src/HOL/Library/AList.thy
 author haftmann Fri Nov 01 18:51:14 2013 +0100 (2013-11-01) changeset 54230 b1d955791529 parent 47397 d654c73e4b12 child 55414 eab03e9cee8a permissions -rw-r--r--
more simplification rules on unary and binary minus
```     1 (*  Title:      HOL/Library/AList.thy
```
```     2     Author:     Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Implementation of Association Lists *}
```
```     6
```
```     7 theory AList
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 text {*
```
```    12   The operations preserve distinctness of keys and
```
```    13   function @{term "clearjunk"} distributes over them. Since
```
```    14   @{term clearjunk} enforces distinctness of keys it can be used
```
```    15   to establish the invariant, e.g. for inductive proofs.
```
```    16 *}
```
```    17
```
```    18 subsection {* @{text update} and @{text updates} *}
```
```    19
```
```    20 primrec update :: "'key \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
```
```    21     "update k v [] = [(k, v)]"
```
```    22   | "update k v (p#ps) = (if fst p = k then (k, v) # ps else p # update k v ps)"
```
```    23
```
```    24 lemma update_conv': "map_of (update k v al)  = (map_of al)(k\<mapsto>v)"
```
```    25   by (induct al) (auto simp add: fun_eq_iff)
```
```    26
```
```    27 corollary update_conv: "map_of (update k v al) k' = ((map_of al)(k\<mapsto>v)) k'"
```
```    28   by (simp add: update_conv')
```
```    29
```
```    30 lemma dom_update: "fst ` set (update k v al) = {k} \<union> fst ` set al"
```
```    31   by (induct al) auto
```
```    32
```
```    33 lemma update_keys:
```
```    34   "map fst (update k v al) =
```
```    35     (if k \<in> set (map fst al) then map fst al else map fst al @ [k])"
```
```    36   by (induct al) simp_all
```
```    37
```
```    38 lemma distinct_update:
```
```    39   assumes "distinct (map fst al)"
```
```    40   shows "distinct (map fst (update k v al))"
```
```    41   using assms by (simp add: update_keys)
```
```    42
```
```    43 lemma update_filter:
```
```    44   "a\<noteq>k \<Longrightarrow> update k v [q\<leftarrow>ps . fst q \<noteq> a] = [q\<leftarrow>update k v ps . fst q \<noteq> a]"
```
```    45   by (induct ps) auto
```
```    46
```
```    47 lemma update_triv: "map_of al k = Some v \<Longrightarrow> update k v al = al"
```
```    48   by (induct al) auto
```
```    49
```
```    50 lemma update_nonempty [simp]: "update k v al \<noteq> []"
```
```    51   by (induct al) auto
```
```    52
```
```    53 lemma update_eqD: "update k v al = update k v' al' \<Longrightarrow> v = v'"
```
```    54 proof (induct al arbitrary: al')
```
```    55   case Nil thus ?case
```
```    56     by (cases al') (auto split: split_if_asm)
```
```    57 next
```
```    58   case Cons thus ?case
```
```    59     by (cases al') (auto split: split_if_asm)
```
```    60 qed
```
```    61
```
```    62 lemma update_last [simp]: "update k v (update k v' al) = update k v al"
```
```    63   by (induct al) auto
```
```    64
```
```    65 text {* Note that the lists are not necessarily the same:
```
```    66         @{term "update k v (update k' v' []) = [(k', v'), (k, v)]"} and
```
```    67         @{term "update k' v' (update k v []) = [(k, v), (k', v')]"}.*}
```
```    68 lemma update_swap: "k\<noteq>k'
```
```    69   \<Longrightarrow> map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"
```
```    70   by (simp add: update_conv' fun_eq_iff)
```
```    71
```
```    72 lemma update_Some_unfold:
```
```    73   "map_of (update k v al) x = Some y \<longleftrightarrow>
```
```    74     x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y"
```
```    75   by (simp add: update_conv' map_upd_Some_unfold)
```
```    76
```
```    77 lemma image_update [simp]:
```
```    78   "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A"
```
```    79   by (simp add: update_conv')
```
```    80
```
```    81 definition updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
```
```    82   "updates ks vs = fold (prod_case update) (zip ks vs)"
```
