src/HOL/Library/Cardinality.thy
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```     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 o 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_nibble: "CARD(nibble) = 16"
```
```   153 unfolding UNIV_nibble by simp
```
```   154
```
```   155 lemma card_UNIV_char: "CARD(char) = 256"
```
```   156 proof -
```
```   157   have "inj (\<lambda>(x, y). Char x y)" by(auto intro: injI)
```
```   158   thus ?thesis unfolding UNIV_char by(simp add: card_image card_nibble)
```
```   159 qed
```
```   160
```
```   161 lemma card_literal: "CARD(String.literal) = 0"
```
```   162 by(simp add: card_eq_0_iff infinite_literal)
```
```   163
```
```   164 subsection \<open>Classes with at least 1 and 2\<close>
```
```   165
```
```   166 text \<open>Class finite already captures "at least 1"\<close>
```
```   167
```
```   168 lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
```
```   169   unfolding neq0_conv [symmetric] by simp
```
```   170
```
```   171 lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
```
```   172   by (simp add: less_Suc_eq_le [symmetric])
```
```   173
```
```   174 text \<open>Class for cardinality "at least 2"\<close>
```
```   175
```
```   176 class card2 = finite +
```
```   177   assumes two_le_card: "2 \<le> CARD('a)"
```
```   178
```
```   179 lemma one_less_card: "Suc 0 < CARD('a::card2)"
```
```   180   using two_le_card [where 'a='a] by simp
```
```   181
```
```   182 lemma one_less_int_card: "1 < int CARD('a::card2)"
```
```   183   using one_less_card [where 'a='a] by simp
```
```   184
```
```   185
```
```   186 subsection \<open>A type class for deciding finiteness of types\<close>
```
```   187
```
```   188 type_synonym 'a finite_UNIV = "('a, bool) phantom"
```
```   189
```
```   190 class finite_UNIV =
```
```   191   fixes finite_UNIV :: "('a, bool) phantom"
```
```   192   assumes finite_UNIV: "finite_UNIV = Phantom('a) (finite (UNIV :: 'a set))"
```
```   193
```
```   194 lemma finite_UNIV_code [code_unfold]:
```
```   195   "finite (UNIV :: 'a :: finite_UNIV set)
```
```   196   \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
```
```   197 by(simp add: finite_UNIV)
```
```   198
```
```   199 subsection \<open>A type class for computing the cardinality of types\<close>
```
```   200
```
```   201 definition is_list_UNIV :: "'a list \<Rightarrow> bool"
```
```   202 where "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)"
```
```   203
```
```   204 lemma is_list_UNIV_iff: "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
```
```   205 by(auto simp add: is_list_UNIV_def Let_def card_eq_0_iff List.card_set[symmetric]
```
```   206    dest: subst[where P="finite", OF _ finite_set] card_eq_UNIV_imp_eq_UNIV)
```
```   207
```
```   208 type_synonym 'a card_UNIV = "('a, nat) phantom"
```
```   209
```
```   210 class card_UNIV = finite_UNIV +
```
```   211   fixes card_UNIV :: "'a card_UNIV"
```
```   212   assumes card_UNIV: "card_UNIV = Phantom('a) CARD('a)"
```
```   213
```
```   214 subsection \<open>Instantiations for \<open>card_UNIV\<close>\<close>
```
```   215
```
```   216 instantiation nat :: card_UNIV begin
```
```   217 definition "finite_UNIV = Phantom(nat) False"
```
```   218 definition "card_UNIV = Phantom(nat) 0"
```
```   219 instance by intro_classes (simp_all add: finite_UNIV_nat_def card_UNIV_nat_def)
```
```   220 end
```
```   221
```
```   222 instantiation int :: card_UNIV begin
```
```   223 definition "finite_UNIV = Phantom(int) False"
```
```   224 definition "card_UNIV = Phantom(int) 0"
```
```   225 instance by intro_classes (simp_all add: card_UNIV_int_def finite_UNIV_int_def infinite_UNIV_int)
```
```   226 end
```
```   227
```
```   228 instantiation natural :: card_UNIV begin
```
```   229 definition "finite_UNIV = Phantom(natural) False"
```
```   230 definition "card_UNIV = Phantom(natural) 0"
```
```   231 instance
```
```   232   by standard
```
