src/HOL/Library/Cardinality.thy
author nipkow
Wed Jan 10 15:25:09 2018 +0100 (19 months ago)
changeset 67399 eab6ce8368fa
parent 67091 1393c2340eec
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ran isabelle update_op on all sources
     1 (*  Title:      HOL/Library/Cardinality.thy
     2     Author:     Brian Huffman, Andreas Lochbihler
     3 *)
     4 
     5 section \<open>Cardinality of types\<close>
     6 
     7 theory Cardinality
     8 imports Phantom_Type
     9 begin
    10 
    11 subsection \<open>Preliminary lemmas\<close>
    12 (* These should be moved elsewhere *)
    13 
    14 lemma (in type_definition) univ:
    15   "UNIV = Abs ` A"
    16 proof
    17   show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
    18   show "UNIV \<subseteq> Abs ` A"
    19   proof
    20     fix x :: 'b
    21     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
    22     moreover have "Rep x \<in> A" by (rule Rep)
    23     ultimately show "x \<in> Abs ` A" by (rule image_eqI)
    24   qed
    25 qed
    26 
    27 lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
    28   by (simp add: univ card_image inj_on_def Abs_inject)
    29 
    30 lemma finite_range_Some: "finite (range (Some :: 'a \<Rightarrow> 'a option)) = finite (UNIV :: 'a set)"
    31 by(auto dest: finite_imageD intro: inj_Some)
    32 
    33 lemma infinite_literal: "\<not> finite (UNIV :: String.literal set)"
    34 proof -
    35   have "inj STR" by(auto intro: injI)
    36   thus ?thesis
    37     by(auto simp add: type_definition.univ[OF type_definition_literal] infinite_UNIV_listI dest: finite_imageD)
    38 qed
    39 
    40 subsection \<open>Cardinalities of types\<close>
    41 
    42 syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")
    43 
    44 translations "CARD('t)" => "CONST card (CONST UNIV :: 't set)"
    45 
    46 print_translation \<open>
    47   let
    48     fun card_univ_tr' ctxt [Const (@{const_syntax UNIV}, Type (_, [T]))] =
    49       Syntax.const @{syntax_const "_type_card"} $ Syntax_Phases.term_of_typ ctxt T
    50   in [(@{const_syntax card}, card_univ_tr')] end
    51 \<close>
    52 
    53 lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a) * CARD('b)"
    54   unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
    55 
    56 lemma card_UNIV_sum: "CARD('a + 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 then CARD('a) + CARD('b) else 0)"
    57 unfolding UNIV_Plus_UNIV[symmetric]
    58 by(auto simp add: card_eq_0_iff card_Plus simp del: UNIV_Plus_UNIV)
    59 
    60 lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
    61 by(simp add: card_UNIV_sum)
    62 
    63 lemma card_UNIV_option: "CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)"
    64 proof -
    65   have "(None :: 'a option) \<notin> range Some" by clarsimp
    66   thus ?thesis
    67     by (simp add: UNIV_option_conv card_eq_0_iff finite_range_Some card_image)
    68 qed
    69 
    70 lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
    71 by(simp add: card_UNIV_option)
    72 
    73 lemma card_UNIV_set: "CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))"
    74 by(simp add: Pow_UNIV[symmetric] card_eq_0_iff card_Pow del: Pow_UNIV)
    75 
    76 lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
    77 by(simp add: card_UNIV_set)
    78 
    79 lemma card_nat [simp]: "CARD(nat) = 0"
    80   by (simp add: card_eq_0_iff)
    81 
    82 lemma card_fun: "CARD('a \<Rightarrow> 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 \<or> CARD('b) = 1 then CARD('b) ^ CARD('a) else 0)"
    83 proof -
    84   {  assume "0 < CARD('a)" and "0 < CARD('b)"
    85     hence fina: "finite (UNIV :: 'a set)" and finb: "finite (UNIV :: 'b set)"
    86       by(simp_all only: card_ge_0_finite)
    87     from finite_distinct_list[OF finb] obtain bs 
    88       where bs: "set bs = (UNIV :: 'b set)" and distb: "distinct bs" by blast
    89     from finite_distinct_list[OF fina] obtain as
    90       where as: "set as = (UNIV :: 'a set)" and dista: "distinct as" by blast
    91     have cb: "CARD('b) = length bs"
    92       unfolding bs[symmetric] distinct_card[OF distb] ..
