src/HOL/Library/DAList_Multiset.thy
 author wenzelm Mon Dec 28 17:43:30 2015 +0100 (2015-12-28) changeset 61952 546958347e05 parent 61585 a9599d3d7610 child 63040 eb4ddd18d635 permissions -rw-r--r--
prefer symbols for "Union", "Inter";
```     1 (*  Title:      HOL/Library/DAList_Multiset.thy
```
```     2     Author:     Lukas Bulwahn, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Multisets partially implemented by association lists\<close>
```
```     6
```
```     7 theory DAList_Multiset
```
```     8 imports Multiset DAList
```
```     9 begin
```
```    10
```
```    11 text \<open>Delete prexisting code equations\<close>
```
```    12
```
```    13 lemma [code, code del]: "{#} = {#}" ..
```
```    14
```
```    15 lemma [code, code del]: "single = single" ..
```
```    16
```
```    17 lemma [code, code del]: "plus = (plus :: 'a multiset \<Rightarrow> _)" ..
```
```    18
```
```    19 lemma [code, code del]: "minus = (minus :: 'a multiset \<Rightarrow> _)" ..
```
```    20
```
```    21 lemma [code, code del]: "inf_subset_mset = (inf_subset_mset :: 'a multiset \<Rightarrow> _)" ..
```
```    22
```
```    23 lemma [code, code del]: "sup_subset_mset = (sup_subset_mset :: 'a multiset \<Rightarrow> _)" ..
```
```    24
```
```    25 lemma [code, code del]: "image_mset = image_mset" ..
```
```    26
```
```    27 lemma [code, code del]: "filter_mset = filter_mset" ..
```
```    28
```
```    29 lemma [code, code del]: "count = count" ..
```
```    30
```
```    31 lemma [code, code del]: "size = (size :: _ multiset \<Rightarrow> nat)" ..
```
```    32
```
```    33 lemma [code, code del]: "msetsum = msetsum" ..
```
```    34
```
```    35 lemma [code, code del]: "msetprod = msetprod" ..
```
```    36
```
```    37 lemma [code, code del]: "set_mset = set_mset" ..
```
```    38
```
```    39 lemma [code, code del]: "sorted_list_of_multiset = sorted_list_of_multiset" ..
```
```    40
```
```    41 lemma [code, code del]: "subset_mset = subset_mset" ..
```
```    42
```
```    43 lemma [code, code del]: "subseteq_mset = subseteq_mset" ..
```
```    44
```
```    45 lemma [code, code del]: "equal_multiset_inst.equal_multiset = equal_multiset_inst.equal_multiset" ..
```
```    46
```
```    47
```
```    48 text \<open>Raw operations on lists\<close>
```
```    49
```
```    50 definition join_raw ::
```
```    51     "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow>
```
```    52       ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
```
```    53   where "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (\<lambda>v'. f k (v', v))) ys xs"
```
```    54
```
```    55 lemma join_raw_Nil [simp]: "join_raw f xs [] = xs"
```
```    56   by (simp add: join_raw_def)
```
```    57
```
```    58 lemma join_raw_Cons [simp]:
```
```    59   "join_raw f xs ((k, v) # ys) = map_default k v (\<lambda>v'. f k (v', v)) (join_raw f xs ys)"
```
```    60   by (simp add: join_raw_def)
```
```    61
```
```    62 lemma map_of_join_raw:
```
```    63   assumes "distinct (map fst ys)"
```
```    64   shows "map_of (join_raw f xs ys) x =
```
```    65     (case map_of xs x of
```
```    66       None \<Rightarrow> map_of ys x
```
```    67     | Some v \<Rightarrow> (case map_of ys x of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (f x (v, v'))))"
```
```    68   using assms
```
```    69   apply (induct ys)
```
```    70   apply (auto simp add: map_of_map_default split: option.split)
```
```    71   apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
```
```    72   apply (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
```
```    73   done
```
```    74
```
```    75 lemma distinct_join_raw:
```
```    76   assumes "distinct (map fst xs)"
```
```    77   shows "distinct (map fst (join_raw f xs ys))"
```
```    78   using assms
```
```    79 proof (induct ys)
```
```    80   case Nil
```
```    81   then show ?case by simp
```
```    82 next
```
```    83   case (Cons y ys)
```
```    84   then show ?case by (cases y) (simp add: distinct_map_default)
```
```    85 qed
```
```    86
```
