# Theory Lattice_Constructions

```(*  Title:      HOL/Library/Lattice_Constructions.thy
Author:     Lukas Bulwahn
*)

section ‹Values extended by a bottom element›

theory Lattice_Constructions
imports Main
begin

datatype 'a bot = Value 'a | Bot

instantiation bot :: (preorder) preorder
begin

definition less_eq_bot where
"x ≤ y ⟷ (case x of Bot ⇒ True | Value x ⇒ (case y of Bot ⇒ False | Value y ⇒ x ≤ y))"

definition less_bot where
"x < y ⟷ (case y of Bot ⇒ False | Value y ⇒ (case x of Bot ⇒ True | Value x ⇒ x < y))"

lemma less_eq_bot_Bot [simp]: "Bot ≤ x"

lemma less_eq_bot_Bot_code [code]: "Bot ≤ x ⟷ True"
by simp

lemma less_eq_bot_Bot_is_Bot: "x ≤ Bot ⟹ x = Bot"
by (cases x) (simp_all add: less_eq_bot_def)

lemma less_eq_bot_Value_Bot [simp, code]: "Value x ≤ Bot ⟷ False"

lemma less_eq_bot_Value [simp, code]: "Value x ≤ Value y ⟷ x ≤ y"

lemma less_bot_Bot [simp, code]: "x < Bot ⟷ False"

lemma less_bot_Bot_is_Value: "Bot < x ⟹ ∃z. x = Value z"
by (cases x) (simp_all add: less_bot_def)

lemma less_bot_Bot_Value [simp]: "Bot < Value x"

lemma less_bot_Bot_Value_code [code]: "Bot < Value x ⟷ True"
by simp

lemma less_bot_Value [simp, code]: "Value x < Value y ⟷ x < y"

instance
by standard
(auto simp add: less_eq_bot_def less_bot_def less_le_not_le elim: order_trans split: bot.splits)

end

instance bot :: (order) order
by standard (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)

instance bot :: (linorder) linorder
by standard (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)

instantiation bot :: (order) bot
begin
definition "bot = Bot"
instance ..
end

instantiation bot :: (top) top
begin
definition "top = Value top"
instance ..
end

instantiation bot :: (semilattice_inf) semilattice_inf
begin

definition inf_bot
where
"inf x y =
(case x of
Bot ⇒ Bot
| Value v ⇒
(case y of
Bot ⇒ Bot
| Value v' ⇒ Value (inf v v')))"

instance
by standard (auto simp add: inf_bot_def less_eq_bot_def split: bot.splits)

end

instantiation bot :: (semilattice_sup) semilattice_sup
begin

definition sup_bot
where
"sup x y =
(case x of
Bot ⇒ y
| Value v ⇒
(case y of
Bot ⇒ x
| Value v' ⇒ Value (sup v v')))"

instance
by standard (auto simp add: sup_bot_def less_eq_bot_def split: bot.splits)

end

instance bot :: (lattice) bounded_lattice_bot

subsection ‹Values extended by a top element›

datatype 'a top = Value 'a | Top

instantiation top :: (preorder) preorder
begin

definition less_eq_top where
"x ≤ y ⟷ (case y of Top ⇒ True | Value y ⇒ (case x of Top ⇒ False | Value x ⇒ x ≤ y))"

definition less_top where
"x < y ⟷ (case x of Top ⇒ False | Value x ⇒ (case y of Top ⇒ True | Value y ⇒ x < y))"

lemma less_eq_top_Top [simp]: "x ≤ Top"

lemma less_eq_top_Top_code [code]: "x ≤ Top ⟷ True"
by simp

lemma less_eq_top_is_Top: "Top ≤ x ⟹ x = Top"
by (cases x) (simp_all add: less_eq_top_def)

lemma less_eq_top_Top_Value [simp, code]: "Top ≤ Value x ⟷ False"

lemma less_eq_top_Value_Value [simp, code]: "Value x ≤ Value y ⟷ x ≤ y"

lemma less_top_Top [simp, code]: "Top < x ⟷ False"

lemma less_top_Top_is_Value: "x < Top ⟹ ∃z. x = Value z"
by (cases x) (simp_all add: less_top_def)

lemma less_top_Value_Top [simp]: "Value x < Top"

lemma less_top_Value_Top_code [code]: "Value x < Top ⟷ True"
by simp

lemma less_top_Value [simp, code]: "Value x < Value y ⟷ x < y"

instance
by standard
(auto simp add: less_eq_top_def less_top_def less_le_not_le elim: order_trans split: top.splits)

end

instance top :: (order) order
by standard (auto simp add: less_eq_top_def less_top_def split: top.splits)

instance top :: (linorder) linorder
by standard (auto simp add: less_eq_top_def less_top_def split: top.splits)

instantiation top :: (order) top
begin
definition "top = Top"
instance ..
