src/HOL/Enum.thy
 author wenzelm Thu May 24 17:25:53 2012 +0200 (2012-05-24) changeset 47988 e4b69e10b990 parent 47231 3ff8c79a9e2f child 48123 104e5fccea12 permissions -rw-r--r--
tuned proofs;
```     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   fixes enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    14   fixes enum_ex  :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    15   assumes UNIV_enum: "UNIV = set enum"
```
```    16     and enum_distinct: "distinct enum"
```
```    17   assumes enum_all : "enum_all P = (\<forall> x. P x)"
```
```    18   assumes enum_ex  : "enum_ex P = (\<exists> x. P x)"
```
```    19 begin
```
```    20
```
```    21 subclass finite proof
```
```    22 qed (simp add: UNIV_enum)
```
```    23
```
```    24 lemma enum_UNIV: "set enum = UNIV" unfolding UNIV_enum ..
```
```    25
```
```    26 lemma in_enum: "x \<in> set enum"
```
```    27   unfolding enum_UNIV by auto
```
```    28
```
```    29 lemma enum_eq_I:
```
```    30   assumes "\<And>x. x \<in> set xs"
```
```    31   shows "set enum = set xs"
```
```    32 proof -
```
```    33   from assms UNIV_eq_I have "UNIV = set xs" by auto
```
```    34   with enum_UNIV show ?thesis by simp
```
```    35 qed
```
```    36
```
```    37 end
```
```    38
```
```    39
```
```    40 subsection {* Equality and order on functions *}
```
```    41
```
```    42 instantiation "fun" :: (enum, equal) equal
```
```    43 begin
```
```    44
```
```    45 definition
```
```    46   "HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
```
```    47
```
```    48 instance proof
```
```    49 qed (simp_all add: equal_fun_def enum_UNIV fun_eq_iff)
```
```    50
```
```    51 end
```
```    52
```
```    53 lemma [code]:
```
```    54   "HOL.equal f g \<longleftrightarrow> enum_all (%x. f x = g x)"
```
```    55 by (auto simp add: equal enum_all fun_eq_iff)
```
```    56
```
```    57 lemma [code nbe]:
```
```    58   "HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"
```
```    59   by (fact equal_refl)
```
```    60
```
```    61 lemma order_fun [code]:
```
```    62   fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
```
```    63   shows "f \<le> g \<longleftrightarrow> enum_all (\<lambda>x. f x \<le> g x)"
```
```    64     and "f < g \<longleftrightarrow> f \<le> g \<and> enum_ex (\<lambda>x. f x \<noteq> g x)"
```
```    65   by (simp_all add: enum_all enum_ex fun_eq_iff le_fun_def order_less_le)
```
```    66
```
```    67
```
```    68 subsection {* Quantifiers *}
```
```    69
```
```    70 lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"
```
```    71   by (simp add: enum_all)
```
```    72
```
```    73 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"
```
```    74   by (simp add: enum_ex)
```
```    75
```
```    76 lemma exists1_code[code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
```
```    77 unfolding list_ex1_iff enum_UNIV by auto
```
```    78
```
```    79
```
```    80 subsection {* Default instances *}
```
```    81
```
```    82 primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
```
```    83   "n_lists 0 xs = [[]]"
```
```    84   | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
```
```    85
```
```    86 lemma n_lists_Nil [simp]: "n_lists n [] = (if n = 0 then [[]] else [])"
```
```    87   by (induct n) simp_all
```
```    88
```
```    89 lemma length_n_lists: "length (n_lists n xs) = length xs ^ n"
```
```    90   by (induct n) (auto simp add: length_concat o_def listsum_triv)
```
```    91
```
```    92 lemma length_n_lists_elem: "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```    93   by (induct n arbitrary: ys) auto
```
```    94
```
```    95 lemma set_n_lists: "set (n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    96 proof (rule set_eqI)
```
```    97   fix ys :: "'a list"
```
```    98   show "ys \<in> set (n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"
```
```    99   proof -
```
```   100     have "ys \<in> set (n_lists n xs) \<Longrightarrow> length ys = n"
```
```   101       by (induct n arbitrary: ys) auto
```
```   102     moreover have "\<And>x. ys \<in> set (n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"
```
```   103       by (induct n arbitrary: ys) auto
```
```   104     moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (n_lists (length ys) xs)"
```
```   105       by (induct ys) auto
```
```   106     ultimately show ?thesis by auto
```
```   107   qed
```
```   108 qed
```
```   109
```
```   110 lemma distinct_n_lists:
```
```   111   assumes "distinct xs"
```
```   112   shows "distinct (n_lists n xs)"
```
```   113 proof (rule card_distinct)
```
```   114   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
```
```   115   have "card (set (n_lists n xs)) = card (set xs) ^ n"
```
```   116   proof (induct n)
```
```   117     case 0 then show ?case by simp
```
```   118   next
```
```   119     case (Suc n)
```
```   120     moreover have "card (\<Union>ys\<in>set (n_lists n xs). (\<lambda>y. y # ys) ` set xs)
```
```   121       = (\<Sum>ys\<in>set (n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
```
```   122       by (rule card_UN_disjoint) auto
```
```   123     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
```
```   124       by (rule card_image) (simp add: inj_on_def)
```
```   125     ultimately show ?case by auto
