src/HOL/Library/Dlist.thy
 author bulwahn Thu Jun 02 08:55:08 2011 +0200 (2011-06-02) changeset 43146 09f74fda1b1d parent 41505 6d19301074cf child 43764 366d5726de09 permissions -rw-r--r--
splitting Dlist theory in Dlist and Dlist_Cset
```     1 (* Author: Florian Haftmann, TU Muenchen *)
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```     2
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```     3 header {* Lists with elements distinct as canonical example for datatype invariants *}
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```     4
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```     5 theory Dlist
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```     6 imports Main More_List
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```     7 begin
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```     8
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```     9 subsection {* The type of distinct lists *}
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```    10
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```    11 typedef (open) 'a dlist = "{xs::'a list. distinct xs}"
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```    12   morphisms list_of_dlist Abs_dlist
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```    13 proof
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```    14   show "[] \<in> ?dlist" by simp
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```    15 qed
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```    16
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```    17 lemma dlist_eq_iff:
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```    18   "dxs = dys \<longleftrightarrow> list_of_dlist dxs = list_of_dlist dys"
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```    19   by (simp add: list_of_dlist_inject)
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```    20
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```    21 lemma dlist_eqI:
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```    22   "list_of_dlist dxs = list_of_dlist dys \<Longrightarrow> dxs = dys"
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```    23   by (simp add: dlist_eq_iff)
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```    24
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```    25 text {* Formal, totalized constructor for @{typ "'a dlist"}: *}
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```    26
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```    27 definition Dlist :: "'a list \<Rightarrow> 'a dlist" where
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```    28   "Dlist xs = Abs_dlist (remdups xs)"
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```    29
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```    30 lemma distinct_list_of_dlist [simp, intro]:
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```    31   "distinct (list_of_dlist dxs)"
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```    32   using list_of_dlist [of dxs] by simp
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```    33
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```    34 lemma list_of_dlist_Dlist [simp]:
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```    35   "list_of_dlist (Dlist xs) = remdups xs"
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```    36   by (simp add: Dlist_def Abs_dlist_inverse)
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```    37
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```    38 lemma remdups_list_of_dlist [simp]:
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```    39   "remdups (list_of_dlist dxs) = list_of_dlist dxs"
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```    40   by simp
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```    41
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```    42 lemma Dlist_list_of_dlist [simp, code abstype]:
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```    43   "Dlist (list_of_dlist dxs) = dxs"
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```    44   by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id)
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```    45
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```    46
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```    47 text {* Fundamental operations: *}
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```    48
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```    49 definition empty :: "'a dlist" where
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```    50   "empty = Dlist []"
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```    51
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```    52 definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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```    53   "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"
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```    54
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```    55 definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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```    56   "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"
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```    57
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```    58 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where
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```    59   "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"
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```    60
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```    61 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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```    62   "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"
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```    63
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```    64
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```    65 text {* Derived operations: *}
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```    66
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```    67 definition null :: "'a dlist \<Rightarrow> bool" where
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```    68   "null dxs = List.null (list_of_dlist dxs)"
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```    69
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```    70 definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where
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```    71   "member dxs = List.member (list_of_dlist dxs)"
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```    72
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```    73 definition length :: "'a dlist \<Rightarrow> nat" where
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```    74   "length dxs = List.length (list_of_dlist dxs)"
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```    75
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```    76 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
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```    77   "fold f dxs = More_List.fold f (list_of_dlist dxs)"
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```    78
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```    79 definition foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
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```    80   "foldr f dxs = List.foldr f (list_of_dlist dxs)"
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```    81
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```    82
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```    83 subsection {* Executable version obeying invariant *}
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```    84
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```    85 lemma list_of_dlist_empty [simp, code abstract]:
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```    86   "list_of_dlist empty = []"
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```    87   by (simp add: empty_def)
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```    88
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```    89 lemma list_of_dlist_insert [simp, code abstract]:
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```    90   "list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)"
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```    91   by (simp add: insert_def)
