src/HOL/Library/Executable_Set.thy
 author chaieb Mon Feb 09 17:21:46 2009 +0000 (2009-02-09) changeset 29847 af32126ee729 parent 29110 476c46e99ada child 30304 d8e4cd2ac2a1 permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Executable_Set.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Stefan Berghofer, TU Muenchen
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```     4 *)
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```     5
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```     6 header {* Implementation of finite sets by lists *}
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```     7
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```     8 theory Executable_Set
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```     9 imports Plain "~~/src/HOL/List"
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```    10 begin
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```    11
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```    12 subsection {* Definitional rewrites *}
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```    13
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```    14 definition subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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```    15   "subset = op \<le>"
```
```    16
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```    17 declare subset_def [symmetric, code unfold]
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```    18
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```    19 lemma [code]: "subset A B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
```
```    20   unfolding subset_def subset_eq ..
```
```    21
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```    22 definition is_empty :: "'a set \<Rightarrow> bool" where
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```    23   "is_empty A \<longleftrightarrow> A = {}"
```
```    24
```
```    25 definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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```    26   [code del]: "eq_set = op ="
```
```    27
```
```    28 lemma [code]: "eq_set A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
```
```    29   unfolding eq_set_def by auto
```
```    30
```
```    31 (* FIXME allow for Stefan's code generator:
```
```    32 declare set_eq_subset[code unfold]
```
```    33 *)
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```    34
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```    35 lemma [code]:
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```    36   "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. x = a)"
```
```    37   unfolding bex_triv_one_point1 ..
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```    38
```
```    39 definition filter_set :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
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```    40   "filter_set P xs = {x\<in>xs. P x}"
```
```    41
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```    42 declare filter_set_def[symmetric, code unfold]
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```    43
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```    44
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```    45 subsection {* Operations on lists *}
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```    46
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```    47 subsubsection {* Basic definitions *}
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```    48
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```    49 definition
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```    50   flip :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c" where
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```    51   "flip f a b = f b a"
```
```    52
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```    53 definition
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```    54   member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
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```    55   "member xs x \<longleftrightarrow> x \<in> set xs"
```
```    56
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```    57 definition
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```    58   insertl :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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```    59   "insertl x xs = (if member xs x then xs else x#xs)"
```
```    60
```
```    61 lemma [code target: List]: "member [] y \<longleftrightarrow> False"
```
```    62   and [code target: List]: "member (x#xs) y \<longleftrightarrow> y = x \<or> member xs y"
```
```    63   unfolding member_def by (induct xs) simp_all
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```    64
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```    65 fun
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```    66   drop_first :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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```    67   "drop_first f [] = []"
```
```    68 | "drop_first f (x#xs) = (if f x then xs else x # drop_first f xs)"
```
```    69 declare drop_first.simps [code del]
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```    70 declare drop_first.simps [code target: List]
```
```    71
```
```    72 declare remove1.simps [code del]
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```    73 lemma [code target: List]:
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```    74   "remove1 x xs = (if member xs x then drop_first (\<lambda>y. y = x) xs else xs)"
```
```    75 proof (cases "member xs x")
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```    76   case False thus ?thesis unfolding member_def by (induct xs) auto
```
```    77 next
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```    78   case True
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```    79   have "remove1 x xs = drop_first (\<lambda>y. y = x) xs" by (induct xs) simp_all
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```    80   with True show ?thesis by simp
```
```    81 qed
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```    82
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```    83 lemma member_nil [simp]:
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```    84   "member [] = (\<lambda>x. False)"
```
```    85 proof (rule ext)
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```    86   fix x
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```    87   show "member [] x = False" unfolding member_def by simp
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```    88 qed
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```    89
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```    90 lemma member_insertl [simp]:
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```    91   "x \<in> set (insertl x xs)"
```
```    92   unfolding insertl_def member_def mem_iff by simp
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```    93
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```    94 lemma insertl_member [simp]:
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```    95   fixes xs x
```
```    96   assumes member: "member xs x"
```
```    97   shows "insertl x xs = xs"
```
```    98   using member unfolding insertl_def by simp
```
```    99
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```   100 lemma insertl_not_member [simp]:
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```   101   fixes xs x
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```   102   assumes member: "\<not> (member xs x)"
```
```   103   shows "insertl x xs = x # xs"
```
```   104   using member unfolding insertl_def by simp
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```   105
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```   106 lemma foldr_remove1_empty [simp]:
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```   107   "foldr remove1 xs [] = []"
```
```   108   by (induct xs) simp_all
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```   109
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```   110
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```   111 subsubsection {* Derived definitions *}
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```   112
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```   113 function unionl :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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```   114 where
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```   115   "unionl [] ys = ys"
```
```   116 | "unionl xs ys = foldr insertl xs ys"
```
```   117 by pat_completeness auto
```
```   118 termination by lexicographic_order
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```   119
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```   120 lemmas unionl_eq = unionl.simps(2)
```
```   121
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```   122 function intersect :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   123 where
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```   124   "intersect [] ys = []"
```
```   125 | "intersect xs [] = []"
```
```   126 | "intersect xs ys = filter (member xs) ys"
```
```   127 by pat_completeness auto
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```   128 termination by lexicographic_order
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```   129
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```   130 lemmas intersect_eq = intersect.simps(3)
```
```   131
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```   132 function subtract :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   133 where
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```   134   "subtract [] ys = ys"
```
```   135 | "subtract xs [] = []"
```
```   136 | "subtract xs ys = foldr remove1 xs ys"
```
```   137 by pat_completeness auto
```
```   138 termination by lexicographic_order
```
```   139
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```   140 lemmas subtract_eq = subtract.simps(3)
```
```   141
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```   142 function map_distinct :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list"
```
```   143 where
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```   144   "map_distinct f [] = []"
```
```   145 | "map_distinct f xs = foldr (insertl o f) xs []"
```
```   146 by pat_completeness auto
```
```   147 termination by lexicographic_order
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```   148
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```   149 lemmas map_distinct_eq = map_distinct.simps(2)
```
```   150
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```   151 function unions :: "'a list list \<Rightarrow> 'a list"
```
```   152 where
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```   153   "unions [] = []"
```
```   154 | "unions xs = foldr unionl xs []"
```
```   155 by pat_completeness auto
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```   156 termination by lexicographic_order
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```   157
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```   158 lemmas unions_eq = unions.simps(2)
```
```   159
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```   160 consts intersects :: "'a list list \<Rightarrow> 'a list"
```
```   161 primrec
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```   162   "intersects (x#xs) = foldr intersect xs x"
```
```   163
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```   164 definition
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```   165   map_union :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
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```   166   "map_union xs f = unions (map f xs)"
```
```   167
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```   168 definition
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```   169   map_inter :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
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```   170   "map_inter xs f = intersects (map f xs)"
```
```   171
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```   172
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```   173 subsection {* Isomorphism proofs *}
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```   174
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```   175 lemma iso_member:
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```   176   "member xs x \<longleftrightarrow> x \<in> set xs"
```
```   177   unfolding member_def mem_iff ..