```
```   126   qed
```
```   127   also have "\<dots> = length xs ^ n" by (simp add: card_length)
```
```   128   finally show "card (set (n_lists n xs)) = length (n_lists n xs)"
```
```   129     by (simp add: length_n_lists)
```
```   130 qed
```
```   131
```
```   132 lemma map_of_zip_enum_is_Some:
```
```   133   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   134   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
```
```   135 proof -
```
```   136   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
```
```   137     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
```
```   138     by (auto intro!: map_of_zip_is_Some)
```
```   139   then show ?thesis using enum_UNIV by auto
```
```   140 qed
```
```   141
```
```   142 lemma map_of_zip_enum_inject:
```
```   143   fixes xs ys :: "'b\<Colon>enum list"
```
```   144   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   145       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   146     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)"
```
```   147   shows "xs = ys"
```
```   148 proof -
```
```   149   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
```
```   150   proof
```
```   151     fix x :: 'a
```
```   152     from length map_of_zip_enum_is_Some obtain y1 y2
```
```   153       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
```
```   154         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
```
```   155     moreover from map_of
```
```   156       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)"
```
```   157       by (auto dest: fun_cong)
```
```   158     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
```
```   159       by simp
```
```   160   qed
```
```   161   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
```
```   162 qed
```
```   163
```
```   164 definition
```
```   165   all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
```
```   166 where
```
```   167   "all_n_lists P n = (\<forall>xs \<in> set (n_lists n enum). P xs)"
```
```   168
```
```   169 lemma [code]:
```
```   170   "all_n_lists P n = (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
```
```   171 unfolding all_n_lists_def enum_all
```
```   172 by (cases n) (auto simp add: enum_UNIV)
```
```   173
```
```   174 definition
```
```   175   ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
```
```   176 where
```
```   177   "ex_n_lists P n = (\<exists>xs \<in> set (n_lists n enum). P xs)"
```
```   178
```
```   179 lemma [code]:
```
```   180   "ex_n_lists P n = (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
```
```   181 unfolding ex_n_lists_def enum_ex
```
```   182 by (cases n) (auto simp add: enum_UNIV)
```
```   183
```
```   184
```
```   185 instantiation "fun" :: (enum, enum) enum
```
```   186 begin
```
```   187
```
```   188 definition
```
```   189   "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
```
```   190
```
```   191 definition
```
```   192   "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
```
```   193
```
```   194 definition
```
```   195   "enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
```
```   196
```
```   197
```
```   198 instance proof
```
```   199   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   200   proof (rule UNIV_eq_I)
```
```   201     fix f :: "'a \<Rightarrow> 'b"
```
```   202     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   203       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   204     then show "f \<in> set enum"
```
```   205       by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
```
```   206   qed
```
```   207 next
```
```   208   from map_of_zip_enum_inject
```
```   209   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   210     by (auto intro!: inj_onI simp add: enum_fun_def
```
```   211       distinct_map distinct_n_lists enum_distinct set_n_lists enum_all)
```
```   212 next
```
```   213   fix P
```
```   214   show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   215   proof
```
```   216     assume "enum_all P"
```
```   217     show "\<forall>x. P x"
```
```   218     proof
```
```   219       fix f :: "'a \<Rightarrow> 'b"
```
```   220       have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   221         by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   222       from `enum_all P` have "P (the \<circ> map_of (zip enum (map f enum)))"
```
```   223         unfolding enum_all_fun_def all_n_lists_def
```
```   224         apply (simp add: set_n_lists)
```
```   225         apply (erule_tac x="map f enum" in allE)
```
```   226         apply (auto intro!: in_enum)
```
```   227         done
```
```   228       from this f show "P f" by auto
```
```   229     qed
```
```   230   next
```
```   231     assume "\<forall>x. P x"
```
```   232     from this show "enum_all P"
```
```   233       unfolding enum_all_fun_def all_n_lists_def by auto
```
```   234   qed
```
```   235 next
```
```   236   fix P
```
```   237   show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   238   proof
```
```   239     assume "enum_ex P"
```
```   240     from this show "\<exists>x. P x"
```
```   241       unfolding enum_ex_fun_def ex_n_lists_def by auto
```
```   242   next
```
```   243     assume "\<exists>x. P x"
```
```   244     from this obtain f where "P f" ..
