src/HOL/Library/Cardinality.thy
 author Andreas Lochbihler Thu Jun 28 09:16:00 2012 +0200 (2012-06-28) changeset 48164 e97369f20c30 parent 48070 02d64fd40852 child 48165 d07a0b9601aa permissions -rw-r--r--
change card_UNIV from itself to phantom type to avoid unnecessary closures in generated code
```     1 (*  Title:      HOL/Library/Cardinality.thy
```
```     2     Author:     Brian Huffman, Andreas Lochbihler
```
```     3 *)
```
```     4
```
```     5 header {* Cardinality of types *}
```
```     6
```
```     7 theory Cardinality
```
```     8 imports Phantom_Type
```
```     9 begin
```
```    10
```
```    11 subsection {* Preliminary lemmas *}
```
```    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
```
```    34 subsection {* Cardinalities of types *}
```
```    35
```
```    36 syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")
```
```    37
```
```    38 translations "CARD('t)" => "CONST card (CONST UNIV \<Colon> 't set)"
```
```    39
```
```    40 typed_print_translation (advanced) {*
```
```    41   let
```
```    42     fun card_univ_tr' ctxt _ [Const (@{const_syntax UNIV}, Type (_, [T, _]))] =
```
```    43       Syntax.const @{syntax_const "_type_card"} \$ Syntax_Phases.term_of_typ ctxt T;
```
```    44   in [(@{const_syntax card}, card_univ_tr')] end
```
```    45 *}
```
```    46
```
```    47 lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a) * CARD('b)"
```
```    48   unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
```
```    49
```
```    50 lemma card_UNIV_sum: "CARD('a + 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 then CARD('a) + CARD('b) else 0)"
```
```    51 unfolding UNIV_Plus_UNIV[symmetric]
```
```    52 by(auto simp add: card_eq_0_iff card_Plus simp del: UNIV_Plus_UNIV)
```
```    53
```
```    54 lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
```
```    55 by(simp add: card_UNIV_sum)
```
```    56
```
```    57 lemma card_UNIV_option: "CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)"
```
```    58 proof -
```
```    59   have "(None :: 'a option) \<notin> range Some" by clarsimp
```
```    60   thus ?thesis
```
```    61     by(simp add: UNIV_option_conv card_eq_0_iff finite_range_Some card_insert_disjoint card_image)
```
```    62 qed
```
```    63
```
```    64 lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
```
```    65 by(simp add: card_UNIV_option)
```
```    66
```
```    67 lemma card_UNIV_set: "CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))"
```
```    68 by(simp add: Pow_UNIV[symmetric] card_eq_0_iff card_Pow del: Pow_UNIV)
```
```    69
```
```    70 lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
```
```    71 by(simp add: card_UNIV_set)
```
```    72
```
```    73 lemma card_nat [simp]: "CARD(nat) = 0"
```
```    74   by (simp add: card_eq_0_iff)
```
```    75
```
```    76 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)"
```
```    77 proof -
```
```    78   {  assume "0 < CARD('a)" and "0 < CARD('b)"
```
```    79     hence fina: "finite (UNIV :: 'a set)" and finb: "finite (UNIV :: 'b set)"
```
```    80       by(simp_all only: card_ge_0_finite)
```
```    81     from finite_distinct_list[OF finb] obtain bs
```
```    82       where bs: "set bs = (UNIV :: 'b set)" and distb: "distinct bs" by blast
```
```    83     from finite_distinct_list[OF fina] obtain as
```
```    84       where as: "set as = (UNIV :: 'a set)" and dista: "distinct as" by blast
```
```    85     have cb: "CARD('b) = length bs"
```
```    86       unfolding bs[symmetric] distinct_card[OF distb] ..
```
```    87     have ca: "CARD('a) = length as"
```
```    88       unfolding as[symmetric] distinct_card[OF dista] ..
