src/HOL/Enum.thy
author haftmann
Fri Jun 12 08:53:23 2015 +0200 (2015-06-12)
changeset 60429 d3d1e185cd63
parent 60352 d46de31a50c4
child 60758 d8d85a8172b5
permissions -rw-r--r--
uniform _ div _ as infix syntax for ring division
     1 (* Author: Florian Haftmann, TU Muenchen *)
     2 
     3 section {* Finite types as explicit enumerations *}
     4 
     5 theory Enum
     6 imports Map Groups_List
     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 declare [[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 
   180 subsubsection {* Bounded accessible part *}
   181 
   182 primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set" 
   183 where
   184   "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
   185 | "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"
   186 
   187 lemma bacc_subseteq_acc:
   188   "bacc r n \<subseteq> Wellfounded.acc r"
   189   by (induct n) (auto intro: acc.intros)
   190 
   191 lemma bacc_mono:
   192   "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"
   193   by (induct rule: dec_induct) auto
   194   
   195 lemma bacc_upper_bound:
   196   "bacc (r :: ('a \<times> 'a) set)  (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"
   197 proof -
   198   have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
   199   moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
   200   moreover have "finite (range (bacc r))" by auto
   201   ultimately show ?thesis
   202    by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
   203      (auto intro: finite_mono_remains_stable_implies_strict_prefix)
   204 qed
   205 
   206 lemma acc_subseteq_bacc:
   207   assumes "finite r"
   208   shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)"
   209 proof
   210   fix x
   211   assume "x : Wellfounded.acc r"
   212   then have "\<exists> n. x : bacc r n"
   213   proof (induct x arbitrary: rule: acc.induct)
   214     case (accI x)
   215     then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
   216     from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
   217     obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
   218     proof
   219       fix y assume y: "(y, x) : r"
   220       with n have "y : bacc r (n y)" by auto
   221       moreover have "n y <= Max ((%(y, x). n y) ` r)"
   222         using y `finite r` by (auto intro!: Max_ge)
   223       note bacc_mono[OF this, of r]
   224       ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
   225     qed
   226     then show ?case
   227       by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
   228   qed
   229   then show "x : (UN n. bacc r n)" by auto
   230 qed
   231 
   232 lemma acc_bacc_eq:
   233   fixes A :: "('a :: finite \<times> 'a) set"
   234   assumes "finite A"
   235   shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))"
   236   using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)
   237 
   238 lemma [code]:
   239   fixes xs :: "('a::finite \<times> 'a) list"
   240   shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"
   241   by (simp add: card_UNIV_def acc_bacc_eq)
   242 
   243 
   244 subsection {* Default instances for @{class enum} *}
   245 
   246 lemma map_of_zip_enum_is_Some:
   247   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
   248   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
   249 proof -
   250   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
   251     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
   252     by (auto intro!: map_of_zip_is_Some)
   253   then show ?thesis using enum_UNIV by auto
   254 qed
   255 
   256 lemma map_of_zip_enum_inject:
   257   fixes xs ys :: "'b\<Colon>enum list"
   258   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
   259       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
   260     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)"
   261   shows "xs = ys"
   262 proof -
   263   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
   264   proof
   265     fix x :: 'a
   266     from length map_of_zip_enum_is_Some obtain y1 y2
   267       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
   268         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
   269     moreover from map_of
   270       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)"
   271       by (auto dest: fun_cong)
   272     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
   273       by simp
   274   qed
   275   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
   276 qed
   277 