```
```   245     have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   246       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   247     from `P f` this have "P (the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum)))"
```
```   248       by auto
```
```   249     from  this show "enum_ex P"
```
```   250       unfolding enum_ex_fun_def ex_n_lists_def
```
```   251       apply (auto simp add: set_n_lists)
```
```   252       apply (rule_tac x="map f enum" in exI)
```
```   253       apply (auto intro!: in_enum)
```
```   254       done
```
```   255   qed
```
```   256 qed
```
```   257
```
```   258 end
```
```   259
```
```   260 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
```
```   261   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (n_lists (length enum_a) enum))"
```
```   262   by (simp add: enum_fun_def Let_def)
```
```   263
```
```   264 lemma enum_all_fun_code [code]:
```
```   265   "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
```
```   266    in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
```
```   267   by (simp add: enum_all_fun_def Let_def)
```
```   268
```
```   269 lemma enum_ex_fun_code [code]:
```
```   270   "enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
```
```   271    in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
```
```   272   by (simp add: enum_ex_fun_def Let_def)
```
```   273
```
```   274 instantiation unit :: enum
```
```   275 begin
```
```   276
```
```   277 definition
```
```   278   "enum = [()]"
```
```   279
```
```   280 definition
```
```   281   "enum_all P = P ()"
```
```   282
```
```   283 definition
```
```   284   "enum_ex P = P ()"
```
```   285
```
```   286 instance proof
```
```   287 qed (auto simp add: enum_unit_def UNIV_unit enum_all_unit_def enum_ex_unit_def intro: unit.exhaust)
```
```   288
```
```   289 end
```
```   290
```
```   291 instantiation bool :: enum
```
```   292 begin
```
```   293
```
```   294 definition
```
```   295   "enum = [False, True]"
```
```   296
```
```   297 definition
```
```   298   "enum_all P = (P False \<and> P True)"
```
```   299
```
```   300 definition
```
```   301   "enum_ex P = (P False \<or> P True)"
```
```   302
```
```   303 instance proof
```
```   304   fix P
```
```   305   show "enum_all (P :: bool \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   306     unfolding enum_all_bool_def by (auto, case_tac x) auto
```
```   307 next
```
```   308   fix P
```
```   309   show "enum_ex (P :: bool \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   310     unfolding enum_ex_bool_def by (auto, case_tac x) auto
```
```   311 qed (auto simp add: enum_bool_def UNIV_bool)
```
```   312
```
```   313 end
```
```   314
```
```   315 primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
```
```   316   "product [] _ = []"
```
```   317   | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
```
```   318
```
```   319 lemma product_list_set:
```
```   320   "set (product xs ys) = set xs \<times> set ys"
```
```   321   by (induct xs) auto
```
```   322
```
```   323 lemma distinct_product:
```
```   324   assumes "distinct xs" and "distinct ys"
```
```   325   shows "distinct (product xs ys)"
```
```   326   using assms by (induct xs)
```
```   327     (auto intro: inj_onI simp add: product_list_set distinct_map)
```
```   328
```
```   329 instantiation prod :: (enum, enum) enum
```
```   330 begin
```
```   331
```
```   332 definition
```
```   333   "enum = product enum enum"
```
```   334
```
```   335 definition
```
```   336   "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
```
```   337
```
```   338 definition
```
```   339   "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
```
```   340
```
```   341
```
```   342 instance by default
```
```   343   (simp_all add: enum_prod_def product_list_set distinct_product
```
```   344     enum_UNIV enum_distinct enum_all_prod_def enum_all enum_ex_prod_def enum_ex)
```
```   345
```
```   346 end
```
```   347
```
```   348 instantiation sum :: (enum, enum) enum
```
```   349 begin
```
```   350
```
```   351 definition
```
```   352   "enum = map Inl enum @ map Inr enum"
```
```   353
```
```   354 definition
```
```   355   "enum_all P = (enum_all (%x. P (Inl x)) \<and> enum_all (%x. P (Inr x)))"
```
```   356
```
```   357 definition
```
```   358   "enum_ex P = (enum_ex (%x. P (Inl x)) \<or> enum_ex (%x. P (Inr x)))"
```
```   359
```
```   360 instance proof
```
```   361   fix P
```
```   362   show "enum_all (P :: ('a + 'b) \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   363     unfolding enum_all_sum_def enum_all
```
```   364     by (auto, case_tac x) auto
```
```   365 next
```
```   366   fix P
```
```   367   show "enum_ex (P :: ('a + 'b) \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   368     unfolding enum_ex_sum_def enum_ex
