Theory Enum

(* Author: Florian Haftmann, TU Muenchen *)

section ‹Finite types as explicit enumerations›

theory Enum
imports Map Groups_List
begin

subsection ‹Class ‹enum››

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

subclass finite proof
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 ‹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 (λx. f x ∈ B) enum_class.enum)"
  unfolding vimage_def Collect_code ..

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

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

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

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

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


subsubsection ‹An executable choice operator›

definition
  [code del]: "enum_the = The"

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

declare [[code abort: enum_the]]

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


subsubsection ‹Equality and order on functions›

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

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

instance proof
qed (simp_all add: equal_fun_def fun_eq_iff enum_UNIV)

end

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

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

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


subsubsection ‹Operations on relations›

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

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

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

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

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

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

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

lemma [code]:
  fixes xs :: "('a::finite × 'a) list"
  shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"
  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 :: 'a::enum list)"
  shows "∃y. map_of (zip (enum :: 'a::enum list) ys) x = Some y"
proof -
  from assms have "x ∈ set (enum :: 'a::enum list) ⟷
    (∃y. map_of (zip (enum :: 'a::enum list) ys) x = Some y)"
    by (auto intro!: map_of_zip_is_Some)
  then show ?thesis using enum_UNIV by auto
qed

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

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

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

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

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

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

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

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

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

instance proof
  show "UNIV = set (enum :: ('a ⇒ 'b) list)"
  proof (rule UNIV_eq_I)
    fix f :: "'a ⇒ 'b"
    have "f = the ∘ map_of (zip (enum :: 'a::enum list) (map f enum))"
      by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
    then show "f ∈ set enum"
      by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
  qed
next
  from map_of_zip_enum_inject
  show "distinct (enum :: ('a ⇒ 'b) list)"
    by (auto intro!: inj_onI simp add: enum_fun_def
      distinct_map distinct_n_lists enum_distinct set_n_lists)
next
  fix P
  show "enum_all (P :: ('a ⇒ 'b) ⇒ bool) = Ball UNIV P"
  proof
    assume "enum_all P"
    show "Ball UNIV P"
    proof
      fix f :: "'a ⇒ 'b"
      have f: "f = the ∘ map_of (zip (enum :: 'a::enum list) (map f enum))"
        by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
      from ‹enum_all P› have "P (the ∘ map_of (zip enum (map f enum)))"
        unfolding enum_all_fun_def all_n_lists_def
        apply (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 ‹P f› this have "P (the ∘ map_of (zip (enum :: 'a::enum list) (map f enum)))"
      by auto
    from  this show "enum_ex P"
      unfolding enum_ex_fun_def ex_n_lists_def
      apply (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 (λys. the ∘ 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 (λbs. P (the ∘ map_of (zip enum_a bs))) (length enum_a))"
  by (simp only: enum_all_fun_def Let_def)

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

instantiation set :: (enum) enum
begin

definition
  "enum = map set (subseqs enum)"

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

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

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

end

instantiation unit :: enum
begin

definition
  "enum = [()]"

definition
  "enum_all P = P ()"

definition
  "enum_ex P = P ()"

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

end

instantiation bool :: enum
begin

definition
  "enum = [False, True]"

definition
  "enum_all P ⟷ P False ∧ P True"

definition
  "enum_ex P ⟷ P False ∨ P True"

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

end

instantiation prod :: (enum, enum) enum
begin

definition
  "enum = List.product enum enum"

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

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

 
instance
  by standard
    (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 (λx. P (Inl x)) ∧ enum_all (λx. P (Inr x))"

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

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

end

instantiation option :: (enum) enum
begin

definition
  "enum = None # map Some enum"

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

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

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

end


subsection ‹Small finite types›

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

datatype (plugins only: code "quickcheck" extraction) finite_1 =
  a1

notation (output) a1  ("a1")

lemma UNIV_finite_1:
  "UNIV = {a1}"
  by (auto intro: finite_1.exhaust)

instantiation finite_1 :: enum
begin

definition
  "enum = [a1]"

definition
  "enum_all P = P a1"

definition
  "enum_ex P = P a1"

