src/HOL/Complete_Lattices.thy
author wenzelm
Sun Jan 03 21:45:34 2016 +0100 (2016-01-03)
changeset 62048 fefd79f6b232
parent 61955 e96292f32c3c
child 62343 24106dc44def
permissions -rw-r--r--
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
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(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
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section \<open>Complete lattices\<close>
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theory Complete_Lattices
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imports Fun
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begin
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notation
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  less_eq (infix "\<sqsubseteq>" 50) and
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  less (infix "\<sqsubset>" 50)
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subsection \<open>Syntactic infimum and supremum operations\<close>
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class Inf =
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  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
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begin
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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 \<open>
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  Note: must use names @{const INFIMUM} and @{const SUPREMUM} here instead of
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  \<open>INF\<close> and \<open>SUP\<close> to allow the following syntax coexist
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  with the plain constant names.
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\<close>
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syntax (ASCII)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
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syntax (output)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
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syntax
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
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translations
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  "\<Sqinter>x y. B"   \<rightleftharpoons> "\<Sqinter>x. \<Sqinter>y. B"
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  "\<Sqinter>x. B"     \<rightleftharpoons> "CONST INFIMUM CONST UNIV (\<lambda>x. B)"
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  "\<Sqinter>x. B"     \<rightleftharpoons> "\<Sqinter>x \<in> CONST UNIV. B"
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  "\<Sqinter>x\<in>A. B"   \<rightleftharpoons> "CONST INFIMUM A (\<lambda>x. B)"
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  "\<Squnion>x y. B"   \<rightleftharpoons> "\<Squnion>x. \<Squnion>y. B"
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  "\<Squnion>x. B"     \<rightleftharpoons> "CONST SUPREMUM CONST UNIV (\<lambda>x. B)"
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  "\<Squnion>x. B"     \<rightleftharpoons> "\<Squnion>x \<in> CONST UNIV. B"
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  "\<Squnion>x\<in>A. B"   \<rightleftharpoons> "CONST SUPREMUM A (\<lambda>x. B)"
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print_translation \<open>
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFIMUM} @{syntax_const "_INF"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPREMUM} @{syntax_const "_SUP"}]
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\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
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subsection \<open>Abstract complete lattices\<close>
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text \<open>A complete lattice always has a bottom and a top,
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so we include them into the following type class,
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along with assumptions that define bottom and top
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in terms of infimum and supremum.\<close>
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class complete_lattice = lattice + Inf + Sup + bot + top +
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  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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  assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
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  assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
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begin
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subclass bounded_lattice
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proof
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  fix a
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  show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
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  show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
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qed
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lemma dual_complete_lattice:
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  "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
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  by (auto intro!: class.complete_lattice.intro dual_lattice)
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    (unfold_locales, (fact Inf_empty Sup_empty
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        Sup_upper Sup_least Inf_lower Inf_greatest)+)
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end
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context complete_lattice
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begin
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lemma 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 \<open>a \<in> A\<close> have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
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  with \<open>a \<sqsubseteq> b\<close> show "\<Sqinter>A \<sqsubseteq> b" by auto
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qed
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lemma INF_mono:
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  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
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  using Inf_mono [of "g ` B" "f ` A"] by auto
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lemma Sup_mono:
hoelzl@41971
   269
  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
haftmann@43741
   270
  shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
haftmann@41082
   271
proof (rule Sup_least)
haftmann@41082
   272
  fix a assume "a \<in> A"
hoelzl@41971
   273
  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
wenzelm@60758
   274
  from \<open>b \<in> B\<close> have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
wenzelm@60758
   275
  with \<open>a \<sqsubseteq> b\<close> show "a \<sqsubseteq> \<Squnion>B" by auto
haftmann@41082
   276
qed
haftmann@32077
   277
haftmann@44041
   278
lemma SUP_mono:
haftmann@44041
   279
  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
haftmann@56166
   280
  using Sup_mono [of "f ` A" "g ` B"] by auto
haftmann@44041
   281
haftmann@44041
   282
lemma INF_superset_mono:
haftmann@44041
   283
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
wenzelm@61799
   284
  \<comment> \<open>The last inclusion is POSITIVE!\<close>
haftmann@44041
   285
  by (blast intro: INF_mono dest: subsetD)
haftmann@44041
   286
haftmann@44041
   287
lemma SUP_subset_mono:
haftmann@44041
   288
  "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
   289
  by (blast intro: SUP_mono dest: subsetD)
haftmann@44041
   290
haftmann@43868
   291
lemma Inf_less_eq:
haftmann@43868
   292
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43868
   293
    and "A \<noteq> {}"
haftmann@43868
   294
  shows "\<Sqinter>A \<sqsubseteq> u"
haftmann@43868
   295
proof -
wenzelm@60758
   296
  from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
wenzelm@60758
   297
  moreover from \<open>v \<in> A\<close> assms(1) have "v \<sqsubseteq> u" by blast
haftmann@43868
   298
