src/HOL/Library/Enum.thy
 author wenzelm Thu Feb 11 23:00:22 2010 +0100 (2010-02-11) changeset 35115 446c5063e4fd parent 33639 603320b93668 child 37601 2a4fb776ca53 permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* Finite types as explicit enumerations *}
```
```     4
```
```     5 theory Enum
```
```     6 imports Map Main
```
```     7 begin
```
```     8
```
```     9 subsection {* Class @{text enum} *}
```
```    10
```
```    11 class enum =
```
```    12   fixes enum :: "'a list"
```
```    13   assumes UNIV_enum: "UNIV = set enum"
```
```    14     and enum_distinct: "distinct enum"
```
```    15 begin
```
```    16
```
```    17 subclass finite proof
```
```    18 qed (simp add: UNIV_enum)
```
```    19
```
```    20 lemma enum_all: "set enum = UNIV" unfolding UNIV_enum ..
```
```    21
```
```    22 lemma in_enum [intro]: "x \<in> set enum"
```
```    23   unfolding enum_all by auto
```
```    24
```
```    25 lemma enum_eq_I:
```
```    26   assumes "\<And>x. x \<in> set xs"
```
```    27   shows "set enum = set xs"
```
```    28 proof -
```
```    29   from assms UNIV_eq_I have "UNIV = set xs" by auto
```
```    30   with enum_all show ?thesis by simp
```
```    31 qed
```
```    32
```
```    33 end
```
```    34
```
```    35
```
```    36 subsection {* Equality and order on functions *}
```
```    37
```
```    38 instantiation "fun" :: (enum, eq) eq
```
```    39 begin
```
```    40
```
```    41 definition
```
```    42   "eq_class.eq f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
```
```    43
```
```    44 instance proof
```
```    45 qed (simp_all add: eq_fun_def enum_all expand_fun_eq)
```
```    46
```
```    47 end
```
```    48
```
```    49 lemma order_fun [code]:
```
```    50   fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
```
```    51   shows "f \<le> g \<longleftrightarrow> list_all (\<lambda>x. f x \<le> g x) enum"
```
```    52     and "f < g \<longleftrightarrow> f \<le> g \<and> \<not> list_all (\<lambda>x. f x = g x) enum"
```
```    53   by (simp_all add: list_all_iff enum_all expand_fun_eq le_fun_def order_less_le)
```
```    54
```
```    55
```
```    56 subsection {* Quantifiers *}
```
```    57
```
```    58 lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> list_all P enum"
```
```    59   by (simp add: list_all_iff enum_all)
```
```    60
```
```    61 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> \<not> list_all (Not o P) enum"
```
```    62   by (simp add: list_all_iff enum_all)
```
```    63
```
```    64
```
```    65 subsection {* Default instances *}
```
```    66
```
```    67 primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
```
```    68   "n_lists 0 xs = [[]]"
```
```    69   | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
```
```    70
```
```    71 lemma n_lists_Nil [simp]: "n_lists n [] = (if n = 0 then [[]] else [])"
```
```    72   by (induct n) simp_all
```
```    73
```
```    74 lemma length_n_lists: "length (n_lists n xs) = length xs ^ n"
```
```    75   by (induct n) (auto simp add: length_concat o_def listsum_triv)
```
```    76
```
```    77 lemma length_n_lists_elem: "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```    78   by (induct n arbitrary: ys) auto
```
```    79
```
```    80 lemma set_n_lists: "set (n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    81 proof (rule set_ext)
```
```    82   fix ys :: "'a list"
```
```    83   show "ys \<in> set (n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    84   proof -
```
```    85     have "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```    86       by (induct n arbitrary: ys) auto
```
```    87     moreover have "\<And>x. ys \<in> set (n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"
```
```    88       by (induct n arbitrary: ys) auto
```
```    89     moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (n_lists (length ys) xs)"
```
```    90       by (induct ys) auto
```
```    91     ultimately show ?thesis by auto
```
```    92   qed
```
```    93 qed
```
```    94
```
```    95 lemma distinct_n_lists:
```
```    96   assumes "distinct xs"
```
```    97   shows "distinct (n_lists n xs)"
```
```    98 proof (rule card_distinct)
```
```    99   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
```
```   100   have "card (set (n_lists n xs)) = card (set xs) ^ n"
```
```   101   proof (induct n)
```
```   102     case 0 then show ?case by simp
```
```   103   next
```
```   104     case (Suc n)
```
```   105     moreover have "card (\<Union>ys\<in>set (n_lists n xs). (\<lambda>y. y # ys) ` set xs)
```
```   106       = (\<Sum>ys\<in>set (n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
```
```   107       by (rule card_UN_disjoint) auto
```
```   108     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
```
```   109       by (rule card_image) (simp add: inj_on_def)
```
```   110     ultimately show ?case by auto
```
```   111   qed
```
```   112   also have "\<dots> = length xs ^ n" by (simp add: card_length)
```
```   113   finally show "card (set (n_lists n xs)) = length (n_lists n xs)"
```
```   114     by (simp add: length_n_lists)
```
```   115 qed
```
```   116
```
```   117 lemma map_of_zip_enum_is_Some:
```
```   118   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   119   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
```
```   120 proof -
```
```   121   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
```
```   122     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
```
```   123     by (auto intro!: map_of_zip_is_Some)
```
```   124   then show ?thesis using enum_all by auto
```
```   125 qed
```
```   126
```
```   127 lemma map_of_zip_enum_inject:
```
```   128   fixes xs ys :: "'b\<Colon>enum list"
```
```   129   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   130       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   131     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)"
```
```   132   shows "xs = ys"
```
```   133 proof -
```
```   134   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
```
```   135   proof
```
```   136     fix x :: 'a
```
```   137     from length map_of_zip_enum_is_Some obtain y1 y2
```
```   138       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
```
```   139         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
