {* Unions of families *}
haftmann@32077
  1074
haftmann@32606
  1075
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@56218
  1076
  "UNION \<equiv> SUPREMUM"
haftmann@32077
  1077
haftmann@43872
  1078
text {*
haftmann@43872
  1079
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
  1080
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
  1081
*}
haftmann@43872
  1082
haftmann@32077
  1083
syntax
wenzelm@35115
  1084
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
  1085
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
  1086
haftmann@32077
  1087
syntax (xsymbols)
wenzelm@35115
  1088
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
  1089
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
  1090
haftmann@32077
  1091
syntax (latex output)
wenzelm@35115
  1092
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
  1093
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
  1094
haftmann@32077
  1095
translations
haftmann@32077
  1096
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
  1097
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
  1098
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
  1099
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
  1100
haftmann@32077
  1101
text {*
haftmann@32077
  1102
  Note the difference between ordinary xsymbol syntax of indexed
wenzelm@53015
  1103
  unions and intersections (e.g.\ @{text"\<Union>a\<^sub>1\<in>A\<^sub>1. B"})
wenzelm@53015
  1104
  and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. The
haftmann@32077
  1105
  former does not make the index expression a subscript of the
haftmann@32077
  1106
  union/intersection symbol because this leads to problems with nested
haftmann@32077
  1107
  subscripts in Proof General.
haftmann@32077
  1108
*}
haftmann@32077
  1109
wenzelm@35115
  1110
print_translation {*
wenzelm@42284
  1111
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
  1112
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
  1113
blanchet@54147
  1114
lemma UNION_eq:
haftmann@32135
  1115
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@56166
  1116
  by (auto intro!: SUP_eqI)
huffman@44920
  1117
haftmann@45960
  1118
lemma bind_UNION [code]:
haftmann@45960
  1119
  "Set.bind A f = UNION A f"
haftmann@45960
  1120
  by (simp add: bind_def UNION_eq)
haftmann@45960
  1121
haftmann@46036
  1122
lemma member_bind [simp]:
haftmann@46036
  1123
  "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036
  1124
  by (simp add: bind_UNION)
haftmann@46036
  1125
haftmann@56166
  1126
lemma Union_image_eq:
haftmann@43817
  1127
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@56166
  1128
  by (fact Sup_image_eq)
huffman@44920
  1129
haftmann@46036
  1130
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@56166
  1131
  using Union_iff [of _ "B ` A"] by simp
wenzelm@11979
  1132
haftmann@43852
  1133
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
  1134
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
  1135
    @{term b} may be flexible. *}
wenzelm@11979
  1136
  by auto
wenzelm@11979
  1137
haftmann@43852
  1138
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@44085
  1139
  by (auto simp add: SUP_def image_def)
clasohm@923
  1140
haftmann@43817
  1141
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
  1142
  by blast
haftmann@32077
  1143
haftmann@43817
  1144
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
  1145
  by (fact SUP_upper)
haftmann@32135
  1146
haftmann@43817
  1147
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
  1148
  by (fact SUP_least)
haftmann@32135
  1149
blanchet@54147
  1150
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
  1151
  by blast
haftmann@32135
  1152
haftmann@43817
  1153
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
  1154
  by blast
haftmann@32135
  1155
blanchet@54147
  1156
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
  1157
  by (fact SUP_empty)
haftmann@32135
  1158
huffman@44920
  1159
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920
  1160
  by (fact SUP_bot) (* already simp *)
haftmann@32135
  1161
haftmann@43817
  1162
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
  1163
  by (fact SUP_absorb)
haftmann@32135
  1164
haftmann@32135
  1165
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
  1166
  by (fact SUP_insert)
haftmann@32135
  1167
haftmann@44085
  1168
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
  1169
  by (fact SUP_union)
haftmann@32135
  1170
haftmann@43967
  1171
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
  1172
  by blast
haftmann@32135
  1173
haftmann@32135
  1174
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
  1175
  by (fact SUP_le_iff)
haftmann@32135
  1176
haftmann@32135
  1177
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
  1178
  by (fact SUP_constant)
haftmann@32135
  1179
haftmann@43944
  1180
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
  1181
  by blast
haftmann@32135
  1182
huffman@44920
  1183
lemma UNION_empty_conv:
haftmann@43817
  1184
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1185
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920
  1186
  by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135
  1187
blanchet@54147
  1188
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1189
  by blast
haftmann@32135
  1190
haftmann@43900
  1191
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
  1192
  by blast
haftmann@32135
  1193
haftmann@43900
  1194
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
  1195
  by blast
haftmann@32135
  1196
haftmann@32135
  1197
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
  1198
  by (auto simp add: split_if_mem2)
haftmann@32135
  1199
haftmann@43817
  1200
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1201