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 = (λ_. a1)"
definition [simp]: "Sup = (λ_. a1)"
definition [simp]: "bot = a1"
definition [simp]: "top = a1"
definition [simp]: "inf = (λ_ _. a1)"
definition [simp]: "sup = (λ_ _. a1)"

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 = a1"
by(cases x) simp

simproc_setup finite_1_eq ("x::finite_1") = ‹
  K (K (fn ct =>
    (case Thm.term_of ct of
      Const (const_name‹a1›, _) => NONE
    | _ => SOME (mk_meta_eq @{thm finite_1_eq}))))
›

instantiation finite_1 :: complete_boolean_algebra
begin
definition [simp]: "(-) = (λ_ _. a1)"
definition [simp]: "uminus = (λ_. a1)"
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 = a1"
definition [simp]: "Groups.one = a1"
definition [simp]: "(+) = (λ_ _. a1)"
definition [simp]: "(*) = (λ_ _. a1)"
definition [simp]: "(mod) = (λ_ _. a1)" 
definition [simp]: "abs = (λ_. a1)"
definition [simp]: "sgn = (λ_. a1)"
definition [simp]: "inverse = (λ_. a1)"
definition [simp]: "divide = (λ_ _. a1)"

instance by intro_classes(simp_all add: less_finite_1_def)
end

declare [[simproc del: finite_1_eq]]
hide_const (open) a1

datatype (plugins only: code "quickcheck" extraction) finite_2 =
  a1 | a2

notation (output) a1  ("a1")
notation (output) a2  ("a2")

lemma UNIV_finite_2:
  "UNIV = {a1, a2}"
  by (auto intro: finite_2.exhaust)

instantiation finite_2 :: enum
begin

definition
  "enum = [a1, a2]"

definition
  "enum_all P ⟷ P a1 ∧ P a2"

definition
  "enum_ex P ⟷ P a1 ∨ P a2"

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 = a1 ∧ y = a2"

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 a1 ∈ A then a1 else a2)"
definition "⨆A = (if a2 ∈ A then a2 else a1)"
definition [simp]: "bot = a1"
definition [simp]: "top = a2"
definition "x ⊓ y = (if x = a1 ∨ y = a1 then a1 else a2)"
definition "x ⊔ y = (if x = a2 ∨ y = a2 then a2 else a1)"

lemma neq_finite_2_a1_iff [simp]: "x ≠ a1 ⟷ x = a2"
by(cases x) simp_all

lemma neq_finite_2_a1_iff' [simp]: "a1 ≠ x ⟷ x = a2"
by(cases x) simp_all

lemma neq_finite_2_a2_iff [simp]: "x ≠ a2 ⟷ x = a1"
by(cases x) simp_all

lemma neq_finite_2_a2_iff' [simp]: "a2 ≠ x ⟷ x = a1"
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 = a1"
definition [simp]: "1 = a2"
definition "x + y = (case (x, y) of (a1, a1) ⇒ a1 | (a2, a2) ⇒ a1 | _ ⇒ a2)"
definition "uminus = (λx :: finite_2. x)"
definition "(-) = ((+) :: finite_2 ⇒ _)"
definition "x * y = (case (x, y) of (a2, a2) ⇒ a2 | _ ⇒ a1)"
definition "inverse = (λx :: finite_2. x)"
definition "divide = ((*) :: finite_2 ⇒ _)"
definition "x mod y = (case (x, y) of (a2, a1) ⇒ a2 | _ ⇒ a1)"
definition "abs = (λx :: finite_2. x)"
definition "sgn = (λx :: finite_2. x)"
instance
  by standard
    (subproofs
      ‹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›)
end

lemma two_finite_2 [simp]:
  "2 = a1"
  by (simp add: numeral.simps plus_finite_2_def)

lemma dvd_finite_2_unfold:
  "x dvd y ⟷ x = a2 ∨ y = a1"
  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 a1 ⇒ 0 | a2 ⇒ 1)"
definition [simp]: "division_segment (x :: finite_2) = 1"
instance
  by standard
    (subproofs
      ‹auto simp add: divide_finite_2_def times_finite_2_def dvd_finite_2_unfold
        split: finite_2.splits›)
end

