renamed theory Complete_Lattice to Complete_Lattices, in accordance with Lattices, Orderings etc.

--- a/src/HOL/Complete_Lattice.thy	Sat Sep 10 00:44:25 2011 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1259 +0,0 @@
- (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
-
-header {* Complete lattices *}
-
-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 sup (op \<ge>) (op >) inf \<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)+)
-
-definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
-  INF_def: "INFI A f = \<Sqinter>(f ` A)"
-
-definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
-  SUP_def: "SUPR A f = \<Squnion>(f ` A)"
-
-text {*
-  Note: must use names @{const INFI} and @{const SUPR} here instead of
-  @{text INF} and @{text SUP} to allow the following syntax coexist
-  with the plain constant names.
-*}
-
-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 INF_foundation_dual [no_atp]:
-  "complete_lattice.SUPR Inf = INFI"
-proof (rule ext)+
-  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
-    by (fact dual_complete_lattice)
-  fix f :: "'b \<Rightarrow> 'a" and A
-  show "complete_lattice.SUPR Inf A f = (\<Sqinter>a\<in>A. f a)"
-    by (simp only: dual.SUP_def INF_def)
-qed
-
-lemma SUP_foundation_dual [no_atp]:
-  "complete_lattice.INFI Sup = SUPR"
-proof (rule ext)+
-  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
-    by (fact dual_complete_lattice)
-  fix f :: "'b \<Rightarrow> 'a" and A
-  show "complete_lattice.INFI Sup A f = (\<Squnion>a\<in>A. f a)"
-    by (simp only: dual.INF_def SUP_def)
-qed
-
-lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
-  by (auto simp add: INF_def intro: Inf_lower)
-
-lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
-  by (auto simp add: INF_def intro: Inf_greatest)
-
-lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
-  by (auto simp add: SUP_def intro: Sup_upper)
-
-lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
-  by (auto simp add: SUP_def intro: Sup_least)
-
-lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
-  using Inf_lower [of u A] by auto
-
-lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
-  using INF_lower [of i A f] by auto
-
-lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
-  using Sup_upper [of u A] by auto
-
-lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
-  using SUP_upper [of i A f] by auto
-
-lemma le_Inf_iff (*[simp]*): "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
-  by (auto intro: Inf_greatest dest: Inf_lower)
-
-lemma le_INF_iff (*[simp]*): "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
-  by (auto simp add: INF_def le_Inf_iff)
-
-lemma Sup_le_iff (*[simp]*): "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
-  by (auto intro: Sup_least dest: Sup_upper)
-
-lemma SUP_le_iff (*[simp]*): "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
-  by (auto simp add: SUP_def Sup_le_iff)
-
-lemma Inf_empty [simp]:
-  "\<Sqinter>{} = \<top>"
-  by (auto intro: antisym Inf_greatest)
-
-lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
-  by (simp add: INF_def)
-
-lemma Sup_empty [simp]:
-  "\<Squnion>{} = \<bottom>"
-  by (auto intro: antisym Sup_least)
-
-lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
-  by (simp add: SUP_def)
-
-lemma Inf_UNIV [simp]:
-  "\<Sqinter>UNIV = \<bottom>"
-  by (auto intro!: antisym Inf_lower)
-
-lemma Sup_UNIV [simp]:
-  "\<Squnion>UNIV = \<top>"
-  by (auto intro!: antisym Sup_upper)
-
-lemma Inf_insert (*[simp]*): "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
-  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
-
-lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
-  by (simp add: INF_def Inf_insert)
-
-lemma Sup_insert (*[simp]*): "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
-  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
-
-lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
-  by (simp add: SUP_def Sup_insert)
-
-lemma INF_image (*[simp]*): "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
-  by (simp add: INF_def image_image)
-
-lemma SUP_image (*[simp]*): "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
-  by (simp add: SUP_def image_image)
-
-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_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
-  by (auto intro: Inf_greatest Inf_lower)
-
-lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
-  by (auto intro: Sup_least Sup_upper)
-
-lemma INF_cong:
-  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
-  by (simp add: INF_def image_def)
-
-lemma SUP_cong:
-  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
-  by (simp add: SUP_def image_def)
-
-lemma Inf_mono:
-  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
-  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>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 "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
-  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
-qed
-
-lemma INF_mono:
-  "(\<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)"
-  by (force intro!: Inf_mono simp: INF_def)
-
-lemma Sup_mono:
-  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
-  shows "\<Squnion>A \<sqsubseteq> \<Squnion>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> \<Squnion>B" by (rule Sup_upper)
-  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
-qed
-
-lemma SUP_mono:
-  "(\<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)"
-  by (force intro!: Sup_mono simp: SUP_def)
-
-lemma INF_superset_mono:
-  "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)"
-  -- {* The last inclusion is POSITIVE! *}
-  by (blast intro: INF_mono dest: subsetD)
-
-lemma SUP_subset_mono:
-  "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)"
-  by (blast intro: SUP_mono dest: subsetD)
-
-lemma Inf_less_eq:
-  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
-    and "A \<noteq> {}"
-  shows "\<Sqinter>A \<sqsubseteq> 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 less_eq_Sup:
-  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
-    and "A \<noteq> {}"
-  shows "u \<sqsubseteq> \<Squnion>A"
-proof -
-  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
-  moreover with assms have "u \<sqsubseteq> v" by blast
-  ultimately show ?thesis by (rule Sup_upper2)
-qed
-
-lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
-  by (auto intro: Inf_greatest Inf_lower)
-
-lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
-  by (auto intro: Sup_least Sup_upper)
-
-lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
-  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
-
-lemma INF_union:
-  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
-  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
-
-lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
-  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
-
-lemma SUP_union:
-  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
-  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
-
-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)"
-  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
-
-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)"
-  by (rule antisym) (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono,
-    rule SUP_least, auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
-
-lemma Inf_top_conv (*[simp]*) [no_atp]:
-  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
-  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
-proof -
-  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
-  proof
-    assume "\<forall>x\<in>A. x = \<top>"
-    then have "A = {} \<or> A = {\<top>}" by auto
-    then show "\<Sqinter>A = \<top>" by (auto simp add: Inf_insert)
-  next
-    assume "\<Sqinter>A = \<top>"
-    show "\<forall>x\<in>A. x = \<top>"
-    proof (rule ccontr)
-      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
-      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
-      then obtain B where "A = insert x B" by blast
-      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
-    qed
-  qed
-  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
-qed
-
-lemma INF_top_conv (*[simp]*):
- "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
- "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
-  by (auto simp add: INF_def Inf_top_conv)
-
-lemma Sup_bot_conv (*[simp]*) [no_atp]:
-  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
-  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
-proof -
-  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
-    by (fact dual_complete_lattice)
-  from dual.Inf_top_conv show ?P and ?Q by simp_all
-qed
-
-lemma SUP_bot_conv (*[simp]*):
- "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
- "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
-  by (auto simp add: SUP_def Sup_bot_conv)
-
-lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
-  by (auto intro: antisym INF_lower INF_greatest)
-
-lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
-  by (auto intro: antisym SUP_upper SUP_least)
-
-lemma INF_top (*[simp]*): "(\<Sqinter>x\<in>A. \<top>) = \<top>"
-  by (cases "A = {}") (simp_all add: INF_empty)
-
-lemma SUP_bot (*[simp]*): "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
-  by (cases "A = {}") (simp_all add: SUP_empty)
-
-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)"
-  by (iprover intro: INF_lower INF_greatest order_trans antisym)
-
-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)"
-  by (iprover intro: SUP_upper SUP_least order_trans antisym)
-
-lemma INF_absorb:
-  assumes "k \<in> I"
-  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
-proof -
-  from assms obtain J where "I = insert k J" by blast
-  then show ?thesis by (simp add: INF_insert)
-qed
-
-lemma SUP_absorb:
-  assumes "k \<in> I"
-  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
-proof -
-  from assms obtain J where "I = insert k J" by blast
-  then show ?thesis by (simp add: SUP_insert)
-qed
-
-lemma INF_constant:
-  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
-  by (simp add: INF_empty)
-
-lemma SUP_constant:
-  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
-  by (simp add: SUP_empty)
-
-lemma less_INF_D:
-  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
-proof -
-  note `y < (\<Sqinter>i\<in>A. f i)`
-  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
-    by (rule INF_lower)
-  finally show "y < f i" .
-qed
-
-lemma SUP_lessD:
-  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
-proof -
-  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
-    by (rule SUP_upper)
-  also note `(\<Squnion>i\<in>A. f i) < y`
-  finally show "f i < y" .
-qed
-
-lemma INF_UNIV_bool_expand:
-  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
-  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