```   233     (auto simp add: finite_UNIV_natural_def card_UNIV_natural_def card_eq_0_iff
```
```   234       type_definition.univ [OF type_definition_natural] natural_eq_iff
```
```   235       dest!: finite_imageD intro: inj_onI)
```
```   236 end
```
```   237
```
```   238 instantiation integer :: card_UNIV begin
```
```   239 definition "finite_UNIV = Phantom(integer) False"
```
```   240 definition "card_UNIV = Phantom(integer) 0"
```
```   241 instance
```
```   242   by standard
```
```   243     (auto simp add: finite_UNIV_integer_def card_UNIV_integer_def card_eq_0_iff
```
```   244       type_definition.univ [OF type_definition_integer] infinite_UNIV_int
```
```   245       dest!: finite_imageD intro: inj_onI)
```
```   246 end
```
```   247
```
```   248 instantiation list :: (type) card_UNIV begin
```
```   249 definition "finite_UNIV = Phantom('a list) False"
```
```   250 definition "card_UNIV = Phantom('a list) 0"
```
```   251 instance by intro_classes (simp_all add: card_UNIV_list_def finite_UNIV_list_def infinite_UNIV_listI)
```
```   252 end
```
```   253
```
```   254 instantiation unit :: card_UNIV begin
```
```   255 definition "finite_UNIV = Phantom(unit) True"
```
```   256 definition "card_UNIV = Phantom(unit) 1"
```
```   257 instance by intro_classes (simp_all add: card_UNIV_unit_def finite_UNIV_unit_def)
```
```   258 end
```
```   259
```
```   260 instantiation bool :: card_UNIV begin
```
```   261 definition "finite_UNIV = Phantom(bool) True"
```
```   262 definition "card_UNIV = Phantom(bool) 2"
```
```   263 instance by(intro_classes)(simp_all add: card_UNIV_bool_def finite_UNIV_bool_def)
```
```   264 end
```
```   265
```
```   266 instantiation nibble :: card_UNIV begin
```
```   267 definition "finite_UNIV = Phantom(nibble) True"
```
```   268 definition "card_UNIV = Phantom(nibble) 16"
```
```   269 instance by(intro_classes)(simp_all add: card_UNIV_nibble_def card_nibble finite_UNIV_nibble_def)
```
```   270 end
```
```   271
```
```   272 instantiation char :: card_UNIV begin
```
```   273 definition "finite_UNIV = Phantom(char) True"
```
```   274 definition "card_UNIV = Phantom(char) 256"
```
```   275 instance by intro_classes (simp_all add: card_UNIV_char_def card_UNIV_char finite_UNIV_char_def)
```
```   276 end
```
```   277
```
```   278 instantiation prod :: (finite_UNIV, finite_UNIV) finite_UNIV begin
```
```   279 definition "finite_UNIV = Phantom('a \<times> 'b)
```
```   280   (of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))"
```
```   281 instance by intro_classes (simp add: finite_UNIV_prod_def finite_UNIV finite_prod)
```
```   282 end
```
```   283
```
```   284 instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   285 definition "card_UNIV = Phantom('a \<times> 'b)
```
```   286   (of_phantom (card_UNIV :: 'a card_UNIV) * of_phantom (card_UNIV :: 'b card_UNIV))"
```
```   287 instance by intro_classes (simp add: card_UNIV_prod_def card_UNIV)
```
```   288 end
```
```   289
```
```   290 instantiation sum :: (finite_UNIV, finite_UNIV) finite_UNIV begin
```
```   291 definition "finite_UNIV = Phantom('a + 'b)
```
```   292   (of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> of_phantom (finite_UNIV :: 'b finite_UNIV))"
```
```   293 instance
```
```   294   by intro_classes (simp add: UNIV_Plus_UNIV[symmetric] finite_UNIV_sum_def finite_UNIV del: UNIV_Plus_UNIV)
```
```   295 end
```
```   296
```
```   297 instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   298 definition "card_UNIV = Phantom('a + 'b)
```
```   299   (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
```
```   300        cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   301    in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)"
```
```   302 instance by intro_classes (auto simp add: card_UNIV_sum_def card_UNIV card_UNIV_sum)
```
```   303 end
```
```   304
```
```   305 instantiation "fun" :: (finite_UNIV, card_UNIV) finite_UNIV begin
