# Theory Enum

theory Enum
imports Map Groups_List
```(* Author: Florian Haftmann, TU Muenchen *)

section ‹Finite types as explicit enumerations›

theory Enum
imports Map Groups_List
begin

subsection ‹Class ‹enum››

class enum =
fixes enum :: "'a list"
fixes enum_all :: "('a ⇒ bool) ⇒ bool"
fixes enum_ex :: "('a ⇒ bool) ⇒ bool"
assumes UNIV_enum: "UNIV = set enum"
and enum_distinct: "distinct enum"
assumes enum_all_UNIV: "enum_all P ⟷ Ball UNIV P"
assumes enum_ex_UNIV: "enum_ex P ⟷ Bex UNIV P"
― ‹tailored towards simple instantiation›
begin

subclass finite proof

lemma enum_UNIV:
"set enum = UNIV"
by (simp only: UNIV_enum)

lemma in_enum: "x ∈ set enum"

lemma enum_eq_I:
assumes "⋀x. x ∈ set xs"
shows "set enum = set xs"
proof -
from assms UNIV_eq_I have "UNIV = set xs" by auto
with enum_UNIV show ?thesis by simp
qed

lemma card_UNIV_length_enum:
"card (UNIV :: 'a set) = length enum"
by (simp add: UNIV_enum distinct_card enum_distinct)

lemma enum_all [simp]:
"enum_all = HOL.All"

lemma enum_ex [simp]:
"enum_ex = HOL.Ex"

end

subsection ‹Implementations using @{class enum}›

subsubsection ‹Unbounded operations and quantifiers›

lemma Collect_code [code]:
"Collect P = set (filter P enum)"

lemma vimage_code [code]:
"f -` B = set (filter (λx. f x ∈ B) enum_class.enum)"
unfolding vimage_def Collect_code ..

definition card_UNIV :: "'a itself ⇒ nat"
where
[code del]: "card_UNIV TYPE('a) = card (UNIV :: 'a set)"

lemma [code]:
"card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
by (simp only: card_UNIV_def enum_UNIV)

lemma all_code [code]: "(∀x. P x) ⟷ enum_all P"
by simp

lemma exists_code [code]: "(∃x. P x) ⟷ enum_ex P"
by simp

lemma exists1_code [code]: "(∃!x. P x) ⟷ list_ex1 P enum"
by (auto simp add: list_ex1_iff enum_UNIV)

subsubsection ‹An executable choice operator›

definition
[code del]: "enum_the = The"

lemma [code]:
"The P = (case filter P enum of [x] ⇒ x | _ ⇒ enum_the P)"
proof -
{
fix a
assume filter_enum: "filter P enum = [a]"
have "The P = a"
proof (rule the_equality)
fix x
assume "P x"
show "x = a"
proof (rule ccontr)
assume "x ≠ a"
from filter_enum obtain us vs
where enum_eq: "enum = us @ [a] @ vs"
and "∀ x ∈ set us. ¬ P x"
and "∀ x ∈ set vs. ¬ P x"
and "P a"
by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
with ‹P x› in_enum[of x, unfolded enum_eq] ‹x ≠ a› show "False" by auto
qed
next
from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
qed
}
from this show ?thesis
unfolding enum_the_def by (auto split: list.split)
qed

declare [[code abort: enum_the]]

code_printing
constant enum_the ⇀ (Eval) "(fn '_ => raise Match)"

subsubsection ‹Equality and order on functions›

instantiation "fun" :: (enum, equal) equal
begin

definition
"HOL.equal f g ⟷ (∀x ∈ set enum. f x = g x)"

instance proof
qed (simp_all add: equal_fun_def fun_eq_iff enum_UNIV)

end

lemma [code]:
"HOL.equal f g ⟷ enum_all (%x. f x = g x)"
by (auto simp add: equal fun_eq_iff)

lemma [code nbe]:
"HOL.equal (f :: _ ⇒ _) f ⟷ True"
by (fact equal_refl)

lemma order_fun [code]:
fixes f g :: "'a::enum ⇒ 'b::order"
shows "f ≤ g ⟷ enum_all (λx. f x ≤ g x)"
and "f < g ⟷ f ≤ g ∧ enum_ex (λx. f x ≠ g x)"
by (simp_all add: fun_eq_iff le_fun_def order_less_le)

subsubsection ‹Operations on relations›

lemma [code]:
"Id = image (λx. (x, x)) (set Enum.enum)"
by (auto intro: imageI in_enum)

lemma tranclp_unfold [code]:
"tranclp r a b ⟷ (a, b) ∈ trancl {(x, y). r x y}"

lemma rtranclp_rtrancl_eq [code]:
"rtranclp r x y ⟷ (x, y) ∈ rtrancl {(x, y). r x y}"

lemma max_ext_eq [code]:
"max_ext R = {(X, Y). finite X ∧ finite Y ∧ Y ≠ {} ∧ (∀x. x ∈ X ⟶ (∃xa ∈ Y. (x, xa) ∈ R))}"

lemma max_extp_eq [code]:
"max_extp r x y ⟷ (x, y) ∈ max_ext {(x, y). r x y}"

lemma mlex_eq [code]:
"f <*mlex*> R = {(x, y). f x < f y ∨ (f x ≤ f y ∧ (x, y) ∈ R)}"

subsubsection ‹Bounded accessible part›

primrec bacc :: "('a × 'a) set ⇒ nat ⇒ 'a set"
where
"bacc r 0 = {x. ∀ y. (y, x) ∉ r}"
| "bacc r (Suc n) = (bacc r n ∪ {x. ∀y. (y, x) ∈ r ⟶ y ∈ bacc r n})"

lemma bacc_subseteq_acc:
"bacc r n ⊆ Wellfounded.acc r"
by (induct n) (auto intro: acc.intros)

lemma bacc_mono:
"n ≤ m ⟹ bacc r n ⊆ bacc r m"
by (induct rule: dec_induct) auto

lemma bacc_upper_bound:
"bacc (r :: ('a × 'a) set)  (card (UNIV :: 'a::finite set)) = (⋃n. bacc r n)"
proof -
have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
moreover have "∀n. bacc r n = bacc r (Suc n) ⟶ bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
moreover have "finite (range (bacc r))" by auto
ultimately show ?thesis
by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
(auto intro: finite_mono_remains_stable_implies_strict_prefix)
qed

lemma acc_subseteq_bacc:
assumes "finite r"
shows "Wellfounded.acc r ⊆ (⋃n. bacc r n)"
proof
fix x
assume "x ∈ Wellfounded.acc r"
then have "∃n. x ∈ bacc r n"
proof (induct x arbitrary: rule: acc.induct)
case (accI x)
then have "∀y. ∃ n. (y, x) ∈ r ⟶ y ∈ bacc r n" by simp
from choice[OF this] obtain n where n: "∀y. (y, x) ∈ r ⟶ y ∈ bacc r (n y)" ..