    93     have ca: "CARD('a) = length as"
    94       unfolding as[symmetric] distinct_card[OF dista] ..
    95     let ?xs = "map (\<lambda>ys. the \<circ> map_of (zip as ys)) (List.n_lists (length as) bs)"
    96     have "UNIV = set ?xs"
    97     proof(rule UNIV_eq_I)
    98       fix f :: "'a \<Rightarrow> 'b"
    99       from as have "f = the \<circ> map_of (zip as (map f as))"
   100         by(auto simp add: map_of_zip_map)
   101       thus "f \<in> set ?xs" using bs by(auto simp add: set_n_lists)
   102     qed
   103     moreover have "distinct ?xs" unfolding distinct_map
   104     proof(intro conjI distinct_n_lists distb inj_onI)
   105       fix xs ys :: "'b list"
   106       assume xs: "xs \<in> set (List.n_lists (length as) bs)"
   107         and ys: "ys \<in> set (List.n_lists (length as) bs)"
   108         and eq: "the \<circ> map_of (zip as xs) = the \<circ> map_of (zip as ys)"
   109       from xs ys have [simp]: "length xs = length as" "length ys = length as"
   110         by(simp_all add: length_n_lists_elem)
   111       have "map_of (zip as xs) = map_of (zip as ys)"
   112       proof
   113         fix x
   114         from as bs have "\<exists>y. map_of (zip as xs) x = Some y" "\<exists>y. map_of (zip as ys) x = Some y"
   115           by(simp_all add: map_of_zip_is_Some[symmetric])
   116         with eq show "map_of (zip as xs) x = map_of (zip as ys) x"
   117           by(auto dest: fun_cong[where x=x])
   118       qed
   119       with dista show "xs = ys" by(simp add: map_of_zip_inject)
   120     qed
   121     hence "card (set ?xs) = length ?xs" by(simp only: distinct_card)
   122     moreover have "length ?xs = length bs ^ length as" by(simp add: length_n_lists)
   123     ultimately have "CARD('a \<Rightarrow> 'b) = CARD('b) ^ CARD('a)" using cb ca by simp }
   124   moreover {
   125     assume cb: "CARD('b) = 1"
   126     then obtain b where b: "UNIV = {b :: 'b}" by(auto simp add: card_Suc_eq)
   127     have eq: "UNIV = {\<lambda>x :: 'a. b ::'b}"
   128     proof(rule UNIV_eq_I)
   129       fix x :: "'a \<Rightarrow> 'b"
   130       { fix y
   131         have "x y \<in> UNIV" ..
   132         hence "x y = b" unfolding b by simp }
   133       thus "x \<in> {\<lambda>x. b}" by(auto)
   134     qed
   135     have "CARD('a \<Rightarrow> 'b) = 1" unfolding eq by simp }
   136   ultimately show ?thesis
   137     by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1)
   138 qed
   139 
   140 corollary finite_UNIV_fun:
   141   "finite (UNIV :: ('a \<Rightarrow> 'b) set) \<longleftrightarrow>
   142    finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set) \<or> CARD('b) = 1"
   143   (is "?lhs \<longleftrightarrow> ?rhs")
   144 proof -
   145   have "?lhs \<longleftrightarrow> CARD('a \<Rightarrow> 'b) > 0" by(simp add: card_gt_0_iff)
   146   also have "\<dots> \<longleftrightarrow> CARD('a) > 0 \<and> CARD('b) > 0 \<or> CARD('b) = 1"
   147     by(simp add: card_fun)
   148   also have "\<dots> = ?rhs" by(simp add: card_gt_0_iff)
   149   finally show ?thesis .