 
hide_const (open) a1 a2

datatype (plugins only: code "quickcheck" extraction) finite_3 =
  a1 | a2 | a3

notation (output) a1  ("a1")
notation (output) a2  ("a2")
notation (output) a3  ("a3")

lemma UNIV_finite_3:
  "UNIV = {a1, a2, a3}"
  by (auto intro: finite_3.exhaust)

instantiation finite_3 :: enum
begin

definition
  "enum = [a1, a2, a3]"

definition
  "enum_all P ⟷ P a1 ∧ P a2 ∧ P a3"

definition
  "enum_ex P ⟷ P a1 ∨ P a2 ∨ P a3"

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 ≠ a1 ⟷ x ∈ {a2, a3}"
  "x ≠ a2 ⟷ x ∈ {a1, a3}"
  "x ≠ a3 ⟷ x ∈ {a1, a2}"
  by (cases x; simp)+

instantiation finite_3 :: linorder
begin

definition less_finite_3 :: "finite_3 ⇒ finite_3 ⇒ bool"
where
  "x < y = (case x of a1 ⇒ y ≠ a1 | a2 ⇒ y = a3 | a3 ⇒ 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 "λ 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 "λ A . (∀ x. x ∈ A ⟶ x ≤ z) ⟶ ⨆A ≤ z" ], simp_all add: Sup_finite_empty Sup_finite_insert)
  from this show " (⋀x. x ∈ A ⟹ x ≤ z) ⟹ ⨆A ≤ z"
    by blast
next
  show "⨅{} = ⊤"
    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 "λ A . a ⊓ (Sup A) = Sup {a ⊓ b | b . b ∈ A}"], simp_all)
  fix x::"'a"
  fix F
  assume "x ∉ F"
  assume [simp]: "a ⊓ ⨆F = ⨆{a ⊓ b |b. b ∈ F}"
  have [simp]: " insert (a ⊓ x) {a ⊓ b |b. b ∈ F} = {a ⊓ b |b. b = x ∨ b ∈ F}"
    by blast
  have "a ⊓ (x ⊔ ⨆F) = a ⊓ x ⊔ a ⊓ ⨆F"
    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 "λA. ⨅(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 = (λ (b::'a) f Y . (if Y = x then b else f Y))"
  have "⋀f b. ∀Y∈F. f Y ∈ Y ⟹ b ∈ x ⟹ insert b (f ` (F ∩ {Y. Y ≠ x})) = insert (fa b f x) (fa b f ` F) ∧ fa b f x ∈ x ∧ (∀Y∈F. fa b f Y ∈ Y)"
    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 ‹x ∉ F› 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 a1 ∈ A then a1 else if a2 ∈ A then a2 else a3)"
definition "⨆A = (if a3 ∈ A then a3 else if a2 ∈ A then a2 else a1)"
definition [simp]: "bot = a1"
definition [simp]: "top = a3"
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 = a1"
definition [simp]: "1 = a2"
definition
  "x + y = (case (x, y) of
     (a1, a1) ⇒ a1 | (a2, a3) ⇒ a1 | (a3, a2) ⇒ a1
   | (a1, a2) ⇒ a2 | (a2, a1) ⇒ a2 | (a3, a3) ⇒ a2
   | _ ⇒ a3)"
definition "- x = (case x of a1 ⇒ a1 | a2 ⇒ a3 | a3 ⇒ a2)"
definition "x - y = x + (- y :: finite_3)"
definition "x * y = (case (x, y) of (a2, a2) ⇒ a2 | (a3, a3) ⇒ a2 | (a2, a3) ⇒ a3 | (a3, a2) ⇒ a3 | _ ⇒ a1)"
definition "inverse = (λx :: finite_3. x)" 
definition "x div y = x * inverse (y :: finite_3)"
definition "x mod y = (case y of a1 ⇒ x | _ ⇒ a1)"
definition "abs = (λx. case x of a3 ⇒ a2 | _ ⇒ x)"
definition "sgn = (λx :: finite_3. x)"
instance
  by standard
    (subproofs
      ‹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›)
end

