(* Author: Florian Haftmann, TU Muenchen *)

header {* Finite types as explicit enumerations *}

theory Enum

imports Map String

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 : "enum_all P = (\<forall> x. P x)"

  assumes enum_ex  : "enum_ex P = (\<exists> x. P x)" 

begin

subclass finite proof

qed (simp add: UNIV_enum)

lemma enum_UNIV: "set enum = UNIV" unfolding UNIV_enum ..

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

  unfolding enum_UNIV by auto

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

end

subsection {* 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 enum_UNIV fun_eq_iff)

end

lemma [code]:

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

by (auto simp add: equal enum_all 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: enum_all enum_ex fun_eq_iff le_fun_def order_less_le)

subsection {* Quantifiers *}

lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"

  by (simp add: enum_all)

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

  by (simp add: enum_ex)

lemma exists1_code[code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"

unfolding list_ex1_iff enum_UNIV by auto

subsection {* Default instances *}

primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where

  "n_lists 0 xs = [[]]"

  | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"

lemma n_lists_Nil [simp]: "n_lists n [] = (if n = 0 then [[]] else [])"

  by (induct n) simp_all

lemma length_n_lists: "length (n_lists n xs) = length xs ^ n"

  by (induct n) (auto simp add: length_concat o_def listsum_triv)

lemma length_n_lists_elem: "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"

  by (induct n arbitrary: ys) auto

lemma set_n_lists: "set (n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"

proof (rule set_eqI)

  fix ys :: "'a list"

  show "ys \<in> set (n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"

  proof -

    have "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"

      by (induct n arbitrary: ys) auto

    moreover have "\<And>x. ys \<in> set (n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"

      by (induct n arbitrary: ys) auto

    moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (n_lists (length ys) xs)"

      by (induct ys) auto

    ultimately show ?thesis by auto

  qed

qed

lemma distinct_n_lists:

  assumes "distinct xs"

  shows "distinct (n_lists n xs)"

proof (rule card_distinct)

  from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)

  have "card (set (n_lists n xs)) = card (set xs) ^ n"

  proof (induct n)

    case 0 then show ?case by simp

  next

    case (Suc n)

    moreover have "card (\<Union>ys\<in>set (n_lists n xs). (\<lambda>y. y # ys) ` set xs)

      = (\<Sum>ys\<in>set (n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"

      by (rule card_UN_disjoint) auto

    moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"

      by (rule card_image) (simp add: inj_on_def)

    ultimately show ?case by auto

  qed

  also have "\<dots> = length xs ^ n" by (simp add: card_length)

  finally show "card (set (n_lists n xs)) = length (n_lists n xs)"

    by (simp add: length_n_lists)

qed

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 = (\<forall>xs \<in> set (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 \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"

where

  "ex_n_lists P n = (\<exists>xs \<in> set (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 (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (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 enum_all)

next

  fix P

  show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = (\<forall>x. P x)"

  proof

    assume "enum_all P"

    show "\<forall>x. P x"

    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 "\<forall>x. P x"

    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) = (\<exists>x. P x)"

  proof

    assume "enum_ex P"

    from this show "\<exists>x. P x"

      unfolding enum_ex_fun_def ex_n_lists_def by auto

  next

    assume "\<exists>x. P x"

    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)) (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 add: 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 add: enum_ex_fun_def Let_def)

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 UNIV_unit enum_all_unit_def enum_ex_unit_def intro: unit.exhaust)

end

instantiation bool :: enum

begin

definition

  "enum = [False, True]"

definition

  "enum_all P = (P False \<and> P True)"

definition

  "enum_ex P = (P False \<or> P True)"

instance proof

  fix P

  show "enum_all (P :: bool \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_bool_def by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: bool \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_bool_def by (auto, case_tac x) auto

qed (auto simp add: enum_bool_def UNIV_bool)

end

primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where

  "product [] _ = []"

  | "product (x#xs) ys = map (Pair x) ys @ product xs ys"

lemma product_list_set:

  "set (product xs ys) = set xs \<times> set ys"

  by (induct xs) auto

lemma distinct_product:

  assumes "distinct xs" and "distinct ys"

  shows "distinct (product xs ys)"

  using assms by (induct xs)

