src/HOL/Set.thy
 changeset 30304 d8e4cd2ac2a1 parent 29901 f4b3f8fbf599 child 30352 047f183c43b0
```--- a/src/HOL/Set.thy	Thu Mar 05 08:23:10 2009 +0100
+++ b/src/HOL/Set.thy	Thu Mar 05 08:23:11 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"
+
+lemma Sup_binary:
+  "\<Squnion>{a, b} = a \<squnion> b"
+
+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 *}
```