```    83
```
```    84 lemma updates_simps [simp]:
```
```    85   "updates [] vs ps = ps"
```
```    86   "updates ks [] ps = ps"
```
```    87   "updates (k#ks) (v#vs) ps = updates ks vs (update k v ps)"
```
```    88   by (simp_all add: updates_def)
```
```    89
```
```    90 lemma updates_key_simp [simp]:
```
```    91   "updates (k # ks) vs ps =
```
```    92     (case vs of [] \<Rightarrow> ps | v # vs \<Rightarrow> updates ks vs (update k v ps))"
```
```    93   by (cases vs) simp_all
```
```    94
```
```    95 lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\<mapsto>]vs)"
```
```    96 proof -
```
```    97   have "map_of \<circ> fold (prod_case update) (zip ks vs) =
```
```    98     fold (\<lambda>(k, v) f. f(k \<mapsto> v)) (zip ks vs) \<circ> map_of"
```
```    99     by (rule fold_commute) (auto simp add: fun_eq_iff update_conv')
```
```   100   then show ?thesis by (auto simp add: updates_def fun_eq_iff map_upds_fold_map_upd foldl_conv_fold split_def)
```
```   101 qed
```
```   102
```
```   103 lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\<mapsto>]vs)) k"
```
```   104   by (simp add: updates_conv')
```
```   105
```
```   106 lemma distinct_updates:
```
```   107   assumes "distinct (map fst al)"
```
```   108   shows "distinct (map fst (updates ks vs al))"
```
```   109 proof -
```
```   110   have "distinct (fold
```
```   111        (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k])
```
```   112        (zip ks vs) (map fst al))"
```
```   113     by (rule fold_invariant [of "zip ks vs" "\<lambda>_. True"]) (auto intro: assms)
```
```   114   moreover have "map fst \<circ> fold (prod_case update) (zip ks vs) =
```
```   115     fold (\<lambda>(k, v) al. if k \<in> set al then al else al @ [k]) (zip ks vs) \<circ> map fst"
```
```   116     by (rule fold_commute) (simp add: update_keys split_def prod_case_beta comp_def)
```
```   117   ultimately show ?thesis by (simp add: updates_def fun_eq_iff)
```
```   118 qed
```
```   119
```
```   120 lemma updates_append1[simp]: "size ks < size vs \<Longrightarrow>
```
```   121   updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)"
```
```   122   by (induct ks arbitrary: vs al) (auto split: list.splits)
```
```   123
```
```   124 lemma updates_list_update_drop[simp]:
```
```   125  "\<lbrakk>size ks \<le> i; i < size vs\<rbrakk>
```
```   126    \<Longrightarrow> updates ks (vs[i:=v]) al = updates ks vs al"
```
```   127   by (induct ks arbitrary: al vs i) (auto split:list.splits nat.splits)
```
```   128
```
```   129 lemma update_updates_conv_if: "
```
```   130  map_of (updates xs ys (update x y al)) =
```
```   131  map_of (if x \<in>  set(take (length ys) xs) then updates xs ys al
```
```   132                                   else (update x y (updates xs ys al)))"
```
```   133   by (simp add: updates_conv' update_conv' map_upd_upds_conv_if)
```
```   134
```
```   135 lemma updates_twist [simp]:
```
```   136  "k \<notin> set ks \<Longrightarrow>
```
```   137   map_of (updates ks vs (update k v al)) = map_of (update k v (updates ks vs al))"
```
```   138   by (simp add: updates_conv' update_conv')
```
```   139
```
```   140 lemma updates_apply_notin[simp]:
```
```   141  "k \<notin> set ks ==> map_of (updates ks vs al) k = map_of al k"
```
```   142   by (simp add: updates_conv)
```
```   143
```
```   144 lemma updates_append_drop[simp]:
```
```   145   "size xs = size ys \<Longrightarrow> updates (xs@zs) ys al = updates xs ys al"
```
```   146   by (induct xs arbitrary: ys al) (auto split: list.splits)
```
```   147
```
```   148 lemma updates_append2_drop[simp]:
```
```   149   "size xs = size ys \<Longrightarrow> updates xs (ys@zs) al = updates xs ys al"
```
```   150   by (induct xs arbitrary: ys al) (auto split: list.splits)
```
```   151
```
```   152
```
```   153 subsection {* @{text delete} *}
```
```   154
```
```   155 definition delete :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
```
```   156   delete_eq: "delete k = filter (\<lambda>(k', _). k \<noteq> k')"
```
```   157
```
```   158 lemma delete_simps [simp]:
