(* Author: Florian Haftmann, TU Muenchen *)

 header {* Finite types as explicit enumerations *}

 theory Enum

 imports Map Groups_List

 begin

 subsection {* Class @{text enum} *}

 class enum =

   fixes enum :: "'a list"

   fixes enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"

   fixes enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"

   assumes UNIV_enum: "UNIV = set enum"

     and enum_distinct: "distinct enum"

   assumes enum_all_UNIV: "enum_all P \<longleftrightarrow> Ball UNIV P"

   assumes enum_ex_UNIV: "enum_ex P \<longleftrightarrow> Bex UNIV P" 

    -- {* tailored towards simple instantiation *}

 begin

 subclass finite proof

 qed (simp add: UNIV_enum)

 lemma enum_UNIV:

   "set enum = UNIV"

   by (simp only: UNIV_enum)

 lemma in_enum: "x \<in> set enum"

   by (simp add: enum_UNIV)

 lemma enum_eq_I:

   assumes "\<And>x. x \<in> 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 {* Implementations using @{class enum} *}

 subsubsection {* Unbounded operations and quantifiers *}

 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 \<Rightarrow> 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]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"

   by simp

 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"

   by simp

 lemma exists1_code [code]: "(\<exists>!x. P x) \<longleftrightarrow> 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 \<noteq> a"

         from filter_enum obtain us vs

           where enum_eq: "enum = us @ [a] @ vs"

           and "\<forall> x \<in> set us. \<not> P x"

           and "\<forall> x \<in> set vs. \<not> 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 \<noteq> 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 \<rightharpoonup> (Eval) "(fn '_ => raise Match)"

 subsubsection {* Equality and order on functions *}

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

 begin

 definition

   "HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> 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 \<longleftrightarrow> enum_all (%x. f x = g x)"

   by (auto simp add: equal fun_eq_iff)

 lemma [code nbe]:

   "HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"

   by (fact equal_refl)

 lemma order_fun [code]:

   fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"

   shows "f \<le> g \<longleftrightarrow> enum_all (\<lambda>x. f x \<le> g x)"

     and "f < g \<longleftrightarrow> f \<le> g \<and> enum_ex (\<lambda>x. f x \<noteq> g x)"

   by (simp_all add: fun_eq_iff le_fun_def order_less_le)

 subsubsection {* Operations on relations *}

 lemma [code]:

   "Id = image (\<lambda>x. (x, x)) (set Enum.enum)"

   by (auto intro: imageI in_enum)

 lemma tranclp_unfold [code]:

   "tranclp r a b \<longleftrightarrow> (a, b) \<in> trancl {(x, y). r x y}"

   by (simp add: trancl_def)

 lemma rtranclp_rtrancl_eq [code]:

   "rtranclp r x y \<longleftrightarrow> (x, y) \<in> rtrancl {(x, y). r x y}"

   by (simp add: rtrancl_def)

 lemma max_ext_eq [code]:

   "max_ext R = {(X, Y). finite X \<and> finite Y \<and> Y \<noteq> {} \<and> (\<forall>x. x \<in> X \<longrightarrow> (\<exists>xa \<in> Y. (x, xa) \<in> R))}"

   by (auto simp add: max_ext.simps)

 lemma max_extp_eq [code]:

   "max_extp r x y \<longleftrightarrow> (x, y) \<in> 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 \<or> (f x \<le> f y \<and> (x, y) \<in> R)}"

   by (auto simp add: mlex_prod_def)

 subsubsection {* Bounded accessible part *}

 primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set" 

 where

   "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"

 | "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"

 lemma bacc_subseteq_acc:

   "bacc r n \<subseteq> Wellfounded.acc r"

   by (induct n) (auto intro: acc.intros)

 lemma bacc_mono:

   "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"

   by (induct rule: dec_induct) auto

 lemma bacc_upper_bound:

   "bacc (r :: ('a \<times> 'a) set)  (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"

 proof -

   have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)

   moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> 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 \<subseteq> (\<Union>n. bacc r n)"

 proof

   fix x

   assume "x : Wellfounded.acc r"

   then have "\<exists> n. x : bacc r n"

   proof (induct x arbitrary: rule: acc.induct)

     case (accI x)

     then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp

     from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..

