src/HOL/Lattices.thy
 author wenzelm Sat, 07 Apr 2012 16:41:59 +0200 changeset 47389 e8552cba702d parent 46884 154dc6ec0041 child 49769 c7c2152322f2 permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
```
(*  Title:      HOL/Lattices.thy
Author:     Tobias Nipkow
*)

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"

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 {* Syntactic infimum and supremum operations *}

class inf =
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)

class sup =
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)

subsection {* Concrete lattices *}

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

class semilattice_inf =  order + inf +
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 semilattice_sup = order + sup +
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_semilattice:
"class.semilattice_inf sup greater_eq greater"
by (rule class.semilattice_inf.intro, rule dual_order)

end

class lattice = semilattice_inf + semilattice_sup

subsubsection {* Intro and elim rules*}

context semilattice_inf
begin

lemma le_infI1:
"a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
by (rule order_trans) auto

lemma le_infI2:
"b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
by (rule order_trans) auto

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

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

lemma le_inf_iff [simp]:
"x \<sqsubseteq> y \<sqinter> z \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
by (blast intro: le_infI elim: le_infE)

lemma le_iff_inf:
"x \<sqsubseteq> y \<longleftrightarrow> x \<sqinter> y = x"
by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1])

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

lemma mono_inf:
fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_inf"
shows "mono f \<Longrightarrow> 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 \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
by (rule order_trans) auto

lemma le_supI2:
"x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
by (rule order_trans) auto

lemma le_supI:
"a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
by (rule sup_least) (* FIXME: duplicate lemma *)

lemma le_supE:
"a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
by (blast intro: order_trans sup_ge1 sup_ge2)

lemma le_sup_iff [simp]:
"x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
by (blast intro: le_supI elim: le_supE)

lemma le_iff_sup:
"x \<sqsubseteq> y \<longleftrightarrow> x \<squnion> y = y"
by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1])

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

lemma mono_sup:
fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_sup"
shows "mono f \<Longrightarrow> 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) (* already simp *)

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

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

lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> 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) (* already simp *)

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

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

lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> 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 sup (op \<ge>) (op >) inf"
by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
(unfold_locales, auto)

lemma inf_sup_absorb [simp]: "x \<sqinter> (x \<squnion> y) = x"
by (blast intro: antisym inf_le1 inf_greatest sup_ge1)

lemma sup_inf_absorb [simp]: "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
also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))"
by (simp add: D inf_commute sup_assoc del: sup_inf_absorb)
also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
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
also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))"
by (simp add: D sup_commute inf_assoc del: inf_sup_absorb)
also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
also have "\<dots> = (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 \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
by (auto simp add: less_le inf_absorb1 intro: le_infI1)

lemma less_infI2:
"b \<sqsubset> x \<Longrightarrow> 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 \<Longrightarrow> x \<sqsubset> a \<squnion> b"
using dual_semilattice
by (rule semilattice_inf.less_infI1)

lemma less_supI2:
"x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
using dual_semilattice
by (rule semilattice_inf.less_infI2)

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)"

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)"

lemma dual_distrib_lattice:
"class.distrib_lattice sup (op \<ge>) (op >) 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]:
"\<bottom> \<sqinter> x = \<bottom>"
by (rule inf_absorb1) simp

lemma inf_bot_right [simp]:
"x \<sqinter> \<bottom> = \<bottom>"
by (rule inf_absorb2) simp

lemma sup_bot_left [simp]:
"\<bottom> \<squnion> x = x"
by (rule sup_absorb2) simp

lemma sup_bot_right [simp]:
"x \<squnion> \<bottom> = x"
by (rule sup_absorb1) simp

lemma sup_eq_bot_iff [simp]:
"x \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"

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> \<longleftrightarrow> x = \<top> \<and> y = \<top>"

end

class bounded_lattice = bounded_lattice_bot + bounded_lattice_top
begin

lemma dual_bounded_lattice:
"class.bounded_lattice sup greater_eq greater inf \<top> \<bottom>"
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 = \<bottom>"
and sup_compl_top: "x \<squnion> - x = \<top>"
assumes diff_eq: "x - y = x \<sqinter> - y"
begin

lemma dual_boolean_algebra:
"class.boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus sup greater_eq greater inf \<top> \<bottom>"
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 [simp]:
"- x \<sqinter> x = \<bottom>"

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

lemma compl_unique:
assumes "x \<sqinter> y = \<bottom>"
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)"
then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)"
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 \<longleftrightarrow> 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]:
"- \<bottom> = \<top>"
proof -
from sup_compl_top have "\<bottom> \<squnion> - \<bottom> = \<top>" .
then show ?thesis by simp
qed

lemma compl_top_eq [simp]:
"- \<top> = \<bottom>"
proof -
from inf_compl_bot have "\<top> \<sqinter> - \<top> = \<bottom>" .