   150 qed
   151 
   152 lemma card_literal: "CARD(String.literal) = 0"
   153 by(simp add: card_eq_0_iff infinite_literal)
   154 
   155 subsection \<open>Classes with at least 1 and 2\<close>
   156 
   157 text \<open>Class finite already captures "at least 1"\<close>
   158 
   159 lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
   160   unfolding neq0_conv [symmetric] by simp
   161 
   162 lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
   163   by (simp add: less_Suc_eq_le [symmetric])
   164 
   165 text \<open>Class for cardinality "at least 2"\<close>
   166 
   167 class card2 = finite + 
   168   assumes two_le_card: "2 \<le> CARD('a)"
   169 
   170 lemma one_less_card: "Suc 0 < CARD('a::card2)"
   171   using two_le_card [where 'a='a] by simp
   172 
   173 lemma one_less_int_card: "1 < int CARD('a::card2)"
   174   using one_less_card [where 'a='a] by simp
   175 
   176 
   177 subsection \<open>A type class for deciding finiteness of types\<close>
   178 
   179 type_synonym 'a finite_UNIV = "('a, bool) phantom"
   180 
   181 class finite_UNIV = 
   182   fixes finite_UNIV :: "('a, bool) phantom"
   183   assumes finite_UNIV: "finite_UNIV = Phantom('a) (finite (UNIV :: 'a set))"
   184 
   185 lemma finite_UNIV_code [code_unfold]:
   186   "finite (UNIV :: 'a :: finite_UNIV set)
   187   \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
   188 by(simp add: finite_UNIV)
   189 
   190 subsection \<open>A type class for computing the cardinality of types\<close>
   191 
   192 definition is_list_UNIV :: "'a list \<Rightarrow> bool"
   193 where "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)"
   194 
   195 lemma is_list_UNIV_iff: "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
   196 by(auto simp add: is_list_UNIV_def Let_def card_eq_0_iff List.card_set[symmetric] 
   197    dest: subst[where P="finite", OF _ finite_set] card_eq_UNIV_imp_eq_UNIV)
   198 
   199 type_synonym 'a card_UNIV = "('a, nat) phantom"
   200 
   201 class card_UNIV = finite_UNIV +
   202   fixes card_UNIV :: "'a card_UNIV"
   203   assumes card_UNIV: "card_UNIV = Phantom('a) CARD('a)"
   204 
   205 subsection \<open>Instantiations for \<open>card_UNIV\<close>\<close>
   206 
   207 instantiation nat :: card_UNIV begin
   208 definition "finite_UNIV = Phantom(nat) False"
   209 definition "card_UNIV = Phantom(nat) 0"
   210 instance by intro_classes (simp_all add: finite_UNIV_nat_def card_UNIV_nat_def)
   211 end
   212 
   213 instantiation int :: card_UNIV begin
   214 definition "finite_UNIV = Phantom(int) False"
   215 definition "card_UNIV = Phantom(int) 0"
   216 instance by intro_classes (simp_all add: card_UNIV_int_def finite_UNIV_int_def infinite_UNIV_int)
   217 end
   218 
   219 instantiation natural :: card_UNIV begin
   220 definition "finite_UNIV = Phantom(natural) False"
   221 definition "card_UNIV = Phantom(natural) 0"
   222 instance
   223   by standard
   224     (auto simp add: finite_UNIV_natural_def card_UNIV_natural_def card_eq_0_iff
   225       type_definition.univ [OF type_definition_natural] natural_eq_iff
   226       dest!: finite_imageD intro: inj_onI)
   227 end
   228 
   229 instantiation integer :: card_UNIV begin
   230 definition "finite_UNIV = Phantom(integer) False"
   231 definition "card_UNIV = Phantom(integer) 0"
   232 instance
   233   by standard
   234     (auto simp add: finite_UNIV_integer_def card_UNIV_integer_def card_eq_0_iff
   235       type_definition.univ [OF type_definition_integer] infinite_UNIV_int
   236       dest!: finite_imageD intro: inj_onI)
   237 end
   238 
   239 instantiation list :: (type) card_UNIV begin
   240 definition "finite_UNIV = Phantom('a list) False"
   241 definition "card_UNIV = Phantom('a list) 0"
