src/HOL/Enum.thy
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 46361 87d5d36a9005 child 47221 7205eb4a0a05 child 47230 6584098d5378 permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
```     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 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)"
```
```   156       by (auto dest: fun_cong)
```
```   157     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
```
```   158       by simp
```
```   159   qed
```
```   160   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
```
```   161 qed
```
```   162
```
```   163 definition
```
```   164   all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
```
```   165 where
```
```   166   "all_n_lists P n = (\<forall>xs \<in> set (n_lists n enum). P xs)"
```
```   167
```
```   168 lemma [code]:
```
```   169   "all_n_lists P n = (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
```
```   170 unfolding all_n_lists_def enum_all
```
```   171 by (cases n) (auto simp add: enum_UNIV)
```
```   172
```
```   173 definition
```
```   174   ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
```
```   175 where
```
```   176   "ex_n_lists P n = (\<exists>xs \<in> set (n_lists n enum). P xs)"
```
```   177
```
```   178 lemma [code]:
```
```   179   "ex_n_lists P n = (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
```
```   180 unfolding ex_n_lists_def enum_ex
```
```   181 by (cases n) (auto simp add: enum_UNIV)
```
```   182
```
```   183
```
```   184 instantiation "fun" :: (enum, enum) enum
```
```   185 begin
```
```   186
```
```   187 definition
```
```   188   "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
```
```   189
```
```   190 definition
```
```   191   "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
```
```   192
```
```   193 definition
```
```   194   "enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
```
```   195
```
```   196
```
```   197 instance proof
```
```   198   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   199   proof (rule UNIV_eq_I)
```
```   200     fix f :: "'a \<Rightarrow> 'b"
```
```   201     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   202       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   203     then show "f \<in> set enum"
```
```   204       by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
```
```   205   qed
```
```   206 next
```
```   207   from map_of_zip_enum_inject
```
```   208   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   209     by (auto intro!: inj_onI simp add: enum_fun_def
```
```   210       distinct_map distinct_n_lists enum_distinct set_n_lists enum_all)
```
```   211 next
```
```   212   fix P
```
```   213   show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   214   proof
```
```   215     assume "enum_all P"
```
```   216     show "\<forall>x. P x"
```
```   217     proof
```
```   218       fix f :: "'a \<Rightarrow> 'b"
```
```   219       have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   220         by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   221       from `enum_all P` have "P (the \<circ> map_of (zip enum (map f enum)))"
```
```   222         unfolding enum_all_fun_def all_n_lists_def
```
```   223         apply (simp add: set_n_lists)
```
```   224         apply (erule_tac x="map f enum" in allE)
```
```   225         apply (auto intro!: in_enum)
```
```   226         done
```
```   227       from this f show "P f" by auto
```
```   228     qed
```
```   229   next
```
```   230     assume "\<forall>x. P x"
```
```   231     from this show "enum_all P"
```
```   232       unfolding enum_all_fun_def all_n_lists_def by auto
```
```   233   qed
```
```   234 next
```
```   235   fix P
```
```   236   show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   237   proof
```
```   238     assume "enum_ex P"
```
```   239     from this show "\<exists>x. P x"
```
```   240       unfolding enum_ex_fun_def ex_n_lists_def by auto
```
```   241   next
```
```   242     assume "\<exists>x. P x"
```
```   243     from this obtain f where "P f" ..