```
```    89     let ?xs = "map (\<lambda>ys. the o map_of (zip as ys)) (Enum.n_lists (length as) bs)"
```
```    90     have "UNIV = set ?xs"
```
```    91     proof(rule UNIV_eq_I)
```
```    92       fix f :: "'a \<Rightarrow> 'b"
```
```    93       from as have "f = the \<circ> map_of (zip as (map f as))"
```
```    94         by(auto simp add: map_of_zip_map)
```
```    95       thus "f \<in> set ?xs" using bs by(auto simp add: set_n_lists)
```
```    96     qed
```
```    97     moreover have "distinct ?xs" unfolding distinct_map
```
```    98     proof(intro conjI distinct_n_lists distb inj_onI)
```
```    99       fix xs ys :: "'b list"
```
```   100       assume xs: "xs \<in> set (Enum.n_lists (length as) bs)"
```
```   101         and ys: "ys \<in> set (Enum.n_lists (length as) bs)"
```
```   102         and eq: "the \<circ> map_of (zip as xs) = the \<circ> map_of (zip as ys)"
```
```   103       from xs ys have [simp]: "length xs = length as" "length ys = length as"
```
```   104         by(simp_all add: length_n_lists_elem)
```
```   105       have "map_of (zip as xs) = map_of (zip as ys)"
```
```   106       proof
```
```   107         fix x
```
```   108         from as bs have "\<exists>y. map_of (zip as xs) x = Some y" "\<exists>y. map_of (zip as ys) x = Some y"
```
```   109           by(simp_all add: map_of_zip_is_Some[symmetric])
```
```   110         with eq show "map_of (zip as xs) x = map_of (zip as ys) x"
```
```   111           by(auto dest: fun_cong[where x=x])
```
```   112       qed
```
```   113       with dista show "xs = ys" by(simp add: map_of_zip_inject)
```
```   114     qed
```
```   115     hence "card (set ?xs) = length ?xs" by(simp only: distinct_card)
```
```   116     moreover have "length ?xs = length bs ^ length as" by(simp add: length_n_lists)
```
```   117     ultimately have "CARD('a \<Rightarrow> 'b) = CARD('b) ^ CARD('a)" using cb ca by simp }
```
```   118   moreover {
```
```   119     assume cb: "CARD('b) = 1"
```
```   120     then obtain b where b: "UNIV = {b :: 'b}" by(auto simp add: card_Suc_eq)
```
```   121     have eq: "UNIV = {\<lambda>x :: 'a. b ::'b}"
```
```   122     proof(rule UNIV_eq_I)
```
```   123       fix x :: "'a \<Rightarrow> 'b"
```
```   124       { fix y
```
```   125         have "x y \<in> UNIV" ..
```
```   126         hence "x y = b" unfolding b by simp }
```
```   127       thus "x \<in> {\<lambda>x. b}" by(auto)
```
```   128     qed
```
```   129     have "CARD('a \<Rightarrow> 'b) = 1" unfolding eq by simp }
```
```   130   ultimately show ?thesis
```
```   131     by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1)
```
```   132 qed
```
```   133
```
```   134 lemma card_nibble: "CARD(nibble) = 16"
```
```   135 unfolding UNIV_nibble by simp
```
```   136
```
```   137 lemma card_UNIV_char: "CARD(char) = 256"
```
```   138 proof -
```
```   139   have "inj (\<lambda>(x, y). Char x y)" by(auto intro: injI)
```
```   140   thus ?thesis unfolding UNIV_char by(simp add: card_image card_nibble)
```
```   141 qed
```
```   142
```
```   143 lemma card_literal: "CARD(String.literal) = 0"
```
```   144 proof -
```
```   145   have "inj STR" by(auto intro: injI)
```
```   146   thus ?thesis by(simp add: type_definition.univ[OF type_definition_literal] card_image infinite_UNIV_listI)
```
```   147 qed
```
```   148
```
```   149 subsection {* Classes with at least 1 and 2  *}
```
```   150
```
```   151 text {* Class finite already captures "at least 1" *}
```
```   152
```
```   153 lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
```
```   154   unfolding neq0_conv [symmetric] by simp
```
```   155
```
```   156 lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
```
```   157   by (simp add: less_Suc_eq_le [symmetric])
```
```   158
```
```   159 text {* Class for cardinality "at least 2" *}