  ultimately show ?thesis by (rule Inf_lower2)
haftmann@43868
   299
qed
haftmann@43868
   300
haftmann@43868
   301
lemma less_eq_Sup:
haftmann@43868
   302
  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
haftmann@43868
   303
    and "A \<noteq> {}"
haftmann@43868
   304
  shows "u \<sqsubseteq> \<Squnion>A"
haftmann@43868
   305
proof -
wenzelm@60758
   306
  from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
wenzelm@60758
   307
  moreover from \<open>v \<in> A\<close> assms(1) have "u \<sqsubseteq> v" by blast
haftmann@43868
   308
  ultimately show ?thesis by (rule Sup_upper2)
haftmann@43868
   309
qed
haftmann@43868
   310
haftmann@56212
   311
lemma SUP_eq:
hoelzl@51328
   312
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
hoelzl@51328
   313
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
haftmann@56166
   314
  shows "(\<Squnion>i\<in>A. f i) = (\<Squnion>j\<in>B. g j)"
hoelzl@51328
   315
  by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
hoelzl@51328
   316
haftmann@56212
   317
lemma INF_eq:
hoelzl@51328
   318
  assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
hoelzl@51328
   319
  assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
haftmann@56166
   320
  shows "(\<Sqinter>i\<in>A. f i) = (\<Sqinter>j\<in>B. g j)"
hoelzl@51328
   321
  by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
hoelzl@51328
   322
haftmann@43899
   323
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43868
   324
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43868
   325
haftmann@43899
   326
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
haftmann@43868
   327
  by (auto intro: Sup_least Sup_upper)
haftmann@43868
   328
haftmann@43868
   329
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43868
   330
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
haftmann@43868
   331
haftmann@44041
   332
lemma INF_union:
haftmann@44041
   333
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@44103
   334
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
haftmann@44041
   335
haftmann@43868
   336
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
haftmann@43868
   337
  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
haftmann@43868
   338
haftmann@44041
   339
lemma SUP_union:
haftmann@44041
   340
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@44103
   341
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
haftmann@44041
   342
haftmann@44041
   343
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
   344
  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041
   345
noschinl@44918
   346
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
   347
proof (rule antisym)
noschinl@44918
   348
  show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
noschinl@44918
   349
next
noschinl@44918
   350
  show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
noschinl@44918
   351
qed
haftmann@44041
   352
blanchet@54147
   353
lemma Inf_top_conv [simp]:
haftmann@43868
   354
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   355
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   356
proof -
haftmann@43868
   357
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   358
  proof
haftmann@43868
   359
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   360
    then have "A = {} \<or> A = {\<top>}" by auto
noschinl@44919
   361
    then show "\<Sqinter>A = \<top>" by auto
haftmann@43868
   362
  next
haftmann@43868
   363
    assume "\<Sqinter>A = \<top>"
haftmann@43868
   364
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43868
   365
    proof (rule ccontr)
haftmann@43868
   366
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868
   367
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868
   368
      then obtain B where "A = insert x B" by blast
wenzelm@60758
   369
      with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp
haftmann@43868
   370
    qed
haftmann@43868
   371
  qed
haftmann@43868
   372
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868
   373
qed
haftmann@43868
   374
noschinl@44918
   375
lemma INF_top_conv [simp]:
haftmann@56166
   376
  "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   377
  "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@56166
   378
  using Inf_top_conv [of "B ` A"] by simp_all
haftmann@44041
   379
blanchet@54147
   380
lemma Sup_bot_conv [simp]:
haftmann@43868
   381
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
haftmann@43868
   382
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
huffman@44920
   383
  using dual_complete_lattice
huffman@44920
   384
  by (rule complete_lattice.Inf_top_conv)+
haftmann@43868
   385
noschinl@44918
   386
lemma SUP_bot_conv [simp]:
haftmann@44041
   387
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041
   388
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@56166
   389
  using Sup_bot_conv [of "B ` A"] by simp_all
haftmann@44041
   390
haftmann@43865
   391
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103
   392
  by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077
   393
haftmann@43870
   394
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103
   395
  by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870
   396
noschinl@44918
   397
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
huffman@44921
   398
  by (cases "A = {}") simp_all
haftmann@43900
   399
noschinl@44918
   400
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
huffman@44921
   401
  by (cases "A = {}") simp_all
haftmann@43900
   402
haftmann@43865
   403
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
   404
  by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865
   405
haftmann@43870
   406
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
   407
  by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870
   408
haftmann@43871
   409
lemma INF_absorb:
haftmann@43868
   410
  assumes "k \<in> I"
haftmann@43868
   411
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   412
proof -
haftmann@43868
   413
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   414
  then show ?thesis by simp
haftmann@43868
   415
qed
haftmann@43868
   416
haftmann@43871
   417
lemma SUP_absorb:
haftmann@43871
   418
  assumes "k \<in> I"
haftmann@43871
   419
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   420
proof -
haftmann@43871
   421
  from assms obtain J where "I = insert k J" by blast
haftmann@56166
   422
  then show ?thesis by simp
haftmann@43871
   423
qed
haftmann@43871
   424
hoelzl@57448
   425
lemma INF_inf_const1:
hoelzl@57448
   426
  "I \<noteq> {} \<Longrightarrow> (INF i:I. inf x (f i)) = inf x (INF i:I. f i)"
hoelzl@57448
   427
  by (intro antisym INF_greatest inf_mono order_refl INF_lower)
hoelzl@57448
   428
     (auto intro: INF_lower2 le_infI2 intro!: INF_mono)
hoelzl@57448
   429
hoelzl@57448
   430
lemma INF_inf_const2:
hoelzl@57448
   431
  "I \<noteq> {} \<Longrightarrow> (INF i:I. inf (f i) x) = inf (INF i:I. f i) x"
hoelzl@57448
   432
  using INF_inf_const1[of I x f] by (simp add: inf_commute)
hoelzl@57448
   433
haftmann@43871
   434
lemma INF_constant:
haftmann@43868
   435
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921
   436
  by simp
haftmann@43868
   437
haftmann@43871
   438
lemma SUP_constant:
haftmann@43871
   439
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921
   440
  by simp
haftmann@43871
   441
haftmann@43943
   442
lemma less_INF_D:
haftmann@43943
   443
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   444
proof -
wenzelm@60758
   445
  note \<open>y < (\<Sqinter>i\<in>A. f i)\<close>
wenzelm@60758
   446
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using \<open>i \<in> A\<close>
haftmann@44103
   447
    by (rule INF_lower)
haftmann@43943
   448
  finally show "y < f i" .
haftmann@43943
   449
qed
haftmann@43943
   450
haftmann@43943
   451
lemma SUP_lessD:
haftmann@43943
   452
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   453
proof -
wenzelm@60758
   454
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using \<open>i \<in> A\<close>
haftmann@44103
   455
    by (rule SUP_upper)
wenzelm@60758
   456
  also note \<open>(\<Squnion>i\<in>A. f i) < y\<close>
haftmann@43943
   457
  finally show "f i < y" .
