(* Title: HOL/Lattices.thy
Author: Tobias Nipkow
*)
header {* Abstract lattices *}
theory Lattices
imports Fun
begin
subsection {* Lattices *}
notation
less_eq (infix "\<sqsubseteq>" 50) and
less (infix "\<sqsubset>" 50)
class lower_semilattice = order +
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
class upper_semilattice = order +
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
begin
text {* Dual lattice *}
lemma dual_lattice:
"lower_semilattice (op \<ge>) (op >) sup"
by (rule lower_semilattice.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)
end
class lattice = lower_semilattice + upper_semilattice
subsubsection {* Intro and elim rules*}
context lower_semilattice
begin
lemma le_infI1[intro]:
assumes "a \<sqsubseteq> x"
shows "a \<sqinter> b \<sqsubseteq> x"
proof (rule order_trans)
from assms show "a \<sqsubseteq> x" .
show "a \<sqinter> b \<sqsubseteq> a" by simp
qed
lemmas (in -) [rule del] = le_infI1
lemma le_infI2[intro]:
assumes "b \<sqsubseteq> x"
shows "a \<sqinter> b \<sqsubseteq> x"
proof (rule order_trans)
from assms show "b \<sqsubseteq> x" .
show "a \<sqinter> b \<sqsubseteq> b" by simp
qed
lemmas (in -) [rule del] = le_infI2
lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
by(blast intro: inf_greatest)
lemmas (in -) [rule del] = le_infI
lemma le_infE [elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
by (blast intro: order_trans)
lemmas (in -) [rule del] = le_infE
lemma le_inf_iff [simp]:
"x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
by blast
lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
by (blast intro: antisym dest: eq_iff [THEN iffD1])
lemma mono_inf:
fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice"
shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B"
by (auto simp add: mono_def intro: Lattices.inf_greatest)
end
context upper_semilattice
begin
lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
by (rule order_trans) auto
lemmas (in -) [rule del] = le_supI1
lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
by (rule order_trans) auto
lemmas (in -) [rule del] = le_supI2
lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
by (blast intro: sup_least)
lemmas (in -) [rule del] = le_supI
lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
by (blast intro: order_trans)
lemmas (in -) [rule del] = le_supE
lemma ge_sup_conv[simp]:
"x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
by blast
lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
by (blast intro: antisym dest: eq_iff [THEN iffD1])
lemma mono_sup:
fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice"
shows "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)"
by (auto simp add: mono_def intro: Lattices.sup_least)
end
subsubsection{* Equational laws *}
context lower_semilattice
begin
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
by (blast intro: antisym)
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
by (blast intro: antisym)
lemma inf_idem[simp]: "x \<sqinter> x = x"
by (blast intro: antisym)
lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
by (blast intro: antisym)
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
by (blast intro: antisym)
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
by (blast intro: antisym)
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
by (blast intro: antisym)
lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
end
context upper_semilattice
begin
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
by (blast intro: antisym)
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
by (blast intro: antisym)
lemma sup_idem[simp]: "x \<squnion> x = x"
by (blast intro: antisym)
lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
by (blast intro: antisym)
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
by (blast intro: antisym)
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
by (blast intro: antisym)
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
by (blast intro: antisym)
lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
end
context lattice
begin
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
by (blast intro: antisym sup_ge1 sup_least inf_le1)
lemmas ACI = inf_ACI sup_ACI
lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
text{* Towards distributivity *}
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
by blast
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
by blast
text{* If you have one of them, you have them all. *}
lemma distrib_imp1:
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
proof-
have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
by(simp add:inf_sup_absorb inf_commute)
also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
finally show ?thesis .
qed
lemma distrib_imp2:
assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
proof-
have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
by(simp add:sup_inf_absorb sup_commute)
also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
finally show ?thesis .
qed
(* seems unused *)
lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
by blast
end
subsection {* Distributive lattices *}
class distrib_lattice = lattice +
assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
context distrib_lattice
begin
lemma sup_inf_distrib2:
"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
by(simp add:ACI sup_inf_distrib1)
lemma inf_sup_distrib1:
"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
by(rule distrib_imp2[OF sup_inf_distrib1])
lemma inf_sup_distrib2:
"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
by(simp add:ACI inf_sup_distrib1)
lemmas distrib =
sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
end
subsection {* Uniqueness of inf and sup *}
lemma (in lower_semilattice) inf_unique:
fixes f (infixl "\<triangle>" 70)
assumes le1: "\<And>x y. x \<triangle> y \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y"
and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z"
shows "x \<sqinter> y = x \<triangle> y"
proof (rule antisym)
show "x \<triangle> y \<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
next
have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest)
show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all
qed
lemma (in upper_semilattice) sup_unique:
fixes f (infixl "\<nabla>" 70)
assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y"
and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x"
shows "x \<squnion> y = x \<nabla> y"
proof (rule antisym)
show "x \<squnion> y \<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
next
have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least)
show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all
qed
subsection {* @{const min}/@{const max} on linear orders as
special case of @{const inf}/@{const sup} *}
lemma (in linorder) distrib_lattice_min_max:
"distrib_lattice (op \<le>) (op <) min max"
proof
have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
by (auto simp add: less_le antisym)
fix x y z
show "max x (min y z) = min (max x y) (max x z)"
unfolding min_def max_def
by auto
qed (auto simp add: min_def max_def not_le less_imp_le)
interpretation min_max!: distrib_lattice "op \<le> :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max
by (rule distrib_lattice_min_max)
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (rule ext)+ (auto intro: antisym)
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (rule ext)+ (auto intro: antisym)
lemmas le_maxI1 = min_max.sup_ge1
lemmas le_maxI2 = min_max.sup_ge2
lemmas max_ac = min_max.sup_assoc min_max.sup_commute
mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
lemmas min_ac = min_max.inf_assoc min_max.inf_commute
mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
text {*
Now we have inherited antisymmetry as an intro-rule on all
linear orders. This is a problem because it applies to bool, which is
undesirable.
*}
lemmas [rule del] = min_max.le_infI min_max.le_supI
min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
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
begin
definition
inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
definition
sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
instance
by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
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 *}
instantiation "fun" :: (type, lattice) lattice
begin
definition
inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
definition
sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
instance
apply intro_classes
unfolding inf_fun_eq sup_fun_eq
apply (auto intro: le_funI)
apply (rule le_funI)
apply (auto dest: le_funD)
apply (rule le_funI)
apply (auto dest: le_funD)
done
end
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 *}
lemmas inf_aci = inf_ACI
lemmas sup_aci = sup_ACI
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)
end