# HG changeset patch
# User haftmann
# Date 1236334257 -3600
# Node ID 6c74ef5a349f51eb23311dce44a078250d86b4da
# Parent  d6bffd97d8d56e1f3c3b25530f31dccc3721444a# Parent  8f4d5eaa98789e18e746fbe27659f247d3a7c9ff
merged

diff -r d6bffd97d8d5 -r 6c74ef5a349f NEWS
--- a/NEWS	Fri Mar 06 09:35:43 2009 +0100
+++ b/NEWS	Fri Mar 06 11:10:57 2009 +0100
@@ -220,6 +220,21 @@
 
 *** HOL ***
 
+* Some set operations are now proper qualified constants with authentic syntax.
+INCOMPATIBILITY:
+
+    op Int ~>   Set.Int
+    op Un ~>    Set.Un
+    INTER ~>    Set.INTER
+    UNION ~>    Set.UNION
+    Inter ~>    Set.Inter
+    Union ~>    Set.Union
+    {} ~>       Set.empty
+    UNIV ~>     Set.UNIV
+
+* Class complete_lattice with operations Inf, Sup, INFI, SUPR now in theory
+Set.
+
 * Auxiliary class "itself" has disappeared -- classes without any parameter
 are treated as expected by the 'class' command.
 
@@ -297,7 +312,7 @@
 avoiding strange error messages.
 
 * Simplifier: simproc for let expressions now unfolds if bound variable
-occurs at most one time in let expression body.  INCOMPATIBILITY.
+occurs at most once in let expression body.  INCOMPATIBILITY.
 
 * New classes "top" and "bot" with corresponding operations "top" and "bot"
 in theory Orderings;  instantiation of class "complete_lattice" requires
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Fun.thy
--- a/src/HOL/Fun.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Fun.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -1,5 +1,4 @@
 (*  Title:      HOL/Fun.thy
-    ID:         $Id$
     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
     Copyright   1994  University of Cambridge
 *)
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Hoare/Hoare.thy
--- a/src/HOL/Hoare/Hoare.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Hoare/Hoare.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -161,9 +161,9 @@
 (* assn_tr' & bexp_tr'*)
 ML{*  
 fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
-  | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
+  | assn_tr' (Const (@{const_name Set.Int}, _) $ (Const ("Collect",_) $ T1) $ 
                                    (Const ("Collect",_) $ T2)) =  
-            Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
+            Syntax.const "Set.Int" $ dest_abstuple T1 $ dest_abstuple T2
   | assn_tr' t = t;
 
 fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Hoare/HoareAbort.thy
--- a/src/HOL/Hoare/HoareAbort.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Hoare/HoareAbort.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -163,9 +163,9 @@
 (* assn_tr' & bexp_tr'*)
 ML{*  
 fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
-  | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
+  | assn_tr' (Const (@{const_name Int},_) $ (Const ("Collect",_) $ T1) $ 
                                    (Const ("Collect",_) $ T2)) =  
-            Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
+            Syntax.const "Set.Int" $ dest_abstuple T1 $ dest_abstuple T2
   | assn_tr' t = t;
 
 fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/HoareParallel/OG_Tran.thy
--- a/src/HOL/HoareParallel/OG_Tran.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/HoareParallel/OG_Tran.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -102,7 +102,7 @@
   "SEM c S \<equiv> \<Union>sem c ` S "
 
 syntax "_Omega" :: "'a com"    ("\<Omega>" 63)
-translations  "\<Omega>" \<rightleftharpoons> "While UNIV UNIV (Basic id)"
+translations  "\<Omega>" \<rightleftharpoons> "While CONST UNIV CONST UNIV (Basic id)"
 
 consts fwhile :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> nat \<Rightarrow> 'a com"
 primrec 
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Lattices.thy
--- a/src/HOL/Lattices.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Lattices.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -5,7 +5,7 @@
 header {* Abstract lattices *}
 
 theory Lattices
-imports Fun
+imports Orderings
 begin
 
 subsection {* Lattices *}
@@ -328,135 +328,6 @@
   min_max.le_infI1 min_max.le_infI2
 
