src/HOL/Enum.thy
 author blanchet Fri Oct 18 10:43:21 2013 +0200 (2013-10-18) changeset 54148 c8cc5ab4a863 parent 53015 a1119cf551e8 child 54295 45a5523d4a63 permissions -rw-r--r--
killed more "no_atp"s
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* Finite types as explicit enumerations *}
```
```     4
```
```     5 theory Enum
```
```     6 imports Map
```
```     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_UNIV: "enum_all P \<longleftrightarrow> Ball UNIV P"
```
```    18   assumes enum_ex_UNIV: "enum_ex P \<longleftrightarrow> Bex UNIV P"
```
```    19    -- {* tailored towards simple instantiation *}
```
```    20 begin
```
```    21
```
```    22 subclass finite proof
```
```    23 qed (simp add: UNIV_enum)
```
```    24
```
```    25 lemma enum_UNIV:
```
```    26   "set enum = UNIV"
```
```    27   by (simp only: UNIV_enum)
```
```    28
```
```    29 lemma in_enum: "x \<in> set enum"
```
```    30   by (simp add: enum_UNIV)
```
```    31
```
```    32 lemma enum_eq_I:
```
```    33   assumes "\<And>x. x \<in> set xs"
```
```    34   shows "set enum = set xs"
```
```    35 proof -
```
```    36   from assms UNIV_eq_I have "UNIV = set xs" by auto
```
```    37   with enum_UNIV show ?thesis by simp
```
```    38 qed
```
```    39
```
```    40 lemma card_UNIV_length_enum:
```
```    41   "card (UNIV :: 'a set) = length enum"
```
```    42   by (simp add: UNIV_enum distinct_card enum_distinct)
```
```    43
```
```    44 lemma enum_all [simp]:
```
```    45   "enum_all = HOL.All"
```
```    46   by (simp add: fun_eq_iff enum_all_UNIV)
```
```    47
```
```    48 lemma enum_ex [simp]:
```
```    49   "enum_ex = HOL.Ex"
```
```    50   by (simp add: fun_eq_iff enum_ex_UNIV)
```
```    51
```
```    52 end
```
```    53
```
```    54
```
```    55 subsection {* Implementations using @{class enum} *}
```
```    56
```
```    57 subsubsection {* Unbounded operations and quantifiers *}
```
```    58
```
```    59 lemma Collect_code [code]:
```
```    60   "Collect P = set (filter P enum)"
```
```    61   by (simp add: enum_UNIV)
```
```    62
```
```    63 lemma vimage_code [code]:
```
```    64   "f -` B = set (filter (%x. f x : B) enum_class.enum)"
```
```    65   unfolding vimage_def Collect_code ..
```
```    66
```
```    67 definition card_UNIV :: "'a itself \<Rightarrow> nat"
```
```    68 where
```
```    69   [code del]: "card_UNIV TYPE('a) = card (UNIV :: 'a set)"
```
```    70
```
```    71 lemma [code]:
```
```    72   "card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
```
```    73   by (simp only: card_UNIV_def enum_UNIV)
```
```    74
```
```    75 lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"
```
```    76   by simp
```
```    77
```
```    78 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"
```
```    79   by simp
```
```    80
```
```    81 lemma exists1_code [code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
```
```    82   by (auto simp add: list_ex1_iff enum_UNIV)
```
```    83
```
```    84
```
```    85 subsubsection {* An executable choice operator *}
```
```    86
```
```    87 definition
```
```    88   [code del]: "enum_the = The"
```
```    89
```
```    90 lemma [code]:
```
```    91   "The P = (case filter P enum of [x] => x | _ => enum_the P)"
```
```    92 proof -
```
```    93   {
```
```    94     fix a
```
```    95     assume filter_enum: "filter P enum = [a]"
```
```    96     have "The P = a"
```
```    97     proof (rule the_equality)
```
```    98       fix x
```
```    99       assume "P x"
```
```   100       show "x = a"
```
```   101       proof (rule ccontr)
```
```   102         assume "x \<noteq> a"
```
```   103         from filter_enum obtain us vs
```
```   104           where enum_eq: "enum = us @ [a] @ vs"
```
```   105           and "\<forall> x \<in> set us. \<not> P x"
```
```   106           and "\<forall> x \<in> set vs. \<not> P x"
```
```   107           and "P a"
```
```   108           by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
```
```   109         with `P x` in_enum[of x, unfolded enum_eq] `x \<noteq> a` show "False" by auto
```
```   110       qed
```
```   111     next
```
```   112       from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
```
```   113     qed
```
```   114   }
```
```   115   from this show ?thesis
```
```   116     unfolding enum_the_def by (auto split: list.split)
```
```   117 qed
```
```   118
```
```   119 code_abort enum_the
```
```   120
```
```   121 code_printing
```
```   122   constant enum_the \<rightharpoonup> (Eval) "(fn '_ => raise Match)"
```
```   123
```
```   124
```
```   125 subsubsection {* Equality and order on functions *}
```
```   126
```
```   127 instantiation "fun" :: (enum, equal) equal
```
```   128 begin
```
```   129
```
```   130 definition
```
```   131   "HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
```
```   132
```
```   133 instance proof
```
```   134 qed (simp_all add: equal_fun_def fun_eq_iff enum_UNIV)
```
```   135
```
```   136 end
```
