(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)

header {* Complete lattices, with special focus on sets *}

theory Complete_Lattice

imports Set

begin

notation

  less_eq (infix "\<sqsubseteq>" 50) and

  less (infix "\<sqsubset>" 50) and

  inf (infixl "\<sqinter>" 70) and

  sup (infixl "\<squnion>" 65) and

  top ("\<top>") and

  bot ("\<bottom>")

subsection {* Syntactic infimum and supremum operations *}

class Inf =

  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)

class Sup =

  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)

subsection {* Abstract complete lattices *}

class complete_lattice = bounded_lattice + Inf + Sup +

  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"

     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"

  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"

     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"

begin

lemma dual_complete_lattice:

  "class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"

  by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)

    (unfold_locales, (fact bot_least top_greatest

        Sup_upper Sup_least Inf_lower Inf_greatest)+)

lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"

  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)

lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"

  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)

lemma Inf_empty [simp]:

  "\<Sqinter>{} = \<top>"

  by (auto intro: antisym Inf_greatest)

lemma Sup_empty [simp]:

  "\<Squnion>{} = \<bottom>"

  by (auto intro: antisym Sup_least)

lemma Inf_UNIV [simp]:

  "\<Sqinter>UNIV = \<bottom>"

  by (simp add: Sup_Inf Sup_empty [symmetric])

lemma Sup_UNIV [simp]:

  "\<Squnion>UNIV = \<top>"

  by (simp add: Inf_Sup Inf_empty [symmetric])

lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"

  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)

lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"

  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)

lemma Inf_singleton [simp]:

  "\<Sqinter>{a} = a"

  by (auto intro: antisym Inf_lower Inf_greatest)

lemma Sup_singleton [simp]:

  "\<Squnion>{a} = a"

  by (auto intro: antisym Sup_upper Sup_least)

lemma Inf_binary:

  "\<Sqinter>{a, b} = a \<sqinter> b"

  by (simp add: Inf_empty Inf_insert)

lemma Sup_binary:

  "\<Squnion>{a, b} = a \<squnion> b"

  by (simp add: Sup_empty Sup_insert)

lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"

  by (auto intro: Inf_greatest dest: Inf_lower)

lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"

  by (auto intro: Sup_least dest: Sup_upper)

lemma Inf_mono:

  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"

  shows "Inf A \<sqsubseteq> Inf B"

proof (rule Inf_greatest)

  fix b assume "b \<in> B"

  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast

  from `a \<in> A` have "Inf A \<sqsubseteq> a" by (rule Inf_lower)

  with `a \<sqsubseteq> b` show "Inf A \<sqsubseteq> b" by auto

qed

lemma Sup_mono:

  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"

  shows "Sup A \<sqsubseteq> Sup B"

proof (rule Sup_least)

  fix a assume "a \<in> A"

  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast

  from `b \<in> B` have "b \<sqsubseteq> Sup B" by (rule Sup_upper)

  with `a \<sqsubseteq> b` show "a \<sqsubseteq> Sup B" by auto

qed

lemma top_le:

  "top \<sqsubseteq> x \<Longrightarrow> x = top"

  by (rule antisym) auto

lemma le_bot:

  "x \<sqsubseteq> bot \<Longrightarrow> x = bot"

  by (rule antisym) auto

lemma not_less_bot[simp]: "\<not> (x \<sqsubset> bot)"

  using bot_least[of x] by (auto simp: le_less)

lemma not_top_less[simp]: "\<not> (top \<sqsubset> x)"

  using top_greatest[of x] by (auto simp: le_less)

lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> Sup A"

  using Sup_upper[of u A] by auto

lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> Inf A \<sqsubseteq> v"

  using Inf_lower[of u A] by auto

definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where

  "INFI A f = \<Sqinter> (f ` A)"

definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where

  "SUPR A f = \<Squnion> (f ` A)"

end

syntax

  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)

  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)

  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)

  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)

syntax (xsymbols)

  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)

  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)

  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)

  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)

translations

  "INF x y. B"   == "INF x. INF y. B"

  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"

  "INF x. B"     == "INF x:CONST UNIV. B"

  "INF x:A. B"   == "CONST INFI A (%x. B)"

  "SUP x y. B"   == "SUP x. SUP y. B"

  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"

  "SUP x. B"     == "SUP x:CONST UNIV. B"

  "SUP x:A. B"   == "CONST SUPR A (%x. B)"

print_translation {*

  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},

    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]

