src/HOL/Quotient_Examples/DList.thy
 author blanchet Wed Feb 12 08:35:57 2014 +0100 (2014-02-12) changeset 55415 05f5fdb8d093 parent 47308 9caab698dbe4 child 58889 5b7a9633cfa8 permissions -rw-r--r--
renamed 'nat_{case,rec}' to '{case,rec}_nat'
```     1 (*  Title:      HOL/Quotient_Examples/DList.thy
```
```     2     Author:     Cezary Kaliszyk, University of Tsukuba
```
```     3
```
```     4 Based on typedef version "Library/Dlist" by Florian Haftmann
```
```     5 and theory morphism version by Maksym Bortin
```
```     6 *)
```
```     7
```
```     8 header {* Lists with distinct elements *}
```
```     9
```
```    10 theory DList
```
```    11 imports "~~/src/HOL/Library/Quotient_List"
```
```    12 begin
```
```    13
```
```    14 text {* Some facts about lists *}
```
```    15
```
```    16 lemma remdups_removeAll_commute[simp]:
```
```    17   "remdups (removeAll x l) = removeAll x (remdups l)"
```
```    18   by (induct l) auto
```
```    19
```
```    20 lemma removeAll_distinct[simp]:
```
```    21   assumes "distinct l"
```
```    22   shows "distinct (removeAll x l)"
```
```    23   using assms by (induct l) simp_all
```
```    24
```
```    25 lemma removeAll_commute:
```
```    26   "removeAll x (removeAll y l) = removeAll y (removeAll x l)"
```
```    27   by (induct l) auto
```
```    28
```
```    29 lemma remdups_hd_notin_tl:
```
```    30   "remdups dl = h # t \<Longrightarrow> h \<notin> set t"
```
```    31   by (induct dl arbitrary: h t)
```
```    32      (case_tac [!] "a \<in> set dl", auto)
```
```    33
```
```    34 lemma remdups_repeat:
```
```    35   "remdups dl = h # t \<Longrightarrow> t = remdups t"
```
```    36   by (induct dl arbitrary: h t, case_tac [!] "a \<in> set dl")
```
```    37      (simp_all, metis remdups_remdups)
```
```    38
```
```    39 lemma remdups_nil_noteq_cons:
```
```    40   "remdups (h # t) \<noteq> remdups []"
```
```    41   "remdups [] \<noteq> remdups (h # t)"
```
```    42   by auto
```
```    43
```
```    44 lemma remdups_eq_map_eq:
```
```    45   assumes "remdups xa = remdups ya"
```
```    46     shows "remdups (map f xa) = remdups (map f ya)"
```
```    47   using assms
```
```    48   by (induct xa ya rule: list_induct2')
```
```    49      (metis (full_types) remdups_nil_noteq_cons(2) remdups_map_remdups)+
```
```    50
```
```    51 lemma remdups_eq_member_eq:
```
```    52   assumes "remdups xa = remdups ya"
```
```    53     shows "List.member xa = List.member ya"
```
```    54   using assms
```
```    55   unfolding fun_eq_iff List.member_def
```
```    56   by (induct xa ya rule: list_induct2')
```
```    57      (metis remdups_nil_noteq_cons set_remdups)+
```
```    58
```
```    59 text {* Setting up the quotient type *}
```
```    60
```
```    61 definition
```
```    62   dlist_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
```
```    63 where
```
```    64   [simp]: "dlist_eq xs ys \<longleftrightarrow> remdups xs = remdups ys"
```
```    65
```
```    66 lemma dlist_eq_reflp:
```
```    67   "reflp dlist_eq"
```
```    68   by (auto intro: reflpI)
```
```    69
```
```    70 lemma dlist_eq_symp:
```
```    71   "symp dlist_eq"
```
```    72   by (auto intro: sympI)
```
```    73
```
```    74 lemma dlist_eq_transp:
```
```    75   "transp dlist_eq"
```
```    76   by (auto intro: transpI)
```
```    77
```
```    78 lemma dlist_eq_equivp:
```
```    79   "equivp dlist_eq"
```
```    80   by (auto intro: equivpI dlist_eq_reflp dlist_eq_symp dlist_eq_transp)
```
```    81
```
```    82 quotient_type
```
```    83   'a dlist = "'a list" / "dlist_eq"
```
```    84   by (rule dlist_eq_equivp)
```
```    85
```
```    86 text {* respectfulness and constant definitions *}
```
```    87
```
```    88 definition [simp]: "card_remdups = length \<circ> remdups"
```
```    89 definition [simp]: "foldr_remdups f xs e = foldr f (remdups xs) e"
```
```    90
```
```    91 quotient_definition empty where "empty :: 'a dlist"
```
```    92   is "Nil" done
```
```    93
```
```    94 quotient_definition insert where "insert :: 'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist"
```
```    95   is "Cons" by (metis (mono_tags) List.insert_def dlist_eq_def remdups.simps(2) set_remdups)
```
```    96
```
```    97 quotient_definition "member :: 'a dlist \<Rightarrow> 'a \<Rightarrow> bool"