```
```   244     have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   245       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   246     from `P f` this have "P (the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum)))"
```
```   247       by auto
```
```   248     from  this show "enum_ex P"
```
```   249       unfolding enum_ex_fun_def ex_n_lists_def
```
```   250       apply (auto simp add: set_n_lists)
```
```   251       apply (rule_tac x="map f enum" in exI)
```
```   252       apply (auto intro!: in_enum)
```
```   253       done
```
```   254   qed
```
```   255 qed
```
```   256
```
```   257 end
```
```   258
```
```   259 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
```
```   260   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (n_lists (length enum_a) enum))"
```
```   261   by (simp add: enum_fun_def Let_def)
```
```   262
```
```   263 lemma enum_all_fun_code [code]:
```
```   264   "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
```
```   265    in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
```
```   266   by (simp add: enum_all_fun_def Let_def)
```
```   267
```
```   268 lemma enum_ex_fun_code [code]:
```
```   269   "enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
```
```   270    in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
```
```   271   by (simp add: enum_ex_fun_def Let_def)
```
```   272
```
```   273 instantiation unit :: enum
```
```   274 begin
```
```   275
```
```   276 definition
```
```   277   "enum = [()]"
```
```   278
```
```   279 definition
```
```   280   "enum_all P = P ()"
```
```   281
```
```   282 definition
```
```   283   "enum_ex P = P ()"
```
```   284
```
```   285 instance proof
```
```   286 qed (auto simp add: enum_unit_def UNIV_unit enum_all_unit_def enum_ex_unit_def intro: unit.exhaust)
```
```   287
```
```   288 end
```
```   289
```
```   290 instantiation bool :: enum
```
```   291 begin
```
```   292
```
```   293 definition
```
```   294   "enum = [False, True]"
```
```   295
```
```   296 definition
```
```   297   "enum_all P = (P False \<and> P True)"
```
```   298
```
```   299 definition
```
```   300   "enum_ex P = (P False \<or> P True)"
```
```   301
```
```   302 instance proof
```
```   303   fix P
```
```   304   show "enum_all (P :: bool \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   305     unfolding enum_all_bool_def by (auto, case_tac x) auto
```
```   306 next
```
```   307   fix P
```
```   308   show "enum_ex (P :: bool \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   309     unfolding enum_ex_bool_def by (auto, case_tac x) auto
```
```   310 qed (auto simp add: enum_bool_def UNIV_bool)
```
```   311
```
```   312 end
```
```   313
```
```   314 primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
```
```   315   "product [] _ = []"
```
```   316   | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
```
```   317
```
```   318 lemma product_list_set:
```
```   319   "set (product xs ys) = set xs \<times> set ys"
```
```   320   by (induct xs) auto
```
```   321
```
```   322 lemma distinct_product:
```
```   323   assumes "distinct xs" and "distinct ys"
```
```   324   shows "distinct (product xs ys)"
```
```   325   using assms by (induct xs)
```
```   326     (auto intro: inj_onI simp add: product_list_set distinct_map)
```
```   327
```
```   328 instantiation prod :: (enum, enum) enum
```
```   329 begin
```
```   330
```
```   331 definition
```
```   332   "enum = product enum enum"
```
```   333
```
```   334 definition
```
```   335   "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
```
```   336
```
```   337 definition
```
```   338   "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
```
```   339
```
```   340
```
```   341 instance by default
```
```   342   (simp_all add: enum_prod_def product_list_set distinct_product
```
```   343     enum_UNIV enum_distinct enum_all_prod_def enum_all enum_ex_prod_def enum_ex)
```
```   344
```
```   345 end
```
```   346
```
```   347 instantiation sum :: (enum, enum) enum
```
```   348 begin
```
```   349
```
```   350 definition
```
```   351   "enum = map Inl enum @ map Inr enum"