obtain n where "⋀y. (y, x) ∈ r ⟹ y ∈ bacc r n"
proof
fix y assume y: "(y, x) ∈ r"
with n have "y ∈ bacc r (n y)" by auto
moreover have "n y <= Max ((λ(y, x). n y) ` r)"
using y ‹finite r› by (auto intro!: Max_ge)
note bacc_mono[OF this, of r]
ultimately show "y ∈ bacc r (Max ((λ(y, x). n y) ` r))" by auto
qed
then show ?case
by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
qed
then show "x ∈ (⋃n. bacc r n)" by auto
qed

lemma acc_bacc_eq:
fixes A :: "('a :: finite × 'a) set"
assumes "finite A"
shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))"
using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)

lemma [code]:
fixes xs :: "('a::finite × 'a) list"
shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"

subsection ‹Default instances for @{class enum}›

lemma map_of_zip_enum_is_Some:
assumes "length ys = length (enum :: 'a::enum list)"
shows "∃y. map_of (zip (enum :: 'a::enum list) ys) x = Some y"
proof -
from assms have "x ∈ set (enum :: 'a::enum list) ⟷
(∃y. map_of (zip (enum :: 'a::enum list) ys) x = Some y)"
by (auto intro!: map_of_zip_is_Some)
then show ?thesis using enum_UNIV by auto
qed

lemma map_of_zip_enum_inject:
fixes xs ys :: "'b::enum list"
assumes length: "length xs = length (enum :: 'a::enum list)"
"length ys = length (enum :: 'a::enum list)"
and map_of: "the ∘ map_of (zip (enum :: 'a::enum list) xs) = the ∘ map_of (zip (enum :: 'a::enum list) ys)"
shows "xs = ys"
proof -
have "map_of (zip (enum :: 'a list) xs) = map_of (zip (enum :: 'a list) ys)"
proof
fix x :: 'a
from length map_of_zip_enum_is_Some obtain y1 y2
where "map_of (zip (enum :: 'a list) xs) x = Some y1"
and "map_of (zip (enum :: 'a list) ys) x = Some y2" by blast
moreover from map_of
have "the (map_of (zip (enum :: 'a::enum list) xs) x) = the (map_of (zip (enum :: 'a::enum list) ys) x)"
by (auto dest: fun_cong)
ultimately show "map_of (zip (enum :: 'a::enum list) xs) x = map_of (zip (enum :: 'a::enum list) ys) x"
by simp
qed
with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
qed

definition all_n_lists :: "(('a :: enum) list ⇒ bool) ⇒ nat ⇒ bool"
where
"all_n_lists P n ⟷ (∀xs ∈ set (List.n_lists n enum). P xs)"

lemma [code]:
"all_n_lists P n ⟷ (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
unfolding all_n_lists_def enum_all
by (cases n) (auto simp add: enum_UNIV)

definition ex_n_lists :: "(('a :: enum) list ⇒ bool) ⇒ nat ⇒ bool"
where
"ex_n_lists P n ⟷ (∃xs ∈ set (List.n_lists n enum). P xs)"

lemma [code]:
"ex_n_lists P n ⟷ (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
unfolding ex_n_lists_def enum_ex
by (cases n) (auto simp add: enum_UNIV)

instantiation "fun" :: (enum, enum) enum
begin

definition
"enum = map (λys. the ∘ map_of (zip (enum::'a list) ys)) (List.n_lists (length (enum::'a::enum list)) enum)"

definition
"enum_all P = all_n_lists (λbs. P (the ∘ map_of (zip enum bs))) (length (enum :: 'a list))"

definition
"enum_ex P = ex_n_lists (λbs. P (the ∘ map_of (zip enum bs))) (length (enum :: 'a list))"

instance proof
show "UNIV = set (enum :: ('a ⇒ 'b) list)"
proof (rule UNIV_eq_I)
fix f :: "'a ⇒ 'b"
have "f = the ∘ map_of (zip (enum :: 'a::enum list) (map f enum))"
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
then show "f ∈ set enum"
by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
qed
next
from map_of_zip_enum_inject
show "distinct (enum :: ('a ⇒ 'b) list)"
by (auto intro!: inj_onI simp add: enum_fun_def
distinct_map distinct_n_lists enum_distinct set_n_lists)
next
fix P
show "enum_all (P :: ('a ⇒ 'b) ⇒ bool) = Ball UNIV P"
proof
assume "enum_all P"
show "Ball UNIV P"
proof
fix f :: "'a ⇒ 'b"
have f: "f = the ∘ map_of (zip (enum :: 'a::enum list) (map f enum))"
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
from ‹enum_all P› have "P (the ∘ map_of (zip enum (map f enum)))"
unfolding enum_all_fun_def all_n_lists_def
apply (erule_tac x="map f enum" in allE)
apply (auto intro!: in_enum)
done
from this f show "P f" by auto
qed
next
assume "Ball UNIV P"
from this show "enum_all P"
unfolding enum_all_fun_def all_n_lists_def by auto
qed
next
fix P
show "enum_ex (P :: ('a ⇒ 'b) ⇒ bool) = Bex UNIV P"
proof
assume "enum_ex P"
from this show "Bex UNIV P"
unfolding enum_ex_fun_def ex_n_lists_def by auto
next
assume "Bex UNIV P"
from this obtain f where "P f" ..
