Theory Lattices

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theory Lattices
imports Groups

(*  Title:      HOL/Lattices.thy
Author: Tobias Nipkow
*)


header {* Abstract lattices *}

theory Lattices
imports Orderings Groups
begin


subsection {* Abstract semilattice *}

text {*
This locales provide a basic structure for interpretation into
bigger structures; extensions require careful thinking, otherwise
undesired effects may occur due to interpretation.
*}


locale semilattice = abel_semigroup +
assumes idem [simp]: "f a a = a"
begin


lemma left_idem [simp]:
"f a (f a b) = f a b"

by (simp add: assoc [symmetric])

end


subsection {* Idempotent semigroup *}

class ab_semigroup_idem_mult = ab_semigroup_mult +
assumes mult_idem: "x * x = x"


sublocale ab_semigroup_idem_mult < times!: semilattice times proof
qed (fact mult_idem)

context ab_semigroup_idem_mult
begin


lemmas mult_left_idem = times.left_idem

end


subsection {* Concrete lattices *}

notation
less_eq (infix "\<sqsubseteq>" 50) and
less (infix "\<sqsubset>" 50) and
top ("\<top>") and
bot ("⊥")


class semilattice_inf = order +
fixes inf :: "'a => 'a => '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 ==> x \<sqsubseteq> z ==> x \<sqsubseteq> y \<sqinter> z"


class semilattice_sup = order +
fixes sup :: "'a => 'a => '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 ==> z \<sqsubseteq> x ==> y \<squnion> z \<sqsubseteq> x"
begin


text {* Dual lattice *}

lemma dual_semilattice:
"class.semilattice_inf (op ≥) (op >) sup"

by (rule class.semilattice_inf.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)


end

class lattice = semilattice_inf + semilattice_sup


subsubsection {* Intro and elim rules*}

context semilattice_inf
begin


lemma le_infI1:
"a \<sqsubseteq> x ==> a \<sqinter> b \<sqsubseteq> x"

by (rule order_trans) auto

lemma le_infI2:
"b \<sqsubseteq> x ==> a \<sqinter> b \<sqsubseteq> x"

by (rule order_trans) auto

lemma le_infI: "x \<sqsubseteq> a ==> x \<sqsubseteq> b ==> x \<sqsubseteq> a \<sqinter> b"
by (rule inf_greatest) (* FIXME: duplicate lemma *)

lemma le_infE: "x \<sqsubseteq> a \<sqinter> b ==> (x \<sqsubseteq> a ==> x \<sqsubseteq> b ==> P) ==> P"
by (blast intro: order_trans inf_le1 inf_le2)

lemma le_inf_iff [simp]:
"x \<sqsubseteq> y \<sqinter> z <-> x \<sqsubseteq> y ∧ x \<sqsubseteq> z"

by (blast intro: le_infI elim: le_infE)

lemma le_iff_inf:
"x \<sqsubseteq> y <-> x \<sqinter> y = x"

by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1])

lemma inf_mono: "a \<sqsubseteq> c ==> b ≤ d ==> a \<sqinter> b \<sqsubseteq> c \<sqinter> d"
by (fast intro: inf_greatest le_infI1 le_infI2)

lemma mono_inf:
fixes f :: "'a => 'b::semilattice_inf"
shows "mono f ==> f (A \<sqinter> B) \<sqsubseteq> f A \<sqinter> f B"

by (auto simp add: mono_def intro: Lattices.inf_greatest)

end

context semilattice_sup
begin


lemma le_supI1:
"x \<sqsubseteq> a ==> x \<sqsubseteq> a \<squnion> b"

by (rule order_trans) auto

lemma le_supI2:
"x \<sqsubseteq> b ==> x \<sqsubseteq> a \<squnion> b"

by (rule order_trans) auto

lemma le_supI:
"a \<sqsubseteq> x ==> b \<sqsubseteq> x ==> a \<squnion> b \<sqsubseteq> x"

by (rule sup_least) (* FIXME: duplicate lemma *)

lemma le_supE:
"a \<squnion> b \<sqsubseteq> x ==> (a \<sqsubseteq> x ==> b \<sqsubseteq> x ==> P) ==> P"

by (blast intro: order_trans sup_ge1 sup_ge2)

lemma le_sup_iff [simp]:
"x \<squnion> y \<sqsubseteq> z <-> x \<sqsubseteq> z ∧ y \<sqsubseteq> z"

by (blast intro: le_supI elim: le_supE)

lemma le_iff_sup:
"x \<sqsubseteq> y <-> x \<squnion> y = y"

by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1])

lemma sup_mono: "a \<sqsubseteq> c ==> b ≤ d ==> a \<squnion> b \<sqsubseteq> c \<squnion> d"
by (fast intro: sup_least le_supI1 le_supI2)