 
-subsection {* Complete lattices *}
-
-class complete_lattice = lattice + bot + top +
-  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
-    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
-  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 Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> 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 \<le> b}"
-  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
-
-lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
-  unfolding Sup_Inf by auto
-
-lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
-  unfolding Inf_Sup by auto
-
-lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
-  by (auto intro: antisym Inf_greatest Inf_lower)
-
-lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
-  by (auto intro: antisym Sup_least Sup_upper)
-
-lemma Inf_singleton [simp]:
-  "\<Sqinter>{a} = a"
-  by (auto intro: antisym Inf_lower Inf_greatest)
-
-lemma Sup_singleton [simp]:
-  "\<Squnion>{a} = a"
-  by (auto intro: antisym Sup_upper Sup_least)
-
-lemma Inf_insert_simp:
-  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
-  by (cases "A = {}") (simp_all, simp add: Inf_insert)
-
-lemma Sup_insert_simp:
-  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
-  by (cases "A = {}") (simp_all, simp add: Sup_insert)
-
-lemma Inf_binary:
-  "\<Sqinter>{a, b} = a \<sqinter> b"
-  by (simp add: Inf_insert_simp)
-
-lemma Sup_binary:
-  "\<Squnion>{a, b} = a \<squnion> b"
-  by (simp add: Sup_insert_simp)
-
-lemma bot_def:
-  "bot = \<Squnion>{}"
-  by (auto intro: antisym Sup_least)
-
-lemma top_def:
-  "top = \<Sqinter>{}"
-  by (auto intro: antisym Inf_greatest)
-
-lemma sup_bot [simp]:
-  "x \<squnion> bot = x"
-  using bot_least [of x] by (simp add: le_iff_sup sup_commute)
-
-lemma inf_top [simp]:
-  "x \<sqinter> top = x"
-  using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
-
-definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
-  "SUPR A f == \<Squnion> (f ` A)"
-
-definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
-  "INFI A f == \<Sqinter> (f ` A)"
-
-end
-
-syntax
-  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
-  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
-  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
-  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
-
-translations
-  "SUP x y. B"   == "SUP x. SUP y. B"
-  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
-  "SUP x. B"     == "SUP x:UNIV. B"
-  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
-  "INF x y. B"   == "INF x. INF y. B"
-  "INF x. B"     == "CONST INFI UNIV (%x. B)"
-  "INF x. B"     == "INF x:UNIV. B"
-  "INF x:A. B"   == "CONST INFI A (%x. B)"
-
-(* To avoid eta-contraction of body: *)
-print_translation {*
-let
-  fun btr' syn (A :: Abs abs :: ts) =
-    let val (x,t) = atomic_abs_tr' abs
-    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
-  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
-in
-[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
-end
-*}
-
-context complete_lattice
-begin
-
-lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
-  by (auto simp add: SUPR_def intro: Sup_upper)
-
-lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
-  by (auto simp add: SUPR_def intro: Sup_least)
-
-lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
-  by (auto simp add: INFI_def intro: Inf_lower)
-
-lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
-  by (auto simp add: INFI_def intro: Inf_greatest)
-
-lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
-  by (auto intro: antisym SUP_leI le_SUPI)
-
-lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
-  by (auto intro: antisym INF_leI le_INFI)
-
-end
-
-
 subsection {* Bool as lattice *}
 
 instantiation bool :: distrib_lattice
@@ -473,28 +344,6 @@
 
 end
 
-instantiation bool :: complete_lattice
-begin
-
-definition
-  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
-
-definition
-  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
-
-instance
-  by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
-
-end
-
-lemma Inf_empty_bool [simp]:
-  "\<Sqinter>{}"
-  unfolding Inf_bool_def by auto
-
-lemma not_Sup_empty_bool [simp]:
-  "\<not> Sup {}"
-  unfolding Sup_bool_def by auto
-
 
 subsection {* Fun as lattice *}
 
@@ -522,85 +371,6 @@
 instance "fun" :: (type, distrib_lattice) distrib_lattice
   by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
 
-instantiation "fun" :: (type, complete_lattice) complete_lattice
-begin
-
-definition
-  Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
-
-definition
-  Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
-
-instance
-  by intro_classes
-    (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
-      intro: Inf_lower Sup_upper Inf_greatest Sup_least)
-
-end
-
-lemma Inf_empty_fun:
-  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
-  by rule (auto simp add: Inf_fun_def)
-
-lemma Sup_empty_fun:
-  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
-  by rule (auto simp add: Sup_fun_def)
-
-
-subsection {* Set as lattice *}
-
-lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
-  apply (rule subset_antisym)
-  apply (rule Int_greatest)
-  apply (rule inf_le1)
-  apply (rule inf_le2)
-  apply (rule inf_greatest)
-  apply (rule Int_lower1)
-  apply (rule Int_lower2)
-  done
-
-lemma sup_set_eq: "A \<squnion> B = A \<union> B"
-  apply (rule subset_antisym)
-  apply (rule sup_least)
-  apply (rule Un_upper1)
-  apply (rule Un_upper2)
-  apply (rule Un_least)
-  apply (rule sup_ge1)
-  apply (rule sup_ge2)
-  done
-
-lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
-  apply (fold inf_set_eq sup_set_eq)
-  apply (erule mono_inf)
-  done
-
-lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
-  apply (fold inf_set_eq sup_set_eq)
-  apply (erule mono_sup)
-  done
-
-lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S"
-  apply (rule subset_antisym)
-  apply (rule Inter_greatest)
-  apply (erule Inf_lower)
-  apply (rule Inf_greatest)
-  apply (erule Inter_lower)
-  done
-
-lemma Sup_set_eq: "\<Squnion>S = \<Union>S"
-  apply (rule subset_antisym)
-  apply (rule Sup_least)
-  apply (erule Union_upper)
-  apply (rule Union_least)
-  apply (erule Sup_upper)
-  done
-  
-lemma top_set_eq: "top = UNIV"
-  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
-
-lemma bot_set_eq: "bot = {}"
-  by (iprover intro!: subset_antisym empty_subsetI bot_least)
-
 
 text {* redundant bindings *}
 
@@ -611,8 +381,6 @@
   less_eq  (infix "\<sqsubseteq>" 50) and
   less (infix "\<sqsubset>" 50) and
   inf  (infixl "\<sqinter>" 70) and
-  sup  (infixl "\<squnion>" 65) and
-  Inf  ("\<Sqinter>_" [900] 900) and
-  Sup  ("\<Squnion>_" [900] 900)
+  sup  (infixl "\<squnion>" 65)
 
 end
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Library/Euclidean_Space.thy
--- a/src/HOL/Library/Euclidean_Space.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Library/Euclidean_Space.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -4199,7 +4199,7 @@
   assumes iB: "independent (B:: (real ^'n) set)"
   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
 proof-
-  from maximal_independent_subset_extend[of B "UNIV"] iB
+  from maximal_independent_subset_extend[of B UNIV] iB
   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
   