have f: "f = the ∘ map_of (zip (enum :: 'a::enum list) (map f enum))"
by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
from ‹P f› this have "P (the ∘ map_of (zip (enum :: 'a::enum list) (map f enum)))"
by auto
from  this show "enum_ex P"
unfolding enum_ex_fun_def ex_n_lists_def
apply (rule_tac x="map f enum" in exI)
apply (auto intro!: in_enum)
done
qed
qed

end

lemma enum_fun_code [code]: "enum = (let enum_a = (enum :: 'a::{enum, equal} list)
in map (λys. the ∘ map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"

lemma enum_all_fun_code [code]:
"enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
in all_n_lists (λbs. P (the ∘ map_of (zip enum_a bs))) (length enum_a))"
by (simp only: enum_all_fun_def Let_def)

lemma enum_ex_fun_code [code]:
"enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
in ex_n_lists (λbs. P (the ∘ map_of (zip enum_a bs))) (length enum_a))"
by (simp only: enum_ex_fun_def Let_def)

instantiation set :: (enum) enum
begin

definition
"enum = map set (subseqs enum)"

definition
"enum_all P ⟷ (∀A∈set enum. P (A::'a set))"

definition
"enum_ex P ⟷ (∃A∈set enum. P (A::'a set))"

instance proof
qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def subseqs_powset distinct_set_subseqs
enum_distinct enum_UNIV)

end

instantiation unit :: enum
begin

definition
"enum = [()]"

definition
"enum_all P = P ()"

definition
"enum_ex P = P ()"

instance proof
qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def)

end

instantiation bool :: enum
begin

definition
"enum = [False, True]"

definition
"enum_all P ⟷ P False ∧ P True"

definition
"enum_ex P ⟷ P False ∨ P True"

instance proof
qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all)

end

instantiation prod :: (enum, enum) enum
begin

definition
"enum = List.product enum enum"

definition
"enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"

definition
"enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"

instance
by standard
enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def)

end

instantiation sum :: (enum, enum) enum
begin

definition
"enum = map Inl enum @ map Inr enum"

definition
"enum_all P ⟷ enum_all (λx. P (Inl x)) ∧ enum_all (λx. P (Inr x))"

definition
"enum_ex P ⟷ enum_ex (λx. P (Inl x)) ∨ enum_ex (λx. P (Inr x))"

instance proof
qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum,
auto simp add: enum_UNIV distinct_map enum_distinct)

end

instantiation option :: (enum) enum
begin

definition
"enum = None # map Some enum"

definition
"enum_all P ⟷ P None ∧ enum_all (λx. P (Some x))"

definition
"enum_ex P ⟷ P None ∨ enum_ex (λx. P (Some x))"

instance proof
qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv,
auto simp add: distinct_map enum_UNIV enum_distinct)

end

subsection ‹Small finite types›

text ‹We define small finite types for use in Quickcheck›

datatype (plugins only: code "quickcheck" extraction) finite_1 =
a⇩1

notation (output) a⇩1  ("a⇩1")

lemma UNIV_finite_1:
"UNIV = {a⇩1}"
by (auto intro: finite_1.exhaust)

instantiation finite_1 :: enum
begin

definition
"enum = [a⇩1]"

definition
"enum_all P = P a⇩1"

definition
"enum_ex P = P a⇩1"

instance proof
qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all)

end

instantiation finite_1 :: linorder
begin

definition less_finite_1 :: "finite_1 ⇒ finite_1 ⇒ bool"
where
"x < (y :: finite_1) ⟷ False"

definition less_eq_finite_1 :: "finite_1 ⇒ finite_1 ⇒ bool"
where
"x ≤ (y :: finite_1) ⟷ True"

instance
apply (intro_classes)
apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
apply (metis finite_1.exhaust)
done

end

instance finite_1 :: "{dense_linorder, wellorder}"

instantiation finite_1 :: complete_lattice
begin

definition [simp]: "Inf = (λ_. a⇩1)"
definition [simp]: "Sup = (λ_. a⇩1)"
definition [simp]: "bot = a⇩1"
definition [simp]: "top = a⇩1"
definition [simp]: "inf = (λ_ _. a⇩1)"
definition [simp]: "sup = (λ_ _. a⇩1)"

end

instance finite_1 :: complete_distrib_lattice
by standard simp_all

instance finite_1 :: complete_linorder ..