   278 definition all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
   279 where
   280   "all_n_lists P n \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)"
   281 
   282 lemma [code]:
   283   "all_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
   284   unfolding all_n_lists_def enum_all
   285   by (cases n) (auto simp add: enum_UNIV)
   286 
   287 definition ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
   288 where
   289   "ex_n_lists P n \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)"
   290 
   291 lemma [code]:
   292   "ex_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
   293   unfolding ex_n_lists_def enum_ex
   294   by (cases n) (auto simp add: enum_UNIV)
   295 
   296 instantiation "fun" :: (enum, enum) enum
   297 begin
   298 
   299 definition
   300   "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)"
   301 
   302 definition
   303   "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
   304 
   305 definition
   306   "enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
   307 
   308 instance proof
   309   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
   310   proof (rule UNIV_eq_I)
   311     fix f :: "'a \<Rightarrow> 'b"
   312     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
   313       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
   314     then show "f \<in> set enum"
   315       by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
   316   qed
   317 next
   318   from map_of_zip_enum_inject
   319   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
   320     by (auto intro!: inj_onI simp add: enum_fun_def
   321       distinct_map distinct_n_lists enum_distinct set_n_lists)
   322 next
   323   fix P
   324   show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Ball UNIV P"
   325   proof
   326     assume "enum_all P"
   327     show "Ball UNIV P"
   328     proof
   329       fix f :: "'a \<Rightarrow> 'b"
   330       have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
   331         by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
   332       from `enum_all P` have "P (the \<circ> map_of (zip enum (map f enum)))"
   333         unfolding enum_all_fun_def all_n_lists_def
   334         apply (simp add: set_n_lists)
   335         apply (erule_tac x="map f enum" in allE)
   336         apply (auto intro!: in_enum)
   337         done
   338       from this f show "P f" by auto
   339     qed
   340   next
   341     assume "Ball UNIV P"
   342     from this show "enum_all P"
   343       unfolding enum_all_fun_def all_n_lists_def by auto
   344   qed
   345 next
   346   fix P
   347   show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Bex UNIV P"
   348   proof
   349     assume "enum_ex P"
   350     from this show "Bex UNIV P"
   351       unfolding enum_ex_fun_def ex_n_lists_def by auto
   352   next
   353     assume "Bex UNIV P"
   354     from this obtain f where "P f" ..
   355     have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
   356       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) 
   357     from `P f` this have "P (the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum)))"
   358       by auto
   359     from  this show "enum_ex P"
   360       unfolding enum_ex_fun_def ex_n_lists_def
   361       apply (auto simp add: set_n_lists)
   362       apply (rule_tac x="map f enum" in exI)
   363       apply (auto intro!: in_enum)
   364       done
   365   qed
   366 qed
   367 
   368 end
   369 
   370 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
   371   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"
   372   by (simp add: enum_fun_def Let_def)
   373 
   374 lemma enum_all_fun_code [code]:
   375   "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
   376    in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
   377   by (simp only: enum_all_fun_def Let_def)
   378 
   379 lemma enum_ex_fun_code [code]:
   380   "enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
   381    in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
   382   by (simp only: enum_ex_fun_def Let_def)
   383 
   384 instantiation set :: (enum) enum
   385 begin
   386 
   387 definition
   388   "enum = map set (sublists enum)"
   389 
   390 definition
   391   "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
   392 
   393 definition
   394   "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
   395 
   396 instance proof
   397 qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
   398   enum_distinct enum_UNIV)
