(* Author: Florian Haftmann, TU Muenchen *)
section \<open>Finite types as explicit enumerations\<close>
theory Enum
imports Map Groups_List
begin
subsection \<open>Class \<open>enum\<close>\<close>
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"
\<comment> \<open>tailored towards simple instantiation\<close>
begin
subclass finite proof
qed (simp add: UNIV_enum)
lemma enum_UNIV:
"set enum = UNIV"
by (simp only: UNIV_enum)
lemma in_enum: "x \ set enum"
by (simp add: enum_UNIV)
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"
by (simp add: fun_eq_iff enum_all_UNIV)
lemma enum_ex [simp]:
"enum_ex = HOL.Ex"
by (simp add: fun_eq_iff enum_ex_UNIV)
end
subsection \<open>Implementations using \<^class>\<open>enum\<close>\<close>
subsubsection \<open>Unbounded operations and quantifiers\<close>
lemma Collect_code [code]:
"Collect P = set (filter P enum)"
by (simp add: enum_UNIV)
lemma vimage_code [code]:
"f -` B = set (filter (\x. 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 \<open>An executable choice operator\<close>
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 \<open>P x\<close> in_enum[of x, unfolded enum_eq] \<open>x \<noteq> a\<close> 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 \<rightharpoonup> (Eval) "(fn '_ => raise Match)"
subsubsection \<open>Equality and order on functions\<close>
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 (\x. f x \ g x)"
and "f < g \ f \ g \ enum_ex (\x. f x \ g x)"
by (simp_all add: fun_eq_iff le_fun_def order_less_le)
subsubsection \<open>Operations on relations\<close>
lemma [code]:
"Id = image (\x. (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}"
by (simp add: trancl_def)
lemma rtranclp_rtrancl_eq [code]:
"rtranclp r x y \ (x, y) \ rtrancl {(x, y). r x y}"
by (simp add: rtrancl_def)
lemma max_ext_eq [code]:
"max_ext R = {(X, Y). finite X \ finite Y \ Y \ {} \ (\x. x \ X \ (\xa \ Y. (x, xa) \ R))}"
by (auto simp add: max_ext.simps)
lemma max_extp_eq [code]:
"max_extp r x y \ (x, y) \ max_ext {(x, y). r x y}"
by (simp add: max_ext_def)
lemma mlex_eq [code]:
"f <*mlex*> R = {(x, y). f x < f y \ (f x \ f y \ (x, y) \ R)}"
by (auto simp add: mlex_prod_def)
subsubsection \<open>Bounded accessible part\<close>
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 ((\(y, x). n y) ` r)"
using y \<open>finite r\<close> by (auto intro!: Max_ge)
note bacc_mono[OF this, of r]
ultimately show "y \ bacc r (Max ((\(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))"
by (simp add: card_UNIV_def acc_bacc_eq)
subsection \<open>Default instances for \<^class>\<open>enum\<close>\<close>
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) \
(\<exists>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 (\ys. the \ map_of (zip (enum::'a list) ys)) (List.n_lists (length (enum::'a::enum list)) enum)"
definition
"enum_all P = all_n_lists (\bs. P (the \ map_of (zip enum bs))) (length (enum :: 'a list))"
definition
"enum_ex P = ex_n_lists (\bs. 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 \<open>enum_all P\<close> have "P (the \<circ> map_of (zip enum (map f enum)))"
unfolding enum_all_fun_def all_n_lists_def
apply (simp add: set_n_lists)
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 \<open>P f\<close> this have "P (the \<circ> 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 (auto simp add: set_n_lists)
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 (\<lambda>ys. the \<circ> map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"
by (simp add: enum_fun_def Let_def)
lemma enum_all_fun_code [code]:
"enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
in all_n_lists (\<lambda>bs. P (the \<circ> 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 (\<lambda>bs. P (the \<circ> 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
