diff --git a/src/HOL/Enum.thy b/src/HOL/Enum.thy
--- a/src/HOL/Enum.thy
+++ b/src/HOL/Enum.thy
@@ -1,1161 +1,1160 @@
 (* 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 \<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" 
    \<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 \<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 \<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 (\<lambda>x. f x \<in> 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 \<open>An executable choice operator\<close>
 
 definition
   [code del]: "enum_the = The"
 
 lemma [code]:
   "The P = (case filter P enum of [x] \<Rightarrow> x | _ \<Rightarrow> 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 \<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 \<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::enum \<Rightarrow> 'b::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 \<open>Operations on relations\<close>
 
 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 \<open>Bounded accessible part\<close>
 
 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 \<in> Wellfounded.acc r"
   then have "\<exists>n. x \<in> bacc r n"
   proof (induct x arbitrary: rule: acc.induct)
     case (accI x)
     then have "\<forall>y. \<exists> n. (y, x) \<in> r \<longrightarrow> y \<in> 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) \<in> r \<Longrightarrow> y \<in> bacc r n"
     proof
       fix y assume y: "(y, x) \<in> r"
       with n have "y \<in> bacc r (n y)" by auto
       moreover have "n y <= Max ((\<lambda>(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 \<in> bacc r (Max ((\<lambda>(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 \<in> (\<Union>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 \<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 "\<exists>y. map_of (zip (enum :: 'a::enum list) ys) x = Some y"
 proof -
   from assms have "x \<in> set (enum :: 'a::enum list) \<longleftrightarrow>
     (\<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 \<circ> map_of (zip (enum :: 'a::enum list) xs) = the \<circ> 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 \<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 \<circ> map_of (zip (enum::'a list) ys)) (List.n_lists (length (enum::'a::enum list)) enum)"
 
 definition
   "enum_all P = all_n_lists (\<lambda>bs. P (the \<circ> map_of (zip enum bs))) (length (enum :: 'a list))"
 
 definition
   "enum_ex P = ex_n_lists (\<lambda>bs. P (the \<circ> map_of (zip enum bs))) (length (enum :: 'a list))"
 
 instance proof
   show "UNIV = set (enum :: ('a \<Rightarrow> 'b) list)"
   proof (rule UNIV_eq_I)
     fix f :: "'a \<Rightarrow> 'b"
     have "f = the \<circ> 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 \<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 :: ('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 :: '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 \<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 :: '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 \<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 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 \<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 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 \<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 \<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 \<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)
+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 = (\<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 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]: "(-) = (\<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, modulo, sgn, inverse}"
 begin
 definition [simp]: "Groups.zero = a\<^sub>1"
 definition [simp]: "Groups.one = a\<^sub>1"
 definition [simp]: "(+) = (\<lambda>_ _. a\<^sub>1)"
 definition [simp]: "(*) = (\<lambda>_ _. a\<^sub>1)"
 definition [simp]: "(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]: "divide = (\<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 (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 \<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(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) \<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 "(-) = ((+) :: 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 "divide = ((*) :: 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 "abs = (\<lambda>x :: finite_2. x)"
 definition "sgn = (\<lambda>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 \<longleftrightarrow> x = a\<^sub>2 \<or> 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 \<Rightarrow> _)"
 definition [simp]: "unit_factor = (id :: finite_2 \<Rightarrow> _)"
 definition [simp]: "euclidean_size x = (case x of a\<^sub>1 \<Rightarrow> 0 | a\<^sub>2 \<Rightarrow> 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 \<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
 
 lemma finite_3_not_eq_unfold:
   "x \<noteq> a\<^sub>1 \<longleftrightarrow> x \<in> {a\<^sub>2, a\<^sub>3}"
   "x \<noteq> a\<^sub>2 \<longleftrightarrow> x \<in> {a\<^sub>1, a\<^sub>3}"
   "x \<noteq> a\<^sub>3 \<longleftrightarrow> x \<in> {a\<^sub>1, a\<^sub>2}"
   by (cases x; simp)+
 