     obtain n where "\<And>y. (y, x) : r \<Longrightarrow> 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 `finite r` 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 : (UN n. bacc r n)" by auto

 qed

 lemma acc_bacc_eq:

   fixes A :: "('a :: finite \<times> '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 \<times> 'a) list"

   shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"

   by (simp add: card_UNIV_def acc_bacc_eq)

 subsection {* Default instances for @{class enum} *}

 lemma map_of_zip_enum_is_Some:

   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"

   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"

 proof -

   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>

     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>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\<Colon>enum list"

   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"

       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"

     and map_of: "the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys)"

   shows "xs = ys"

 proof -

   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"

   proof

     fix x :: 'a

     from length map_of_zip_enum_is_Some obtain y1 y2

       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"

         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast

     moreover from map_of

       have "the (map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x) = the (map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x)"

       by (auto dest: fun_cong)

     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>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 \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"

 where

   "all_n_lists P n \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)"

 lemma [code]:

   "all_n_lists P n \<longleftrightarrow> (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 \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"

 where

   "ex_n_lists P n \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)"

 lemma [code]:

   "ex_n_lists P n \<longleftrightarrow> (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 (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (List.n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"

 definition

   "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"

 definition

   "enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"

 instance proof

   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"

   proof (rule UNIV_eq_I)

     fix f :: "'a \<Rightarrow> 'b"

     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"

       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)

     then show "f \<in> 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 \<Colon> ('a \<Rightarrow> '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 \<Rightarrow> 'b) \<Rightarrow> bool) = Ball UNIV P"

   proof

     assume "enum_all P"

     show "Ball UNIV P"

     proof

       fix f :: "'a \<Rightarrow> 'b"

       have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>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 \<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 \<Rightarrow> 'b) \<Rightarrow> 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 \<circ> map_of (zip (enum \<Colon> 'a\<Colon>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 \<circ> map_of (zip (enum \<Colon> 'a\<Colon>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 \<Colon> 'a\<Colon>{enum, equal} list)

   in map (\<lambda>ys. the o 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 o 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 o 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 (sublists enum)"

 definition

   "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"

 definition

   "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"

 instance proof

 qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists

   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 \<longleftrightarrow> P False \<and> P True"

 definition

   "enum_ex P \<longleftrightarrow> P False \<or> 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 default

   (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 \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))"

 definition

   "enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>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 \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))"

 definition

   "enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>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 {* Small finite types *}

 text {* We define small finite types for the use in Quickcheck *}

 datatype_new 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 \<Rightarrow> finite_1 \<Rightarrow> bool"

 where

   "x < (y :: finite_1) \<longleftrightarrow> False"

 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"

 where

   "x \<le> (y :: finite_1) \<longleftrightarrow> 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}"

 by intro_classes (simp_all add: less_finite_1_def)

 instantiation finite_1 :: complete_lattice

 begin

 definition [simp]: "Inf = (\<lambda>_. a\<^sub>1)"

 definition [simp]: "Sup = (\<lambda>_. a\<^sub>1)"

 definition [simp]: "bot = a\<^sub>1"

 definition [simp]: "top = a\<^sub>1"

 definition [simp]: "inf = (\<lambda>_ _. a\<^sub>1)"

 definition [simp]: "sup = (\<lambda>_ _. a\<^sub>1)"

 instance by intro_classes(simp_all add: less_eq_finite_1_def)

 end

 instance finite_1 :: complete_distrib_lattice

 by intro_classes(simp_all add: INF_def SUP_def)

 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") = {*

   fn _ => fn _ => fn ct => case term_of ct of

     Const (@{const_name a\<^sub>1}, _) => NONE

   | _ => SOME (mk_meta_eq @{thm finite_1_eq})