(simp_all add: enum_prod_def distinct_product
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 (\x. P (Inl x)) \ enum_all (\x. P (Inr x))"
definition
"enum_ex P \ enum_ex (\x. P (Inl x)) \ enum_ex (\x. 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 (\x. P (Some x))"
definition
"enum_ex P \ P None \ enum_ex (\x. 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 \<open>Small finite types\<close>
text \<open>We define small finite types for use in Quickcheck\<close>
datatype (plugins only: code "quickcheck" extraction) finite_1 =
a\<^sub>1
notation (output) a\<^sub>1 ("a\<^sub>1")
lemma UNIV_finite_1:
"UNIV = {a\<^sub>1}"
by (auto intro: finite_1.exhaust)
instantiation finite_1 :: enum
begin
definition
"enum = [a\<^sub>1]"
definition
"enum_all P = P a\<^sub>1"
definition
"enum_ex P = P a\<^sub>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 (full_types) finite_1.exhaust)
done
end
instance finite_1 :: "{dense_linorder, wellorder}"
by intro_classes (simp_all add: less_finite_1_def)
instantiation finite_1 :: complete_lattice
begin
definition [simp]: "Inf = (\_. a\<^sub>1)"
definition [simp]: "Sup = (\_. a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>1"
definition [simp]: "inf = (\_ _. a\<^sub>1)"
definition [simp]: "sup = (\_ _. a\<^sub>1)"
instance by intro_classes(simp_all add: less_eq_finite_1_def)
end
instance finite_1 :: complete_distrib_lattice
by standard simp_all
instance finite_1 :: complete_linorder ..
lemma finite_1_eq: "x = a\<^sub>1"
by(cases x) simp
simproc_setup finite_1_eq ("x::finite_1") = \<open>
fn _ => fn _ => fn ct =>
(case Thm.term_of ct of
Const (\<^const_name>\<open>a\<^sub>1\<close>, _) => NONE
| _ => SOME (mk_meta_eq @{thm finite_1_eq}))
\<close>
instantiation finite_1 :: complete_boolean_algebra
begin
definition [simp]: "(-) = (\_ _. a\<^sub>1)"
definition [simp]: "uminus = (\_. a\<^sub>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\<^sub>1"
definition [simp]: "Groups.one = a\<^sub>1"
definition [simp]: "(+) = (\_ _. a\<^sub>1)"
definition [simp]: "(*) = (\_ _. a\<^sub>1)"
definition [simp]: "(mod) = (\_ _. a\<^sub>1)"
definition [simp]: "abs = (\_. a\<^sub>1)"
definition [simp]: "sgn = (\_. a\<^sub>1)"
definition [simp]: "inverse = (\_. a\<^sub>1)"
definition [simp]: "divide = (\_ _. a\<^sub>1)"
instance by intro_classes(simp_all add: less_finite_1_def)
end
declare [[simproc del: finite_1_eq]]
hide_const (open) a\<^sub>1
datatype (plugins only: code "quickcheck" extraction) finite_2 =
a\<^sub>1 | a\<^sub>2
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
lemma UNIV_finite_2:
"UNIV = {a\<^sub>1, a\<^sub>2}"
by (auto intro: finite_2.exhaust)
instantiation finite_2 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2]"
definition
"enum_all P \ P a\<^sub>1 \ P a\<^sub>2"
definition
"enum_ex P \ P a\<^sub>1 \ P a\<^sub>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\<^sub>1 \ y = a\<^sub>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
by(rule wf_wellorderI)(simp add: less_finite_2_def, intro_classes)
instantiation finite_2 :: complete_lattice
begin
definition "\A = (if a\<^sub>1 \ A then a\<^sub>1 else a\<^sub>2)"
definition "\A = (if a\<^sub>2 \ A then a\<^sub>2 else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>2"
definition "x \ y = (if x = a\<^sub>1 \ y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)"
definition "x \ y = (if x = a\<^sub>2 \ y = a\<^sub>2 then a\<^sub>2 else a\<^sub>1)"
lemma neq_finite_2_a\<^sub>1_iff [simp]: "x \<noteq> a\<^sub>1 \<longleftrightarrow> x = a\<^sub>2"
by(cases x) simp_all
lemma neq_finite_2_a\<^sub>1_iff' [simp]: "a\<^sub>1 \<noteq> x \<longleftrightarrow> x = a\<^sub>2"
by(cases x) simp_all
lemma neq_finite_2_a\<^sub>2_iff [simp]: "x \<noteq> a\<^sub>2 \<longleftrightarrow> x = a\<^sub>1"
