--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Enum.thy Mon Nov 22 11:34:55 2010 +0100
@@ -0,0 +1,379 @@
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* Finite types as explicit enumerations *}
+
+theory Enum
+imports Map Main
+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 [intro]: "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 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)
+
+
+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)
+ then show "f \<in> set enum"
+ by (auto simp add: enum_fun_def set_n_lists)
+ 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
+
+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
+
+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
+
+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
+
+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_type finite_1 finite_2 finite_3 finite_4 finite_5
+
+end
--- a/src/HOL/Library/Enum.thy Mon Nov 22 11:34:54 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,379 +0,0 @@
-(* Author: Florian Haftmann, TU Muenchen *)
-
-header {* Finite types as explicit enumerations *}
-
-theory Enum
-imports Map Main
-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 [intro]: "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 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)
-
-
-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)
- then show "f \<in> set enum"
- by (auto simp add: enum_fun_def set_n_lists)
- 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
-
-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
-
-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
-
-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
-
-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_type finite_1 finite_2 finite_3 finite_4 finite_5
-
-end