```    87 definition "subtract_entries_raw xs ys = foldr (\<lambda>(k, v). AList.map_entry k (\<lambda>v'. v' - v)) ys xs"
```
```    88
```
```    89 lemma map_of_subtract_entries_raw:
```
```    90   assumes "distinct (map fst ys)"
```
```    91   shows "map_of (subtract_entries_raw xs ys) x =
```
```    92     (case map_of xs x of
```
```    93       None \<Rightarrow> None
```
```    94     | Some v \<Rightarrow> (case map_of ys x of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (v - v')))"
```
```    95   using assms
```
```    96   unfolding subtract_entries_raw_def
```
```    97   apply (induct ys)
```
```    98   apply auto
```
```    99   apply (simp split: option.split)
```
```   100   apply (simp add: map_of_map_entry)
```
```   101   apply (auto split: option.split)
```
```   102   apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
```
```   103   apply (metis map_of_eq_None_iff option.simps(4) option.simps(5))
```
```   104   done
```
```   105
```
```   106 lemma distinct_subtract_entries_raw:
```
```   107   assumes "distinct (map fst xs)"
```
```   108   shows "distinct (map fst (subtract_entries_raw xs ys))"
```
```   109   using assms
```
```   110   unfolding subtract_entries_raw_def
```
```   111   by (induct ys) (auto simp add: distinct_map_entry)
```
```   112
```
```   113
```
```   114 text \<open>Operations on alists with distinct keys\<close>
```
```   115
```
```   116 lift_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
```
```   117   is join_raw
```
```   118   by (simp add: distinct_join_raw)
```
```   119
```
```   120 lift_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
```
```   121   is subtract_entries_raw
```
```   122   by (simp add: distinct_subtract_entries_raw)
```
```   123
```
```   124
```
```   125 text \<open>Implementing multisets by means of association lists\<close>
```
```   126
```
```   127 definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat"
```
```   128   where "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
```
```   129
```
```   130 lemma count_of_multiset: "count_of xs \<in> multiset"
```
```   131 proof -
```
```   132   let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0::nat | Some n \<Rightarrow> n)}"
```
```   133   have "?A \<subseteq> dom (map_of xs)"
```
```   134   proof
```
```   135     fix x
```
```   136     assume "x \<in> ?A"
```
```   137     then have "0 < (case map_of xs x of None \<Rightarrow> 0::nat | Some n \<Rightarrow> n)"
```
```   138       by simp
```
```   139     then have "map_of xs x \<noteq> None"
```
```   140       by (cases "map_of xs x") auto
```
```   141     then show "x \<in> dom (map_of xs)"
```
```   142       by auto
```
```   143   qed
```
```   144   with finite_dom_map_of [of xs] have "finite ?A"
```
```   145     by (auto intro: finite_subset)
```
```   146   then show ?thesis
```
```   147     by (simp add: count_of_def fun_eq_iff multiset_def)
```
```   148 qed
```
```   149
```
```   150 lemma count_simps [simp]:
```
```   151   "count_of [] = (\<lambda>_. 0)"
```
```   152   "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
```
```   153   by (simp_all add: count_of_def fun_eq_iff)
```
```   154
```
```   155 lemma count_of_empty: "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
```
```   156   by (induct xs) (simp_all add: count_of_def)
```
```   157
```
```   158 lemma count_of_filter: "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
```
```   159   by (induct xs) auto
```
```   160
```
```   161 lemma count_of_map_default [simp]:
```
```   162   "count_of (map_default x b (\<lambda>x. x + b) xs) y =
```
```   163     (if x = y then count_of xs x + b else count_of xs y)"
```
```   164   unfolding count_of_def by (simp add: map_of_map_default split: option.split)
```
```   165
```
```   166 lemma count_of_join_raw:
```
```   167   "distinct (map fst ys) \<Longrightarrow>
```
```   168     count_of xs x + count_of ys x = count_of (join_raw (\<lambda>x (x, y). x + y) xs ys) x"
```
```   169   unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
```
```   170
```
```   171 lemma count_of_subtract_entries_raw:
```
