src/HOL/Library/Executable_Set.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 30664 167da873c3b3 child 31847 7de0e20ca24d permissions -rw-r--r--
cleaned up some proofs
```     1 (*  Title:      HOL/Library/Executable_Set.thy
```
```     2     Author:     Stefan Berghofer, TU Muenchen
```
```     3 *)
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```     4
```
```     5 header {* Implementation of finite sets by lists *}
```
```     6
```
```     7 theory Executable_Set
```
```     8 imports Main
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```     9 begin
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```    10
```
```    11 subsection {* Definitional rewrites *}
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```    12
```
```    13 definition subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
```
```    14   "subset = op \<le>"
```
```    15
```
```    16 declare subset_def [symmetric, code unfold]
```
```    17
```
```    18 lemma [code]: "subset A B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
```
```    19   unfolding subset_def subset_eq ..
```
```    20
```
```    21 definition is_empty :: "'a set \<Rightarrow> bool" where
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```    22   "is_empty A \<longleftrightarrow> A = {}"
```
```    23
```
```    24 definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
```
```    25   [code del]: "eq_set = op ="
```
```    26
```
```    27 lemma [code]: "eq_set A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
```
```    28   unfolding eq_set_def by auto
```
```    29
```
```    30 (* FIXME allow for Stefan's code generator:
```
```    31 declare set_eq_subset[code unfold]
```
```    32 *)
```
```    33
```
```    34 lemma [code]:
```
```    35   "a \<in> A \<longleftrightarrow> (\<exists>x\<in>A. x = a)"
```
```    36   unfolding bex_triv_one_point1 ..
```
```    37
```
```    38 definition filter_set :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
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```    39   "filter_set P xs = {x\<in>xs. P x}"
```
```    40
```
```    41 declare filter_set_def[symmetric, code unfold]
```
```    42
```
```    43
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```    44 subsection {* Operations on lists *}
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```    45
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```    46 subsubsection {* Basic definitions *}
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```    47
```
```    48 definition
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```    49   flip :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c" where
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```    50   "flip f a b = f b a"
```
```    51
```
```    52 definition
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```    53   member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
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```    54   "member xs x \<longleftrightarrow> x \<in> set xs"
```
```    55
```
```    56 definition
```
```    57   insertl :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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```    58   "insertl x xs = (if member xs x then xs else x#xs)"
```
```    59
```
```    60 lemma [code target: List]: "member [] y \<longleftrightarrow> False"
```
```    61   and [code target: List]: "member (x#xs) y \<longleftrightarrow> y = x \<or> member xs y"
```
```    62   unfolding member_def by (induct xs) simp_all
```
```    63
```
```    64 fun
```
```    65   drop_first :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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```    66   "drop_first f [] = []"
```
```    67 | "drop_first f (x#xs) = (if f x then xs else x # drop_first f xs)"
```
```    68 declare drop_first.simps [code del]
```
```    69 declare drop_first.simps [code target: List]
```
```    70
```
```    71 declare remove1.simps [code del]
```
```    72 lemma [code target: List]:
```
```    73   "remove1 x xs = (if member xs x then drop_first (\<lambda>y. y = x) xs else xs)"
```
```    74 proof (cases "member xs x")
```
```    75   case False thus ?thesis unfolding member_def by (induct xs) auto
```
```    76 next
```
```    77   case True
```
```    78   have "remove1 x xs = drop_first (\<lambda>y. y = x) xs" by (induct xs) simp_all
```
```    79   with True show ?thesis by simp
```
```    80 qed
```
```    81
```