```
```   159   "delete k [] = []"
```
```   160   "delete k (p#ps) = (if fst p = k then delete k ps else p # delete k ps)"
```
```   161   by (auto simp add: delete_eq)
```
```   162
```
```   163 lemma delete_conv': "map_of (delete k al) = (map_of al)(k := None)"
```
```   164   by (induct al) (auto simp add: fun_eq_iff)
```
```   165
```
```   166 corollary delete_conv: "map_of (delete k al) k' = ((map_of al)(k := None)) k'"
```
```   167   by (simp add: delete_conv')
```
```   168
```
```   169 lemma delete_keys:
```
```   170   "map fst (delete k al) = removeAll k (map fst al)"
```
```   171   by (simp add: delete_eq removeAll_filter_not_eq filter_map split_def comp_def)
```
```   172
```
```   173 lemma distinct_delete:
```
```   174   assumes "distinct (map fst al)"
```
```   175   shows "distinct (map fst (delete k al))"
```
```   176   using assms by (simp add: delete_keys distinct_removeAll)
```
```   177
```
```   178 lemma delete_id [simp]: "k \<notin> fst ` set al \<Longrightarrow> delete k al = al"
```
```   179   by (auto simp add: image_iff delete_eq filter_id_conv)
```
```   180
```
```   181 lemma delete_idem: "delete k (delete k al) = delete k al"
```
```   182   by (simp add: delete_eq)
```
```   183
```
```   184 lemma map_of_delete [simp]:
```
```   185     "k' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'"
```
```   186   by (simp add: delete_conv')
```
```   187
```
```   188 lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)"
```
```   189   by (auto simp add: delete_eq)
```
```   190
```
```   191 lemma dom_delete_subset: "fst ` set (delete k al) \<subseteq> fst ` set al"
```
```   192   by (auto simp add: delete_eq)
```
```   193
```
```   194 lemma delete_update_same:
```
```   195   "delete k (update k v al) = delete k al"
```
```   196   by (induct al) simp_all
```
```   197
```
```   198 lemma delete_update:
```
```   199   "k \<noteq> l \<Longrightarrow> delete l (update k v al) = update k v (delete l al)"
```
```   200   by (induct al) simp_all
```
```   201
```
```   202 lemma delete_twist: "delete x (delete y al) = delete y (delete x al)"
```
```   203   by (simp add: delete_eq conj_commute)
```
```   204
```
```   205 lemma length_delete_le: "length (delete k al) \<le> length al"
```
```   206   by (simp add: delete_eq)
```
```   207
```
```   208
```
```   209 subsection {* @{text restrict} *}
```
```   210
```
```   211 definition restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
```
```   212   restrict_eq: "restrict A = filter (\<lambda>(k, v). k \<in> A)"
```
```   213
```
```   214 lemma restr_simps [simp]:
```
```   215   "restrict A [] = []"
```
```   216   "restrict A (p#ps) = (if fst p \<in> A then p # restrict A ps else restrict A ps)"
```
```   217   by (auto simp add: restrict_eq)
```
```   218
```
```   219 lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)"
```
```   220 proof
```
```   221   fix k
```
```   222   show "map_of (restrict A al) k = ((map_of al)|` A) k"
```
```   223     by (induct al) (simp, cases "k \<in> A", auto)
```
```   224 qed
```
```   225
```
```   226 corollary restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k"
```
```   227   by (simp add: restr_conv')
```
```   228
```
```   229 lemma distinct_restr:
```
```   230   "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))"
```
```   231   by (induct al) (auto simp add: restrict_eq)
```
```   232
```
```   233 lemma restr_empty [simp]:
```
```   234   "restrict {} al = []"
```
```   235   "restrict A [] = []"
```
```   236   by (induct al) (auto simp add: restrict_eq)
```
```   237
```
```   238 lemma restr_in [simp]: "x \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x"
```
```   239   by (simp add: restr_conv')
```
```   240
```
```   241 lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None"
```
```   242   by (simp add: restr_conv')
```
```   243
```
```   244 lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> A"
```
```   245   by (induct al) (auto simp add: restrict_eq)
```
```   246
```
```   247 lemma restr_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) al"