haftmann@43943
   458
qed
haftmann@43943
   459
haftmann@43873
   460
lemma INF_UNIV_bool_expand:
haftmann@43868
   461
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@56166
   462
  by (simp add: UNIV_bool inf_commute)
haftmann@43868
   463
haftmann@43873
   464
lemma SUP_UNIV_bool_expand:
haftmann@43871
   465
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@56166
   466
  by (simp add: UNIV_bool sup_commute)
haftmann@43871
   467
hoelzl@51328
   468
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@51328
   469
  by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
hoelzl@51328
   470
haftmann@56218
   471
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f"
haftmann@56166
   472
  using Inf_le_Sup [of "f ` A"] by simp
hoelzl@51328
   473
hoelzl@54414
   474
lemma INF_eq_const:
haftmann@56218
   475
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x"
hoelzl@54414
   476
  by (auto intro: INF_eqI)
hoelzl@54414
   477
haftmann@56248
   478
lemma SUP_eq_const:
haftmann@56248
   479
  "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x"
haftmann@56248
   480
  by (auto intro: SUP_eqI)
hoelzl@54414
   481
hoelzl@54414
   482
lemma INF_eq_iff:
haftmann@56218
   483
  "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
   484
  using INF_eq_const [of I f c] INF_lower [of _ I f]
haftmann@56248
   485
  by (auto intro: antisym cong del: strong_INF_cong)
haftmann@56248
   486
haftmann@56248
   487
lemma SUP_eq_iff:
haftmann@56248
   488
  "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
   489
  using SUP_eq_const [of I f c] SUP_upper [of _ I f]
haftmann@56248
   490
  by (auto intro: antisym cong del: strong_SUP_cong)
hoelzl@54414
   491
haftmann@32077
   492
end
haftmann@32077
   493
haftmann@44024
   494
class complete_distrib_lattice = complete_lattice +
haftmann@44039
   495
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   496
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   497
begin
haftmann@44024
   498
haftmann@44039
   499
lemma sup_INF:
haftmann@44039
   500
  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@56166
   501
  by (simp only: INF_def sup_Inf image_image)
haftmann@44039
   502
haftmann@44039
   503
lemma inf_SUP:
haftmann@44039
   504
  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@56166
   505
  by (simp only: SUP_def inf_Sup image_image)
haftmann@44039
   506
haftmann@44032
   507
lemma dual_complete_distrib_lattice:
krauss@44845
   508
  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024
   509
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   510
  apply (fact dual_complete_lattice)
haftmann@44024
   511
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032
   512
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032
   513
  done
haftmann@44024
   514
haftmann@44322
   515
subclass distrib_lattice proof
haftmann@44024
   516
  fix a b c
haftmann@44024
   517
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
noschinl@44919
   518
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
haftmann@44024
   519
qed
haftmann@44024
   520
haftmann@44039
   521
lemma Inf_sup:
haftmann@44039
   522
  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039
   523
  by (simp add: sup_Inf sup_commute)
haftmann@44039
   524
haftmann@44039
   525
lemma Sup_inf:
haftmann@44039
   526
  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039
   527
  by (simp add: inf_Sup inf_commute)
haftmann@44039
   528
haftmann@44039
   529
lemma INF_sup: 
haftmann@44039
   530
  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039
   531
  by (simp add: sup_INF sup_commute)
haftmann@44039
   532
haftmann@44039
   533
lemma SUP_inf:
haftmann@44039
   534
  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039
   535
  by (simp add: inf_SUP inf_commute)
haftmann@44039
   536
haftmann@44039
   537
lemma Inf_sup_eq_top_iff:
haftmann@44039
   538
  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039
   539
  by (simp only: Inf_sup INF_top_conv)
haftmann@44039
   540
haftmann@44039
   541
lemma Sup_inf_eq_bot_iff:
haftmann@44039
   542
  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039
   543
  by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039
   544
haftmann@44039
   545
lemma INF_sup_distrib2:
haftmann@44039
   546
  "(\<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
   547
  by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039
   548
haftmann@44039
   549
lemma SUP_inf_distrib2:
haftmann@44039
   550
  "(\<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
   551
  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039
   552
haftmann@56074
   553
context
haftmann@56074
   554
  fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
haftmann@56074
   555
  assumes "mono f"
haftmann@56074
   556
begin
haftmann@56074
   557
haftmann@56074
   558
lemma mono_Inf:
haftmann@56074
   559
  shows "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
wenzelm@60758
   560
  using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
haftmann@56074
   561
haftmann@56074
   562
lemma mono_Sup:
haftmann@56074
   563
  shows "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
wenzelm@60758
   564
  using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
haftmann@56074
   565
hoelzl@60172
   566
lemma mono_INF:
hoelzl@60172
   567
  "f (INF i : I. A i) \<le> (INF x : I. f (A x))"
wenzelm@60758
   568
  by (intro complete_lattice_class.INF_greatest monoD[OF \<open>mono f\<close>] INF_lower)
hoelzl@60172
   569
hoelzl@60172
   570
lemma mono_SUP:
hoelzl@60172
   571
  "(SUP x : I. f (A x)) \<le> f (SUP i : I. A i)"
wenzelm@60758
   572
  by (intro complete_lattice_class.SUP_least monoD[OF \<open>mono f\<close>] SUP_upper)
hoelzl@60172
   573
haftmann@56074
   574
end
haftmann@56074
   575
haftmann@44024
   576
end
haftmann@44024
   577
haftmann@44032
   578
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   579
begin
haftmann@43873
   580
haftmann@43943
   581
lemma dual_complete_boolean_algebra:
krauss@44845
   582
  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   583
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   584
haftmann@43873
   585
lemma uminus_Inf:
haftmann@43873
   586
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   587
proof (rule antisym)
haftmann@43873
   588
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   589
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   590
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   591
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   592
qed
haftmann@43873
   593
haftmann@44041
   594
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@56166
   595
  by (simp only: INF_def SUP_def uminus_Inf image_image)
haftmann@44041
   596
haftmann@43873
   597
lemma uminus_Sup:
haftmann@43873
   598
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   599
proof -
haftmann@56166
   600
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_INF)
haftmann@43873
   601
  then show ?thesis by simp
haftmann@43873
   602
qed
haftmann@43873
   603
  
haftmann@43873
   604
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@56166
   605
  by (simp only: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   606
haftmann@43873
   607
end
haftmann@43873
   608
haftmann@43940
   609
class complete_linorder = linorder + complete_lattice
haftmann@43940
   610
begin