lemma two_finite_3 [simp]:
  "2 = a3"
  by (simp add: numeral.simps plus_finite_3_def)

lemma dvd_finite_3_unfold:
  "x dvd y ⟷ x = a2 ∨ x = a3 ∨ y = a1"
  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 a3 ⇒ a2 | _ ⇒ x)"
definition [simp]: "unit_factor = (id :: finite_3 ⇒ _)"
definition [simp]: "euclidean_size x = (case x of a1 ⇒ 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
    ‹auto simp add: divide_finite_3_def times_finite_3_def
      dvd_finite_3_unfold inverse_finite_3_def plus_finite_3_def
      split: finite_3.splits›)
end

hide_const (open) a1 a2 a3

datatype (plugins only: code "quickcheck" extraction) finite_4 =
  a1 | a2 | a3 | a4

notation (output) a1  ("a1")
notation (output) a2  ("a2")
notation (output) a3  ("a3")
notation (output) a4  ("a4")

lemma UNIV_finite_4:
  "UNIV = {a1, a2, a3, a4}"
  by (auto intro: finite_4.exhaust)

instantiation finite_4 :: enum
begin

definition
  "enum = [a1, a2, a3, a4]"

definition
  "enum_all P ⟷ P a1 ∧ P a2 ∧ P a3 ∧ P a4"

definition
  "enum_ex P ⟷ P a1 ∨ P a2 ∨ P a3 ∨ P a4"

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

end

instantiation finite_4 :: finite_distrib_lattice begin

text ‹term‹a1› $<$ term‹a2›,term‹a3› $<$ term‹a4›,
  but term‹a2› and term‹a3› are incomparable.›

definition
  "x < y ⟷ (case (x, y) of
     (a1, a1) ⇒ False | (a1, _) ⇒ True
   |  (a2, a4) ⇒ True
   |  (a3, a4) ⇒ True  | _ ⇒ False)"

definition 
  "x ≤ y ⟷ (case (x, y) of
     (a1, _) ⇒ True
   | (a2, a2) ⇒ True | (a2, a4) ⇒ True
   | (a3, a3) ⇒ True | (a3, a4) ⇒ True
   | (a4, a4) ⇒ True | _ ⇒ False)"

definition
  "⨅A = (if a1 ∈ A ∨ a2 ∈ A ∧ a3 ∈ A then a1 else if a2 ∈ A then a2 else if a3 ∈ A then a3 else a4)"
definition
  "⨆A = (if a4 ∈ A ∨ a2 ∈ A ∧ a3 ∈ A then a4 else if a2 ∈ A then a2 else if a3 ∈ A then a3 else a1)"
definition [simp]: "bot = a1"
definition [simp]: "top = a4"
definition
  "x ⊓ y = (case (x, y) of
     (a1, _) ⇒ a1 | (_, a1) ⇒ a1 | (a2, a3) ⇒ a1 | (a3, a2) ⇒ a1
   | (a2, _) ⇒ a2 | (_, a2) ⇒ a2
   | (a3, _) ⇒ a3 | (_, a3) ⇒ a3
   | _ ⇒ a4)"
definition
  "x ⊔ y = (case (x, y) of
     (a4, _) ⇒ a4 | (_, a4) ⇒ a4 | (a2, a3) ⇒ a4 | (a3, a2) ⇒ a4
  | (a2, _) ⇒ a2 | (_, a2) ⇒ a2
  | (a3, _) ⇒ a3 | (_, a3) ⇒ a3
  | _ ⇒ a1)"

instance
  by standard
    (subproofs
      ‹auto simp add: less_finite_4_def less_eq_finite_4_def Inf_finite_4_def Sup_finite_4_def 
        inf_finite_4_def sup_finite_4_def split: finite_4.splits›)
end

instance finite_4 :: complete_lattice ..

instance finite_4 :: complete_distrib_lattice ..