lemma finite_1_eq: "x = a⇩1"
by(cases x) simp

simproc_setup finite_1_eq ("x::finite_1") = ‹
fn _ => fn _ => fn ct =>
(case Thm.term_of ct of
Const (@{const_name a⇩1}, _) => NONE
| _ => SOME (mk_meta_eq @{thm finite_1_eq}))
›

instantiation finite_1 :: complete_boolean_algebra
begin
definition [simp]: "(-) = (λ_ _. a⇩1)"
definition [simp]: "uminus = (λ_. a⇩1)"
instance by intro_classes simp_all
end

instantiation finite_1 ::
"{linordered_ring_strict, linordered_comm_semiring_strict, ordered_comm_ring,
ordered_cancel_comm_monoid_diff, comm_monoid_mult, ordered_ring_abs,
one, modulo, sgn, inverse}"
begin
definition [simp]: "Groups.zero = a⇩1"
definition [simp]: "Groups.one = a⇩1"
definition [simp]: "(+) = (λ_ _. a⇩1)"
definition [simp]: "( * ) = (λ_ _. a⇩1)"
definition [simp]: "(mod) = (λ_ _. a⇩1)"
definition [simp]: "abs = (λ_. a⇩1)"
definition [simp]: "sgn = (λ_. a⇩1)"
definition [simp]: "inverse = (λ_. a⇩1)"
definition [simp]: "divide = (λ_ _. a⇩1)"

end

declare [[simproc del: finite_1_eq]]
hide_const (open) a⇩1

datatype (plugins only: code "quickcheck" extraction) finite_2 =
a⇩1 | a⇩2

notation (output) a⇩1  ("a⇩1")
notation (output) a⇩2  ("a⇩2")

lemma UNIV_finite_2:
"UNIV = {a⇩1, a⇩2}"
by (auto intro: finite_2.exhaust)

instantiation finite_2 :: enum
begin

definition
"enum = [a⇩1, a⇩2]"

definition
"enum_all P ⟷ P a⇩1 ∧ P a⇩2"

definition
"enum_ex P ⟷ P a⇩1 ∨ P a⇩2"

instance proof
qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all)

end

instantiation finite_2 :: linorder
begin

definition less_finite_2 :: "finite_2 ⇒ finite_2 ⇒ bool"
where
"x < y ⟷ x = a⇩1 ∧ y = a⇩2"

definition less_eq_finite_2 :: "finite_2 ⇒ finite_2 ⇒ bool"
where
"x ≤ y ⟷ x = y ∨ x < (y :: finite_2)"

instance
apply (intro_classes)
apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
apply (metis finite_2.nchotomy)+
done

end

instance finite_2 :: wellorder

instantiation finite_2 :: complete_lattice
begin

definition "⨅A = (if a⇩1 ∈ A then a⇩1 else a⇩2)"
definition "⨆A = (if a⇩2 ∈ A then a⇩2 else a⇩1)"
definition [simp]: "bot = a⇩1"
definition [simp]: "top = a⇩2"
definition "x ⊓ y = (if x = a⇩1 ∨ y = a⇩1 then a⇩1 else a⇩2)"
definition "x ⊔ y = (if x = a⇩2 ∨ y = a⇩2 then a⇩2 else a⇩1)"

lemma neq_finite_2_a⇩1_iff [simp]: "x ≠ a⇩1 ⟷ x = a⇩2"
by(cases x) simp_all

lemma neq_finite_2_a⇩1_iff' [simp]: "a⇩1 ≠ x ⟷ x = a⇩2"
by(cases x) simp_all

lemma neq_finite_2_a⇩2_iff [simp]: "x ≠ a⇩2 ⟷ x = a⇩1"
by(cases x) simp_all

lemma neq_finite_2_a⇩2_iff' [simp]: "a⇩2 ≠ x ⟷ x = a⇩1"
by(cases x) simp_all

instance
proof
fix x :: finite_2 and A
assume "x ∈ A"
then show "⨅A ≤ x" "x ≤ ⨆A"
by(case_tac [!] x)(auto simp add: less_eq_finite_2_def less_finite_2_def Inf_finite_2_def Sup_finite_2_def)
qed(auto simp add: less_eq_finite_2_def less_finite_2_def inf_finite_2_def sup_finite_2_def Inf_finite_2_def Sup_finite_2_def)
end

instance finite_2 :: complete_linorder ..

instance finite_2 :: complete_distrib_lattice ..