haftmann@43940
   611
haftmann@43943
   612
lemma dual_complete_linorder:
krauss@44845
   613
  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943
   614
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   615
haftmann@51386
   616
lemma complete_linorder_inf_min: "inf = min"
haftmann@51540
   617
  by (auto intro: antisym simp add: min_def fun_eq_iff)
haftmann@51386
   618
haftmann@51386
   619
lemma complete_linorder_sup_max: "sup = max"
haftmann@51540
   620
  by (auto intro: antisym simp add: max_def fun_eq_iff)
haftmann@51386
   621
noschinl@44918
   622
lemma Inf_less_iff:
haftmann@43940
   623
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   624
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   625
noschinl@44918
   626
lemma INF_less_iff:
haftmann@44041
   627
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@56166
   628
  using Inf_less_iff [of "f ` A"] by simp
haftmann@44041
   629
noschinl@44918
   630
lemma less_Sup_iff:
haftmann@43940
   631
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   632
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   633
noschinl@44918
   634
lemma less_SUP_iff:
haftmann@43940
   635
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@56166
   636
  using less_Sup_iff [of _ "f ` A"] by simp
haftmann@43940
   637
noschinl@44918
   638
lemma Sup_eq_top_iff [simp]:
haftmann@43943
   639
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   640
proof
haftmann@43943
   641
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   642
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   643
  proof (intro allI impI)
haftmann@43943
   644
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   645
      unfolding less_Sup_iff by auto
haftmann@43943
   646
  qed
haftmann@43943
   647
next
haftmann@43943
   648
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   649
  show "\<Squnion>A = \<top>"
haftmann@43943
   650
  proof (rule ccontr)
haftmann@43943
   651
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   652
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   653
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   654
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   655
      using * unfolding less_Sup_iff by auto
haftmann@43943
   656
    then show False by auto
haftmann@43943
   657
  qed
haftmann@43943
   658
qed
haftmann@43943
   659
noschinl@44918
   660
lemma SUP_eq_top_iff [simp]:
haftmann@44041
   661
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
haftmann@56166
   662
  using Sup_eq_top_iff [of "f ` A"] by simp
haftmann@44041
   663
noschinl@44918
   664
lemma Inf_eq_bot_iff [simp]:
haftmann@43943
   665
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920
   666
  using dual_complete_linorder
huffman@44920
   667
  by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943
   668
noschinl@44918
   669
lemma INF_eq_bot_iff [simp]:
haftmann@43967
   670
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
haftmann@56166
   671
  using Inf_eq_bot_iff [of "f ` A"] by simp
hoelzl@51328
   672
hoelzl@51328
   673
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
hoelzl@51328
   674
proof safe
hoelzl@51328
   675
  fix y assume "x \<ge> \<Sqinter>A" "y > x"
hoelzl@51328
   676
  then have "y > \<Sqinter>A" by auto
hoelzl@51328
   677
  then show "\<exists>a\<in>A. y > a"
hoelzl@51328
   678
    unfolding Inf_less_iff .
hoelzl@51328
   679
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
hoelzl@51328
   680
hoelzl@51328
   681
lemma INF_le_iff:
haftmann@56218
   682
  "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
haftmann@56166
   683
  using Inf_le_iff [of "f ` A"] by simp
haftmann@56166
   684
haftmann@56166
   685
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
haftmann@56166
   686
proof safe
haftmann@56166
   687
  fix y assume "x \<le> \<Squnion>A" "y < x"
haftmann@56166
   688
  then have "y < \<Squnion>A" by auto
haftmann@56166
   689
  then show "\<exists>a\<in>A. y < a"
haftmann@56166
   690
    unfolding less_Sup_iff .
haftmann@56166
   691
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
haftmann@56166
   692
haftmann@56218
   693
lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
haftmann@56166
   694
  using le_Sup_iff [of _ "f ` A"] by simp
hoelzl@51328
   695
haftmann@51386
   696
subclass complete_distrib_lattice
haftmann@51386
   697
proof
haftmann@51386
   698
  fix a and B
haftmann@51386
   699
  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
   700
    by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
haftmann@51386
   701
      (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
haftmann@51386
   702
        le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
haftmann@51386
   703
qed
haftmann@51386
   704
haftmann@43940
   705
end
haftmann@43940
   706
hoelzl@51341
   707
wenzelm@60758
   708
subsection \<open>Complete lattice on @{typ bool}\<close>
haftmann@32077
   709
haftmann@44024
   710
instantiation bool :: complete_lattice
haftmann@32077
   711
begin
haftmann@32077
   712
haftmann@32077
   713
definition
haftmann@46154
   714
  [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077
   715
haftmann@32077
   716
definition
haftmann@46154
   717
  [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077
   718
haftmann@32077
   719
instance proof
haftmann@44322
   720
qed (auto intro: bool_induct)
haftmann@32077
   721
haftmann@32077
   722
end
haftmann@32077
   723
haftmann@49905
   724
lemma not_False_in_image_Ball [simp]:
haftmann@49905
   725
  "False \<notin> P ` A \<longleftrightarrow> Ball A P"
haftmann@49905
   726
  by auto
haftmann@49905
   727
haftmann@49905
   728
lemma True_in_image_Bex [simp]:
haftmann@49905
   729
  "True \<in> P ` A \<longleftrightarrow> Bex A P"
haftmann@49905
   730
  by auto
haftmann@49905
   731
haftmann@43873
   732
lemma INF_bool_eq [simp]:
haftmann@56218
   733
  "INFIMUM = Ball"
haftmann@49905
   734
  by (simp add: fun_eq_iff INF_def)
haftmann@32120
   735
haftmann@43873
   736
lemma SUP_bool_eq [simp]:
haftmann@56218
   737
  "SUPREMUM = Bex"
haftmann@49905
   738
  by (simp add: fun_eq_iff SUP_def)
haftmann@32120
   739
haftmann@44032
   740
instance bool :: complete_boolean_algebra proof
haftmann@44322
   741
qed (auto intro: bool_induct)
haftmann@44024
   742
haftmann@46631
   743
wenzelm@60758
   744
subsection \<open>Complete lattice on @{typ "_ \<Rightarrow> _"}\<close>
haftmann@46631
   745
nipkow@57197
   746
instantiation "fun" :: (type, Inf) Inf
haftmann@32077
   747
begin
haftmann@32077
   748
haftmann@32077
   749
definition
haftmann@44024
   750
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   751
noschinl@46882
   752
lemma Inf_apply [simp, code]:
haftmann@44024
   753
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   754
  by (simp add: Inf_fun_def)
haftmann@32077
   755
nipkow@57197
   756
instance ..
nipkow@57197
   757
nipkow@57197
   758
end
nipkow@57197
   759
nipkow@57197
   760
instantiation "fun" :: (type, Sup) Sup
nipkow@57197
   761
begin
nipkow@57197
   762
haftmann@32077
   763
definition
haftmann@44024
   764
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   765
noschinl@46882
   766
lemma Sup_apply [simp, code]:
haftmann@44024
   767
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   768
  by (simp add: Sup_fun_def)
haftmann@32077
   769
nipkow@57197
   770
instance ..