by(cases x) simp_all
lemma neq_finite_2_a\<^sub>2_iff' [simp]: "a\<^sub>2 \<noteq> x \<longleftrightarrow> x = a\<^sub>1"
by(cases x) simp_all
instance
proof
fix x :: finite_2 and A
assume "x \ A"
then show "\A \ x" "x \ \A"
by(cases 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\<^sub>1"
definition [simp]: "1 = a\<^sub>2"
definition "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \ a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \ a\<^sub>1 | _ \ a\<^sub>2)"
definition "uminus = (\x :: finite_2. x)"
definition "(-) = ((+) :: finite_2 \ _)"
definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \ a\<^sub>2 | _ \ a\<^sub>1)"
definition "inverse = (\x :: finite_2. x)"
definition "divide = ((*) :: finite_2 \ _)"
definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \ a\<^sub>2 | _ \ a\<^sub>1)"
definition "abs = (\x :: finite_2. x)"
definition "sgn = (\x :: finite_2. x)"
instance
by standard
(subproofs
\<open>simp_all add: plus_finite_2_def uminus_finite_2_def minus_finite_2_def
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\<close>)
end
lemma two_finite_2 [simp]:
"2 = a\<^sub>1"
by (simp add: numeral.simps plus_finite_2_def)
lemma dvd_finite_2_unfold:
"x dvd y \ x = a\<^sub>2 \ y = a\<^sub>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\<^sub>1 \ 0 | a\<^sub>2 \ 1)"
definition [simp]: "division_segment (x :: finite_2) = 1"
instance
by standard
(subproofs
\<open>auto simp add: divide_finite_2_def times_finite_2_def dvd_finite_2_unfold
split: finite_2.splits\<close>)
end
hide_const (open) a\<^sub>1 a\<^sub>2
datatype (plugins only: code "quickcheck" extraction) finite_3 =
a\<^sub>1 | a\<^sub>2 | a\<^sub>3
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
notation (output) a\<^sub>3 ("a\<^sub>3")
lemma UNIV_finite_3:
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}"
by (auto intro: finite_3.exhaust)
instantiation finite_3 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
definition
"enum_all P \ P a\<^sub>1 \ P a\<^sub>2 \ P a\<^sub>3"
definition
"enum_ex P \ P a\<^sub>1 \ P a\<^sub>2 \ P a\<^sub>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\<^sub>1 \ x \ {a\<^sub>2, a\<^sub>3}"
"x \ a\<^sub>2 \ x \ {a\<^sub>1, a\<^sub>3}"
"x \ a\<^sub>3 \ x \ {a\<^sub>1, a\<^sub>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\<^sub>1 \ y \ a\<^sub>1 | a\<^sub>2 \ y = a\<^sub>3 | a\<^sub>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"
by (rule finite_induct [of A "\ A . (\ x. x \ A \ z \ x) \ z \ \A"])
(simp_all add: Inf_finite_empty Inf_finite_insert)
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"
by (simp add: local.inf.absorb_iff2)
have "(\ x. x \ A \ x \ z) \ \A \ z"
by (rule finite_induct [of A "\ 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 "\{} = \"
by (simp add: Inf_finite_empty top_finite_def)
show " \{} = \"
by (simp add: Sup_finite_empty bot_finite_def)
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 "\ 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"
by (simp add: inf_sup_distrib1)
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: "\(Sup ` A) \ \(Inf ` {f ` A |f. \Y\A. f Y \ Y})"
proof (rule finite_induct [of A "\A. \(Sup ` A) \ \(Inf ` {f ` A |f. \Y\A. f Y \ Y})"], 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 = (\ (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)"
by (auto simp add: fa_def)
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) \ \(Inf ` {insert (f x) (f ` F) |f. f x \ x \ (\Y\F. f Y \ Y)})"
using B apply (rule SUP_upper2)
using \<open>x \<notin> F\<close> apply (simp_all add: fa_def Inf_union_distrib)
apply (simp add: image_mono Inf_superset_mono inf.coboundedI2)
done
assume "\(Sup ` F) \ \(Inf ` {f ` F |f. \Y\F. f Y \ Y})"