```
```   369     by (auto, case_tac x) auto
```
```   370 qed (auto simp add: enum_UNIV enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)
```
```   371
```
```   372 end
```
```   373
```
```   374 instantiation nibble :: enum
```
```   375 begin
```
```   376
```
```   377 definition
```
```   378   "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
```
```   379     Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
```
```   380
```
```   381 definition
```
```   382   "enum_all P = (P Nibble0 \<and> P Nibble1 \<and> P Nibble2 \<and> P Nibble3 \<and> P Nibble4 \<and> P Nibble5 \<and> P Nibble6 \<and> P Nibble7
```
```   383      \<and> P Nibble8 \<and> P Nibble9 \<and> P NibbleA \<and> P NibbleB \<and> P NibbleC \<and> P NibbleD \<and> P NibbleE \<and> P NibbleF)"
```
```   384
```
```   385 definition
```
```   386   "enum_ex P = (P Nibble0 \<or> P Nibble1 \<or> P Nibble2 \<or> P Nibble3 \<or> P Nibble4 \<or> P Nibble5 \<or> P Nibble6 \<or> P Nibble7
```
```   387      \<or> P Nibble8 \<or> P Nibble9 \<or> P NibbleA \<or> P NibbleB \<or> P NibbleC \<or> P NibbleD \<or> P NibbleE \<or> P NibbleF)"
```
```   388
```
```   389 instance proof
```
```   390   fix P
```
```   391   show "enum_all (P :: nibble \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   392     unfolding enum_all_nibble_def
```
```   393     by (auto, case_tac x) auto
```
```   394 next
```
```   395   fix P
```
```   396   show "enum_ex (P :: nibble \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   397     unfolding enum_ex_nibble_def
```
```   398     by (auto, case_tac x) auto
```
```   399 qed (simp_all add: enum_nibble_def UNIV_nibble)
```
```   400
```
```   401 end
```
```   402
```
```   403 instantiation char :: enum
```
```   404 begin
```
```   405
```
```   406 definition
```
```   407   "enum = map (split Char) (product enum enum)"
```
```   408
```
```   409 lemma enum_chars [code]:
```
```   410   "enum = chars"
```
```   411   unfolding enum_char_def chars_def enum_nibble_def by simp
```
```   412
```
```   413 definition
```
```   414   "enum_all P = list_all P chars"
```
```   415
```
```   416 definition
```
```   417   "enum_ex P = list_ex P chars"
```
```   418
```
```   419 lemma set_enum_char: "set (enum :: char list) = UNIV"
```
```   420     by (auto intro: char.exhaust simp add: enum_char_def product_list_set enum_UNIV full_SetCompr_eq [symmetric])
```
```   421
```
```   422 instance proof
```
```   423   fix P
```
```   424   show "enum_all (P :: char \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   425     unfolding enum_all_char_def enum_chars[symmetric]
```
```   426     by (auto simp add: list_all_iff set_enum_char)
```
```   427 next
```
```   428   fix P
```
```   429   show "enum_ex (P :: char \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   430     unfolding enum_ex_char_def enum_chars[symmetric]
```
```   431     by (auto simp add: list_ex_iff set_enum_char)
```
```   432 next
```
```   433   show "distinct (enum :: char list)"
```
```   434     by (auto intro: inj_onI simp add: enum_char_def product_list_set distinct_map distinct_product enum_distinct)
```
```   435 qed (auto simp add: set_enum_char)
```
```   436
```
```   437 end
```
```   438
```
```   439 instantiation option :: (enum) enum
```
```   440 begin
```
```   441
```
```   442 definition
```
```   443   "enum = None # map Some enum"
```
```   444
```
```   445 definition
```
```   446   "enum_all P = (P None \<and> enum_all (%x. P (Some x)))"
```
```   447
```
```   448 definition
```
```   449   "enum_ex P = (P None \<or> enum_ex (%x. P (Some x)))"
```
```   450
```
```   451 instance proof
```
```   452   fix P
```
```   453   show "enum_all (P :: 'a option \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   454     unfolding enum_all_option_def enum_all
```
```   455     by (auto, case_tac x) auto
```
```   456 next
```
```   457   fix P
```
```   458   show "enum_ex (P :: 'a option \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   459     unfolding enum_ex_option_def enum_ex
```
```   460     by (auto, case_tac x) auto
```
```   461 qed (auto simp add: enum_UNIV enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
```
```   462 end
```
```   463
```
```   464 primrec sublists :: "'a list \<Rightarrow> 'a list list" where
```
```   465   "sublists [] = [[]]"
```
```   466   | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
```
```   467
```
```   468 lemma length_sublists:
```
```   469   "length (sublists xs) = 2 ^ length xs"
```
```   470   by (induct xs) (simp_all add: Let_def)
```
```   471
```
```   472 lemma sublists_powset:
```
```   473   "set ` set (sublists xs) = Pow (set xs)"
```
```   474 proof -
```
```   475   have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