lemma mono_sup:
fixes f :: "'a => 'b::semilattice_sup"
shows "mono f ==> f A \<squnion> f B \<sqsubseteq> f (A \<squnion> B)"

by (auto simp add: mono_def intro: Lattices.sup_least)

end


subsubsection {* Equational laws *}

sublocale semilattice_inf < inf!: semilattice inf
proof
fix a b c
show "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
by (rule antisym) (auto intro: le_infI1 le_infI2)
show "a \<sqinter> b = b \<sqinter> a"
by (rule antisym) auto
show "a \<sqinter> a = a"
by (rule antisym) auto
qed

context semilattice_inf
begin


lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
by (fact inf.assoc)

lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
by (fact inf.commute)

lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
by (fact inf.left_commute)

lemma inf_idem: "x \<sqinter> x = x"
by (fact inf.idem)

lemma inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
by (fact inf.left_idem)

lemma inf_absorb1: "x \<sqsubseteq> y ==> x \<sqinter> y = x"
by (rule antisym) auto

lemma inf_absorb2: "y \<sqsubseteq> x ==> x \<sqinter> y = y"
by (rule antisym) auto

lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem

end

sublocale semilattice_sup < sup!: semilattice sup
proof
fix a b c
show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
by (rule antisym) (auto intro: le_supI1 le_supI2)
show "a \<squnion> b = b \<squnion> a"
by (rule antisym) auto
show "a \<squnion> a = a"
by (rule antisym) auto
qed

context semilattice_sup
begin


lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
by (fact sup.assoc)

lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
by (fact sup.commute)

lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
by (fact sup.left_commute)

lemma sup_idem: "x \<squnion> x = x"
by (fact sup.idem)

lemma sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
by (fact sup.left_idem)

lemma sup_absorb1: "y \<sqsubseteq> x ==> x \<squnion> y = x"
by (rule antisym) auto

lemma sup_absorb2: "x \<sqsubseteq> y ==> x \<squnion> y = y"
by (rule antisym) auto

lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem

end

context lattice
begin


lemma dual_lattice:
"class.lattice (op ≥) (op >) sup inf"

by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
(unfold_locales, auto)


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 inf_sup_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 (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)

lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)

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 "… = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
also have "… = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
by(simp add:inf_sup_absorb inf_commute)
also have "… = (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 "… = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
also have "… = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
by(simp add:sup_inf_absorb sup_commute)
also have "… = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
finally show ?thesis .
qed

end

subsubsection {* Strict order *}

context semilattice_inf
begin


lemma less_infI1:
"a \<sqsubset> x ==> a \<sqinter> b \<sqsubset> x"

by (auto simp add: less_le inf_absorb1 intro: le_infI1)

lemma less_infI2:
"b \<sqsubset> x ==> a \<sqinter> b \<sqsubset> x"

by (auto simp add: less_le inf_absorb2 intro: le_infI2)

end

context semilattice_sup
begin


lemma less_supI1:
"x \<sqsubset> a ==> x \<sqsubset> a \<squnion> b"

proof -
interpret dual: semilattice_inf "op ≥" "op >" sup
by (fact dual_semilattice)
assume "x \<sqsubset> a"
then show "x \<sqsubset> a \<squnion> b"
by (fact dual.less_infI1)
qed

lemma less_supI2:
"x \<sqsubset> b ==> x \<sqsubset> a \<squnion> b"

proof -
interpret dual: semilattice_inf "op ≥" "op >" sup
by (fact dual_semilattice)
assume "x \<sqsubset> b"
then show "x \<sqsubset> a \<squnion> b"
by (fact dual.less_infI2)
qed

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: inf_sup_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: inf_sup_aci inf_sup_distrib1)

lemma dual_distrib_lattice:
"class.distrib_lattice (op ≥) (op >) sup inf"

by (rule class.distrib_lattice.intro, rule dual_lattice)
(unfold_locales, fact inf_sup_distrib1)


lemmas sup_inf_distrib =
sup_inf_distrib1 sup_inf_distrib2


lemmas inf_sup_distrib =
inf_sup_distrib1 inf_sup_distrib2


lemmas distrib =
sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2


end


subsection {* Bounded lattices and boolean algebras *}

class bounded_lattice_bot = lattice + bot
begin


lemma inf_bot_left [simp]:
"⊥ \<sqinter> x = ⊥"

by (rule inf_absorb1) simp

lemma inf_bot_right [simp]:
"x \<sqinter> ⊥ = ⊥"

by (rule inf_absorb2) simp

lemma sup_bot_left [simp]:
"⊥ \<squnion> x = x"

by (rule sup_absorb2) simp

lemma sup_bot_right [simp]:
"x \<squnion> ⊥ = x"

by (rule sup_absorb1) simp

lemma sup_eq_bot_iff [simp]:
"x \<squnion> y = ⊥ <-> x = ⊥ ∧ y = ⊥"