instantiation finite_2 :: "{field, idom_abs_sgn, idom_modulo}" begin
definition [simp]: "0 = a⇩1"
definition [simp]: "1 = a⇩2"
definition "x + y = (case (x, y) of (a⇩1, a⇩1) ⇒ a⇩1 | (a⇩2, a⇩2) ⇒ a⇩1 | _ ⇒ a⇩2)"
definition "uminus = (λx :: finite_2. x)"
definition "(-) = ((+) :: finite_2 ⇒ _)"
definition "x * y = (case (x, y) of (a⇩2, a⇩2) ⇒ a⇩2 | _ ⇒ a⇩1)"
definition "inverse = (λx :: finite_2. x)"
definition "divide = (( * ) :: finite_2 ⇒ _)"
definition "x mod y = (case (x, y) of (a⇩2, a⇩1) ⇒ a⇩2 | _ ⇒ a⇩1)"
definition "abs = (λx :: finite_2. x)"
definition "sgn = (λx :: finite_2. x)"
instance
by standard
times_finite_2_def
inverse_finite_2_def divide_finite_2_def modulo_finite_2_def
abs_finite_2_def sgn_finite_2_def
split: finite_2.splits)
end

lemma two_finite_2 [simp]:
"2 = a⇩1"

lemma dvd_finite_2_unfold:
"x dvd y ⟷ x = a⇩2 ∨ y = a⇩1"
by (auto simp add: dvd_def times_finite_2_def split: finite_2.splits)

instantiation finite_2 :: "{normalization_semidom, unique_euclidean_semiring}" begin
definition [simp]: "normalize = (id :: finite_2 ⇒ _)"
definition [simp]: "unit_factor = (id :: finite_2 ⇒ _)"
definition [simp]: "euclidean_size x = (case x of a⇩1 ⇒ 0 | a⇩2 ⇒ 1)"
definition [simp]: "division_segment (x :: finite_2) = 1"
instance
by standard
(auto simp add: divide_finite_2_def times_finite_2_def dvd_finite_2_unfold
split: finite_2.splits)
end

hide_const (open) a⇩1 a⇩2

datatype (plugins only: code "quickcheck" extraction) finite_3 =
a⇩1 | a⇩2 | a⇩3

notation (output) a⇩1  ("a⇩1")
notation (output) a⇩2  ("a⇩2")
notation (output) a⇩3  ("a⇩3")

lemma UNIV_finite_3:
"UNIV = {a⇩1, a⇩2, a⇩3}"
by (auto intro: finite_3.exhaust)

instantiation finite_3 :: enum
begin

definition
"enum = [a⇩1, a⇩2, a⇩3]"

definition
"enum_all P ⟷ P a⇩1 ∧ P a⇩2 ∧ P a⇩3"

definition
"enum_ex P ⟷ P a⇩1 ∨ P a⇩2 ∨ P a⇩3"

instance proof
qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all)

end

lemma finite_3_not_eq_unfold:
"x ≠ a⇩1 ⟷ x ∈ {a⇩2, a⇩3}"
"x ≠ a⇩2 ⟷ x ∈ {a⇩1, a⇩3}"
"x ≠ a⇩3 ⟷ x ∈ {a⇩1, a⇩2}"
by (cases x; simp)+

instantiation finite_3 :: linorder
begin

definition less_finite_3 :: "finite_3 ⇒ finite_3 ⇒ bool"
where
"x < y = (case x of a⇩1 ⇒ y ≠ a⇩1 | a⇩2 ⇒ y = a⇩3 | a⇩3 ⇒ False)"

definition less_eq_finite_3 :: "finite_3 ⇒ finite_3 ⇒ bool"
where
"x ≤ y ⟷ x = y ∨ x < (y :: finite_3)"

instance proof (intro_classes)
qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)

end

instance finite_3 :: wellorder
proof(rule wf_wellorderI)
have "inv_image less_than (case_finite_3 0 1 2) = {(x, y). x < y}"
by(auto simp add: less_finite_3_def split: finite_3.splits)
from this[symmetric] show "wf …" by simp
qed intro_classes

class finite_lattice = finite +  lattice + Inf + Sup  + bot + top +
assumes Inf_finite_empty: "Inf {} = Sup UNIV"
assumes Inf_finite_insert: "Inf (insert a A) = a ⊓ Inf A"
assumes Sup_finite_empty: "Sup {} = Inf UNIV"
assumes Sup_finite_insert: "Sup (insert a A) = a ⊔ Sup A"
assumes bot_finite_def: "bot = Inf UNIV"
assumes top_finite_def: "top = Sup UNIV"
begin

subclass complete_lattice
proof
fix x A
show "x ∈ A ⟹ ⨅A ≤ x"
by (metis Set.set_insert abel_semigroup.commute local.Inf_finite_insert local.inf.abel_semigroup_axioms local.inf.left_idem local.inf.orderI)
show "x ∈ A ⟹ x ≤ ⨆A"
by (metis Set.set_insert insert_absorb2 local.Sup_finite_insert local.sup.absorb_iff2)
next
fix A z
have "⨆ UNIV = z ⊔ ⨆UNIV"
by (subst Sup_finite_insert [symmetric], simp add: insert_UNIV)
from this have [simp]: "z ≤ ⨆UNIV"
using local.le_iff_sup by auto
have "(∀ x. x ∈ A ⟶ z ≤ x) ⟶ z ≤ ⨅A"
