instance for functions, explicit characters

--- a/src/HOL/Library/Enum.thy	Thu Mar 27 19:04:37 2008 +0100
+++ b/src/HOL/Library/Enum.thy	Thu Mar 27 19:04:38 2008 +0100
@@ -11,11 +11,17 @@
 
 subsection {* Class @{text enum} *}
 
-class enum = finite + -- FIXME
+class enum = itself +
   fixes enum :: "'a list"
-  assumes enum_all: "set enum = UNIV"
+  assumes UNIV_enum [code func]: "UNIV = set enum"
+    and enum_distinct: "distinct enum"
 begin
 
+lemma finite_enum: "finite (UNIV \<Colon> 'a set)"
+  unfolding UNIV_enum ..
+
+lemma enum_all: "set enum = UNIV" unfolding UNIV_enum ..
+
 lemma in_enum [intro]: "x \<in> set enum"
   unfolding enum_all by auto
 
@@ -32,8 +38,6 @@
 
 subsection {* Equality and order on functions *}
 
-declare eq_fun [code func del] order_fun [code func del]
-
 instance "fun" :: (enum, eq) eq ..
 
 lemma eq_fun [code func]:
@@ -50,6 +54,120 @@
 
 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 map_compose [symmetric] 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_ext)
+  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_map:
+  fixes f :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>enum"
+  shows "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
+  by (induct xs) (simp_all add: expand_fun_eq)
+
+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
+  [code func del]: "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 expand_fun_eq)
+    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 [code func]:
+  "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>('a\<Colon>{enum, eq}) list) ys)) (n_lists (length (enum\<Colon>'a\<Colon>{enum, eq} list)) enum)"
+  unfolding enum_fun_def ..
+
 instantiation unit :: enum
 begin
 
@@ -57,7 +175,7 @@
   "enum = [()]"
 
 instance by default
-  (simp add: enum_unit_def UNIV_unit)
+  (simp_all add: enum_unit_def UNIV_unit)
 
 end
 
@@ -68,7 +186,7 @@
   "enum = [False, True]"
 
 instance by default
-  (simp add: enum_bool_def UNIV_bool)
+  (simp_all add: enum_bool_def UNIV_bool)
 
 end
 
@@ -80,6 +198,12 @@
   "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 * :: (enum, enum) enum
 begin
 
@@ -87,7 +211,7 @@
   "enum = product enum enum"
 
 instance by default
-  (simp add: enum_prod_def product_list_set enum_all)
+  (simp_all add: enum_prod_def product_list_set distinct_product enum_all enum_distinct)
 
 end
 
@@ -98,7 +222,7 @@
   "enum = map Inl enum @ map Inr enum"
 
 instance by default
-  (auto simp add: enum_all enum_sum_def, case_tac x, auto)
+  (auto simp add: enum_all enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)
 
 end
 
@@ -106,14 +230,31 @@
   "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 (map set (sublists xs)) = Pow (set xs)"
+  "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)
-  show ?thesis
+  have "set (map set (sublists xs)) = Pow (set xs)"
     by (induct xs)
-     (simp_all add: aux Let_def Pow_insert Un_commute)
+      (simp_all add: aux Let_def Pow_insert Un_commute)
+  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 set :: (enum) enum
@@ -123,7 +264,7 @@
   "enum = map set (sublists enum)"
 
 instance by default
-  (simp add: enum_set_def sublists_powset enum_all del: set_map)
+  (simp_all add: enum_set_def enum_all sublists_powset distinct_set_sublists enum_distinct)
 
 end
 
@@ -135,7 +276,7 @@
     Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
 
 instance by default
-  (simp add: enum_nibble_def UNIV_nibble)
+  (simp_all add: enum_nibble_def UNIV_nibble)
 