   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Library/Executable_Set.thy
--- a/src/HOL/Library/Executable_Set.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Library/Executable_Set.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -178,7 +178,7 @@
 
 lemma iso_insert:
   "set (insertl x xs) = insert x (set xs)"
-  unfolding insertl_def iso_member by (simp add: Set.insert_absorb)
+  unfolding insertl_def iso_member by (simp add: insert_absorb)
 
 lemma iso_remove1:
   assumes distnct: "distinct xs"
@@ -261,7 +261,7 @@
 subsubsection {* const serializations *}
 
 consts_code
-  "{}" ("{*[]*}")
+  "Set.empty" ("{*[]*}")
   insert ("{*insertl*}")
   is_empty ("{*null*}")
   "op \<union>" ("{*unionl*}")
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Library/Finite_Cartesian_Product.thy
--- a/src/HOL/Library/Finite_Cartesian_Product.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Library/Finite_Cartesian_Product.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -15,7 +15,7 @@
 definition "dimindex (S:: 'a set) = (if finite (UNIV::'a set) then card (UNIV:: 'a set) else 1)"
 
 syntax "_type_dimindex" :: "type => nat" ("(1DIM/(1'(_')))")
-translations "DIM(t)" => "CONST dimindex (UNIV :: t set)"
+translations "DIM(t)" => "CONST dimindex (CONST UNIV :: t set)"
 
 lemma dimindex_nonzero: "dimindex S \<noteq>  0"
 unfolding dimindex_def 
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Library/LaTeXsugar.thy
--- a/src/HOL/Library/LaTeXsugar.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Library/LaTeXsugar.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -30,7 +30,7 @@
 
 (* empty set *)
 notation (latex)
-  "{}" ("\<emptyset>")
+  "Set.empty" ("\<emptyset>")
 
 (* insert *)
 translations 
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Nominal/nominal_inductive2.ML
--- a/src/HOL/Nominal/nominal_inductive2.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Nominal/nominal_inductive2.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -75,7 +75,7 @@
   | add_binders thy i (Abs (_, _, t)) bs = add_binders thy (i + 1) t bs
   | add_binders thy i _ bs = bs;
 
-fun mk_set T [] = Const ("{}", HOLogic.mk_setT T)
+fun mk_set T [] = Const (@{const_name Set.empty}, HOLogic.mk_setT T)
   | mk_set T (x :: xs) =
       Const ("insert", T --> HOLogic.mk_setT T --> HOLogic.mk_setT T) $ x $
         mk_set T xs;
@@ -201,7 +201,7 @@
           | add_set (t, T) ((u, U) :: ps) =
               if T = U then
                 let val S = HOLogic.mk_setT T
-                in (Const (@{const_name "op Un"}, S --> S --> S) $ u $ t, T) :: ps
+                in (Const (@{const_name "Un"}, S --> S --> S) $ u $ t, T) :: ps
                 end
               else (u, U) :: add_set (t, T) ps
       in
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Nominal/nominal_package.ML
--- a/src/HOL/Nominal/nominal_package.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Nominal/nominal_package.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -1010,8 +1010,8 @@
             (augment_sort thy8 pt_cp_sort
               (HOLogic.mk_Trueprop (HOLogic.mk_eq
                 (supp c,
-                 if null dts then Const ("{}", HOLogic.mk_setT atomT)
-                 else foldr1 (HOLogic.mk_binop "op Un") (map supp args2)))))
+                 if null dts then Const (@{const_name Set.empty}, HOLogic.mk_setT atomT)
+                 else foldr1 (HOLogic.mk_binop @{const_name Un}) (map supp args2)))))
             (fn _ =>
               simp_tac (HOL_basic_ss addsimps (supp_def ::
                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Set.thy
--- a/src/HOL/Set.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Set.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -1,12 +1,11 @@
 (*  Title:      HOL/Set.thy
-    ID:         $Id$
     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
 *)
 
 header {* Set theory for higher-order logic *}
 
 theory Set
-imports Orderings
+imports Lattices
 begin
 
 text {* A set in HOL is simply a predicate. *}
@@ -19,36 +18,21 @@
 types 'a set = "'a => bool"
 
 consts
-  "{}"          :: "'a set"                             ("{}")
-  UNIV          :: "'a set"
+  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
+  "op :"        :: "'a => 'a set => bool"                -- "membership"
   insert        :: "'a => 'a set => 'a set"
-  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
-  "op Int"      :: "'a set => 'a set => 'a set"          (infixl "Int" 70)
-  "op Un"       :: "'a set => 'a set => 'a set"          (infixl "Un" 65)
-  UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
-  INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
-  Union         :: "'a set set => 'a set"                -- "union of a set"
-  Inter         :: "'a set set => 'a set"                -- "intersection of a set"
-  Pow           :: "'a set => 'a set set"                -- "powerset"
   Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
   Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
   Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
+  Pow           :: "'a set => 'a set set"                -- "powerset"
   image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
-  "op :"        :: "'a => 'a set => bool"                -- "membership"
+
+local
 
 notation
   "op :"  ("op :") and
   "op :"  ("(_/ : _)" [50, 51] 50)
 