    (auto intro: inj_onI simp add: product_list_set distinct_map)

instantiation prod :: (enum, enum) enum

begin

definition

  "enum = 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 product_list_set distinct_product

    enum_UNIV enum_distinct enum_all_prod_def enum_all enum_ex_prod_def enum_ex)

end

instantiation sum :: (enum, enum) enum

begin

definition

  "enum = map Inl enum @ map Inr enum"

definition

  "enum_all P = (enum_all (%x. P (Inl x)) \<and> enum_all (%x. P (Inr x)))"

definition

  "enum_ex P = (enum_ex (%x. P (Inl x)) \<or> enum_ex (%x. P (Inr x)))"

instance proof

  fix P

  show "enum_all (P :: ('a + 'b) \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_sum_def enum_all

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: ('a + 'b) \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_sum_def enum_ex

    by (auto, case_tac x) auto

qed (auto simp add: enum_UNIV enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)

end

primrec sublists :: "'a list \<Rightarrow> 'a list list" where

  "sublists [] = [[]]"

  | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"

lemma length_sublists:

  "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"

  by (induct xs) (simp_all add: Let_def)

lemma sublists_powset:

  "set ` set (sublists xs) = Pow (set xs)"

proof -

  have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"

    by (auto simp add: image_def)

  have "set (map set (sublists xs)) = Pow (set xs)"

    by (induct xs)

      (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)

  then show ?thesis by simp

qed

lemma distinct_set_sublists:

  assumes "distinct xs"

  shows "distinct (map set (sublists xs))"

proof (rule card_distinct)

  have "finite (set xs)" by rule

  then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)

  with assms distinct_card [of xs]

    have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp

  then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"

    by (simp add: sublists_powset length_sublists)

qed

instantiation nibble :: enum

begin

definition

  "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,

    Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"

definition

  "enum_all P = (P Nibble0 \<and> P Nibble1 \<and> P Nibble2 \<and> P Nibble3 \<and> P Nibble4 \<and> P Nibble5 \<and> P Nibble6 \<and> P Nibble7

     \<and> P Nibble8 \<and> P Nibble9 \<and> P NibbleA \<and> P NibbleB \<and> P NibbleC \<and> P NibbleD \<and> P NibbleE \<and> P NibbleF)"

definition

  "enum_ex P = (P Nibble0 \<or> P Nibble1 \<or> P Nibble2 \<or> P Nibble3 \<or> P Nibble4 \<or> P Nibble5 \<or> P Nibble6 \<or> P Nibble7

     \<or> P Nibble8 \<or> P Nibble9 \<or> P NibbleA \<or> P NibbleB \<or> P NibbleC \<or> P NibbleD \<or> P NibbleE \<or> P NibbleF)"

instance proof

  fix P

  show "enum_all (P :: nibble \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_nibble_def

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: nibble \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_nibble_def

    by (auto, case_tac x) auto

qed (simp_all add: enum_nibble_def UNIV_nibble)

end

instantiation char :: enum

begin

definition

  "enum = map (split Char) (product enum enum)"

lemma enum_chars [code]:

  "enum = chars"

  unfolding enum_char_def chars_def enum_nibble_def by simp

definition

  "enum_all P = list_all P chars"

definition

  "enum_ex P = list_ex P chars"

lemma set_enum_char: "set (enum :: char list) = UNIV"

    by (auto intro: char.exhaust simp add: enum_char_def product_list_set enum_UNIV full_SetCompr_eq [symmetric])

instance proof

  fix P

  show "enum_all (P :: char \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_char_def enum_chars[symmetric]

    by (auto simp add: list_all_iff set_enum_char)

next

  fix P

  show "enum_ex (P :: char \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_char_def enum_chars[symmetric]

    by (auto simp add: list_ex_iff set_enum_char)

next

  show "distinct (enum :: char list)"

    by (auto intro: inj_onI simp add: enum_char_def product_list_set distinct_map distinct_product enum_distinct)

qed (auto simp add: set_enum_char)

end

instantiation option :: (enum) enum

begin

definition

  "enum = None # map Some enum"

definition

  "enum_all P = (P None \<and> enum_all (%x. P (Some x)))"

definition

  "enum_ex P = (P None \<or> enum_ex (%x. P (Some x)))"

instance proof

  fix P

  show "enum_all (P :: 'a option \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_option_def enum_all

