src/HOL/Complete_Lattices.thy
author wenzelm
Fri Jun 26 10:20:33 2015 +0200 (2015-06-26)
changeset 60585 48fdff264eb2
parent 60307 75e1aa7a450e
child 60758 d8d85a8172b5
permissions -rw-r--r--
tuned whitespace;
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(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
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section {* 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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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
hoelzl@57448
   418
lemma INF_inf_const1:
hoelzl@57448
   419
  "I \<noteq> {} \<Longrightarrow> (INF i:I. inf x (f i)) = inf x (INF i:I. f i)"
hoelzl@57448
   420
  by (intro antisym INF_greatest inf_mono order_refl INF_lower)
hoelzl@57448
   421
     (auto intro: INF_lower2 le_infI2 intro!: INF_mono)
hoelzl@57448
   422
hoelzl@57448
   423
lemma INF_inf_const2:
hoelzl@57448
   424
  "I \<noteq> {} \<Longrightarrow> (INF i:I. inf (f i) x) = inf (INF i:I. f i) x"
hoelzl@57448
   425
  using INF_inf_const1[of I x f] by (simp add: inf_commute)
hoelzl@57448
   426
haftmann@43871
   427
lemma INF_constant:
haftmann@43868
   428
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921
   429
  by simp
haftmann@43868
   430
haftmann@43871
   431
lemma SUP_constant:
haftmann@43871
   432
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921
   433
  by simp
haftmann@43871
   434
haftmann@43943
   435
lemma less_INF_D:
haftmann@43943
   436
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   437
proof -
haftmann@43943
   438
  note `y < (\<Sqinter>i\<in>A. f i)`
haftmann@43943
   439
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
haftmann@44103
   440
    by (rule INF_lower)
haftmann@43943
   441
  finally show "y < f i" .
haftmann@43943
   442
qed
haftmann@43943
   443
haftmann@43943
   444
lemma SUP_lessD:
haftmann@43943
   445
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   446
proof -
haftmann@43943
   447
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
haftmann@44103
   448
    by (rule SUP_upper)
haftmann@43943
   449
  also note `(\<Squnion>i\<in>A. f i) < y`
haftmann@43943
   450
  finally show "f i < y" .
haftmann@43943
   451
qed
haftmann@43943
   452
haftmann@43873
   453
lemma INF_UNIV_bool_expand:
haftmann@43868
   454
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@56166
   455
  by (simp add: UNIV_bool inf_commute)
haftmann@43868
   456
haftmann@43873
   457
lemma SUP_UNIV_bool_expand:
haftmann@43871
   458
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@56166
   459
  by (simp add: UNIV_bool sup_commute)
haftmann@43871
   460
hoelzl@51328
   461
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@51328
   462
  by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
hoelzl@51328
   463
haftmann@56218
   464
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f"
haftmann@56166
   465
  using Inf_le_Sup [of "f ` A"] by simp
hoelzl@51328
   466
hoelzl@54414
   467
lemma INF_eq_const:
haftmann@56218
   468
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x"
hoelzl@54414
   469
  by (auto intro: INF_eqI)
hoelzl@54414
   470
haftmann@56248
   471
lemma SUP_eq_const:
haftmann@56248
   472
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x"
haftmann@56248
   473
  by (auto intro: SUP_eqI)
hoelzl@54414
   474
hoelzl@54414
   475
lemma INF_eq_iff:
haftmann@56218
   476
  "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
   477
  using INF_eq_const [of I f c] INF_lower [of _ I f]
haftmann@56248
   478
  by (auto intro: antisym cong del: strong_INF_cong)
haftmann@56248
   479
haftmann@56248
   480
lemma SUP_eq_iff:
haftmann@56248
   481
  "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
   482
  using SUP_eq_const [of I f c] SUP_upper [of _ I f]
haftmann@56248
   483
  by (auto intro: antisym cong del: strong_SUP_cong)
hoelzl@54414
   484
haftmann@32077
   485
end
haftmann@32077
   486
haftmann@44024
   487
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   488
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   489
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   490
begin
haftmann@44024
   491
haftmann@44039
   492
lemma sup_INF:
haftmann@44039
   493
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@56166
   494
  by (simp only: INF_def sup_Inf image_image)
haftmann@44039
   495
haftmann@44039
   496
lemma inf_SUP:
haftmann@44039
   497
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@56166
   498
  by (simp only: SUP_def inf_Sup image_image)
haftmann@44039
   499
haftmann@44032
   500
lemma dual_complete_distrib_lattice:
krauss@44845
   501
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   502
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   503
  apply (fact dual_complete_lattice)
haftmann@44024
   504
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032
   505
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032
   506
  done
haftmann@44024
   507
haftmann@44322
   508
subclass distrib_lattice proof
haftmann@44024
   509
  fix a b c
haftmann@44024
   510
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
noschinl@44919
   511
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
haftmann@44024
   512
qed
haftmann@44024
   513
haftmann@44039
   514
lemma Inf_sup:
haftmann@44039
   515
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   516
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   517
haftmann@44039
   518
lemma Sup_inf:
haftmann@44039
   519
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   520
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   521
haftmann@44039
   522
lemma INF_sup: 
haftmann@44039
   523
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   524
  by (simp add: sup_INF sup_commute)
haftmann@44039
   525
haftmann@44039
   526
lemma SUP_inf:
haftmann@44039
   527
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   528
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   529
haftmann@44039
   530
lemma Inf_sup_eq_top_iff:
haftmann@44039
   531
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   532
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   533
haftmann@44039
   534
lemma Sup_inf_eq_bot_iff:
haftmann@44039
   535
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   536
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   537
haftmann@44039
   538
lemma INF_sup_distrib2:
haftmann@44039
   539