```
```   248   by (induct al) (auto simp add: restrict_eq)
```
```   249
```
```   250 lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al"
```
```   251   by (induct al) (auto simp add: restrict_eq)
```
```   252
```
```   253 lemma restr_update[simp]:
```
```   254  "map_of (restrict D (update x y al)) =
```
```   255   map_of ((if x \<in> D then (update x y (restrict (D-{x}) al)) else restrict D al))"
```
```   256   by (simp add: restr_conv' update_conv')
```
```   257
```
```   258 lemma restr_delete [simp]:
```
```   259   "(delete x (restrict D al)) =
```
```   260     (if x \<in> D then restrict (D - {x}) al else restrict D al)"
```
```   261 apply (simp add: delete_eq restrict_eq)
```
```   262 apply (auto simp add: split_def)
```
```   263 proof -
```
```   264   have "\<And>y. y \<noteq> x \<longleftrightarrow> x \<noteq> y" by auto
```
```   265   then show "[p \<leftarrow> al. fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al. fst p \<in> D \<and> fst p \<noteq> x]"
```
```   266     by simp
```
```   267   assume "x \<notin> D"
```
```   268   then have "\<And>y. y \<in> D \<longleftrightarrow> y \<in> D \<and> x \<noteq> y" by auto
```
```   269   then show "[p \<leftarrow> al . fst p \<in> D \<and> x \<noteq> fst p] = [p \<leftarrow> al . fst p \<in> D]"
```
```   270     by simp
```
```   271 qed
```
```   272
```
```   273 lemma update_restr:
```
```   274  "map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
```
```   275   by (simp add: update_conv' restr_conv') (rule fun_upd_restrict)
```
```   276
```
```   277 lemma update_restr_conv [simp]:
```
```   278  "x \<in> D \<Longrightarrow>
```
```   279  map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
```
```   280   by (simp add: update_conv' restr_conv')
```
```   281
```
```   282 lemma restr_updates [simp]: "
```
```   283  \<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk>
```
```   284  \<Longrightarrow> map_of (restrict D (updates xs ys al)) =
```
```   285      map_of (updates xs ys (restrict (D - set xs) al))"
```
```   286   by (simp add: updates_conv' restr_conv')
```
```   287
```
```   288 lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)"
```
```   289   by (induct ps) auto
```
```   290
```
```   291
```
```   292 subsection {* @{text clearjunk} *}
```
```   293
```
```   294 function clearjunk  :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
```
```   295     "clearjunk [] = []"
```
```   296   | "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
```
```   297   by pat_completeness auto
```
```   298 termination by (relation "measure length")
```
```   299   (simp_all add: less_Suc_eq_le length_delete_le)
```
```   300
```
```   301 lemma map_of_clearjunk:
```
```   302   "map_of (clearjunk al) = map_of al"
```
```   303   by (induct al rule: clearjunk.induct)
```
```   304     (simp_all add: fun_eq_iff)
```
```   305
```
```   306 lemma clearjunk_keys_set:
```
```   307   "set (map fst (clearjunk al)) = set (map fst al)"
```
```   308   by (induct al rule: clearjunk.induct)
```
```   309     (simp_all add: delete_keys)
```
```   310
```
```   311 lemma dom_clearjunk:
```
```   312   "fst ` set (clearjunk al) = fst ` set al"
```
```   313   using clearjunk_keys_set by simp
```
```   314
```
```   315 lemma distinct_clearjunk [simp]:
```
```   316   "distinct (map fst (clearjunk al))"
```
```   317   by (induct al rule: clearjunk.induct)
```
```   318     (simp_all del: set_map add: clearjunk_keys_set delete_keys)
```
```   319
```
```   320 lemma ran_clearjunk:
```
```   321   "ran (map_of (clearjunk al)) = ran (map_of al)"
```
```   322   by (simp add: map_of_clearjunk)
```
```   323
```
```   324 lemma ran_map_of:
```
```   325   "ran (map_of al) = snd ` set (clearjunk al)"
```
```   326 proof -
```
```   327   have "ran (map_of al) = ran (map_of (clearjunk al))"
```
```   328     by (simp add: ran_clearjunk)
```
```   329   also have "\<dots> = snd ` set (clearjunk al)"
```
```   330     by (simp add: ran_distinct)
```
```   331   finally show ?thesis .