   399 
   400 end
   401 
   402 instantiation unit :: enum
   403 begin
   404 
   405 definition
   406   "enum = [()]"
   407 
   408 definition
   409   "enum_all P = P ()"
   410 
   411 definition
   412   "enum_ex P = P ()"
   413 
   414 instance proof
   415 qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def)
   416 
   417 end
   418 
   419 instantiation bool :: enum
   420 begin
   421 
   422 definition
   423   "enum = [False, True]"
   424 
   425 definition
   426   "enum_all P \<longleftrightarrow> P False \<and> P True"
   427 
   428 definition
   429   "enum_ex P \<longleftrightarrow> P False \<or> P True"
   430 
   431 instance proof
   432 qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all)
   433 
   434 end
   435 
   436 instantiation prod :: (enum, enum) enum
   437 begin
   438 
   439 definition
   440   "enum = List.product enum enum"
   441 
   442 definition
   443   "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
   444 
   445 definition
   446   "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
   447 
   448  
   449 instance by default
   450   (simp_all add: enum_prod_def distinct_product
   451     enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def)
   452 
   453 end
   454 
   455 instantiation sum :: (enum, enum) enum
   456 begin
   457 
   458 definition
   459   "enum = map Inl enum @ map Inr enum"
   460 
   461 definition
   462   "enum_all P \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))"
   463 
   464 definition
   465   "enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x))"
   466 
   467 instance proof
   468 qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum,
   469   auto simp add: enum_UNIV distinct_map enum_distinct)
   470 
   471 end
   472 
   473 instantiation option :: (enum) enum
   474 begin
   475 
   476 definition
   477   "enum = None # map Some enum"
   478 
   479 definition
   480   "enum_all P \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))"
   481 
   482 definition
   483   "enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>x. P (Some x))"
   484 
   485 instance proof
   486 qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv,
   487   auto simp add: distinct_map enum_UNIV enum_distinct)
   488 
   489 end
   490 
   491 
   492 subsection {* Small finite types *}
   493 
   494 text {* We define small finite types for use in Quickcheck *}
   495 
   496 datatype (plugins only: code "quickcheck" extraction) finite_1 =
   497   a\<^sub>1
   498 
   499 notation (output) a\<^sub>1  ("a\<^sub>1")
   500 
   501 lemma UNIV_finite_1:
   502   "UNIV = {a\<^sub>1}"
   503   by (auto intro: finite_1.exhaust)
   504 
   505 instantiation finite_1 :: enum
   506 begin
   507 
   508 definition
   509   "enum = [a\<^sub>1]"
   510 
   511 definition
   512   "enum_all P = P a\<^sub>1"
   513 
   514 definition
   515   "enum_ex P = P a\<^sub>1"
   516 
   517 instance proof
   518 qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all)
   519 
   520 end
   521 
   522 instantiation finite_1 :: linorder
   523 begin
   524 
   525 definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
   526 where
   527   "x < (y :: finite_1) \<longleftrightarrow> False"
   528 
   529 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
   530 where
   531   "x \<le> (y :: finite_1) \<longleftrightarrow> True"
   532 
   533 instance
   534 apply (intro_classes)
   535 apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
   536 apply (metis finite_1.exhaust)
   537 done
   538 
   539 end
   540 
   541 instance finite_1 :: "{dense_linorder, wellorder}"
   542 by intro_classes (simp_all add: less_finite_1_def)
   543 
   544 instantiation finite_1 :: complete_lattice
   545 begin
   546 
   547 definition [simp]: "Inf = (\<lambda>_. a\<^sub>1)"
   548 definition [simp]: "Sup = (\<lambda>_. a\<^sub>1)"
   549 definition [simp]: "bot = a\<^sub>1"
   550 definition [simp]: "top = a\<^sub>1"
   551 definition [simp]: "inf = (\<lambda>_ _. a\<^sub>1)"
   552 definition [simp]: "sup = (\<lambda>_ _. a\<^sub>1)"
   553 
   554 instance by intro_classes(simp_all add: less_eq_finite_1_def)
   555 end
   556 
   557 instance finite_1 :: complete_distrib_lattice
   558 by intro_classes(simp_all add: INF_def SUP_def)
   559 
   560 instance finite_1 :: complete_linorder ..
   561 
   562 lemma finite_1_eq: "x = a\<^sub>1"
   563 by(cases x) simp
   564 
   565 simproc_setup finite_1_eq ("x::finite_1") = {*
   566   fn _ => fn _ => fn ct =>
   567     (case Thm.term_of ct of