```   137
```
```   138 lemma [code]:
```
```   139   "HOL.equal f g \<longleftrightarrow> enum_all (%x. f x = g x)"
```
```   140   by (auto simp add: equal fun_eq_iff)
```
```   141
```
```   142 lemma [code nbe]:
```
```   143   "HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"
```
```   144   by (fact equal_refl)
```
```   145
```
```   146 lemma order_fun [code]:
```
```   147   fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
```
```   148   shows "f \<le> g \<longleftrightarrow> enum_all (\<lambda>x. f x \<le> g x)"
```
```   149     and "f < g \<longleftrightarrow> f \<le> g \<and> enum_ex (\<lambda>x. f x \<noteq> g x)"
```
```   150   by (simp_all add: fun_eq_iff le_fun_def order_less_le)
```
```   151
```
```   152
```
```   153 subsubsection {* Operations on relations *}
```
```   154
```
```   155 lemma [code]:
```
```   156   "Id = image (\<lambda>x. (x, x)) (set Enum.enum)"
```
```   157   by (auto intro: imageI in_enum)
```
```   158
```
```   159 lemma tranclp_unfold [code]:
```
```   160   "tranclp r a b \<longleftrightarrow> (a, b) \<in> trancl {(x, y). r x y}"
```
```   161   by (simp add: trancl_def)
```
```   162
```
```   163 lemma rtranclp_rtrancl_eq [code]:
```
```   164   "rtranclp r x y \<longleftrightarrow> (x, y) \<in> rtrancl {(x, y). r x y}"
```
```   165   by (simp add: rtrancl_def)
```
```   166
```
```   167 lemma max_ext_eq [code]:
```
```   168   "max_ext R = {(X, Y). finite X \<and> finite Y \<and> Y \<noteq> {} \<and> (\<forall>x. x \<in> X \<longrightarrow> (\<exists>xa \<in> Y. (x, xa) \<in> R))}"
```
```   169   by (auto simp add: max_ext.simps)
```
```   170
```
```   171 lemma max_extp_eq [code]:
```
```   172   "max_extp r x y \<longleftrightarrow> (x, y) \<in> max_ext {(x, y). r x y}"
```
```   173   by (simp add: max_ext_def)
```
```   174
```
```   175 lemma mlex_eq [code]:
```
```   176   "f <*mlex*> R = {(x, y). f x < f y \<or> (f x \<le> f y \<and> (x, y) \<in> R)}"
```
```   177   by (auto simp add: mlex_prod_def)
```
```   178
```
```   179 lemma [code]:
```
```   180   fixes xs :: "('a::finite \<times> 'a) list"
```
```   181   shows "acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"
```
```   182   by (simp add: card_UNIV_def acc_bacc_eq)
```
```   183
```
```   184 lemma [code]:
```
```   185   "accp r = (\<lambda>x. x \<in> acc {(x, y). r x y})"
```
```   186   by (simp add: acc_def)
```
```   187
```
```   188
```
```   189 subsection {* Default instances for @{class enum} *}
```
```   190
```
```   191 lemma map_of_zip_enum_is_Some:
```
```   192   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   193   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
```
```   194 proof -
```
```   195   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
```
```   196     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
```
```   197     by (auto intro!: map_of_zip_is_Some)
```
```   198   then show ?thesis using enum_UNIV by auto
```
```   199 qed
```
```   200
```
```   201 lemma map_of_zip_enum_inject:
```
```   202   fixes xs ys :: "'b\<Colon>enum list"
```
```   203   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   204       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
```
```   205     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)"
```
```   206   shows "xs = ys"
```
```   207 proof -
```
```   208   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
```
```   209   proof
```
```   210     fix x :: 'a
```
```   211     from length map_of_zip_enum_is_Some obtain y1 y2
```
```   212       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
```
```   213         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
```
```   214     moreover from map_of
```
```   215       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)"
```
```   216       by (auto dest: fun_cong)
```
```   217     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
```
```   218       by simp
```
```   219   qed
```
```   220   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
```
```   221 qed
```
```   222
```
```   223 definition all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
```
```   224 where
```
```   225   "all_n_lists P n \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)"
```
```   226
```
```   227 lemma [code]:
```
```   228   "all_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
```
```   229   unfolding all_n_lists_def enum_all
```
```   230   by (cases n) (auto simp add: enum_UNIV)
```
```   231
```
```   232 definition ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
```
```   233 where
```
```   234   "ex_n_lists P n \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)"
```
```   235
```
```   236 lemma [code]:
```
```   237   "ex_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
```
```   238   unfolding ex_n_lists_def enum_ex
```
```   239   by (cases n) (auto simp add: enum_UNIV)
```
```   240
```
```   241 instantiation "fun" :: (enum, enum) enum
```
```   242 begin
```
```   243
```