```
```   160
```
```   161 class card2 = finite +
```
```   162   assumes two_le_card: "2 \<le> CARD('a)"
```
```   163
```
```   164 lemma one_less_card: "Suc 0 < CARD('a::card2)"
```
```   165   using two_le_card [where 'a='a] by simp
```
```   166
```
```   167 lemma one_less_int_card: "1 < int CARD('a::card2)"
```
```   168   using one_less_card [where 'a='a] by simp
```
```   169
```
```   170 subsection {* A type class for computing the cardinality of types *}
```
```   171
```
```   172 definition is_list_UNIV :: "'a list \<Rightarrow> bool"
```
```   173 where [code del]: "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)"
```
```   174
```
```   175 lemma is_list_UNIV_iff: "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
```
```   176 by(auto simp add: is_list_UNIV_def Let_def card_eq_0_iff List.card_set[symmetric]
```
```   177    dest: subst[where P="finite", OF _ finite_set] card_eq_UNIV_imp_eq_UNIV)
```
```   178
```
```   179 type_synonym 'a card_UNIV = "('a, nat) phantom"
```
```   180
```
```   181 class card_UNIV =
```
```   182   fixes card_UNIV :: "'a card_UNIV"
```
```   183   assumes card_UNIV: "card_UNIV = Phantom('a) CARD('a)"
```
```   184
```
```   185 lemma card_UNIV_code [code_unfold]:
```
```   186   "CARD('a :: card_UNIV) = of_phantom (card_UNIV :: 'a card_UNIV)"
```
```   187 by(simp add: card_UNIV)
```
```   188
```
```   189 lemma is_list_UNIV_code [code]:
```
```   190   "is_list_UNIV (xs :: 'a list) =
```
```   191   (let c = CARD('a :: card_UNIV) in if c = 0 then False else size (remdups xs) = c)"
```
```   192 by(rule is_list_UNIV_def)
```
```   193
```
```   194 lemma finite_UNIV_conv_card_UNIV [code_unfold]:
```
```   195   "finite (UNIV :: 'a :: card_UNIV set) \<longleftrightarrow>
```
```   196   of_phantom (card_UNIV :: 'a card_UNIV) > 0"
```
```   197 by(simp add: card_UNIV card_gt_0_iff)
```
```   198
```
```   199 subsection {* Instantiations for @{text "card_UNIV"} *}
```
```   200
```
```   201 instantiation nat :: card_UNIV begin
```
```   202 definition "card_UNIV = Phantom(nat) 0"
```
```   203 instance by intro_classes (simp add: card_UNIV_nat_def)
```
```   204 end
```
```   205
```
```   206 instantiation int :: card_UNIV begin
```
```   207 definition "card_UNIV = Phantom(int) 0"
```
```   208 instance by intro_classes (simp add: card_UNIV_int_def infinite_UNIV_int)
```
```   209 end
```
```   210
```
```   211 instantiation list :: (type) card_UNIV begin
```
```   212 definition "card_UNIV = Phantom('a list) 0"
```
```   213 instance by intro_classes (simp add: card_UNIV_list_def infinite_UNIV_listI)
```
```   214 end
```
```   215
```
```   216 instantiation unit :: card_UNIV begin
```
```   217 definition "card_UNIV = Phantom(unit) 1"
```
```   218 instance by intro_classes (simp add: card_UNIV_unit_def card_UNIV_unit)
```
```   219 end
```
```   220
```
```   221 instantiation bool :: card_UNIV begin
```
```   222 definition "card_UNIV = Phantom(bool) 2"
```
```   223 instance by(intro_classes)(simp add: card_UNIV_bool_def card_UNIV_bool)
```
```   224 end
```
```   225
```
```   226 instantiation char :: card_UNIV begin
```
```   227 definition "card_UNIV = Phantom(char) 256"
```
```   228 instance by intro_classes (simp add: card_UNIV_char_def card_UNIV_char)
```
```   229 end
```
```   230
```
```   231 instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   232 definition "card_UNIV =
```
```   233   Phantom('a \<times> 'b) (of_phantom (card_UNIV :: 'a card_UNIV) * of_phantom (card_UNIV :: 'b card_UNIV))"
```
```   234 instance by intro_classes (simp add: card_UNIV_prod_def card_UNIV)
```
```   235 end
```
```   236
```
```   237 instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   238 definition "card_UNIV = Phantom('a + 'b)
```