end

instantiation top :: (bot) bot
begin
definition "bot = Value bot"
instance ..
end

instantiation top :: (semilattice_inf) semilattice_inf
begin

definition inf_top
where
"inf x y =
(case x of
Top ⇒ y
| Value v ⇒
(case y of
Top ⇒ x
| Value v' ⇒ Value (inf v v')))"

instance
by standard (auto simp add: inf_top_def less_eq_top_def split: top.splits)

end

instantiation top :: (semilattice_sup) semilattice_sup
begin

definition sup_top
where
"sup x y =
(case x of
Top ⇒ Top
| Value v ⇒
(case y of
Top ⇒ Top
| Value v' ⇒ Value (sup v v')))"

instance
by standard (auto simp add: sup_top_def less_eq_top_def split: top.splits)

end

instance top :: (lattice) bounded_lattice_top

subsection ‹Values extended by a top and a bottom element›

datatype 'a flat_complete_lattice = Value 'a | Bot | Top

instantiation flat_complete_lattice :: (type) order
begin

definition less_eq_flat_complete_lattice
where
"x ≤ y ≡
(case x of
Bot ⇒ True
| Value v1 ⇒
(case y of
Bot ⇒ False
| Value v2 ⇒ v1 = v2
| Top ⇒ True)
| Top ⇒ y = Top)"

definition less_flat_complete_lattice
where
"x < y =
(case x of
Bot ⇒ y ≠ Bot
| Value v1 ⇒ y = Top
| Top ⇒ False)"

lemma [simp]: "Bot ≤ y"
unfolding less_eq_flat_complete_lattice_def by auto

lemma [simp]: "y ≤ Top"
unfolding less_eq_flat_complete_lattice_def by (auto split: flat_complete_lattice.splits)

lemma greater_than_two_values:
assumes "a ≠ b" "Value a ≤ z" "Value b ≤ z"
shows "z = Top"
using assms
by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)

lemma lesser_than_two_values:
assumes "a ≠ b" "z ≤ Value a" "z ≤ Value b"
shows "z = Bot"
using assms
by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)

instance
by standard
split: flat_complete_lattice.splits)

end

instantiation flat_complete_lattice :: (type) bot
begin
definition "bot = Bot"
instance ..
end

instantiation flat_complete_lattice :: (type) top
begin
definition "top = Top"
instance ..
end

instantiation flat_complete_lattice :: (type) lattice
begin

definition inf_flat_complete_lattice
where
"inf x y =
(case x of
Bot ⇒ Bot
| Value v1 ⇒
(case y of
Bot ⇒ Bot
| Value v2 ⇒ if v1 = v2 then x else Bot
| Top ⇒ x)
| Top ⇒ y)"

definition sup_flat_complete_lattice
where
"sup x y =
(case x of
Bot ⇒ y
| Value v1 ⇒
(case y of
Bot ⇒ x
| Value v2 ⇒ if v1 = v2 then x else Top
| Top ⇒ Top)
| Top ⇒ Top)"

instance
by standard
less_eq_flat_complete_lattice_def split: flat_complete_lattice.splits)

end

instantiation flat_complete_lattice :: (type) complete_lattice
begin

definition Sup_flat_complete_lattice
where
"Sup A =
(if A = {} ∨ A = {Bot} then Bot
else if ∃v. A - {Bot} = {Value v} then Value (THE v. A - {Bot} = {Value v})
else Top)"

definition Inf_flat_complete_lattice
where
"Inf A =
(if A = {} ∨ A = {Top} then Top
else if ∃v. A - {Top} = {Value v} then Value (THE v. A - {Top} = {Value v})
else Bot)"

instance
proof
fix x :: "'a flat_complete_lattice"
fix A
assume "x ∈ A"
{
fix v
assume "A - {Top} = {Value v}"
then have "(THE v. A - {Top} = {Value v}) = v"
by (auto intro!: the1_equality)
moreover
from ‹x ∈ A› ‹A - {Top} = {Value v}› have "x = Top ∨ x = Value v"
by auto