```
```   352
```
```   353 definition
```
```   354   "enum_all P = (enum_all (%x. P (Inl x)) \<and> enum_all (%x. P (Inr x)))"
```
```   355
```
```   356 definition
```
```   357   "enum_ex P = (enum_ex (%x. P (Inl x)) \<or> enum_ex (%x. P (Inr x)))"
```
```   358
```
```   359 instance proof
```
```   360   fix P
```
```   361   show "enum_all (P :: ('a + 'b) \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   362     unfolding enum_all_sum_def enum_all
```
```   363     by (auto, case_tac x) auto
```
```   364 next
```
```   365   fix P
```
```   366   show "enum_ex (P :: ('a + 'b) \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   367     unfolding enum_ex_sum_def enum_ex
```
```   368     by (auto, case_tac x) auto
```
```   369 qed (auto simp add: enum_UNIV enum_sum_def, case_tac x, auto intro: inj_onI simp add: distinct_map enum_distinct)
```
```   370
```
```   371 end
```
```   372
```
```   373 instantiation nibble :: enum
```
```   374 begin
```
```   375
```
```   376 definition
```
```   377   "enum = [Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
```
```   378     Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF]"
```
```   379
```
```   380 definition
```
```   381   "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
```
```   382      \<and> P Nibble8 \<and> P Nibble9 \<and> P NibbleA \<and> P NibbleB \<and> P NibbleC \<and> P NibbleD \<and> P NibbleE \<and> P NibbleF)"
```
```   383
```
```   384 definition
```
```   385   "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
```
```   386      \<or> P Nibble8 \<or> P Nibble9 \<or> P NibbleA \<or> P NibbleB \<or> P NibbleC \<or> P NibbleD \<or> P NibbleE \<or> P NibbleF)"
```
```   387
```
```   388 instance proof
```
```   389   fix P
```
```   390   show "enum_all (P :: nibble \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   391     unfolding enum_all_nibble_def
```
```   392     by (auto, case_tac x) auto
```
```   393 next
```
```   394   fix P
```
```   395   show "enum_ex (P :: nibble \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   396     unfolding enum_ex_nibble_def
```
```   397     by (auto, case_tac x) auto
```
```   398 qed (simp_all add: enum_nibble_def UNIV_nibble)
```
```   399
```
```   400 end
```
```   401
```
```   402 instantiation char :: enum
```
```   403 begin
```
```   404
```
```   405 definition
```
```   406   "enum = map (split Char) (product enum enum)"
```
```   407
```
```   408 lemma enum_chars [code]:
```
```   409   "enum = chars"
```
```   410   unfolding enum_char_def chars_def enum_nibble_def by simp
```
```   411
```
```   412 definition
```
```   413   "enum_all P = list_all P chars"
```
```   414
```
```   415 definition
```
```   416   "enum_ex P = list_ex P chars"
```
```   417
```
```   418 lemma set_enum_char: "set (enum :: char list) = UNIV"
```
```   419     by (auto intro: char.exhaust simp add: enum_char_def product_list_set enum_UNIV full_SetCompr_eq [symmetric])
```
```   420
```
```   421 instance proof
```
```   422   fix P
```
```   423   show "enum_all (P :: char \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   424     unfolding enum_all_char_def enum_chars[symmetric]
```
```   425     by (auto simp add: list_all_iff set_enum_char)
```
```   426 next
```
```   427   fix P
```
```   428   show "enum_ex (P :: char \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   429     unfolding enum_ex_char_def enum_chars[symmetric]
```
```   430     by (auto simp add: list_ex_iff set_enum_char)
```
```   431 next
```
```   432   show "distinct (enum :: char list)"
```
```   433     by (auto intro: inj_onI simp add: enum_char_def product_list_set distinct_map distinct_product enum_distinct)
```
```   434 qed (auto simp add: set_enum_char)
```
```   435
```
```   436 end
```
```   437
```
```   438 instantiation option :: (enum) enum
```
```   439 begin
```
```   440
```
```   441 definition
```
```   442   "enum = None # map Some enum"
```
```   443
```
```   444 definition
```
```   445   "enum_all P = (P None \<and> enum_all (%x. P (Some x)))"
```
```   446
```
```   447 definition
```
```   448   "enum_ex P = (P None \<or> enum_ex (%x. P (Some x)))"
