src/HOL/Library/DAList_Multiset.thy
author Andreas Lochbihler
Fri Sep 20 10:09:16 2013 +0200 (2013-09-20)
changeset 53745 788730ab7da4
parent 51623 1194b438426a
child 55808 488c3e8282c8
permissions -rw-r--r--
prefer Code.abort over code_abort
     1 (*  Title:      HOL/Library/DAList_Multiset.thy
     2     Author:     Lukas Bulwahn, TU Muenchen
     3 *)
     4 
     5 header {* Multisets partially implemented by association lists *}
     6 
     7 theory DAList_Multiset
     8 imports Multiset DAList
     9 begin
    10 
    11 text {* Delete prexisting code equations *}
    12 
    13 lemma [code, code del]:
    14   "{#} = {#}"
    15   ..
    16 
    17 lemma [code, code del]:
    18   "single = single"
    19   ..
    20 
    21 lemma [code, code del]:
    22   "plus = (plus :: 'a multiset \<Rightarrow> _)"
    23   ..
    24 
    25 lemma [code, code del]:
    26   "minus = (minus :: 'a multiset \<Rightarrow> _)"
    27   ..
    28 
    29 lemma [code, code del]:
    30   "inf = (inf :: 'a multiset \<Rightarrow> _)"
    31   ..
    32 
    33 lemma [code, code del]:
    34   "sup = (sup :: 'a multiset \<Rightarrow> _)"
    35   ..
    36 
    37 lemma [code, code del]:
    38   "image_mset = image_mset"
    39   ..
    40 
    41 lemma [code, code del]:
    42   "Multiset.filter = Multiset.filter"
    43   ..
    44 
    45 lemma [code, code del]:
    46   "count = count"
    47   ..
    48 
    49 lemma [code, code del]:
    50   "mcard = mcard"
    51   ..
    52 
    53 lemma [code, code del]:
    54   "msetsum = msetsum"
    55   ..
    56 
    57 lemma [code, code del]:
    58   "msetprod = msetprod"
    59   ..
    60 
    61 lemma [code, code del]:
    62   "set_of = set_of"
    63   ..
    64 
    65 lemma [code, code del]:
    66   "sorted_list_of_multiset = sorted_list_of_multiset"
    67   ..
    68 
    69 
    70 text {* Raw operations on lists *}
    71 
    72 definition join_raw :: "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
    73 where
    74   "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
    75 
    76 lemma join_raw_Nil [simp]:
    77   "join_raw f xs [] = xs"
    78 by (simp add: join_raw_def)
    79 
    80 lemma join_raw_Cons [simp]:
    81   "join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
    82 by (simp add: join_raw_def)
    83 
    84 lemma map_of_join_raw:
    85   assumes "distinct (map fst ys)"
    86   shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v =>
    87     (case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
    88 using assms
    89 apply (induct ys)
    90 apply (auto simp add: map_of_map_default split: option.split)
    91 apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
    92 by (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
    93 
    94 lemma distinct_join_raw:
    95   assumes "distinct (map fst xs)"
    96   shows "distinct (map fst (join_raw f xs ys))"
    97 using assms
    98 proof (induct ys)
    99   case (Cons y ys)
   100   thus ?case by (cases y) (simp add: distinct_map_default)
   101 qed auto
   102 
   103 definition
   104   "subtract_entries_raw xs ys = foldr (%(k, v). AList.map_entry k (%v'. v' - v)) ys xs"
   105 
   106 lemma map_of_subtract_entries_raw:
   107   assumes "distinct (map fst ys)"
   108   shows "map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None => None | Some v =>
   109     (case map_of ys x of None => Some v | Some v' => Some (v - v')))"
   110 using assms unfolding subtract_entries_raw_def
   111 apply (induct ys)
   112 apply auto
   113 apply (simp split: option.split)
   114 apply (simp add: map_of_map_entry)
   115 apply (auto split: option.split)
   116 apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
   117 by (metis map_of_eq_None_iff option.simps(4) option.simps(5))
   118 
   119 lemma distinct_subtract_entries_raw:
   120   assumes "distinct (map fst xs)"
   121   shows "distinct (map fst (subtract_entries_raw xs ys))"
   122 using assms
   123 unfolding subtract_entries_raw_def by (induct ys) (auto simp add: distinct_map_entry)
   124 
   125 
   126 text {* Operations on alists with distinct keys *}
   127 
   128 lift_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist" 
   129 is join_raw
   130 by (simp add: distinct_join_raw)
   131 
   132 lift_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
   133 is subtract_entries_raw 
   134 by (simp add: distinct_subtract_entries_raw)
   135 
   136 
   137 text {* Implementing multisets by means of association lists *}
   138 