-local
-
-
-subsection {* Additional concrete syntax *}
-
-abbreviation
-  range :: "('a => 'b) => 'b set" where -- "of function"
-  "range f == f ` UNIV"
-
 abbreviation
   "not_mem x A == ~ (x : A)" -- "non-membership"
 
@@ -57,32 +41,51 @@
   not_mem  ("(_/ ~: _)" [50, 51] 50)
 
 notation (xsymbols)
-  "op Int"  (infixl "\<inter>" 70) and
-  "op Un"  (infixl "\<union>" 65) and
   "op :"  ("op \<in>") and
   "op :"  ("(_/ \<in> _)" [50, 51] 50) and
   not_mem  ("op \<notin>") and
-  not_mem  ("(_/ \<notin> _)" [50, 51] 50) and
-  Union  ("\<Union>_" [90] 90) and
-  Inter  ("\<Inter>_" [90] 90)
+  not_mem  ("(_/ \<notin> _)" [50, 51] 50)
 
 notation (HTML output)
-  "op Int"  (infixl "\<inter>" 70) and
-  "op Un"  (infixl "\<union>" 65) and
   "op :"  ("op \<in>") and
   "op :"  ("(_/ \<in> _)" [50, 51] 50) and
   not_mem  ("op \<notin>") and
   not_mem  ("(_/ \<notin> _)" [50, 51] 50)
 
 syntax
+  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
+
+translations
+  "{x. P}"      == "Collect (%x. P)"
+
+definition empty :: "'a set" ("{}") where
+  "empty \<equiv> {x. False}"
+
+definition UNIV :: "'a set" where
+  "UNIV \<equiv> {x. True}"
+
+syntax
   "@Finset"     :: "args => 'a set"                       ("{(_)}")
-  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
-  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
-  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
-  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
-  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
-  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
-  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
+
+translations
+  "{x, xs}"     == "insert x {xs}"
+  "{x}"         == "insert x {}"
+
+definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
+  "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"
+
+definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
+  "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"
+
+notation (xsymbols)
+  "Int"  (infixl "\<inter>" 70) and
+  "Un"  (infixl "\<union>" 65)
+
+notation (HTML output)
+  "Int"  (infixl "\<inter>" 70) and
+  "Un"  (infixl "\<union>" 65)
+
+syntax
   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
@@ -93,24 +96,6 @@
   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
 
-translations
-  "{x, xs}"     == "insert x {xs}"
-  "{x}"         == "insert x {}"
-  "{x. P}"      == "Collect (%x. P)"
-  "{x:A. P}"    => "{x. x:A & P}"
-  "UN x y. B"   == "UN x. UN y. B"
-  "UN x. B"     == "UNION UNIV (%x. B)"
-  "UN x. B"     == "UN x:UNIV. B"
-  "INT x y. B"  == "INT x. INT y. B"
-  "INT x. B"    == "INTER UNIV (%x. B)"
-  "INT x. B"    == "INT x:UNIV. B"
-  "UN x:A. B"   == "UNION A (%x. B)"
-  "INT x:A. B"  == "INTER A (%x. B)"
-  "ALL x:A. P"  == "Ball A (%x. P)"
-  "EX x:A. P"   == "Bex A (%x. P)"
-  "EX! x:A. P"  == "Bex1 A (%x. P)"
-  "LEAST x:A. P" => "LEAST x. x:A & P"
-
 syntax (xsymbols)
   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
@@ -122,26 +107,71 @@
   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
 
+translations
+  "ALL x:A. P"  == "Ball A (%x. P)"
+  "EX x:A. P"   == "Bex A (%x. P)"
+  "EX! x:A. P"  == "Bex1 A (%x. P)"
+  "LEAST x:A. P" => "LEAST x. x:A & P"
+
+definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+  "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"
+
+definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+  "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"
+
+definition Inter :: "'a set set \<Rightarrow> 'a set" where
+  "Inter S \<equiv> INTER S (\<lambda>x. x)"
+
+definition Union :: "'a set set \<Rightarrow> 'a set" where
+  "Union S \<equiv> UNION S (\<lambda>x. x)"
+
+notation (xsymbols)
+  Inter  ("\<Inter>_" [90] 90) and
+  Union  ("\<Union>_" [90] 90)
+
+
+subsection {* Additional concrete syntax *}
+
+syntax
+  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
+  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
+  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
+  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
+  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
+  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
+
 syntax (xsymbols)
   "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
+  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
-  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
+  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
-  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
 
 syntax (latex output)
+  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
-  "@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, 10] 10)
   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
-  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
-
-text{*
+
+translations
+  "{x:A. P}"    => "{x. x:A & P}"
+  "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)"
+  "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. *}
+  subscripts in Proof General.
+*}
 
 abbreviation
   subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
@@ -183,6 +213,10 @@
   supset_eq  ("op \<supseteq>") and
   supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)
 
+abbreviation
+  range :: "('a => 'b) => 'b set" where -- "of function"
+  "range f == f ` UNIV"
+
 
 subsubsection "Bounded quantifiers"
 
@@ -280,12 +314,12 @@
 (* To avoid eta-contraction of body: *)
 print_translation {*
 let
-  fun btr' syn [A,Abs abs] =
-    let val (x,t) = atomic_abs_tr' abs
+  fun btr' syn [A, Abs abs] =
+    let val (x, t) = atomic_abs_tr' abs
     in Syntax.const syn $ x $ A $ t end
 in
-[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"),
- ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")]
+[(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"),
+ (@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")]
 end
 *}
 