instantiation finite_4 :: complete_boolean_algebra begin
definition "- x = (case x of a1 ⇒ a4 | a2 ⇒ a3 | a3 ⇒ a2 | a4 ⇒ a1)"
definition "x - y = x ⊓ - (y :: finite_4)"
instance
  by standard
    (subproofs
      ‹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) a1 a2 a3 a4

datatype (plugins only: code "quickcheck" extraction) finite_5 =
  a1 | a2 | a3 | a4 | a5

notation (output) a1  ("a1")
notation (output) a2  ("a2")
notation (output) a3  ("a3")
notation (output) a4  ("a4")
notation (output) a5  ("a5")

lemma UNIV_finite_5:
  "UNIV = {a1, a2, a3, a4, a5}"
  by (auto intro: finite_5.exhaust)

instantiation finite_5 :: enum
begin

definition
  "enum = [a1, a2, a3, a4, a5]"

definition
  "enum_all P ⟷ P a1 ∧ P a2 ∧ P a3 ∧ P a4 ∧ P a5"

definition
  "enum_ex P ⟷ P a1 ∨ P a2 ∨ P a3 ∨ P a4 ∨ P a5"

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

end

instantiation finite_5 :: finite_lattice
begin

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

definition
  "x < y ⟷ (case (x, y) of
     (a1, a1) ⇒ False | (a1, _) ⇒ True
   | (a2, a3) ⇒ True  | (a2, a5) ⇒ True
   | (a3, a5) ⇒ True
   | (a4, a5) ⇒ True  | _ ⇒ False)"

definition
  "x ≤ y ⟷ (case (x, y) of
     (a1, _) ⇒ True
   | (a2, a2) ⇒ True | (a2, a3) ⇒ True | (a2, a5) ⇒ True
   | (a3, a3) ⇒ True | (a3, a5) ⇒ True
   | (a4, a4) ⇒ True | (a4, a5) ⇒ True
   | (a5, a5) ⇒ True | _ ⇒ False)"

definition
  "⨅A = 
  (if a1 ∈ A ∨ a4 ∈ A ∧ (a2 ∈ A ∨ a3 ∈ A) then a1
   else if a2 ∈ A then a2
   else if a3 ∈ A then a3
   else if a4 ∈ A then a4
   else a5)"
definition
  "⨆A = 
  (if a5 ∈ A ∨ a4 ∈ A ∧ (a2 ∈ A ∨ a3 ∈ A) then a5
   else if a3 ∈ A then a3
   else if a2 ∈ A then a2
   else if a4 ∈ A then a4
   else a1)"
definition [simp]: "bot = a1"
definition [simp]: "top = a5"
definition
  "x ⊓ y = (case (x, y) of
     (a1, _) ⇒ a1 | (_, a1) ⇒ a1 | (a2, a4) ⇒ a1 | (a4, a2) ⇒ a1 | (a3, a4) ⇒ a1 | (a4, a3) ⇒ a1
   | (a2, _) ⇒ a2 | (_, a2) ⇒ a2
   | (a3, _) ⇒ a3 | (_, a3) ⇒ a3
   | (a4, _) ⇒ a4 | (_, a4) ⇒ a4
   | _ ⇒ a5)"
definition
  "x ⊔ y = (case (x, y) of
     (a5, _) ⇒ a5 | (_, a5) ⇒ a5 | (a2, a4) ⇒ a5 | (a4, a2) ⇒ a5 | (a3, a4) ⇒ a5 | (a4, a3) ⇒ a5
   | (a3, _) ⇒ a3 | (_, a3) ⇒ a3
   | (a2, _) ⇒ a2 | (_, a2) ⇒ a2
   | (a4, _) ⇒ a4 | (_, a4) ⇒ a4
   | _ ⇒ a1)"

instance
  by standard
    (subproofs
      ‹auto simp add: less_eq_finite_5_def less_finite_5_def inf_finite_5_def sup_finite_5_def 
        Inf_finite_5_def Sup_finite_5_def split: finite_5.splits if_split_asm›)
end


instance  finite_5 :: complete_lattice ..


hide_const (open) a1 a2 a3 a4 a5


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