```
```   332 qed
```
```   333
```
```   334 lemma clearjunk_update:
```
```   335   "clearjunk (update k v al) = update k v (clearjunk al)"
```
```   336   by (induct al rule: clearjunk.induct)
```
```   337     (simp_all add: delete_update)
```
```   338
```
```   339 lemma clearjunk_updates:
```
```   340   "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
```
```   341 proof -
```
```   342   have "clearjunk \<circ> fold (prod_case update) (zip ks vs) =
```
```   343     fold (prod_case update) (zip ks vs) \<circ> clearjunk"
```
```   344     by (rule fold_commute) (simp add: clearjunk_update prod_case_beta o_def)
```
```   345   then show ?thesis by (simp add: updates_def fun_eq_iff)
```
```   346 qed
```
```   347
```
```   348 lemma clearjunk_delete:
```
```   349   "clearjunk (delete x al) = delete x (clearjunk al)"
```
```   350   by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist)
```
```   351
```
```   352 lemma clearjunk_restrict:
```
```   353  "clearjunk (restrict A al) = restrict A (clearjunk al)"
```
```   354   by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist)
```
```   355
```
```   356 lemma distinct_clearjunk_id [simp]:
```
```   357   "distinct (map fst al) \<Longrightarrow> clearjunk al = al"
```
```   358   by (induct al rule: clearjunk.induct) auto
```
```   359
```
```   360 lemma clearjunk_idem:
```
```   361   "clearjunk (clearjunk al) = clearjunk al"
```
```   362   by simp
```
```   363
```
```   364 lemma length_clearjunk:
```
```   365   "length (clearjunk al) \<le> length al"
```
```   366 proof (induct al rule: clearjunk.induct [case_names Nil Cons])
```
```   367   case Nil then show ?case by simp
```
```   368 next
```
```   369   case (Cons kv al)
```
```   370   moreover have "length (delete (fst kv) al) \<le> length al" by (fact length_delete_le)
```
```   371   ultimately have "length (clearjunk (delete (fst kv) al)) \<le> length al" by (rule order_trans)
```
```   372   then show ?case by simp
```
```   373 qed
```
```   374
```
```   375 lemma delete_map:
```
```   376   assumes "\<And>kv. fst (f kv) = fst kv"
```
```   377   shows "delete k (map f ps) = map f (delete k ps)"
```
```   378   by (simp add: delete_eq filter_map comp_def split_def assms)
```
```   379
```
```   380 lemma clearjunk_map:
```
```   381   assumes "\<And>kv. fst (f kv) = fst kv"
```
```   382   shows "clearjunk (map f ps) = map f (clearjunk ps)"
```
```   383   by (induct ps rule: clearjunk.induct [case_names Nil Cons])
```
```   384     (simp_all add: clearjunk_delete delete_map assms)
```
```   385
```
```   386
```
```   387 subsection {* @{text map_ran} *}
```
```   388
```
```   389 definition map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
```
```   390   "map_ran f = map (\<lambda>(k, v). (k, f k v))"
```
```   391
```
```   392 lemma map_ran_simps [simp]:
```
```   393   "map_ran f [] = []"
```
```   394   "map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps"
```
```   395   by (simp_all add: map_ran_def)
```
```   396
```
```   397 lemma dom_map_ran:
```
```   398   "fst ` set (map_ran f al) = fst ` set al"
```
```   399   by (simp add: map_ran_def image_image split_def)
```
```   400
```
```   401 lemma map_ran_conv:
```
```   402   "map_of (map_ran f al) k = Option.map (f k) (map_of al k)"
```
```   403   by (induct al) auto
```
```   404
```
```   405 lemma distinct_map_ran:
```
```   406   "distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))"
```
```   407   by (simp add: map_ran_def split_def comp_def)
```
```   408
```
```   409 lemma map_ran_filter:
```
```   410   "map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]"
```
```   411   by (simp add: map_ran_def filter_map split_def comp_def)
```
```   412
```
```   413 lemma clearjunk_map_ran:
```
```   414   "clearjunk (map_ran f al) = map_ran f (clearjunk al)"
```
```   415   by (simp add: map_ran_def split_def clearjunk_map)
```
```   416
```
```   417
```
```   418 subsection {* @{text merge} *}
```
```   419
```
```   420 definition merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" where