  "(\<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
   540
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   541
haftmann@44039
   542
lemma SUP_inf_distrib2:
haftmann@44039
   543
  "(\<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
   544
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   545
haftmann@56074
   546
context
haftmann@56074
   547
  fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
haftmann@56074
   548
  assumes "mono f"
haftmann@56074
   549
begin
haftmann@56074
   550
haftmann@56074
   551
lemma mono_Inf:
haftmann@56074
   552
  shows "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
haftmann@56074
   553
  using `mono f` by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
haftmann@56074
   554
haftmann@56074
   555
lemma mono_Sup:
haftmann@56074
   556
  shows "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
haftmann@56074
   557
  using `mono f` by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
haftmann@56074
   558
hoelzl@60172
   559
lemma mono_INF:
hoelzl@60172
   560
  "f (INF i : I. A i) \<le> (INF x : I. f (A x))"
hoelzl@60172
   561
  by (intro complete_lattice_class.INF_greatest monoD[OF `mono f`] INF_lower)
hoelzl@60172
   562
hoelzl@60172
   563
lemma mono_SUP:
hoelzl@60172
   564
  "(SUP x : I. f (A x)) \<le> f (SUP i : I. A i)"
hoelzl@60172
   565
  by (intro complete_lattice_class.SUP_least monoD[OF `mono f`] SUP_upper)
hoelzl@60172
   566
haftmann@56074
   567
end
haftmann@56074
   568
haftmann@44024
   569
end
haftmann@44024
   570
haftmann@44032
   571
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   572
begin
haftmann@43873
   573
haftmann@43943
   574
lemma dual_complete_boolean_algebra:
krauss@44845
   575
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   576
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   577
haftmann@43873
   578
lemma uminus_Inf:
haftmann@43873
   579
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   580
proof (rule antisym)
haftmann@43873
   581
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   582
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   583
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   584
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   585
qed
haftmann@43873
   586
haftmann@44041
   587
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@56166
   588
  by (simp only: INF_def SUP_def uminus_Inf image_image)
haftmann@44041
   589
haftmann@43873
   590
lemma uminus_Sup:
haftmann@43873
   591
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   592
proof -
haftmann@56166
   593
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_INF)
haftmann@43873
   594
  then show ?thesis by simp
haftmann@43873
   595
qed
haftmann@43873
   596
  
haftmann@43873
   597
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@56166
   598
  by (simp only: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   599
haftmann@43873
   600
end
haftmann@43873
   601
haftmann@43940
   602
class complete_linorder = linorder + complete_lattice
haftmann@43940
   603
begin
haftmann@43940
   604
haftmann@43943
   605
lemma dual_complete_linorder:
krauss@44845
   606
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   607
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   608
haftmann@51386
   609
lemma complete_linorder_inf_min: "inf = min"
haftmann@51540
   610
  by (auto intro: antisym simp add: min_def fun_eq_iff)
haftmann@51386
   611
haftmann@51386
   612
lemma complete_linorder_sup_max: "sup = max"
haftmann@51540
   613
  by (auto intro: antisym simp add: max_def fun_eq_iff)
haftmann@51386
   614
noschinl@44918
   615
lemma Inf_less_iff:
haftmann@43940
   616
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   617
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   618
noschinl@44918
   619
lemma INF_less_iff:
haftmann@44041
   620
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@56166
   621
  using Inf_less_iff [of "f ` A"] by simp
haftmann@44041
   622
noschinl@44918
   623
lemma less_Sup_iff:
haftmann@43940
   624
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   625
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   626
noschinl@44918
   627
lemma less_SUP_iff:
haftmann@43940
   628
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@56166
   629
  using less_Sup_iff [of _ "f ` A"] by simp
haftmann@43940
   630
noschinl@44918
   631
lemma Sup_eq_top_iff [simp]:
haftmann@43943
   632
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   633
proof
haftmann@43943
   634
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   635
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   636
  proof (intro allI impI)
haftmann@43943
   637
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   638
      unfolding less_Sup_iff by auto
haftmann@43943
   639
  qed
haftmann@43943
   640
next
haftmann@43943
   641
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   642
  show "\<Squnion>A = \<top>"
haftmann@43943
   643
  proof (rule ccontr)
haftmann@43943
   644
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   645
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   646
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   647
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   648
      using * unfolding less_Sup_iff by auto
haftmann@43943
   649
    then show False by auto
haftmann@43943
   650
  qed
haftmann@43943
   651
qed
haftmann@43943
   652
noschinl@44918
   653
lemma SUP_eq_top_iff [simp]:
haftmann@44041
   654
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
haftmann@56166
   655
  using Sup_eq_top_iff [of "f ` A"] by simp
haftmann@44041
   656
noschinl@44918
   657
lemma Inf_eq_bot_iff [simp]:
haftmann@43943
   658
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920
   659
  using dual_complete_linorder
huffman@44920
   660
  by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943
   661
noschinl@44918
   662
lemma INF_eq_bot_iff [simp]:
haftmann@43967
   663
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
haftmann@56166
   664
  using Inf_eq_bot_iff [of "f ` A"] by simp
hoelzl@51328
   665
hoelzl@51328
   666
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
hoelzl@51328
   667
proof safe
hoelzl@51328
   668
  fix y assume "x \<ge> \<Sqinter>A" "y > x"
hoelzl@51328
   669
  then have "y > \<Sqinter>A" by auto
hoelzl@51328
   670
  then show "\<exists>a\<in>A. y > a"
hoelzl@51328
   671
    unfolding Inf_less_iff .