 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
 
 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 \<sqinter> Inf A"
   assumes Sup_finite_empty: "Sup {} = Inf UNIV"
   assumes Sup_finite_insert: "Sup (insert a A) = a \<squnion> 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 \<in> A \<Longrightarrow> \<Sqinter>A \<le> 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 \<in> A \<Longrightarrow> x \<le> \<Squnion>A"
     by (metis Set.set_insert insert_absorb2 local.Sup_finite_insert local.sup.absorb_iff2)
 next
   fix A z
   have "\<Squnion> UNIV = z \<squnion> \<Squnion>UNIV"
     by (subst Sup_finite_insert [symmetric], simp add: insert_UNIV)
   from this have [simp]: "z \<le> \<Squnion>UNIV"
     using local.le_iff_sup by auto
   have "(\<forall> x. x \<in> A \<longrightarrow> z \<le> x) \<longrightarrow> z \<le> \<Sqinter>A"
-    apply (rule finite_induct [of A "\<lambda> A . (\<forall> x. x \<in> A \<longrightarrow> z \<le> x) \<longrightarrow> z \<le> \<Sqinter>A"])
-    by (simp_all add: Inf_finite_empty Inf_finite_insert)
+    by (rule finite_induct [of A "\<lambda> A . (\<forall> x. x \<in> A \<longrightarrow> z \<le> x) \<longrightarrow> z \<le> \<Sqinter>A"])
+      (simp_all add: Inf_finite_empty Inf_finite_insert)
   from this show "(\<And>x. x \<in> A \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> \<Sqinter>A"
     by simp
 
   have "\<Sqinter> UNIV = z \<sqinter> \<Sqinter>UNIV"
     by (subst Inf_finite_insert [symmetric], simp add: insert_UNIV)
   from this have [simp]: "\<Sqinter>UNIV \<le> z"
     by (simp add: local.inf.absorb_iff2)
   have "(\<forall> x. x \<in> A \<longrightarrow> x \<le> z) \<longrightarrow> \<Squnion>A \<le> z"
     by (rule finite_induct [of A "\<lambda> A . (\<forall> x. x \<in> A \<longrightarrow> x \<le> z) \<longrightarrow> \<Squnion>A \<le> z" ], simp_all add: Sup_finite_empty Sup_finite_insert)
   from this show " (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> \<Squnion>A \<le> z"
     by blast
 next
   show "\<Sqinter>{} = \<top>"
     by (simp add: Inf_finite_empty top_finite_def)
   show " \<Squnion>{} = \<bottom>"
     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 \<sqinter> (Sup A) = Sup {a \<sqinter> b | b . b \<in> A}"
 proof (rule finite_induct [of A "\<lambda> A . a \<sqinter> (Sup A) = Sup {a \<sqinter> b | b . b \<in> A}"], simp_all)
   fix x::"'a"
   fix F
   assume "x \<notin> F"
   assume [simp]: "a \<sqinter> \<Squnion>F = \<Squnion>{a \<sqinter> b |b. b \<in> F}"
   have [simp]: " insert (a \<sqinter> x) {a \<sqinter> b |b. b \<in> F} = {a \<sqinter> b |b. b = x \<or> b \<in> F}"
     by blast
   have "a \<sqinter> (x \<squnion> \<Squnion>F) = a \<sqinter> x \<squnion> a \<sqinter> \<Squnion>F"
     by (simp add: inf_sup_distrib1)
   also have "... = a \<sqinter> x \<squnion> \<Squnion>{a \<sqinter> b |b. b \<in> F}"
     by simp
   also have "... = \<Squnion>{a \<sqinter> b |b. b = x \<or> b \<in> F}"
     by (unfold Sup_insert[THEN sym], simp)
   finally show "a \<sqinter> (x \<squnion> \<Squnion>F) = \<Squnion>{a \<sqinter> b |b. b = x \<or> b \<in> F}"
     by simp
 qed
 
