src/HOL/Enum.thy
 author bulwahn Mon Nov 22 11:34:57 2010 +0100 (2010-11-22) changeset 40651 9752ba7348b5 parent 40650 d40b347d5b0b child 40652 7bdfc1d6b143 permissions -rw-r--r--
adding code equation for function equality; adding some instantiations for the finite types
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* Finite types as explicit enumerations *}
```
```     4
```
```     5 theory Enum
```
```     6 imports Map String
```
```     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, equal) equal
```
```    39 begin
```
```    40
```
```    41 definition
```
```    42   "HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
```
```    43
```
```    44 instance proof
```
```    45 qed (simp_all add: equal_fun_def enum_all fun_eq_iff)
```
```    46
```
```    47 end
```
```    48
```
```    49 lemma [code nbe]:
```
```    50   "HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"
```
```    51   by (fact equal_refl)
```
```    52
```
```    53 lemma [code]:
```
```    54   "HOL.equal f g \<longleftrightarrow>  list_all (%x. f x = g x) enum"
```
```    55 by (auto simp add: list_all_iff enum_all equal fun_eq_iff)
```
```    56
```
```    57 lemma order_fun [code]:
```
```    58   fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
```
```    59   shows "f \<le> g \<longleftrightarrow> list_all (\<lambda>x. f x \<le> g x) enum"
```
```    60     and "f < g \<longleftrightarrow> f \<le> g \<and> list_ex (\<lambda>x. f x \<noteq> g x) enum"
```
```    61   by (simp_all add: list_all_iff list_ex_iff enum_all fun_eq_iff le_fun_def order_less_le)
```
```    62
```
```    63
```
```    64 subsection {* Quantifiers *}
```
```    65
```
```    66 lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> list_all P enum"
```
```    67   by (simp add: list_all_iff enum_all)
```
```    68
```
```    69 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> list_ex P enum"
```
```    70   by (simp add: list_ex_iff enum_all)
```
```    71
```
```    72
```
```    73 subsection {* Default instances *}
```
```    74
```
```    75 primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
```
```    76   "n_lists 0 xs = [[]]"
```
```    77   | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
```
```    78
```
```    79 lemma n_lists_Nil [simp]: "n_lists n [] = (if n = 0 then [[]] else [])"
```
```    80   by (induct n) simp_all
```
```    81
```
```    82 lemma length_n_lists: "length (n_lists n xs) = length xs ^ n"
```
```    83   by (induct n) (auto simp add: length_concat o_def listsum_triv)
```
```    84
```
```    85 lemma length_n_lists_elem: "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```    86   by (induct n arbitrary: ys) auto
```
```    87
```
```    88 lemma set_n_lists: "set (n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    89 proof (rule set_eqI)
```
```    90   fix ys :: "'a list"
```
```    91   show "ys \<in> set (n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    92   proof -
```
```    93     have "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```    94       by (induct n arbitrary: ys) auto
```
```    95     moreover have "\<And>x. ys \<in> set (n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"
```
```    96       by (induct n arbitrary: ys) auto
```
```    97     moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (n_lists (length ys) xs)"
```
```    98       by (induct ys) auto
```
```    99     ultimately show ?thesis by auto
```
```   100   qed
```
```   101 qed
```
```   102
```
```   103 lemma distinct_n_lists:
```
```   104   assumes "distinct xs"
```
```   105   shows "distinct (n_lists n xs)"
```
```   106 proof (rule card_distinct)
```
```   107   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
```
```   108   have "card (set (n_lists n xs)) = card (set xs) ^ n"
```
```   109   proof (induct n)
```
```   110     case 0 then show ?case by simp
```
```   111   next
```
```   112     case (Suc n)
```
```   113     moreover have "card (\<Union>ys\<in>set (n_lists n xs). (\<lambda>y. y # ys) ` set xs)
```
```   114       = (\<Sum>ys\<in>set (n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
```
```   115       by (rule card_UN_disjoint) auto
```
```   116     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
```
```   117       by (rule card_image) (simp add: inj_on_def)
```
```   118     ultimately show ?case by auto
```
```   119   qed
```
```   120   also have "\<dots> = length xs ^ n" by (simp add: card_length)
```
```   121   finally show "card (set (n_lists n xs)) = length (n_lists n xs)"
```
```   122     by (simp add: length_n_lists)
```
```   123 qed
```
```   124
```
```   125 lemma map_of_zip_enum_is_Some:
```
```   126   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   127   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
```
```   128 proof -
```
```   129   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
```
```   130     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
```
```   131     by (auto intro!: map_of_zip_is_Some)
```
```   132   then show ?thesis using enum_all by auto
```
```   133 qed
```
```   134
```
```   135 lemma map_of_zip_enum_inject:
```
```   136   fixes xs ys :: "'b\<Colon>enum list"
```
```   137   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   138       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   139     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)"
```
```   140   shows "xs = ys"
```
```   141 proof -
```