hoelzl@51328
   672
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
hoelzl@51328
   673
hoelzl@51328
   674
lemma INF_le_iff:
haftmann@56218
   675
  "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
haftmann@56166
   676
  using Inf_le_iff [of "f ` A"] by simp
haftmann@56166
   677
haftmann@56166
   678
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
haftmann@56166
   679
proof safe
haftmann@56166
   680
  fix y assume "x \<le> \<Squnion>A" "y < x"
haftmann@56166
   681
  then have "y < \<Squnion>A" by auto
haftmann@56166
   682
  then show "\<exists>a\<in>A. y < a"
haftmann@56166
   683
    unfolding less_Sup_iff .
haftmann@56166
   684
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
haftmann@56166
   685
haftmann@56218
   686
lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
haftmann@56166
   687
  using le_Sup_iff [of _ "f ` A"] by simp
hoelzl@51328
   688
haftmann@51386
   689
subclass complete_distrib_lattice
haftmann@51386
   690
proof
haftmann@51386
   691
  fix a and B
haftmann@51386
   692
  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
   693
    by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
haftmann@51386
   694
      (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
haftmann@51386
   695
        le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
haftmann@51386
   696
qed
haftmann@51386
   697
haftmann@43940
   698
end
haftmann@43940
   699
hoelzl@51341
   700
haftmann@46631
   701
subsection {* Complete lattice on @{typ bool} *}
haftmann@32077
   702
haftmann@44024
   703
instantiation bool :: complete_lattice
haftmann@32077
   704
begin
haftmann@32077
   705
haftmann@32077
   706
definition
haftmann@46154
   707
  [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   708
haftmann@32077
   709
definition
haftmann@46154
   710
  [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   711
haftmann@32077
   712
instance proof
haftmann@44322
   713
qed (auto intro: bool_induct)
haftmann@32077
   714
haftmann@32077
   715
end
haftmann@32077
   716
haftmann@49905
   717
lemma not_False_in_image_Ball [simp]:
haftmann@49905
   718
  "False \<notin> P ` A \<longleftrightarrow> Ball A P"
haftmann@49905
   719
  by auto
haftmann@49905
   720
haftmann@49905
   721
lemma True_in_image_Bex [simp]:
haftmann@49905
   722
  "True \<in> P ` A \<longleftrightarrow> Bex A P"
haftmann@49905
   723
  by auto
haftmann@49905
   724
haftmann@43873
   725
lemma INF_bool_eq [simp]:
haftmann@56218
   726
  "INFIMUM = Ball"
haftmann@49905
   727
  by (simp add: fun_eq_iff INF_def)
haftmann@32120
   728
haftmann@43873
   729
lemma SUP_bool_eq [simp]:
haftmann@56218
   730
  "SUPREMUM = Bex"
haftmann@49905
   731
  by (simp add: fun_eq_iff SUP_def)
haftmann@32120
   732
haftmann@44032
   733
instance bool :: complete_boolean_algebra proof
haftmann@44322
   734
qed (auto intro: bool_induct)
haftmann@44024
   735
haftmann@46631
   736
haftmann@46631
   737
subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
haftmann@46631
   738
nipkow@57197
   739
instantiation "fun" :: (type, Inf) Inf
haftmann@32077
   740
begin
haftmann@32077
   741
haftmann@32077
   742
definition
haftmann@44024
   743
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   744
noschinl@46882
   745
lemma Inf_apply [simp, code]:
haftmann@44024
   746
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   747
  by (simp add: Inf_fun_def)
haftmann@32077
   748
nipkow@57197
   749
instance ..
nipkow@57197
   750
nipkow@57197
   751
end
nipkow@57197
   752
nipkow@57197
   753
instantiation "fun" :: (type, Sup) Sup
nipkow@57197
   754
begin
nipkow@57197
   755
haftmann@32077
   756
definition
haftmann@44024
   757
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   758
noschinl@46882
   759
lemma Sup_apply [simp, code]:
haftmann@44024
   760
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   761
  by (simp add: Sup_fun_def)
haftmann@32077
   762
nipkow@57197
   763
instance ..