   568       Const (@{const_name a\<^sub>1}, _) => NONE
   569     | _ => SOME (mk_meta_eq @{thm finite_1_eq}))
   570 *}
   571 
   572 instantiation finite_1 :: complete_boolean_algebra
   573 begin
   574 definition [simp]: "op - = (\<lambda>_ _. a\<^sub>1)"
   575 definition [simp]: "uminus = (\<lambda>_. a\<^sub>1)"
   576 instance by intro_classes simp_all
   577 end
   578 
   579 instantiation finite_1 :: 
   580   "{linordered_ring_strict, linordered_comm_semiring_strict, ordered_comm_ring,
   581     ordered_cancel_comm_monoid_diff, comm_monoid_mult, ordered_ring_abs,
   582     one, Divides.div, sgn_if, inverse}"
   583 begin
   584 definition [simp]: "Groups.zero = a\<^sub>1"
   585 definition [simp]: "Groups.one = a\<^sub>1"
   586 definition [simp]: "op + = (\<lambda>_ _. a\<^sub>1)"
   587 definition [simp]: "op * = (\<lambda>_ _. a\<^sub>1)"
   588 definition [simp]: "op mod = (\<lambda>_ _. a\<^sub>1)" 
   589 definition [simp]: "abs = (\<lambda>_. a\<^sub>1)"
   590 definition [simp]: "sgn = (\<lambda>_. a\<^sub>1)"
   591 definition [simp]: "inverse = (\<lambda>_. a\<^sub>1)"
   592 definition [simp]: "divide = (\<lambda>_ _. a\<^sub>1)"
   593 
   594 instance by intro_classes(simp_all add: less_finite_1_def)
   595 end
   596 
   597 declare [[simproc del: finite_1_eq]]
   598 hide_const (open) a\<^sub>1
   599 
   600 datatype (plugins only: code "quickcheck" extraction) finite_2 =
   601   a\<^sub>1 | a\<^sub>2
   602 
   603 notation (output) a\<^sub>1  ("a\<^sub>1")
   604 notation (output) a\<^sub>2  ("a\<^sub>2")
   605 
   606 lemma UNIV_finite_2:
   607   "UNIV = {a\<^sub>1, a\<^sub>2}"
   608   by (auto intro: finite_2.exhaust)
   609 
   610 instantiation finite_2 :: enum
   611 begin
   612 
   613 definition
   614   "enum = [a\<^sub>1, a\<^sub>2]"
   615 
   616 definition
   617   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2"
   618 
   619 definition
   620   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2"
   621 
   622 instance proof
   623 qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all)
   624 
   625 end
   626 
   627 instantiation finite_2 :: linorder
   628 begin
   629 
   630 definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
   631 where
   632   "x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2"
   633 
   634 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
   635 where
   636   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)"
   637 
   638 instance
   639 apply (intro_classes)
   640 apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
   641 apply (metis finite_2.nchotomy)+
   642 done
   643 
   644 end
   645 
   646 instance finite_2 :: wellorder
   647 by(rule wf_wellorderI)(simp add: less_finite_2_def, intro_classes)
   648 
   649 instantiation finite_2 :: complete_lattice
   650 begin
   651 
   652 definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else a\<^sub>2)"
   653 definition "\<Squnion>A = (if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
   654 definition [simp]: "bot = a\<^sub>1"
   655 definition [simp]: "top = a\<^sub>2"
   656 definition "x \<sqinter> y = (if x = a\<^sub>1 \<or> y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)"
   657 definition "x \<squnion> y = (if x = a\<^sub>2 \<or> y = a\<^sub>2 then a\<^sub>2 else a\<^sub>1)"
   658 
   659 lemma neq_finite_2_a\<^sub>1_iff [simp]: "x \<noteq> a\<^sub>1 \<longleftrightarrow> x = a\<^sub>2"
   660 by(cases x) simp_all
   661 
   662 lemma neq_finite_2_a\<^sub>1_iff' [simp]: "a\<^sub>1 \<noteq> x \<longleftrightarrow> x = a\<^sub>2"
   663 by(cases x) simp_all
   664 
   665 lemma neq_finite_2_a\<^sub>2_iff [simp]: "x \<noteq> a\<^sub>2 \<longleftrightarrow> x = a\<^sub>1"
   666 by(cases x) simp_all
   667 
   668 lemma neq_finite_2_a\<^sub>2_iff' [simp]: "a\<^sub>2 \<noteq> x \<longleftrightarrow> x = a\<^sub>1"
   669 by(cases x) simp_all
   670 
   671 instance
   672 proof
   673   fix x :: finite_2 and A
   674   assume "x \<in> A"
   675   then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
   676     by(case_tac [!] x)(auto simp add: less_eq_finite_2_def less_finite_2_def Inf_finite_2_def Sup_finite_2_def)
   677 qed(auto simp add: less_eq_finite_2_def less_finite_2_def inf_finite_2_def sup_finite_2_def Inf_finite_2_def Sup_finite_2_def)
   678 end
   679 
   680 instance finite_2 :: complete_distrib_lattice
   681 by(intro_classes)(auto simp add: INF_def SUP_def sup_finite_2_def inf_finite_2_def Inf_finite_2_def Sup_finite_2_def)
   682 
   683 instance finite_2 :: complete_linorder ..