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```   178
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```   179 lemma iso_insert:
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```   180   "set (insertl x xs) = insert x (set xs)"
```
```   181   unfolding insertl_def iso_member by (simp add: Set.insert_absorb)
```
```   182
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```   183 lemma iso_remove1:
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```   184   assumes distnct: "distinct xs"
```
```   185   shows "set (remove1 x xs) = set xs - {x}"
```
```   186   using distnct set_remove1_eq by auto
```
```   187
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```   188 lemma iso_union:
```
```   189   "set (unionl xs ys) = set xs \<union> set ys"
```
```   190   unfolding unionl_eq
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```   191   by (induct xs arbitrary: ys) (simp_all add: iso_insert)
```
```   192
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```   193 lemma iso_intersect:
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```   194   "set (intersect xs ys) = set xs \<inter> set ys"
```
```   195   unfolding intersect_eq Int_def by (simp add: Int_def iso_member) auto
```
```   196
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```   197 definition
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```   198   subtract' :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
```
```   199   "subtract' = flip subtract"
```
```   200
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```   201 lemma iso_subtract:
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```   202   fixes ys
```
```   203   assumes distnct: "distinct ys"
```
```   204   shows "set (subtract' ys xs) = set ys - set xs"
```
```   205     and "distinct (subtract' ys xs)"
```
```   206   unfolding subtract'_def flip_def subtract_eq
```
```   207   using distnct by (induct xs arbitrary: ys) auto
```
```   208
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```   209 lemma iso_map_distinct:
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```   210   "set (map_distinct f xs) = image f (set xs)"
```
```   211   unfolding map_distinct_eq by (induct xs) (simp_all add: iso_insert)
```
```   212
```
```   213 lemma iso_unions:
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```   214   "set (unions xss) = \<Union> set (map set xss)"
```
```   215   unfolding unions_eq
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```   216 proof (induct xss)
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```   217   case Nil show ?case by simp
```
```   218 next
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```   219   case (Cons xs xss) thus ?case by (induct xs) (simp_all add: iso_insert)
```
```   220 qed
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```   221
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```   222 lemma iso_intersects:
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```   223   "set (intersects (xs#xss)) = \<Inter> set (map set (xs#xss))"
```
```   224   by (induct xss) (simp_all add: Int_def iso_member, auto)
```
```   225
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```   226 lemma iso_UNION:
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```   227   "set (map_union xs f) = UNION (set xs) (set o f)"
```
```   228   unfolding map_union_def iso_unions by simp
```
```   229
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```   230 lemma iso_INTER:
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```   231   "set (map_inter (x#xs) f) = INTER (set (x#xs)) (set o f)"
```
```   232   unfolding map_inter_def iso_intersects by (induct xs) (simp_all add: iso_member, auto)
```
```   233
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```   234 definition
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```   235   Blall :: "'a list \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   236   "Blall = flip list_all"
```
```   237 definition
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```   238   Blex :: "'a list \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   239   "Blex = flip list_ex"
```
```   240
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```   241 lemma iso_Ball:
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```   242   "Blall xs f = Ball (set xs) f"
```
```   243   unfolding Blall_def flip_def by (induct xs) simp_all
```
```   244
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```   245 lemma iso_Bex:
```
```   246   "Blex xs f = Bex (set xs) f"
```
```   247   unfolding Blex_def flip_def by (induct xs) simp_all
```
```   248
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```   249 lemma iso_filter:
```
```   250   "set (filter P xs) = filter_set P (set xs)"
```
```   251   unfolding filter_set_def by (induct xs) auto
```
```   252
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```   253 subsection {* code generator setup *}
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```   254
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```   255 ML {*
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```   256 nonfix inter;
```
```   257 nonfix union;
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```   258 nonfix subset;
```
```   259 *}
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```   260
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```   261 subsubsection {* const serializations *}
```
```   262
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```   263 consts_code
```
```   264   "{}" ("{*[]*}")
```
```   265   insert ("{*insertl*}")
```
```   266   is_empty ("{*null*}")
```
```   267   "op \<union>" ("{*unionl*}")
```
```   268   "op \<inter>" ("{*intersect*}")
```
```   269   "op - \<Colon> 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" ("{* flip subtract *}")
```
```   270   image ("{*map_distinct*}")
```
```   271   Union ("{*unions*}")
```
```   272   Inter ("{*intersects*}")
```
```   273   UNION ("{*map_union*}")
```
```   274   INTER ("{*map_inter*}")
```
```   275   Ball ("{*Blall*}")
```
```   276   Bex ("{*Blex*}")
```
```   277   filter_set ("{*filter*}")
```
```   278   fold ("{* foldl o flip *}")
```
```   279
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```   280 end
```