```   172   "distinct (map fst ys) \<Longrightarrow>
```
```   173     count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
```
```   174   unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split)
```
```   175
```
```   176
```
```   177 text \<open>Code equations for multiset operations\<close>
```
```   178
```
```   179 definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset"
```
```   180   where "Bag xs = Abs_multiset (count_of (DAList.impl_of xs))"
```
```   181
```
```   182 code_datatype Bag
```
```   183
```
```   184 lemma count_Bag [simp, code]: "count (Bag xs) = count_of (DAList.impl_of xs)"
```
```   185   by (simp add: Bag_def count_of_multiset)
```
```   186
```
```   187 lemma Mempty_Bag [code]: "{#} = Bag (DAList.empty)"
```
```   188   by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)
```
```   189
```
```   190 lemma single_Bag [code]: "{#x#} = Bag (DAList.update x 1 DAList.empty)"
```
```   191   by (simp add: multiset_eq_iff alist.Alist_inverse update.rep_eq empty.rep_eq)
```
```   192
```
```   193 lemma union_Bag [code]: "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
```
```   194   by (rule multiset_eqI)
```
```   195     (simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def)
```
```   196
```
```   197 lemma minus_Bag [code]: "Bag xs - Bag ys = Bag (subtract_entries xs ys)"
```
```   198   by (rule multiset_eqI)
```
```   199     (simp add: count_of_subtract_entries_raw alist.Alist_inverse
```
```   200       distinct_subtract_entries_raw subtract_entries_def)
```
```   201
```
```   202 lemma filter_Bag [code]: "filter_mset P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)"
```
```   203   by (rule multiset_eqI) (simp add: count_of_filter DAList.filter.rep_eq)
```
```   204
```
```   205
```
```   206 lemma mset_eq [code]: "HOL.equal (m1::'a::equal multiset) m2 \<longleftrightarrow> m1 \<le># m2 \<and> m2 \<le># m1"
```
```   207   by (metis equal_multiset_def subset_mset.eq_iff)
```
```   208
```
```   209 text \<open>By default the code for \<open><\<close> is @{prop"xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> xs = ys"}.
```
```   210 With equality implemented by \<open>\<le>\<close>, this leads to three calls of  \<open>\<le>\<close>.
```
```   211 Here is a more efficient version:\<close>
```
```   212 lemma mset_less[code]: "xs <# (ys :: 'a multiset) \<longleftrightarrow> xs \<le># ys \<and> \<not> ys \<le># xs"
```
```   213   by (rule subset_mset.less_le_not_le)
```
```   214
```
```   215 lemma mset_less_eq_Bag0:
```
```   216   "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)"
```
```   217     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   218 proof
```
```   219   assume ?lhs
```
```   220   then show ?rhs by (auto simp add: subseteq_mset_def)
```
```   221 next
```
```   222   assume ?rhs
```
```   223   show ?lhs
```
```   224   proof (rule mset_less_eqI)
```
```   225     fix x
```
```   226     from \<open>?rhs\<close> have "count_of (DAList.impl_of xs) x \<le> count A x"
```
```   227       by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
```
```   228     then show "count (Bag xs) x \<le> count A x" by (simp add: subset_mset_def)
```
```   229   qed
```
```   230 qed
```
```   231
```
```   232 lemma mset_less_eq_Bag [code]:
```
```   233   "Bag xs \<le># (A :: 'a multiset) \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). n \<le> count A x)"
```
```   234 proof -
```
```   235   {
```
```   236     fix x n
```
```   237     assume "(x,n) \<in> set (DAList.impl_of xs)"
```
```   238     then have "count_of (DAList.impl_of xs) x = n"
```
```   239     proof transfer
```
```   240       fix x n
```
```   241       fix xs :: "('a \<times> nat) list"
```
```   242       show "(distinct \<circ> map fst) xs \<Longrightarrow> (x, n) \<in> set xs \<Longrightarrow> count_of xs x = n"
```
```   243       proof (induct xs)
```
```   244         case Nil
```
```   245         then show ?case by simp
```
```   246       next
```
```   247         case (Cons ym ys)
```
```   248         obtain y m where ym: "ym = (y,m)" by force