@@ -373,15 +407,7 @@
 end
 
 defs
-  Un_def:       "A Un B         == {x. x:A | x:B}"
-  Int_def:      "A Int B        == {x. x:A & x:B}"
-  INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
-  UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
-  Inter_def:    "Inter S        == (INT x:S. x)"
-  Union_def:    "Union S        == (UN x:S. x)"
   Pow_def:      "Pow A          == {B. B <= A}"
-  empty_def:    "{}             == {x. False}"
-  UNIV_def:     "UNIV           == {x. True}"
   insert_def:   "insert a B     == {x. x=a} Un B"
   image_def:    "f`A            == {y. EX x:A. y = f(x)}"
 
@@ -1048,12 +1074,12 @@
   to use term-nets to associate patterns with rules.  Also, if a rule fails to
   apply, then the formula should be kept.
   [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
-   ("op Int", [IntD1,IntD2]),
+   ("Int", [IntD1,IntD2]),
    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
  *)
 
 ML {*
-  val mksimps_pairs = [("Ball", @{thms bspec})] @ mksimps_pairs;
+  val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
 *}
 declaration {* fn _ =>
   Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
@@ -2160,9 +2186,7 @@
 
 subsection {* Getting the Contents of a Singleton Set *}
 
-definition
-  contents :: "'a set \<Rightarrow> 'a"
-where
+definition contents :: "'a set \<Rightarrow> 'a" where
   [code del]: "contents X = (THE x. X = {x})"
 
 lemma contents_eq [simp]: "contents {x} = x"
@@ -2215,6 +2239,255 @@
   unfolding vimage_def Collect_def mem_def ..
 
 
+subsection {* Complete lattices *}
+
+notation
+  less_eq  (infix "\<sqsubseteq>" 50) and
+  less (infix "\<sqsubset>" 50) and
+  inf  (infixl "\<sqinter>" 70) and
+  sup  (infixl "\<squnion>" 65)
+
+class complete_lattice = lattice + bot + top +
+  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
+    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
+  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 Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> 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 \<le> b}"
+  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
+
+lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
+  unfolding Sup_Inf by auto
+
+lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
+  unfolding Inf_Sup by auto
+
+lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
+  by (auto intro: antisym Inf_greatest Inf_lower)
+
+lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
+  by (auto intro: antisym Sup_least Sup_upper)
+
+lemma Inf_singleton [simp]:
+  "\<Sqinter>{a} = a"
+  by (auto intro: antisym Inf_lower Inf_greatest)
+
+lemma Sup_singleton [simp]:
+  "\<Squnion>{a} = a"
+  by (auto intro: antisym Sup_upper Sup_least)
+
+lemma Inf_insert_simp:
+  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
+  by (cases "A = {}") (simp_all, simp add: Inf_insert)
+
+lemma Sup_insert_simp:
+  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
+  by (cases "A = {}") (simp_all, simp add: Sup_insert)
+
+lemma Inf_binary:
+  "\<Sqinter>{a, b} = a \<sqinter> b"
+  by (simp add: Inf_insert_simp)
+
+lemma Sup_binary:
+  "\<Squnion>{a, b} = a \<squnion> b"
+  by (simp add: Sup_insert_simp)
+
+lemma bot_def:
+  "bot = \<Squnion>{}"
+  by (auto intro: antisym Sup_least)
+
+lemma top_def:
+  "top = \<Sqinter>{}"
+  by (auto intro: antisym Inf_greatest)
+
+lemma sup_bot [simp]:
+  "x \<squnion> bot = x"
+  using bot_least [of x] by (simp add: le_iff_sup sup_commute)
+
+lemma inf_top [simp]:
+  "x \<sqinter> top = x"
+  using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
+
+definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
+  "SUPR A f == \<Squnion> (f ` A)"
+
+definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
+  "INFI A f == \<Sqinter> (f ` A)"
+
+end
+
+syntax
+  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
+  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
+  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
+  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
+
+translations
+  "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)"
+  "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)"
+
+(* To avoid eta-contraction of body: *)
+print_translation {*
+let
+  fun btr' syn (A :: Abs abs :: ts) =
+    let val (x,t) = atomic_abs_tr' abs
+    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
+  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
+in
+[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
+end
+*}
+
+context complete_lattice
+begin
+
+lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
+  by (auto simp add: SUPR_def intro: Sup_upper)
+
+lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
+  by (auto simp add: SUPR_def intro: Sup_least)
+
+lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
+  by (auto simp add: INFI_def intro: Inf_lower)
+
+lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
+  by (auto simp add: INFI_def intro: Inf_greatest)
+
+lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
+  by (auto intro: antisym SUP_leI le_SUPI)
+
+lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
+  by (auto intro: antisym INF_leI le_INFI)
+
+end
+
+
+subsection {* Bool as complete lattice *}
+
+instantiation bool :: complete_lattice
+begin
+
+definition
+  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
+
+definition
+  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
+
+instance
+  by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
+
+end
+
+lemma Inf_empty_bool [simp]:
+  "\<Sqinter>{}"
+  unfolding Inf_bool_def by auto
+
+lemma not_Sup_empty_bool [simp]:
+  "\<not> Sup {}"
+  unfolding Sup_bool_def by auto
+
+
+subsection {* Fun as complete lattice *}
+
+instantiation "fun" :: (type, complete_lattice) complete_lattice
+begin
+
+definition
+  Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
+
+definition
+  Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
+
+instance
+  by intro_classes
+    (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
+      intro: Inf_lower Sup_upper Inf_greatest Sup_least)
+
+end
+
+lemma Inf_empty_fun:
+  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
+  by rule (auto simp add: Inf_fun_def)
+
+lemma Sup_empty_fun:
+  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
+  by rule (auto simp add: Sup_fun_def)
+
+
+subsection {* Set as lattice *}
+
+lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
+  apply (rule subset_antisym)
+  apply (rule Int_greatest)
+  apply (rule inf_le1)
+  apply (rule inf_le2)
+  apply (rule inf_greatest)
+  apply (rule Int_lower1)
+  apply (rule Int_lower2)
+  done
+
+lemma sup_set_eq: "A \<squnion> B = A \<union> B"
+  apply (rule subset_antisym)
+  apply (rule sup_least)
+  apply (rule Un_upper1)
+  apply (rule Un_upper2)
+  apply (rule Un_least)
+  apply (rule sup_ge1)
+  apply (rule sup_ge2)
+  done
+
+lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
+  apply (fold inf_set_eq sup_set_eq)
+  apply (erule mono_inf)
+  done
+
+lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
+  apply (fold inf_set_eq sup_set_eq)
+  apply (erule mono_sup)
+  done
+
+lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S"
+  apply (rule subset_antisym)
+  apply (rule Inter_greatest)
+  apply (erule Inf_lower)
+  apply (rule Inf_greatest)
+  apply (erule Inter_lower)
+  done
+
+lemma Sup_set_eq: "\<Squnion>S = \<Union>S"
+  apply (rule subset_antisym)
+  apply (rule Sup_least)
+  apply (erule Union_upper)
+  apply (rule Union_least)
+  apply (erule Sup_upper)
+  done
+  
+lemma top_set_eq: "top = UNIV"
+  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
+
+lemma bot_set_eq: "bot = {}"
+  by (iprover intro!: subset_antisym empty_subsetI bot_least)
+
+no_notation
+  less_eq  (infix "\<sqsubseteq>" 50) and
+  less (infix "\<sqsubset>" 50) and
+  inf  (infixl "\<sqinter>" 70) and
+  sup  (infixl "\<squnion>" 65) and
+  Inf  ("\<Sqinter>_" [900] 900) and
+  Sup  ("\<Squnion>_" [900] 900)
+
 