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: 'a option \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_option_def enum_ex

    by (auto, case_tac x) auto

qed (auto simp add: enum_UNIV enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)

end

subsection {* Small finite types *}

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

datatype finite_1 = a\<^isub>1

notation (output) a\<^isub>1  ("a\<^isub>1")

instantiation finite_1 :: enum

begin

definition

  "enum = [a\<^isub>1]"

definition

  "enum_all P = P a\<^isub>1"

definition

  "enum_ex P = P a\<^isub>1"

instance proof

  fix P

  show "enum_all (P :: finite_1 \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_finite_1_def

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: finite_1 \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_finite_1_def

    by (auto, case_tac x) auto

qed (auto simp add: enum_finite_1_def intro: finite_1.exhaust)

end

instantiation finite_1 :: linorder

begin

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

where

  "less_eq_finite_1 x y = True"

definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"

where

  "less_finite_1 x y = False"

instance

apply (intro_classes)

apply (auto simp add: less_finite_1_def less_eq_finite_1_def)

apply (metis finite_1.exhaust)

done

end

hide_const (open) a\<^isub>1

datatype finite_2 = a\<^isub>1 | a\<^isub>2

notation (output) a\<^isub>1  ("a\<^isub>1")

notation (output) a\<^isub>2  ("a\<^isub>2")

instantiation finite_2 :: enum

begin

definition

  "enum = [a\<^isub>1, a\<^isub>2]"

definition

  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2)"

definition

  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2)"

instance proof

  fix P

  show "enum_all (P :: finite_2 \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_finite_2_def

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: finite_2 \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_finite_2_def

    by (auto, case_tac x) auto

qed (auto simp add: enum_finite_2_def intro: finite_2.exhaust)

end

instantiation finite_2 :: linorder

begin

definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"

where

  "less_finite_2 x y = ((x = a\<^isub>1) & (y = a\<^isub>2))"

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

where

  "less_eq_finite_2 x y = ((x = y) \<or> (x < y))"

instance

apply (intro_classes)

apply (auto simp add: less_finite_2_def less_eq_finite_2_def)

apply (metis finite_2.distinct finite_2.nchotomy)+

done

end

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

datatype finite_3 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3

notation (output) a\<^isub>1  ("a\<^isub>1")

notation (output) a\<^isub>2  ("a\<^isub>2")

notation (output) a\<^isub>3  ("a\<^isub>3")

instantiation finite_3 :: enum

begin

definition

  "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3]"

definition

  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3)"

definition

  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3)"

instance proof

  fix P

  show "enum_all (P :: finite_3 \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_finite_3_def

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: finite_3 \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_finite_3_def

    by (auto, case_tac x) auto

qed (auto simp add: enum_finite_3_def intro: finite_3.exhaust)

end

instantiation finite_3 :: linorder

begin

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

where

  "less_finite_3 x y = (case x of a\<^isub>1 => (y \<noteq> a\<^isub>1)

     | a\<^isub>2 => (y = a\<^isub>3)| a\<^isub>3 => False)"

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

where

  "less_eq_finite_3 x y = ((x = y) \<or> (x < y))"

instance proof (intro_classes)

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

end

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

datatype finite_4 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4

notation (output) a\<^isub>1  ("a\<^isub>1")

notation (output) a\<^isub>2  ("a\<^isub>2")

notation (output) a\<^isub>3  ("a\<^isub>3")

notation (output) a\<^isub>4  ("a\<^isub>4")

instantiation finite_4 :: enum

begin

definition

  "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4]"

definition

  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3 \<and> P a\<^isub>4)"

definition

  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3 \<or> P a\<^isub>4)"

instance proof

  fix P

  show "enum_all (P :: finite_4 \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_finite_4_def