```   239   (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
```
```   240        cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   241    in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)"
```
```   242 instance by intro_classes (auto simp add: card_UNIV_sum_def card_UNIV card_UNIV_sum)
```
```   243 end
```
```   244
```
```   245 instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
```
```   246 definition "card_UNIV = Phantom('a \<Rightarrow> 'b)
```
```   247   (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
```
```   248        cb = of_phantom (card_UNIV :: 'b card_UNIV)
```
```   249    in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
```
```   250 instance by intro_classes (simp add: card_UNIV_fun_def card_UNIV Let_def card_fun)
```
```   251 end
```
```   252
```
```   253 instantiation option :: (card_UNIV) card_UNIV begin
```
```   254 definition "card_UNIV = Phantom('a option)
```
```   255   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c \<noteq> 0 then Suc c else 0)"
```
```   256 instance by intro_classes (simp add: card_UNIV_option_def card_UNIV card_UNIV_option)
```
```   257 end
```
```   258
```
```   259 instantiation String.literal :: card_UNIV begin
```
```   260 definition "card_UNIV = Phantom(String.literal) 0"
```
```   261 instance by intro_classes (simp add: card_UNIV_literal_def card_literal)
```
```   262 end
```
```   263
```
```   264 instantiation set :: (card_UNIV) card_UNIV begin
```
```   265 definition "card_UNIV = Phantom('a set)
```
```   266   (let c = of_phantom (card_UNIV :: 'a card_UNIV) in if c = 0 then 0 else 2 ^ c)"
```
```   267 instance by intro_classes (simp add: card_UNIV_set_def card_UNIV_set card_UNIV)
```
```   268 end
```
```   269
```
```   270
```
```   271 lemma UNIV_finite_1: "UNIV = set [finite_1.a\<^isub>1]"
```
```   272 by(auto intro: finite_1.exhaust)
```
```   273
```
```   274 lemma UNIV_finite_2: "UNIV = set [finite_2.a\<^isub>1, finite_2.a\<^isub>2]"
```
```   275 by(auto intro: finite_2.exhaust)
```
```   276
```
```   277 lemma UNIV_finite_3: "UNIV = set [finite_3.a\<^isub>1, finite_3.a\<^isub>2, finite_3.a\<^isub>3]"
```
```   278 by(auto intro: finite_3.exhaust)
```
```   279
```
```   280 lemma UNIV_finite_4: "UNIV = set [finite_4.a\<^isub>1, finite_4.a\<^isub>2, finite_4.a\<^isub>3, finite_4.a\<^isub>4]"
```
```   281 by(auto intro: finite_4.exhaust)
```
```   282
```
```   283 lemma UNIV_finite_5:
```
```   284   "UNIV = set [finite_5.a\<^isub>1, finite_5.a\<^isub>2, finite_5.a\<^isub>3, finite_5.a\<^isub>4, finite_5.a\<^isub>5]"
```
```   285 by(auto intro: finite_5.exhaust)
```
```   286
```
```   287 instantiation Enum.finite_1 :: card_UNIV begin
```
```   288 definition "card_UNIV = Phantom(Enum.finite_1) 1"
```
```   289 instance by intro_classes (simp add: UNIV_finite_1 card_UNIV_finite_1_def)
```
```   290 end
```
```   291
```
```   292 instantiation Enum.finite_2 :: card_UNIV begin
```
```   293 definition "card_UNIV = Phantom(Enum.finite_2) 2"
```
```   294 instance by intro_classes (simp add: UNIV_finite_2 card_UNIV_finite_2_def)
```
```   295 end
```
```   296
```
```   297 instantiation Enum.finite_3 :: card_UNIV begin
```
```   298 definition "card_UNIV = Phantom(Enum.finite_3) 3"
```
```   299 instance by intro_classes (simp add: UNIV_finite_3 card_UNIV_finite_3_def)
```
```   300 end
```
```   301
```
```   302 instantiation Enum.finite_4 :: card_UNIV begin
```
```   303 definition "card_UNIV = Phantom(Enum.finite_4) 4"
```
```   304 instance by intro_classes (simp add: UNIV_finite_4 card_UNIV_finite_4_def)
```
```   305 end
```
```   306
```
```   307 instantiation Enum.finite_5 :: card_UNIV begin
```
```   308 definition "card_UNIV = Phantom(Enum.finite_5) 5"
```
```   309 instance by intro_classes (simp add: UNIV_finite_5 card_UNIV_finite_5_def)
```