 end
 
@@ -143,31 +284,83 @@
 begin
 
 definition
-  "enum = map (split Char) (product enum enum)"
+  [code func del]: "enum = map (split Char) (product enum enum)"
+
+lemma enum_char [code func]:
+  "enum = [Char Nibble0 Nibble0, Char Nibble0 Nibble1, Char Nibble0 Nibble2,
+  Char Nibble0 Nibble3, Char Nibble0 Nibble4, Char Nibble0 Nibble5,
+  Char Nibble0 Nibble6, Char Nibble0 Nibble7, Char Nibble0 Nibble8,
+  Char Nibble0 Nibble9, Char Nibble0 NibbleA, Char Nibble0 NibbleB,
+  Char Nibble0 NibbleC, Char Nibble0 NibbleD, Char Nibble0 NibbleE,
+  Char Nibble0 NibbleF, Char Nibble1 Nibble0, Char Nibble1 Nibble1,
+  Char Nibble1 Nibble2, Char Nibble1 Nibble3, Char Nibble1 Nibble4,
+  Char Nibble1 Nibble5, Char Nibble1 Nibble6, Char Nibble1 Nibble7,
+  Char Nibble1 Nibble8, Char Nibble1 Nibble9, Char Nibble1 NibbleA,
+  Char Nibble1 NibbleB, Char Nibble1 NibbleC, Char Nibble1 NibbleD,
+  Char Nibble1 NibbleE, Char Nibble1 NibbleF, CHR '' '', CHR ''!'',
+  Char Nibble2 Nibble2, CHR ''#'', CHR ''$'', CHR ''%'', CHR ''&'',
+  Char Nibble2 Nibble7, CHR ''('', CHR '')'', CHR ''*'', CHR ''+'', CHR '','',
+  CHR ''-'', CHR ''.'', CHR ''/'', CHR ''0'', CHR ''1'', CHR ''2'', CHR ''3'',
+  CHR ''4'', CHR ''5'', CHR ''6'', CHR ''7'', CHR ''8'', CHR ''9'', CHR '':'',
+  CHR '';'', CHR ''<'', CHR ''='', CHR ''>'', CHR ''?'', CHR ''@'', CHR ''A'',
+  CHR ''B'', CHR ''C'', CHR ''D'', CHR ''E'', CHR ''F'', CHR ''G'', CHR ''H'',
+  CHR ''I'', CHR ''J'', CHR ''K'', CHR ''L'', CHR ''M'', CHR ''N'', CHR ''O'',
+  CHR ''P'', CHR ''Q'', CHR ''R'', CHR ''S'', CHR ''T'', CHR ''U'', CHR ''V'',
+  CHR ''W'', CHR ''X'', CHR ''Y'', CHR ''Z'', CHR ''['', Char Nibble5 NibbleC,
+  CHR '']'', CHR ''^'', CHR ''_'', Char Nibble6 Nibble0, CHR ''a'', CHR ''b'',
+  CHR ''c'', CHR ''d'', CHR ''e'', CHR ''f'', CHR ''g'', CHR ''h'', CHR ''i'',
+  CHR ''j'', CHR ''k'', CHR ''l'', CHR ''m'', CHR ''n'', CHR ''o'', CHR ''p'',
+  CHR ''q'', CHR ''r'', CHR ''s'', CHR ''t'', CHR ''u'', CHR ''v'', CHR ''w'',
+  CHR ''x'', CHR ''y'', CHR ''z'', CHR ''{'', CHR ''|'', CHR ''}'', CHR ''~'',
+  Char Nibble7 NibbleF, Char Nibble8 Nibble0, Char Nibble8 Nibble1,
+  Char Nibble8 Nibble2, Char Nibble8 Nibble3, Char Nibble8 Nibble4,
+  Char Nibble8 Nibble5, Char Nibble8 Nibble6, Char Nibble8 Nibble7,
+  Char Nibble8 Nibble8, Char Nibble8 Nibble9, Char Nibble8 NibbleA,
+  Char Nibble8 NibbleB, Char Nibble8 NibbleC, Char Nibble8 NibbleD,
+  Char Nibble8 NibbleE, Char Nibble8 NibbleF, Char Nibble9 Nibble0,
+  Char Nibble9 Nibble1, Char Nibble9 Nibble2, Char Nibble9 Nibble3,
+  Char Nibble9 Nibble4, Char Nibble9 Nibble5, Char Nibble9 Nibble6,
+  Char Nibble9 Nibble7, Char Nibble9 Nibble8, Char Nibble9 Nibble9,
+  Char Nibble9 NibbleA, Char Nibble9 NibbleB, Char Nibble9 NibbleC,
+  Char Nibble9 NibbleD, Char Nibble9 NibbleE, Char Nibble9 NibbleF,
+  Char NibbleA Nibble0, Char NibbleA Nibble1, Char NibbleA Nibble2,