```    82 lemma member_nil [simp]:
```
```    83   "member [] = (\<lambda>x. False)"
```
```    84 proof (rule ext)
```
```    85   fix x
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```    86   show "member [] x = False" unfolding member_def by simp
```
```    87 qed
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```    88
```
```    89 lemma member_insertl [simp]:
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```    90   "x \<in> set (insertl x xs)"
```
```    91   unfolding insertl_def member_def mem_iff by simp
```
```    92
```
```    93 lemma insertl_member [simp]:
```
```    94   fixes xs x
```
```    95   assumes member: "member xs x"
```
```    96   shows "insertl x xs = xs"
```
```    97   using member unfolding insertl_def by simp
```
```    98
```
```    99 lemma insertl_not_member [simp]:
```
```   100   fixes xs x
```
```   101   assumes member: "\<not> (member xs x)"
```
```   102   shows "insertl x xs = x # xs"
```
```   103   using member unfolding insertl_def by simp
```
```   104
```
```   105 lemma foldr_remove1_empty [simp]:
```
```   106   "foldr remove1 xs [] = []"
```
```   107   by (induct xs) simp_all
```
```   108
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```   109
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```   110 subsubsection {* Derived definitions *}
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```   111
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```   112 function unionl :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   113 where
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```   114   "unionl [] ys = ys"
```
```   115 | "unionl xs ys = foldr insertl xs ys"
```
```   116 by pat_completeness auto
```
```   117 termination by lexicographic_order
```
```   118
```
```   119 lemmas unionl_eq = unionl.simps(2)
```
```   120
```
```   121 function intersect :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   122 where
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```   123   "intersect [] ys = []"
```
```   124 | "intersect xs [] = []"
```
```   125 | "intersect xs ys = filter (member xs) ys"
```
```   126 by pat_completeness auto
```
```   127 termination by lexicographic_order
```
```   128
```
```   129 lemmas intersect_eq = intersect.simps(3)
```
```   130
```
```   131 function subtract :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   132 where
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```   133   "subtract [] ys = ys"
```
```   134 | "subtract xs [] = []"
```
```   135 | "subtract xs ys = foldr remove1 xs ys"
```
```   136 by pat_completeness auto
```
```   137 termination by lexicographic_order
```
```   138
```
```   139 lemmas subtract_eq = subtract.simps(3)
```
```   140
```
```   141 function map_distinct :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list"
```
```   142 where
```
```   143   "map_distinct f [] = []"
```
```   144 | "map_distinct f xs = foldr (insertl o f) xs []"
```
```   145 by pat_completeness auto
```
```   146 termination by lexicographic_order
```
```   147
```
```   148 lemmas map_distinct_eq = map_distinct.simps(2)
```
```   149
```
```   150 function unions :: "'a list list \<Rightarrow> 'a list"
```
```   151 where
```
```   152   "unions [] = []"
```
```   153 | "unions xs = foldr unionl xs []"
```
```   154 by pat_completeness auto
```
```   155 termination by lexicographic_order
```
```   156
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```   157 lemmas unions_eq = unions.simps(2)
```
```   158
```
```   159 consts intersects :: "'a list list \<Rightarrow> 'a list"
```
```   160 primrec
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```   161   "intersects (x#xs) = foldr intersect xs x"
```
```   162
```
```   163 definition
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```   164   map_union :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
```
```   165   "map_union xs f = unions (map f xs)"
```
```   166
```
```   167 definition
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```   168   map_inter :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
```
```   169   "map_inter xs f = intersects (map f xs)"
```
```   170
```
```   171
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```   172 subsection {* Isomorphism proofs *}
```
```   173
```
```   174 lemma iso_member:
```
```   175   "member xs x \<longleftrightarrow> x \<in> set xs"
```
```   176   unfolding member_def mem_iff ..
