src/HOL/Library/Enum.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 31482 7288382fd549 child 31596 c96d7e5df659 permissions -rw-r--r--
cleaned up some proofs
```     1 (* Author: Florian Haftmann, TU Muenchen
```
```     2 *)
```
```     3
```
```     4 header {* Finite types as explicit enumerations *}
```
```     5
```
```     6 theory Enum
```
```     7 imports Map Main
```
```     8 begin
```
```     9
```
```    10 subsection {* Class @{text enum} *}
```
```    11
```
```    12 class enum =
```
```    13   fixes enum :: "'a list"
```
```    14   assumes UNIV_enum [code]: "UNIV = set enum"
```
```    15     and enum_distinct: "distinct enum"
```
```    16 begin
```
```    17
```
```    18 subclass finite proof
```
```    19 qed (simp add: UNIV_enum)
```
```    20
```
```    21 lemma enum_all: "set enum = UNIV" unfolding UNIV_enum ..
```
```    22
```
```    23 lemma in_enum [intro]: "x \<in> set enum"
```
```    24   unfolding enum_all by auto
```
```    25
```
```    26 lemma enum_eq_I:
```
```    27   assumes "\<And>x. x \<in> set xs"
```
```    28   shows "set enum = set xs"
```
```    29 proof -
```
```    30   from assms UNIV_eq_I have "UNIV = set xs" by auto
```
```    31   with enum_all show ?thesis by simp
```
```    32 qed
```
```    33
```
```    34 end
```
```    35
```
```    36
```
```    37 subsection {* Equality and order on functions *}
```
```    38
```
```    39 instantiation "fun" :: (enum, eq) eq
```
```    40 begin
```
```    41
```
```    42 definition
```
```    43   "eq_class.eq f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
```
```    44
```
```    45 instance proof
```
```    46 qed (simp_all add: eq_fun_def enum_all expand_fun_eq)
```
```    47
```
```    48 end
```
```    49
```
```    50 lemma order_fun [code]:
```
```    51   fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
```
```    52   shows "f \<le> g \<longleftrightarrow> list_all (\<lambda>x. f x \<le> g x) enum"
```
```    53     and "f < g \<longleftrightarrow> f \<le> g \<and> \<not> list_all (\<lambda>x. f x = g x) enum"
```
```    54   by (simp_all add: list_all_iff enum_all expand_fun_eq le_fun_def order_less_le)
```
```    55
```
```    56
```
```    57 subsection {* Quantifiers *}
```
```    58
```
```    59 lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> list_all P enum"
```
```    60   by (simp add: list_all_iff enum_all)
```
```    61
```
```    62 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> \<not> list_all (Not o P) enum"
```
```    63   by (simp add: list_all_iff enum_all)
```
```    64
```
```    65
```
```    66 subsection {* Default instances *}
```
```    67
```
```    68 primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
```
```    69   "n_lists 0 xs = [[]]"
```
```    70   | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
```
```    71
```
```    72 lemma n_lists_Nil [simp]: "n_lists n [] = (if n = 0 then [[]] else [])"
```
```    73   by (induct n) simp_all
```
```    74
```
```    75 lemma length_n_lists: "length (n_lists n xs) = length xs ^ n"
```
```    76   by (induct n) (auto simp add: length_concat map_compose [symmetric] o_def listsum_triv)
```
```    77
```
```    78 lemma length_n_lists_elem: "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```    79   by (induct n arbitrary: ys) auto
```
```    80
```
```    81 lemma set_n_lists: "set (n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    82 proof (rule set_ext)
```
```    83   fix ys :: "'a list"
```
```    84   show "ys \<in> set (n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    85   proof -
```
```    86     have "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```    87       by (induct n arbitrary: ys) auto
```
```    88     moreover have "\<And>x. ys \<in> set (n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"
```
```    89       by (induct n arbitrary: ys) auto
```
```    90     moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (n_lists (length ys) xs)"
```
```    91       by (induct ys) auto
```
```    92     ultimately show ?thesis by auto
```
```    93   qed
```
```    94 qed
```
```    95
```
```    96 lemma distinct_n_lists:
```
```    97   assumes "distinct xs"
```
```    98   shows "distinct (n_lists n xs)"
```
```    99 proof (rule card_distinct)
```