nipkow@57197
   764
nipkow@57197
   765
end
nipkow@57197
   766
nipkow@57197
   767
instantiation "fun" :: (type, complete_lattice) complete_lattice
nipkow@57197
   768
begin
nipkow@57197
   769
haftmann@32077
   770
instance proof
noschinl@46884
   771
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   772
haftmann@32077
   773
end
haftmann@32077
   774
noschinl@46882
   775
lemma INF_apply [simp]:
haftmann@41080
   776
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@56166
   777
  using Inf_apply [of "f ` A"] by (simp add: comp_def)
hoelzl@38705
   778
noschinl@46882
   779
lemma SUP_apply [simp]:
haftmann@41080
   780
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@56166
   781
  using Sup_apply [of "f ` A"] by (simp add: comp_def)
haftmann@32077
   782
haftmann@44024
   783
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
haftmann@56166
   784
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf fun_eq_iff image_image
haftmann@56166
   785
  simp del: Inf_image_eq Sup_image_eq)
haftmann@44024
   786
haftmann@43873
   787
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   788
haftmann@46631
   789
haftmann@46631
   790
subsection {* Complete lattice on unary and binary predicates *}
haftmann@46631
   791
haftmann@56742
   792
lemma Inf1_I: 
haftmann@56742
   793
  "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a"
noschinl@46884
   794
  by auto
haftmann@46631
   795
haftmann@56742
   796
lemma INF1_I:
haftmann@56742
   797
  "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   798
  by simp
haftmann@56742
   799
haftmann@56742
   800
lemma INF2_I:
haftmann@56742
   801
  "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   802
  by simp
haftmann@56742
   803
haftmann@56742
   804
lemma Inf2_I: 
haftmann@56742
   805
  "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b"
noschinl@46884
   806
  by auto
haftmann@46631
   807
haftmann@56742
   808
lemma Inf1_D:
haftmann@56742
   809
  "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a"
noschinl@46884
   810
  by auto
haftmann@46631
   811
haftmann@56742
   812
lemma INF1_D:
haftmann@56742
   813
  "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
haftmann@56742
   814
  by simp
haftmann@56742
   815
haftmann@56742
   816
lemma Inf2_D:
haftmann@56742
   817
  "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b"
noschinl@46884
   818
  by auto
haftmann@46631
   819
haftmann@56742
   820
lemma INF2_D:
haftmann@56742
   821
  "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
haftmann@56742
   822
  by simp
haftmann@56742
   823
haftmann@56742
   824
lemma Inf1_E:
haftmann@56742
   825
  assumes "(\<Sqinter>A) a"
haftmann@56742
   826
  obtains "P a" | "P \<notin> A"
haftmann@56742
   827
  using assms by auto
haftmann@46631
   828
haftmann@56742
   829
lemma INF1_E:
haftmann@56742
   830
  assumes "(\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   831
  obtains "B a b" | "a \<notin> A"
haftmann@56742
   832
  using assms by auto
haftmann@56742
   833
haftmann@56742
   834
lemma Inf2_E:
haftmann@56742
   835
  assumes "(\<Sqinter>A) a b"
haftmann@56742
   836
  obtains "r a b" | "r \<notin> A"
haftmann@56742
   837
  using assms by auto
haftmann@56742
   838
haftmann@56742
   839
lemma INF2_E:
haftmann@56742
   840
  assumes "(\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   841
  obtains "B a b c" | "a \<notin> A"
haftmann@56742
   842
  using assms by auto
haftmann@56742
   843
haftmann@56742
   844
lemma Sup1_I:
haftmann@56742
   845
  "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a"
noschinl@46884
   846
  by auto
haftmann@46631
   847
haftmann@56742
   848
lemma SUP1_I:
haftmann@56742
   849
  "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
haftmann@56742
   850
  by auto
haftmann@56742
   851
haftmann@56742
   852
lemma Sup2_I:
haftmann@56742
   853
  "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b"
haftmann@56742
   854
  by auto
haftmann@56742
   855
haftmann@56742
   856
lemma SUP2_I:
haftmann@56742
   857
  "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
noschinl@46884
   858
  by auto
haftmann@46631
   859
haftmann@56742
   860
lemma Sup1_E:
haftmann@56742
   861
  assumes "(\<Squnion>A) a"
haftmann@56742
   862
  obtains P where "P \<in> A" and "P a"
haftmann@56742
   863
  using assms by auto
haftmann@56742
   864
haftmann@56742
   865
lemma SUP1_E:
haftmann@56742
   866
  assumes "(\<Squnion>x\<in>A. B x) b"
haftmann@56742
   867
  obtains x where "x \<in> A" and "B x b"
haftmann@56742
   868
  using assms by auto
haftmann@46631
   869
haftmann@56742
   870
lemma Sup2_E:
haftmann@56742
   871
  assumes "(\<Squnion>A) a b"
haftmann@56742
   872
  obtains r where "r \<in> A" "r a b"
haftmann@56742
   873
  using assms by auto
haftmann@56742
   874
haftmann@56742
   875
lemma SUP2_E:
haftmann@56742
   876
  assumes "(\<Squnion>x\<in>A. B x) b c"
haftmann@56742
   877
  obtains x where "x \<in> A" "B x b c"
haftmann@56742
   878
  using assms by auto
haftmann@46631
   879
haftmann@46631
   880
haftmann@46631
   881
subsection {* Complete lattice on @{typ "_ set"} *}
haftmann@46631
   882
haftmann@45960
   883
instantiation "set" :: (type) complete_lattice
haftmann@45960
   884
begin
haftmann@45960
   885
haftmann@45960
   886
definition
haftmann@45960
   887
  "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   888
haftmann@45960
   889
definition
haftmann@45960
   890
  "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   891
haftmann@45960
   892
instance proof
haftmann@51386
   893
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
haftmann@45960
   894
haftmann@45960
   895
end
haftmann@45960
   896
haftmann@45960
   897
instance "set" :: (type) complete_boolean_algebra
haftmann@45960
   898
proof
haftmann@45960
   899
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
haftmann@45960
   900
  
haftmann@32077
   901
haftmann@46631
   902
subsubsection {* Inter *}
haftmann@41082
   903
haftmann@41082
   904
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   905
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   906
  
haftmann@41082
   907
notation (xsymbols)
haftmann@52141
   908
  Inter  ("\<Inter>_" [900] 900)
haftmann@41082
   909
haftmann@41082
   910
lemma Inter_eq:
haftmann@41082
   911
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   912
proof (rule set_eqI)
haftmann@41082
   913
  fix x
haftmann@41082
   914
  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
   915
    by auto
haftmann@41082
   916
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960
   917
    by (simp add: Inf_set_def image_def)
haftmann@41082
   918
qed
haftmann@41082
   919
blanchet@54147
   920
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   921
  by (unfold Inter_eq) blast
haftmann@41082
   922
haftmann@43741
   923
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   924