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) = \<bottom>"
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>"
qed

lemma compl_sup [simp]:
"- (x \<squnion> y) = - x \<sqinter> - y"
using dual_boolean_algebra
by (rule boolean_algebra.compl_inf)

lemma compl_mono:
"x \<sqsubseteq> y \<Longrightarrow> - 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 [simp]:
"- x \<sqsubseteq> - y \<longleftrightarrow> y \<sqsubseteq> x"
by (auto dest: compl_mono)

lemma compl_le_swap1:
assumes "y \<sqsubseteq> - x" shows "x \<sqsubseteq> -y"
proof -
from assms have "- (- x) \<sqsubseteq> - y" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed

lemma compl_le_swap2:
assumes "- y \<sqsubseteq> x" shows "- x \<sqsubseteq> y"
proof -
from assms have "- x \<sqsubseteq> - (- y)" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed

lemma compl_less_compl_iff: (* TODO: declare [simp] ? *)
"- x \<sqsubset> - y \<longleftrightarrow> y \<sqsubset> x"

lemma compl_less_swap1:
assumes "y \<sqsubset> - x" shows "x \<sqsubset> - y"
proof -
from assms have "- (- x) \<sqsubset> - y" by (simp only: compl_less_compl_iff)
then show ?thesis by simp
qed

lemma compl_less_swap2:
assumes "- y \<sqsubset> x" shows "- x \<sqsubset> y"
proof -
from assms have "- x \<sqsubset> - (- y)" by (simp only: compl_less_compl_iff)
then show ?thesis by simp
qed

end

subsection {* Uniqueness of inf and sup *}

lemma (in semilattice_inf) inf_unique:
fixes f (infixl "\<triangle>" 70)
assumes le1: "\<And>x y. x \<triangle> y \<sqsubseteq> x" and le2: "\<And>x y. x \<triangle> y \<sqsubseteq> y"
and greatest: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> 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: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> 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 "\<nabla>" 70)
assumes ge1 [simp]: "\<And>x y. x \<sqsubseteq> x \<nabla> y" and ge2: "\<And>x y. y \<sqsubseteq> x \<nabla> y"
and least: "\<And>x y z. y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<nabla> z \<sqsubseteq> x"
shows "x \<squnion> y = x \<nabla> y"
proof (rule antisym)
show "x \<squnion> y \<sqsubseteq> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
next
have leI: "\<And>x y z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<nabla> y \<sqsubseteq> z" by (blast intro: least)
show "x \<nabla> 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 min less_eq less 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 \<Colon> 'a\<Colon>{semilattice_inf, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (rule ext)+ (auto intro: antisym)

lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{semilattice_sup, 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 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 {* Lattice on @{typ bool} *}

instantiation bool :: boolean_algebra
begin

definition
bool_Compl_def [simp]: "uminus = Not"

definition
bool_diff_def [simp]: "A - B \<longleftrightarrow> A \<and> \<not> B"

definition
[simp]: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"

definition
[simp]: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"

instance proof
qed auto

end

lemma sup_boolI1:
"P \<Longrightarrow> P \<squnion> Q"
by simp

lemma sup_boolI2:
"Q \<Longrightarrow> P \<squnion> Q"
by simp

lemma sup_boolE:
"P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
by auto

subsection {* Lattice on @{typ "_ \<Rightarrow> _"} *}

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

definition
"f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"

lemma inf_apply [simp] (* CANDIDATE [code] *):
"(f \<sqinter> g) x = f x \<sqinter> g x"

definition
"f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"

lemma sup_apply [simp] (* CANDIDATE [code] *):
"(f \<squnion> g) x = f x \<squnion> g x"

instance proof

end

instance "fun" :: (type, distrib_lattice) distrib_lattice proof
qed (rule ext, simp add: sup_inf_distrib1)

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

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

definition
fun_Compl_def: "- A = (\<lambda>x. - A x)"

lemma uminus_apply [simp] (* CANDIDATE [code] *):
"(- A) x = - (A x)"

instance ..

end

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

definition
fun_diff_def: "A - B = (\<lambda>x. A x - B x)"

lemma minus_apply [simp] (* CANDIDATE [code] *):
"(A - B) x = A x - B x"

instance ..

end

instance "fun" :: (type, boolean_algebra) boolean_algebra proof
qed (rule ext, simp_all add: inf_compl_bot sup_compl_top diff_eq)+

subsection {* Lattice on unary and binary predicates *}

lemma inf1I: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x"

lemma inf2I: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y"

lemma inf1E: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P"

lemma inf2E: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P"

lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x"

lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y"

lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x"

lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y"

lemma sup1I1: "A x \<Longrightarrow> (A \<squnion> B) x"

lemma sup2I1: "A x y \<Longrightarrow> (A \<squnion> B) x y"

lemma sup1I2: "B x \<Longrightarrow> (A \<squnion> B) x"

lemma sup2I2: "B x y \<Longrightarrow> (A \<squnion> B) x y"

lemma sup1E: "(A \<squnion> B) x \<Longrightarrow> (A x \<Longrightarrow> P) \<Longrightarrow> (B x \<Longrightarrow> P) \<Longrightarrow> P"

lemma sup2E: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P"

text {*
\medskip Classical introduction rule: no commitment to @{text A} vs
@{text B}.
*}

lemma sup1CI: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x"