-
-lemma SUP_UNIV_bool_expand:
-  "(\<Squnion>b. A b) = A True \<squnion> A False"
-  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
-
-end
-
-class complete_distrib_lattice = complete_lattice +
-  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
-  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
-begin
-
-lemma sup_INF:
-  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
-  by (simp add: INF_def sup_Inf image_image)
-
-lemma inf_SUP:
-  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
-  by (simp add: SUP_def inf_Sup image_image)
-
-lemma dual_complete_distrib_lattice:
-  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
-  apply (rule class.complete_distrib_lattice.intro)
-  apply (fact dual_complete_lattice)
-  apply (rule class.complete_distrib_lattice_axioms.intro)
-  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
-  done
-
-subclass distrib_lattice proof
-  fix a b c
-  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
-  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def Inf_insert)
-qed
-
-lemma Inf_sup:
-  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
-  by (simp add: sup_Inf sup_commute)
-
-lemma Sup_inf:
-  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
-  by (simp add: inf_Sup inf_commute)
-
-lemma INF_sup: 
-  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
-  by (simp add: sup_INF sup_commute)
-
-lemma SUP_inf:
-  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
-  by (simp add: inf_SUP inf_commute)
-
-lemma Inf_sup_eq_top_iff:
-  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
-  by (simp only: Inf_sup INF_top_conv)
-
-lemma Sup_inf_eq_bot_iff:
-  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
-  by (simp only: Sup_inf SUP_bot_conv)
-
-lemma INF_sup_distrib2:
-  "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)"
-  by (subst INF_commute) (simp add: sup_INF INF_sup)
-
-lemma SUP_inf_distrib2:
-  "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)"
-  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
-
-end
-
-class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
-begin
-
-lemma dual_complete_boolean_algebra:
-  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
-  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
-
-lemma uminus_Inf:
-  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
-proof (rule antisym)
-  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
-    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
-  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
-    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
-qed
-
-lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
-  by (simp add: INF_def SUP_def uminus_Inf image_image)
-
-lemma uminus_Sup:
-  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
-proof -
-  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
-  then show ?thesis by simp
-qed
-  
-lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
-  by (simp add: INF_def SUP_def uminus_Sup image_image)
-
-end
-
-class complete_linorder = linorder + complete_lattice
-begin
-
-lemma dual_complete_linorder:
-  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
-  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
-
-lemma Inf_less_iff (*[simp]*):
-  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
-  unfolding not_le [symmetric] le_Inf_iff by auto
-
-lemma INF_less_iff (*[simp]*):
-  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
-  unfolding INF_def Inf_less_iff by auto
-
-lemma less_Sup_iff (*[simp]*):
-  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
-  unfolding not_le [symmetric] Sup_le_iff by auto
-
-lemma less_SUP_iff (*[simp]*):
-  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
-  unfolding SUP_def less_Sup_iff by auto
-
-lemma Sup_eq_top_iff (*[simp]*):
-  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
-proof
-  assume *: "\<Squnion>A = \<top>"
-  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
-  proof (intro allI impI)
-    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
-      unfolding less_Sup_iff by auto
-  qed
-next
-  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
-  show "\<Squnion>A = \<top>"
-  proof (rule ccontr)
-    assume "\<Squnion>A \<noteq> \<top>"
-    with top_greatest [of "\<Squnion>A"]
-    have "\<Squnion>A < \<top>" unfolding le_less by auto
-    then have "\<Squnion>A < \<Squnion>A"
-      using * unfolding less_Sup_iff by auto
-    then show False by auto
-  qed
-qed
-
-lemma SUP_eq_top_iff (*[simp]*):
-  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
-  unfolding SUP_def Sup_eq_top_iff by auto
-
-lemma Inf_eq_bot_iff (*[simp]*):
-  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
-proof -
-  interpret dual: complete_linorder Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
-    by (fact dual_complete_linorder)
-  from dual.Sup_eq_top_iff show ?thesis .
-qed
-
-lemma INF_eq_bot_iff (*[simp]*):
-  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
-  unfolding INF_def Inf_eq_bot_iff by auto
-
-end
-
-
-subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
-
-instantiation bool :: complete_lattice
-begin
-
-definition
-  [simp]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
-
-definition
-  [simp]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
-
-instance proof
-qed (auto intro: bool_induct)
-
-end
-
-lemma INF_bool_eq [simp]:
-  "INFI = Ball"
-proof (rule ext)+
-  fix A :: "'a set"
-  fix P :: "'a \<Rightarrow> bool"
-  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
-    by (auto simp add: INF_def)
-qed
-
-lemma SUP_bool_eq [simp]:
-  "SUPR = Bex"
-proof (rule ext)+
-  fix A :: "'a set"
-  fix P :: "'a \<Rightarrow> bool"
-  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
-    by (auto simp add: SUP_def)
-qed
-
-instance bool :: complete_boolean_algebra proof
-qed (auto intro: bool_induct)
-
-instantiation "fun" :: (type, complete_lattice) complete_lattice
-begin
-
-definition
-  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
-
-lemma Inf_apply:
-  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
-  by (simp add: Inf_fun_def)
-
-definition
-  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
-
-lemma Sup_apply:
-  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
-  by (simp add: Sup_fun_def)
-
-instance proof
-qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_lower INF_greatest SUP_upper SUP_least)
-
-end
-
-lemma INF_apply:
-  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
-  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
-
-lemma SUP_apply:
-  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
-  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
-
-instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
-qed (auto simp add: inf_apply sup_apply Inf_apply Sup_apply INF_def SUP_def inf_Sup sup_Inf image_image)
-
-instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
-
-
-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) (simp add: mem_def)
-qed
-
-lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
-  by (unfold Inter_eq) blast
-
-lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<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 \<in> X"} can hold when
-  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
-*}
-
-lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
-  by auto
-
-lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
-  -- {* ``Classical'' elimination rule -- does not require proving
-    @{prop "X \<in> C"}. *}
-  by (unfold Inter_eq) blast
-
-lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
-  by (fact Inf_lower)
-
-lemma Inter_subset:
-  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
-  by (fact Inf_less_eq)
-
-lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
-  by (fact Inf_greatest)
-
-lemma Inter_empty: "\<Inter>{} = UNIV"
-  by (fact Inf_empty) (* already simp *)
-
-lemma Inter_UNIV: "\<Inter>UNIV = {}"
-  by (fact Inf_UNIV) (* already simp *)
-
-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 (fact less_eq_Inf_inter)
-
-lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
-  by (fact Inf_union_distrib)
-
-lemma Inter_UNIV_conv [simp, no_atp]:
-  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
-  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
-  by (fact Inf_top_conv)+
-
-lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
-  by (fact Inf_superset_mono)
-
-
-subsection {* Intersections of families *}
-
-abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
-  "INTER \<equiv> INFI"
-
-text {*
-  Note: must use name @{const INTER} here instead of @{text INT}
-  to allow the following syntax coexist with the plain constant name.
-*}
-
-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>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
-  by (auto simp add: INF_def)
-
-lemma Inter_image_eq [simp]:
-  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
-  by (rule sym) (fact INF_def)
-
-lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
-  by (auto simp add: INF_def image_def)
-
-lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
-  by (auto simp add: INF_def image_def)
-
-lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
-  by auto
-
-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"
-  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
-  by (auto simp add: INF_def image_def)
-
-lemma INT_cong [cong]:
-  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
-  by (fact INF_cong)
-
-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 \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
-  by (fact INF_lower)
-
-lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
-  by (fact INF_greatest)
-
-lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
-  by (fact INF_empty)
-
-lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
-  by (fact INF_absorb)
-
-lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<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 (fact INF_insert)
-
-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 (fact INF_union)
-
-lemma INT_insert_distrib:
-  "u \<in> A \<Longrightarrow> (\<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 (fact INF_constant)
-
-lemma INTER_UNIV_conv [simp]:
- "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
- "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
-  by (fact INF_top_conv)+
-
-lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
-  by (fact INF_UNIV_bool_expand)
-
-lemma INT_anti_mono:
-  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
-  -- {* The last inclusion is POSITIVE! *}
-  by (fact INF_superset_mono)
-
-lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
-  by blast
-
-lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>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) (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 \<Longrightarrow> B \<subseteq> \<Union>A"
-  by (fact Sup_upper)
-
-lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
-  by (fact Sup_least)
-
-lemma Union_empty [simp]: "\<Union>{} = {}"
-  by (fact Sup_empty)
-
-lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
-  by (fact Sup_UNIV)
-
-lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
-  by (fact Sup_insert)
-
-lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
-  by (fact Sup_union_distrib)
-
-lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
-  by (fact Sup_inter_less_eq)
-
-lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
-  by (fact Sup_bot_conv)
-
-lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
-  by (fact Sup_bot_conv)
-
-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 \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
-  by (fact Sup_subset_mono)
-
-
-subsection {* Unions of families *}
-
-abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
-  "UNION \<equiv> SUPR"
-
-text {*
-  Note: must use name @{const UNION} here instead of @{text UN}
-  to allow the following syntax coexist with the plain constant name.
-*}
-
-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 [no_atp]:
-  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
-  by (auto simp add: SUP_def)
-  
-lemma Union_image_eq [simp]:
-  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
-  by (auto simp add: UNION_eq)
-  
-lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
-  by (auto simp add: SUP_def image_def)
-
-lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>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 \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
-  by (auto simp add: SUP_def image_def)
-
-lemma UN_cong [cong]:
-  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
-  by (fact SUP_cong)
-
-lemma strong_UN_cong:
-  "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
-  by (unfold simp_implies_def) (fact UN_cong)
-
-lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
-  by blast
-
-lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
-  by (fact SUP_upper)
-
-lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
-  by (fact SUP_least)
-
-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 \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
-  by blast
-
-lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}"
-  by (fact SUP_empty)
-
-lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
-  by (fact SUP_bot)
-
-lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
-  by (fact SUP_absorb)
-
-lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
-  by (fact SUP_insert)
-
-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 (fact SUP_union)
-
-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 UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
-  by (fact SUP_constant)
-
-lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
-  by blast
-
-lemma UNION_empty_conv[simp]:
-  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
-  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
-  by (fact SUP_bot_conv)+
-
-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) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
-  by blast
-
-lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<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. A b) = (A True \<union> A False)"
-  by (fact SUP_UNIV_bool_expand)
-
-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 \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
-    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
-  by (fact SUP_subset_mono)
-
-lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
-  by blast
-
-lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
-  by blast
-
-lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
-  -- {* NOT suitable for rewriting *}
-  by blast
-
-lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
-  by blast
-
-
-subsection {* Distributive laws *}
-
-lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
-  by (fact inf_Sup)
-
-lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
-  by (fact sup_Inf)
-
-lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
-  by (fact Sup_inf)
-
-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)"
-  by (rule sym) (rule INF_inf_distrib)
-
-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)"
-  by (rule sym) (rule SUP_sup_distrib)
-
-lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
-  by (simp only: INT_Int_distrib INF_def)
-
-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 (simp only: UN_Un_distrib SUP_def)
-
-lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
-  by (fact sup_INF)
-
-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 (fact inf_SUP)
-
-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 (fact SUP_inf_distrib2)
-
-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 (fact INF_sup_distrib2)
-
-lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
-  by (fact Sup_inf_eq_bot_iff)
-
-
-subsection {* Complement *}
-
-lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
-  by (fact uminus_INF)
-
-lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
-  by (fact uminus_SUP)
-
-
-subsection {* Miniscoping and maxiscoping *}
-
-text {* \medskip Miniscoping: pushing in quantifiers and big Unions
-           and Intersections. *}
-
-lemma UN_simps [simp]:
-  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
-  "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
-  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
-  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
-  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
-  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
-  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
-  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
-  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
-  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
-  by auto
-
-lemma INT_simps [simp]:
-  "\<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)"
-  "\<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))"
-  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
-  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
-  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
-  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
-  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
-  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
-  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
-  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
-  by auto
-
-lemma UN_ball_bex_simps [simp, no_atp]:
-  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
-  "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
-  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
-  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
-  by auto
-
-
-text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
-
-lemma UN_extend_simps:
-  "\<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)))"
-  "\<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))"
-  "\<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))"
-  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
-  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
-  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
-  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
-  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
-  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
-  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
-  by auto
-
-lemma INT_extend_simps:
-  "\<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))"
-  "\<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))"
-  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
-  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
-  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
-  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
-  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
-  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
-  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
-  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
-  by auto
-
-
-text {* Legacy names *}
-
-lemma (in complete_lattice) Inf_singleton [simp]:
-  "\<Sqinter>{a} = a"
-  by (simp add: Inf_insert)
-
-lemma (in complete_lattice) Sup_singleton [simp]:
-  "\<Squnion>{a} = a"
-  by (simp add: Sup_insert)
-
-lemma (in complete_lattice) Inf_binary:
-  "\<Sqinter>{a, b} = a \<sqinter> b"
-  by (simp add: Inf_insert)
-
-lemma (in complete_lattice) Sup_binary:
-  "\<Squnion>{a, b} = a \<squnion> b"
-  by (simp add: Sup_insert)
-
-lemmas (in complete_lattice) INFI_def = INF_def
-lemmas (in complete_lattice) SUPR_def = SUP_def
-lemmas (in complete_lattice) INF_leI = INF_lower
-lemmas (in complete_lattice) INF_leI2 = INF_lower2
-lemmas (in complete_lattice) le_INFI = INF_greatest
-lemmas (in complete_lattice) le_SUPI = SUP_upper
-lemmas (in complete_lattice) le_SUPI2 = SUP_upper2
-lemmas (in complete_lattice) SUP_leI = SUP_least
-lemmas (in complete_lattice) less_INFD = less_INF_D
-
-lemmas INFI_apply = INF_apply
-lemmas SUPR_apply = SUP_apply
-
-text {* Grep and put to news from here *}
-
-lemma (in complete_lattice) INF_eq:
-  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
-  by (simp add: INF_def image_def)
-
-lemma (in complete_lattice) SUP_eq:
-  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
-  by (simp add: SUP_def image_def)
-
-lemma (in complete_lattice) INF_subset:
-  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
-  by (rule INF_superset_mono) auto
-
-lemma (in complete_lattice) INF_UNIV_range:
-  "(\<Sqinter>x. f x) = \<Sqinter>range f"
-  by (fact INF_def)
-
-lemma (in complete_lattice) SUP_UNIV_range:
-  "(\<Squnion>x. f x) = \<Squnion>range f"
-  by (fact SUP_def)
-
-lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
-  by (simp add: Inf_insert)
-
-lemma INTER_eq_Inter_image:
-  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
-  by (fact INF_def)
-  
-lemma Inter_def:
-  "\<Inter>S = (\<Inter>x\<in>S. x)"
-  by (simp add: INTER_eq_Inter_image image_def)
-
-lemmas INTER_def = INTER_eq
-lemmas UNION_def = UNION_eq
-
-lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
-  by (fact INF_eq)
-
-lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
-  by blast
-
-lemma UNION_eq_Union_image:
-  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
-  by (fact SUP_def)
-
-lemma Union_def:
-  "\<Union>S = (\<Union>x\<in>S. x)"
-  by (simp add: UNION_eq_Union_image image_def)
-
-lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
-  by blast
-
-lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
-  by (fact SUP_eq)
-
-
-text {* Finally *}
-
-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

