(* 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"
assumes UNIV_enum: "UNIV = set enum"
and enum_distinct: "distinct enum"
begin
subclass finite proof
qed (simp add: UNIV_enum)
lemma enum_all: "set enum = UNIV" unfolding UNIV_enum ..
lemma in_enum: "x \<in> set enum"
unfolding enum_all 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_all 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_all fun_eq_iff)
end
lemma [code nbe]:
"HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"
by (fact equal_refl)
lemma [code]:
"HOL.equal f g \<longleftrightarrow> list_all (%x. f x = g x) enum"
by (auto simp add: list_all_iff enum_all equal fun_eq_iff)
lemma order_fun [code]:
fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
shows "f \<le> g \<longleftrightarrow> list_all (\<lambda>x. f x \<le> g x) enum"
and "f < g \<longleftrightarrow> f \<le> g \<and> list_ex (\<lambda>x. f x \<noteq> g x) enum"
by (simp_all add: list_all_iff list_ex_iff enum_all fun_eq_iff le_fun_def order_less_le)
subsection {* Quantifiers *}
lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> list_all P enum"
by (simp add: list_all_iff enum_all)
lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> list_ex P enum"
by (simp add: list_ex_iff enum_all)
lemma exists1_code[code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
unfolding list_ex1_iff enum_all 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_all 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
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)"
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)
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)
instantiation unit :: enum
begin
definition
"enum = [()]"
instance proof
qed (simp_all add: enum_unit_def UNIV_unit)
end
instantiation bool :: enum
begin
definition
"enum = [False, True]"
instance proof
qed (simp_all 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"
instance by default
(simp_all add: enum_prod_def product_list_set distinct_product enum_all enum_distinct)
end
instantiation sum :: (enum, enum) enum
begin
definition
"enum = map Inl enum @ map Inr enum"
instance by default
(auto simp add: enum_all 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]"
instance proof
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
instance proof
qed (auto intro: char.exhaust injI simp add: enum_char_def product_list_set enum_all full_SetCompr_eq [symmetric]
distinct_map distinct_product enum_distinct)
end
instantiation option :: (enum) enum
begin
definition
"enum = None # map Some enum"
instance proof
qed (auto simp add: enum_all 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
instantiation finite_1 :: enum
begin
definition
"enum = [a\<^isub>1]"
instance proof
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 a\<^isub>1
datatype finite_2 = a\<^isub>1 | a\<^isub>2
instantiation finite_2 :: enum
begin
definition
"enum = [a\<^isub>1, a\<^isub>2]"
instance proof
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 a\<^isub>1 a\<^isub>2
datatype finite_3 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3
instantiation finite_3 :: enum
begin
definition
"enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3]"
instance proof
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 a\<^isub>1 a\<^isub>2 a\<^isub>3
datatype finite_4 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4
instantiation finite_4 :: enum
begin
definition
"enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4]"
instance proof
qed (auto simp add: enum_finite_4_def intro: finite_4.exhaust)
end
hide_const 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
instantiation finite_5 :: enum
begin
definition
"enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4, a\<^isub>5]"
instance proof
qed (auto simp add: enum_finite_5_def intro: finite_5.exhaust)
end
hide_const a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5
hide_type finite_1 finite_2 finite_3 finite_4 finite_5
hide_const (open) enum n_lists product
end