```
```   140     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)"
```
```   141       by (auto dest: fun_cong)
```
```   142     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
```
```   143       by simp
```
```   144   qed
```
```   145   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
```
```   146 qed
```
```   147
```
```   148 instantiation "fun" :: (enum, enum) enum
```
```   149 begin
```
```   150
```
```   151 definition
```
```   152   [code 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)"
```
```   153
```
```   154 instance proof
```
```   155   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   156   proof (rule UNIV_eq_I)
```
```   157     fix f :: "'a \<Rightarrow> 'b"
```
```   158     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   159       by (auto simp add: map_of_zip_map expand_fun_eq)
```
```   160     then show "f \<in> set enum"
```
```   161       by (auto simp add: enum_fun_def set_n_lists)
```
```   162   qed
```
```   163 next
```
```   164   from map_of_zip_enum_inject
```
```   165   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   166     by (auto intro!: inj_onI simp add: enum_fun_def
```
```   167       distinct_map distinct_n_lists enum_distinct set_n_lists enum_all)
```
```   168 qed
```
```   169
```
```   170 end
```
```   171
```
```   172 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, eq} list)
```
```   173   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (n_lists (length enum_a) enum))"
```
```   174   by (simp add: enum_fun_def Let_def)
```
```   175
```
```   176 instantiation unit :: enum
```
```   177 begin
```
```   178
```
```   179 definition
```
```   180   "enum = [()]"
```
```   181
```
```   182 instance proof
```
```   183 qed (simp_all add: enum_unit_def UNIV_unit)
```
```   184
```
```   185 end
```
```   186
```
```   187 instantiation bool :: enum
```
```   188 begin
```
```   189
```
```   190 definition
```
```   191   "enum = [False, True]"
```
```   192
```
```   193 instance proof
```
```   194 qed (simp_all add: enum_bool_def UNIV_bool)
```
```   195
```
```   196 end
```
```   197
```
```   198 primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
```
```   199   "product [] _ = []"
```
```   200   | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
```
```   201
```
```   202 lemma product_list_set:
```
```   203   "set (product xs ys) = set xs \<times> set ys"
```
```   204   by (induct xs) auto
```
```   205
```
```   206 lemma distinct_product:
```
```   207   assumes "distinct xs" and "distinct ys"
```
```   208   shows "distinct (product xs ys)"
```
```   209   using assms by (induct xs)
```
```   210     (auto intro: inj_onI simp add: product_list_set distinct_map)
```
```   211
```
```   212 instantiation * :: (enum, enum) enum
```
```   213 begin
```
```   214
```
```   215 definition
```
```   216   "enum = product enum enum"
```
```   217
```
```   218 instance by default
```
```   219   (simp_all add: enum_prod_def product_list_set distinct_product enum_all enum_distinct)
```
```   220
```
```   221 end
```
```   222
```
```   223 instantiation "+" :: (enum, enum) enum
```
```   224 begin
```
```   225
```
```   226 definition
```
```   227   "enum = map Inl enum @ map Inr enum"
```
```   228
```
```   229 instance by default
```
```   230   (auto simp add: enum_all enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)
```
```   231
```
```   232 end
```
```   233
```
```   234 primrec sublists :: "'a list \<Rightarrow> 'a list list" where
```
```   235   "sublists [] = [[]]"
```
```   236   | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
```
```   237
```
```   238 lemma length_sublists:
```
```   239   "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
```
```   240   by (induct xs) (simp_all add: Let_def)
```
```   241
```
```   242 lemma sublists_powset:
```
```   243   "set ` set (sublists xs) = Pow (set xs)"
```
```   244 proof -
```
```   245   have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
```
```   246     by (auto simp add: image_def)
```
```   247   have "set (map set (sublists xs)) = Pow (set xs)"
```
```   248     by (induct xs)
```
```   249       (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
```
```   250   then show ?thesis by simp
```
```   251 qed
```
```   252
```
```   253 lemma distinct_set_sublists:
```
```   254   assumes "distinct xs"
```
```   255   shows "distinct (map set (sublists xs))"
```
```   256 proof (rule card_distinct)
```
```   257   have "finite (set xs)" by rule
```
```   258   then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
```
```   259   with assms distinct_card [of xs]
```
```   260     have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
```
```   261   then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
```
```   262     by (simp add: sublists_powset length_sublists)
```
```   263 qed
```
```   264
```
```   265 instantiation nibble :: enum
```
```   266 begin
```
```   267
```
```   268 definition
```
```   269   "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
```
```   270     Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
```
```   271
```
```   272 instance proof
```
```   273 qed (simp_all add: enum_nibble_def UNIV_nibble)
```
```   274
```
```   275 end
```
```   276
```
```   277 instantiation char :: enum
```
```   278 begin
```
```   279
```
```   280 definition
```
```   281   [code del]: "enum = map (split Char) (product enum enum)"
```
```   282
```
```   283 lemma enum_chars [code]:
```
```   284   "enum = chars"
```
```   285   unfolding enum_char_def chars_def enum_nibble_def by simp
```
```   286
```
```   287 instance proof
```
```   288 qed (auto intro: char.exhaust injI simp add: enum_char_def product_list_set enum_all full_SetCompr_eq [symmetric]
```
```   289   distinct_map distinct_product enum_distinct)
```
```   290
```
```   291 end
```
```   292
```
```   293 instantiation option :: (enum) enum
```
```   294 begin
```
```   295
```
```   296 definition
```
```   297   "enum = None # map Some enum"
```
```   298
```
```   299 instance proof
```
```   300 qed (auto simp add: enum_all enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
```
```   301
```
```   302 end
```
```   303
```
```   304 end
```