```   244 definition
```
```   245   "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (List.n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
```
```   246
```
```   247 definition
```
```   248   "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
```
```   249
```
```   250 definition
```
```   251   "enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
```
```   252
```
```   253 instance proof
```
```   254   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   255   proof (rule UNIV_eq_I)
```
```   256     fix f :: "'a \<Rightarrow> 'b"
```
```   257     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   258       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   259     then show "f \<in> set enum"
```
```   260       by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
```
```   261   qed
```
```   262 next
```
```   263   from map_of_zip_enum_inject
```
```   264   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
```
```   265     by (auto intro!: inj_onI simp add: enum_fun_def
```
```   266       distinct_map distinct_n_lists enum_distinct set_n_lists)
```
```   267 next
```
```   268   fix P
```
```   269   show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Ball UNIV P"
```
```   270   proof
```
```   271     assume "enum_all P"
```
```   272     show "Ball UNIV P"
```
```   273     proof
```
```   274       fix f :: "'a \<Rightarrow> 'b"
```
```   275       have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   276         by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   277       from `enum_all P` have "P (the \<circ> map_of (zip enum (map f enum)))"
```
```   278         unfolding enum_all_fun_def all_n_lists_def
```
```   279         apply (simp add: set_n_lists)
```
```   280         apply (erule_tac x="map f enum" in allE)
```
```   281         apply (auto intro!: in_enum)
```
```   282         done
```
```   283       from this f show "P f" by auto
```
```   284     qed
```
```   285   next
```
```   286     assume "Ball UNIV P"
```
```   287     from this show "enum_all P"
```
```   288       unfolding enum_all_fun_def all_n_lists_def by auto
```
```   289   qed
```
```   290 next
```
```   291   fix P
```
```   292   show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Bex UNIV P"
```
```   293   proof
```
```   294     assume "enum_ex P"
```
```   295     from this show "Bex UNIV P"
```
```   296       unfolding enum_ex_fun_def ex_n_lists_def by auto
```
```   297   next
```
```   298     assume "Bex UNIV P"
```
```   299     from this obtain f where "P f" ..
```
```   300     have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
```
```   301       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
```
```   302     from `P f` this have "P (the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum)))"
```
```   303       by auto
```
```   304     from  this show "enum_ex P"
```
```   305       unfolding enum_ex_fun_def ex_n_lists_def
```
```   306       apply (auto simp add: set_n_lists)
```
```   307       apply (rule_tac x="map f enum" in exI)
```
```   308       apply (auto intro!: in_enum)
```
```   309       done
```
```   310   qed
```
```   311 qed
```
```   312
```
```   313 end
```
```   314
```
```   315 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
```
```   316   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"
```
```   317   by (simp add: enum_fun_def Let_def)
```
```   318
```
```   319 lemma enum_all_fun_code [code]:
```
```   320   "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
```
```   321    in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
```
```   322   by (simp only: enum_all_fun_def Let_def)
```
```   323
```
```   324 lemma enum_ex_fun_code [code]:
```
```   325   "enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
```
```   326    in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
```
```   327   by (simp only: enum_ex_fun_def Let_def)
```
```   328
```
```   329 instantiation set :: (enum) enum
```
```   330 begin
```
```   331
```
```   332 definition
```
```   333   "enum = map set (sublists enum)"
```
```   334
```
```   335 definition
```
```   336   "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
```
```   337
```
```   338 definition
```
```   339   "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
```
```   340
```
```   341 instance proof
```
```   342 qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
```
```   343   enum_distinct enum_UNIV)
```
```   344
```
```   345 end
```
```   346
```
```   347 instantiation unit :: enum
```
```   348 begin
```
```   349
```
```   350 definition
```
```   351   "enum = [()]"
```
```   352
```
```   353 definition
```
```   354   "enum_all P = P ()"
```
```   355
```
```   356 definition
```
```   357   "enum_ex P = P ()"
```
```   358
```
```   359 instance proof
```
```   360 qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def)
```
```   361
```
```   362 end
```
```   363
```
```   364 instantiation bool :: enum
```