```
```   306 definition "finite_UNIV = Phantom('a \<Rightarrow> 'b)
```
```   307   (let cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   308    in cb = 1 \<or> of_phantom (finite_UNIV :: 'a finite_UNIV) \<and> cb \<noteq> 0)"
```
```   309 instance
```
```   310   by intro_classes (auto simp add: finite_UNIV_fun_def Let_def card_UNIV finite_UNIV finite_UNIV_fun card_gt_0_iff)
```
```   311 end
```
```   312
```
```   313 instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   314 definition "card_UNIV = Phantom('a \<Rightarrow> 'b)
```
```   315   (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
```
```   316        cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   317    in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
```
```   318 instance by intro_classes (simp add: card_UNIV_fun_def card_UNIV Let_def card_fun)
```
```   319 end
```
```   320
```
```   321 instantiation option :: (finite_UNIV) finite_UNIV begin
```
```   322 definition "finite_UNIV = Phantom('a option) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
```
```   323 instance by intro_classes (simp add: finite_UNIV_option_def finite_UNIV)
```
```   324 end
```
```   325
```
```   326 instantiation option :: (card_UNIV) card_UNIV begin
```
```   327 definition "card_UNIV = Phantom('a option)
```
```   328   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c \<noteq> 0 then Suc c else 0)"
```
```   329 instance by intro_classes (simp add: card_UNIV_option_def card_UNIV card_UNIV_option)
```
```   330 end
```
```   331
```
```   332 instantiation String.literal :: card_UNIV begin
```
```   333 definition "finite_UNIV = Phantom(String.literal) False"
```
```   334 definition "card_UNIV = Phantom(String.literal) 0"
```
```   335 instance
```
```   336   by intro_classes (simp_all add: card_UNIV_literal_def finite_UNIV_literal_def infinite_literal card_literal)
```
```   337 end
```
```   338
```
```   339 instantiation set :: (finite_UNIV) finite_UNIV begin
```
```   340 definition "finite_UNIV = Phantom('a set) (of_phantom (finite_UNIV :: 'a finite_UNIV))"
```
```   341 instance by intro_classes (simp add: finite_UNIV_set_def finite_UNIV Finite_Set.finite_set)
```
```   342 end
```
```   343
```
```   344 instantiation set :: (card_UNIV) card_UNIV begin
```
```   345 definition "card_UNIV = Phantom('a set)
```
```   346   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c = 0 then 0 else 2 ^ c)"
```
```   347 instance by intro_classes (simp add: card_UNIV_set_def card_UNIV_set card_UNIV)
```
```   348 end
```
```   349
```
```   350 lemma UNIV_finite_1: "UNIV = set [finite_1.a\<^sub>1]"
```
```   351 by(auto intro: finite_1.exhaust)
```
```   352
```
```   353 lemma UNIV_finite_2: "UNIV = set [finite_2.a\<^sub>1, finite_2.a\<^sub>2]"
```
```   354 by(auto intro: finite_2.exhaust)
```
```   355
```
```   356 lemma UNIV_finite_3: "UNIV = set [finite_3.a\<^sub>1, finite_3.a\<^sub>2, finite_3.a\<^sub>3]"
```
```   357 by(auto intro: finite_3.exhaust)
```
```   358
```
```   359 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]"
```
```   360 by(auto intro: finite_4.exhaust)
```
```   361
```
```   362 lemma UNIV_finite_5:
```
```   363   "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]"
```
```   364 by(auto intro: finite_5.exhaust)
```
```   365
```
```   366 instantiation Enum.finite_1 :: card_UNIV begin
```
```   367 definition "finite_UNIV = Phantom(Enum.finite_1) True"
```
```   368 definition "card_UNIV = Phantom(Enum.finite_1) 1"
```
```   369 instance
```
```   370   by intro_classes (simp_all add: UNIV_finite_1 card_UNIV_finite_1_def finite_UNIV_finite_1_def)
```
```   371 end
```
```   372
```
```   373 instantiation Enum.finite_2 :: card_UNIV begin
```
```   374 definition "finite_UNIV = Phantom(Enum.finite_2) True"
```
```   375 definition "card_UNIV = Phantom(Enum.finite_2) 2"
```
```   376 instance
```
```   377   by intro_classes (simp_all add: UNIV_finite_2 card_UNIV_finite_2_def finite_UNIV_finite_2_def)