```
```   476     by (auto simp add: image_def)
```
```   477   have "set (map set (sublists xs)) = Pow (set xs)"
```
```   478     by (induct xs)
```
```   479       (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
```
```   480   then show ?thesis by simp
```
```   481 qed
```
```   482
```
```   483 lemma distinct_set_sublists:
```
```   484   assumes "distinct xs"
```
```   485   shows "distinct (map set (sublists xs))"
```
```   486 proof (rule card_distinct)
```
```   487   have "finite (set xs)" by rule
```
```   488   then have "card (Pow (set xs)) = 2 ^ card (set xs)" by (rule card_Pow)
```
```   489   with assms distinct_card [of xs]
```
```   490     have "card (Pow (set xs)) = 2 ^ length xs" by simp
```
```   491   then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
```
```   492     by (simp add: sublists_powset length_sublists)
```
```   493 qed
```
```   494
```
```   495 instantiation set :: (enum) enum
```
```   496 begin
```
```   497
```
```   498 definition
```
```   499   "enum = map set (sublists enum)"
```
```   500
```
```   501 definition
```
```   502   "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
```
```   503
```
```   504 definition
```
```   505   "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
```
```   506
```
```   507 instance proof
```
```   508 qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
```
```   509   enum_distinct enum_UNIV)
```
```   510
```
```   511 end
```
```   512
```
```   513
```
```   514 subsection {* Small finite types *}
```
```   515
```
```   516 text {* We define small finite types for the use in Quickcheck *}
```
```   517
```
```   518 datatype finite_1 = a\<^isub>1
```
```   519
```
```   520 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   521
```
```   522 instantiation finite_1 :: enum
```
```   523 begin
```
```   524
```
```   525 definition
```
```   526   "enum = [a\<^isub>1]"
```
```   527
```
```   528 definition
```
```   529   "enum_all P = P a\<^isub>1"
```
```   530
```
```   531 definition
```
```   532   "enum_ex P = P a\<^isub>1"
```
```   533
```
```   534 instance proof
```
```   535   fix P
```
```   536   show "enum_all (P :: finite_1 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   537     unfolding enum_all_finite_1_def
```
```   538     by (auto, case_tac x) auto
```
```   539 next
```
```   540   fix P
```
```   541   show "enum_ex (P :: finite_1 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   542     unfolding enum_ex_finite_1_def
```
```   543     by (auto, case_tac x) auto
```
```   544 qed (auto simp add: enum_finite_1_def intro: finite_1.exhaust)
```
```   545
```
```   546 end
```
```   547
```
```   548 instantiation finite_1 :: linorder
```
```   549 begin
```
```   550
```
```   551 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   552 where
```
```   553   "less_eq_finite_1 x y = True"
```
```   554
```
```   555 definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   556 where
```
```   557   "less_finite_1 x y = False"
```
```   558
```
```   559 instance
```
```   560 apply (intro_classes)
```
```   561 apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
```
```   562 apply (metis finite_1.exhaust)
```
```   563 done
```
```   564
```
```   565 end
```
```   566
```
```   567 hide_const (open) a\<^isub>1
```
```   568
```
```   569 datatype finite_2 = a\<^isub>1 | a\<^isub>2
```
```   570
```
```   571 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   572 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   573
```
```   574 instantiation finite_2 :: enum
```
```   575 begin
```
```   576
```
```   577 definition
```
```   578   "enum = [a\<^isub>1, a\<^isub>2]"
```
```   579
```
```   580 definition
```
```   581   "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2)"
```
```   582
```
```   583 definition
```
```   584   "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2)"
```
```   585
```
```   586 instance proof
```
```   587   fix P
```
```   588   show "enum_all (P :: finite_2 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   589     unfolding enum_all_finite_2_def
```
```   590     by (auto, case_tac x) auto
```
```   591 next
```
```   592   fix P
```
```   593   show "enum_ex (P :: finite_2 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   594     unfolding enum_ex_finite_2_def
```
```   595     by (auto, case_tac x) auto
```
```   596 qed (auto simp add: enum_finite_2_def intro: finite_2.exhaust)
```
```   597
```
```   598 end
```
```   599
```
```   600 instantiation finite_2 :: linorder
```
```   601 begin
```
```   602
```
```   603 definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   604 where
```
```   605   "less_finite_2 x y = ((x = a\<^isub>1) & (y = a\<^isub>2))"
```
```   606
```