   684 
   685 instantiation finite_2 :: "{field, abs_if, ring_div, sgn_if, semiring_div}" begin
   686 definition [simp]: "0 = a\<^sub>1"
   687 definition [simp]: "1 = a\<^sub>2"
   688 definition "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
   689 definition "uminus = (\<lambda>x :: finite_2. x)"
   690 definition "op - = (op + :: finite_2 \<Rightarrow> _)"
   691 definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
   692 definition "inverse = (\<lambda>x :: finite_2. x)"
   693 definition "divide = (op * :: finite_2 \<Rightarrow> _)"
   694 definition "abs = (\<lambda>x :: finite_2. x)"
   695 definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
   696 definition "sgn = (\<lambda>x :: finite_2. x)"
   697 instance
   698 by intro_classes
   699   (simp_all add: plus_finite_2_def uminus_finite_2_def minus_finite_2_def times_finite_2_def
   700        inverse_finite_2_def divide_finite_2_def abs_finite_2_def mod_finite_2_def sgn_finite_2_def
   701      split: finite_2.splits)
   702 end
   703 
   704 lemma two_finite_2 [simp]:
   705   "2 = a\<^sub>1"
   706   by (simp add: numeral.simps plus_finite_2_def)
   707   
   708 hide_const (open) a\<^sub>1 a\<^sub>2
   709 
   710 datatype (plugins only: code "quickcheck" extraction) finite_3 =
   711   a\<^sub>1 | a\<^sub>2 | a\<^sub>3
   712 
   713 notation (output) a\<^sub>1  ("a\<^sub>1")
   714 notation (output) a\<^sub>2  ("a\<^sub>2")
   715 notation (output) a\<^sub>3  ("a\<^sub>3")
   716 
   717 lemma UNIV_finite_3:
   718   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}"
   719   by (auto intro: finite_3.exhaust)
   720 
   721 instantiation finite_3 :: enum
   722 begin
   723 
   724 definition
   725   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
   726 
   727 definition
   728   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3"
   729 
   730 definition
   731   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3"
   732 
   733 instance proof
   734 qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all)
   735 
   736 end
   737 
   738 instantiation finite_3 :: linorder
   739 begin
   740 
   741 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
   742 where
   743   "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)"
   744 
   745 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
   746 where
   747   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)"
   748 
   749 instance proof (intro_classes)
   750 qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
   751 
   752 end
   753 
   754 instance finite_3 :: wellorder
   755 proof(rule wf_wellorderI)
   756   have "inv_image less_than (case_finite_3 0 1 2) = {(x, y). x < y}"
   757     by(auto simp add: less_finite_3_def split: finite_3.splits)
   758   from this[symmetric] show "wf \<dots>" by simp
   759 qed intro_classes
   760 
   761 instantiation finite_3 :: complete_lattice
   762 begin
   763 
   764 definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>3)"
   765 definition "\<Squnion>A = (if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
   766 definition [simp]: "bot = a\<^sub>1"
   767 definition [simp]: "top = a\<^sub>3"
   768 definition [simp]: "inf = (min :: finite_3 \<Rightarrow> _)"
   769 definition [simp]: "sup = (max :: finite_3 \<Rightarrow> _)"
   770 
   771 instance
   772 proof
   773   fix x :: finite_3 and A
   774   assume "x \<in> A"
   775   then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
   776     by(case_tac [!] x)(auto simp add: Inf_finite_3_def Sup_finite_3_def less_eq_finite_3_def less_finite_3_def)
   777 next
   778   fix A and z :: finite_3
   779   assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
   780   then show "z \<le> \<Sqinter>A"
   781     by(cases z)(auto simp add: Inf_finite_3_def less_eq_finite_3_def less_finite_3_def)
   782 next
   783   fix A and z :: finite_3
   784   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
   785   show "\<Squnion>A \<le> z"
   786     by(auto simp add: Sup_finite_3_def less_eq_finite_3_def less_finite_3_def dest: *)
   787 qed(auto simp add: Inf_finite_3_def Sup_finite_3_def)
   788 end
   789 
   790 instance finite_3 :: complete_distrib_lattice
   791 proof
   792   fix a :: finite_3 and B
   793   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
   794   proof(cases a "\<Sqinter>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
   795     case a\<^sub>2_a\<^sub>3
   796     then have "\<And>x. x \<in> B \<Longrightarrow> x = a\<^sub>3"
   797       by(case_tac x)(auto simp add: Inf_finite_3_def split: split_if_asm)
   798     then show ?thesis using a\<^sub>2_a\<^sub>3
   799       by(auto simp add: INF_def Inf_finite_3_def max_def less_eq_finite_3_def less_finite_3_def split: split_if_asm)
   800   qed(auto simp add: INF_def Inf_finite_3_def max_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
   801   show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
   802     by(cases a "\<Squnion>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
   803       (auto simp add: SUP_def Sup_finite_3_def min_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
   804 qed
   805 
   806 instance finite_3 :: complete_linorder ..