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complete_Lattices.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -0,0 +1,1259 @@
+ (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
+
+header {* Complete lattices *}
+
+theory Complete_Lattices
+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 sup (op \<ge>) (op >) inf \<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)+)
+
+definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
+  INF_def: "INFI A f = \<Sqinter>(f ` A)"
+
+definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
+  SUP_def: "SUPR A f = \<Squnion>(f ` A)"
+
+text {*
+  Note: must use names @{const INFI} and @{const SUPR} here instead of
+  @{text INF} and @{text SUP} to allow the following syntax coexist
+  with the plain constant names.
+*}
+
+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 INF_foundation_dual [no_atp]:
+  "complete_lattice.SUPR Inf = INFI"
+proof (rule ext)+
+  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
+    by (fact dual_complete_lattice)
+  fix f :: "'b \<Rightarrow> 'a" and A
+  show "complete_lattice.SUPR Inf A f = (\<Sqinter>a\<in>A. f a)"
+    by (simp only: dual.SUP_def INF_def)
+qed
+
+lemma SUP_foundation_dual [no_atp]:
+  "complete_lattice.INFI Sup = SUPR"
+proof (rule ext)+
+  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
+    by (fact dual_complete_lattice)
+  fix f :: "'b \<Rightarrow> 'a" and A
+  show "complete_lattice.INFI Sup A f = (\<Squnion>a\<in>A. f a)"
+    by (simp only: dual.INF_def SUP_def)
+qed
+
+lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
+  by (auto simp add: INF_def intro: Inf_lower)
+
+lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
+  by (auto simp add: INF_def intro: Inf_greatest)
+
+lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
+  by (auto simp add: SUP_def intro: Sup_upper)
+
+lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
+  by (auto simp add: SUP_def intro: Sup_least)
+
+lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
+  using Inf_lower [of u A] by auto
+
+lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
+  using INF_lower [of i A f] by auto
+
+lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
+  using Sup_upper [of u A] by auto
+
+lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
+  using SUP_upper [of i A f] by auto
+
+lemma le_Inf_iff (*[simp]*): "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
+  by (auto intro: Inf_greatest dest: Inf_lower)
+
+lemma le_INF_iff (*[simp]*): "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
+  by (auto simp add: INF_def le_Inf_iff)
+
+lemma Sup_le_iff (*[simp]*): "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
+  by (auto intro: Sup_least dest: Sup_upper)
+
+lemma SUP_le_iff (*[simp]*): "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
+  by (auto simp add: SUP_def Sup_le_iff)
+
+lemma Inf_empty [simp]:
+  "\<Sqinter>{} = \<top>"
+  by (auto intro: antisym Inf_greatest)
+
+lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
+  by (simp add: INF_def)
+
+lemma Sup_empty [simp]:
+  "\<Squnion>{} = \<bottom>"
+  by (auto intro: antisym Sup_least)
+
+lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
+  by (simp add: SUP_def)
+
+lemma Inf_UNIV [simp]:
+  "\<Sqinter>UNIV = \<bottom>"
+  by (auto intro!: antisym Inf_lower)
+
+lemma Sup_UNIV [simp]:
+  "\<Squnion>UNIV = \<top>"
+  by (auto intro!: antisym Sup_upper)
+
+lemma Inf_insert (*[simp]*): "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
+  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
+
+lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
+  by (simp add: INF_def Inf_insert)
+
+lemma Sup_insert (*[simp]*): "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
+  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
+
+lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
+  by (simp add: SUP_def Sup_insert)
+
+lemma INF_image (*[simp]*): "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
+  by (simp add: INF_def image_image)
+
+lemma SUP_image (*[simp]*): "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
+  by (simp add: SUP_def image_image)
+
+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_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
+  by (auto intro: Inf_greatest Inf_lower)
+
+lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
+  by (auto intro: Sup_least Sup_upper)
+
+lemma INF_cong:
+  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
+  by (simp add: INF_def image_def)
+
+lemma SUP_cong:
+  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
+  by (simp add: SUP_def image_def)
+
+lemma Inf_mono:
+  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
+  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>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 "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
+  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
+qed
+
+lemma INF_mono:
+  "(\<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)"
+  by (force intro!: Inf_mono simp: INF_def)
+
+lemma Sup_mono:
+  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
+  shows "\<Squnion>A \<sqsubseteq> \<Squnion>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> \<Squnion>B" by (rule Sup_upper)
+  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
+qed
+
+lemma SUP_mono:
+  "(\<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)"
+  by (force intro!: Sup_mono simp: SUP_def)
+
+lemma INF_superset_mono:
+  "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)"
+  -- {* The last inclusion is POSITIVE! *}
+  by (blast intro: INF_mono dest: subsetD)
+
+lemma SUP_subset_mono:
+  "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)"
+  by (blast intro: SUP_mono dest: subsetD)
+
+lemma Inf_less_eq:
+  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
+    and "A \<noteq> {}"
+  shows "\<Sqinter>A \<sqsubseteq> 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 less_eq_Sup:
+  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
+    and "A \<noteq> {}"
+  shows "u \<sqsubseteq> \<Squnion>A"
+proof -
+  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
+  moreover with assms have "u \<sqsubseteq> v" by blast
+  ultimately show ?thesis by (rule Sup_upper2)
+qed
+
+lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
+  by (auto intro: Inf_greatest Inf_lower)
+
+lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
+  by (auto intro: Sup_least Sup_upper)
+
+lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
+  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
+
+lemma INF_union:
+  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
+  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
+
+lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
+  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
+
+lemma SUP_union:
+  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
+  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
+
+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)"
+  by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
+
+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)"
+  by (rule antisym) (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono,
+    rule SUP_least, auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
+
+lemma Inf_top_conv (*[simp]*) [no_atp]:
+  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
+  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
+proof -
+  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
+  proof
+    assume "\<forall>x\<in>A. x = \<top>"
+    then have "A = {} \<or> A = {\<top>}" by auto
+    then show "\<Sqinter>A = \<top>" by (auto simp add: Inf_insert)
+  next
+    assume "\<Sqinter>A = \<top>"
+    show "\<forall>x\<in>A. x = \<top>"
+    proof (rule ccontr)
+      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
+      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
+      then obtain B where "A = insert x B" by blast
+      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
+    qed
+  qed
+  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
+qed
+
+lemma INF_top_conv (*[simp]*):
+ "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
+ "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
+  by (auto simp add: INF_def Inf_top_conv)
+
+lemma Sup_bot_conv (*[simp]*) [no_atp]:
+  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
+  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
+proof -
+  interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
+    by (fact dual_complete_lattice)
+  from dual.Inf_top_conv show ?P and ?Q by simp_all
+qed
+
+lemma SUP_bot_conv (*[simp]*):
+ "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
+ "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
+  by (auto simp add: SUP_def Sup_bot_conv)
+
+lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
+  by (auto intro: antisym INF_lower INF_greatest)
+
+lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
+  by (auto intro: antisym SUP_upper SUP_least)
+
+lemma INF_top (*[simp]*): "(\<Sqinter>x\<in>A. \<top>) = \<top>"
+  by (cases "A = {}") (simp_all add: INF_empty)
+
+lemma SUP_bot (*[simp]*): "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
+  by (cases "A = {}") (simp_all add: SUP_empty)
+
+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)"
+  by (iprover intro: INF_lower INF_greatest order_trans antisym)
+
+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)"
+  by (iprover intro: SUP_upper SUP_least order_trans antisym)
+
+lemma INF_absorb:
+  assumes "k \<in> I"
+  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
+proof -
+  from assms obtain J where "I = insert k J" by blast
+  then show ?thesis by (simp add: INF_insert)
+qed
+
+lemma SUP_absorb:
+  assumes "k \<in> I"
+  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
+proof -
+  from assms obtain J where "I = insert k J" by blast
+  then show ?thesis by (simp add: SUP_insert)
+qed
+
+lemma INF_constant:
+  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
+  by (simp add: INF_empty)
+
+lemma SUP_constant:
+  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
+  by (simp add: SUP_empty)
+
+lemma less_INF_D:
+  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
+proof -
+  note `y < (\<Sqinter>i\<in>A. f i)`
+  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
+    by (rule INF_lower)
+  finally show "y < f i" .
+qed
+
+lemma SUP_lessD:
+  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
+proof -
+  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
+    by (rule SUP_upper)
+  also note `(\<Squnion>i\<in>A. f i) < y`
+  finally show "f i < y" .
+qed
+
+lemma INF_UNIV_bool_expand:
+  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
+  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
+
+lemma SUP_UNIV_bool_expand:
+  "(\<Squnion>b. A b) = A True \<squnion> A False"
+  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
+
+end
+
+class complete_distrib_lattice = complete_lattice +
+  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
+  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
+begin
+
+lemma sup_INF:
+  "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
+  by (simp add: INF_def sup_Inf image_image)
+
+lemma inf_SUP:
+  "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
+  by (simp add: SUP_def inf_Sup image_image)
+
+lemma dual_complete_distrib_lattice:
+  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
+  apply (rule class.complete_distrib_lattice.intro)
+  apply (fact dual_complete_lattice)
+  apply (rule class.complete_distrib_lattice_axioms.intro)
+  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
+  done
+
+subclass distrib_lattice proof
+  fix a b c
+  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
+  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def Inf_insert)
+qed
+
+lemma Inf_sup:
+  "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
+  by (simp add: sup_Inf sup_commute)
+
+lemma Sup_inf:
+  "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
+  by (simp add: inf_Sup inf_commute)
+
+lemma INF_sup: 
+  "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
+  by (simp add: sup_INF sup_commute)
+
+lemma SUP_inf:
+  "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
+  by (simp add: inf_SUP inf_commute)
+
+lemma Inf_sup_eq_top_iff:
+  "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
+  by (simp only: Inf_sup INF_top_conv)
+
+lemma Sup_inf_eq_bot_iff:
+  "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
+  by (simp only: Sup_inf SUP_bot_conv)
+
+lemma INF_sup_distrib2:
+  "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)"
+  by (subst INF_commute) (simp add: sup_INF INF_sup)
+
+lemma SUP_inf_distrib2:
+  "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)"
+  by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
+
+end
+
+class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
+begin
+
+lemma dual_complete_boolean_algebra:
+  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
+  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
+
+lemma uminus_Inf:
+  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
+proof (rule antisym)
+  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
+    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
+  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
+    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
+qed
+
+lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
+  by (simp add: INF_def SUP_def uminus_Inf image_image)
+
+lemma uminus_Sup:
+  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
+proof -
+  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
+  then show ?thesis by simp
+qed
+  
+lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
+  by (simp add: INF_def SUP_def uminus_Sup image_image)
+
+end
+
+class complete_linorder = linorder + complete_lattice
+begin
+
+lemma dual_complete_linorder:
+  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
+  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
+
+lemma Inf_less_iff (*[simp]*):
+  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
+  unfolding not_le [symmetric] le_Inf_iff by auto
+
+lemma INF_less_iff (*[simp]*):
+  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
+  unfolding INF_def Inf_less_iff by auto
+
+lemma less_Sup_iff (*[simp]*):
+  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
+  unfolding not_le [symmetric] Sup_le_iff by auto
+
+lemma less_SUP_iff (*[simp]*):
+  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
+  unfolding SUP_def less_Sup_iff by auto
+
+lemma Sup_eq_top_iff (*[simp]*):
+  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
+proof
+  assume *: "\<Squnion>A = \<top>"
+  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
+  proof (intro allI impI)
+    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
+      unfolding less_Sup_iff by auto
+  qed
+next
+  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
+  show "\<Squnion>A = \<top>"
+  proof (rule ccontr)
+    assume "\<Squnion>A \<noteq> \<top>"
+    with top_greatest [of "\<Squnion>A"]
+    have "\<Squnion>A < \<top>" unfolding le_less by auto
+    then have "\<Squnion>A < \<Squnion>A"
+      using * unfolding less_Sup_iff by auto
+    then show False by auto
+  qed
+qed
+
+lemma SUP_eq_top_iff (*[simp]*):
+  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
+  unfolding SUP_def Sup_eq_top_iff by auto
+
+lemma Inf_eq_bot_iff (*[simp]*):
+  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
+proof -
+  interpret dual: complete_linorder Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
+    by (fact dual_complete_linorder)
+  from dual.Sup_eq_top_iff show ?thesis .
+qed
+
+lemma INF_eq_bot_iff (*[simp]*):
+  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
+  unfolding INF_def Inf_eq_bot_iff by auto
+
+end
+
+
+subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
+
+instantiation bool :: complete_lattice
+begin
+
+definition
+  [simp]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
+
+definition
+  [simp]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
+
+instance proof
+qed (auto intro: bool_induct)
+
+end
+
+lemma INF_bool_eq [simp]:
+  "INFI = Ball"
+proof (rule ext)+
+  fix A :: "'a set"
+  fix P :: "'a \<Rightarrow> bool"
+  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
+    by (auto simp add: INF_def)
+qed
+
+lemma SUP_bool_eq [simp]:
+  "SUPR = Bex"
+proof (rule ext)+
+  fix A :: "'a set"
+  fix P :: "'a \<Rightarrow> bool"
+  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
+    by (auto simp add: SUP_def)
+qed
+
+instance bool :: complete_boolean_algebra proof
+qed (auto intro: bool_induct)
+
+instantiation "fun" :: (type, complete_lattice) complete_lattice
+begin
+
+definition
+  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
+
+lemma Inf_apply:
+  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
+  by (simp add: Inf_fun_def)
+
+definition
+  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
+
+lemma Sup_apply:
+  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
+  by (simp add: Sup_fun_def)
+
+instance proof
+qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_lower INF_greatest SUP_upper SUP_least)
+
+end
+
+lemma INF_apply:
+  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
+  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
+
+lemma SUP_apply:
+  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
+  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
+
+instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
+qed (auto simp add: inf_apply sup_apply Inf_apply Sup_apply INF_def SUP_def inf_Sup sup_Inf image_image)
+
+instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
+
+
+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) (simp add: mem_def)
+qed
+
+lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
+  by (unfold Inter_eq) blast
+
+lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<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 \<in> X"} can hold when
+  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
+*}
+
+lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
+  by auto
+
+lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
+  -- {* ``Classical'' elimination rule -- does not require proving
+    @{prop "X \<in> C"}. *}
+  by (unfold Inter_eq) blast
+
+lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
+  by (fact Inf_lower)
+
+lemma Inter_subset:
+  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
+  by (fact Inf_less_eq)
+
+lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
+  by (fact Inf_greatest)
+
+lemma Inter_empty: "\<Inter>{} = UNIV"
+  by (fact Inf_empty) (* already simp *)
+
+lemma Inter_UNIV: "\<Inter>UNIV = {}"
+  by (fact Inf_UNIV) (* already simp *)
+
+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 (fact less_eq_Inf_inter)
+
+lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
+  by (fact Inf_union_distrib)
+
+lemma Inter_UNIV_conv [simp, no_atp]:
+  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
+  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
+  by (fact Inf_top_conv)+
+
+lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
+  by (fact Inf_superset_mono)
+
+
+subsection {* Intersections of families *}
+
+abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+  "INTER \<equiv> INFI"
+
+text {*
+  Note: must use name @{const INTER} here instead of @{text INT}
+  to allow the following syntax coexist with the plain constant name.
+*}
+
+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>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
+  by (auto simp add: INF_def)
+
+lemma Inter_image_eq [simp]:
+  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
+  by (rule sym) (fact INF_def)
+
+lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
+  by (auto simp add: INF_def image_def)
+
+lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
+  by (auto simp add: INF_def image_def)
+
+lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
+  by auto
+
+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"
+  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
+  by (auto simp add: INF_def image_def)
+
+lemma INT_cong [cong]:
+  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
+  by (fact INF_cong)
+
+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 \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
+  by (fact INF_lower)
+
+lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
+  by (fact INF_greatest)
+
+lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
+  by (fact INF_empty)
+
+lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
+  by (fact INF_absorb)
+
+lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<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 (fact INF_insert)
+
+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 (fact INF_union)
+
+lemma INT_insert_distrib:
+  "u \<in> A \<Longrightarrow> (\<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 (fact INF_constant)
+
+lemma INTER_UNIV_conv [simp]:
+ "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
+ "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
+  by (fact INF_top_conv)+
+
+lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
+  by (fact INF_UNIV_bool_expand)
+
+lemma INT_anti_mono:
+  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
+  -- {* The last inclusion is POSITIVE! *}
+  by (fact INF_superset_mono)
+
+lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
+  by blast
+
+lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>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) (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 \<Longrightarrow> B \<subseteq> \<Union>A"