```   607 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   608 where
```
```   609   "less_eq_finite_2 x y = ((x = y) \<or> (x < y))"
```
```   610
```
```   611
```
```   612 instance
```
```   613 apply (intro_classes)
```
```   614 apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
```
```   615 apply (metis finite_2.distinct finite_2.nchotomy)+
```
```   616 done
```
```   617
```
```   618 end
```
```   619
```
```   620 hide_const (open) a\<^isub>1 a\<^isub>2
```
```   621
```
```   622
```
```   623 datatype finite_3 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3
```
```   624
```
```   625 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   626 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   627 notation (output) a\<^isub>3  ("a\<^isub>3")
```
```   628
```
```   629 instantiation finite_3 :: enum
```
```   630 begin
```
```   631
```
```   632 definition
```
```   633   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3]"
```
```   634
```
```   635 definition
```
```   636   "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3)"
```
```   637
```
```   638 definition
```
```   639   "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3)"
```
```   640
```
```   641 instance proof
```
```   642   fix P
```
```   643   show "enum_all (P :: finite_3 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   644     unfolding enum_all_finite_3_def
```
```   645     by (auto, case_tac x) auto
```
```   646 next
```
```   647   fix P
```
```   648   show "enum_ex (P :: finite_3 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   649     unfolding enum_ex_finite_3_def
```
```   650     by (auto, case_tac x) auto
```
```   651 qed (auto simp add: enum_finite_3_def intro: finite_3.exhaust)
```
```   652
```
```   653 end
```
```   654
```
```   655 instantiation finite_3 :: linorder
```
```   656 begin
```
```   657
```
```   658 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   659 where
```
```   660   "less_finite_3 x y = (case x of a\<^isub>1 => (y \<noteq> a\<^isub>1)
```
```   661      | a\<^isub>2 => (y = a\<^isub>3)| a\<^isub>3 => False)"
```
```   662
```
```   663 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   664 where
```
```   665   "less_eq_finite_3 x y = ((x = y) \<or> (x < y))"
```
```   666
```
```   667
```
```   668 instance proof (intro_classes)
```
```   669 qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
```
```   670
```
```   671 end
```
```   672
```
```   673 hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3
```
```   674
```
```   675
```
```   676 datatype finite_4 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4
```
```   677
```
```   678 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   679 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   680 notation (output) a\<^isub>3  ("a\<^isub>3")
```
```   681 notation (output) a\<^isub>4  ("a\<^isub>4")
```
```   682
```
```   683 instantiation finite_4 :: enum
```
```   684 begin
```
```   685
```
```   686 definition
```
```   687   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4]"
```
```   688
```
```   689 definition
```
```   690   "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3 \<and> P a\<^isub>4)"
```
```   691
```
```   692 definition
```
```   693   "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3 \<or> P a\<^isub>4)"
```
```   694
```
```   695 instance proof
```
```   696   fix P
```
```   697   show "enum_all (P :: finite_4 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   698     unfolding enum_all_finite_4_def
```
```   699     by (auto, case_tac x) auto
```
```   700 next
```
```   701   fix P
```
```   702   show "enum_ex (P :: finite_4 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   703     unfolding enum_ex_finite_4_def
```
```   704     by (auto, case_tac x) auto
```
```   705 qed (auto simp add: enum_finite_4_def intro: finite_4.exhaust)
```
```   706
```
```   707 end
```
```   708
```
```   709 hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4
```
```   710
```
```   711
```
```   712 datatype finite_5 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4 | a\<^isub>5
```
```   713
```
```   714 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   715 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   716 notation (output) a\<^isub>3  ("a\<^isub>3")
```
```   717 notation (output) a\<^isub>4  ("a\<^isub>4")
```
```   718 notation (output) a\<^isub>5  ("a\<^isub>5")
```
```   719
```
```   720 instantiation finite_5 :: enum
```
```   721 begin
```
```   722
```
```   723 definition
```
```   724   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4, a\<^isub>5]"
```
```   725
```
```   726 definition
```
```   727   "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3 \<and> P a\<^isub>4 \<and> P a\<^isub>5)"
```
```   728
```
```   729 definition