 lemma finite_Inf_Sup: "\<Sqinter>(Sup ` A) \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})"
 proof (rule finite_induct [of A "\<lambda>A. \<Sqinter>(Sup ` A) \<le> \<Squnion>(Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})"], simp_all add: finite_UnionD)
   fix x::"'a set"
   fix F
   assume "x \<notin> F"
   have [simp]: "{\<Squnion>x \<sqinter> b |b . b \<in> Inf ` {f ` F |f. \<forall>Y\<in>F. f Y \<in> Y} } = {\<Squnion>x \<sqinter> (Inf (f ` F)) |f  . (\<forall>Y\<in>F. f Y \<in> Y)}"
     by auto
   define fa where "fa = (\<lambda> (b::'a) f Y . (if Y = x then b else f Y))"
   have "\<And>f b. \<forall>Y\<in>F. f Y \<in> Y \<Longrightarrow> b \<in> x \<Longrightarrow> insert b (f ` (F \<inter> {Y. Y \<noteq> x})) = insert (fa b f x) (fa b f ` F) \<and> fa b f x \<in> x \<and> (\<forall>Y\<in>F. fa b f Y \<in> Y)"
     by (auto simp add: fa_def)
   from this have B: "\<And>f b. \<forall>Y\<in>F. f Y \<in> Y \<Longrightarrow> b \<in> x \<Longrightarrow> fa b f ` ({x} \<union> F) \<in> {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)}"
     by blast
   have [simp]: "\<And>f b. \<forall>Y\<in>F. f Y \<in> Y \<Longrightarrow> b \<in> x \<Longrightarrow> b \<sqinter> (\<Sqinter>x\<in>F. f x)  \<le> \<Squnion>(Inf ` {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> 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 "\<Sqinter>(Sup ` F) \<le> \<Squnion>(Inf ` {f ` F |f. \<forall>Y\<in>F. f Y \<in> Y})"
 
   from this have "\<Squnion>x \<sqinter> \<Sqinter>(Sup ` F) \<le> \<Squnion>x \<sqinter> \<Squnion>(Inf ` {f ` F |f. \<forall>Y\<in>F. f Y \<in> Y})"
-    apply simp
-    using inf.coboundedI2 by blast
+    using inf.coboundedI2 by auto
   also have "... = Sup {\<Squnion>x \<sqinter> (Inf (f ` F)) |f  .  (\<forall>Y\<in>F. f Y \<in> Y)}"
     by (simp add: finite_inf_Sup)
 
   also have "... = Sup {Sup {Inf (f ` F) \<sqinter> b | b . b \<in> x} |f  .  (\<forall>Y\<in>F. f Y \<in> Y)}"
-    apply (subst inf_commute)
-    by (simp add: finite_inf_Sup)
+    by (subst inf_commute) (simp add: finite_inf_Sup)
 
   also have "... \<le> \<Squnion>(Inf ` {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)})"
     apply (rule Sup_least, clarsimp)+
-    by (subst inf_commute, simp)
+    apply (subst inf_commute, simp)
+    done
 
   finally show "\<Squnion>x \<sqinter> \<Sqinter>(Sup ` F) \<le> \<Squnion>(Inf ` {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)})"
     by simp
 qed
 
 subclass complete_distrib_lattice
   by (standard, rule finite_Inf_Sup)
 end
 
 instantiation finite_3 :: finite_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
 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 \<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 div y = x * inverse (y :: finite_3)"
 definition "x mod y = (case y of a\<^sub>1 \<Rightarrow> x | _ \<Rightarrow> a\<^sub>1)"
 definition "abs = (\<lambda>x. case x of a\<^sub>3 \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> x)"
 definition "sgn = (\<lambda>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 \<longleftrightarrow> x = a\<^sub>2 \<or> x = a\<^sub>3 \<or> 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 \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> x)"
 definition [simp]: "unit_factor = (id :: finite_3 \<Rightarrow> _)"
 definition [simp]: "euclidean_size x = (case x of a\<^sub>1 \<Rightarrow> 0 | _ \<Rightarrow> 1)"
 definition [simp]: "division_segment (x :: finite_3) = 1"
 instance
 proof
   fix x :: finite_3
   assume "x \<noteq> 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 \<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 :: 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 \<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
   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 \<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 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 \<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 :: finite_lattice
 begin
 
 text \<open>The non-distributive pentagon lattice $N_5$\<close>
 
 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
   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