src/HOL/Library/Dlist.thy
 author haftmann Mon Sep 27 14:13:22 2010 +0200 (2010-09-27) changeset 39727 5dab9549c80d parent 39380 5a2662c1e44a child 39915 ecf97cf3d248 permissions -rw-r--r--
lemma remdups_list_of_dlist
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     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 Fset
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```     7 begin
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```     8
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```     9 section {* The type of distinct lists *}
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```    10
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```    11 typedef (open) 'a dlist = "{xs::'a list. distinct xs}"
```
```    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:
```
```    18   "dxs = dys \<longleftrightarrow> list_of_dlist dxs = list_of_dlist dys"
```
```    19   by (simp add: list_of_dlist_inject)
```
```    20
```
```    21 lemma dlist_eqI:
```
```    22   "list_of_dlist dxs = list_of_dlist dys \<Longrightarrow> dxs = dys"
```
```    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)"
```
```    32   using list_of_dlist [of dxs] by simp
```
```    33
```
```    34 lemma list_of_dlist_Dlist [simp]:
```
```    35   "list_of_dlist (Dlist xs) = remdups xs"
```
```    36   by (simp add: Dlist_def Abs_dlist_inverse)
```
```    37
```
```    38 lemma remdups_list_of_dlist [simp]:
```
```    39   "remdups (list_of_dlist dxs) = list_of_dlist dxs"
```
```    40   by simp
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```    41
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```    42 lemma Dlist_list_of_dlist [simp, code abstype]:
```
```    43   "Dlist (list_of_dlist dxs) = dxs"
```
```    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))"
```
```    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))"
```
```    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)))"
```
```    60
```
```    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))"
```
```    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)"
```
```    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)"
```
```    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)"
```
```    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)"
```
```    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 section {* Executable version obeying invariant *}
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```    84
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```    85 lemma list_of_dlist_empty [simp, code abstract]:
```
```    86   "list_of_dlist empty = []"
```
```    87   by (simp add: empty_def)
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```    88
```
```    89 lemma list_of_dlist_insert [simp, code abstract]:
```
```    90   "list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)"
```
```    91   by (simp add: insert_def)
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```    92
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```    93 lemma list_of_dlist_remove [simp, code abstract]:
```
```    94   "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"
```
```    95   by (simp add: remove_def)
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```    96
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```    97 lemma list_of_dlist_map [simp, code abstract]:
```
```    98   "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"
```
```    99   by (simp add: map_def)
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```   100
```
```   101 lemma list_of_dlist_filter [simp, code abstract]:
```
```   102   "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"
```
```   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"
```
```   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 section {* 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)"
```
```   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 rule: distinct_induct)
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```   144     case Nil from empty show ?case by (simp add: empty_def)
```
```   145   next
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```   146     case (insert 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))" .
```
```   150     with insert 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"
```
```   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 section {* Implementation of sets by distinct lists -- canonical! *}
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```   177
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```   178 definition Set :: "'a dlist \<Rightarrow> 'a fset" where
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```   179   "Set dxs = Fset.Set (list_of_dlist dxs)"
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```   180
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```   181 definition Coset :: "'a dlist \<Rightarrow> 'a fset" where
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```   182   "Coset dxs = Fset.Coset (list_of_dlist dxs)"
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```   183
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```   184 code_datatype Set Coset
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```   185
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```   186 declare member_code [code del]
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```   187 declare is_empty_Set [code del]
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```   188 declare empty_Set [code del]
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```   189 declare UNIV_Set [code del]
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```   190 declare insert_Set [code del]
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```   191 declare remove_Set [code del]
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```   192 declare compl_Set [code del]
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```   193 declare compl_Coset [code del]
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```   194 declare map_Set [code del]
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```   195 declare filter_Set [code del]
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```   196 declare forall_Set [code del]
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```   197 declare exists_Set [code del]
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```   198 declare card_Set [code del]
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```   199 declare inter_project [code del]
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```   200 declare subtract_remove [code del]
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```   201 declare union_insert [code del]