by (simp add: eq_iff)

end

class bounded_lattice_top = lattice + top
begin


lemma sup_top_left [simp]:
"\<top> \<squnion> x = \<top>"

by (rule sup_absorb1) simp

lemma sup_top_right [simp]:
"x \<squnion> \<top> = \<top>"

by (rule sup_absorb2) simp

lemma inf_top_left [simp]:
"\<top> \<sqinter> x = x"

by (rule inf_absorb2) simp

lemma inf_top_right [simp]:
"x \<sqinter> \<top> = x"

by (rule inf_absorb1) simp

lemma inf_eq_top_iff [simp]:
"x \<sqinter> y = \<top> <-> x = \<top> ∧ y = \<top>"

by (simp add: eq_iff)

end

class bounded_lattice = bounded_lattice_bot + bounded_lattice_top
begin


lemma dual_bounded_lattice:
"class.bounded_lattice (op ≥) (op >) (op \<squnion>) (op \<sqinter>) \<top> ⊥"

by unfold_locales (auto simp add: less_le_not_le)

end

class boolean_algebra = distrib_lattice + bounded_lattice + minus + uminus +
assumes inf_compl_bot: "x \<sqinter> - x = ⊥"
and sup_compl_top: "x \<squnion> - x = \<top>"
assumes diff_eq: "x - y = x \<sqinter> - y"
begin


lemma dual_boolean_algebra:
"class.boolean_algebra (λx y. x \<squnion> - y) uminus (op ≥) (op >) (op \<squnion>) (op \<sqinter>) \<top> ⊥"

by (rule class.boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
(unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)


lemma compl_inf_bot:
"- x \<sqinter> x = ⊥"

by (simp add: inf_commute inf_compl_bot)

lemma compl_sup_top:
"- x \<squnion> x = \<top>"

by (simp add: sup_commute sup_compl_top)

lemma compl_unique:
assumes "x \<sqinter> y = ⊥"
and "x \<squnion> y = \<top>"
shows "- x = y"

proof -
have "(x \<sqinter> - x) \<squnion> (- x \<sqinter> y) = (x \<sqinter> y) \<squnion> (- x \<sqinter> y)"
using inf_compl_bot assms(1) by simp
then have "(- x \<sqinter> x) \<squnion> (- x \<sqinter> y) = (y \<sqinter> x) \<squnion> (y \<sqinter> - x)"
by (simp add: inf_commute)
then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)"
by (simp add: inf_sup_distrib1)
then have "- x \<sqinter> \<top> = y \<sqinter> \<top>"
using sup_compl_top assms(2) by simp
then show "- x = y" by simp
qed

lemma double_compl [simp]:
"- (- x) = x"

using compl_inf_bot compl_sup_top by (rule compl_unique)

lemma compl_eq_compl_iff [simp]:
"- x = - y <-> x = y"

proof
assume "- x = - y"
then have "- (- x) = - (- y)" by (rule arg_cong)
then show "x = y" by simp
next
assume "x = y"
then show "- x = - y" by simp
qed

lemma compl_bot_eq [simp]:
"- ⊥ = \<top>"

proof -
from sup_compl_top have "⊥ \<squnion> - ⊥ = \<top>" .
then show ?thesis by simp
qed

lemma compl_top_eq [simp]:
"- \<top> = ⊥"

proof -
from inf_compl_bot have "\<top> \<sqinter> - \<top> = ⊥" .
then show ?thesis by simp
qed

lemma compl_inf [simp]:
"- (x \<sqinter> y) = - x \<squnion> - y"

proof (rule compl_unique)
have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
by (simp only: inf_sup_distrib inf_aci)
then show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = ⊥"
by (simp add: inf_compl_bot)
next
have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
by (simp only: sup_inf_distrib sup_aci)
then show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = \<top>"
by (simp add: sup_compl_top)
qed

lemma compl_sup [simp]:
"- (x \<squnion> y) = - x \<sqinter> - y"

proof -
interpret boolean_algebra "λx y. x \<squnion> - y" uminus "op ≥" "op >" "op \<squnion>" "op \<sqinter>" \<top>
by (rule dual_boolean_algebra)
then show ?thesis by simp
qed

lemma compl_mono:
"x \<sqsubseteq> y ==> - y \<sqsubseteq> - x"

proof -
assume "x \<sqsubseteq> y"
then have "x \<squnion> y = y" by (simp only: le_iff_sup)
then have "- (x \<squnion> y) = - y" by simp
then have "- x \<sqinter> - y = - y" by simp
then have "- y \<sqinter> - x = - y" by (simp only: inf_commute)
then show "- y \<sqsubseteq> - x" by (simp only: le_iff_inf)
qed

lemma compl_le_compl_iff: (* TODO: declare [simp] ? *)
"- x ≤ - y <-> y ≤ x"