 *}

 instantiation finite_1 :: complete_boolean_algebra begin

 definition [simp]: "op - = (\<lambda>_ _. a\<^sub>1)"

 definition [simp]: "uminus = (\<lambda>_. 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, Divides.div, sgn_if, inverse}"

 begin

 definition [simp]: "Groups.zero = a\<^sub>1"

 definition [simp]: "Groups.one = a\<^sub>1"

 definition [simp]: "op + = (\<lambda>_ _. a\<^sub>1)"

 definition [simp]: "op * = (\<lambda>_ _. a\<^sub>1)"

 definition [simp]: "op div = (\<lambda>_ _. a\<^sub>1)" 

 definition [simp]: "op mod = (\<lambda>_ _. a\<^sub>1)" 

 definition [simp]: "abs = (\<lambda>_. a\<^sub>1)"

 definition [simp]: "sgn = (\<lambda>_. a\<^sub>1)"

 definition [simp]: "inverse = (\<lambda>_. a\<^sub>1)"

 definition [simp]: "op / = (\<lambda>_ _. 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_new 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 \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2"

 definition

   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> 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 \<Rightarrow> finite_2 \<Rightarrow> bool"

 where

   "x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2"

 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"

 where

   "x \<le> y \<longleftrightarrow> x = y \<or> 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 "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else a\<^sub>2)"

 definition "\<Squnion>A = (if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"

 definition [simp]: "bot = a\<^sub>1"

 definition [simp]: "top = a\<^sub>2"

 definition "x \<sqinter> y = (if x = a\<^sub>1 \<or> y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)"

 definition "x \<squnion> y = (if x = a\<^sub>2 \<or> 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 \<in> A"

   then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>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_distrib_lattice

 by(intro_classes)(auto simp add: INF_def SUP_def sup_finite_2_def inf_finite_2_def Inf_finite_2_def Sup_finite_2_def)

 instance finite_2 :: complete_linorder ..

 instantiation finite_2 :: "{field_inverse_zero, abs_if, ring_div, semiring_div_parity, sgn_if}" 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>2) \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"

 definition "uminus = (\<lambda>x :: finite_2. x)"

 definition "op - = (op + :: finite_2 \<Rightarrow> _)"

 definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"

 definition "inverse = (\<lambda>x :: finite_2. x)"

 definition "op / = (op * :: finite_2 \<Rightarrow> _)"

 definition "abs = (\<lambda>x :: finite_2. x)"

 definition "op div = (op / :: finite_2 \<Rightarrow> _)"

 definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"

 definition "sgn = (\<lambda>x :: finite_2. x)"

 instance

 by intro_classes

   (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 abs_finite_2_def div_finite_2_def mod_finite_2_def sgn_finite_2_def

      split: finite_2.splits)

 end

 hide_const (open) a\<^sub>1 a\<^sub>2

 datatype_new 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 \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3"

 definition

   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> 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

 instantiation finite_3 :: linorder

 begin

 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"

 where

   "x < y = (case x of a\<^sub>1 \<Rightarrow> y \<noteq> a\<^sub>1 | a\<^sub>2 \<Rightarrow> y = a\<^sub>3 | a\<^sub>3 \<Rightarrow> False)"

 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"

 where

   "x \<le> y \<longleftrightarrow> x = y \<or> 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 \<dots>" by simp

 qed intro_classes

 instantiation finite_3 :: complete_lattice

 begin

 definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>3)"

 definition "\<Squnion>A = (if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>2 \<in> 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 \<Rightarrow> _)"

 definition [simp]: "sup = (max :: finite_3 \<Rightarrow> _)"

 instance

 proof

   fix x :: finite_3 and A

   assume "x \<in> A"

   then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"

     by(case_tac [!] x)(auto simp add: Inf_finite_3_def Sup_finite_3_def less_eq_finite_3_def less_finite_3_def)

 next

   fix A and z :: finite_3

   assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"

   then show "z \<le> \<Sqinter>A"

     by(cases z)(auto simp add: Inf_finite_3_def less_eq_finite_3_def less_finite_3_def)

 next

   fix A and z :: finite_3

   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"

   show "\<Squnion>A \<le> z"

     by(auto simp add: Sup_finite_3_def less_eq_finite_3_def less_finite_3_def dest: *)

 qed(auto simp add: Inf_finite_3_def Sup_finite_3_def)

 end

 instance finite_3 :: complete_distrib_lattice

 proof

   fix a :: finite_3 and B

   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"

   proof(cases a "\<Sqinter>B" rule: finite_3.exhaust[case_product finite_3.exhaust])

     case a\<^sub>2_a\<^sub>3

     then have "\<And>x. x \<in> B \<Longrightarrow> x = a\<^sub>3"

       by(case_tac x)(auto simp add: Inf_finite_3_def split: split_if_asm)

     then show ?thesis using a\<^sub>2_a\<^sub>3

       by(auto simp add: INF_def Inf_finite_3_def max_def less_eq_finite_3_def less_finite_3_def split: split_if_asm)

   qed(auto simp add: INF_def Inf_finite_3_def max_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)

   show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"

     by(cases a "\<Squnion>B" rule: finite_3.exhaust[case_product finite_3.exhaust])