 subsection {* Basic ML bindings *}
 
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/SizeChange/sct.ML
--- a/src/HOL/SizeChange/sct.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/SizeChange/sct.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -266,12 +266,12 @@
 
 fun mk_edge m G n = HOLogic.mk_prod (m, HOLogic.mk_prod (G, n))
 
-fun mk_set T [] = Const ("{}", HOLogic.mk_setT T)
-  | mk_set T (x :: xs) = Const ("insert",
+fun mk_set T [] = Const (@{const_name Set.empty}, HOLogic.mk_setT T)
+  | mk_set T (x :: xs) = Const (@{const_name insert},
       T --> HOLogic.mk_setT T --> HOLogic.mk_setT T) $ x $ mk_set T xs
 
-fun dest_set (Const ("{}", _)) = []
-  | dest_set (Const ("insert", _) $ x $ xs) = x :: dest_set xs
+fun dest_set (Const (@{const_name Set.empty}, _)) = []
+  | dest_set (Const (@{const_name insert}, _) $ x $ xs) = x :: dest_set xs
 
 val in_graph_tac =
     simp_tac (HOL_basic_ss addsimps has_edge_simps) 1
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/Qelim/cooper.ML
--- a/src/HOL/Tools/Qelim/cooper.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/Qelim/cooper.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -428,8 +428,8 @@
    in
     fun provein x S = 
      case term_of S of
-        Const("{}",_) => error "Unexpected error in Cooper please email Amine Chaieb"
-      | Const("insert",_)$y$_ => 
+        Const(@{const_name Set.empty}, _) => error "Unexpected error in Cooper please email Amine Chaieb"
+      | Const(@{const_name insert}, _) $ y $ _ => 
          let val (cy,S') = Thm.dest_binop S
          in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1
          else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2) 
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/Qelim/ferrante_rackoff.ML
--- a/src/HOL/Tools/Qelim/ferrante_rackoff.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/Qelim/ferrante_rackoff.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -99,8 +99,8 @@
    in
     fun provein x S =
      case term_of S of
-        Const("{}",_) => raise CTERM ("provein : not a member!", [S])
-      | Const("insert",_)$y$_ =>
+        Const(@{const_name Set.empty}, _) => raise CTERM ("provein : not a member!", [S])
+      | Const(@{const_name insert}, _) $ y $_ =>
          let val (cy,S') = Thm.dest_binop S
          in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1
          else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2)
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/Qelim/langford.ML
--- a/src/HOL/Tools/Qelim/langford.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/Qelim/langford.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -14,9 +14,9 @@
 val dest_set =
  let 
   fun h acc ct = 
-   case (term_of ct) of
-     Const("{}",_) => acc
-   | Const("insert",_)$_$t => h (Thm.dest_arg1 ct :: acc) (Thm.dest_arg ct)
+   case term_of ct of
+     Const (@{const_name Set.empty}, _) => acc
+   | Const (@{const_name insert}, _) $ _ $ t => h (Thm.dest_arg1 ct :: acc) (Thm.dest_arg ct)
 in h [] end;
 