apply (rule finite_induct [of A "λ A . (∀ x. x ∈ A ⟶ z ≤ x) ⟶ z ≤ ⨅A"])
from this show "(⋀x. x ∈ A ⟹ z ≤ x) ⟹ z ≤ ⨅A"
by simp

have "⨅ UNIV = z ⊓ ⨅UNIV"
by (subst Inf_finite_insert [symmetric], simp add: insert_UNIV)
from this have [simp]: "⨅UNIV ≤ z"
have "(∀ x. x ∈ A ⟶ x ≤ z) ⟶ ⨆A ≤ z"
by (rule finite_induct [of A "λ A . (∀ x. x ∈ A ⟶ x ≤ z) ⟶ ⨆A ≤ z" ], simp_all add: Sup_finite_empty Sup_finite_insert)
from this show " (⋀x. x ∈ A ⟹ x ≤ z) ⟹ ⨆A ≤ z"
by blast
next
show "⨅{} = ⊤"
show " ⨆{} = ⊥"
qed
end

class finite_distrib_lattice = finite_lattice + distrib_lattice
begin
lemma finite_inf_Sup: "a ⊓ (Sup A) = Sup {a ⊓ b | b . b ∈ A}"
proof (rule finite_induct [of A "λ A . a ⊓ (Sup A) = Sup {a ⊓ b | b . b ∈ A}"], simp_all)
fix x::"'a"
fix F
assume "x ∉ F"
assume [simp]: "a ⊓ ⨆F = ⨆{a ⊓ b |b. b ∈ F}"
have [simp]: " insert (a ⊓ x) {a ⊓ b |b. b ∈ F} = {a ⊓ b |b. b = x ∨ b ∈ F}"
by blast
have "a ⊓ (x ⊔ ⨆F) = a ⊓ x ⊔ a ⊓ ⨆F"
also have "... = a ⊓ x ⊔ ⨆{a ⊓ b |b. b ∈ F}"
by simp
also have "... = ⨆{a ⊓ b |b. b = x ∨ b ∈ F}"
by (unfold Sup_insert[THEN sym], simp)
finally show "a ⊓ (x ⊔ ⨆F) = ⨆{a ⊓ b |b. b = x ∨ b ∈ F}"
by simp
qed

lemma finite_Inf_Sup: "INFIMUM A Sup ≤ SUPREMUM {f ` A |f. ∀Y∈A. f Y ∈ Y} Inf"
proof (rule  finite_induct [of A "λ A .INFIMUM A Sup ≤ SUPREMUM {f ` A |f. ∀Y∈A. f Y ∈ Y} Inf"], simp_all add: finite_UnionD)
fix x::"'a set"
fix F
assume "x ∉ F"
have [simp]: "{⨆x ⊓ b |b . b ∈ Inf ` {f ` F |f. ∀Y∈F. f Y ∈ Y} } = {⨆x ⊓ (Inf (f ` F)) |f  . (∀Y∈F. f Y ∈ Y)}"
by auto
define fa where "fa = (λ (b::'a) f Y . (if Y = x then b else f Y))"
have "⋀f b. ∀Y∈F. f Y ∈ Y ⟹ b ∈ x ⟹ insert b (f ` (F ∩ {Y. Y ≠ x})) = insert (fa b f x) (fa b f ` F) ∧ fa b f x ∈ x ∧ (∀Y∈F. fa b f Y ∈ Y)"
from this have B: "⋀f b. ∀Y∈F. f Y ∈ Y ⟹ b ∈ x ⟹ fa b f ` ({x} ∪ F) ∈ {insert (f x) (f ` F) |f. f x ∈ x ∧ (∀Y∈F. f Y ∈ Y)}"
by blast
have [simp]: "⋀f b. ∀Y∈F. f Y ∈ Y ⟹ b ∈ x ⟹ b ⊓ (⨅x∈F. f x)  ≤ SUPREMUM {insert (f x) (f ` F) |f. f x ∈ x ∧ (∀Y∈F. f Y ∈ Y)} Inf"
using B apply (rule SUP_upper2, simp_all)
by (simp_all add: INF_greatest Inf_lower inf.coboundedI2 fa_def)

assume "INFIMUM F Sup ≤ SUPREMUM {f ` F |f. ∀Y∈F. f Y ∈ Y} Inf"

from this have "⨆x ⊓ INFIMUM F Sup ≤ ⨆x ⊓ SUPREMUM {f ` F |f. ∀Y∈F. f Y ∈ Y} Inf"
apply simp
using inf.coboundedI2 by blast
also have "... = Sup {⨆x ⊓ (Inf (f ` F)) |f  .  (∀Y∈F. f Y ∈ Y)}"

also have "... = Sup {Sup {Inf (f ` F) ⊓ b | b . b ∈ x} |f  .  (∀Y∈F. f Y ∈ Y)}"
apply (subst inf_commute)

also have "... ≤ SUPREMUM {insert (f x) (f ` F) |f. f x ∈ x ∧ (∀Y∈F. f Y ∈ Y)} Inf"
apply (rule Sup_least, clarsimp)+
by (subst inf_commute, simp)

finally show "⨆x ⊓ INFIMUM F Sup ≤ SUPREMUM {insert (f x) (f ` F) |f. f x ∈ x ∧ (∀Y∈F. f Y ∈ Y)} Inf"
by simp
qed

subclass complete_distrib_lattice
by (standard, rule finite_Inf_Sup)
end

instantiation finite_3 :: finite_lattice
begin

definition "⨅A = (if a⇩1 ∈ A then a⇩1 else if a⇩2 ∈ A then a⇩2 else a⇩3)"
definition "⨆A = (if a⇩3 ∈ A then a⇩3 else if a⇩2 ∈ A then a⇩2 else a⇩1)"
definition [simp]: "bot = a⇩1"
definition [simp]: "top = a⇩3"
definition [simp]: "inf = (min :: finite_3 ⇒ _)"
definition [simp]: "sup = (max :: finite_3 ⇒ _)"

instance
proof
qed (auto simp add: Inf_finite_3_def Sup_finite_3_def max_def min_def less_eq_finite_3_def less_finite_3_def split: finite_3.split)
end

instance finite_3 :: complete_lattice ..

instance finite_3 :: finite_distrib_lattice
proof
qed (auto simp add: min_def max_def)

instance finite_3 :: complete_distrib_lattice ..

instance finite_3 :: complete_linorder ..