  by (simp add: Inter_eq)
haftmann@41082
   925
haftmann@41082
   926
text {*
haftmann@41082
   927
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   928
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   929
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   930
*}
haftmann@41082
   931
haftmann@43741
   932
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   933
  by auto
haftmann@41082
   934
haftmann@43741
   935
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   936
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   937
    @{prop "X \<in> C"}. *}
haftmann@41082
   938
  by (unfold Inter_eq) blast
haftmann@41082
   939
haftmann@43741
   940
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   941
  by (fact Inf_lower)
haftmann@43740
   942
haftmann@41082
   943
lemma Inter_subset:
haftmann@43755
   944
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   945
  by (fact Inf_less_eq)
haftmann@41082
   946
haftmann@43755
   947
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   948
  by (fact Inf_greatest)
haftmann@41082
   949
huffman@44067
   950
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   951
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   952
huffman@44067
   953
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   954
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   955
huffman@44920
   956
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920
   957
  by (fact Inf_insert) (* already simp *)
haftmann@41082
   958
haftmann@41082
   959
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   960
  by (fact less_eq_Inf_inter)
haftmann@41082
   961
haftmann@41082
   962
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   963
  by (fact Inf_union_distrib)
haftmann@43756
   964
blanchet@54147
   965
lemma Inter_UNIV_conv [simp]:
haftmann@43741
   966
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   967
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   968
  by (fact Inf_top_conv)+
haftmann@41082
   969
haftmann@43741
   970
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   971
  by (fact Inf_superset_mono)
haftmann@41082
   972
haftmann@41082
   973
haftmann@46631
   974
subsubsection {* Intersections of families *}
haftmann@41082
   975
haftmann@41082
   976
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@56218
   977
  "INTER \<equiv> INFIMUM"
haftmann@41082
   978
haftmann@43872
   979
text {*
haftmann@43872
   980
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   981
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   982
*}
haftmann@43872
   983
haftmann@41082
   984
syntax
haftmann@41082
   985
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   986
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   987
haftmann@41082
   988
syntax (xsymbols)
haftmann@41082
   989
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   990
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   991
haftmann@41082
   992
syntax (latex output)
haftmann@41082
   993
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   994
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   995
haftmann@41082
   996
translations
haftmann@41082
   997
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   998
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   999
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
  1000
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
  1001
haftmann@41082
  1002
print_translation {*
wenzelm@42284
  1003
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
  1004
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
  1005
haftmann@44085
  1006
lemma INTER_eq:
haftmann@41082
  1007
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@56166
  1008
  by (auto intro!: INF_eqI)
haftmann@41082
  1009
haftmann@56166
  1010
lemma Inter_image_eq:
haftmann@56166
  1011
  "\<Inter>(B ` A) = (\<Inter>x\<in>A. B x)"
haftmann@56166
  1012
  by (fact Inf_image_eq)
haftmann@41082
  1013
haftmann@43817
  1014
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@56166
  1015
  using Inter_iff [of _ "B ` A"] by simp
haftmann@41082
  1016
haftmann@43817
  1017
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
  1018
  by (auto simp add: INF_def image_def)
haftmann@41082
  1019
haftmann@43852
  1020
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
  1021
  by auto
haftmann@41082
  1022
haftmann@43852
  1023
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
  1024
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@44085
  1025
  by (auto simp add: INF_def image_def)
haftmann@41082
  1026
haftmann@41082
  1027
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
  1028
  by blast
haftmann@41082
  1029
haftmann@41082
  1030
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
  1031
  by blast
haftmann@41082
  1032
haftmann@43817
  1033
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
  1034
  by (fact INF_lower)
haftmann@41082
  1035
haftmann@43817
  1036
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
  1037
  by (fact INF_greatest)
haftmann@41082
  1038
huffman@44067
  1039
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
  1040
  by (fact INF_empty)
haftmann@43854
  1041
haftmann@43817
  1042
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
  1043
  by (fact INF_absorb)
haftmann@41082
  1044
haftmann@43854
  1045
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
  1046
  by (fact le_INF_iff)
haftmann@41082
  1047
haftmann@41082
  1048
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
  1049
  by (fact INF_insert)
haftmann@43865
  1050
haftmann@43865
  1051
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
  1052
  by (fact INF_union)
haftmann@43865
  1053
haftmann@43865
  1054
lemma INT_insert_distrib:
haftmann@43865
  1055
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
  1056
  by blast
haftmann@43854
  1057
haftmann@41082
  1058
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
  1059
  by (fact INF_constant)
haftmann@43865
  1060
huffman@44920
  1061
lemma INTER_UNIV_conv:
haftmann@43817
  1062
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
  1063
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920
  1064
  by (fact INF_top_conv)+ (* already simp *)
haftmann@43865
  1065
haftmann@43865
  1066
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
  1067
  by (fact INF_UNIV_bool_expand)
haftmann@43865
  1068
haftmann@43865
  1069
lemma INT_anti_mono:
haftmann@43900
  1070
  "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
  1071
  -- {* The last inclusion is POSITIVE! *}
haftmann@43940
  1072
  by (fact INF_superset_mono)
haftmann@41082
  1073
haftmann@41082
  1074
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
  1075
  by blast
haftmann@41082
  1076
haftmann@43817
  1077
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
  1078
  by blast