```
```   449
```
```   450 instance proof
```
```   451   fix P
```
```   452   show "enum_all (P :: 'a option \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   453     unfolding enum_all_option_def enum_all
```
```   454     by (auto, case_tac x) auto
```
```   455 next
```
```   456   fix P
```
```   457   show "enum_ex (P :: 'a option \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   458     unfolding enum_ex_option_def enum_ex
```
```   459     by (auto, case_tac x) auto
```
```   460 qed (auto simp add: enum_UNIV enum_option_def, rule option.exhaust, auto intro: simp add: distinct_map enum_distinct)
```
```   461 end
```
```   462
```
```   463 primrec sublists :: "'a list \<Rightarrow> 'a list list" where
```
```   464   "sublists [] = [[]]"
```
```   465   | "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
```
```   466
```
```   467 lemma length_sublists:
```
```   468   "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
```
```   469   by (induct xs) (simp_all add: Let_def)
```
```   470
```
```   471 lemma sublists_powset:
```
```   472   "set ` set (sublists xs) = Pow (set xs)"
```
```   473 proof -
```
```   474   have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
```
```   475     by (auto simp add: image_def)
```
```   476   have "set (map set (sublists xs)) = Pow (set xs)"
```
```   477     by (induct xs)
```
```   478       (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
```
```   479   then show ?thesis by simp
```
```   480 qed
```
```   481
```
```   482 lemma distinct_set_sublists:
```
```   483   assumes "distinct xs"
```
```   484   shows "distinct (map set (sublists xs))"
```
```   485 proof (rule card_distinct)
```
```   486   have "finite (set xs)" by rule
```
```   487   then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
```
```   488   with assms distinct_card [of xs]
```
```   489     have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
```
```   490   then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
```
```   491     by (simp add: sublists_powset length_sublists)
```
```   492 qed
```
```   493
```
```   494 instantiation set :: (enum) enum
```
```   495 begin
```
```   496
```
```   497 definition
```
```   498   "enum = map set (sublists enum)"
```
```   499
```
```   500 definition
```
```   501   "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
```
```   502
```
```   503 definition
```
```   504   "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
```
```   505
```
```   506 instance proof
```
```   507 qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
```
```   508   enum_distinct enum_UNIV)
```
```   509
```
```   510 end
```
```   511
```
```   512
```
```   513 subsection {* Small finite types *}
```
```   514
```
```   515 text {* We define small finite types for the use in Quickcheck *}
```
```   516
```
```   517 datatype finite_1 = a\<^isub>1
```
```   518
```
```   519 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   520
```
```   521 instantiation finite_1 :: enum
```
```   522 begin
```
```   523
```
```   524 definition
```
```   525   "enum = [a\<^isub>1]"
```
```   526
```
```   527 definition
```
```   528   "enum_all P = P a\<^isub>1"
```
```   529
```
```   530 definition
```
```   531   "enum_ex P = P a\<^isub>1"
```
```   532
```
```   533 instance proof
```
```   534   fix P
```
```   535   show "enum_all (P :: finite_1 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   536     unfolding enum_all_finite_1_def
```
```   537     by (auto, case_tac x) auto
```
```   538 next
```
```   539   fix P
```
```   540   show "enum_ex (P :: finite_1 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   541     unfolding enum_ex_finite_1_def
```
```   542     by (auto, case_tac x) auto
```
```   543 qed (auto simp add: enum_finite_1_def intro: finite_1.exhaust)
```
```   544
```
```   545 end
```
```   546
```
```   547 instantiation finite_1 :: linorder
```
```   548 begin
```
```   549
```
```   550 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   551 where
```
```   552   "less_eq_finite_1 x y = True"
```
```   553
```