nipkow@57197
   771
nipkow@57197
   772
end
nipkow@57197
   773
nipkow@57197
   774
instantiation "fun" :: (type, complete_lattice) complete_lattice
nipkow@57197
   775
begin
nipkow@57197
   776
haftmann@32077
   777
instance proof
noschinl@46884
   778
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077
   779
haftmann@32077
   780
end
haftmann@32077
   781
noschinl@46882
   782
lemma INF_apply [simp]:
haftmann@41080
   783
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@56166
   784
  using Inf_apply [of "f ` A"] by (simp add: comp_def)
hoelzl@38705
   785
noschinl@46882
   786
lemma SUP_apply [simp]:
haftmann@41080
   787
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@56166
   788
  using Sup_apply [of "f ` A"] by (simp add: comp_def)
haftmann@32077
   789
haftmann@44024
   790
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
haftmann@56166
   791
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf fun_eq_iff image_image
haftmann@56166
   792
  simp del: Inf_image_eq Sup_image_eq)
haftmann@44024
   793
haftmann@43873
   794
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   795
haftmann@46631
   796
wenzelm@60758
   797
subsection \<open>Complete lattice on unary and binary predicates\<close>
haftmann@46631
   798
haftmann@56742
   799
lemma Inf1_I: 
haftmann@56742
   800
  "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a"
noschinl@46884
   801
  by auto
haftmann@46631
   802
haftmann@56742
   803
lemma INF1_I:
haftmann@56742
   804
  "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   805
  by simp
haftmann@56742
   806
haftmann@56742
   807
lemma INF2_I:
haftmann@56742
   808
  "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   809
  by simp
haftmann@56742
   810
haftmann@56742
   811
lemma Inf2_I: 
haftmann@56742
   812
  "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b"
noschinl@46884
   813
  by auto
haftmann@46631
   814
haftmann@56742
   815
lemma Inf1_D:
haftmann@56742
   816
  "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a"
noschinl@46884
   817
  by auto
haftmann@46631
   818
haftmann@56742
   819
lemma INF1_D:
haftmann@56742
   820
  "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
haftmann@56742
   821
  by simp
haftmann@56742
   822
haftmann@56742
   823
lemma Inf2_D:
haftmann@56742
   824
  "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b"
noschinl@46884
   825
  by auto
haftmann@46631
   826
haftmann@56742
   827
lemma INF2_D:
haftmann@56742
   828
  "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
haftmann@56742
   829
  by simp
haftmann@56742
   830
haftmann@56742
   831
lemma Inf1_E:
haftmann@56742
   832
  assumes "(\<Sqinter>A) a"
haftmann@56742
   833
  obtains "P a" | "P \<notin> A"
haftmann@56742
   834
  using assms by auto
haftmann@46631
   835
haftmann@56742
   836
lemma INF1_E:
haftmann@56742
   837
  assumes "(\<Sqinter>x\<in>A. B x) b"
haftmann@56742
   838
  obtains "B a b" | "a \<notin> A"
haftmann@56742
   839
  using assms by auto
haftmann@56742
   840
haftmann@56742
   841
lemma Inf2_E:
haftmann@56742
   842
  assumes "(\<Sqinter>A) a b"
haftmann@56742
   843
  obtains "r a b" | "r \<notin> A"
haftmann@56742
   844
  using assms by auto
haftmann@56742
   845
haftmann@56742
   846
lemma INF2_E:
haftmann@56742
   847
  assumes "(\<Sqinter>x\<in>A. B x) b c"
haftmann@56742
   848
  obtains "B a b c" | "a \<notin> A"
haftmann@56742
   849
  using assms by auto
haftmann@56742
   850
haftmann@56742
   851
lemma Sup1_I:
haftmann@56742
   852
  "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a"
noschinl@46884
   853
  by auto
haftmann@46631
   854
haftmann@56742
   855
lemma SUP1_I:
haftmann@56742
   856
  "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
haftmann@56742
   857
  by auto
haftmann@56742
   858
haftmann@56742
   859
lemma Sup2_I:
haftmann@56742
   860
  "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b"
haftmann@56742
   861
  by auto
haftmann@56742
   862
haftmann@56742
   863
lemma SUP2_I:
haftmann@56742
   864
  "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
noschinl@46884
   865
  by auto
haftmann@46631
   866
haftmann@56742
   867
lemma Sup1_E:
haftmann@56742
   868
  assumes "(\<Squnion>A) a"
haftmann@56742
   869
  obtains P where "P \<in> A" and "P a"
haftmann@56742
   870
  using assms by auto
haftmann@56742
   871
haftmann@56742
   872
lemma SUP1_E:
haftmann@56742
   873
  assumes "(\<Squnion>x\<in>A. B x) b"
haftmann@56742
   874
  obtains x where "x \<in> A" and "B x b"
haftmann@56742
   875
  using assms by auto
haftmann@46631
   876
haftmann@56742
   877
lemma Sup2_E:
haftmann@56742
   878
  assumes "(\<Squnion>A) a b"
haftmann@56742
   879
  obtains r where "r \<in> A" "r a b"
haftmann@56742
   880
  using assms by auto
haftmann@56742
   881
haftmann@56742
   882
lemma SUP2_E:
haftmann@56742
   883
  assumes "(\<Squnion>x\<in>A. B x) b c"
haftmann@56742
   884
  obtains x where "x \<in> A" "B x b c"
haftmann@56742
   885
  using assms by auto
haftmann@46631
   886
haftmann@46631
   887
wenzelm@60758
   888
subsection \<open>Complete lattice on @{typ "_ set"}\<close>
haftmann@46631
   889
haftmann@45960
   890
instantiation "set" :: (type) complete_lattice
haftmann@45960
   891
begin
haftmann@45960
   892
haftmann@45960
   893
definition
haftmann@45960
   894
  "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   895
haftmann@45960
   896
definition
haftmann@45960
   897
  "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960
   898
haftmann@45960
   899
instance proof
haftmann@51386
   900
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
haftmann@45960
   901
haftmann@45960
   902
end
haftmann@45960
   903
haftmann@45960
   904
instance "set" :: (type) complete_boolean_algebra
haftmann@45960
   905
proof
haftmann@45960
   906
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
haftmann@45960
   907
  
haftmann@32077
   908
wenzelm@60758
   909
subsubsection \<open>Inter\<close>
haftmann@41082
   910
wenzelm@61952
   911
abbreviation Inter :: "'a set set \<Rightarrow> 'a set"  ("\<Inter>_" [900] 900)
wenzelm@61952
   912
  where "\<Inter>S \<equiv> \<Sqinter>S"
haftmann@41082
   913
  
haftmann@41082
   914
lemma Inter_eq:
haftmann@41082
   915
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   916
proof (rule set_eqI)
haftmann@41082
   917
  fix x
haftmann@41082
   918
  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
   919
    by auto
haftmann@41082
   920
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960
   921
    by (simp add: Inf_set_def image_def)
haftmann@41082
   922
qed
haftmann@41082
   923
blanchet@54147
   924
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   925
  by (unfold Inter_eq) blast
haftmann@41082
   926
haftmann@43741
   927
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   928
  by (simp add: Inter_eq)
haftmann@41082
   929
wenzelm@60758
   930
text \<open>
haftmann@41082
   931
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   932
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
wenzelm@61799
   933
  @{prop "X \<in> C"} does not!  This rule is analogous to \<open>spec\<close>.