```
```   730   "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3 \<or> P a\<^isub>4 \<or> P a\<^isub>5)"
```
```   731
```
```   732 instance proof
```
```   733   fix P
```
```   734   show "enum_all (P :: finite_5 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   735     unfolding enum_all_finite_5_def
```
```   736     by (auto, case_tac x) auto
```
```   737 next
```
```   738   fix P
```
```   739   show "enum_ex (P :: finite_5 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   740     unfolding enum_ex_finite_5_def
```
```   741     by (auto, case_tac x) auto
```
```   742 qed (auto simp add: enum_finite_5_def intro: finite_5.exhaust)
```
```   743
```
```   744 end
```
```   745
```
```   746 hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5
```
```   747
```
```   748 subsection {* An executable THE operator on finite types *}
```
```   749
```
```   750 definition
```
```   751   [code del]: "enum_the P = The P"
```
```   752
```
```   753 lemma [code]:
```
```   754   "The P = (case filter P enum of [x] => x | _ => enum_the P)"
```
```   755 proof -
```
```   756   {
```
```   757     fix a
```
```   758     assume filter_enum: "filter P enum = [a]"
```
```   759     have "The P = a"
```
```   760     proof (rule the_equality)
```
```   761       fix x
```
```   762       assume "P x"
```
```   763       show "x = a"
```
```   764       proof (rule ccontr)
```
```   765         assume "x \<noteq> a"
```
```   766         from filter_enum obtain us vs
```
```   767           where enum_eq: "enum = us @ [a] @ vs"
```
```   768           and "\<forall> x \<in> set us. \<not> P x"
```
```   769           and "\<forall> x \<in> set vs. \<not> P x"
```
```   770           and "P a"
```
```   771           by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
```
```   772         with `P x` in_enum[of x, unfolded enum_eq] `x \<noteq> a` show "False" by auto
```
```   773       qed
```
```   774     next
```
```   775       from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
```
```   776     qed
```
```   777   }
```
```   778   from this show ?thesis
```
```   779     unfolding enum_the_def by (auto split: list.split)
```
```   780 qed
```
```   781
```
```   782 code_abort enum_the
```
```   783 code_const enum_the (Eval "(fn p => raise Match)")
```
```   784
```
```   785 subsection {* Further operations on finite types *}
```
```   786
```
```   787 lemma [code]:
```
```   788   "Collect P = set (filter P enum)"
```
```   789 by (auto simp add: enum_UNIV)
```
```   790
```
```   791 lemma tranclp_unfold [code, no_atp]:
```
```   792   "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
```
```   793 by (simp add: trancl_def)
```
```   794
```
```   795 lemma rtranclp_rtrancl_eq[code, no_atp]:
```
```   796   "rtranclp r x y = ((x, y) : rtrancl {(x, y). r x y})"
```
```   797 unfolding rtrancl_def by auto
```
```   798
```
```   799 lemma max_ext_eq[code]:
```
```   800   "max_ext R = {(X, Y). finite X & finite Y & Y ~={} & (ALL x. x : X --> (EX xa : Y. (x, xa) : R))}"
```
```   801 by (auto simp add: max_ext.simps)
```
```   802
```
```   803 lemma max_extp_eq[code]:
```
```   804   "max_extp r x y = ((x, y) : max_ext {(x, y). r x y})"
```
```   805 unfolding max_ext_def by auto
```
```   806
```
```   807 lemma mlex_eq[code]:
```
```   808   "f <*mlex*> R = {(x, y). f x < f y \<or> (f x <= f y \<and> (x, y) : R)}"
```
```   809 unfolding mlex_prod_def by auto
```
```   810
```
```   811 subsection {* Executable accessible part *}
```
```   812 (* FIXME: should be moved somewhere else !? *)
```
```   813
```
```   814 subsubsection {* Finite monotone eventually stable sequences *}
```
```   815
```
```   816 lemma finite_mono_remains_stable_implies_strict_prefix:
```
```   817   fixes f :: "nat \<Rightarrow> 'a::order"
```
```   818   assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
```
```   819   shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
```
```   820   using assms
```
```   821 proof -
```
```   822   have "\<exists>n. f n = f (Suc n)"
```
```   823   proof (rule ccontr)
```
```   824     assume "\<not> ?thesis"
```
```   825     then have "\<And>n. f n \<noteq> f (Suc n)" by auto
```
```   826     then have "\<And>n. f n < f (Suc n)"
```
```   827       using  `mono f` by (auto simp: le_less mono_iff_le_Suc)
```
```   828     with lift_Suc_mono_less_iff[of f]
```
```   829     have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
```
```   830     then have "inj f"
```
```   831       by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
```
```   832     with `finite (range f)` have "finite (UNIV::nat set)"
```
```   833       by (rule finite_imageD)
```
```   834     then show False by simp
```
```   835   qed
```
```   836   then obtain n where n: "f n = f (Suc n)" ..