```   554 definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   555 where
```
```   556   "less_finite_1 x y = False"
```
```   557
```
```   558 instance
```
```   559 apply (intro_classes)
```
```   560 apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
```
```   561 apply (metis finite_1.exhaust)
```
```   562 done
```
```   563
```
```   564 end
```
```   565
```
```   566 hide_const (open) a\<^isub>1
```
```   567
```
```   568 datatype finite_2 = a\<^isub>1 | a\<^isub>2
```
```   569
```
```   570 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   571 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   572
```
```   573 instantiation finite_2 :: enum
```
```   574 begin
```
```   575
```
```   576 definition
```
```   577   "enum = [a\<^isub>1, a\<^isub>2]"
```
```   578
```
```   579 definition
```
```   580   "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2)"
```
```   581
```
```   582 definition
```
```   583   "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2)"
```
```   584
```
```   585 instance proof
```
```   586   fix P
```
```   587   show "enum_all (P :: finite_2 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   588     unfolding enum_all_finite_2_def
```
```   589     by (auto, case_tac x) auto
```
```   590 next
```
```   591   fix P
```
```   592   show "enum_ex (P :: finite_2 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   593     unfolding enum_ex_finite_2_def
```
```   594     by (auto, case_tac x) auto
```
```   595 qed (auto simp add: enum_finite_2_def intro: finite_2.exhaust)
```
```   596
```
```   597 end
```
```   598
```
```   599 instantiation finite_2 :: linorder
```
```   600 begin
```
```   601
```
```   602 definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   603 where
```
```   604   "less_finite_2 x y = ((x = a\<^isub>1) & (y = a\<^isub>2))"
```
```   605
```
```   606 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   607 where
```
```   608   "less_eq_finite_2 x y = ((x = y) \<or> (x < y))"
```
```   609
```
```   610
```
```   611 instance
```
```   612 apply (intro_classes)
```
```   613 apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
```
```   614 apply (metis finite_2.distinct finite_2.nchotomy)+
```
```   615 done
```
```   616
```
```   617 end
```
```   618
```
```   619 hide_const (open) a\<^isub>1 a\<^isub>2
```
```   620
```
```   621
```
```   622 datatype finite_3 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3
```
```   623
```
```   624 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   625 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   626 notation (output) a\<^isub>3  ("a\<^isub>3")
```
```   627
```
```   628 instantiation finite_3 :: enum
```
```   629 begin
```
```   630
```
```   631 definition
```
```   632   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3]"
```
```   633
```
```   634 definition
```
```   635   "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3)"
```
```   636
```
```   637 definition
```
```   638   "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3)"
```
```   639
```
```   640 instance proof
```
```   641   fix P
```
```   642   show "enum_all (P :: finite_3 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   643     unfolding enum_all_finite_3_def
```
```   644     by (auto, case_tac x) auto
```
```   645 next
```
```   646   fix P
```
```   647   show "enum_ex (P :: finite_3 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   648     unfolding enum_ex_finite_3_def
```
```   649     by (auto, case_tac x) auto
```
```   650 qed (auto simp add: enum_finite_3_def intro: finite_3.exhaust)
```
```   651
```
```   652 end
```
```   653
```
```   654 instantiation finite_3 :: linorder
```
```   655 begin
```
```   656
```
```   657 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   658 where
```
```   659   "less_finite_3 x y = (case x of a\<^isub>1 => (y \<noteq> a\<^isub>1)
```
```   660      | a\<^isub>2 => (y = a\<^isub>3)| a\<^isub>3 => False)"
```
```   661
```
```   662 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   663 where
```
```   664   "less_eq_finite_3 x y = ((x = y) \<or> (x < y))"
```
```   665
```
```   666
```