 fun prove_finite cT u = 
@@ -48,11 +48,11 @@
        in (ne, f) end
 
      val qe = case (term_of L, term_of U) of 
-      (Const("{}",_),_) =>  
+      (Const (@{const_name Set.empty}, _),_) =>  
         let
           val (neU,fU) = proveneF U 
         in simp_rule (transitive ths (dlo_qeth_nolb OF [neU, fU])) end
-    | (_,Const("{}",_)) =>  
+    | (_,Const (@{const_name Set.empty}, _)) =>  
         let
           val (neL,fL) = proveneF L
         in simp_rule (transitive ths (dlo_qeth_noub OF [neL, fL])) end
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/datatype_rep_proofs.ML
--- a/src/HOL/Tools/datatype_rep_proofs.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/datatype_rep_proofs.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -461,7 +461,7 @@
           (if i < length newTs then Const ("True", HOLogic.boolT)
            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
-               Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
+               Const (iso_name, isoT) $ Const (@{const_name UNIV}, HOLogic.mk_setT T)))
       end;
 
     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/function_package/decompose.ML
--- a/src/HOL/Tools/function_package/decompose.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/function_package/decompose.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -30,7 +30,7 @@
           if is_some (Termination.get_chain D c1 c2) then D else
           let
             val goal = HOLogic.mk_eq (HOLogic.mk_binop @{const_name "Relation.rel_comp"} (c1, c2),
-                                      Const (@{const_name "{}"}, fastype_of c1))
+                                      Const (@{const_name Set.empty}, fastype_of c1))
                        |> HOLogic.mk_Trueprop (* "C1 O C2 = {}" *)
 
             val chain = case FundefLib.try_proof (cterm_of thy goal) chain_tac of
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/function_package/fundef_lib.ML
--- a/src/HOL/Tools/function_package/fundef_lib.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/function_package/fundef_lib.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -139,7 +139,7 @@
 
    Here, + can be any binary operation that is AC.
 
-   cn - The name of the binop-constructor (e.g. @{const_name "op Un"})
+   cn - The name of the binop-constructor (e.g. @{const_name Un})
    ac - the AC rewrite rules for cn
    is - the list of indices of the expressions that should become the first part
         (e.g. [0,1,3] in the above example)
@@ -168,8 +168,8 @@
 
 (* instance for unions *)
 fun regroup_union_conv t =
-    regroup_conv (@{const_name "{}"})
-                  @{const_name "op Un"}
+    regroup_conv (@{const_name Set.empty})
+                  @{const_name Un}
        (map (fn t => t RS eq_reflection) (@{thms "Un_ac"} @
                                           @{thms "Un_empty_right"} @
                                           @{thms "Un_empty_left"})) t
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/function_package/lexicographic_order.ML
--- a/src/HOL/Tools/function_package/lexicographic_order.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/function_package/lexicographic_order.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -24,7 +24,7 @@
     let 
         val relT = HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT))
         val mlexT = (domT --> HOLogic.natT) --> relT --> relT
-        fun mk_ms [] = Const (@{const_name "{}"}, relT)
+        fun mk_ms [] = Const (@{const_name Set.empty}, relT)
           | mk_ms (f::fs) = 
             Const (@{const_name "mlex_prod"}, mlexT) $ f $ mk_ms fs
     in
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/function_package/scnp_reconstruct.ML
--- a/src/HOL/Tools/function_package/scnp_reconstruct.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/function_package/scnp_reconstruct.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -142,7 +142,7 @@
 
 val setT = HOLogic.mk_setT
 
-fun mk_set T [] = Const (@{const_name "{}"}, setT T)
+fun mk_set T [] = Const (@{const_name Set.empty}, setT T)
   | mk_set T (x :: xs) =
       Const (@{const_name insert}, T --> setT T --> setT T) $
             x $ mk_set T xs
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/function_package/termination.ML
--- a/src/HOL/Tools/function_package/termination.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/function_package/termination.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -145,7 +145,7 @@
 
 fun mk_sum_skel rel =
   let
-    val cs = FundefLib.dest_binop_list @{const_name "op Un"} rel
+    val cs = FundefLib.dest_binop_list @{const_name Un} rel
     fun collect_pats (Const ("Collect", _) $ Abs (_, _, c)) =
       let
         val (Const ("op &", _) $ (Const ("op =", _) $ _ $ (Const ("Pair", _) $ r $ l)) $ Gam)
@@ -233,7 +233,7 @@
 fun CALLS tac i st =
   if Thm.no_prems st then all_tac st
   else case Thm.term_of (Thm.cprem_of st i) of
-    (_ $ (_ $ rel)) => tac (FundefLib.dest_binop_list @{const_name "op Un"} rel, i) st
+    (_ $ (_ $ rel)) => tac (FundefLib.dest_binop_list @{const_name Un} rel, i) st
   |_ => no_tac st
 
 type ttac = (data -> int -> tactic) -> (data -> int -> tactic) -> data -> int -> tactic
@@ -291,9 +291,9 @@
 
       val relation =
           if null ineqs then
-              Const (@{const_name "{}"}, fastype_of rel)
+              Const (@{const_name Set.empty}, fastype_of rel)
           else
-              foldr1 (HOLogic.mk_binop @{const_name "op Un"}) (map mk_compr ineqs)
+              foldr1 (HOLogic.mk_binop @{const_name Un}) (map mk_compr ineqs)
 
       fun solve_membership_tac i =
           (EVERY' (replicate (i - 2) (rtac @{thm UnI2}))  (* pick the right component of the union *)
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/hologic.ML
--- a/src/HOL/Tools/hologic.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/hologic.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -225,7 +225,7 @@
 fun dest_mem (Const ("op :", _) $ x $ A) = (x, A)
   | dest_mem t = raise TERM ("dest_mem", [t]);
 