```
```   249         note Cons = Cons[unfolded ym]
```
```   250         show ?case
```
```   251         proof (cases "x = y")
```
```   252           case False
```
```   253           with Cons show ?thesis
```
```   254             unfolding ym by auto
```
```   255         next
```
```   256           case True
```
```   257           with Cons(2-3) have "m = n" by force
```
```   258           with True show ?thesis
```
```   259             unfolding ym by auto
```
```   260         qed
```
```   261       qed
```
```   262     qed
```
```   263   }
```
```   264   then show ?thesis
```
```   265     unfolding mset_less_eq_Bag0 by auto
```
```   266 qed
```
```   267
```
```   268 declare multiset_inter_def [code]
```
```   269 declare sup_subset_mset_def [code]
```
```   270 declare mset.simps [code]
```
```   271
```
```   272
```
```   273 fun fold_impl :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<times> nat) list \<Rightarrow> 'b"
```
```   274 where
```
```   275   "fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)"
```
```   276 | "fold_impl fn e [] = e"
```
```   277
```
```   278 context
```
```   279 begin
```
```   280
```
```   281 qualified definition fold :: "('a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a, nat) alist \<Rightarrow> 'b"
```
```   282   where "fold f e al = fold_impl f e (DAList.impl_of al)"
```
```   283
```
```   284 end
```
```   285
```
```   286 context comp_fun_commute
```
```   287 begin
```
```   288
```
```   289 lemma DAList_Multiset_fold:
```
```   290   assumes fn: "\<And>a n x. fn a n x = (f a ^^ n) x"
```
```   291   shows "fold_mset f e (Bag al) = DAList_Multiset.fold fn e al"
```
```   292   unfolding DAList_Multiset.fold_def
```
```   293 proof (induct al)
```
```   294   fix ys
```
```   295   let ?inv = "{xs :: ('a \<times> nat) list. (distinct \<circ> map fst) xs}"
```
```   296   note cs[simp del] = count_simps
```
```   297   have count[simp]: "\<And>x. count (Abs_multiset (count_of x)) = count_of x"
```
```   298     by (rule Abs_multiset_inverse[OF count_of_multiset])
```
```   299   assume ys: "ys \<in> ?inv"
```
```   300   then show "fold_mset f e (Bag (Alist ys)) = fold_impl fn e (DAList.impl_of (Alist ys))"
```
```   301     unfolding Bag_def unfolding Alist_inverse[OF ys]
```
```   302   proof (induct ys arbitrary: e rule: list.induct)
```
```   303     case Nil
```
```   304     show ?case
```
```   305       by (rule trans[OF arg_cong[of _ "{#}" "fold_mset f e", OF multiset_eqI]])
```
```   306          (auto, simp add: cs)
```
```   307   next
```
```   308     case (Cons pair ys e)
```
```   309     obtain a n where pair: "pair = (a,n)"
```
```   310       by force
```
```   311     from fn[of a n] have [simp]: "fn a n = (f a ^^ n)"
```
```   312       by auto
```
```   313     have inv: "ys \<in> ?inv"
```
```   314       using Cons(2) by auto
```
```   315     note IH = Cons(1)[OF inv]
```
```   316     def Ys \<equiv> "Abs_multiset (count_of ys)"
```
```   317     have id: "Abs_multiset (count_of ((a, n) # ys)) = ((op + {# a #}) ^^ n) Ys"
```
```   318       unfolding Ys_def
```
```   319     proof (rule multiset_eqI, unfold count)
```
```   320       fix c
```
```   321       show "count_of ((a, n) # ys) c =
```
```   322         count ((op + {#a#} ^^ n) (Abs_multiset (count_of ys))) c" (is "?l = ?r")
```
```   323       proof (cases "c = a")
```
```   324         case False
```
```   325         then show ?thesis
```
```   326           unfolding cs by (induct n) auto
```
```   327       next
```
```   328         case True
```
```   329         then have "?l = n" by (simp add: cs)
```
```   330         also have "n = ?r" unfolding True
```
```   331         proof (induct n)
```
```   332           case 0
```
```   333           from Cons(2)[unfolded pair] have "a \<notin> fst ` set ys" by auto
```
```   334           then show ?case by (induct ys) (simp, auto simp: cs)
```
```   335         next
```
```   336           case Suc
```
```   337           then show ?case by simp
```
```   338         qed
```
```   339         finally show ?thesis .