+  by (fact Sup_upper)
+
+lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
+  by (fact Sup_least)
+
+lemma Union_empty [simp]: "\<Union>{} = {}"
+  by (fact Sup_empty)
+
+lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
+  by (fact Sup_UNIV)
+
+lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
+  by (fact Sup_insert)
+
+lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
+  by (fact Sup_union_distrib)
+
+lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
+  by (fact Sup_inter_less_eq)
+
+lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
+  by (fact Sup_bot_conv)
+
+lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
+  by (fact Sup_bot_conv)
+
+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 \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
+  by (fact Sup_subset_mono)
+
+
+subsection {* Unions of families *}
+
+abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+  "UNION \<equiv> SUPR"
+
+text {*
+  Note: must use name @{const UNION} here instead of @{text UN}
+  to allow the following syntax coexist with the plain constant name.
+*}
+
+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 [no_atp]:
+  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
+  by (auto simp add: SUP_def)
+  
+lemma Union_image_eq [simp]:
+  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
+  by (auto simp add: UNION_eq)
+  
+lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
+  by (auto simp add: SUP_def image_def)
+
+lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>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 \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
+  by (auto simp add: SUP_def image_def)
+
+lemma UN_cong [cong]:
+  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
+  by (fact SUP_cong)
+
+lemma strong_UN_cong:
+  "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
+  by (unfold simp_implies_def) (fact UN_cong)
+
+lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
+  by blast
+
+lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
+  by (fact SUP_upper)
+
+lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
+  by (fact SUP_least)
+
+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 \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
+  by blast
+
+lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}"
+  by (fact SUP_empty)
+
+lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
+  by (fact SUP_bot)
+
+lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
+  by (fact SUP_absorb)
+
+lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
+  by (fact SUP_insert)
+
+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 (fact SUP_union)
+
+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 UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
+  by (fact SUP_constant)
+
+lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
+  by blast
+
+lemma UNION_empty_conv[simp]:
+  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
+  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
+  by (fact SUP_bot_conv)+
+
+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) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
+  by blast
+
+lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<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. A b) = (A True \<union> A False)"
+  by (fact SUP_UNIV_bool_expand)
+
+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 \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
+    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
+  by (fact SUP_subset_mono)
+
+lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
+  by blast
+
+lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
+  by blast
+
+lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
+  -- {* NOT suitable for rewriting *}
+  by blast
+
+lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
+  by blast
+
+
+subsection {* Distributive laws *}
+
+lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
+  by (fact inf_Sup)
+
+lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
+  by (fact sup_Inf)
+
+lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
+  by (fact Sup_inf)
+
+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)"
+  by (rule sym) (rule INF_inf_distrib)
+
+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)"
+  by (rule sym) (rule SUP_sup_distrib)
+
+lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
+  by (simp only: INT_Int_distrib INF_def)
+
+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 (simp only: UN_Un_distrib SUP_def)
+
+lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
+  by (fact sup_INF)
+
+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 (fact inf_SUP)
+
+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 (fact SUP_inf_distrib2)
+
+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 (fact INF_sup_distrib2)
+
+lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
+  by (fact Sup_inf_eq_bot_iff)
+
+
+subsection {* Complement *}
+
+lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
+  by (fact uminus_INF)
+
+lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
+  by (fact uminus_SUP)
+
+
+subsection {* Miniscoping and maxiscoping *}
+
+text {* \medskip Miniscoping: pushing in quantifiers and big Unions
+           and Intersections. *}
+
+lemma UN_simps [simp]:
+  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
+  "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
+  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
+  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
+  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
+  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
+  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
+  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
+  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
+  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
+  by auto
+
+lemma INT_simps [simp]:
+  "\<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)"
+  "\<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))"
+  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
+  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
+  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
+  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
+  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
+  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
+  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
+  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
+  by auto
+
+lemma UN_ball_bex_simps [simp, no_atp]:
+  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
+  "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
+  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
+  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
+  by auto
+
+
+text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
+
+lemma UN_extend_simps:
+  "\<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)))"
+  "\<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))"
+  "\<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))"
+  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
+  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
+  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
+  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
+  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
+  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
+  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
+  by auto
+
+lemma INT_extend_simps:
+  "\<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))"
+  "\<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))"
+  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
+  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
+  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
+  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
+  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
+  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
+  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
+  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
+  by auto
+
+
+text {* Legacy names *}
+
+lemma (in complete_lattice) Inf_singleton [simp]:
+  "\<Sqinter>{a} = a"
+  by (simp add: Inf_insert)
+
+lemma (in complete_lattice) Sup_singleton [simp]:
+  "\<Squnion>{a} = a"
+  by (simp add: Sup_insert)
+
+lemma (in complete_lattice) Inf_binary:
+  "\<Sqinter>{a, b} = a \<sqinter> b"
+  by (simp add: Inf_insert)
+
+lemma (in complete_lattice) Sup_binary:
+  "\<Squnion>{a, b} = a \<squnion> b"
+  by (simp add: Sup_insert)
+
+lemmas (in complete_lattice) INFI_def = INF_def
+lemmas (in complete_lattice) SUPR_def = SUP_def
+lemmas (in complete_lattice) INF_leI = INF_lower
+lemmas (in complete_lattice) INF_leI2 = INF_lower2
+lemmas (in complete_lattice) le_INFI = INF_greatest
+lemmas (in complete_lattice) le_SUPI = SUP_upper
+lemmas (in complete_lattice) le_SUPI2 = SUP_upper2
+lemmas (in complete_lattice) SUP_leI = SUP_least
+lemmas (in complete_lattice) less_INFD = less_INF_D
+
+lemmas INFI_apply = INF_apply
+lemmas SUPR_apply = SUP_apply
+
+text {* Grep and put to news from here *}
+
+lemma (in complete_lattice) INF_eq:
+  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
+  by (simp add: INF_def image_def)
+
+lemma (in complete_lattice) SUP_eq:
+  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
+  by (simp add: SUP_def image_def)
+
+lemma (in complete_lattice) INF_subset:
+  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
+  by (rule INF_superset_mono) auto
+
+lemma (in complete_lattice) INF_UNIV_range:
+  "(\<Sqinter>x. f x) = \<Sqinter>range f"
+  by (fact INF_def)
+
+lemma (in complete_lattice) SUP_UNIV_range:
+  "(\<Squnion>x. f x) = \<Squnion>range f"
+  by (fact SUP_def)
+
+lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
+  by (simp add: Inf_insert)
+
+lemma INTER_eq_Inter_image:
+  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
+  by (fact INF_def)
+  
+lemma Inter_def:
+  "\<Inter>S = (\<Inter>x\<in>S. x)"
+  by (simp add: INTER_eq_Inter_image image_def)
+
+lemmas INTER_def = INTER_eq
+lemmas UNION_def = UNION_eq
+
+lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
+  by (fact INF_eq)
+
+lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
+  by blast
+
+lemma UNION_eq_Union_image:
+  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
+  by (fact SUP_def)
+
+lemma Union_def:
+  "\<Union>S = (\<Union>x\<in>S. x)"
+  by (simp add: UNION_eq_Union_image image_def)
+
+lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
+  by blast
+
+lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
+  by (fact SUP_eq)
+
+
+text {* Finally *}
+
+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