```   142   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
```
```   143   proof
```
```   144     fix x :: 'a
```
```   145     from length map_of_zip_enum_is_Some obtain y1 y2
```
```   146       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
```
```   147         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
```
```   148     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)"
```
```   149       by (auto dest: fun_cong)
```
```   150     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
```
```   151       by simp
```
```   152   qed
```
```   153   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
```
```   154 qed
```
```   155
```
```   156 instantiation "fun" :: (enum, enum) enum
```
```   157 begin
```
```   158
```
```   159 definition
```
```   160   "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
```
```   161
```
```   162 instance proof
```
```   163   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   164   proof (rule UNIV_eq_I)
```
```   165     fix f :: "'a \<Rightarrow> 'b"
```
```   166     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   167       by (auto simp add: map_of_zip_map fun_eq_iff)
```
```   168     then show "f \<in> set enum"
```
```   169       by (auto simp add: enum_fun_def set_n_lists)
```
```   170   qed
```
```   171 next
```
```   172   from map_of_zip_enum_inject
```
```   173   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   174     by (auto intro!: inj_onI simp add: enum_fun_def
```
```   175       distinct_map distinct_n_lists enum_distinct set_n_lists enum_all)
```
```   176 qed
```
```   177
```
```   178 end
```
```   179
```
```   180 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
```
```   181   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (n_lists (length enum_a) enum))"
```
```   182   by (simp add: enum_fun_def Let_def)
```
```   183
```
```   184 instantiation unit :: enum
```
```   185 begin
```
```   186
```
```   187 definition
```
```   188   "enum = [()]"
```
```   189
```
```   190 instance proof
```
```   191 qed (simp_all add: enum_unit_def UNIV_unit)
```
```   192
```
```   193 end
```
```   194
```
```   195 instantiation bool :: enum
```
```   196 begin
```
```   197
```
```   198 definition
```
```   199   "enum = [False, True]"
```
```   200
```
```   201 instance proof
```
```   202 qed (simp_all add: enum_bool_def UNIV_bool)
```
```   203
```
```   204 end
```
```   205
```
```   206 primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
```
```   207   "product [] _ = []"
```
```   208   | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
```
```   209
```
```   210 lemma product_list_set:
```
```   211   "set (product xs ys) = set xs \<times> set ys"
```
```   212   by (induct xs) auto
```
```   213
```
```   214 lemma distinct_product:
```
```   215   assumes "distinct xs" and "distinct ys"
```
```   216   shows "distinct (product xs ys)"
```
```   217   using assms by (induct xs)
```
```   218     (auto intro: inj_onI simp add: product_list_set distinct_map)
```
```   219
```
```   220 instantiation prod :: (enum, enum) enum
```
```   221 begin
```
```   222
```
```   223 definition
```
```   224   "enum = product enum enum"
```
```   225
```
```   226 instance by default
```
```   227   (simp_all add: enum_prod_def product_list_set distinct_product enum_all enum_distinct)
```
```   228
```
```   229 end
```
```   230
```
```   231 instantiation sum :: (enum, enum) enum
```
```   232 begin
```
```   233
```
```   234 definition
```
```   235   "enum = map Inl enum @ map Inr enum"
```
```   236
```
```   237 instance by default
```
```   238   (auto simp add: enum_all enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)
```
```   239
```
```   240 end
```
```   241
```
```   242 primrec sublists :: "'a list \<Rightarrow> 'a list list" where
```
```   243   "sublists [] = [[]]"
```
```   244   | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
```
```   245
```
```   246 lemma length_sublists:
```
```   247   "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
```
```   248   by (induct xs) (simp_all add: Let_def)
```
```   249
```
```   250 lemma sublists_powset:
```
```   251   "set ` set (sublists xs) = Pow (set xs)"
```
```   252 proof -
```
```   253   have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
```
```   254     by (auto simp add: image_def)
```
```   255   have "set (map set (sublists xs)) = Pow (set xs)"
```
```   256     by (induct xs)
```
```   257       (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
```
```   258   then show ?thesis by simp
```
```   259 qed
```
```   260
```
```   261 lemma distinct_set_sublists:
```
```   262   assumes "distinct xs"
```
```   263   shows "distinct (map set (sublists xs))"
```
```   264 proof (rule card_distinct)
```
```   265   have "finite (set xs)" by rule
```
```   266   then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
```
```   267   with assms distinct_card [of xs]
```
```   268     have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
```
```   269   then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
```
```   270     by (simp add: sublists_powset length_sublists)
```
```   271 qed
```
```   272
```
```   273 instantiation nibble :: enum
```
```   274 begin
```
```   275
```
```   276 definition
```
```   277   "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
```
```   278     Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
```
```   279
```
```   280 instance proof