```   667 instance proof (intro_classes)
```
```   668 qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
```
```   669
```
```   670 end
```
```   671
```
```   672 hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3
```
```   673
```
```   674
```
```   675 datatype finite_4 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4
```
```   676
```
```   677 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   678 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   679 notation (output) a\<^isub>3  ("a\<^isub>3")
```
```   680 notation (output) a\<^isub>4  ("a\<^isub>4")
```
```   681
```
```   682 instantiation finite_4 :: enum
```
```   683 begin
```
```   684
```
```   685 definition
```
```   686   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4]"
```
```   687
```
```   688 definition
```
```   689   "enum_all P = (P a\<^isub>1 \<and> P a\<^isub>2 \<and> P a\<^isub>3 \<and> P a\<^isub>4)"
```
```   690
```
```   691 definition
```
```   692   "enum_ex P = (P a\<^isub>1 \<or> P a\<^isub>2 \<or> P a\<^isub>3 \<or> P a\<^isub>4)"
```
```   693
```
```   694 instance proof
```
```   695   fix P
```
```   696   show "enum_all (P :: finite_4 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   697     unfolding enum_all_finite_4_def
```
```   698     by (auto, case_tac x) auto
```
```   699 next
```
```   700   fix P
```
```   701   show "enum_ex (P :: finite_4 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   702     unfolding enum_ex_finite_4_def
```
```   703     by (auto, case_tac x) auto
```
```   704 qed (auto simp add: enum_finite_4_def intro: finite_4.exhaust)
```
```   705
```
```   706 end
```
```   707
```
```   708 hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4
```
```   709
```
```   710
```
```   711 datatype finite_5 = a\<^isub>1 | a\<^isub>2 | a\<^isub>3 | a\<^isub>4 | a\<^isub>5
```
```   712
```
```   713 notation (output) a\<^isub>1  ("a\<^isub>1")
```
```   714 notation (output) a\<^isub>2  ("a\<^isub>2")
```
```   715 notation (output) a\<^isub>3  ("a\<^isub>3")
```
```   716 notation (output) a\<^isub>4  ("a\<^isub>4")
```
```   717 notation (output) a\<^isub>5  ("a\<^isub>5")
```
```   718
```
```   719 instantiation finite_5 :: enum
```
```   720 begin
```
```   721
```
```   722 definition
```
```   723   "enum = [a\<^isub>1, a\<^isub>2, a\<^isub>3, a\<^isub>4, a\<^isub>5]"
```
```   724
```
```   725 definition
```
```   726   "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)"
```
```   727
```
```   728 definition
```
```   729   "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)"
```
```   730
```
```   731 instance proof
```
```   732   fix P
```
```   733   show "enum_all (P :: finite_5 \<Rightarrow> bool) = (\<forall>x. P x)"
```
```   734     unfolding enum_all_finite_5_def
```
```   735     by (auto, case_tac x) auto
```
```   736 next
```
```   737   fix P
```
```   738   show "enum_ex (P :: finite_5 \<Rightarrow> bool) = (\<exists>x. P x)"
```
```   739     unfolding enum_ex_finite_5_def
```
```   740     by (auto, case_tac x) auto
```
```   741 qed (auto simp add: enum_finite_5_def intro: finite_5.exhaust)
```
```   742
```
```   743 end
```
```   744
```
```   745 hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5
```
```   746
```
```   747 subsection {* An executable THE operator on finite types *}
```
```   748
```
```   749 definition
```
```   750   [code del]: "enum_the P = The P"
```
```   751
```
```   752 lemma [code]:
```
```   753   "The P = (case filter P enum of [x] => x | _ => enum_the P)"
```
```   754 proof -
```
```   755   {
```
```   756     fix a
```
```   757     assume filter_enum: "filter P enum = [a]"
```
```   758     have "The P = a"
```
```   759     proof (rule the_equality)
```
```   760       fix x
```
```   761       assume "P x"
```
```   762       show "x = a"
```
```   763       proof (rule ccontr)
```
```   764         assume "x \<noteq> a"
```
```   765         from filter_enum obtain us vs
```
```   766           where enum_eq: "enum = us @ [a] @ vs"
```
```   767           and "\<forall> x \<in> set us. \<not> P x"
```
```   768           and "\<forall> x \<in> set vs. \<not> P x"