   242 instance by intro_classes (simp_all add: card_UNIV_list_def finite_UNIV_list_def infinite_UNIV_listI)
   243 end
   244 
   245 instantiation unit :: card_UNIV begin
   246 definition "finite_UNIV = Phantom(unit) True"
   247 definition "card_UNIV = Phantom(unit) 1"
   248 instance by intro_classes (simp_all add: card_UNIV_unit_def finite_UNIV_unit_def)
   249 end
   250 
   251 instantiation bool :: card_UNIV begin
   252 definition "finite_UNIV = Phantom(bool) True"
   253 definition "card_UNIV = Phantom(bool) 2"
   254 instance by(intro_classes)(simp_all add: card_UNIV_bool_def finite_UNIV_bool_def)
   255 end
   256 
   257 instantiation char :: card_UNIV begin
   258 definition "finite_UNIV = Phantom(char) True"
   259 definition "card_UNIV = Phantom(char) 256"
   260 instance by intro_classes (simp_all add: card_UNIV_char_def card_UNIV_char finite_UNIV_char_def)
   261 end
   262 
   263 instantiation prod :: (finite_UNIV, finite_UNIV) finite_UNIV begin
   264 definition "finite_UNIV = Phantom('a \<times> 'b) 
   265   (of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))"
   266 instance by intro_classes (simp add: finite_UNIV_prod_def finite_UNIV finite_prod)
   267 end
   268 
   269 instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin
   270 definition "card_UNIV = Phantom('a \<times> 'b) 
   271   (of_phantom (card_UNIV :: 'a card_UNIV) * of_phantom (card_UNIV :: 'b card_UNIV))"
   272 instance by intro_classes (simp add: card_UNIV_prod_def card_UNIV)
   273 end
   274 
   275 instantiation sum :: (finite_UNIV, finite_UNIV) finite_UNIV begin
   276 definition "finite_UNIV = Phantom('a + 'b)
   277   (of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))"
   278 instance
   279   by intro_classes (simp add: UNIV_Plus_UNIV[symmetric] finite_UNIV_sum_def finite_UNIV del: UNIV_Plus_UNIV)
   280 end
   281 
   282 instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
   283 definition "card_UNIV = Phantom('a + 'b)
   284   (let ca = of_phantom (card_UNIV :: 'a card_UNIV); 
   285        cb = of_phantom (card_UNIV :: 'b card_UNIV)
   286    in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)"
   287 instance by intro_classes (auto simp add: card_UNIV_sum_def card_UNIV card_UNIV_sum)
   288 end
   289 
   290 instantiation "fun" :: (finite_UNIV, card_UNIV) finite_UNIV begin
   291 definition "finite_UNIV = Phantom('a \<Rightarrow> 'b)
   292   (let cb = of_phantom (card_UNIV :: 'b card_UNIV)
   293    in cb = 1 \<or> of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> cb \<noteq> 0)"
   294 instance
   295   by intro_classes (auto simp add: finite_UNIV_fun_def Let_def card_UNIV finite_UNIV finite_UNIV_fun card_gt_0_iff)
   296 end
   297 
   298 instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
   299 definition "card_UNIV = Phantom('a \<Rightarrow> 'b)
   300   (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
   301        cb = of_phantom (card_UNIV :: 'b card_UNIV)
   302    in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
   303 instance by intro_classes (simp add: card_UNIV_fun_def card_UNIV Let_def card_fun)
   304 end
   305 
   306 instantiation option :: (finite_UNIV) finite_UNIV begin
   307 definition "finite_UNIV = Phantom('a option) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
   308 instance by intro_classes (simp add: finite_UNIV_option_def finite_UNIV)
   309 end
   310 
   311 instantiation option :: (card_UNIV) card_UNIV begin
   312 definition "card_UNIV = Phantom('a option)
   313   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c \<noteq> 0 then Suc c else 0)"
   314 instance by intro_classes (simp add: card_UNIV_option_def card_UNIV card_UNIV_option)
   315 end
   316 
   317 instantiation String.literal :: card_UNIV begin