haftmann@41082
  1079
haftmann@41082
  1080
haftmann@46631
  1081
subsubsection {* Union *}
haftmann@32115
  1082
haftmann@32587
  1083
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
  1084
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
  1085
haftmann@32115
  1086
notation (xsymbols)
haftmann@52141
  1087
  Union  ("\<Union>_" [900] 900)
haftmann@32115
  1088
haftmann@32135
  1089
lemma Union_eq:
haftmann@32135
  1090
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
  1091
proof (rule set_eqI)
haftmann@32115
  1092
  fix x
haftmann@32135
  1093
  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
  1094
    by auto
haftmann@32135
  1095
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960
  1096
    by (simp add: Sup_set_def image_def)
haftmann@32115
  1097
qed
haftmann@32115
  1098
blanchet@54147
  1099
lemma Union_iff [simp]:
haftmann@32115
  1100
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
  1101
  by (unfold Union_eq) blast
haftmann@32115
  1102
haftmann@32115
  1103
lemma UnionI [intro]:
haftmann@32115
  1104
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
  1105
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
  1106
    @{term A} may be flexible. *}
haftmann@32115
  1107
  by auto
haftmann@32115
  1108
haftmann@32115
  1109
lemma UnionE [elim!]:
haftmann@43817
  1110
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
  1111
  by auto
haftmann@32115
  1112
haftmann@43817
  1113
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
  1114
  by (fact Sup_upper)
haftmann@32135
  1115
haftmann@43817
  1116
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
  1117
  by (fact Sup_least)
haftmann@32135
  1118
huffman@44920
  1119
lemma Union_empty: "\<Union>{} = {}"
huffman@44920
  1120
  by (fact Sup_empty) (* already simp *)
haftmann@32135
  1121
huffman@44920
  1122
lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920
  1123
  by (fact Sup_UNIV) (* already simp *)
haftmann@32135
  1124
huffman@44920
  1125
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920
  1126
  by (fact Sup_insert) (* already simp *)
haftmann@32135
  1127
haftmann@43817
  1128
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
  1129
  by (fact Sup_union_distrib)
haftmann@32135
  1130
haftmann@32135
  1131
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
  1132
  by (fact Sup_inter_less_eq)
haftmann@32135
  1133
blanchet@54147
  1134
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1135
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1136
blanchet@54147
  1137
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1138
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1139
haftmann@32135
  1140
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
  1141
  by blast
haftmann@32135
  1142
haftmann@32135
  1143
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
  1144
  by blast
haftmann@32135
  1145
haftmann@43817
  1146
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
  1147
  by (fact Sup_subset_mono)
haftmann@32135
  1148
haftmann@32115
  1149
haftmann@46631
  1150
subsubsection {* Unions of families *}
haftmann@32077
  1151
haftmann@32606
  1152
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@56218
  1153
  "UNION \<equiv> SUPREMUM"
haftmann@32077
  1154
haftmann@43872
  1155
text {*
haftmann@43872
  1156
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
  1157
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
  1158
*}
haftmann@43872
  1159
haftmann@32077
  1160
syntax
wenzelm@35115
  1161
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
  1162
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
  1163
haftmann@32077
  1164
syntax (xsymbols)
wenzelm@35115
  1165
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
  1166
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
  1167
haftmann@32077
  1168
syntax (latex output)
wenzelm@35115
  1169
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
  1170
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
  1171
haftmann@32077
  1172
translations
haftmann@32077
  1173
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
  1174
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
  1175
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
  1176
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
  1177
haftmann@32077
  1178
text {*
haftmann@32077
  1179
  Note the difference between ordinary xsymbol syntax of indexed
wenzelm@53015
  1180
  unions and intersections (e.g.\ @{text"\<Union>a\<^sub>1\<in>A\<^sub>1. B"})
wenzelm@53015
  1181
  and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. The
haftmann@32077
  1182
  former does not make the index expression a subscript of the
haftmann@32077
  1183
  union/intersection symbol because this leads to problems with nested
haftmann@32077
  1184
  subscripts in Proof General.
haftmann@32077
  1185
*}
haftmann@32077
  1186
wenzelm@35115
  1187
print_translation {*
wenzelm@42284
  1188
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
  1189
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
  1190
blanchet@54147
  1191
lemma UNION_eq:
haftmann@32135
  1192
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@56166
  1193
  by (auto intro!: SUP_eqI)
huffman@44920
  1194
haftmann@45960
  1195
lemma bind_UNION [code]:
haftmann@45960
  1196
  "Set.bind A f = UNION A f"
haftmann@45960
  1197
  by (simp add: bind_def UNION_eq)
haftmann@45960
  1198
haftmann@46036
  1199
lemma member_bind [simp]:
haftmann@46036
  1200
  "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036
  1201
  by (simp add: bind_UNION)
haftmann@46036
  1202
haftmann@56166
  1203
lemma Union_image_eq:
haftmann@43817
  1204
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@56166
  1205
  by (fact Sup_image_eq)
huffman@44920
  1206
wenzelm@60585
  1207
lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
lp15@60307
  1208
  by blast
lp15@60307
  1209
haftmann@46036
  1210
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@56166
  1211
  using Union_iff [of _ "B ` A"] by simp
wenzelm@11979
  1212
haftmann@43852
  1213
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
  1214
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
  1215
    @{term b} may be flexible. *}
wenzelm@11979
  1216
  by auto
wenzelm@11979
  1217
haftmann@43852
  1218
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
  1219
  by (auto simp add: SUP_def image_def)
clasohm@923
  1220
haftmann@43817
  1221
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
  1222
  by blast
haftmann@32077
  1223
haftmann@43817
  1224
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
  1225
  by (fact SUP_upper)
haftmann@32135
  1226
haftmann@43817
  1227
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
  1228
  by (fact SUP_least)
haftmann@32135
  1229
blanchet@54147
  1230
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