```
```   421   "merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs"
```
```   422
```
```   423 lemma merge_simps [simp]:
```
```   424   "merge qs [] = qs"
```
```   425   "merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)"
```
```   426   by (simp_all add: merge_def split_def)
```
```   427
```
```   428 lemma merge_updates:
```
```   429   "merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs"
```
```   430   by (simp add: merge_def updates_def foldr_conv_fold zip_rev zip_map_fst_snd)
```
```   431
```
```   432 lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys"
```
```   433   by (induct ys arbitrary: xs) (auto simp add: dom_update)
```
```   434
```
```   435 lemma distinct_merge:
```
```   436   assumes "distinct (map fst xs)"
```
```   437   shows "distinct (map fst (merge xs ys))"
```
```   438 using assms by (simp add: merge_updates distinct_updates)
```
```   439
```
```   440 lemma clearjunk_merge:
```
```   441   "clearjunk (merge xs ys) = merge (clearjunk xs) ys"
```
```   442   by (simp add: merge_updates clearjunk_updates)
```
```   443
```
```   444 lemma merge_conv':
```
```   445   "map_of (merge xs ys) = map_of xs ++ map_of ys"
```
```   446 proof -
```
```   447   have "map_of \<circ> fold (prod_case update) (rev ys) =
```
```   448     fold (\<lambda>(k, v) m. m(k \<mapsto> v)) (rev ys) \<circ> map_of"
```
```   449     by (rule fold_commute) (simp add: update_conv' prod_case_beta split_def fun_eq_iff)
```
```   450   then show ?thesis
```
```   451     by (simp add: merge_def map_add_map_of_foldr foldr_conv_fold fun_eq_iff)
```
```   452 qed
```
```   453
```
```   454 corollary merge_conv:
```
```   455   "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"
```
```   456   by (simp add: merge_conv')
```
```   457
```
```   458 lemma merge_empty: "map_of (merge [] ys) = map_of ys"
```
```   459   by (simp add: merge_conv')
```
```   460
```
```   461 lemma merge_assoc[simp]: "map_of (merge m1 (merge m2 m3)) =
```
```   462                            map_of (merge (merge m1 m2) m3)"
```
```   463   by (simp add: merge_conv')
```
```   464
```
```   465 lemma merge_Some_iff:
```
```   466  "(map_of (merge m n) k = Some x) =
```
```   467   (map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x)"
```
```   468   by (simp add: merge_conv' map_add_Some_iff)
```
```   469
```
```   470 lemmas merge_SomeD [dest!] = merge_Some_iff [THEN iffD1]
```
```   471
```
```   472 lemma merge_find_right[simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v"
```
```   473   by (simp add: merge_conv')
```
```   474
```
```   475 lemma merge_None [iff]:
```
```   476   "(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)"
```
```   477   by (simp add: merge_conv')
```
```   478
```
```   479 lemma merge_upd[simp]:
```
```   480   "map_of (merge m (update k v n)) = map_of (update k v (merge m n))"
```
```   481   by (simp add: update_conv' merge_conv')
```
```   482
```
```   483 lemma merge_updatess[simp]:
```
```   484   "map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))"
```
```   485   by (simp add: updates_conv' merge_conv')
```
```   486
```
```   487 lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)"
```
```   488   by (simp add: merge_conv')
```
```   489
```
```   490
```
```   491 subsection {* @{text compose} *}
```
```   492
```
```   493 function compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" where
```
```   494     "compose [] ys = []"
```
```   495   | "compose (x#xs) ys = (case map_of ys (snd x)
```
```   496        of None \<Rightarrow> compose (delete (fst x) xs) ys
```
```   497         | Some v \<Rightarrow> (fst x, v) # compose xs ys)"
```
```   498   by pat_completeness auto
```
```   499 termination by (relation "measure (length \<circ> fst)")
```
```   500   (simp_all add: less_Suc_eq_le length_delete_le)
```
```   501
```
```   502 lemma compose_first_None [simp]:
```
```   503   assumes "map_of xs k = None"
```
```   504   shows "map_of (compose xs ys) k = None"
```
```   505 using assms by (induct xs ys rule: compose.induct)
