src/HOL/Library/Cardinality.thy
 author Manuel Eberl Mon Mar 26 16:14:16 2018 +0200 (19 months ago) changeset 67951 655aa11359dc parent 67443 3abf6a722518 child 68011 fb6469cdf094 permissions -rw-r--r--
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
```     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> \<open>test code setup\<close>
```
```   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
```