   318 definition "finite_UNIV = Phantom(String.literal) False"
   319 definition "card_UNIV = Phantom(String.literal) 0"
   320 instance
   321   by intro_classes (simp_all add: card_UNIV_literal_def finite_UNIV_literal_def infinite_literal card_literal)
   322 end
   323 
   324 instantiation set :: (finite_UNIV) finite_UNIV begin
   325 definition "finite_UNIV = Phantom('a set) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
   326 instance by intro_classes (simp add: finite_UNIV_set_def finite_UNIV Finite_Set.finite_set)
   327 end
   328 
   329 instantiation set :: (card_UNIV) card_UNIV begin
   330 definition "card_UNIV = Phantom('a set)
   331   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c = 0 then 0 else 2 ^ c)"
   332 instance by intro_classes (simp add: card_UNIV_set_def card_UNIV_set card_UNIV)
   333 end
   334 
   335 lemma UNIV_finite_1: "UNIV = set [finite_1.a\<^sub>1]"
   336 by(auto intro: finite_1.exhaust)
   337 
   338 lemma UNIV_finite_2: "UNIV = set [finite_2.a\<^sub>1, finite_2.a\<^sub>2]"
   339 by(auto intro: finite_2.exhaust)
   340 
   341 lemma UNIV_finite_3: "UNIV = set [finite_3.a\<^sub>1, finite_3.a\<^sub>2, finite_3.a\<^sub>3]"
   342 by(auto intro: finite_3.exhaust)
   343 
   344 lemma UNIV_finite_4: "UNIV = set [finite_4.a\<^sub>1, finite_4.a\<^sub>2, finite_4.a\<^sub>3, finite_4.a\<^sub>4]"
   345 by(auto intro: finite_4.exhaust)
   346 
   347 lemma UNIV_finite_5:
   348   "UNIV = set [finite_5.a\<^sub>1, finite_5.a\<^sub>2, finite_5.a\<^sub>3, finite_5.a\<^sub>4, finite_5.a\<^sub>5]"
   349 by(auto intro: finite_5.exhaust)
   350 
   351 instantiation Enum.finite_1 :: card_UNIV begin
   352 definition "finite_UNIV = Phantom(Enum.finite_1) True"
   353 definition "card_UNIV = Phantom(Enum.finite_1) 1"
   354 instance
   355   by intro_classes (simp_all add: UNIV_finite_1 card_UNIV_finite_1_def finite_UNIV_finite_1_def)
   356 end
   357 
   358 instantiation Enum.finite_2 :: card_UNIV begin
   359 definition "finite_UNIV = Phantom(Enum.finite_2) True"
   360 definition "card_UNIV = Phantom(Enum.finite_2) 2"
   361 instance
   362   by intro_classes (simp_all add: UNIV_finite_2 card_UNIV_finite_2_def finite_UNIV_finite_2_def)
   363 end
   364 
   365 instantiation Enum.finite_3 :: card_UNIV begin
   366 definition "finite_UNIV = Phantom(Enum.finite_3) True"
   367 definition "card_UNIV = Phantom(Enum.finite_3) 3"
   368 instance
   369   by intro_classes (simp_all add: UNIV_finite_3 card_UNIV_finite_3_def finite_UNIV_finite_3_def)
   370 end
   371 
   372 instantiation Enum.finite_4 :: card_UNIV begin
   373 definition "finite_UNIV = Phantom(Enum.finite_4) True"
   374 definition "card_UNIV = Phantom(Enum.finite_4) 4"
   375 instance
   376   by intro_classes (simp_all add: UNIV_finite_4 card_UNIV_finite_4_def finite_UNIV_finite_4_def)
   377 end
   378 
   379 instantiation Enum.finite_5 :: card_UNIV begin
   380 definition "finite_UNIV = Phantom(Enum.finite_5) True"
   381 definition "card_UNIV = Phantom(Enum.finite_5) 5"
   382 instance
   383   by intro_classes (simp_all add: UNIV_finite_5 card_UNIV_finite_5_def finite_UNIV_finite_5_def)
   384 end
   385 
   386 subsection \<open>Code setup for sets\<close>
   387 
   388 text \<open>
   389   Implement @{term "CARD('a)"} via @{term card_UNIV} and provide
   390   implementations for @{term "finite"}, @{term "card"}, @{term "(\<subseteq>)"}, 
   391   and @{term "(=)"}if the calling context already provides @{class finite_UNIV}
   392   and @{class card_UNIV} instances. If we implemented the latter
   393   always via @{term card_UNIV}, we would require instances of essentially all 
   394   element types, i.e., a lot of instantiation proofs and -- at run time --
   395   possibly slow dictionary constructions.