```
```   506   (auto split: option.splits split_if_asm)
```
```   507
```
```   508 lemma compose_conv:
```
```   509   shows "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
```
```   510 proof (induct xs ys rule: compose.induct)
```
```   511   case 1 then show ?case by simp
```
```   512 next
```
```   513   case (2 x xs ys) show ?case
```
```   514   proof (cases "map_of ys (snd x)")
```
```   515     case None with 2
```
```   516     have hyp: "map_of (compose (delete (fst x) xs) ys) k =
```
```   517                (map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k"
```
```   518       by simp
```
```   519     show ?thesis
```
```   520     proof (cases "fst x = k")
```
```   521       case True
```
```   522       from True delete_notin_dom [of k xs]
```
```   523       have "map_of (delete (fst x) xs) k = None"
```
```   524         by (simp add: map_of_eq_None_iff)
```
```   525       with hyp show ?thesis
```
```   526         using True None
```
```   527         by simp
```
```   528     next
```
```   529       case False
```
```   530       from False have "map_of (delete (fst x) xs) k = map_of xs k"
```
```   531         by simp
```
```   532       with hyp show ?thesis
```
```   533         using False None
```
```   534         by (simp add: map_comp_def)
```
```   535     qed
```
```   536   next
```
```   537     case (Some v)
```
```   538     with 2
```
```   539     have "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
```
```   540       by simp
```
```   541     with Some show ?thesis
```
```   542       by (auto simp add: map_comp_def)
```
```   543   qed
```
```   544 qed
```
```   545
```
```   546 lemma compose_conv':
```
```   547   shows "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)"
```
```   548   by (rule ext) (rule compose_conv)
```
```   549
```
```   550 lemma compose_first_Some [simp]:
```
```   551   assumes "map_of xs k = Some v"
```
```   552   shows "map_of (compose xs ys) k = map_of ys v"
```
```   553 using assms by (simp add: compose_conv)
```
```   554
```
```   555 lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs"
```
```   556 proof (induct xs ys rule: compose.induct)
```
```   557   case 1 thus ?case by simp
```
```   558 next
```
```   559   case (2 x xs ys)
```
```   560   show ?case
```
```   561   proof (cases "map_of ys (snd x)")
```
```   562     case None
```
```   563     with "2.hyps"
```
```   564     have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)"
```
```   565       by simp
```
```   566     also
```
```   567     have "\<dots> \<subseteq> fst ` set xs"
```
```   568       by (rule dom_delete_subset)
```
```   569     finally show ?thesis
```
```   570       using None
```
```   571       by auto
```
```   572   next
```
```   573     case (Some v)
```
```   574     with "2.hyps"
```
```   575     have "fst ` set (compose xs ys) \<subseteq> fst ` set xs"
```
```   576       by simp
```
```   577     with Some show ?thesis
```
```   578       by auto
```
```   579   qed
```
```   580 qed
```
```   581
```
```   582 lemma distinct_compose:
```
```   583  assumes "distinct (map fst xs)"
```
```   584  shows "distinct (map fst (compose xs ys))"
```
```   585 using assms
```
```   586 proof (induct xs ys rule: compose.induct)
```
```   587   case 1 thus ?case by simp
```
```   588 next
```
```   589   case (2 x xs ys)
```
```   590   show ?case
```
```   591   proof (cases "map_of ys (snd x)")
```
```   592     case None
```
```   593     with 2 show ?thesis by simp
```
```   594   next
```
```   595     case (Some v)
```
```   596     with 2 dom_compose [of xs ys] show ?thesis
```
```   597       by (auto)
```
```   598   qed
```
```   599 qed
```
```   600
```
```   601 lemma compose_delete_twist: "(compose (delete k xs) ys) = delete k (compose xs ys)"
```
```   602 proof (induct xs ys rule: compose.induct)
```
```   603   case 1 thus ?case by simp
```
```   604 next
```
```   605   case (2 x xs ys)
```
```   606   show ?case
```
```   607   proof (cases "map_of ys (snd x)")
```
```   608     case None
```
