src/HOL/Library/Dlist.thy
 author haftmann Mon Sep 13 16:43:23 2010 +0200 (2010-09-13 ago) changeset 39380 5a2662c1e44a parent 38857 97775f3e8722 child 39727 5dab9549c80d permissions -rw-r--r--
established emerging canonical names *_eqI and *_eq_iff
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* Lists with elements distinct as canonical example for datatype invariants *}
```
```     4
```
```     5 theory Dlist
```
```     6 imports Main Fset
```
```     7 begin
```
```     8
```
```     9 section {* The type of distinct lists *}
```
```    10
```
```    11 typedef (open) 'a dlist = "{xs::'a list. distinct xs}"
```
```    12   morphisms list_of_dlist Abs_dlist
```
```    13 proof
```
```    14   show "[] \<in> ?dlist" by simp
```
```    15 qed
```
```    16
```
```    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)
```
```    24
```
```    25 text {* Formal, totalized constructor for @{typ "'a dlist"}: *}
```
```    26
```
```    27 definition Dlist :: "'a list \<Rightarrow> 'a dlist" where
```
```    28   "Dlist xs = Abs_dlist (remdups xs)"
```
```    29
```
```    30 lemma distinct_list_of_dlist [simp, intro]:
```
```    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 Dlist_list_of_dlist [simp, code abstype]:
```
```    39   "Dlist (list_of_dlist dxs) = dxs"
```
```    40   by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id)
```
```    41
```
```    42
```
```    43 text {* Fundamental operations: *}
```
```    44
```
```    45 definition empty :: "'a dlist" where
```
```    46   "empty = Dlist []"
```
```    47
```
```    48 definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
```
```    49   "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"
```
```    50
```
```    51 definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
```
```    52   "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"
```
```    53
```
```    54 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where
```
```    55   "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"
```
```    56
```
```    57 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
```
```    58   "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"
```
```    59
```
```    60
```
```    61 text {* Derived operations: *}
```
```    62
```
```    63 definition null :: "'a dlist \<Rightarrow> bool" where
```
```    64   "null dxs = List.null (list_of_dlist dxs)"
```
```    65
```
```    66 definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where
```
```    67   "member dxs = List.member (list_of_dlist dxs)"
```
```    68
```
```    69 definition length :: "'a dlist \<Rightarrow> nat" where
```
```    70   "length dxs = List.length (list_of_dlist dxs)"
```
```    71
```
```    72 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
```
```    73   "fold f dxs = More_List.fold f (list_of_dlist dxs)"
```
```    74
```
```    75 definition foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
```
```    76   "foldr f dxs = List.foldr f (list_of_dlist dxs)"
```
```    77
```
```    78
```
```    79 section {* Executable version obeying invariant *}
```
```    80
```
```    81 lemma list_of_dlist_empty [simp, code abstract]:
```
```    82   "list_of_dlist empty = []"
```
```    83   by (simp add: empty_def)
```
```    84
```
```    85 lemma list_of_dlist_insert [simp, code abstract]:
```
```    86   "list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)"
```
```    87   by (simp add: insert_def)
```
```    88
```
```    89 lemma list_of_dlist_remove [simp, code abstract]:
```
```    90   "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"
```
```    91   by (simp add: remove_def)
```
```    92
```
```    93 lemma list_of_dlist_map [simp, code abstract]:
```
```    94   "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"
```
```    95   by (simp add: map_def)
```
```    96
```
```    97 lemma list_of_dlist_filter [simp, code abstract]:
```
```    98   "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"
```
```    99   by (simp add: filter_def)
```
```   100
```
```   101
```
```   102 text {* Explicit executable conversion *}
```
```   103
```
```   104 definition dlist_of_list [simp]:
```
```   105   "dlist_of_list = Dlist"
```
```   106
```
```   107 lemma [code abstract]:
```
```   108   "list_of_dlist (dlist_of_list xs) = remdups xs"
```
```   109   by simp
```
```   110
```
```   111
```
```   112 text {* Equality *}
```
```   113
```
```   114 instantiation dlist :: (equal) equal
```
```   115 begin
```
```   116
```
```   117 definition "HOL.equal dxs dys \<longleftrightarrow> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)"
```
```   118
```
```   119 instance proof
```
```   120 qed (simp add: equal_dlist_def equal list_of_dlist_inject)
```
```   121
```
```   122 end
```
```   123
```
```   124 lemma [code nbe]:
```
```   125   "HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> True"
```
```   126   by (fact equal_refl)
```
```   127
```
```   128
```
```   129 section {* Induction principle and case distinction *}
```
```   130
```
```   131 lemma dlist_induct [case_names empty insert, induct type: dlist]:
```
```   132   assumes empty: "P empty"
```
```   133   assumes insrt: "\<And>x dxs. \<not> member dxs x \<Longrightarrow> P dxs \<Longrightarrow> P (insert x dxs)"
```
```   134   shows "P dxs"
```
```   135 proof (cases dxs)
```
```   136   case (Abs_dlist xs)
```
```   137   then have "distinct xs" and dxs: "dxs = Dlist xs" by (simp_all add: Dlist_def distinct_remdups_id)
```
```   138   from `distinct xs` have "P (Dlist xs)"
```
```   139   proof (induct xs rule: distinct_induct)
```
```   140     case Nil from empty show ?case by (simp add: empty_def)
```
```   141   next
```
```   142     case (insert x xs)
```
```   143     then have "\<not> member (Dlist xs) x" and "P (Dlist xs)"
```
```   144       by (simp_all add: member_def List.member_def)
```
```   145     with insrt have "P (insert x (Dlist xs))" .