  1231
  by blast
haftmann@32135
  1232
haftmann@43817
  1233
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
  1234
  by blast
haftmann@32135
  1235
blanchet@54147
  1236
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
  1237
  by (fact SUP_empty)
haftmann@32135
  1238
huffman@44920
  1239
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920
  1240
  by (fact SUP_bot) (* already simp *)
haftmann@32135
  1241
haftmann@43817
  1242
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
  1243
  by (fact SUP_absorb)
haftmann@32135
  1244
haftmann@32135
  1245
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
  1246
  by (fact SUP_insert)
haftmann@32135
  1247
haftmann@44085
  1248
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
  1249
  by (fact SUP_union)
haftmann@32135
  1250
haftmann@43967
  1251
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
  1252
  by blast
haftmann@32135
  1253
haftmann@32135
  1254
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
  1255
  by (fact SUP_le_iff)
haftmann@32135
  1256
haftmann@32135
  1257
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
  1258
  by (fact SUP_constant)
haftmann@32135
  1259
haftmann@43944
  1260
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
  1261
  by blast
haftmann@32135
  1262
huffman@44920
  1263
lemma UNION_empty_conv:
haftmann@43817
  1264
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1265
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920
  1266
  by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135
  1267
blanchet@54147
  1268
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1269
  by blast
haftmann@32135
  1270
haftmann@43900
  1271
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
  1272
  by blast
haftmann@32135
  1273
haftmann@43900
  1274
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
  1275
  by blast
haftmann@32135
  1276
haftmann@32135
  1277
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
  1278
  by (auto simp add: split_if_mem2)
haftmann@32135
  1279
haftmann@43817
  1280
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1281
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1282
haftmann@32135
  1283
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1284
  by blast
haftmann@32135
  1285
haftmann@32135
  1286
lemma UN_mono:
haftmann@43817
  1287
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1288
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1289
  by (fact SUP_subset_mono)
haftmann@32135
  1290
haftmann@43817
  1291
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1292
  by blast
haftmann@32135
  1293
haftmann@43817
  1294
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1295
  by blast
haftmann@32135
  1296
haftmann@43817
  1297
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
  1298
  -- {* NOT suitable for rewriting *}
haftmann@32135
  1299
  by blast
haftmann@32135
  1300
haftmann@43817
  1301
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1302
  by blast
haftmann@32135
  1303
haftmann@45013
  1304
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013
  1305
  by blast
haftmann@45013
  1306
wenzelm@11979
  1307
haftmann@46631
  1308
subsubsection {* Distributive laws *}
wenzelm@12897
  1309
wenzelm@12897
  1310
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1311
  by (fact inf_Sup)
wenzelm@12897
  1312
haftmann@44039
  1313
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1314
  by (fact sup_Inf)
haftmann@44039
  1315
wenzelm@12897
  1316
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1317
  by (fact Sup_inf)
haftmann@44039
  1318
haftmann@44039
  1319
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
  1320
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1321
haftmann@44039
  1322
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
  1323
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1324
haftmann@56166
  1325
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" -- {* FIXME drop *}
haftmann@56166
  1326
  by (simp add: INT_Int_distrib)
wenzelm@12897
  1327
haftmann@56166
  1328
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" -- {* FIXME drop *}
wenzelm@12897
  1329
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1330
  -- {* Union of a family of unions *}
haftmann@56166
  1331
  by (simp add: UN_Un_distrib)
wenzelm@12897
  1332
haftmann@44039
  1333
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1334
  by (fact sup_INF)
wenzelm@12897
  1335
wenzelm@12897
  1336
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1337
  -- {* Halmos, Naive Set Theory, page 35. *}
haftmann@44039
  1338
  by (fact inf_SUP)
wenzelm@12897
  1339
wenzelm@12897
  1340
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
  1341
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1342
wenzelm@12897
  1343
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
  1344
  by (fact INF_sup_distrib2)
haftmann@44039
  1345
haftmann@44039
  1346
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1347
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1348
wenzelm@12897
  1349
haftmann@56015
  1350
subsection {* Injections and bijections *}
haftmann@56015
  1351
haftmann@56015
  1352
lemma inj_on_Inter:
haftmann@56015
  1353
  "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
haftmann@56015
  1354
  unfolding inj_on_def by blast
haftmann@56015
  1355
haftmann@56015
  1356
lemma inj_on_INTER:
haftmann@56015
  1357
  "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
  1358
  unfolding inj_on_def by blast
haftmann@56015
  1359
haftmann@56015
  1360
lemma inj_on_UNION_chain:
haftmann@56015
  1361
  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
  1362
         INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
wenzelm@60585
  1363
  shows "inj_on f (\<Union>i \<in> I. A i)"
haftmann@56015
  1364
proof -
haftmann@56015
  1365
  {
haftmann@56015
  1366
    fix i j x y
haftmann@56015
  1367
    assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
haftmann@56015
  1368
      and ***: "f x = f y"
haftmann@56015
  1369
    have "x = y"
haftmann@56015
  1370
    proof -
haftmann@56015
  1371
      {
haftmann@56015
  1372
        assume "A i \<le> A j"
haftmann@56015
  1373
        with ** have "x \<in> A j" by auto
haftmann@56015
  1374
        with INJ * ** *** have ?thesis
haftmann@56015
  1375
        by(auto simp add: inj_on_def)
haftmann@56015
  1376
      }
haftmann@56015
  1377
      moreover
haftmann@56015
  1378
      {
haftmann@56015
  1379
        assume "A j \<le> A i"
haftmann@56015
  1380
        with ** have "y \<in> A i" by auto
haftmann@56015
  1381
        with INJ * ** *** have ?thesis
haftmann@56015
  1382
        by(auto simp add: inj_on_def)
haftmann@56015
  1383
      }
haftmann@56015
  1384
      ultimately show ?thesis using CH * by blast
haftmann@56015
  1385
    qed
haftmann@56015