```
```   769           and "P a"
```
```   770           by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
```
```   771         with `P x` in_enum[of x, unfolded enum_eq] `x \<noteq> a` show "False" by auto
```
```   772       qed
```
```   773     next
```
```   774       from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
```
```   775     qed
```
```   776   }
```
```   777   from this show ?thesis
```
```   778     unfolding enum_the_def by (auto split: list.split)
```
```   779 qed
```
```   780
```
```   781 code_abort enum_the
```
```   782 code_const enum_the (Eval "(fn p => raise Match)")
```
```   783
```
```   784 subsection {* Further operations on finite types *}
```
```   785
```
```   786 lemma [code]:
```
```   787   "Collect P = set (filter P enum)"
```
```   788 by (auto simp add: enum_UNIV)
```
```   789
```
```   790 lemma tranclp_unfold [code, no_atp]:
```
```   791   "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
```
```   792 by (simp add: trancl_def)
```
```   793
```
```   794 lemma rtranclp_rtrancl_eq[code, no_atp]:
```
```   795   "rtranclp r x y = ((x, y) : rtrancl {(x, y). r x y})"
```
```   796 unfolding rtrancl_def by auto
```
```   797
```
```   798 lemma max_ext_eq[code]:
```
```   799   "max_ext R = {(X, Y). finite X & finite Y & Y ~={} & (ALL x. x : X --> (EX xa : Y. (x, xa) : R))}"
```
```   800 by (auto simp add: max_ext.simps)
```
```   801
```
```   802 lemma max_extp_eq[code]:
```
```   803   "max_extp r x y = ((x, y) : max_ext {(x, y). r x y})"
```
```   804 unfolding max_ext_def by auto
```
```   805
```
```   806 lemma mlex_eq[code]:
```
```   807   "f <*mlex*> R = {(x, y). f x < f y \<or> (f x <= f y \<and> (x, y) : R)}"
```
```   808 unfolding mlex_prod_def by auto
```
```   809
```
```   810 subsection {* Executable accessible part *}
```
```   811 (* FIXME: should be moved somewhere else !? *)
```
```   812
```
```   813 subsubsection {* Finite monotone eventually stable sequences *}
```
```   814
```
```   815 lemma finite_mono_remains_stable_implies_strict_prefix:
```
```   816   fixes f :: "nat \<Rightarrow> 'a::order"
```
```   817   assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
```
```   818   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)"
```
```   819   using assms
```
```   820 proof -
```
```   821   have "\<exists>n. f n = f (Suc n)"
```
```   822   proof (rule ccontr)
```
```   823     assume "\<not> ?thesis"
```
```   824     then have "\<And>n. f n \<noteq> f (Suc n)" by auto
```
```   825     then have "\<And>n. f n < f (Suc n)"
```
```   826       using  `mono f` by (auto simp: le_less mono_iff_le_Suc)
```
```   827     with lift_Suc_mono_less_iff[of f]
```
```   828     have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
```
```   829     then have "inj f"
```
```   830       by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
```
```   831     with `finite (range f)` have "finite (UNIV::nat set)"
```
```   832       by (rule finite_imageD)
```
```   833     then show False by simp
```
```   834   qed
```
```   835   then guess n .. note n = this
```
```   836   def N \<equiv> "LEAST n. f n = f (Suc n)"
```
```   837   have N: "f N = f (Suc N)"
```
```   838     unfolding N_def using n by (rule LeastI)
```
```   839   show ?thesis
```
```   840   proof (intro exI[of _ N] conjI allI impI)
```
```   841     fix n assume "N \<le> n"
```
```   842     then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
```
```   843     proof (induct rule: dec_induct)
```
```   844       case (step n) then show ?case
```
```   845         using eq[rule_format, of "n - 1"] N
```
```   846         by (cases n) (auto simp add: le_Suc_eq)
```
```   847     qed simp
```
```   848     from this[of n] `N \<le> n` show "f N = f n" by auto
```
```   849   next
```
```   850     fix n m :: nat assume "m < n" "n \<le> N"
```
```   851     then show "f m < f n"
```
```   852     proof (induct rule: less_Suc_induct[consumes 1])
```
```   853       case (1 i)
```
```   854       then have "i < N" by simp
```
```   855       then have "f i \<noteq> f (Suc i)"
```
```   856         unfolding N_def by (rule not_less_Least)