```   310 end
```
```   311
```
```   312 subsection {* Code setup for sets *}
```
```   313
```
```   314 lemma card_Compl:
```
```   315   "finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)"
```
```   316 by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest)
```
```   317
```
```   318 context fixes xs :: "'a :: card_UNIV list"
```
```   319 begin
```
```   320
```
```   321 definition card' :: "'a set \<Rightarrow> nat"
```
```   322 where [simp, code del, code_abbrev]: "card' = card"
```
```   323
```
```   324 lemma card'_code [code]:
```
```   325   "card' (set xs) = length (remdups xs)"
```
```   326   "card' (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
```
```   327 by(simp_all add: List.card_set card_Compl card_UNIV)
```
```   328
```
```   329 lemma card'_UNIV [code_unfold]:
```
```   330   "card' (UNIV :: 'a :: card_UNIV set) = of_phantom (card_UNIV :: 'a card_UNIV)"
```
```   331 by(simp add: card_UNIV)
```
```   332
```
```   333 definition finite' :: "'a set \<Rightarrow> bool"
```
```   334 where [simp, code del, code_abbrev]: "finite' = finite"
```
```   335
```
```   336 lemma finite'_code [code]:
```
```   337   "finite' (set xs) \<longleftrightarrow> True"
```
```   338   "finite' (List.coset xs) \<longleftrightarrow> CARD('a) > 0"
```
```   339 by(simp_all add: card_gt_0_iff)
```
```   340
```
```   341 definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   342 where [simp, code del, code_abbrev]: "subset' = op \<subseteq>"
```
```   343
```
```   344 lemma subset'_code [code]:
```
```   345   "subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)"
```
```   346   "subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)"
```
```   347   "subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)"
```
```   348 by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card])
```
```   349   (metis finite_compl finite_set rev_finite_subset)
```
```   350
```
```   351 definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   352 where [simp, code del, code_abbrev]: "eq_set = op ="
```
```   353
```
```   354 lemma eq_set_code [code]:
```
```   355   fixes ys
```
```   356   defines "rhs \<equiv>
```
```   357   let n = CARD('a)
```
```   358   in if n = 0 then False else
```
```   359         let xs' = remdups xs; ys' = remdups ys
```
```   360         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')"
```
```   361   shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1)
```
```   362   and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2)
```
```   363   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)" (is ?thesis3)
```
```   364   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)" (is ?thesis4)
```
```   365 proof -
```
```   366   show ?thesis1 (is "?lhs \<longleftrightarrow> ?rhs")
```
```   367   proof
```
```   368     assume ?lhs thus ?rhs
```
```   369       by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
```
```   370   next
```
```   371     assume ?rhs
```
```   372     moreover 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
```
```   373     ultimately show ?lhs
```
```   374       by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"] dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm)
```
```   375   qed
```
```   376   thus ?thesis2 unfolding eq_set_def by blast
```
```   377   show ?thesis3 ?thesis4 unfolding eq_set_def List.coset_def by blast+
```
```   378 qed
```
```   379
```
```   380 end
```
```   381
```
```   382 notepad begin (* test code setup *)
```
```   383 have "List.coset [True] = set [False] \<and> List.coset [] \<subseteq> List.set [True, False] \<and> finite (List.coset [True])"
```
```   384   by eval
```
```   385 end
```
```   386
```
```   387 hide_const (open) card' finite' subset' eq_set
```
```   388
```
```   389 end
```