   807 
   808 instantiation finite_3 :: "{field, abs_if, ring_div, semiring_div, sgn_if}" begin
   809 definition [simp]: "0 = a\<^sub>1"
   810 definition [simp]: "1 = a\<^sub>2"
   811 definition
   812   "x + y = (case (x, y) of
   813      (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
   814    | (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2
   815    | _ \<Rightarrow> a\<^sub>3)"
   816 definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2)"
   817 definition "x - y = x + (- y :: finite_3)"
   818 definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
   819 definition "inverse = (\<lambda>x :: finite_3. x)" 
   820 definition "x div y = x * inverse (y :: finite_3)"
   821 definition "abs = (\<lambda>x :: finite_3. x)"
   822 definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
   823 definition "sgn = (\<lambda>x. case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
   824 instance
   825 by intro_classes
   826   (simp_all add: plus_finite_3_def uminus_finite_3_def minus_finite_3_def times_finite_3_def
   827        inverse_finite_3_def divide_finite_3_def abs_finite_3_def mod_finite_3_def sgn_finite_3_def
   828        less_finite_3_def
   829      split: finite_3.splits)
   830 end
   831 
   832 
   833 
   834 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3
   835 
   836 datatype (plugins only: code "quickcheck" extraction) finite_4 =
   837   a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4
   838 
   839 notation (output) a\<^sub>1  ("a\<^sub>1")
   840 notation (output) a\<^sub>2  ("a\<^sub>2")
   841 notation (output) a\<^sub>3  ("a\<^sub>3")
   842 notation (output) a\<^sub>4  ("a\<^sub>4")
   843 
   844 lemma UNIV_finite_4:
   845   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}"
   846   by (auto intro: finite_4.exhaust)
   847 
   848 instantiation finite_4 :: enum
   849 begin
   850 
   851 definition
   852   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
   853 
   854 definition
   855   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4"
   856 
   857 definition
   858   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4"
   859 
   860 instance proof
   861 qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all)
   862 
   863 end
   864 
   865 instantiation finite_4 :: complete_lattice begin
   866 
   867 text {* @{term a\<^sub>1} $<$ @{term a\<^sub>2},@{term a\<^sub>3} $<$ @{term a\<^sub>4},
   868   but @{term a\<^sub>2} and @{term a\<^sub>3} are incomparable. *}
   869 
   870 definition
   871   "x < y \<longleftrightarrow> (case (x, y) of
   872      (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
   873    |  (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
   874    |  (a\<^sub>3, a\<^sub>4) \<Rightarrow> True  | _ \<Rightarrow> False)"
   875 
   876 definition 
   877   "x \<le> y \<longleftrightarrow> (case (x, y) of
   878      (a\<^sub>1, _) \<Rightarrow> True
   879    | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
   880    | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True
   881    | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)"
   882 
   883 definition
   884   "\<Sqinter>A = (if a\<^sub>1 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>4)"
   885 definition
   886   "\<Squnion>A = (if a\<^sub>4 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>4 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>1)"
   887 definition [simp]: "bot = a\<^sub>1"
   888 definition [simp]: "top = a\<^sub>4"
   889 definition
   890   "x \<sqinter> y = (case (x, y) of
   891      (a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
   892    | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
   893    | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
   894    | _ \<Rightarrow> a\<^sub>4)"
   895 definition
   896   "x \<squnion> y = (case (x, y) of
   897      (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>4 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>4
   898   | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
   899   | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
   900   | _ \<Rightarrow> a\<^sub>1)"
   901 
   902 instance
   903 proof
   904   fix A and z :: finite_4
   905   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
   906   show "\<Squnion>A \<le> z"
   907     by(auto simp add: Sup_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
   908 next
   909   fix A and z :: finite_4
   910   assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
   911   show "z \<le> \<Sqinter>A"
   912     by(auto simp add: Inf_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
   913 qed(auto simp add: less_finite_4_def less_eq_finite_4_def Inf_finite_4_def Sup_finite_4_def inf_finite_4_def sup_finite_4_def split: finite_4.splits)
   914 
   915 end
   916 
   917 instance finite_4 :: complete_distrib_lattice
   918 proof
   919   fix a :: finite_4 and B
   920   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
   921     by(cases a "\<Sqinter>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
   922       (auto simp add: sup_finite_4_def Inf_finite_4_def INF_def split: finite_4.splits split_if_asm)
   923   show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