*} -- {* to avoid eta-contraction of body *}

context complete_lattice

begin

lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"

  by (simp add: SUPR_def cong: image_cong)

lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"

  by (simp add: INFI_def cong: image_cong)

lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)"

  by (auto simp add: SUPR_def intro: Sup_upper)

lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (SUP i:A. M i)"

  using le_SUPI[of i A M] by auto

lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u"

  by (auto simp add: SUPR_def intro: Sup_least)

lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i"

  by (auto simp add: INFI_def intro: Inf_lower)

lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> u"

  using INF_leI[of i A M] by auto

lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"

  by (auto simp add: INFI_def intro: Inf_greatest)

lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"

  unfolding SUPR_def by (auto simp add: Sup_le_iff)

lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"

  unfolding INFI_def by (auto simp add: le_Inf_iff)

lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"

  by (auto intro: antisym INF_leI le_INFI)

lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"

  by (auto intro: antisym SUP_leI le_SUPI)

lemma INF_mono:

  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)"

  by (force intro!: Inf_mono simp: INFI_def)

lemma SUP_mono:

  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (SUP n:A. f n) \<le> (SUP n:B. g n)"

  by (force intro!: Sup_mono simp: SUPR_def)

lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"

  by (intro INF_mono) auto

lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f"

  by (intro SUP_mono) auto

lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"

  by (iprover intro: INF_leI le_INFI order_trans antisym)

lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"

  by (iprover intro: SUP_leI le_SUPI order_trans antisym)

end

lemma Inf_less_iff:

  fixes a :: "'a\<Colon>{complete_lattice,linorder}"

  shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"

  unfolding not_le[symmetric] le_Inf_iff by auto

lemma less_Sup_iff:

  fixes a :: "'a\<Colon>{complete_lattice,linorder}"

  shows "a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"

  unfolding not_le[symmetric] Sup_le_iff by auto

lemma INF_less_iff:

  fixes a :: "'a::{complete_lattice,linorder}"

  shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"

  unfolding INFI_def Inf_less_iff by auto

lemma less_SUP_iff:

  fixes a :: "'a::{complete_lattice,linorder}"

  shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"

  unfolding SUPR_def less_Sup_iff by auto

subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}

instantiation bool :: complete_lattice

begin

definition

  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"

definition

  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"

instance proof

qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)

end

lemma INFI_bool_eq [simp]:

  "INFI = Ball"

proof (rule ext)+

  fix A :: "'a set"

  fix P :: "'a \<Rightarrow> bool"

  show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)"

    by (auto simp add: Ball_def INFI_def Inf_bool_def)

qed

lemma SUPR_bool_eq [simp]:

  "SUPR = Bex"

proof (rule ext)+

  fix A :: "'a set"

  fix P :: "'a \<Rightarrow> bool"

  show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"

    by (auto simp add: Bex_def SUPR_def Sup_bool_def)

qed

instantiation "fun" :: (type, complete_lattice) complete_lattice

begin

definition

  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"

lemma Inf_apply:

  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"

  by (simp add: Inf_fun_def)

definition

  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"

lemma Sup_apply:

  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"

  by (simp add: Sup_fun_def)

instance proof

qed (auto simp add: le_fun_def Inf_apply Sup_apply

  intro: Inf_lower Sup_upper Inf_greatest Sup_least)

end

lemma INFI_apply:

  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"

  by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)

lemma SUPR_apply:

  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"

  by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)

subsection {* Inter *}

abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where

  "Inter S \<equiv> \<Sqinter>S"

notation (xsymbols)

  Inter  ("\<Inter>_" [90] 90)

lemma Inter_eq:

  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"

proof (rule set_eqI)

  fix x

  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"

    by auto

  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"

    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)

qed

lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"

  by (unfold Inter_eq) blast

lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"

  by (simp add: Inter_eq)

text {*

  \medskip A ``destruct'' rule -- every @{term X} in @{term C}

  contains @{term A} as an element, but @{prop "A:X"} can hold when

  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.