```
```   281 qed (simp_all add: enum_nibble_def UNIV_nibble)
```
```   282
```
```   283 end
```
```   284
```
```   285 instantiation char :: enum
```
```   286 begin
```
```   287
```
```   288 definition
```
```   289   "enum = map (split Char) (product enum enum)"
```
```   290
```
```   291 lemma enum_chars [code]:
```
```   292   "enum = chars"
```
```   293   unfolding enum_char_def chars_def enum_nibble_def by simp
```
```   294
```
```   295 instance proof
```
```   296 qed (auto intro: char.exhaust injI simp add: enum_char_def product_list_set enum_all full_SetCompr_eq [symmetric]
```
```   297   distinct_map distinct_product enum_distinct)
```
```   298
```
```   299 end
```
```   300
```
```   301 instantiation option :: (enum) enum
```
```   302 begin
```
```   303
```
```   304 definition
```
```   305   "enum = None # map Some enum"
```
```   306
```
```   307 instance proof
```
```   308 qed (auto simp add: enum_all enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
```
```   309
```
```   310 end
```
```   311
```
```   312 subsection {* Small finite types *}
```
```   313
```
```   314 text {* We define small finite types for the use in Quickcheck *}
```
```   315
```
```   316 datatype finite_1 = a\<^isub>1
```
```   317
```
```   318 instantiation finite_1 :: enum
```
```   319 begin
```
```   320
```
```   321 definition
```
```   322   "enum = [a\<^isub>1]"
```
```   323
```
```   324 instance proof
```
```   325 qed (auto simp add: enum_finite_1_def intro: finite_1.exhaust)
```
```   326
```
```   327 end
```
```   328
```
```   329 instantiation finite_1 :: linorder
```
```   330 begin
```
```   331
```
```   332 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   333 where
```
```   334   "less_eq_finite_1 x y = True"
```
```   335
```
```   336 definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   337 where
```
```   338   "less_finite_1 x y = False"
```
```   339
```
```   340 instance
```
```   341 apply (intro_classes)
```
```   342 apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
```
```   343 apply (metis finite_1.exhaust)
```
```   344 done
```
```   345
```
```   346 end
```
```   347
```
```   348 datatype finite_2 = a\<^isub>1 | a\<^isub>2
```
```   349
```
```   350 instantiation finite_2 :: enum
```
```   351 begin
```
```   352
```
```   353 definition
```
```   354   "enum = [a\<^isub>1, a\<^isub>2]"
```
```   355
```
```   356 instance proof
```
```   357 qed (auto simp add: enum_finite_2_def intro: finite_2.exhaust)
```
```   358
```
```   359 end
```
```   360
```
```   361 instantiation finite_2 :: linorder
```
```   362 begin
```
```   363
```
```   364 definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   365 where
```
```   366   "less_finite_2 x y = ((x = a\<^isub>1) & (y = a\<^isub>2))"
```
```   367
```
```   368 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   369 where
```
```   370   "less_eq_finite_2 x y = ((x = y) \<or> (x < y))"
```
```   371
```
```   372
```
```   373 instance
```
```   374 apply (intro_classes)
```
```   375 apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
```
```   376 apply (metis finite_2.distinct finite_2.nchotomy)+
```
```   377 done
```
```   378
```
```   379 end
```
```   380
```
```   381
```
```   382 datatype finite_3 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3
```
```   383
```
```   384 instantiation finite_3 :: enum
```
```   385 begin
```
```   386
```
```   387 definition
```
```   388   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3]"
```
```   389
```
```   390 instance proof
```
```   391 qed (auto simp add: enum_finite_3_def intro: finite_3.exhaust)
```
```   392
```
```   393 end
```
```   394
```
```   395 instantiation finite_3 :: linorder
```
```   396 begin
```
```   397
```
```   398 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   399 where
```
```   400   "less_finite_3 x y = (case x of a\<^isub>1 => (y \<noteq> a\<^isub>1)
```
```   401      | a\<^isub>2 => (y = a\<^isub>3)| a\<^isub>3 => False)"
```
```   402
```
```   403 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   404 where
```
```   405   "less_eq_finite_3 x y = ((x = y) \<or> (x < y))"
```
```   406
```
```   407
```
```   408 instance proof (intro_classes)
```
```   409 qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
```
```   410
```
```   411 end
```
```   412
```
```   413
```
```   414 datatype finite_4 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4
```
```   415
```
```   416 instantiation finite_4 :: enum
```
```   417 begin
```
```   418
```
```   419 definition
```
```   420   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4]"
```
```   421
```
```   422 instance proof
```
```   423 qed (auto simp add: enum_finite_4_def intro: finite_4.exhaust)
```
```   424
```
```   425 end
```
```   426
```
```   427
```
```   428
```
```   429 datatype finite_5 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4 | a\<^isub>5
```
```   430
```
```   431 instantiation finite_5 :: enum
```
```   432 begin
```
```   433
```
```   434 definition
```
```   435   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4, a\<^isub>5]"
```
```   436
```
```   437 instance proof
```
```   438 qed (auto simp add: enum_finite_5_def intro: finite_5.exhaust)
```
```   439
```
```   440 end
```
```   441
```
```   442 hide_type finite_1 finite_2 finite_3 finite_4 finite_5
```
```   443 hide_const (open) n_lists product
```
```   444
```
```   445 end
```