```
```   857       with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
```
```   858     qed auto
```
```   859   qed
```
```   860 qed
```
```   861
```
```   862 lemma finite_mono_strict_prefix_implies_finite_fixpoint:
```
```   863   fixes f :: "nat \<Rightarrow> 'a set"
```
```   864   assumes S: "\<And>i. f i \<subseteq> S" "finite S"
```
```   865     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)"
```
```   866   shows "f (card S) = (\<Union>n. f n)"
```
```   867 proof -
```
```   868   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
```
```   869
```
```   870   { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
```
```   871     proof (induct i)
```
```   872       case 0 then show ?case by simp
```
```   873     next
```
```   874       case (Suc i)
```
```   875       with inj[rule_format, of "Suc i" i]
```
```   876       have "(f i) \<subset> (f (Suc i))" by auto
```
```   877       moreover have "finite (f (Suc i))" using S by (rule finite_subset)
```
```   878       ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
```
```   879       with Suc show ?case using inj by auto
```
```   880     qed
```
```   881   }
```
```   882   then have "N \<le> card (f N)" by simp
```
```   883   also have "\<dots> \<le> card S" using S by (intro card_mono)
```
```   884   finally have "f (card S) = f N" using eq by auto
```
```   885   then show ?thesis using eq inj[rule_format, of N]
```
```   886     apply auto
```
```   887     apply (case_tac "n < N")
```
```   888     apply (auto simp: not_less)
```
```   889     done
```
```   890 qed
```
```   891
```
```   892 subsubsection {* Bounded accessible part *}
```
```   893
```
```   894 fun bacc :: "('a * 'a) set => nat => 'a set"
```
```   895 where
```
```   896   "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
```
```   897 | "bacc r (Suc n) = (bacc r n Un {x. \<forall> y. (y, x) : r --> y : bacc r n})"
```
```   898
```
```   899 lemma bacc_subseteq_acc:
```
```   900   "bacc r n \<subseteq> acc r"
```
```   901 by (induct n) (auto intro: acc.intros)
```
```   902
```
```   903 lemma bacc_mono:
```
```   904   "n <= m ==> bacc r n \<subseteq> bacc r m"
```
```   905 by (induct rule: dec_induct) auto
```
```   906
```
```   907 lemma bacc_upper_bound:
```
```   908   "bacc (r :: ('a * 'a) set)  (card (UNIV :: ('a :: enum) set)) = (UN n. bacc r n)"
```
```   909 proof -
```
```   910   have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
```
```   911   moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
```
```   912   moreover have "finite (range (bacc r))" by auto
```
```   913   ultimately show ?thesis
```
```   914    by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
```
```   915      (auto intro: finite_mono_remains_stable_implies_strict_prefix  simp add: enum_UNIV)
```
```   916 qed
```
```   917
```
```   918 lemma acc_subseteq_bacc:
```
```   919   assumes "finite r"
```
```   920   shows "acc r \<subseteq> (UN n. bacc r n)"
```
```   921 proof
```
```   922   fix x
```
```   923   assume "x : acc r"
```
```   924   from this have "\<exists> n. x : bacc r n"
```
```   925   proof (induct x arbitrary: n rule: acc.induct)
```
```   926     case (accI x)
```
```   927     from accI have "\<forall> y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
```
```   928     from choice[OF this] guess n .. note n = this
```
```   929     have "\<exists> n. \<forall>y. (y, x) : r --> y : bacc r n"
```
```   930     proof (safe intro!: exI[of _ "Max ((%(y,x). n y)`r)"])
```
```   931       fix y assume y: "(y, x) : r"
```
```   932       with n have "y : bacc r (n y)" by auto
```
```   933       moreover have "n y <= Max ((%(y, x). n y) ` r)"
```
```   934         using y `finite r` by (auto intro!: Max_ge)
```
```   935       note bacc_mono[OF this, of r]
```
```   936       ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
```
```   937     qed
```
```   938     from this guess n ..
```
```   939     from this show ?case
```
```   940       by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
```
```   941   qed
```
```   942   thus "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
```