+  Char NibbleA Nibble3, Char NibbleA Nibble4, Char NibbleA Nibble5,
+  Char NibbleA Nibble6, Char NibbleA Nibble7, Char NibbleA Nibble8,
+  Char NibbleA Nibble9, Char NibbleA NibbleA, Char NibbleA NibbleB,
+  Char NibbleA NibbleC, Char NibbleA NibbleD, Char NibbleA NibbleE,
+  Char NibbleA NibbleF, Char NibbleB Nibble0, Char NibbleB Nibble1,
+  Char NibbleB Nibble2, Char NibbleB Nibble3, Char NibbleB Nibble4,
+  Char NibbleB Nibble5, Char NibbleB Nibble6, Char NibbleB Nibble7,
+  Char NibbleB Nibble8, Char NibbleB Nibble9, Char NibbleB NibbleA,
+  Char NibbleB NibbleB, Char NibbleB NibbleC, Char NibbleB NibbleD,
+  Char NibbleB NibbleE, Char NibbleB NibbleF, Char NibbleC Nibble0,
+  Char NibbleC Nibble1, Char NibbleC Nibble2, Char NibbleC Nibble3,
+  Char NibbleC Nibble4, Char NibbleC Nibble5, Char NibbleC Nibble6,
+  Char NibbleC Nibble7, Char NibbleC Nibble8, Char NibbleC Nibble9,
+  Char NibbleC NibbleA, Char NibbleC NibbleB, Char NibbleC NibbleC,
+  Char NibbleC NibbleD, Char NibbleC NibbleE, Char NibbleC NibbleF,
+  Char NibbleD Nibble0, Char NibbleD Nibble1, Char NibbleD Nibble2,
+  Char NibbleD Nibble3, Char NibbleD Nibble4, Char NibbleD Nibble5,
+  Char NibbleD Nibble6, Char NibbleD Nibble7, Char NibbleD Nibble8,
+  Char NibbleD Nibble9, Char NibbleD NibbleA, Char NibbleD NibbleB,
+  Char NibbleD NibbleC, Char NibbleD NibbleD, Char NibbleD NibbleE,
+  Char NibbleD NibbleF, Char NibbleE Nibble0, Char NibbleE Nibble1,
+  Char NibbleE Nibble2, Char NibbleE Nibble3, Char NibbleE Nibble4,
+  Char NibbleE Nibble5, Char NibbleE Nibble6, Char NibbleE Nibble7,
+  Char NibbleE Nibble8, Char NibbleE Nibble9, Char NibbleE NibbleA,
+  Char NibbleE NibbleB, Char NibbleE NibbleC, Char NibbleE NibbleD,
+  Char NibbleE NibbleE, Char NibbleE NibbleF, Char NibbleF Nibble0,
+  Char NibbleF Nibble1, Char NibbleF Nibble2, Char NibbleF Nibble3,
+  Char NibbleF Nibble4, Char NibbleF Nibble5, Char NibbleF Nibble6,
+  Char NibbleF Nibble7, Char NibbleF Nibble8, Char NibbleF Nibble9,
+  Char NibbleF NibbleA, Char NibbleF NibbleB, Char NibbleF NibbleC,
+  Char NibbleF NibbleD, Char NibbleF NibbleE, Char NibbleF NibbleF]"
+  unfolding enum_char_def enum_nibble_def by simp
 
 instance by default
-  (auto intro: char.exhaust simp add: enum_char_def product_list_set enum_all full_SetCompr_eq [symmetric])
+  (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 "fun" :: (enum, enum) enum
-begin
-
-
-definition
-  "enum 
-
-lemma inj_graph: "inj (%f. {(x, y). y = f x})"
-  by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
-
-instance "fun" :: (finite, finite) finite
-proof
-  show "finite (UNIV :: ('a => 'b) set)"
-  proof (rule finite_imageD)
-    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
-    show "finite (range ?graph)" by (rule finite)
-    show "inj ?graph" by (rule inj_graph)
-  qed
-qed*)
-
 end
\ No newline at end of file