wenzelm@60758
   934
\<close>
haftmann@41082
   935
haftmann@43741
   936
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   937
  by auto
haftmann@41082
   938
haftmann@43741
   939
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@61799
   940
  \<comment> \<open>``Classical'' elimination rule -- does not require proving
wenzelm@60758
   941
    @{prop "X \<in> C"}.\<close>
haftmann@41082
   942
  by (unfold Inter_eq) blast
haftmann@41082
   943
haftmann@43741
   944
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   945
  by (fact Inf_lower)
haftmann@43740
   946
haftmann@41082
   947
lemma Inter_subset:
haftmann@43755
   948
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   949
  by (fact Inf_less_eq)
haftmann@41082
   950
wenzelm@61952
   951
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A"
haftmann@43740
   952
  by (fact Inf_greatest)
haftmann@41082
   953
huffman@44067
   954
lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067
   955
  by (fact Inf_empty) (* already simp *)
haftmann@41082
   956
huffman@44067
   957
lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067
   958
  by (fact Inf_UNIV) (* already simp *)
haftmann@41082
   959
huffman@44920
   960
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920
   961
  by (fact Inf_insert) (* already simp *)
haftmann@41082
   962
haftmann@41082
   963
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   964
  by (fact less_eq_Inf_inter)
haftmann@41082
   965
haftmann@41082
   966
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   967
  by (fact Inf_union_distrib)
haftmann@43756
   968
blanchet@54147
   969
lemma Inter_UNIV_conv [simp]:
haftmann@43741
   970
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   971
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   972
  by (fact Inf_top_conv)+
haftmann@41082
   973
haftmann@43741
   974
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   975
  by (fact Inf_superset_mono)
haftmann@41082
   976
haftmann@41082
   977
wenzelm@60758
   978
subsubsection \<open>Intersections of families\<close>
haftmann@41082
   979
wenzelm@61955
   980
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
wenzelm@61955
   981
  where "INTER \<equiv> INFIMUM"
haftmann@41082
   982
wenzelm@60758
   983
text \<open>
wenzelm@61799
   984
  Note: must use name @{const INTER} here instead of \<open>INT\<close>
haftmann@43872
   985
  to allow the following syntax coexist with the plain constant name.
wenzelm@60758
   986
\<close>
haftmann@43872
   987
wenzelm@61955
   988
syntax (ASCII)
wenzelm@61955
   989
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3INT _./ _)" [0, 10] 10)
wenzelm@61955
   990
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   991
haftmann@41082
   992
syntax (latex output)
wenzelm@61955
   993
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
wenzelm@61955
   994
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
wenzelm@61955
   995
wenzelm@61955
   996
syntax
wenzelm@61955
   997
  "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
wenzelm@61955
   998
  "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   999
haftmann@41082
  1000
translations
wenzelm@61955
  1001
  "\<Inter>x y. B"  \<rightleftharpoons> "\<Inter>x. \<Inter>y. B"
wenzelm@61955
  1002
  "\<Inter>x. B"    \<rightleftharpoons> "CONST INTER CONST UNIV (\<lambda>x. B)"
wenzelm@61955
  1003
  "\<Inter>x. B"    \<rightleftharpoons> "\<Inter>x \<in> CONST UNIV. B"
wenzelm@61955
  1004
  "\<Inter>x\<in>A. B"  \<rightleftharpoons> "CONST INTER A (\<lambda>x. B)"
haftmann@41082
  1005
wenzelm@60758
  1006
print_translation \<open>
wenzelm@42284
  1007
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
wenzelm@61799
  1008
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
haftmann@41082
  1009
haftmann@44085
  1010
lemma INTER_eq:
haftmann@41082
  1011
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@56166
  1012
  by (auto intro!: INF_eqI)
haftmann@41082
  1013
haftmann@56166
  1014
lemma Inter_image_eq:
haftmann@56166
  1015
  "\<Inter>(B ` A) = (\<Inter>x\<in>A. B x)"
haftmann@56166
  1016
  by (fact Inf_image_eq)
haftmann@41082
  1017
haftmann@43817
  1018
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@56166
  1019
  using Inter_iff [of _ "B ` A"] by simp
haftmann@41082
  1020
haftmann@43817
  1021
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
  1022
  by (auto simp add: INF_def image_def)
haftmann@41082
  1023
haftmann@43852
  1024
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
  1025
  by auto
haftmann@41082
  1026
haftmann@43852
  1027
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@61799
  1028
  \<comment> \<open>"Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}.\<close>
haftmann@44085
  1029
  by (auto simp add: INF_def image_def)
haftmann@41082
  1030
haftmann@41082
  1031
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
  1032
  by blast
haftmann@41082
  1033
haftmann@41082
  1034
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
  1035
  by blast
haftmann@41082
  1036
haftmann@43817
  1037
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103
  1038
  by (fact INF_lower)
haftmann@41082
  1039
haftmann@43817
  1040
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103
  1041
  by (fact INF_greatest)
haftmann@41082
  1042
huffman@44067
  1043
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085
  1044
  by (fact INF_empty)
haftmann@43854
  1045
haftmann@43817
  1046
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
  1047
  by (fact INF_absorb)
haftmann@41082
  1048
haftmann@43854
  1049
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
  1050
  by (fact le_INF_iff)
haftmann@41082
  1051
haftmann@41082
  1052
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
  1053
  by (fact INF_insert)
haftmann@43865
  1054
haftmann@43865
  1055
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
  1056
  by (fact INF_union)
haftmann@43865
  1057
haftmann@43865
  1058