```   365 begin
```
```   366
```
```   367 definition
```
```   368   "enum = [False, True]"
```
```   369
```
```   370 definition
```
```   371   "enum_all P \<longleftrightarrow> P False \<and> P True"
```
```   372
```
```   373 definition
```
```   374   "enum_ex P \<longleftrightarrow> P False \<or> P True"
```
```   375
```
```   376 instance proof
```
```   377 qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all)
```
```   378
```
```   379 end
```
```   380
```
```   381 instantiation prod :: (enum, enum) enum
```
```   382 begin
```
```   383
```
```   384 definition
```
```   385   "enum = List.product enum enum"
```
```   386
```
```   387 definition
```
```   388   "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
```
```   389
```
```   390 definition
```
```   391   "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
```
```   392
```
```   393
```
```   394 instance by default
```
```   395   (simp_all add: enum_prod_def product_list_set distinct_product
```
```   396     enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def)
```
```   397
```
```   398 end
```
```   399
```
```   400 instantiation sum :: (enum, enum) enum
```
```   401 begin
```
```   402
```
```   403 definition
```
```   404   "enum = map Inl enum @ map Inr enum"
```
```   405
```
```   406 definition
```
```   407   "enum_all P \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))"
```
```   408
```
```   409 definition
```
```   410   "enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x))"
```
```   411
```
```   412 instance proof
```
```   413 qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum,
```
```   414   auto simp add: enum_UNIV distinct_map enum_distinct)
```
```   415
```
```   416 end
```
```   417
```
```   418 instantiation option :: (enum) enum
```
```   419 begin
```
```   420
```
```   421 definition
```
```   422   "enum = None # map Some enum"
```
```   423
```
```   424 definition
```
```   425   "enum_all P \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))"
```
```   426
```
```   427 definition
```
```   428   "enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>x. P (Some x))"
```
```   429
```
```   430 instance proof
```
```   431 qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv,
```
```   432   auto simp add: distinct_map enum_UNIV enum_distinct)
```
```   433
```
```   434 end
```
```   435
```
```   436
```
```   437 subsection {* Small finite types *}
```
```   438
```
```   439 text {* We define small finite types for the use in Quickcheck *}
```
```   440
```
```   441 datatype finite_1 = a\<^sub>1
```
```   442
```
```   443 notation (output) a\<^sub>1  ("a\<^sub>1")
```
```   444
```
```   445 lemma UNIV_finite_1:
```
```   446   "UNIV = {a\<^sub>1}"
```
```   447   by (auto intro: finite_1.exhaust)
```
```   448
```
```   449 instantiation finite_1 :: enum
```
```   450 begin
```
```   451
```
```   452 definition
```
```   453   "enum = [a\<^sub>1]"
```
```   454
```
```   455 definition
```
```   456   "enum_all P = P a\<^sub>1"
```
```   457
```
```   458 definition
```
```   459   "enum_ex P = P a\<^sub>1"
```
```   460
```
```   461 instance proof
```
```   462 qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all)
```
```   463
```
```   464 end
```
```   465
```
```   466 instantiation finite_1 :: linorder
```
```   467 begin
```
```   468
```
```   469 definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   470 where
```
```   471   "x < (y :: finite_1) \<longleftrightarrow> False"
```
```   472
```
```   473 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
```
```   474 where
```
```   475   "x \<le> (y :: finite_1) \<longleftrightarrow> True"
```
```   476
```
```   477 instance
```
```   478 apply (intro_classes)
```
```   479 apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
```
```   480 apply (metis finite_1.exhaust)
```
```   481 done
```
```   482
```
```   483 end
```
```   484
```
```   485 hide_const (open) a\<^sub>1
```
```   486
```
```   487 datatype finite_2 = a\<^sub>1 | a\<^sub>2
```
```   488
```
```   489 notation (output) a\<^sub>1  ("a\<^sub>1")
```
```   490 notation (output) a\<^sub>2  ("a\<^sub>2")
```
```   491
```
```   492 lemma UNIV_finite_2:
```
```   493   "UNIV = {a\<^sub>1, a\<^sub>2}"
```
```   494   by (auto intro: finite_2.exhaust)
```
```   495
```
```   496 instantiation finite_2 :: enum
```
```   497 begin
```
```   498
```
```   499 definition
```
```   500   "enum = [a\<^sub>1, a\<^sub>2]"
```
```   501
```
```   502 definition
```
```   503   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2"
```
```   504
```
```   505 definition
```
```   506   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2"
```
```   507
```
```   508 instance proof
```
```   509 qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all)
```
```   510
```
```   511 end
```
```   512
```
```   513 instantiation finite_2 :: linorder
```
```   514 begin
```
```   515
```
```   516 definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   517 where