instantiation finite_3 :: "{field, idom_abs_sgn, idom_modulo}" begin
definition [simp]: "0 = a⇩1"
definition [simp]: "1 = a⇩2"
definition
"x + y = (case (x, y) of
(a⇩1, a⇩1) ⇒ a⇩1 | (a⇩2, a⇩3) ⇒ a⇩1 | (a⇩3, a⇩2) ⇒ a⇩1
| (a⇩1, a⇩2) ⇒ a⇩2 | (a⇩2, a⇩1) ⇒ a⇩2 | (a⇩3, a⇩3) ⇒ a⇩2
| _ ⇒ a⇩3)"
definition "- x = (case x of a⇩1 ⇒ a⇩1 | a⇩2 ⇒ a⇩3 | a⇩3 ⇒ a⇩2)"
definition "x - y = x + (- y :: finite_3)"
definition "x * y = (case (x, y) of (a⇩2, a⇩2) ⇒ a⇩2 | (a⇩3, a⇩3) ⇒ a⇩2 | (a⇩2, a⇩3) ⇒ a⇩3 | (a⇩3, a⇩2) ⇒ a⇩3 | _ ⇒ a⇩1)"
definition "inverse = (λx :: finite_3. x)"
definition "x div y = x * inverse (y :: finite_3)"
definition "x mod y = (case y of a⇩1 ⇒ x | _ ⇒ a⇩1)"
definition "abs = (λx. case x of a⇩3 ⇒ a⇩2 | _ ⇒ x)"
definition "sgn = (λx :: finite_3. x)"
instance
by standard
times_finite_3_def
inverse_finite_3_def divide_finite_3_def modulo_finite_3_def
abs_finite_3_def sgn_finite_3_def
less_finite_3_def
split: finite_3.splits)
end

lemma two_finite_3 [simp]:
"2 = a⇩3"

lemma dvd_finite_3_unfold:
"x dvd y ⟷ x = a⇩2 ∨ x = a⇩3 ∨ y = a⇩1"
by (cases x) (auto simp add: dvd_def times_finite_3_def split: finite_3.splits)

instantiation finite_3 :: "{normalization_semidom, unique_euclidean_semiring}" begin
definition [simp]: "normalize x = (case x of a⇩3 ⇒ a⇩2 | _ ⇒ x)"
definition [simp]: "unit_factor = (id :: finite_3 ⇒ _)"
definition [simp]: "euclidean_size x = (case x of a⇩1 ⇒ 0 | _ ⇒ 1)"
definition [simp]: "division_segment (x :: finite_3) = 1"
instance proof
fix x :: finite_3
assume "x ≠ 0"
then show "is_unit (unit_factor x)"
by (cases x) (simp_all add: dvd_finite_3_unfold)
qed (auto simp add: divide_finite_3_def times_finite_3_def
dvd_finite_3_unfold inverse_finite_3_def plus_finite_3_def
split: finite_3.splits)
end

hide_const (open) a⇩1 a⇩2 a⇩3

datatype (plugins only: code "quickcheck" extraction) finite_4 =
a⇩1 | a⇩2 | a⇩3 | a⇩4

notation (output) a⇩1  ("a⇩1")
notation (output) a⇩2  ("a⇩2")
notation (output) a⇩3  ("a⇩3")
notation (output) a⇩4  ("a⇩4")

lemma UNIV_finite_4:
"UNIV = {a⇩1, a⇩2, a⇩3, a⇩4}"
by (auto intro: finite_4.exhaust)

instantiation finite_4 :: enum
begin

definition
"enum = [a⇩1, a⇩2, a⇩3, a⇩4]"

definition
"enum_all P ⟷ P a⇩1 ∧ P a⇩2 ∧ P a⇩3 ∧ P a⇩4"

definition
"enum_ex P ⟷ P a⇩1 ∨ P a⇩2 ∨ P a⇩3 ∨ P a⇩4"

instance proof
qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all)

end

instantiation finite_4 :: finite_distrib_lattice begin

text ‹@{term a⇩1} \$<\$ @{term a⇩2},@{term a⇩3} \$<\$ @{term a⇩4},
but @{term a⇩2} and @{term a⇩3} are incomparable.›

definition
"x < y ⟷ (case (x, y) of
(a⇩1, a⇩1) ⇒ False | (a⇩1, _) ⇒ True
|  (a⇩2, a⇩4) ⇒ True
|  (a⇩3, a⇩4) ⇒ True  | _ ⇒ False)"

definition
"x ≤ y ⟷ (case (x, y) of
(a⇩1, _) ⇒ True
| (a⇩2, a⇩2) ⇒ True | (a⇩2, a⇩4) ⇒ True
| (a⇩3, a⇩3) ⇒ True | (a⇩3, a⇩4) ⇒ True
| (a⇩4, a⇩4) ⇒ True | _ ⇒ False)"

definition
"⨅A = (if a⇩1 ∈ A ∨ a⇩2 ∈ A ∧ a⇩3 ∈ A then a⇩1 else if a⇩2 ∈ A then a⇩2 else if a⇩3 ∈ A then a⇩3 else a⇩4)"
definition
"⨆A = (if a⇩4 ∈ A ∨ a⇩2 ∈ A ∧ a⇩3 ∈ A then a⇩4 else if a⇩2 ∈ A then a⇩2 else if a⇩3 ∈ A then a⇩3 else a⇩1)"
definition [simp]: "bot = a⇩1"
definition [simp]: "top = a⇩4"
definition
"x ⊓ y = (case (x, y) of
(a⇩1, _) ⇒ a⇩1 | (_, a⇩1) ⇒ a⇩1 | (a⇩2, a⇩3) ⇒ a⇩1 | (a⇩3, a⇩2) ⇒ a⇩1
| (a⇩2, _) ⇒ a⇩2 | (_, a⇩2) ⇒ a⇩2
| (a⇩3, _) ⇒ a⇩3 | (_, a⇩3) ⇒ a⇩3
| _ ⇒ a⇩4)"
definition
"x ⊔ y = (case (x, y) of
(a⇩4, _) ⇒ a⇩4 | (_, a⇩4) ⇒ a⇩4 | (a⇩2, a⇩3) ⇒ a⇩4 | (a⇩3, a⇩2) ⇒ a⇩4
| (a⇩2, _) ⇒ a⇩2 | (_, a⇩2) ⇒ a⇩2
| (a⇩3, _) ⇒ a⇩3 | (_, a⇩3) ⇒ a⇩3
| _ ⇒ a⇩1)"

instance proof
qed(auto simp add: less_finite_4_def less_eq_finite_4_def Inf_finite_4_def Sup_finite_4_def
inf_finite_4_def sup_finite_4_def split: finite_4.splits)
end

instance finite_4 :: complete_lattice ..