```   100   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
```
```   101   have "card (set (n_lists n xs)) = card (set xs) ^ n"
```
```   102   proof (induct n)
```
```   103     case 0 then show ?case by simp
```
```   104   next
```
```   105     case (Suc n)
```
```   106     moreover have "card (\<Union>ys\<in>set (n_lists n xs). (\<lambda>y. y # ys) ` set xs)
```
```   107       = (\<Sum>ys\<in>set (n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
```
```   108       by (rule card_UN_disjoint) auto
```
```   109     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
```
```   110       by (rule card_image) (simp add: inj_on_def)
```
```   111     ultimately show ?case by auto
```
```   112   qed
```
```   113   also have "\<dots> = length xs ^ n" by (simp add: card_length)
```
```   114   finally show "card (set (n_lists n xs)) = length (n_lists n xs)"
```
```   115     by (simp add: length_n_lists)
```
```   116 qed
```
```   117
```
```   118 lemma map_of_zip_map: (*FIXME move to Map.thy*)
```
```   119   "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
```
```   120   by (induct xs) (simp_all add: expand_fun_eq)
```
```   121
```
```   122 lemma map_of_zip_enum_is_Some:
```
```   123   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   124   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
```
```   125 proof -
```
```   126   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
```
```   127     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
```
```   128     by (auto intro!: map_of_zip_is_Some)
```
```   129   then show ?thesis using enum_all by auto
```
```   130 qed
```
```   131
```
```   132 lemma map_of_zip_enum_inject:
```
```   133   fixes xs ys :: "'b\<Colon>enum list"
```
```   134   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   135       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   136     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)"
```
```   137   shows "xs = ys"
```
```   138 proof -
```
```   139   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
```
```   140   proof
```
```   141     fix x :: 'a
```
```   142     from length map_of_zip_enum_is_Some obtain y1 y2
```
```   143       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
```
```   144         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
```
```   145     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)"
```
```   146       by (auto dest: fun_cong)
```
```   147     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
```
```   148       by simp
```
```   149   qed
```
```   150   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
```
```   151 qed
```
```   152
```
```   153 instantiation "fun" :: (enum, enum) enum
```
```   154 begin
```
```   155
```
```   156 definition
```
```   157   [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)"
```
```   158
```
```   159 instance proof
```
```   160   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   161   proof (rule UNIV_eq_I)
```
```   162     fix f :: "'a \<Rightarrow> 'b"
```
```   163     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   164       by (auto simp add: map_of_zip_map expand_fun_eq)
```
```   165     then show "f \<in> set enum"
```
```   166       by (auto simp add: enum_fun_def set_n_lists)
```
```   167   qed
```
```   168 next
```
```   169   from map_of_zip_enum_inject
```
```   170   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   171     by (auto intro!: inj_onI simp add: enum_fun_def
```
```   172       distinct_map distinct_n_lists enum_distinct set_n_lists enum_all)
```
```   173 qed
```
```   174
```
```   175 end
```
```   176
```
```   177 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, eq} list)
```
```   178   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (n_lists (length enum_a) enum))"
```
```   179   by (simp add: enum_fun_def Let_def)
```
```   180
```
```   181 instantiation unit :: enum
```
```   182 begin
```
```   183
```
```   184 definition
```
```   185   "enum = [()]"
```
```   186
```
```   187 instance proof
```
```   188 qed (simp_all add: enum_unit_def UNIV_unit)
```
```   189
```
```   190 end
```
```   191
```
```   192 instantiation bool :: enum
```
```   193 begin
```
```   194
```
```   195 definition
```
```   196   "enum = [False, True]"