   139 definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
   140   "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
   141 
   142 lemma count_of_multiset:
   143   "count_of xs \<in> multiset"
   144 proof -
   145   let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
   146   have "?A \<subseteq> dom (map_of xs)"
   147   proof
   148     fix x
   149     assume "x \<in> ?A"
   150     then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
   151     then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
   152     then show "x \<in> dom (map_of xs)" by auto
   153   qed
   154   with finite_dom_map_of [of xs] have "finite ?A"
   155     by (auto intro: finite_subset)
   156   then show ?thesis
   157     by (simp add: count_of_def fun_eq_iff multiset_def)
   158 qed
   159 
   160 lemma count_simps [simp]:
   161   "count_of [] = (\<lambda>_. 0)"
   162   "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
   163   by (simp_all add: count_of_def fun_eq_iff)
   164 
   165 lemma count_of_empty:
   166   "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
   167   by (induct xs) (simp_all add: count_of_def)
   168 
   169 lemma count_of_filter:
   170   "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
   171   by (induct xs) auto
   172 
   173 lemma count_of_map_default [simp]:
   174   "count_of (map_default x b (%x. x + b) xs) y = (if x = y then count_of xs x + b else count_of xs y)"
   175 unfolding count_of_def by (simp add: map_of_map_default split: option.split)
   176 
   177 lemma count_of_join_raw:
   178   "distinct (map fst ys) ==> count_of xs x + count_of ys x = count_of (join_raw (%x (x, y). x + y) xs ys) x"
   179 unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
   180 
   181 lemma count_of_subtract_entries_raw:
   182   "distinct (map fst ys) ==> count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
   183 unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split)
   184 
   185 
   186 text {* Code equations for multiset operations *}
   187 
   188 definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset" where
   189   "Bag xs = Abs_multiset (count_of (DAList.impl_of xs))"
   190 
   191 code_datatype Bag
   192 
   193 lemma count_Bag [simp, code]:
   194   "count (Bag xs) = count_of (DAList.impl_of xs)"
   195   by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
   196 
   197 lemma Mempty_Bag [code]:
   198   "{#} = Bag (DAList.empty)"
   199   by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)
   200 
   201 lemma single_Bag [code]:
   202   "{#x#} = Bag (DAList.update x 1 DAList.empty)"
   203   by (simp add: multiset_eq_iff alist.Alist_inverse update.rep_eq empty.rep_eq)
   204 
   205 lemma union_Bag [code]:
   206   "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
   207 by (rule multiset_eqI) (simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def)
   208 
   209 lemma minus_Bag [code]:
   210   "Bag xs - Bag ys = Bag (subtract_entries xs ys)"
   211 by (rule multiset_eqI)
   212   (simp add: count_of_subtract_entries_raw alist.Alist_inverse distinct_subtract_entries_raw subtract_entries_def)
   213 
   214 lemma filter_Bag [code]:
   215   "Multiset.filter P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)"
   216 by (rule multiset_eqI) (simp add: count_of_filter DAList.filter.rep_eq)
   217 
   218 lemma mset_less_eq_Bag [code]:
   219   "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). count_of (DAList.impl_of xs) x \<le> count A x)"
   220     (is "?lhs \<longleftrightarrow> ?rhs")
   221 proof
   222   assume ?lhs then show ?rhs
   223     by (auto simp add: mset_le_def)
   224 next
   225   assume ?rhs
   226   show ?lhs
   227   proof (rule mset_less_eqI)
   228     fix x
   229     from `?rhs` have "count_of (DAList.impl_of xs) x \<le> count A x"
   230       by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
   231     then show "count (Bag xs) x \<le> count A x"
   232       by (simp add: mset_le_def)
   233   qed
   234 qed
   235 
   236 declare multiset_inter_def [code]
   237 declare sup_multiset_def [code]
   238 declare multiset_of.simps [code]
   239 
   240 instantiation multiset :: (exhaustive) exhaustive
   241 begin
   242 
   243 definition exhaustive_multiset :: "('a multiset \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool * term list) option"
   244 where
   245   "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (\<lambda>xs. f (Bag xs)) i"
   246 
   247 instance ..
   248 
   249 end
   250 
   251 end
   252