lemma INT_insert_distrib:
haftmann@43865
  1059
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
  1060
  by blast
haftmann@43854
  1061
haftmann@41082
  1062
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
  1063
  by (fact INF_constant)
haftmann@43865
  1064
huffman@44920
  1065
lemma INTER_UNIV_conv:
haftmann@43817
  1066
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
  1067
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920
  1068
  by (fact INF_top_conv)+ (* already simp *)
haftmann@43865
  1069
haftmann@43865
  1070
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
  1071
  by (fact INF_UNIV_bool_expand)
haftmann@43865
  1072
haftmann@43865
  1073
lemma INT_anti_mono:
haftmann@43900
  1074
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
wenzelm@61799
  1075
  \<comment> \<open>The last inclusion is POSITIVE!\<close>
haftmann@43940
  1076
  by (fact INF_superset_mono)
haftmann@41082
  1077
haftmann@41082
  1078
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
  1079
  by blast
haftmann@41082
  1080
haftmann@43817
  1081
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
  1082
  by blast
haftmann@41082
  1083
haftmann@41082
  1084
wenzelm@60758
  1085
subsubsection \<open>Union\<close>
haftmann@32115
  1086
wenzelm@61952
  1087
abbreviation Union :: "'a set set \<Rightarrow> 'a set"  ("\<Union>_" [900] 900)
wenzelm@61952
  1088
  where "\<Union>S \<equiv> \<Squnion>S"
haftmann@32115
  1089
haftmann@32135
  1090
lemma Union_eq:
haftmann@32135
  1091
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
  1092
proof (rule set_eqI)
haftmann@32115
  1093
  fix x
haftmann@32135
  1094
  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
  1095
    by auto
haftmann@32135
  1096
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960
  1097
    by (simp add: Sup_set_def image_def)
haftmann@32115
  1098
qed
haftmann@32115
  1099
blanchet@54147
  1100
lemma Union_iff [simp]:
haftmann@32115
  1101
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
  1102
  by (unfold Union_eq) blast
haftmann@32115
  1103
haftmann@32115
  1104
lemma UnionI [intro]:
haftmann@32115
  1105
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
wenzelm@61799
  1106
  \<comment> \<open>The order of the premises presupposes that @{term C} is rigid;
wenzelm@60758
  1107
    @{term A} may be flexible.\<close>
haftmann@32115
  1108
  by auto
haftmann@32115
  1109
haftmann@32115
  1110
lemma UnionE [elim!]:
haftmann@43817
  1111
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
  1112
  by auto
haftmann@32115
  1113
haftmann@43817
  1114
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
  1115
  by (fact Sup_upper)
haftmann@32135
  1116
haftmann@43817
  1117
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
  1118
  by (fact Sup_least)
haftmann@32135
  1119
huffman@44920
  1120
lemma Union_empty: "\<Union>{} = {}"
huffman@44920
  1121
  by (fact Sup_empty) (* already simp *)
haftmann@32135
  1122
huffman@44920
  1123
lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920
  1124
  by (fact Sup_UNIV) (* already simp *)
haftmann@32135
  1125
huffman@44920
  1126
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920
  1127
  by (fact Sup_insert) (* already simp *)
haftmann@32135
  1128
haftmann@43817
  1129
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
  1130
  by (fact Sup_union_distrib)
haftmann@32135
  1131
haftmann@32135
  1132
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
  1133
  by (fact Sup_inter_less_eq)
haftmann@32135
  1134
blanchet@54147
  1135
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1136
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1137
blanchet@54147
  1138
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920
  1139
  by (fact Sup_bot_conv) (* already simp *)
haftmann@32135
  1140
haftmann@32135
  1141
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
  1142
  by blast
haftmann@32135
  1143
haftmann@32135
  1144
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
  1145
  by blast
haftmann@32135
  1146
haftmann@43817
  1147
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
  1148
  by (fact Sup_subset_mono)
haftmann@32135
  1149
haftmann@32115
  1150
wenzelm@60758
  1151
subsubsection \<open>Unions of families\<close>
haftmann@32077
  1152
wenzelm@61955
  1153
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
wenzelm@61955
  1154
  where "UNION \<equiv> SUPREMUM"
haftmann@32077
  1155
wenzelm@60758
  1156
text \<open>
wenzelm@61799
  1157
  Note: must use name @{const UNION} here instead of \<open>UN\<close>
haftmann@43872
  1158
  to allow the following syntax coexist with the plain constant name.
wenzelm@60758
  1159
\<close>
haftmann@43872
  1160
wenzelm@61955
  1161
syntax (ASCII)
wenzelm@35115
  1162
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
  1163
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
  1164
haftmann@32077
  1165
syntax (latex output)
wenzelm@35115
  1166
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
  1167
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
  1168
wenzelm@61955
  1169
syntax
wenzelm@61955
  1170
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
wenzelm@61955
  1171
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
wenzelm@61955
  1172
haftmann@32077
  1173
translations
wenzelm@61955
  1174
  "\<Union>x y. B"   \<rightleftharpoons> "\<Union>x. \<Union>y. B"
wenzelm@61955
  1175
  "\<Union>x. B"     \<rightleftharpoons> "CONST UNION CONST UNIV (\<lambda>x. B)"
wenzelm@61955
  1176
  "\<Union>x. B"     \<rightleftharpoons> "\<Union>x \<in> CONST UNIV. B"
wenzelm@61955
  1177
  "\<Union>x\<in>A. B"   \<rightleftharpoons> "CONST UNION A (\<lambda>x. B)"
haftmann@32077
  1178
wenzelm@60758
  1179
text \<open>
wenzelm@61955
  1180
  Note the difference between ordinary syntax of indexed
wenzelm@61799
  1181
  unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>)
wenzelm@61955
  1182
  and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}.