   924     by(cases a "\<Squnion>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
   925       (auto simp add: inf_finite_4_def Sup_finite_4_def SUP_def split: finite_4.splits split_if_asm)
   926 qed
   927 
   928 instantiation finite_4 :: complete_boolean_algebra begin
   929 definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>4 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2 | a\<^sub>4 \<Rightarrow> a\<^sub>1)"
   930 definition "x - y = x \<sqinter> - (y :: finite_4)"
   931 instance
   932 by intro_classes
   933   (simp_all add: inf_finite_4_def sup_finite_4_def uminus_finite_4_def minus_finite_4_def split: finite_4.splits)
   934 end
   935 
   936 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4
   937 
   938 datatype (plugins only: code "quickcheck" extraction) finite_5 =
   939   a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5
   940 
   941 notation (output) a\<^sub>1  ("a\<^sub>1")
   942 notation (output) a\<^sub>2  ("a\<^sub>2")
   943 notation (output) a\<^sub>3  ("a\<^sub>3")
   944 notation (output) a\<^sub>4  ("a\<^sub>4")
   945 notation (output) a\<^sub>5  ("a\<^sub>5")
   946 
   947 lemma UNIV_finite_5:
   948   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}"
   949   by (auto intro: finite_5.exhaust)
   950 
   951 instantiation finite_5 :: enum
   952 begin
   953 
   954 definition
   955   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
   956 
   957 definition
   958   "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"
   959 
   960 definition
   961   "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"
   962 
   963 instance proof
   964 qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all)
   965 
   966 end
   967 
   968 instantiation finite_5 :: complete_lattice
   969 begin
   970 
   971 text {* The non-distributive pentagon lattice $N_5$ *}
   972 
   973 definition
   974   "x < y \<longleftrightarrow> (case (x, y) of
   975      (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
   976    | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True  | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
   977    | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
   978    | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True  | _ \<Rightarrow> False)"
   979 
   980 definition
   981   "x \<le> y \<longleftrightarrow> (case (x, y) of
   982      (a\<^sub>1, _) \<Rightarrow> True
   983    | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
   984    | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
   985    | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True
   986    | (a\<^sub>5, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)"
   987 
   988 definition
   989   "\<Sqinter>A = 
   990   (if a\<^sub>1 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>1
   991    else if a\<^sub>2 \<in> A then a\<^sub>2
   992    else if a\<^sub>3 \<in> A then a\<^sub>3
   993    else if a\<^sub>4 \<in> A then a\<^sub>4
   994    else a\<^sub>5)"
   995 definition
   996   "\<Squnion>A = 
   997   (if a\<^sub>5 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>5
   998    else if a\<^sub>3 \<in> A then a\<^sub>3
   999    else if a\<^sub>2 \<in> A then a\<^sub>2
  1000    else if a\<^sub>4 \<in> A then a\<^sub>4
  1001    else a\<^sub>1)"
  1002 definition [simp]: "bot = a\<^sub>1"
  1003 definition [simp]: "top = a\<^sub>5"
  1004 definition
  1005   "x \<sqinter> y = (case (x, y) of
  1006      (a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>1
  1007    | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
  1008    | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
  1009    | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
  1010    | _ \<Rightarrow> a\<^sub>5)"
  1011 definition
  1012   "x \<squnion> y = (case (x, y) of
  1013      (a\<^sub>5, _) \<Rightarrow> a\<^sub>5 | (_, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>5 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>5
  1014    | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
  1015    | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
  1016    | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
  1017    | _ \<Rightarrow> a\<^sub>1)"
  1018 
  1019 instance 
  1020 proof intro_classes
  1021   fix A and z :: finite_5
  1022   assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
  1023   show "z \<le> \<Sqinter>A"
  1024     by(auto simp add: less_eq_finite_5_def Inf_finite_5_def split: finite_5.splits split_if_asm dest!: *)
  1025 next
  1026   fix A and z :: finite_5
  1027   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
  1028   show "\<Squnion>A \<le> z"
  1029     by(auto simp add: less_eq_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm dest!: *)
  1030 qed(auto simp add: less_eq_finite_5_def less_finite_5_def inf_finite_5_def sup_finite_5_def Inf_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm)
  1031 
  1032 end
  1033 
  1034 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5
  1035 
  1036 
  1037 subsection {* Closing up *}
  1038 
  1039 hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
  1040 hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl
  1041 
  1042 end