```   609     with 2 have
```
```   610       hyp: "compose (delete k (delete (fst x) xs)) ys =
```
```   611             delete k (compose (delete (fst x) xs) ys)"
```
```   612       by simp
```
```   613     show ?thesis
```
```   614     proof (cases "fst x = k")
```
```   615       case True
```
```   616       with None hyp
```
```   617       show ?thesis
```
```   618         by (simp add: delete_idem)
```
```   619     next
```
```   620       case False
```
```   621       from None False hyp
```
```   622       show ?thesis
```
```   623         by (simp add: delete_twist)
```
```   624     qed
```
```   625   next
```
```   626     case (Some v)
```
```   627     with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" by simp
```
```   628     with Some show ?thesis
```
```   629       by simp
```
```   630   qed
```
```   631 qed
```
```   632
```
```   633 lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys"
```
```   634   by (induct xs ys rule: compose.induct)
```
```   635      (auto simp add: map_of_clearjunk split: option.splits)
```
```   636
```
```   637 lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys"
```
```   638   by (induct xs rule: clearjunk.induct)
```
```   639      (auto split: option.splits simp add: clearjunk_delete delete_idem
```
```   640                compose_delete_twist)
```
```   641
```
```   642 lemma compose_empty [simp]:
```
```   643  "compose xs [] = []"
```
```   644   by (induct xs) (auto simp add: compose_delete_twist)
```
```   645
```
```   646 lemma compose_Some_iff:
```
```   647   "(map_of (compose xs ys) k = Some v) =
```
```   648      (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = Some v)"
```
```   649   by (simp add: compose_conv map_comp_Some_iff)
```
```   650
```
```   651 lemma map_comp_None_iff:
```
```   652   "(map_of (compose xs ys) k = None) =
```
```   653     (map_of xs k = None \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None)) "
```
```   654   by (simp add: compose_conv map_comp_None_iff)
```
```   655
```
```   656 subsection {* @{text map_entry} *}
```
```   657
```
```   658 fun map_entry :: "'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
```
```   659 where
```
```   660   "map_entry k f [] = []"
```
```   661 | "map_entry k f (p # ps) = (if fst p = k then (k, f (snd p)) # ps else p # map_entry k f ps)"
```
```   662
```
```   663 lemma map_of_map_entry:
```
```   664   "map_of (map_entry k f xs) = (map_of xs)(k := case map_of xs k of None => None | Some v' => Some (f v'))"
```
```   665 by (induct xs) auto
```
```   666
```
```   667 lemma dom_map_entry:
```
```   668   "fst ` set (map_entry k f xs) = fst ` set xs"
```
```   669 by (induct xs) auto
```
```   670
```
```   671 lemma distinct_map_entry:
```
```   672   assumes "distinct (map fst xs)"
```
```   673   shows "distinct (map fst (map_entry k f xs))"
```
```   674 using assms by (induct xs) (auto simp add: dom_map_entry)
```
```   675
```
```   676 subsection {* @{text map_default} *}
```
```   677
```
```   678 fun map_default :: "'key \<Rightarrow> 'val \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
```
```   679 where
```
```   680   "map_default k v f [] = [(k, v)]"
```
```   681 | "map_default k v f (p # ps) = (if fst p = k then (k, f (snd p)) # ps else p # map_default k v f ps)"
```
```   682
```
```   683 lemma map_of_map_default:
```
```   684   "map_of (map_default k v f xs) = (map_of xs)(k := case map_of xs k of None => Some v | Some v' => Some (f v'))"
```
```   685 by (induct xs) auto
```
```   686
```
```   687 lemma dom_map_default:
```
```   688   "fst ` set (map_default k v f xs) = insert k (fst ` set xs)"
```
```   689 by (induct xs) auto
```
```   690
```
```   691 lemma distinct_map_default:
```
```   692   assumes "distinct (map fst xs)"
```
```   693   shows "distinct (map fst (map_default k v f xs))"
```
```   694 using assms by (induct xs) (auto simp add: dom_map_default)
```
```   695
```
```   696 hide_const (open) update updates delete restrict clearjunk merge compose map_entry
```
```   697
```
```   698 end
```