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1202
haftmann@32135
  1203
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1204
  by blast
haftmann@32135
  1205
haftmann@32135
  1206
lemma UN_mono:
haftmann@43817
  1207
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1208
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1209
  by (fact SUP_subset_mono)
haftmann@32135
  1210
haftmann@43817
  1211
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1212
  by blast
haftmann@32135
  1213
haftmann@43817
  1214
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1215
  by blast
haftmann@32135
  1216
haftmann@43817
  1217
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
  1218
  -- {* NOT suitable for rewriting *}
haftmann@32135
  1219
  by blast
haftmann@32135
  1220
haftmann@43817
  1221
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1222
  by blast
haftmann@32135
  1223
haftmann@45013
  1224
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013
  1225
  by blast
haftmann@45013
  1226
wenzelm@11979
  1227
haftmann@46631
  1228
subsubsection {* Distributive laws *}
wenzelm@12897
  1229
wenzelm@12897
  1230
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1231
  by (fact inf_Sup)
wenzelm@12897
  1232
haftmann@44039
  1233
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1234
  by (fact sup_Inf)
haftmann@44039
  1235
wenzelm@12897
  1236
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1237
  by (fact Sup_inf)
haftmann@44039
  1238
haftmann@44039
  1239
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
  1240
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1241
haftmann@44039
  1242
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
  1243
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1244
haftmann@56166
  1245
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" -- {* FIXME drop *}
haftmann@56166
  1246
  by (simp add: INT_Int_distrib)
wenzelm@12897
  1247
haftmann@56166
  1248
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" -- {* FIXME drop *}
wenzelm@12897
  1249
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1250
  -- {* Union of a family of unions *}
haftmann@56166
  1251
  by (simp add: UN_Un_distrib)
wenzelm@12897
  1252
haftmann@44039
  1253
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1254
  by (fact sup_INF)
wenzelm@12897
  1255
wenzelm@12897
  1256
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1257
  -- {* Halmos, Naive Set Theory, page 35. *}
haftmann@44039
  1258
  by (fact inf_SUP)
wenzelm@12897
  1259
wenzelm@12897
  1260
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
  1261
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1262
wenzelm@12897
  1263
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
  1264
  by (fact INF_sup_distrib2)
haftmann@44039
  1265
haftmann@44039
  1266
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1267
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1268
wenzelm@12897
  1269
haftmann@56015
  1270
subsection {* Injections and bijections *}
haftmann@56015
  1271
haftmann@56015
  1272
lemma inj_on_Inter:
haftmann@56015
  1273
  "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
haftmann@56015
  1274
  unfolding inj_on_def by blast
haftmann@56015
  1275
haftmann@56015
  1276
lemma inj_on_INTER:
haftmann@56015
  1277
  "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@56015
  1278
  unfolding inj_on_def by blast
haftmann@56015
  1279
haftmann@56015
  1280
lemma inj_on_UNION_chain:
haftmann@56015
  1281
  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
haftmann@56015
  1282
         INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
haftmann@56015
  1283
  shows "inj_on f (\<Union> i \<in> I. A i)"
haftmann@56015
  1284
proof -
haftmann@56015
  1285
  {
haftmann@56015
  1286
    fix i j x y
haftmann@56015
  1287
    assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
haftmann@56015
  1288
      and ***: "f x = f y"
haftmann@56015
  1289
    have "x = y"
haftmann@56015
  1290
    proof -
haftmann@56015
  1291
      {
haftmann@56015
  1292
        assume "A i \<le> A j"
haftmann@56015
  1293
        with ** have "x \<in> A j" by auto
haftmann@56015
  1294
        with INJ * ** *** have ?thesis
haftmann@56015
  1295
        by(auto simp add: inj_on_def)
haftmann@56015
  1296
      }
haftmann@56015
  1297
      moreover
haftmann@56015
  1298
      {
haftmann@56015
  1299
        assume "A j \<le> A i"
haftmann@56015
  1300
        with ** have "y \<in> A i" by auto
haftmann@56015
  1301
        with INJ * ** *** have ?thesis
haftmann@56015
  1302
        by(auto simp add: inj_on_def)
haftmann@56015
  1303
      }
haftmann@56015
  1304
      ultimately show ?thesis using CH * by blast
haftmann@56015
  1305
    qed
haftmann@56015
  1306
  }
haftmann@56015
  1307
  then show ?thesis by (unfold inj_on_def UNION_eq) auto
haftmann@56015
  1308
qed
haftmann@56015
  1309
haftmann@56015
  1310
lemma bij_betw_UNION_chain:
haftmann@56015
  1311
  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
haftmann@56015
  1312
         BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
haftmann@56015
  1313
  shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
haftmann@56015
  1314
proof (unfold bij_betw_def, auto)
haftmann@56015
  1315
  have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
haftmann@56015
  1316
  using BIJ bij_betw_def[of f] by auto
haftmann@56015
  1317
  thus "inj_on f (\<Union> i \<in> I. A i)"
haftmann@56015
  1318
  using CH inj_on_UNION_chain[of I A f] by auto
haftmann@56015
  1319
next
haftmann@56015
  1320
  fix i x
haftmann@56015
  1321
  assume *: "i \<in> I" "x \<in> A i"
haftmann@56015
  1322
  hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
haftmann@56015
  1323
  thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
haftmann@56015
  1324
next