*}

lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"

  by auto

lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"

  -- {* ``Classical'' elimination rule -- does not require proving

    @{prop "X:C"}. *}

  by (unfold Inter_eq) blast

lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"

  by (fact Inf_lower)

lemma (in complete_lattice) Inf_less_eq:

  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"

    and "A \<noteq> {}"

  shows "\<Sqinter>A \<le> u"

proof -

  from `A \<noteq> {}` obtain v where "v \<in> A" by blast

  moreover with assms have "v \<sqsubseteq> u" by blast

  ultimately show ?thesis by (rule Inf_lower2)

qed

lemma Inter_subset:

  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"

  by (fact Inf_less_eq)

lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"

  by (fact Inf_greatest)

lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"

  by (fact Inf_binary [symmetric])

lemma Inter_empty [simp]: "\<Inter>{} = UNIV"

  by (fact Inf_empty)

lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"

  by (fact Inf_UNIV)

lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"

  by (fact Inf_insert)

lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"

  by blast

lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"

  by blast

lemma Inter_UNIV_conv [simp,no_atp]:

  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"

  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"

  by blast+

lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"

  by blast

subsection {* Intersections of families *}

abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

  "INTER \<equiv> INFI"

syntax

  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)

  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)

syntax (xsymbols)

  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)

  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)

syntax (latex output)

  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)

  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)

translations

  "INT x y. B"  == "INT x. INT y. B"

  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"

  "INT x. B"    == "INT x:CONST UNIV. B"

  "INT x:A. B"  == "CONST INTER A (%x. B)"

print_translation {*

  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]

*} -- {* to avoid eta-contraction of body *}

lemma INTER_eq_Inter_image:

  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"

  by (fact INFI_def)

lemma Inter_def:

  "\<Inter>S = (\<Inter>x\<in>S. x)"

  by (simp add: INTER_eq_Inter_image image_def)

lemma INTER_def:

  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"

  by (auto simp add: INTER_eq_Inter_image Inter_eq)

lemma Inter_image_eq [simp]:

  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"

  by (rule sym) (fact INTER_eq_Inter_image)

lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"

  by (unfold INTER_def) blast

lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"

  by (unfold INTER_def) blast

lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"

  by auto

lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"

  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}

  by (unfold INTER_def) blast

lemma INT_cong [cong]:

    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"

  by (simp add: INTER_def)

lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"

  by blast

lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"

  by blast

lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"

  by (fact INF_leI)

lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"

  by (fact le_INFI)

lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"

  by blast

lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"

  by blast

lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"

  by (fact le_INF_iff)

lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"

  by blast

lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"

  by blast

lemma INT_insert_distrib:

    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"

  by blast

lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"

  by auto

lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"

  -- {* Look: it has an \emph{existential} quantifier *}

  by blast

lemma INTER_UNIV_conv[simp]:

 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"

 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"

by blast+

lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"

  by (auto intro: bool_induct)

lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"

  by blast

lemma INT_anti_mono:

  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>

    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"

  -- {* The last inclusion is POSITIVE! *}

  by (blast dest: subsetD)

lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"

  by blast

subsection {* Union *}

abbreviation Union :: "'a set set \<Rightarrow> 'a set" where

  "Union S \<equiv> \<Squnion>S"

notation (xsymbols)

  Union  ("\<Union>_" [90] 90)

lemma Union_eq:

  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"

proof (rule set_eqI)

  fix x

  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"

    by auto

  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"

    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)

qed

lemma Union_iff [simp, no_atp]:

  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"

  by (unfold Union_eq) blast

lemma UnionI [intro]:

  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"

  -- {* The order of the premises presupposes that @{term C} is rigid;

    @{term A} may be flexible. *}

  by auto

lemma UnionE [elim!]:

  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"

  by auto

lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"

  by (iprover intro: subsetI UnionI)

lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"

  by (iprover intro: subsetI elim: UnionE dest: subsetD)

lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"

  by blast

lemma Union_empty [simp]: "Union({}) = {}"

  by blast

lemma Union_UNIV [simp]: "Union UNIV = UNIV"

  by blast

lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"

  by blast

lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"

  by blast

lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"

  by blast

lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"

  by blast

lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"

  by blast

lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"

  by blast

lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"

  by blast

lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"

  by blast

lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"

  by blast

subsection {* Unions of families *}

abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

  "UNION \<equiv> SUPR"

syntax

  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)

  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)

syntax (xsymbols)

  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)

  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)

syntax (latex output)

  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)

  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)

translations

  "UN x y. B"   == "UN x. UN y. B"

  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"

  "UN x. B"     == "UN x:CONST UNIV. B"

  "UN x:A. B"   == "CONST UNION A (%x. B)"

text {*

  Note the difference between ordinary xsymbol syntax of indexed

  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})

  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The

  former does not make the index expression a subscript of the

  union/intersection symbol because this leads to problems with nested

  subscripts in Proof General.