```
```   378 end
```
```   379
```
```   380 instantiation Enum.finite_3 :: card_UNIV begin
```
```   381 definition "finite_UNIV = Phantom(Enum.finite_3) True"
```
```   382 definition "card_UNIV = Phantom(Enum.finite_3) 3"
```
```   383 instance
```
```   384   by intro_classes (simp_all add: UNIV_finite_3 card_UNIV_finite_3_def finite_UNIV_finite_3_def)
```
```   385 end
```
```   386
```
```   387 instantiation Enum.finite_4 :: card_UNIV begin
```
```   388 definition "finite_UNIV = Phantom(Enum.finite_4) True"
```
```   389 definition "card_UNIV = Phantom(Enum.finite_4) 4"
```
```   390 instance
```
```   391   by intro_classes (simp_all add: UNIV_finite_4 card_UNIV_finite_4_def finite_UNIV_finite_4_def)
```
```   392 end
```
```   393
```
```   394 instantiation Enum.finite_5 :: card_UNIV begin
```
```   395 definition "finite_UNIV = Phantom(Enum.finite_5) True"
```
```   396 definition "card_UNIV = Phantom(Enum.finite_5) 5"
```
```   397 instance
```
```   398   by intro_classes (simp_all add: UNIV_finite_5 card_UNIV_finite_5_def finite_UNIV_finite_5_def)
```
```   399 end
```
```   400
```
```   401 subsection \<open>Code setup for sets\<close>
```
```   402
```
```   403 text \<open>
```
```   404   Implement @{term "CARD('a)"} via @{term card_UNIV} and provide
```
```   405   implementations for @{term "finite"}, @{term "card"}, @{term "op \<subseteq>"},
```
```   406   and @{term "op ="}if the calling context already provides @{class finite_UNIV}
```
```   407   and @{class card_UNIV} instances. If we implemented the latter
```
```   408   always via @{term card_UNIV}, we would require instances of essentially all
```
```   409   element types, i.e., a lot of instantiation proofs and -- at run time --
```
```   410   possibly slow dictionary constructions.
```
```   411 \<close>
```
```   412
```
```   413 context
```
```   414 begin
```
```   415
```
```   416 qualified definition card_UNIV' :: "'a card_UNIV"
```
```   417 where [code del]: "card_UNIV' = Phantom('a) CARD('a)"
```
```   418
```
```   419 lemma CARD_code [code_unfold]:
```
```   420   "CARD('a) = of_phantom (card_UNIV' :: 'a card_UNIV)"
```
```   421 by(simp add: card_UNIV'_def)
```
```   422
```
```   423 lemma card_UNIV'_code [code]:
```
```   424   "card_UNIV' = card_UNIV"
```
```   425 by(simp add: card_UNIV card_UNIV'_def)
```
```   426
```
```   427 end
```
```   428
```
```   429 lemma card_Compl:
```
```   430   "finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)"
```
```   431 by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest)
```
```   432
```
```   433 context fixes xs :: "'a :: finite_UNIV list"
```
```   434 begin
```
```   435
```
```   436 qualified definition finite' :: "'a set \<Rightarrow> bool"
```
```   437 where [simp, code del, code_abbrev]: "finite' = finite"
```
```   438
```
```   439 lemma finite'_code [code]:
```
```   440   "finite' (set xs) \<longleftrightarrow> True"
```
```   441   "finite' (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
```
```   442 by(simp_all add: card_gt_0_iff finite_UNIV)
```
```   443
```
```   444 end
```
```   445
```
```   446 context fixes xs :: "'a :: card_UNIV list"
```
```   447 begin
```
```   448
```
```   449 qualified definition card' :: "'a set \<Rightarrow> nat"
```
```   450 where [simp, code del, code_abbrev]: "card' = card"
```
```   451
```
```   452 lemma card'_code [code]:
```
```   453   "card' (set xs) = length (remdups xs)"
```
```   454   "card' (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
```
```   455 by(simp_all add: List.card_set card_Compl card_UNIV)
```
```   456
```
```   457
```
```   458 qualified definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   459 where [simp, code del, code_abbrev]: "subset' = op \<subseteq>"
```
```   460
```
```   461 lemma subset'_code [code]:
```
```   462   "subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)"
```