wenzelm@60758
  1183
\<close>
haftmann@32077
  1184
wenzelm@60758
  1185
print_translation \<open>
wenzelm@42284
  1186
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@61799
  1187
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
haftmann@32077
  1188
blanchet@54147
  1189
lemma UNION_eq:
haftmann@32135
  1190
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@56166
  1191
  by (auto intro!: SUP_eqI)
huffman@44920
  1192
haftmann@45960
  1193
lemma bind_UNION [code]:
haftmann@45960
  1194
  "Set.bind A f = UNION A f"
haftmann@45960
  1195
  by (simp add: bind_def UNION_eq)
haftmann@45960
  1196
haftmann@46036
  1197
lemma member_bind [simp]:
haftmann@46036
  1198
  "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036
  1199
  by (simp add: bind_UNION)
haftmann@46036
  1200
haftmann@56166
  1201
lemma Union_image_eq:
haftmann@43817
  1202
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@56166
  1203
  by (fact Sup_image_eq)
huffman@44920
  1204
wenzelm@60585
  1205
lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
lp15@60307
  1206
  by blast
lp15@60307
  1207
haftmann@46036
  1208
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@56166
  1209
  using Union_iff [of _ "B ` A"] by simp
wenzelm@11979
  1210
haftmann@43852
  1211
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@61799
  1212
  \<comment> \<open>The order of the premises presupposes that @{term A} is rigid;
wenzelm@60758
  1213
    @{term b} may be flexible.\<close>
wenzelm@11979
  1214
  by auto
wenzelm@11979
  1215
haftmann@43852
  1216
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
  1217
  by (auto simp add: SUP_def image_def)
clasohm@923
  1218
haftmann@43817
  1219
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
  1220
  by blast
haftmann@32077
  1221
haftmann@43817
  1222
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103
  1223
  by (fact SUP_upper)
haftmann@32135
  1224
haftmann@43817
  1225
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103
  1226
  by (fact SUP_least)
haftmann@32135
  1227
blanchet@54147
  1228
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
  1229
  by blast
haftmann@32135
  1230
haftmann@43817
  1231
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
  1232
  by blast
haftmann@32135
  1233
blanchet@54147
  1234
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085
  1235
  by (fact SUP_empty)
haftmann@32135
  1236
huffman@44920
  1237
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920
  1238
  by (fact SUP_bot) (* already simp *)
haftmann@32135
  1239
haftmann@43817
  1240
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
  1241
  by (fact SUP_absorb)
haftmann@32135
  1242
haftmann@32135
  1243
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
  1244
  by (fact SUP_insert)
haftmann@32135
  1245
haftmann@44085
  1246
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
  1247
  by (fact SUP_union)
haftmann@32135
  1248
haftmann@43967
  1249
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
  1250
  by blast
haftmann@32135
  1251
haftmann@32135
  1252
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
  1253
  by (fact SUP_le_iff)
haftmann@32135
  1254
haftmann@32135
  1255
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
  1256
  by (fact SUP_constant)
haftmann@32135
  1257
haftmann@43944
  1258
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
  1259
  by blast
haftmann@32135
  1260
huffman@44920
  1261
lemma UNION_empty_conv:
haftmann@43817
  1262
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1263
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920
  1264
  by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135
  1265
blanchet@54147
  1266
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1267
  by blast
haftmann@32135
  1268
haftmann@43900
  1269
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
  1270
  by blast
haftmann@32135
  1271
haftmann@43900
  1272
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
  1273
  by blast
haftmann@32135
  1274
haftmann@32135
  1275
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
  1276
  by (auto simp add: split_if_mem2)
haftmann@32135
  1277
haftmann@43817
  1278
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1279
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1280
haftmann@32135
  1281
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1282
  by blast
haftmann@32135
  1283
haftmann@32135
  1284
lemma UN_mono:
haftmann@43817
  1285
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1286
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1287
  by (fact SUP_subset_mono)
haftmann@32135
  1288
haftmann@43817
  1289
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1290
  by blast
haftmann@32135
  1291
haftmann@43817
  1292
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1293
  by blast
haftmann@32135
  1294
haftmann@43817
  1295
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
wenzelm@61799
  1296
  \<comment> \<open>NOT suitable for rewriting\<close>
haftmann@32135
  1297
  by blast
haftmann@32135
  1298
haftmann@43817
  1299
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1300
  by blast
haftmann@32135
  1301
haftmann@45013
  1302
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013
  1303
  by blast
haftmann@45013
  1304
wenzelm@11979
  1305
wenzelm@60758
  1306
subsubsection \<open>Distributive laws\<close>
wenzelm@12897
  1307
wenzelm@12897
  1308
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1309
  by (fact inf_Sup)
wenzelm@12897
  1310
haftmann@44039
  1311
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039
  1312
  by (fact sup_Inf)
haftmann@44039
  1313
wenzelm@12897
  1314
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039
  1315
  by (fact Sup_inf)
haftmann@44039
  1316
haftmann@44039
  1317
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
  1318
  by (rule sym) (rule INF_inf_distrib)
haftmann@44039
  1319
haftmann@44039
  1320
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
  1321
  by (rule sym) (rule SUP_sup_distrib)
haftmann@44039
  1322
wenzelm@61799
  1323
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" \<comment> \<open>FIXME drop\<close>
haftmann@56166
  1324
  by (simp add: INT_Int_distrib)
wenzelm@12897
  1325
wenzelm@61799
  1326
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" \<comment> \<open>FIXME drop\<close>
wenzelm@61799
  1327
  \<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close>
wenzelm@61799
  1328
  \<comment> \<open>Union of a family of unions\<close>
haftmann@56166
  1329
  by (simp add: UN_Un_distrib)
wenzelm@12897
  1330
haftmann@44039
  1331
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039
  1332
  by (fact sup_INF)
wenzelm@12897
  1333
wenzelm@12897
  1334
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@61799
  1335
  \<comment> \<open>Halmos, Naive Set Theory, page 35.\<close>
haftmann@44039
  1336
  by (fact inf_SUP)
wenzelm@12897
  1337
wenzelm@12897
  1338
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
  1339
  by (fact SUP_inf_distrib2)
wenzelm@12897
  1340
wenzelm@12897
  1341
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
  1342
  by (fact INF_sup_distrib2)
haftmann@44039
  1343
haftmann@44039
  1344
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039
  1345
  by (fact Sup_inf_eq_bot_iff)
wenzelm@12897
  1346
Andreas@61630
  1347
lemma SUP_UNION: "(SUP x:(UN y:A. g y). f x) = (SUP y:A. SUP x:g y. f x :: _ :: complete_lattice)"
Andreas@61630
  1348
by(rule order_antisym)(blast intro: SUP_least SUP_upper2)+
wenzelm@12897
  1349
wenzelm@60758
  1350
subsection \<open>Injections and bijections\<close>
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
wenzelm@60758
  1432
subsubsection \<open>Complement\<close>
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
wenzelm@60758
  1441
subsubsection \<open>Miniscoping and maxiscoping\<close>
wenzelm@12897
  1442
wenzelm@60758
  1443
text \<open>\medskip Miniscoping: pushing in quantifiers and big Unions
wenzelm@60758
  1444
           and Intersections.\<close>
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
wenzelm@60758
  1480
text \<open>\medskip Maxiscoping: pulling out big Unions and Intersections.\<close>
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
wenzelm@60758
  1508
text \<open>Finally\<close>
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
wenzelm@61799
  1517
  \<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close>
wenzelm@21669
  1518
wenzelm@11979
  1519
end
haftmann@49905
  1520