```
```   837   def N \<equiv> "LEAST n. f n = f (Suc n)"
```
```   838   have N: "f N = f (Suc N)"
```
```   839     unfolding N_def using n by (rule LeastI)
```
```   840   show ?thesis
```
```   841   proof (intro exI[of _ N] conjI allI impI)
```
```   842     fix n assume "N \<le> n"
```
```   843     then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
```
```   844     proof (induct rule: dec_induct)
```
```   845       case (step n) then show ?case
```
```   846         using eq[rule_format, of "n - 1"] N
```
```   847         by (cases n) (auto simp add: le_Suc_eq)
```
```   848     qed simp
```
```   849     from this[of n] `N \<le> n` show "f N = f n" by auto
```
```   850   next
```
```   851     fix n m :: nat assume "m < n" "n \<le> N"
```
```   852     then show "f m < f n"
```
```   853     proof (induct rule: less_Suc_induct[consumes 1])
```
```   854       case (1 i)
```
```   855       then have "i < N" by simp
```
```   856       then have "f i \<noteq> f (Suc i)"
```
```   857         unfolding N_def by (rule not_less_Least)
```
```   858       with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
```
```   859     qed auto
```
```   860   qed
```
```   861 qed
```
```   862
```
```   863 lemma finite_mono_strict_prefix_implies_finite_fixpoint:
```
```   864   fixes f :: "nat \<Rightarrow> 'a set"
```
```   865   assumes S: "\<And>i. f i \<subseteq> S" "finite S"
```
```   866     and inj: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
```
```   867   shows "f (card S) = (\<Union>n. f n)"
```
```   868 proof -
```
```   869   from inj obtain N where inj: "(\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n)" and eq: "(\<forall>n\<ge>N. f N = f n)" by auto
```
```   870
```
```   871   { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
```
```   872     proof (induct i)
```
```   873       case 0 then show ?case by simp
```
```   874     next
```
```   875       case (Suc i)
```
```   876       with inj[rule_format, of "Suc i" i]
```
```   877       have "(f i) \<subset> (f (Suc i))" by auto
```
```   878       moreover have "finite (f (Suc i))" using S by (rule finite_subset)
```
```   879       ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
```
```   880       with Suc show ?case using inj by auto
```
```   881     qed
```
```   882   }
```
```   883   then have "N \<le> card (f N)" by simp
```
```   884   also have "\<dots> \<le> card S" using S by (intro card_mono)
```
```   885   finally have "f (card S) = f N" using eq by auto
```
```   886   then show ?thesis using eq inj[rule_format, of N]
```
```   887     apply auto
```
```   888     apply (case_tac "n < N")
```
```   889     apply (auto simp: not_less)
```
```   890     done
```
```   891 qed
```
```   892
```
```   893 subsubsection {* Bounded accessible part *}
```
```   894
```
```   895 fun bacc :: "('a * 'a) set => nat => 'a set"
```
```   896 where
```
```   897   "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
```
```   898 | "bacc r (Suc n) = (bacc r n Un {x. \<forall> y. (y, x) : r --> y : bacc r n})"
```
```   899
```
```   900 lemma bacc_subseteq_acc:
```
```   901   "bacc r n \<subseteq> acc r"
```
```   902 by (induct n) (auto intro: acc.intros)
```
```   903
```
```   904 lemma bacc_mono:
```
```   905   "n <= m ==> bacc r n \<subseteq> bacc r m"
```
```   906 by (induct rule: dec_induct) auto
```
```   907
```
```   908 lemma bacc_upper_bound:
```
```   909   "bacc (r :: ('a * 'a) set)  (card (UNIV :: ('a :: enum) set)) = (UN n. bacc r n)"
```
```   910 proof -
```
```   911   have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
```
```   912   moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
```
```   913   moreover have "finite (range (bacc r))" by auto
```
```   914   ultimately show ?thesis
```
```   915    by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
```
```   916      (auto intro: finite_mono_remains_stable_implies_strict_prefix  simp add: enum_UNIV)
```
```   917 qed
```
```   918
```
```   919 lemma acc_subseteq_bacc:
```
```   920   assumes "finite r"
```
```   921   shows "acc r \<subseteq> (UN n. bacc r n)"
```
```   922 proof
```
```   923   fix x
```
```   924   assume "x : acc r"
```
```   925   then have "\<exists> n. x : bacc r n"
```
```   926   proof (induct x arbitrary: rule: acc.induct)
```
```   927     case (accI x)
```
```   928     then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
```
```   929     from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
```
```   930     obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
```
```   931     proof
```
```   932       fix y assume y: "(y, x) : r"
```
```   933       with n have "y : bacc r (n y)" by auto
```
```   934       moreover have "n y <= Max ((%(y, x). n y) ` r)"
```
```   935         using y `finite r` by (auto intro!: Max_ge)
```
```   936       note bacc_mono[OF this, of r]
```
```   937       ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
```
```   938     qed
```
```   939     then show ?case
```
```   940       by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
```
```   941   qed
```
```   942   then show "x : (UN n. bacc r n)" by auto
```
```   943 qed
```
```   944
```
```   945 lemma acc_bacc_eq: "acc ((set xs) :: (('a :: enum) * 'a) set) = bacc (set xs) (card (UNIV :: 'a set))"
```
```   946 by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound finite_set order_eq_iff)
```
```   947
```
```   948 definition
```
```   949   [code del]: "card_UNIV = card UNIV"
```
```   950
```
```   951 lemma [code]:
```
```   952   "card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
```
```   953 unfolding card_UNIV_def enum_UNIV ..
```
```   954
```
```   955 declare acc_bacc_eq[folded card_UNIV_def, code]
```
```   956
```
```   957 lemma [code_unfold]: "accp r = (%x. x : acc {(x, y). r x y})"
```
```   958 unfolding acc_def by simp
```
```   959
```
```   960 subsection {* Closing up *}
```
```   961
```
```   962 hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
```
```   963 hide_const (open) enum enum_all enum_ex n_lists all_n_lists ex_n_lists product ntrancl
```
```   964
```
```   965 end
```