--- a/src/HOL/Fun.thy	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Fun.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -6,7 +6,7 @@
 header {* Notions about functions *}
 
 theory Fun
-imports Complete_Lattice
+imports Complete_Lattices
 uses ("Tools/enriched_type.ML")
 begin
 

--- a/src/HOL/Import/Generate-HOLLight/GenHOLLight.thy	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Import/Generate-HOLLight/GenHOLLight.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -146,9 +146,9 @@
   GSPEC > Set.Collect
   SETSPEC > HOLLightCompat.SETSPEC
   UNION > Lattices.sup_class.sup :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
-  UNIONS > Complete_Lattice.Sup_class.Sup :: "(('a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+  UNIONS > Complete_Lattices.Sup_class.Sup :: "(('a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
   INTER > Lattices.inf_class.inf :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
-  INTERS > Complete_Lattice.Inf_class.Inf :: "(('a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+  INTERS > Complete_Lattices.Inf_class.Inf :: "(('a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
   DIFF > Groups.minus_class.minus :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
   SUBSET > Orderings.ord_class.less_eq :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   PSUBSET > Orderings.ord_class.less :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"

--- a/src/HOL/Import/HOLLight/hollight.imp	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Import/HOLLight/hollight.imp	Sat Sep 10 10:29:24 2011 +0200
@@ -266,7 +266,7 @@
   "ZBOT" > "HOLLight.hollight.ZBOT"
   "WF" > "Wellfounded.wfP"
   "UNIV" > "Orderings.top_class.top" :: "'a => bool"
-  "UNIONS" > "Complete_Lattice.Sup_class.Sup" :: "(('a => bool) => bool) => 'a => bool"
+  "UNIONS" > "Complete_Lattices.Sup_class.Sup" :: "(('a => bool) => bool) => 'a => bool"
   "UNION" > "Lattices.sup_class.sup" :: "('a => bool) => ('a => bool) => 'a => bool"
   "UNCURRY" > "HOLLight.hollight.UNCURRY"
   "TL" > "List.tl"
@@ -316,7 +316,7 @@
   "ITLIST2" > "HOLLightList.fold2"
   "ITLIST" > "List.foldr"
   "ISO" > "HOLLight.hollight.ISO"
-  "INTERS" > "Complete_Lattice.Inf_class.Inf" :: "(('a => bool) => bool) => 'a => bool"
+  "INTERS" > "Complete_Lattices.Inf_class.Inf" :: "(('a => bool) => bool) => 'a => bool"
   "INTER" > "Lattices.inf_class.inf" :: "('a => bool) => ('a => bool) => 'a => bool"
   "INSERT" > "Set.insert"
   "INR" > "Sum_Type.Inr"
@@ -584,15 +584,15 @@
   "UNION_COMM" > "Lattices.lattice_class.inf_sup_aci_5"
   "UNION_ASSOC" > "Lattices.lattice_class.inf_sup_aci_6"
   "UNION_ACI" > "HOLLight.hollight.UNION_ACI"
-  "UNIONS_UNION" > "Complete_Lattice.Union_Un_distrib"
+  "UNIONS_UNION" > "Complete_Lattices.Union_Un_distrib"
   "UNIONS_SUBSET" > "HOLLight.hollight.UNIONS_SUBSET"
   "UNIONS_INTERS" > "HOLLight.hollight.UNIONS_INTERS"
-  "UNIONS_INSERT" > "Complete_Lattice.Union_insert"
+  "UNIONS_INSERT" > "Complete_Lattices.Union_insert"
   "UNIONS_IMAGE" > "HOLLight.hollight.UNIONS_IMAGE"
   "UNIONS_GSPEC" > "HOLLight.hollight.UNIONS_GSPEC"
-  "UNIONS_2" > "Complete_Lattice.Un_eq_Union"
-  "UNIONS_1" > "Complete_Lattice.complete_lattice_class.Sup_singleton"
-  "UNIONS_0" > "Complete_Lattice.Union_empty"
+  "UNIONS_2" > "Complete_Lattices.Un_eq_Union"
+  "UNIONS_1" > "Complete_Lattices.complete_lattice_class.Sup_singleton"
+  "UNIONS_0" > "Complete_Lattices.Union_empty"
   "UNCURRY_def" > "HOLLight.hollight.UNCURRY_def"
   "TYDEF_recspace" > "HOLLight.hollight.TYDEF_recspace"
   "TYDEF_real" > "HOLLight.hollight.TYDEF_real"
@@ -779,7 +779,7 @@
   "SUB_0" > "HOLLight.hollight.SUB_0"
   "SUBSET_UNIV" > "Orderings.top_class.top_greatest"
   "SUBSET_UNION_ABSORPTION" > "Lattices.semilattice_sup_class.le_iff_sup"
-  "SUBSET_UNIONS" > "Complete_Lattice.Union_mono"
+  "SUBSET_UNIONS" > "Complete_Lattices.Union_mono"
   "SUBSET_UNION" > "HOLLight.hollight.SUBSET_UNION"
   "SUBSET_TRANS" > "Orderings.order_trans_rules_23"
   "SUBSET_RESTRICT" > "HOLLight.hollight.SUBSET_RESTRICT"
@@ -1544,7 +1544,7 @@
   "ISO_FUN" > "HOLLight.hollight.ISO_FUN"
   "IN_UNIV" > "Set.UNIV_I"
   "IN_UNIONS" > "HOLLight.hollight.IN_UNIONS"
-  "IN_UNION" > "Complete_Lattice.mem_simps_3"
+  "IN_UNION" > "Complete_Lattices.mem_simps_3"
   "IN_SUPPORT" > "HOLLight.hollight.IN_SUPPORT"
   "IN_SING" > "Set.singleton_iff"
   "IN_SET_OF_LIST" > "HOLLightList.IN_SET_OF_LIST"
@@ -1552,13 +1552,13 @@
   "IN_NUMSEG_0" > "HOLLight.hollight.IN_NUMSEG_0"
   "IN_NUMSEG" > "SetInterval.ord_class.atLeastAtMost_iff"
   "IN_INTERS" > "HOLLight.hollight.IN_INTERS"
-  "IN_INTER" > "Complete_Lattice.mem_simps_4"
-  "IN_INSERT" > "Complete_Lattice.mem_simps_1"
+  "IN_INTER" > "Complete_Lattices.mem_simps_4"
+  "IN_INSERT" > "Complete_Lattices.mem_simps_1"
   "IN_IMAGE" > "HOLLight.hollight.IN_IMAGE"
   "IN_ELIM_THM" > "HOLLight.hollight.IN_ELIM_THM"
   "IN_ELIM_PAIR_THM" > "HOLLight.hollight.IN_ELIM_PAIR_THM"
   "IN_DISJOINT" > "HOLLight.hollight.IN_DISJOINT"
-  "IN_DIFF" > "Complete_Lattice.mem_simps_6"
+  "IN_DIFF" > "Complete_Lattices.mem_simps_6"
   "IN_DELETE_EQ" > "HOLLight.hollight.IN_DELETE_EQ"
   "IN_DELETE" > "HOLLight.hollight.IN_DELETE"
   "IN_CROSS" > "HOLLight.hollight.IN_CROSS"
@@ -1594,12 +1594,12 @@
   "INTER_ASSOC" > "Lattices.lattice_class.inf_sup_aci_2"
   "INTER_ACI" > "HOLLight.hollight.INTER_ACI"
   "INTERS_UNIONS" > "HOLLight.hollight.INTERS_UNIONS"
-  "INTERS_INSERT" > "Complete_Lattice.Inter_insert"
+  "INTERS_INSERT" > "Complete_Lattices.Inter_insert"
   "INTERS_IMAGE" > "HOLLight.hollight.INTERS_IMAGE"
   "INTERS_GSPEC" > "HOLLight.hollight.INTERS_GSPEC"
-  "INTERS_2" > "Complete_Lattice.Int_eq_Inter"
-  "INTERS_1" > "Complete_Lattice.complete_lattice_class.Inf_singleton"
-  "INTERS_0" > "Complete_Lattice.Inter_empty"
+  "INTERS_2" > "Complete_Lattices.Int_eq_Inter"
+  "INTERS_1" > "Complete_Lattices.complete_lattice_class.Inf_singleton"
+  "INTERS_0" > "Complete_Lattices.Inter_empty"
   "INSERT_UNIV" > "HOLLight.hollight.INSERT_UNIV"
   "INSERT_UNION_EQ" > "Set.Un_insert_left"
   "INSERT_UNION" > "HOLLight.hollight.INSERT_UNION"