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```   202 declare Infimum_inf [code del]
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```   203 declare Supremum_sup [code del]
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```   204
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```   205 lemma Set_Dlist [simp]:
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```   206   "Set (Dlist xs) = Fset (set xs)"
```
```   207   by (rule fset_eqI) (simp add: Set_def)
```
```   208
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```   209 lemma Coset_Dlist [simp]:
```
```   210   "Coset (Dlist xs) = Fset (- set xs)"
```
```   211   by (rule fset_eqI) (simp add: Coset_def)
```
```   212
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```   213 lemma member_Set [simp]:
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```   214   "Fset.member (Set dxs) = List.member (list_of_dlist dxs)"
```
```   215   by (simp add: Set_def member_set)
```
```   216
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```   217 lemma member_Coset [simp]:
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```   218   "Fset.member (Coset dxs) = Not \<circ> List.member (list_of_dlist dxs)"
```
```   219   by (simp add: Coset_def member_set not_set_compl)
```
```   220
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```   221 lemma Set_dlist_of_list [code]:
```
```   222   "Fset.Set xs = Set (dlist_of_list xs)"
```
```   223   by (rule fset_eqI) simp
```
```   224
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```   225 lemma Coset_dlist_of_list [code]:
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```   226   "Fset.Coset xs = Coset (dlist_of_list xs)"
```
```   227   by (rule fset_eqI) simp
```
```   228
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```   229 lemma is_empty_Set [code]:
```
```   230   "Fset.is_empty (Set dxs) \<longleftrightarrow> null dxs"
```
```   231   by (simp add: null_def List.null_def member_set)
```
```   232
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```   233 lemma bot_code [code]:
```
```   234   "bot = Set empty"
```
```   235   by (simp add: empty_def)
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```   236
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```   237 lemma top_code [code]:
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```   238   "top = Coset empty"
```
```   239   by (simp add: empty_def)
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```   240
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```   241 lemma insert_code [code]:
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```   242   "Fset.insert x (Set dxs) = Set (insert x dxs)"
```
```   243   "Fset.insert x (Coset dxs) = Coset (remove x dxs)"
```
```   244   by (simp_all add: insert_def remove_def member_set not_set_compl)
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```   245
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```   246 lemma remove_code [code]:
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```   247   "Fset.remove x (Set dxs) = Set (remove x dxs)"
```
```   248   "Fset.remove x (Coset dxs) = Coset (insert x dxs)"
```
```   249   by (auto simp add: insert_def remove_def member_set not_set_compl)
```
```   250
```
```   251 lemma member_code [code]:
```
```   252   "Fset.member (Set dxs) = member dxs"
```
```   253   "Fset.member (Coset dxs) = Not \<circ> member dxs"
```
```   254   by (simp_all add: member_def)
```
```   255
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```   256 lemma compl_code [code]:
```
```   257   "- Set dxs = Coset dxs"
```
```   258   "- Coset dxs = Set dxs"
```
```   259   by (rule fset_eqI, simp add: member_set not_set_compl)+
```
```   260
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```   261 lemma map_code [code]:
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```   262   "Fset.map f (Set dxs) = Set (map f dxs)"
```
```   263   by (rule fset_eqI) (simp add: member_set)
```
```   264
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```   265 lemma filter_code [code]:
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```   266   "Fset.filter f (Set dxs) = Set (filter f dxs)"
```
```   267   by (rule fset_eqI) (simp add: member_set)
```
```   268
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```   269 lemma forall_Set [code]:
```
```   270   "Fset.forall P (Set xs) \<longleftrightarrow> list_all P (list_of_dlist xs)"
```
```   271   by (simp add: member_set list_all_iff)
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```   272
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```   273 lemma exists_Set [code]:
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```   274   "Fset.exists P (Set xs) \<longleftrightarrow> list_ex P (list_of_dlist xs)"
```
```   275   by (simp add: member_set list_ex_iff)
```
```   276
```
```   277 lemma card_code [code]:
```
```   278   "Fset.card (Set dxs) = length dxs"
```
```   279   by (simp add: length_def member_set distinct_card)
```
```   280
```
```   281 lemma inter_code [code]:
```
```   282   "inf A (Set xs) = Set (filter (Fset.member A) xs)"
```
```   283   "inf A (Coset xs) = foldr Fset.remove xs A"
```
```   284   by (simp_all only: Set_def Coset_def foldr_def inter_project list_of_dlist_filter)
```
```   285
```
```   286 lemma subtract_code [code]:
```
```   287   "A - Set xs = foldr Fset.remove xs A"
```
```   288   "A - Coset xs = Set (filter (Fset.member A) xs)"
```
```   289   by (simp_all only: Set_def Coset_def foldr_def subtract_remove list_of_dlist_filter)
```
```   290
```
```   291 lemma union_code [code]:
```
```   292   "sup (Set xs) A = foldr Fset.insert xs A"
```
```   293   "sup (Coset xs) A = Coset (filter (Not \<circ> Fset.member A) xs)"
```
```   294   by (simp_all only: Set_def Coset_def foldr_def union_insert list_of_dlist_filter)
```
```   295
```
```   296 context complete_lattice
```
```   297 begin
```
```   298
```
```   299 lemma Infimum_code [code]:
```
```   300   "Infimum (Set As) = foldr inf As top"
```
```   301   by (simp only: Set_def Infimum_inf foldr_def inf.commute)
```
```   302
```
```   303 lemma Supremum_code [code]:
```
```   304   "Supremum (Set As) = foldr sup As bot"
```
```   305   by (simp only: Set_def Supremum_sup foldr_def sup.commute)
```
```   306
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```   307 end
```
```   308
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```   309
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```   310 hide_const (open) member fold foldr empty insert remove map filter null member length fold
```
```   311
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```   312 end
```