from this have "\x \ \(Sup ` F) \ \x \ \(Inf ` {f ` F |f. \Y\F. f Y \ Y})"
using inf.coboundedI2 by auto
also have "... = Sup {\x \ (Inf (f ` F)) |f . (\Y\F. f Y \ Y)}"
by (simp add: finite_inf_Sup)
also have "... = Sup {Sup {Inf (f ` F) \ b | b . b \ x} |f . (\Y\F. f Y \ Y)}"
by (subst inf_commute) (simp add: finite_inf_Sup)
also have "... \ \(Inf ` {insert (f x) (f ` F) |f. f x \ x \ (\Y\F. f Y \ Y)})"
apply (rule Sup_least, clarsimp)+
apply (subst inf_commute, simp)
done
finally show "\x \ \(Sup ` F) \ \(Inf ` {insert (f x) (f ` F) |f. f x \ x \ (\Y\F. f Y \ Y)})"
by simp
qed
subclass complete_distrib_lattice
by (standard, rule finite_Inf_Sup)
end
instantiation finite_3 :: finite_lattice
begin
definition "\A = (if a\<^sub>1 \ A then a\<^sub>1 else if a\<^sub>2 \ A then a\<^sub>2 else a\<^sub>3)"
definition "\A = (if a\<^sub>3 \ A then a\<^sub>3 else if a\<^sub>2 \ A then a\<^sub>2 else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>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\<^sub>1"
definition [simp]: "1 = a\<^sub>2"
definition
"x + y = (case (x, y) of
(a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
| (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2
| _ \<Rightarrow> a\<^sub>3)"
definition "- x = (case x of a\<^sub>1 \ a\<^sub>1 | a\<^sub>2 \ a\<^sub>3 | a\<^sub>3 \ a\<^sub>2)"
definition "x - y = x + (- y :: finite_3)"
definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \ a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \ a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \ a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \ a\<^sub>3 | _ \ a\<^sub>1)"
definition "inverse = (\x :: finite_3. x)"
definition "x div y = x * inverse (y :: finite_3)"
definition "x mod y = (case y of a\<^sub>1 \ x | _ \ a\<^sub>1)"
definition "abs = (\x. case x of a\<^sub>3 \ a\<^sub>2 | _ \ x)"
definition "sgn = (\x :: finite_3. x)"
instance
by standard
(subproofs
\<open>simp_all add: plus_finite_3_def uminus_finite_3_def minus_finite_3_def
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\<close>)
end
lemma two_finite_3 [simp]:
"2 = a\<^sub>3"
by (simp add: numeral.simps plus_finite_3_def)
lemma dvd_finite_3_unfold:
"x dvd y \ x = a\<^sub>2 \ x = a\<^sub>3 \ y = a\<^sub>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\<^sub>3 \ a\<^sub>2 | _ \ x)"
definition [simp]: "unit_factor = (id :: finite_3 \ _)"
definition [simp]: "euclidean_size x = (case x of a\<^sub>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
(subproofs
\<open>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\<close>)
end
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3
datatype (plugins only: code "quickcheck" extraction) finite_4 =
a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
notation (output) a\<^sub>3 ("a\<^sub>3")
notation (output) a\<^sub>4 ("a\<^sub>4")
lemma UNIV_finite_4:
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}"
by (auto intro: finite_4.exhaust)
instantiation finite_4 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
definition
"enum_all P \ P a\<^sub>1 \ P a\<^sub>2 \ P a\<^sub>3 \ P a\<^sub>4"
definition
"enum_ex P \ P a\<^sub>1 \ P a\<^sub>2 \ P a\<^sub>3 \ P a\<^sub>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 \<open>\<^term>\<open>a\<^sub>1\<close> $<$ \<^term>\<open>a\<^sub>2\<close>,\<^term>\<open>a\<^sub>3\<close> $<$ \<^term>\<open>a\<^sub>4\<close>,
but \<^term>\<open>a\<^sub>2\<close> and \<^term>\<open>a\<^sub>3\<close> are incomparable.\<close>
definition
"x < y \ (case (x, y) of
(a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"x \ y \ (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True
| (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"\A = (if a\<^sub>1 \ A \ a\<^sub>2 \ A \ a\<^sub>3 \ A then a\<^sub>1 else if a\<^sub>2 \ A then a\<^sub>2 else if a\<^sub>3 \ A then a\<^sub>3 else a\<^sub>4)"