  1386
  }
haftmann@56015
  1387
  then show ?thesis by (unfold inj_on_def UNION_eq) auto
haftmann@56015
  1388
qed
haftmann@56015
  1389
haftmann@56015
  1390
lemma bij_betw_UNION_chain:
haftmann@56015
  1391
  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
  1392
         BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
wenzelm@60585
  1393
  shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)"
haftmann@56015
  1394
proof (unfold bij_betw_def, auto)
haftmann@56015
  1395
  have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
haftmann@56015
  1396
  using BIJ bij_betw_def[of f] by auto
wenzelm@60585
  1397
  thus "inj_on f (\<Union>i \<in> I. A i)"
haftmann@56015
  1398
  using CH inj_on_UNION_chain[of I A f] by auto
haftmann@56015
  1399
next
haftmann@56015
  1400
  fix i x
haftmann@56015
  1401
  assume *: "i \<in> I" "x \<in> A i"
haftmann@56015
  1402
  hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
haftmann@56015
  1403
  thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
haftmann@56015
  1404
next
haftmann@56015
  1405
  fix i x'
haftmann@56015
  1406
  assume *: "i \<in> I" "x' \<in> A' i"
haftmann@56015
  1407
  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
haftmann@56015
  1408
  then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
haftmann@56015
  1409
    using * by blast
haftmann@56015
  1410
  then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by blast
haftmann@56015
  1411
qed
haftmann@56015
  1412
haftmann@56015
  1413
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
haftmann@56015
  1414
lemma image_INT:
haftmann@56015
  1415
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
haftmann@56015
  1416
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
haftmann@56015
  1417
apply (simp add: inj_on_def, blast)
haftmann@56015
  1418
done
haftmann@56015
  1419
haftmann@56015
  1420
(*Compare with image_INT: no use of inj_on, and if f is surjective then
haftmann@56015
  1421
  it doesn't matter whether A is empty*)
haftmann@56015
  1422
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
haftmann@56015
  1423
apply (simp add: bij_def)
haftmann@56015
  1424
apply (simp add: inj_on_def surj_def, blast)
haftmann@56015
  1425
done
haftmann@56015
  1426
haftmann@56015
  1427
lemma UNION_fun_upd:
haftmann@56015
  1428
  "UNION J (A(i:=B)) = (UNION (J-{i}) A \<union> (if i\<in>J then B else {}))"
haftmann@56015
  1429
by (auto split: if_splits)
haftmann@56015
  1430
haftmann@56015
  1431
haftmann@46631
  1432
subsubsection {* Complement *}
haftmann@32135
  1433
haftmann@43873
  1434
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1435
  by (fact uminus_INF)
wenzelm@12897
  1436
haftmann@43873
  1437
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1438
  by (fact uminus_SUP)
wenzelm@12897
  1439
wenzelm@12897
  1440
haftmann@46631
  1441
subsubsection {* Miniscoping and maxiscoping *}
wenzelm@12897
  1442
paulson@13860
  1443
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1444
           and Intersections. *}
wenzelm@12897
  1445
wenzelm@12897
  1446
lemma UN_simps [simp]:
haftmann@43817
  1447
  "\<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
  1448
  "\<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
  1449
  "\<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
  1450
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1451
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1452
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1453
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1454
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1455
  "\<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
  1456
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1457
  by auto
wenzelm@12897
  1458
wenzelm@12897
  1459
lemma INT_simps [simp]:
haftmann@44032
  1460
  "\<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
  1461
  "\<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
  1462
  "\<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
  1463
  "\<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
  1464
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1465
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1466
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1467
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1468
  "\<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
  1469
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1470
  by auto
wenzelm@12897
  1471
blanchet@54147
  1472
lemma UN_ball_bex_simps [simp]:
haftmann@43852
  1473
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1474
  "\<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
  1475
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1476
  "\<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
  1477
  by auto
wenzelm@12897
  1478
haftmann@43943
  1479
paulson@13860
  1480
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1481
paulson@13860
  1482
lemma UN_extend_simps:
haftmann@43817
  1483
  "\<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
  1484
  "\<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
  1485
  "\<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
  1486
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1487
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1488
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1489
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1490
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1491
  "\<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
  1492
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1493
  by auto
paulson@13860
  1494
paulson@13860
  1495
lemma INT_extend_simps:
haftmann@43852
  1496
  "\<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
  1497
  "\<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
  1498
  "\<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
  1499
  "\<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
  1500
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1501
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1502
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1503
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1504
  "\<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
  1505
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1506
  by auto
paulson@13860
  1507
haftmann@43872
  1508
text {* Finally *}
haftmann@43872
  1509
haftmann@32135
  1510
no_notation
haftmann@46691
  1511
  less_eq (infix "\<sqsubseteq>" 50) and
haftmann@46691
  1512
  less (infix "\<sqsubset>" 50)
haftmann@32135
  1513
haftmann@30596
  1514
lemmas mem_simps =
haftmann@30596
  1515
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1516
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1517
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1518
wenzelm@11979
  1519
end
haftmann@49905
  1520