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: finite_4 \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_finite_4_def

    by (auto, case_tac x) auto

qed (auto simp add: enum_finite_4_def intro: finite_4.exhaust)

end

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

datatype finite_5 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4 | a\<^isub>5

notation (output) a\<^isub>1  ("a\<^isub>1")

notation (output) a\<^isub>2  ("a\<^isub>2")

notation (output) a\<^isub>3  ("a\<^isub>3")

notation (output) a\<^isub>4  ("a\<^isub>4")

notation (output) a\<^isub>5  ("a\<^isub>5")

instantiation finite_5 :: enum

begin

definition

  "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4, a\<^isub>5]"

definition

  "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3 \<and> P a\<^isub>4 \<and> P a\<^isub>5)"

definition

  "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3 \<or> P a\<^isub>4 \<or> P a\<^isub>5)"

instance proof

  fix P

  show "enum_all (P :: finite_5 \<Rightarrow> bool) = (\<forall>x. P x)"

    unfolding enum_all_finite_5_def

    by (auto, case_tac x) auto

next

  fix P

  show "enum_ex (P :: finite_5 \<Rightarrow> bool) = (\<exists>x. P x)"

    unfolding enum_ex_finite_5_def

    by (auto, case_tac x) auto

qed (auto simp add: enum_finite_5_def intro: finite_5.exhaust)

end

subsection {* An executable THE operator on finite types *}

definition

  [code del]: "enum_the P = The P"

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

subsection {* An executable card operator on finite types *}

lemma

  [code]: "card R = length (filter R enum)"

by (simp add: distinct_length_filter[OF enum_distinct] enum_UNIV Collect_def)

subsection {* An executable (reflexive) transitive closure on finite relations *}

text {* Definitions could be moved to Transitive Closure theory if they are of more general use. *}

definition ntrancl :: "('a * 'a => bool) => nat => ('a * 'a => bool)"

where

 [code del]: "ntrancl R n = (UN i : {i. 0 < i & i <= (Suc n)}. R ^^ i)"

lemma [code]:

  "ntrancl (R :: 'a * 'a => bool) 0 = R"

proof

  show "R <= ntrancl R 0"

    unfolding ntrancl_def by fastforce

next

  { 

    fix i have "(0 < i & i <= Suc 0) = (i = 1)" by auto

  }

  from this show "ntrancl R 0 <= R"

    unfolding ntrancl_def by auto

qed

lemma [code]:

  "ntrancl (R :: 'a * 'a => bool) (Suc n) = (ntrancl R n) O (Id Un R)"

proof

  {

    fix a b

    assume "(a, b) : ntrancl R (Suc n)"

    from this obtain i where "0 < i" "i <= Suc (Suc n)" "(a, b) : R ^^ i"

      unfolding ntrancl_def by auto

    have "(a, b) : ntrancl R n O (Id Un R)"

    proof (cases "i = 1")

      case True

      from this `(a, b) : R ^^ i` show ?thesis

        unfolding ntrancl_def by auto

    next

      case False

      from this `0 < i` obtain j where j: "i = Suc j" "0 < j"

        by (cases i) auto

      from this `(a, b) : R ^^ i` obtain c where c1: "(a, c) : R ^^ j" and c2:"(c, b) : R"

        by auto

      from c1 j `i <= Suc (Suc n)` have "(a, c): ntrancl R n"

        unfolding ntrancl_def by fastforce

      from this c2 show ?thesis by fastforce

    qed

  }

  from this show "ntrancl R (Suc n) <= ntrancl R n O (Id Un R)" by auto

next

  show "ntrancl R n O (Id Un R) <= ntrancl R (Suc n)"

    unfolding ntrancl_def by fastforce

qed

lemma [code]: "trancl (R :: ('a :: finite) * 'a => bool) = ntrancl R (card R - 1)"

by (cases "card R") (auto simp add: trancl_finite_eq_rel_pow rel_pow_empty ntrancl_def)

(* a copy of Nitpick.rtrancl_unfold, should be moved to Transitive_Closure *)

lemma [code]: "r^* = (r^+)^="

by simp

subsection {* Closing up *}

code_abort enum_the

hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5

hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5

hide_const (open) enum enum_all enum_ex n_lists all_n_lists ex_n_lists product ntrancl

end