```
```    98   is "List.member" by (metis dlist_eq_def remdups_eq_member_eq)
```
```    99
```
```   100 quotient_definition foldr where "foldr :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   101   is "foldr_remdups" by auto
```
```   102
```
```   103 quotient_definition "remove :: 'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist"
```
```   104   is "removeAll" by force
```
```   105
```
```   106 quotient_definition card where "card :: 'a dlist \<Rightarrow> nat"
```
```   107   is "card_remdups" by fastforce
```
```   108
```
```   109 quotient_definition map where "map :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist"
```
```   110   is "List.map :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" by (metis dlist_eq_def remdups_eq_map_eq)
```
```   111
```
```   112 quotient_definition filter where "filter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist"
```
```   113   is "List.filter" by (metis dlist_eq_def remdups_filter)
```
```   114
```
```   115 quotient_definition "list_of_dlist :: 'a dlist \<Rightarrow> 'a list"
```
```   116   is "remdups" by simp
```
```   117
```
```   118 text {* lifted theorems *}
```
```   119
```
```   120 lemma dlist_member_insert:
```
```   121   "member dl x \<Longrightarrow> insert x dl = dl"
```
```   122   by descending (simp add: List.member_def)
```
```   123
```
```   124 lemma dlist_member_insert_eq:
```
```   125   "member (insert y dl) x = (x = y \<or> member dl x)"
```
```   126   by descending (simp add: List.member_def)
```
```   127
```
```   128 lemma dlist_insert_idem:
```
```   129   "insert x (insert x dl) = insert x dl"
```
```   130   by descending simp
```
```   131
```
```   132 lemma dlist_insert_not_empty:
```
```   133   "insert x dl \<noteq> empty"
```
```   134   by descending auto
```
```   135
```
```   136 lemma not_dlist_member_empty:
```
```   137   "\<not> member empty x"
```
```   138   by descending (simp add: List.member_def)
```
```   139
```
```   140 lemma not_dlist_member_remove:
```
```   141   "\<not> member (remove x dl) x"
```
```   142   by descending (simp add: List.member_def)
```
```   143
```
```   144 lemma dlist_in_remove:
```
```   145   "a \<noteq> b \<Longrightarrow> member (remove b dl) a = member dl a"
```
```   146   by descending (simp add: List.member_def)
```
```   147
```
```   148 lemma dlist_not_memb_remove:
```
```   149   "\<not> member dl x \<Longrightarrow> remove x dl = dl"
```
```   150   by descending (simp add: List.member_def)
```
```   151
```
```   152 lemma dlist_no_memb_remove_insert:
```
```   153 "\<not> member dl x \<Longrightarrow> remove x (insert x dl) = dl"
```
```   154   by descending (simp add: List.member_def)
```
```   155
```
```   156 lemma dlist_remove_empty:
```
```   157   "remove x empty = empty"
```
```   158   by descending simp
```
```   159
```
```   160 lemma dlist_remove_insert_commute:
```
```   161   "a \<noteq> b \<Longrightarrow> remove a (insert b dl) = insert b (remove a dl)"
```
```   162   by descending simp
```
```   163
```
```   164 lemma dlist_remove_commute:
```
```   165 "remove a (remove b dl) = remove b (remove a dl)"
```
```   166   by (lifting removeAll_commute)
```
```   167
```
```   168 lemma dlist_foldr_empty:
```
```   169   "foldr f empty e = e"
```
```   170   by descending simp
```
```   171
```
```   172 lemma dlist_no_memb_foldr:
```
```   173   assumes "\<not> member dl x"
```
```   174   shows "foldr f (insert x dl) e = f x (foldr f dl e)"
```
```   175   using assms by descending (simp add: List.member_def)
```
```   176
```
```   177 lemma dlist_foldr_insert_not_memb:
```
```   178   assumes "\<not>member t h"
```
```   179   shows "foldr f (insert h t) e = f h (foldr f t e)"
```
```   180   using assms by descending (simp add: List.member_def)
```
```   181
```
```   182 lemma list_of_dlist_empty[simp]:
```
```   183   "list_of_dlist empty = []"
```
```   184   by descending simp
```
```   185
```
```   186 lemma list_of_dlist_insert[simp]:
```
```   187   "list_of_dlist (insert x xs) =
```
```   188     (if member xs x then (remdups (list_of_dlist xs))
```
```   189     else x # (remdups (list_of_dlist xs)))"
```
```   190   by descending (simp add: List.member_def remdups_remdups)
```
```   191
```
```   192 lemma list_of_dlist_remove[simp]:
```
```   193   "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"
```
```   194   by descending (simp add: distinct_remove1_removeAll)
```
```   195
```
```   196 lemma list_of_dlist_map[simp]:
```
```   197   "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"
```
```   198   by descending (simp add: remdups_map_remdups)
```
```   199
```
```   200 lemma list_of_dlist_filter [simp]:
```
```   201   "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"
```
```   202   by descending (simp add: remdups_filter)
```
```   203
```
```   204 lemma dlist_map_empty:
```
```   205   "map f empty = empty"
```
```   206   by descending simp
```
```   207
```
```   208 lemma dlist_map_insert:
```
```   209   "map f (insert x xs) = insert (f x) (map f xs)"
```
```   210   by descending simp
```
```   211
```
```   212 lemma dlist_eq_iff:
```
```   213   "dxs = dys \<longleftrightarrow> list_of_dlist dxs = list_of_dlist dys"
```
```   214   by descending simp
```
```   215
```
```   216 lemma dlist_eqI:
```
```   217   "list_of_dlist dxs = list_of_dlist dys \<Longrightarrow> dxs = dys"
```
```   218   by (simp add: dlist_eq_iff)
```
```   219
```
```   220 abbreviation
```
```   221   "dlist xs \<equiv> abs_dlist xs"
```
```   222
```
```   223 lemma distinct_list_of_dlist [simp, intro]:
```
```   224   "distinct (list_of_dlist dxs)"
```
```   225   by descending simp
```
```   226
```
```   227 lemma list_of_dlist_dlist [simp]:
```
```   228   "list_of_dlist (dlist xs) = remdups xs"
```
```   229   unfolding list_of_dlist_def map_fun_apply id_def
```
```   230   by (metis Quotient3_rep_abs[OF Quotient3_dlist] dlist_eq_def)
```
```   231
```
```   232 lemma remdups_list_of_dlist [simp]:
```
```   233   "remdups (list_of_dlist dxs) = list_of_dlist dxs"
```
```   234   by simp
```
```   235
```
```   236 lemma dlist_list_of_dlist [simp, code abstype]:
```
```   237   "dlist (list_of_dlist dxs) = dxs"
```
```   238   by (simp add: list_of_dlist_def)
```
```   239   (metis Quotient3_def Quotient3_dlist dlist_eqI list_of_dlist_dlist remdups_list_of_dlist)
```
```   240
```
```   241 lemma dlist_filter_simps:
```
```   242   "filter P empty = empty"
```
```   243   "filter P (insert x xs) = (if P x then insert x (filter P xs) else filter P xs)"
```
```   244   by (lifting filter.simps)
```
```   245
```
```   246 lemma dlist_induct:
```
```   247   assumes "P empty"
```
```   248       and "\<And>a dl. P dl \<Longrightarrow> P (insert a dl)"
```
```   249     shows "P dl"
```
```   250   using assms by descending (drule list.induct, simp)
```
```   251
```
```   252 lemma dlist_induct_stronger:
```
```   253   assumes a1: "P empty"
```
```   254   and     a2: "\<And>x dl. \<lbrakk>\<not>member dl x; P dl\<rbrakk> \<Longrightarrow> P (insert x dl)"
```
```   255   shows "P dl"
```
```   256 proof(induct dl rule: dlist_induct)
```
```   257   show "P empty" using a1 by simp
```
```   258 next
```
```   259   fix x dl
```
```   260   assume "P dl"
```
```   261   then show "P (insert x dl)" using a2
```
```   262     by (cases "member dl x") (simp_all add: dlist_member_insert)
```
```   263 qed
```
```   264
```
```   265 lemma dlist_card_induct:
```
```   266   assumes "\<And>xs. (\<And>ys. card ys < card xs \<Longrightarrow> P ys) \<Longrightarrow> P xs"
```
```   267     shows "P xs"
```
```   268   using assms
```
```   269   by descending (rule measure_induct [of card_remdups], blast)
```
```   270
```
```   271 lemma dlist_cases:
```
```   272   "dl = empty \<or> (\<exists>h t. dl = insert h t \<and> \<not> member t h)"
```
```   273   apply (descending, simp add: List.member_def)
```
```   274   by (metis list.exhaust remdups_eq_nil_iff remdups_hd_notin_tl remdups_repeat)
```
```   275
```
```   276 lemma dlist_exhaust:
```
```   277   assumes "y = empty \<Longrightarrow> P"
```
```   278       and "\<And>a dl. y = insert a dl \<Longrightarrow> P"
```
```   279     shows "P"
```
```   280   using assms by (lifting list.exhaust)
```
```   281
```
```   282 lemma dlist_exhaust_stronger:
```
```   283   assumes "y = empty \<Longrightarrow> P"
```
```   284       and "\<And>a dl. y = insert a dl \<Longrightarrow> \<not> member dl a \<Longrightarrow> P"
```
```   285     shows "P"
```
```   286   using assms by (metis dlist_cases)
```
```   287
```
```   288 end
```