--- a/src/HOL/Inductive.thy	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Inductive.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -5,7 +5,7 @@
 header {* Knaster-Tarski Fixpoint Theorem and inductive definitions *}
 
 theory Inductive 
-imports Complete_Lattice
+imports Complete_Lattices
 uses
   ("Tools/inductive.ML")
   "Tools/dseq.ML"

--- a/src/HOL/IsaMakefile	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/IsaMakefile	Sat Sep 10 10:29:24 2011 +0200
@@ -171,7 +171,7 @@
   $(SRC)/Tools/Metis/metis.ML \
   $(SRC)/Tools/rat.ML \
   ATP.thy \
-  Complete_Lattice.thy \
+  Complete_Lattices.thy \
   Complete_Partial_Order.thy \
   Datatype.thy \
   Extraction.thy \

--- a/src/HOL/Library/Executable_Set.thy	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Library/Executable_Set.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -199,17 +199,17 @@
   by simp
 
 definition Inf :: "'a::complete_lattice set \<Rightarrow> 'a" where
-  [simp]: "Inf = Complete_Lattice.Inf"
+  [simp]: "Inf = Complete_Lattices.Inf"
 
 lemma [code_unfold]:
-  "Complete_Lattice.Inf = Inf"
+  "Complete_Lattices.Inf = Inf"
   by simp
 
 definition Sup :: "'a::complete_lattice set \<Rightarrow> 'a" where
-  [simp]: "Sup = Complete_Lattice.Sup"
+  [simp]: "Sup = Complete_Lattices.Sup"
 
 lemma [code_unfold]:
-  "Complete_Lattice.Sup = Sup"
+  "Complete_Lattices.Sup = Sup"
   by simp
 
 definition Inter :: "'a set set \<Rightarrow> 'a set" where

--- a/src/HOL/Library/Lattice_Syntax.thy	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Library/Lattice_Syntax.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -3,7 +3,7 @@
 header {* Pretty syntax for lattice operations *}
 
 theory Lattice_Syntax
-imports Complete_Lattice
+imports Complete_Lattices
 begin
 
 notation

--- a/src/HOL/Main.thy	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Main.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -21,7 +21,7 @@
   "Sup A \<longleftrightarrow> (\<exists>x\<in>A. x)"
   by auto
 
-declare Complete_Lattice.Inf_bool_def [simp del]
-declare Complete_Lattice.Sup_bool_def [simp del]
+declare Complete_Lattices.Inf_bool_def [simp del]
+declare Complete_Lattices.Sup_bool_def [simp del]
 
 end

--- a/src/HOL/Quotient_Examples/Cset.thy	Sat Sep 10 00:44:25 2011 +0200
+++ b/src/HOL/Quotient_Examples/Cset.thy	Sat Sep 10 10:29:24 2011 +0200
@@ -89,7 +89,7 @@
 quotient_definition Sup where "Sup :: ('a :: Sup) Cset.set \<Rightarrow> 'a"
 is "Sup_class.Sup :: ('a :: Sup) set \<Rightarrow> 'a"
 quotient_definition UNION where "UNION :: 'a Cset.set \<Rightarrow> ('a \<Rightarrow> 'b Cset.set) \<Rightarrow> 'b Cset.set"
-is "Complete_Lattice.UNION :: 'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
+is "Complete_Lattices.UNION :: 'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
 
 hide_const (open) is_empty insert remove map filter forall exists card
   set subset psubset inter union empty UNIV uminus minus Inf Sup UNION