```   463   "subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)"
```
```   464   "subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)"
```
```   465 by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card])
```
```   466   (metis finite_compl finite_set rev_finite_subset)
```
```   467
```
```   468 qualified definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   469 where [simp, code del, code_abbrev]: "eq_set = op ="
```
```   470
```
```   471 lemma eq_set_code [code]:
```
```   472   fixes ys
```
```   473   defines "rhs \<equiv>
```
```   474   let n = CARD('a)
```
```   475   in if n = 0 then False else
```
```   476         let xs' = remdups xs; ys' = remdups ys
```
```   477         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')"
```
```   478   shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs"
```
```   479   and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs"
```
```   480   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)"
```
```   481   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)"
```
```   482 proof goal_cases
```
```   483   {
```
```   484     case 1
```
```   485     show ?case (is "?lhs \<longleftrightarrow> ?rhs")
```
```   486     proof
```
```   487       show ?rhs if ?lhs
```
```   488         using that
```
```   489         by (auto simp add: rhs_def Let_def List.card_set[symmetric]
```
```   490           card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV
```
```   491           Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
```
```   492       show ?lhs if ?rhs
```
```   493       proof -
```
```   494         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
```
```   495         with that show ?thesis
```
```   496           by (auto simp add: rhs_def Let_def List.card_set[symmetric]
```
```   497             card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"]
```
```   498             dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm)
```
```   499       qed
```
```   500     qed
```
```   501   }
```
```   502   moreover
```
```   503   case 2
```
```   504   ultimately show ?case unfolding eq_set_def by blast
```
```   505 next
```
```   506   case 3
```
```   507   show ?case unfolding eq_set_def List.coset_def by blast
```
```   508 next
```
```   509   case 4
```
```   510   show ?case unfolding eq_set_def List.coset_def by blast
```
```   511 qed
```
```   512
```
```   513 end
```
```   514
```
```   515 text \<open>
```
```   516   Provide more informative exceptions than Match for non-rewritten cases.
```
```   517   If generated code raises one these exceptions, then a code equation calls
```
```   518   the mentioned operator for an element type that is not an instance of
```
```   519   @{class card_UNIV} and is therefore not implemented via @{term card_UNIV}.
```
```   520   Constrain the element type with sort @{class card_UNIV} to change this.
```
```   521 \<close>
```
```   522
```
```   523 lemma card_coset_error [code]:
```
```   524   "card (List.coset xs) =
```
```   525    Code.abort (STR ''card (List.coset _) requires type class instance card_UNIV'')
```
```   526      (\<lambda>_. card (List.coset xs))"
```
```   527 by(simp)
```
```   528
```
```   529 lemma coset_subseteq_set_code [code]:
```
```   530   "List.coset xs \<subseteq> set ys \<longleftrightarrow>
```
```   531   (if xs = [] \<and> ys = [] then False
```
```   532    else Code.abort
```
```   533      (STR ''subset_eq (List.coset _) (List.set _) requires type class instance card_UNIV'')
```
```   534      (\<lambda>_. List.coset xs \<subseteq> set ys))"
```
```   535 by simp
```
```   536
```
```   537 notepad begin \<comment> "test code setup"
```
```   538 have "List.coset [True] = set [False] \<and>
```
```   539       List.coset [] \<subseteq> List.set [True, False] \<and>
```
```   540       finite (List.coset [True])"
```
```   541   by eval
```
```   542 end
```
```   543
```
```   544 end
```