instance finite_4 :: complete_distrib_lattice ..

instantiation finite_4 :: complete_boolean_algebra begin
definition "- x = (case x of a⇩1 ⇒ a⇩4 | a⇩2 ⇒ a⇩3 | a⇩3 ⇒ a⇩2 | a⇩4 ⇒ a⇩1)"
definition "x - y = x ⊓ - (y :: finite_4)"
instance
by intro_classes
(simp_all add: inf_finite_4_def sup_finite_4_def uminus_finite_4_def minus_finite_4_def
split: finite_4.splits)
end

hide_const (open) a⇩1 a⇩2 a⇩3 a⇩4

datatype (plugins only: code "quickcheck" extraction) finite_5 =
a⇩1 | a⇩2 | a⇩3 | a⇩4 | a⇩5

notation (output) a⇩1  ("a⇩1")
notation (output) a⇩2  ("a⇩2")
notation (output) a⇩3  ("a⇩3")
notation (output) a⇩4  ("a⇩4")
notation (output) a⇩5  ("a⇩5")

lemma UNIV_finite_5:
"UNIV = {a⇩1, a⇩2, a⇩3, a⇩4, a⇩5}"
by (auto intro: finite_5.exhaust)

instantiation finite_5 :: enum
begin

definition
"enum = [a⇩1, a⇩2, a⇩3, a⇩4, a⇩5]"

definition
"enum_all P ⟷ P a⇩1 ∧ P a⇩2 ∧ P a⇩3 ∧ P a⇩4 ∧ P a⇩5"

definition
"enum_ex P ⟷ P a⇩1 ∨ P a⇩2 ∨ P a⇩3 ∨ P a⇩4 ∨ P a⇩5"

instance proof
qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all)

end

instantiation finite_5 :: finite_lattice
begin

text ‹The non-distributive pentagon lattice \$N_5\$›

definition
"x < y ⟷ (case (x, y) of
(a⇩1, a⇩1) ⇒ False | (a⇩1, _) ⇒ True
| (a⇩2, a⇩3) ⇒ True  | (a⇩2, a⇩5) ⇒ True
| (a⇩3, a⇩5) ⇒ True
| (a⇩4, a⇩5) ⇒ True  | _ ⇒ False)"

definition
"x ≤ y ⟷ (case (x, y) of
(a⇩1, _) ⇒ True
| (a⇩2, a⇩2) ⇒ True | (a⇩2, a⇩3) ⇒ True | (a⇩2, a⇩5) ⇒ True
| (a⇩3, a⇩3) ⇒ True | (a⇩3, a⇩5) ⇒ True
| (a⇩4, a⇩4) ⇒ True | (a⇩4, a⇩5) ⇒ True
| (a⇩5, a⇩5) ⇒ True | _ ⇒ False)"

definition
"⨅A =
(if a⇩1 ∈ A ∨ a⇩4 ∈ A ∧ (a⇩2 ∈ A ∨ a⇩3 ∈ A) then a⇩1
else if a⇩2 ∈ A then a⇩2
else if a⇩3 ∈ A then a⇩3
else if a⇩4 ∈ A then a⇩4
else a⇩5)"
definition
"⨆A =
(if a⇩5 ∈ A ∨ a⇩4 ∈ A ∧ (a⇩2 ∈ A ∨ a⇩3 ∈ A) then a⇩5
else if a⇩3 ∈ A then a⇩3
else if a⇩2 ∈ A then a⇩2
else if a⇩4 ∈ A then a⇩4
else a⇩1)"
definition [simp]: "bot = a⇩1"
definition [simp]: "top = a⇩5"
definition
"x ⊓ y = (case (x, y) of
(a⇩1, _) ⇒ a⇩1 | (_, a⇩1) ⇒ a⇩1 | (a⇩2, a⇩4) ⇒ a⇩1 | (a⇩4, a⇩2) ⇒ a⇩1 | (a⇩3, a⇩4) ⇒ a⇩1 | (a⇩4, a⇩3) ⇒ a⇩1
| (a⇩2, _) ⇒ a⇩2 | (_, a⇩2) ⇒ a⇩2
| (a⇩3, _) ⇒ a⇩3 | (_, a⇩3) ⇒ a⇩3
| (a⇩4, _) ⇒ a⇩4 | (_, a⇩4) ⇒ a⇩4
| _ ⇒ a⇩5)"
definition
"x ⊔ y = (case (x, y) of
(a⇩5, _) ⇒ a⇩5 | (_, a⇩5) ⇒ a⇩5 | (a⇩2, a⇩4) ⇒ a⇩5 | (a⇩4, a⇩2) ⇒ a⇩5 | (a⇩3, a⇩4) ⇒ a⇩5 | (a⇩4, a⇩3) ⇒ a⇩5
| (a⇩3, _) ⇒ a⇩3 | (_, a⇩3) ⇒ a⇩3
| (a⇩2, _) ⇒ a⇩2 | (_, a⇩2) ⇒ a⇩2
| (a⇩4, _) ⇒ a⇩4 | (_, a⇩4) ⇒ a⇩4
| _ ⇒ a⇩1)"

instance
proof
qed (auto simp add: less_eq_finite_5_def less_finite_5_def inf_finite_5_def sup_finite_5_def
Inf_finite_5_def Sup_finite_5_def split: finite_5.splits if_split_asm)
end

instance  finite_5 :: complete_lattice ..

hide_const (open) a⇩1 a⇩2 a⇩3 a⇩4 a⇩5

subsection ‹Closing up›

hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl

end
```