```
```   146     with insert show ?case by (simp add: insert_def distinct_remdups_id)
```
```   147   qed
```
```   148   with dxs show "P dxs" by simp
```
```   149 qed
```
```   150
```
```   151 lemma dlist_case [case_names empty insert, cases type: dlist]:
```
```   152   assumes empty: "dxs = empty \<Longrightarrow> P"
```
```   153   assumes insert: "\<And>x dys. \<not> member dys x \<Longrightarrow> dxs = insert x dys \<Longrightarrow> P"
```
```   154   shows P
```
```   155 proof (cases dxs)
```
```   156   case (Abs_dlist xs)
```
```   157   then have dxs: "dxs = Dlist xs" and distinct: "distinct xs"
```
```   158     by (simp_all add: Dlist_def distinct_remdups_id)
```
```   159   show P proof (cases xs)
```
```   160     case Nil with dxs have "dxs = empty" by (simp add: empty_def)
```
```   161     with empty show P .
```
```   162   next
```
```   163     case (Cons x xs)
```
```   164     with dxs distinct have "\<not> member (Dlist xs) x"
```
```   165       and "dxs = insert x (Dlist xs)"
```
```   166       by (simp_all add: member_def List.member_def insert_def distinct_remdups_id)
```
```   167     with insert show P .
```
```   168   qed
```
```   169 qed
```
```   170
```
```   171
```
```   172 section {* Implementation of sets by distinct lists -- canonical! *}
```
```   173
```
```   174 definition Set :: "'a dlist \<Rightarrow> 'a fset" where
```
```   175   "Set dxs = Fset.Set (list_of_dlist dxs)"
```
```   176
```
```   177 definition Coset :: "'a dlist \<Rightarrow> 'a fset" where
```
```   178   "Coset dxs = Fset.Coset (list_of_dlist dxs)"
```
```   179
```
```   180 code_datatype Set Coset
```
```   181
```
```   182 declare member_code [code del]
```
```   183 declare is_empty_Set [code del]
```
```   184 declare empty_Set [code del]
```
```   185 declare UNIV_Set [code del]
```
```   186 declare insert_Set [code del]
```
```   187 declare remove_Set [code del]
```
```   188 declare compl_Set [code del]
```
```   189 declare compl_Coset [code del]
```
```   190 declare map_Set [code del]
```
```   191 declare filter_Set [code del]
```
```   192 declare forall_Set [code del]
```
```   193 declare exists_Set [code del]
```
```   194 declare card_Set [code del]
```
```   195 declare inter_project [code del]
```
```   196 declare subtract_remove [code del]
```
```   197 declare union_insert [code del]
```
```   198 declare Infimum_inf [code del]
```
```   199 declare Supremum_sup [code del]
```
```   200
```
```   201 lemma Set_Dlist [simp]:
```
```   202   "Set (Dlist xs) = Fset (set xs)"
```
```   203   by (rule fset_eqI) (simp add: Set_def)
```
```   204
```
```   205 lemma Coset_Dlist [simp]:
```
```   206   "Coset (Dlist xs) = Fset (- set xs)"
```
```   207   by (rule fset_eqI) (simp add: Coset_def)
```
```   208
```
```   209 lemma member_Set [simp]:
```
```   210   "Fset.member (Set dxs) = List.member (list_of_dlist dxs)"
```
```   211   by (simp add: Set_def member_set)
```
```   212
```
```   213 lemma member_Coset [simp]:
```
```   214   "Fset.member (Coset dxs) = Not \<circ> List.member (list_of_dlist dxs)"
```
```   215   by (simp add: Coset_def member_set not_set_compl)
```
```   216
```
```   217 lemma Set_dlist_of_list [code]:
```
```   218   "Fset.Set xs = Set (dlist_of_list xs)"
```
```   219   by (rule fset_eqI) simp
```
```   220
```
```   221 lemma Coset_dlist_of_list [code]:
```
```   222   "Fset.Coset xs = Coset (dlist_of_list xs)"
```
```   223   by (rule fset_eqI) simp
```
```   224
```
```   225 lemma is_empty_Set [code]:
```
```   226   "Fset.is_empty (Set dxs) \<longleftrightarrow> null dxs"
```
