# HG changeset patch
# User bulwahn
# Date 1290422095 -3600
# Node ID dc1b5aa908ff23282b42b4ec35d305728f8a4cef
# Parent  1598ec648b0d5117e9e9ee34669e934b9504ae46
moving Enum theory from HOL/Library to HOL

diff -r 1598ec648b0d -r dc1b5aa908ff src/HOL/Enum.thy
--- /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
diff -r 1598ec648b0d -r dc1b5aa908ff src/HOL/Library/Enum.thy
--- 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