```
```   197
```
```   198 instance proof
```
```   199 qed (simp_all add: enum_bool_def UNIV_bool)
```
```   200
```
```   201 end
```
```   202
```
```   203 primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
```
```   204   "product [] _ = []"
```
```   205   | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
```
```   206
```
```   207 lemma product_list_set:
```
```   208   "set (product xs ys) = set xs \<times> set ys"
```
```   209   by (induct xs) auto
```
```   210
```
```   211 lemma distinct_product:
```
```   212   assumes "distinct xs" and "distinct ys"
```
```   213   shows "distinct (product xs ys)"
```
```   214   using assms by (induct xs)
```
```   215     (auto intro: inj_onI simp add: product_list_set distinct_map)
```
```   216
```
```   217 instantiation * :: (enum, enum) enum
```
```   218 begin
```
```   219
```
```   220 definition
```
```   221   "enum = product enum enum"
```
```   222
```
```   223 instance by default
```
```   224   (simp_all add: enum_prod_def product_list_set distinct_product enum_all enum_distinct)
```
```   225
```
```   226 end
```
```   227
```
```   228 instantiation "+" :: (enum, enum) enum
```
```   229 begin
```
```   230
```
```   231 definition
```
```   232   "enum = map Inl enum @ map Inr enum"
```
```   233
```
```   234 instance by default
```
```   235   (auto simp add: enum_all enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)
```
```   236
```
```   237 end
```
```   238
```
```   239 primrec sublists :: "'a list \<Rightarrow> 'a list list" where
```
```   240   "sublists [] = [[]]"
```
```   241   | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
```
```   242
```
```   243 lemma length_sublists:
```
```   244   "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
```
```   245   by (induct xs) (simp_all add: Let_def)
```
```   246
```
```   247 lemma sublists_powset:
```
```   248   "set ` set (sublists xs) = Pow (set xs)"
```
```   249 proof -
```
```   250   have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
```
```   251     by (auto simp add: image_def)
```
```   252   have "set (map set (sublists xs)) = Pow (set xs)"
```
```   253     by (induct xs)
```
```   254       (simp_all add: aux Let_def Pow_insert Un_commute)
```
```   255   then show ?thesis by simp
```
```   256 qed
```
```   257
```
```   258 lemma distinct_set_sublists:
```
```   259   assumes "distinct xs"
```
```   260   shows "distinct (map set (sublists xs))"
```
```   261 proof (rule card_distinct)
```
```   262   have "finite (set xs)" by rule
```
```   263   then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
```
```   264   with assms distinct_card [of xs]
```
```   265     have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
```
```   266   then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
```
```   267     by (simp add: sublists_powset length_sublists)
```
```   268 qed
```
```   269
```
```   270 instantiation nibble :: enum
```
```   271 begin
```
```   272
```
```   273 definition
```
```   274   "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
```
```   275     Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
```
```   276
```
```   277 instance proof
```
```   278 qed (simp_all add: enum_nibble_def UNIV_nibble)
```
```   279
```
```   280 end
```
```   281
```
```   282 instantiation char :: enum
```
```   283 begin
```
```   284
```
```   285 definition
```
```   286   [code del]: "enum = map (split Char) (product enum enum)"
```
```   287
```
```   288 lemma enum_chars [code]:
```
```   289   "enum = chars"
```
```   290   unfolding enum_char_def chars_def enum_nibble_def by simp
```
```   291
```
```   292 instance proof
```
```   293 qed (auto intro: char.exhaust injI simp add: enum_char_def product_list_set enum_all full_SetCompr_eq [symmetric]
```
```   294   distinct_map distinct_product enum_distinct)
```
```   295
```
```   296 end
```
```   297
```
```   298 instantiation option :: (enum) enum
```
```   299 begin
```
```   300
```
```   301 definition
```
```   302   "enum = None # map Some enum"
```
```   303
```
```   304 instance proof
```
```   305 qed (auto simp add: enum_all enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
```
```   306
```
```   307 end
```
```   308
```
```   309 end
```