```   227   by (simp add: null_def List.null_def member_set)
```
```   228
```
```   229 lemma bot_code [code]:
```
```   230   "bot = Set empty"
```
```   231   by (simp add: empty_def)
```
```   232
```
```   233 lemma top_code [code]:
```
```   234   "top = Coset empty"
```
```   235   by (simp add: empty_def)
```
```   236
```
```   237 lemma insert_code [code]:
```
```   238   "Fset.insert x (Set dxs) = Set (insert x dxs)"
```
```   239   "Fset.insert x (Coset dxs) = Coset (remove x dxs)"
```
```   240   by (simp_all add: insert_def remove_def member_set not_set_compl)
```
```   241
```
```   242 lemma remove_code [code]:
```
```   243   "Fset.remove x (Set dxs) = Set (remove x dxs)"
```
```   244   "Fset.remove x (Coset dxs) = Coset (insert x dxs)"
```
```   245   by (auto simp add: insert_def remove_def member_set not_set_compl)
```
```   246
```
```   247 lemma member_code [code]:
```
```   248   "Fset.member (Set dxs) = member dxs"
```
```   249   "Fset.member (Coset dxs) = Not \<circ> member dxs"
```
```   250   by (simp_all add: member_def)
```
```   251
```
```   252 lemma compl_code [code]:
```
```   253   "- Set dxs = Coset dxs"
```
```   254   "- Coset dxs = Set dxs"
```
```   255   by (rule fset_eqI, simp add: member_set not_set_compl)+
```
```   256
```
```   257 lemma map_code [code]:
```
```   258   "Fset.map f (Set dxs) = Set (map f dxs)"
```
```   259   by (rule fset_eqI) (simp add: member_set)
```
```   260
```
```   261 lemma filter_code [code]:
```
```   262   "Fset.filter f (Set dxs) = Set (filter f dxs)"
```
```   263   by (rule fset_eqI) (simp add: member_set)
```
```   264
```
```   265 lemma forall_Set [code]:
```
```   266   "Fset.forall P (Set xs) \<longleftrightarrow> list_all P (list_of_dlist xs)"
```
```   267   by (simp add: member_set list_all_iff)
```
```   268
```
```   269 lemma exists_Set [code]:
```
```   270   "Fset.exists P (Set xs) \<longleftrightarrow> list_ex P (list_of_dlist xs)"
```
```   271   by (simp add: member_set list_ex_iff)
```
```   272
```
```   273 lemma card_code [code]:
```
```   274   "Fset.card (Set dxs) = length dxs"
```
```   275   by (simp add: length_def member_set distinct_card)
```
```   276
```
```   277 lemma inter_code [code]:
```
```   278   "inf A (Set xs) = Set (filter (Fset.member A) xs)"
```
```   279   "inf A (Coset xs) = foldr Fset.remove xs A"
```
```   280   by (simp_all only: Set_def Coset_def foldr_def inter_project list_of_dlist_filter)
```
```   281
```
```   282 lemma subtract_code [code]:
```
```   283   "A - Set xs = foldr Fset.remove xs A"
```
```   284   "A - Coset xs = Set (filter (Fset.member A) xs)"
```
```   285   by (simp_all only: Set_def Coset_def foldr_def subtract_remove list_of_dlist_filter)
```
```   286
```
```   287 lemma union_code [code]:
```
```   288   "sup (Set xs) A = foldr Fset.insert xs A"
```
```   289   "sup (Coset xs) A = Coset (filter (Not \<circ> Fset.member A) xs)"
```
```   290   by (simp_all only: Set_def Coset_def foldr_def union_insert list_of_dlist_filter)
```
```   291
```
```   292 context complete_lattice
```
```   293 begin
```
```   294
```
```   295 lemma Infimum_code [code]:
```
```   296   "Infimum (Set As) = foldr inf As top"
```
```   297   by (simp only: Set_def Infimum_inf foldr_def inf.commute)
```
```   298
```
```   299 lemma Supremum_code [code]:
```
```   300   "Supremum (Set As) = foldr sup As bot"
```
```   301   by (simp only: Set_def Supremum_sup foldr_def sup.commute)
```
```   302
```
```   303 end
```
```   304
```
```   305
```
```   306 hide_const (open) member fold foldr empty insert remove map filter null member length fold
```
```   307
```
```   308 end
```