-fun mk_UNIV T = Const ("UNIV", mk_setT T);
+fun mk_UNIV T = Const ("Set.UNIV", mk_setT T);
 
 
 (* binary operations and relations *)
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/inductive_set_package.ML
--- a/src/HOL/Tools/inductive_set_package.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/inductive_set_package.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -73,8 +73,8 @@
         in Drule.instantiate' [] (rev (map (SOME o cterm_of thy o Var) vs))
           (p (fold (Logic.all o Var) vs t) f)
         end;
-      fun mkop "op &" T x = SOME (Const ("op Int", T --> T --> T), x)
-        | mkop "op |" T x = SOME (Const ("op Un", T --> T --> T), x)
+      fun mkop "op &" T x = SOME (Const (@{const_name "Int"}, T --> T --> T), x)
+        | mkop "op |" T x = SOME (Const (@{const_name "Un"}, T --> T --> T), x)
         | mkop _ _ _ = NONE;
       fun mk_collect p T t =
         let val U = HOLogic.dest_setT T
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/refute.ML
--- a/src/HOL/Tools/refute.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/refute.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -2094,7 +2094,7 @@
           val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)
           val HOLogic_setT  = HOLogic.mk_setT HOLogic_prodT
           (* Term.term *)
-          val HOLogic_empty_set = Const (@{const_name "{}"}, HOLogic_setT)
+          val HOLogic_empty_set = Const (@{const_name Set.empty}, HOLogic_setT)
           val HOLogic_insert    =
             Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)
         in
@@ -3159,7 +3159,7 @@
         val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)
         val HOLogic_setT  = HOLogic.mk_setT HOLogic_prodT
         (* Term.term *)
-        val HOLogic_empty_set = Const (@{const_name "{}"}, HOLogic_setT)
+        val HOLogic_empty_set = Const (@{const_name Set.empty}, HOLogic_setT)
         val HOLogic_insert    =
           Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)
       in
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/Tools/res_clause.ML
--- a/src/HOL/Tools/res_clause.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/Tools/res_clause.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -98,21 +98,21 @@
 
 (*Provide readable names for the more common symbolic functions*)
 val const_trans_table =
-      Symtab.make [("op =", "equal"),
+      Symtab.make [(@{const_name "op ="}, "equal"),
                    (@{const_name HOL.less_eq}, "lessequals"),
                    (@{const_name HOL.less}, "less"),
-                   ("op &", "and"),
-                   ("op |", "or"),
+                   (@{const_name "op &"}, "and"),
+                   (@{const_name "op |"}, "or"),
                    (@{const_name HOL.plus}, "plus"),
                    (@{const_name HOL.minus}, "minus"),
                    (@{const_name HOL.times}, "times"),
                    (@{const_name Divides.div}, "div"),
                    (@{const_name HOL.divide}, "divide"),
-                   ("op -->", "implies"),
-                   ("{}", "emptyset"),
-                   ("op :", "in"),
-                   ("op Un", "union"),
-                   ("op Int", "inter"),
+                   (@{const_name "op -->"}, "implies"),
+                   (@{const_name Set.empty}, "emptyset"),
+                   (@{const_name "op :"}, "in"),
+                   (@{const_name Un}, "union"),
+                   (@{const_name Int}, "inter"),
                    ("List.append", "append"),
                    ("ATP_Linkup.fequal", "fequal"),
                    ("ATP_Linkup.COMBI", "COMBI"),
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOL/UNITY/Union.thy
--- a/src/HOL/UNITY/Union.thy	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOL/UNITY/Union.thy	Fri Mar 06 11:10:57 2009 +0100
@@ -43,7 +43,7 @@
 translations
   "JN x : A. B"   == "JOIN A (%x. B)"
   "JN x y. B"   == "JN x. JN y. B"
-  "JN x. B"     == "JOIN UNIV (%x. B)"
+  "JN x. B"     == "JOIN CONST UNIV (%x. B)"
 
 syntax (xsymbols)
   SKIP     :: "'a program"                              ("\<bottom>")
diff -r d6bffd97d8d5 -r 6c74ef5a349f src/HOLCF/IOA/meta_theory/ioa_package.ML
--- a/src/HOLCF/IOA/meta_theory/ioa_package.ML	Fri Mar 06 09:35:43 2009 +0100
+++ b/src/HOLCF/IOA/meta_theory/ioa_package.ML	Fri Mar 06 11:10:57 2009 +0100
@@ -159,7 +159,8 @@
  (constructor_head thy atypstr a)::(constructor_head_list thy atypstr r);
 
 (* producing an action set *)
-fun action_set_string thy atyp [] = "{}" |
+(*FIXME*)
+fun action_set_string thy atyp [] = "Set.empty" |
 action_set_string thy atyp [a] = constr_set_string thy atyp (strip_quote a) |
 action_set_string thy atyp (a::r) = (constr_set_string thy atyp (strip_quote a)) ^
          " Un " ^ (action_set_string thy atyp r);