definition
"\A = (if a\<^sub>4 \ A \ a\<^sub>2 \ A \ a\<^sub>3 \ A then a\<^sub>4 else if a\<^sub>2 \ A then a\<^sub>2 else if a\<^sub>3 \ A then a\<^sub>3 else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>4"
definition
"x \ y = (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| _ \<Rightarrow> a\<^sub>4)"
definition
"x \ y = (case (x, y) of
(a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>4 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>4
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| _ \<Rightarrow> a\<^sub>1)"
instance
by standard
(subproofs
\<open>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\<close>)
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\<^sub>1 \ a\<^sub>4 | a\<^sub>2 \ a\<^sub>3 | a\<^sub>3 \ a\<^sub>2 | a\<^sub>4 \ a\<^sub>1)"
definition "x - y = x \ - (y :: finite_4)"
instance
by standard
(subproofs
\<open>simp_all add: inf_finite_4_def sup_finite_4_def uminus_finite_4_def minus_finite_4_def
split: finite_4.splits\<close>)
end
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4
datatype (plugins only: code "quickcheck" extraction) finite_5 =
a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5
notation (output) a\<^sub>1 ("a\<^sub>1")
notation (output) a\<^sub>2 ("a\<^sub>2")
notation (output) a\<^sub>3 ("a\<^sub>3")
notation (output) a\<^sub>4 ("a\<^sub>4")
notation (output) a\<^sub>5 ("a\<^sub>5")
lemma UNIV_finite_5:
"UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}"
by (auto intro: finite_5.exhaust)
instantiation finite_5 :: enum
begin
definition
"enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
definition
"enum_all P \ P a\<^sub>1 \ P a\<^sub>2 \ P a\<^sub>3 \ P a\<^sub>4 \ P a\<^sub>5"
definition
"enum_ex P \ P a\<^sub>1 \ P a\<^sub>2 \ P a\<^sub>3 \ P a\<^sub>4 \ P a\<^sub>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 \<open>The non-distributive pentagon lattice $N_5$\<close>
definition
"x < y \ (case (x, y) of
(a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>4, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"x \ y \ (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> True
| (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True
| (a\<^sub>5, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)"
definition
"\A =
(if a\<^sub>1 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>1
else if a\<^sub>2 \<in> A then a\<^sub>2
else if a\<^sub>3 \<in> A then a\<^sub>3
else if a\<^sub>4 \<in> A then a\<^sub>4
else a\<^sub>5)"
definition
"\A =
(if a\<^sub>5 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>5
else if a\<^sub>3 \<in> A then a\<^sub>3
else if a\<^sub>2 \<in> A then a\<^sub>2
else if a\<^sub>4 \<in> A then a\<^sub>4
else a\<^sub>1)"
definition [simp]: "bot = a\<^sub>1"
definition [simp]: "top = a\<^sub>5"
definition
"x \ y = (case (x, y) of
(a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>1
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
| _ \<Rightarrow> a\<^sub>5)"
definition
"x \ y = (case (x, y) of
(a\<^sub>5, _) \<Rightarrow> a\<^sub>5 | (_, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>5 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>5
| (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
| (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
| (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
| _ \<Rightarrow> a\<^sub>1)"
instance
by standard
(subproofs
\<open>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\<close>)
end
instance finite_5 :: complete_lattice ..
hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5
subsection \<open>Closing up\<close>
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
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