ultimately have "Value (THE v. A - {Top} = {Value v}) ≤ x"
by auto
}
with ‹x ∈ A› show "Inf A ≤ x"
unfolding Inf_flat_complete_lattice_def
by fastforce
next
fix z :: "'a flat_complete_lattice"
fix A
show "z ≤ Inf A" if z: "⋀x. x ∈ A ⟹ z ≤ x"
proof -
consider "A = {} ∨ A = {Top}"
| "A ≠ {}" "A ≠ {Top}" "∃v. A - {Top} = {Value v}"
| "A ≠ {}" "A ≠ {Top}" "¬ (∃v. A - {Top} = {Value v})"
by blast
then show ?thesis
proof cases
case 1
then have "Inf A = Top"
unfolding Inf_flat_complete_lattice_def by auto
then show ?thesis by simp
next
case 2
then obtain v where v1: "A - {Top} = {Value v}"
by auto
then have v2: "(THE v. A - {Top} = {Value v}) = v"
by (auto intro!: the1_equality)
from 2 v2 have Inf: "Inf A = Value v"
unfolding Inf_flat_complete_lattice_def by simp
from v1 have "Value v ∈ A" by blast
then have "z ≤ Value v" by (rule z)
with Inf show ?thesis by simp
next
case 3
then have Inf: "Inf A = Bot"
unfolding Inf_flat_complete_lattice_def by auto
have "z ≤ Bot"
proof (cases "A - {Top} = {Bot}")
case True
then have "Bot ∈ A" by blast
then show ?thesis by (rule z)
next
case False
from 3 obtain a1 where a1: "a1 ∈ A - {Top}"
by auto
from 3 False a1 obtain a2 where "a2 ∈ A - {Top} ∧ a1 ≠ a2"
by (cases a1) auto
with a1 z[of "a1"] z[of "a2"] show ?thesis
apply (cases a1)
apply auto
apply (cases a2)
apply auto
apply (auto dest!: lesser_than_two_values)
done
qed
with Inf show ?thesis by simp
qed
qed
next
fix x :: "'a flat_complete_lattice"
fix A
assume "x ∈ A"
{
fix v
assume "A - {Bot} = {Value v}"
then have "(THE v. A - {Bot} = {Value v}) = v"
by (auto intro!: the1_equality)
moreover
from ‹x ∈ A› ‹A - {Bot} = {Value v}› have "x = Bot ∨ x = Value v"
by auto
ultimately have "x ≤ Value (THE v. A - {Bot} = {Value v})"
by auto
}
with ‹x ∈ A› show "x ≤ Sup A"
unfolding Sup_flat_complete_lattice_def
by fastforce
next
fix z :: "'a flat_complete_lattice"
fix A
show "Sup A ≤ z" if z: "⋀x. x ∈ A ⟹ x ≤ z"
proof -
consider "A = {} ∨ A = {Bot}"
| "A ≠ {}" "A ≠ {Bot}" "∃v. A - {Bot} = {Value v}"
| "A ≠ {}" "A ≠ {Bot}" "¬ (∃v. A - {Bot} = {Value v})"
by blast
then show ?thesis
proof cases
case 1
then have "Sup A = Bot"
unfolding Sup_flat_complete_lattice_def by auto
then show ?thesis by simp
next
case 2
then obtain v where v1: "A - {Bot} = {Value v}"
by auto
then have v2: "(THE v. A - {Bot} = {Value v}) = v"
by (auto intro!: the1_equality)
from 2 v2 have Sup: "Sup A = Value v"
unfolding Sup_flat_complete_lattice_def by simp
from v1 have "Value v ∈ A" by blast
then have "Value v ≤ z" by (rule z)
with Sup show ?thesis by simp
next
case 3
then have Sup: "Sup A = Top"
unfolding Sup_flat_complete_lattice_def by auto
have "Top ≤ z"
proof (cases "A - {Bot} = {Top}")
case True
then have "Top ∈ A" by blast
then show ?thesis by (rule z)
next
case False
from 3 obtain a1 where a1: "a1 ∈ A - {Bot}"
by auto
from 3 False a1 obtain a2 where "a2 ∈ A - {Bot} ∧ a1 ≠ a2"
by (cases a1) auto
with a1 z[of "a1"] z[of "a2"] show ?thesis
apply (cases a1)
apply auto
apply (cases a2)
apply (auto dest!: greater_than_two_values)
done
qed
with Sup show ?thesis by simp
qed
qed
next
show "Inf {} = (top :: 'a flat_complete_lattice)"