   396 \<close>
   397 
   398 context
   399 begin
   400 
   401 qualified definition card_UNIV' :: "'a card_UNIV"
   402 where [code del]: "card_UNIV' = Phantom('a) CARD('a)"
   403 
   404 lemma CARD_code [code_unfold]:
   405   "CARD('a) = of_phantom (card_UNIV' :: 'a card_UNIV)"
   406 by(simp add: card_UNIV'_def)
   407 
   408 lemma card_UNIV'_code [code]:
   409   "card_UNIV' = card_UNIV"
   410 by(simp add: card_UNIV card_UNIV'_def)
   411 
   412 end
   413 
   414 lemma card_Compl:
   415   "finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)"
   416 by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest)
   417 
   418 context fixes xs :: "'a :: finite_UNIV list"
   419 begin
   420 
   421 qualified definition finite' :: "'a set \<Rightarrow> bool"
   422 where [simp, code del, code_abbrev]: "finite' = finite"
   423 
   424 lemma finite'_code [code]:
   425   "finite' (set xs) \<longleftrightarrow> True"
   426   "finite' (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
   427 by(simp_all add: card_gt_0_iff finite_UNIV)
   428 
   429 end
   430 
   431 context fixes xs :: "'a :: card_UNIV list"
   432 begin
   433 
   434 qualified definition card' :: "'a set \<Rightarrow> nat" 
   435 where [simp, code del, code_abbrev]: "card' = card"
   436  
   437 lemma card'_code [code]:
   438   "card' (set xs) = length (remdups xs)"
   439   "card' (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
   440 by(simp_all add: List.card_set card_Compl card_UNIV)
   441 
   442 
   443 qualified definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   444 where [simp, code del, code_abbrev]: "subset' = (\<subseteq>)"
   445 
   446 lemma subset'_code [code]:
   447   "subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)"
   448   "subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)"
   449   "subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)"
   450 by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card])
   451   (metis finite_compl finite_set rev_finite_subset)
   452 
   453 qualified definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   454 where [simp, code del, code_abbrev]: "eq_set = (=)"
   455 
   456 lemma eq_set_code [code]:
   457   fixes ys
   458   defines "rhs \<equiv> 
   459   let n = CARD('a)
   460   in if n = 0 then False else 
   461         let xs' = remdups xs; ys' = remdups ys 
   462         in length xs' + length ys' = n \<and> (\<forall>x \<in> set xs'. x \<notin> set ys') \<and> (\<forall>y \<in> set ys'. y \<notin> set xs')"
   463   shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs"
   464   and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs"
   465   and "eq_set (set xs) (set ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)"
   466   and "eq_set (List.coset xs) (List.coset ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)"
   467 proof goal_cases
   468   {
   469     case 1
   470     show ?case (is "?lhs \<longleftrightarrow> ?rhs")
   471     proof
   472       show ?rhs if ?lhs
   473         using that
   474         by (auto simp add: rhs_def Let_def List.card_set[symmetric]
   475           card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV
   476           Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
   477       show ?lhs if ?rhs
   478       proof - 
   479         have "\<lbrakk> \<forall>y\<in>set xs. y \<notin> set ys; \<forall>x\<in>set ys. x \<notin> set xs \<rbrakk> \<Longrightarrow> set xs \<inter> set ys = {}" by blast
   480         with that show ?thesis
   481           by (auto simp add: rhs_def Let_def List.card_set[symmetric]
   482             card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"]
   483             dest: card_eq_UNIV_imp_eq_UNIV split: if_split_asm)
   484       qed
   485     qed
   486   }
   487   moreover
   488   case 2
   489   ultimately show ?case unfolding eq_set_def by blast
   490 next
   491   case 3
   492   show ?case unfolding eq_set_def List.coset_def by blast
   493 next
   494   case 4
   495   show ?case unfolding eq_set_def List.coset_def by blast
   496 qed
   497 
   498 end
   499 
   500 text \<open>
   501   Provide more informative exceptions than Match for non-rewritten cases.
   502   If generated code raises one these exceptions, then a code equation calls
   503   the mentioned operator for an element type that is not an instance of
   504   @{class card_UNIV} and is therefore not implemented via @{term card_UNIV}.
   505   Constrain the element type with sort @{class card_UNIV} to change this.
   506 \<close>
   507 
   508 lemma card_coset_error [code]:
   509   "card (List.coset xs) = 
   510    Code.abort (STR ''card (List.coset _) requires type class instance card_UNIV'')
   511      (\<lambda>_. card (List.coset xs))"
   512 by(simp)
   513 
   514 lemma coset_subseteq_set_code [code]:
   515   "List.coset xs \<subseteq> set ys \<longleftrightarrow> 
   516   (if xs = [] \<and> ys = [] then False 
   517    else Code.abort
   518      (STR ''subset_eq (List.coset _) (List.set _) requires type class instance card_UNIV'')
   519      (\<lambda>_. List.coset xs \<subseteq> set ys))"
   520 by simp
   521 
   522 notepad begin \<comment> "test code setup"
   523 have "List.coset [True] = set [False] \<and> 
   524       List.coset [] \<subseteq> List.set [True, False] \<and> 
   525       finite (List.coset [True])"
   526   by eval
   527 end
   528 
   529 end