```
```   518   "x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2"
```
```   519
```
```   520 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
```
```   521 where
```
```   522   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)"
```
```   523
```
```   524 instance
```
```   525 apply (intro_classes)
```
```   526 apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
```
```   527 apply (metis finite_2.nchotomy)+
```
```   528 done
```
```   529
```
```   530 end
```
```   531
```
```   532 hide_const (open) a\<^sub>1 a\<^sub>2
```
```   533
```
```   534 datatype finite_3 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3
```
```   535
```
```   536 notation (output) a\<^sub>1  ("a\<^sub>1")
```
```   537 notation (output) a\<^sub>2  ("a\<^sub>2")
```
```   538 notation (output) a\<^sub>3  ("a\<^sub>3")
```
```   539
```
```   540 lemma UNIV_finite_3:
```
```   541   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}"
```
```   542   by (auto intro: finite_3.exhaust)
```
```   543
```
```   544 instantiation finite_3 :: enum
```
```   545 begin
```
```   546
```
```   547 definition
```
```   548   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
```
```   549
```
```   550 definition
```
```   551   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3"
```
```   552
```
```   553 definition
```
```   554   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3"
```
```   555
```
```   556 instance proof
```
```   557 qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all)
```
```   558
```
```   559 end
```
```   560
```
```   561 instantiation finite_3 :: linorder
```
```   562 begin
```
```   563
```
```   564 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   565 where
```
```   566   "x < y = (case x of a\<^sub>1 \<Rightarrow> y \<noteq> a\<^sub>1 | a\<^sub>2 \<Rightarrow> y = a\<^sub>3 | a\<^sub>3 \<Rightarrow> False)"
```
```   567
```
```   568 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
```
```   569 where
```
```   570   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)"
```
```   571
```
```   572 instance proof (intro_classes)
```
```   573 qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
```
```   574
```
```   575 end
```
```   576
```
```   577 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3
```
```   578
```
```   579 datatype finite_4 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4
```
```   580
```
```   581 notation (output) a\<^sub>1  ("a\<^sub>1")
```
```   582 notation (output) a\<^sub>2  ("a\<^sub>2")
```
```   583 notation (output) a\<^sub>3  ("a\<^sub>3")
```
```   584 notation (output) a\<^sub>4  ("a\<^sub>4")
```
```   585
```
```   586 lemma UNIV_finite_4:
```
```   587   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}"
```
```   588   by (auto intro: finite_4.exhaust)
```
```   589
```
```   590 instantiation finite_4 :: enum
```
```   591 begin
```
```   592
```
```   593 definition
```
```   594   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
```
```   595
```
```   596 definition
```
```   597   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4"
```
```   598
```
```   599 definition
```
```   600   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4"
```
```   601
```
```   602 instance proof
```
```   603 qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all)
```
```   604
```
```   605 end
```
```   606
```
```   607 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4
```
```   608
```
```   609
```
```   610 datatype finite_5 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5
```
```   611
```
```   612 notation (output) a\<^sub>1  ("a\<^sub>1")
```
```   613 notation (output) a\<^sub>2  ("a\<^sub>2")
```
```   614 notation (output) a\<^sub>3  ("a\<^sub>3")
```
```   615 notation (output) a\<^sub>4  ("a\<^sub>4")
```
```   616 notation (output) a\<^sub>5  ("a\<^sub>5")
```
```   617
```
```   618 lemma UNIV_finite_5:
```
```   619   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}"
```
```   620   by (auto intro: finite_5.exhaust)
```
```   621
```
```   622 instantiation finite_5 :: enum
```
```   623 begin
```
```   624
```
```   625 definition
```
```   626   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
```
```   627
```
```   628 definition
```
```   629   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4 \<and> P a\<^sub>5"
```
```   630
```
```   631 definition
```
```   632   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4 \<or> P a\<^sub>5"
```
```   633
```
```   634 instance proof
```
```   635 qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all)
```
```   636
```
```   637 end
```
```   638
```
```   639 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5
```
```   640
```
```   641
```
```   642 subsection {* Closing up *}
```
```   643
```
```   644 hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
```
```   645 hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl
```
```   646
```
```   647 end
```
```   648
```