by (auto dest: compl_mono)

end


subsection {* Uniqueness of inf and sup *}

lemma (in semilattice_inf) inf_unique:
fixes f (infixl "\<triangle>" 70)
assumes le1: "!!x y. x \<triangle> y \<sqsubseteq> x" and le2: "!!x y. x \<triangle> y \<sqsubseteq> y"
and greatest: "!!x y z. x \<sqsubseteq> y ==> x \<sqsubseteq> z ==> x \<sqsubseteq> y \<triangle> z"
shows "x \<sqinter> y = x \<triangle> y"

proof (rule antisym)
show "x \<triangle> y \<sqsubseteq> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
next
have leI: "!!x y z. x \<sqsubseteq> y ==> x \<sqsubseteq> z ==> x \<sqsubseteq> y \<triangle> z" by (blast intro: greatest)
show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
qed

lemma (in semilattice_sup) sup_unique:
fixes f (infixl "∇" 70)
assumes ge1 [simp]: "!!x y. x \<sqsubseteq> x ∇ y" and ge2: "!!x y. y \<sqsubseteq> x ∇ y"
and least: "!!x y z. y \<sqsubseteq> x ==> z \<sqsubseteq> x ==> y ∇ z \<sqsubseteq> x"
shows "x \<squnion> y = x ∇ y"

proof (rule antisym)
show "x \<squnion> y \<sqsubseteq> x ∇ y" by (rule le_supI) (rule ge1, rule ge2)
next
have leI: "!!x y z. x \<sqsubseteq> z ==> y \<sqsubseteq> z ==> x ∇ y \<sqsubseteq> z" by (blast intro: least)
show "x ∇ y \<sqsubseteq> 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} *}


sublocale linorder < min_max!: distrib_lattice less_eq less min max
proof
fix x y z
show "max x (min y z) = min (max x y) (max x z)"
by (auto simp add: min_def max_def)
qed (auto simp add: min_def max_def not_le less_imp_le)

lemma inf_min: "inf = (min :: 'a::{semilattice_inf, linorder} => 'a => 'a)"
by (rule ext)+ (auto intro: antisym)

lemma sup_max: "sup = (max :: 'a::{semilattice_sup, linorder} => 'a => 'a)"
by (rule ext)+ (auto intro: antisym)

lemmas le_maxI1 = min_max.sup_ge1
lemmas le_maxI2 = min_max.sup_ge2

lemmas min_ac = min_max.inf_assoc min_max.inf_commute
min_max.inf.left_commute


lemmas max_ac = min_max.sup_assoc min_max.sup_commute
min_max.sup.left_commute



subsection {* Bool as lattice *}

instantiation bool :: boolean_algebra
begin


definition
bool_Compl_def: "uminus = Not"


definition
bool_diff_def: "A - B <-> A ∧ ¬ B"


definition
inf_bool_eq: "P \<sqinter> Q <-> P ∧ Q"


definition
sup_bool_eq: "P \<squnion> Q <-> P ∨ Q"


instance proof
qed (simp_all add: inf_bool_eq sup_bool_eq le_bool_def
bot_bool_eq top_bool_eq bool_Compl_def bool_diff_def, auto)


end

lemma sup_boolI1:
"P ==> P \<squnion> Q"

by (simp add: sup_bool_eq)

lemma sup_boolI2:
"Q ==> P \<squnion> Q"

by (simp add: sup_bool_eq)

lemma sup_boolE:
"P \<squnion> Q ==> (P ==> R) ==> (Q ==> R) ==> R"

by (auto simp add: sup_bool_eq)


subsection {* Fun as lattice *}

instantiation "fun" :: (type, lattice) lattice
begin


definition
inf_fun_eq [code del]: "f \<sqinter> g = (λx. f x \<sqinter> g x)"


definition
sup_fun_eq [code del]: "f \<squnion> g = (λx. f x \<squnion> g x)"


instance proof
qed (simp_all add: le_fun_def inf_fun_eq sup_fun_eq)

end

instance "fun" :: (type, distrib_lattice) distrib_lattice
proof
qed (simp_all add: inf_fun_eq sup_fun_eq sup_inf_distrib1)

instance "fun" :: (type, bounded_lattice) bounded_lattice ..

instantiation "fun" :: (type, uminus) uminus
begin


definition
fun_Compl_def: "- A = (λx. - A x)"


instance ..

end

instantiation "fun" :: (type, minus) minus
begin


definition
fun_diff_def: "A - B = (λx. A x - B x)"


instance ..

end

instance "fun" :: (type, boolean_algebra) boolean_algebra
proof
qed (simp_all add: inf_fun_eq sup_fun_eq bot_fun_eq top_fun_eq fun_Compl_def fun_diff_def
inf_compl_bot sup_compl_top diff_eq)



no_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 ("⊥")


end