       (auto simp add: SUP_def Sup_finite_3_def min_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)

 qed

 instance finite_3 :: complete_linorder ..

 instantiation finite_3 :: "{field_inverse_zero, abs_if, ring_div, semiring_div, sgn_if}" 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 \<Rightarrow> a\<^sub>1 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2)"

 definition "x - y = x + (- y :: finite_3)"

 definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"

 definition "inverse = (\<lambda>x :: finite_3. x)" 

 definition "x / y = x * inverse (y :: finite_3)"

 definition "abs = (\<lambda>x :: finite_3. x)"

 definition "op div = (op / :: finite_3 \<Rightarrow> _)"

 definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"

 definition "sgn = (\<lambda>x. case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"

 instance

 by intro_classes

   (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 abs_finite_3_def div_finite_3_def mod_finite_3_def sgn_finite_3_def

        less_finite_3_def

      split: finite_3.splits)

 end

 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3

 datatype_new 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 \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4"

 definition

   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> 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 :: complete_lattice begin

 text {* @{term a\<^sub>1} $<$ @{term a\<^sub>2},@{term a\<^sub>3} $<$ @{term a\<^sub>4},

   but @{term a\<^sub>2} and @{term a\<^sub>3} are incomparable. *}

 definition

   "x < y \<longleftrightarrow> (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 \<le> y \<longleftrightarrow> (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

   "\<Sqinter>A = (if a\<^sub>1 \<in> A \<or> a\<^sub>2 \<in> A \<and> 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 a\<^sub>4)"

 definition

   "\<Squnion>A = (if a\<^sub>4 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>4 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>1)"

 definition [simp]: "bot = a\<^sub>1"

 definition [simp]: "top = a\<^sub>4"

 definition

   "x \<sqinter> 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 \<squnion> 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

 proof

   fix A and z :: finite_4

   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"

   show "\<Squnion>A \<le> z"

     by(auto simp add: Sup_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)

 next

   fix A and z :: finite_4

   assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"

   show "z \<le> \<Sqinter>A"

     by(auto simp add: Inf_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)

 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_distrib_lattice

 proof

   fix a :: finite_4 and B

   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"

     by(cases a "\<Sqinter>B" rule: finite_4.exhaust[case_product finite_4.exhaust])

       (auto simp add: sup_finite_4_def Inf_finite_4_def INF_def split: finite_4.splits split_if_asm)

   show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"

     by(cases a "\<Squnion>B" rule: finite_4.exhaust[case_product finite_4.exhaust])

       (auto simp add: inf_finite_4_def Sup_finite_4_def SUP_def split: finite_4.splits split_if_asm)

 qed

 instantiation finite_4 :: complete_boolean_algebra begin

 definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>4 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2 | a\<^sub>4 \<Rightarrow> a\<^sub>1)"

 definition "x - y = x \<sqinter> - (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\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4

 datatype_new 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 \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4 \<and> P a\<^sub>5"

 definition

   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4 \<or> 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 :: complete_lattice

 begin

 text {* The non-distributive pentagon lattice $N_5$ *}

 definition

   "x < y \<longleftrightarrow> (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 \<le> y \<longleftrightarrow> (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

   "\<Sqinter>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

   "\<Squnion>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 \<sqinter> 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 \<squnion> 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 

 proof intro_classes

   fix A and z :: finite_5

   assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"

   show "z \<le> \<Sqinter>A"

     by(auto simp add: less_eq_finite_5_def Inf_finite_5_def split: finite_5.splits split_if_asm dest!: *)

 next

   fix A and z :: finite_5

   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"

   show "\<Squnion>A \<le> z"

     by(auto simp add: less_eq_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm dest!: *)

 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 split_if_asm)

 end

 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>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