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```    92
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```    93 lemma list_of_dlist_remove [simp, code abstract]:
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```    94   "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"
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```    95   by (simp add: remove_def)
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```    96
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```    97 lemma list_of_dlist_map [simp, code abstract]:
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```    98   "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"
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```    99   by (simp add: map_def)
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```   100
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```   101 lemma list_of_dlist_filter [simp, code abstract]:
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```   102   "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"
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```   103   by (simp add: filter_def)
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```   104
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```   105
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```   106 text {* Explicit executable conversion *}
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```   107
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```   108 definition dlist_of_list [simp]:
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```   109   "dlist_of_list = Dlist"
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```   110
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```   111 lemma [code abstract]:
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```   112   "list_of_dlist (dlist_of_list xs) = remdups xs"
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```   113   by simp
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```   114
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```   115
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```   116 text {* Equality *}
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```   117
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```   118 instantiation dlist :: (equal) equal
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```   119 begin
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```   120
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```   121 definition "HOL.equal dxs dys \<longleftrightarrow> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)"
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```   122
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```   123 instance proof
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```   124 qed (simp add: equal_dlist_def equal list_of_dlist_inject)
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```   125
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```   126 end
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```   127
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```   128 lemma [code nbe]:
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```   129   "HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> True"
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```   130   by (fact equal_refl)
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```   131
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```   132
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```   133 subsection {* Induction principle and case distinction *}
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```   134
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```   135 lemma dlist_induct [case_names empty insert, induct type: dlist]:
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```   136   assumes empty: "P empty"
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```   137   assumes insrt: "\<And>x dxs. \<not> member dxs x \<Longrightarrow> P dxs \<Longrightarrow> P (insert x dxs)"
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```   138   shows "P dxs"
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```   139 proof (cases dxs)
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```   140   case (Abs_dlist xs)
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```   141   then have "distinct xs" and dxs: "dxs = Dlist xs" by (simp_all add: Dlist_def distinct_remdups_id)
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```   142   from `distinct xs` have "P (Dlist xs)"
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```   143   proof (induct xs)
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```   144     case Nil from empty show ?case by (simp add: empty_def)
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```   145   next
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```   146     case (Cons x xs)
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```   147     then have "\<not> member (Dlist xs) x" and "P (Dlist xs)"
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```   148       by (simp_all add: member_def List.member_def)
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```   149     with insrt have "P (insert x (Dlist xs))" .
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```   150     with Cons show ?case by (simp add: insert_def distinct_remdups_id)
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```   151   qed
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```   152   with dxs show "P dxs" by simp
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```   153 qed
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```   154
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```   155 lemma dlist_case [case_names empty insert, cases type: dlist]:
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```   156   assumes empty: "dxs = empty \<Longrightarrow> P"
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```   157   assumes insert: "\<And>x dys. \<not> member dys x \<Longrightarrow> dxs = insert x dys \<Longrightarrow> P"
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```   158   shows P
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```   159 proof (cases dxs)
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```   160   case (Abs_dlist xs)
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```   161   then have dxs: "dxs = Dlist xs" and distinct: "distinct xs"
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```   162     by (simp_all add: Dlist_def distinct_remdups_id)
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```   163   show P proof (cases xs)
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```   164     case Nil with dxs have "dxs = empty" by (simp add: empty_def)
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```   165     with empty show P .
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```   166   next
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```   167     case (Cons x xs)
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```   168     with dxs distinct have "\<not> member (Dlist xs) x"
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```   169       and "dxs = insert x (Dlist xs)"
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```   170       by (simp_all add: member_def List.member_def insert_def distinct_remdups_id)
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```   171     with insert show P .
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```   172   qed
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```   173 qed
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```   174
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```   175
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```   176 subsection {* Functorial structure *}
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```   177
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```   178 enriched_type map: map
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```   179   by (simp_all add: List.map.id remdups_map_remdups fun_eq_iff dlist_eq_iff)
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```   180
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```   181
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```   182 hide_const (open) member fold foldr empty insert remove map filter null member length fold
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```   183
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```   184 end
```