*}

print_translation {*

  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]

*} -- {* to avoid eta-contraction of body *}

lemma UNION_eq_Union_image:

  "(\<Union>x\<in>A. B x) = \<Union>(B`A)"

  by (fact SUPR_def)

lemma Union_def:

  "\<Union>S = (\<Union>x\<in>S. x)"

  by (simp add: UNION_eq_Union_image image_def)

lemma UNION_def [no_atp]:

  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"

  by (auto simp add: UNION_eq_Union_image Union_eq)

lemma Union_image_eq [simp]:

  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"

  by (rule sym) (fact UNION_eq_Union_image)

lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"

  by (unfold UNION_def) blast

lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"

  -- {* The order of the premises presupposes that @{term A} is rigid;

    @{term b} may be flexible. *}

  by auto

lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"

  by (unfold UNION_def) blast

lemma UN_cong [cong]:

    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"

  by (simp add: UNION_def)

lemma strong_UN_cong:

    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"

  by (simp add: UNION_def simp_implies_def)

lemma image_eq_UN: "f`A = (UN x:A. {f x})"

  by blast

lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"

  by (fact le_SUPI)

lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"

  by (iprover intro: subsetI elim: UN_E dest: subsetD)

lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"

  by blast

lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"

  by blast

lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"

  by blast

lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"

  by blast

lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"

  by blast

lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"

  by auto

lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"

  by blast

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)"

  by blast

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)"

  by blast

lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"

  by (fact SUP_le_iff)

lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"

  by blast

lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"

  by auto

lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"

  by blast

lemma UNION_empty_conv[simp]:

  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"

  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"

by blast+

lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"

  by blast

lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"

  by blast

lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"

  by blast

lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"

  by (auto simp add: split_if_mem2)

lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"

  by (auto intro: bool_contrapos)

lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"

  by blast

lemma UN_mono:

  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>

    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"

  by (blast dest: subsetD)

lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"

  by blast

lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"

  by blast

lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"

  -- {* NOT suitable for rewriting *}

  by blast

lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"

by blast

subsection {* Distributive laws *}

lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"

  by blast

lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"

  by blast

lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"

  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}

  -- {* Union of a family of unions *}

  by blast

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)"

  -- {* Equivalent version *}

  by blast

lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"

  by blast

lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"

  by blast

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)"

  -- {* Equivalent version *}

  by blast

lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"

  -- {* Halmos, Naive Set Theory, page 35. *}

  by blast

lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"

  by blast

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)"

  by blast

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)"

  by blast

subsection {* Complement *}

lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"

  by blast

lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"

  by blast

subsection {* Miniscoping and maxiscoping *}

text {* \medskip Miniscoping: pushing in quantifiers and big Unions

           and Intersections. *}

lemma UN_simps [simp]:

  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"

  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"

  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"

  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"

  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"

  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"

  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"

  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"

  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"

  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"

  by auto

lemma INT_simps [simp]:

  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"

  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"

  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"

  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"

  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"

  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"

  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"

  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"

  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"

  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"

  by auto

lemma ball_simps [simp,no_atp]:

  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"

  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"

  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"

  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"

  "!!P. (ALL x:{}. P x) = True"

  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"

  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"

  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"

  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"

  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"

  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"

  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"

  by auto

lemma bex_simps [simp,no_atp]:

  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"

  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"

  "!!P. (EX x:{}. P x) = False"

  "!!P. (EX x:UNIV. P x) = (EX x. P x)"

  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"

  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"

  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"

  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"

  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"

  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"

  by auto

lemma ball_conj_distrib:

  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"

  by blast

lemma bex_disj_distrib:

  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"

  by blast

text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}

lemma UN_extend_simps:

  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"

  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"

  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"

  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"

  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"

  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"

  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"

  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"

  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"

  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"

  by auto

lemma INT_extend_simps:

  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"

  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"

  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"

  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"

  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"

  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"

  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"

  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"

  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"

  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"

  by auto

no_notation

  less_eq  (infix "\<sqsubseteq>" 50) and

  less (infix "\<sqsubset>" 50) and

  bot ("\<bottom>") and

  top ("\<top>") and

  inf  (infixl "\<sqinter>" 70) and

  sup  (infixl "\<squnion>" 65) and

  Inf  ("\<Sqinter>_" [900] 900) and

  Sup  ("\<Squnion>_" [900] 900)

no_syntax (xsymbols)

  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)

  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)

  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)

  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)

lemmas mem_simps =

  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff

  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff

  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}

end