```
```   177
```
```   178 lemma iso_insert:
```
```   179   "set (insertl x xs) = insert x (set xs)"
```
```   180   unfolding insertl_def iso_member by (simp add: insert_absorb)
```
```   181
```
```   182 lemma iso_remove1:
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```   183   assumes distnct: "distinct xs"
```
```   184   shows "set (remove1 x xs) = set xs - {x}"
```
```   185   using distnct set_remove1_eq by auto
```
```   186
```
```   187 lemma iso_union:
```
```   188   "set (unionl xs ys) = set xs \<union> set ys"
```
```   189   unfolding unionl_eq
```
```   190   by (induct xs arbitrary: ys) (simp_all add: iso_insert)
```
```   191
```
```   192 lemma iso_intersect:
```
```   193   "set (intersect xs ys) = set xs \<inter> set ys"
```
```   194   unfolding intersect_eq Int_def by (simp add: Int_def iso_member) auto
```
```   195
```
```   196 definition
```
```   197   subtract' :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
```
```   198   "subtract' = flip subtract"
```
```   199
```
```   200 lemma iso_subtract:
```
```   201   fixes ys
```
```   202   assumes distnct: "distinct ys"
```
```   203   shows "set (subtract' ys xs) = set ys - set xs"
```
```   204     and "distinct (subtract' ys xs)"
```
```   205   unfolding subtract'_def flip_def subtract_eq
```
```   206   using distnct by (induct xs arbitrary: ys) auto
```
```   207
```
```   208 lemma iso_map_distinct:
```
```   209   "set (map_distinct f xs) = image f (set xs)"
```
```   210   unfolding map_distinct_eq by (induct xs) (simp_all add: iso_insert)
```
```   211
```
```   212 lemma iso_unions:
```
```   213   "set (unions xss) = \<Union> set (map set xss)"
```
```   214   unfolding unions_eq
```
```   215 proof (induct xss)
```
```   216   case Nil show ?case by simp
```
```   217 next
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```   218   case (Cons xs xss) thus ?case by (induct xs) (simp_all add: iso_insert)
```
```   219 qed
```
```   220
```
```   221 lemma iso_intersects:
```
```   222   "set (intersects (xs#xss)) = \<Inter> set (map set (xs#xss))"
```
```   223   by (induct xss) (simp_all add: Int_def iso_member, auto)
```
```   224
```
```   225 lemma iso_UNION:
```
```   226   "set (map_union xs f) = UNION (set xs) (set o f)"
```
```   227   unfolding map_union_def iso_unions by simp
```
```   228
```
```   229 lemma iso_INTER:
```
```   230   "set (map_inter (x#xs) f) = INTER (set (x#xs)) (set o f)"
```
```   231   unfolding map_inter_def iso_intersects by (induct xs) (simp_all add: iso_member, auto)
```
```   232
```
```   233 definition
```
```   234   Blall :: "'a list \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   235   "Blall = flip list_all"
```
```   236 definition
```
```   237   Blex :: "'a list \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   238   "Blex = flip list_ex"
```
```   239
```
```   240 lemma iso_Ball:
```
```   241   "Blall xs f = Ball (set xs) f"
```
```   242   unfolding Blall_def flip_def by (induct xs) simp_all
```
```   243
```
```   244 lemma iso_Bex:
```
```   245   "Blex xs f = Bex (set xs) f"
```
```   246   unfolding Blex_def flip_def by (induct xs) simp_all
```
```   247
```
```   248 lemma iso_filter:
```
```   249   "set (filter P xs) = filter_set P (set xs)"
```
```   250   unfolding filter_set_def by (induct xs) auto
```
```   251
```
```   252 subsection {* code generator setup *}
```
```   253
```
```   254 ML {*
```
```   255 nonfix inter;
```
```   256 nonfix union;
```
```   257 nonfix subset;
```
```   258 *}
```
```   259
```
```   260 subsubsection {* const serializations *}
```
```   261
```
```   262 consts_code
```
```   263   "Set.empty" ("{*[]*}")
```
```   264   insert ("{*insertl*}")
```
```   265   is_empty ("{*null*}")
```
```   266   "op \<union>" ("{*unionl*}")
```
```   267   "op \<inter>" ("{*intersect*}")
```
```   268   "op - \<Colon> 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" ("{* flip subtract *}")
```
```   269   image ("{*map_distinct*}")
```
```   270   Union ("{*unions*}")
```
```   271   Inter ("{*intersects*}")
```
```   272   UNION ("{*map_union*}")
```
```   273   INTER ("{*map_inter*}")
```
```   274   Ball ("{*Blall*}")
```
```   275   Bex ("{*Blex*}")
```
```   276   filter_set ("{*filter*}")
```
```   277   fold ("{* foldl o flip *}")
```
```   278
```
```   279 end
```