haftmann@56015
  1325
  fix i x'
haftmann@56015
  1326
  assume *: "i \<in> I" "x' \<in> A' i"
haftmann@56015
  1327
  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
haftmann@56015
  1328
  then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
haftmann@56015
  1329
    using * by blast
haftmann@56015
  1330
  then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by blast
haftmann@56015
  1331
qed
haftmann@56015
  1332
haftmann@56015
  1333
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
haftmann@56015
  1334
lemma image_INT:
haftmann@56015
  1335
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
haftmann@56015
  1336
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
haftmann@56015
  1337
apply (simp add: inj_on_def, blast)
haftmann@56015
  1338
done
haftmann@56015
  1339
haftmann@56015
  1340
(*Compare with image_INT: no use of inj_on, and if f is surjective then
haftmann@56015
  1341
  it doesn't matter whether A is empty*)
haftmann@56015
  1342
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
haftmann@56015
  1343
apply (simp add: bij_def)
haftmann@56015
  1344
apply (simp add: inj_on_def surj_def, blast)
haftmann@56015
  1345
done
haftmann@56015
  1346
haftmann@56015
  1347
lemma UNION_fun_upd:
haftmann@56015
  1348
  "UNION J (A(i:=B)) = (UNION (J-{i}) A \<union> (if i\<in>J then B else {}))"
haftmann@56015
  1349
by (auto split: if_splits)
haftmann@56015
  1350
haftmann@56015
  1351
haftmann@46631
  1352
subsubsection {* Complement *}
haftmann@32135
  1353
haftmann@43873
  1354
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1355
  by (fact uminus_INF)
wenzelm@12897
  1356
haftmann@43873
  1357
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1358
  by (fact uminus_SUP)
wenzelm@12897
  1359
wenzelm@12897
  1360
haftmann@46631
  1361
subsubsection {* Miniscoping and maxiscoping *}
wenzelm@12897
  1362
paulson@13860
  1363
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1364
           and Intersections. *}
wenzelm@12897
  1365
wenzelm@12897
  1366
lemma UN_simps [simp]:
haftmann@43817
  1367
  "\<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
  1368
  "\<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
  1369
  "\<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
  1370
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1371
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1372
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1373
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1374
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1375
  "\<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
  1376
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1377
  by auto
wenzelm@12897
  1378
wenzelm@12897
  1379
lemma INT_simps [simp]:
haftmann@44032
  1380
  "\<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
  1381
  "\<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
  1382
  "\<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
  1383
  "\<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
  1384
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1385
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1386
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1387
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1388
  "\<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
  1389
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1390
  by auto
wenzelm@12897
  1391
blanchet@54147
  1392
lemma UN_ball_bex_simps [simp]:
haftmann@43852
  1393
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1394
  "\<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
  1395
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1396
  "\<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
  1397
  by auto
wenzelm@12897
  1398
haftmann@43943
  1399
paulson@13860
  1400
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1401
paulson@13860
  1402
lemma UN_extend_simps:
haftmann@43817
  1403
  "\<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
  1404
  "\<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
  1405
  "\<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
  1406
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1407
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1408
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1409
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1410
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1411
  "\<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
  1412
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1413
  by auto
paulson@13860
  1414
paulson@13860
  1415
lemma INT_extend_simps:
haftmann@43852
  1416
  "\<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
  1417
  "\<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
  1418
  "\<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
  1419
  "\<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
  1420
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1421
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1422
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1423
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1424
  "\<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
  1425
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1426
  by auto
paulson@13860
  1427
haftmann@43872
  1428
text {* Finally *}
haftmann@43872
  1429
haftmann@32135
  1430
no_notation
haftmann@46691
  1431
  less_eq (infix "\<sqsubseteq>" 50) and
haftmann@46691
  1432
  less (infix "\<sqsubset>" 50)
haftmann@32135
  1433
haftmann@30596
  1434
lemmas mem_simps =
haftmann@30596
  1435
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1436
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1437
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1438
wenzelm@11979
  1439
end
haftmann@49905
  1440