```
```   340       qed
```
```   341     qed
```
```   342     show ?case
```
```   343       unfolding pair
```
```   344       apply (simp add: IH[symmetric])
```
```   345       unfolding id Ys_def[symmetric]
```
```   346       apply (induct n)
```
```   347       apply (auto simp: fold_mset_fun_left_comm[symmetric])
```
```   348       done
```
```   349   qed
```
```   350 qed
```
```   351
```
```   352 end
```
```   353
```
```   354 context
```
```   355 begin
```
```   356
```
```   357 private lift_definition single_alist_entry :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) alist" is "\<lambda>a b. [(a, b)]"
```
```   358   by auto
```
```   359
```
```   360 lemma image_mset_Bag [code]:
```
```   361   "image_mset f (Bag ms) =
```
```   362     DAList_Multiset.fold (\<lambda>a n m. Bag (single_alist_entry (f a) n) + m) {#} ms"
```
```   363   unfolding image_mset_def
```
```   364 proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1])
```
```   365   fix a n m
```
```   366   show "Bag (single_alist_entry (f a) n) + m = ((op + \<circ> single \<circ> f) a ^^ n) m" (is "?l = ?r")
```
```   367   proof (rule multiset_eqI)
```
```   368     fix x
```
```   369     have "count ?r x = (if x = f a then n + count m x else count m x)"
```
```   370       by (induct n) auto
```
```   371     also have "\<dots> = count ?l x"
```
```   372       by (simp add: single_alist_entry.rep_eq)
```
```   373     finally show "count ?l x = count ?r x" ..
```
```   374   qed
```
```   375 qed
```
```   376
```
```   377 end
```
```   378
```
```   379 (* we cannot use (\<lambda>a n. op + (a * n)) for folding, since * is not defined
```
```   380    in comm_monoid_add *)
```
```   381 lemma msetsum_Bag[code]: "msetsum (Bag ms) = DAList_Multiset.fold (\<lambda>a n. ((op + a) ^^ n)) 0 ms"
```
```   382   unfolding msetsum.eq_fold
```
```   383   apply (rule comp_fun_commute.DAList_Multiset_fold)
```
```   384   apply unfold_locales
```
```   385   apply (auto simp: ac_simps)
```
```   386   done
```
```   387
```
```   388 (* we cannot use (\<lambda>a n. op * (a ^ n)) for folding, since ^ is not defined
```
```   389    in comm_monoid_mult *)
```
```   390 lemma msetprod_Bag[code]: "msetprod (Bag ms) = DAList_Multiset.fold (\<lambda>a n. ((op * a) ^^ n)) 1 ms"
```
```   391   unfolding msetprod.eq_fold
```
```   392   apply (rule comp_fun_commute.DAList_Multiset_fold)
```
```   393   apply unfold_locales
```
```   394   apply (auto simp: ac_simps)
```
```   395   done
```
```   396
```
```   397 lemma size_fold: "size A = fold_mset (\<lambda>_. Suc) 0 A" (is "_ = fold_mset ?f _ _")
```
```   398 proof -
```
```   399   interpret comp_fun_commute ?f by standard auto
```
```   400   show ?thesis by (induct A) auto
```
```   401 qed
```
```   402
```
```   403 lemma size_Bag[code]: "size (Bag ms) = DAList_Multiset.fold (\<lambda>a n. op + n) 0 ms"
```
```   404   unfolding size_fold
```
```   405 proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, simp)
```
```   406   fix a n x
```
```   407   show "n + x = (Suc ^^ n) x"
```
```   408     by (induct n) auto
```
```   409 qed
```
```   410
```
```   411
```
```   412 lemma set_mset_fold: "set_mset A = fold_mset insert {} A" (is "_ = fold_mset ?f _ _")
```
```   413 proof -
```
```   414   interpret comp_fun_commute ?f by standard auto
```
```   415   show ?thesis by (induct A) auto
```
```   416 qed
```
```   417
```
```   418 lemma set_mset_Bag[code]:
```
```   419   "set_mset (Bag ms) = DAList_Multiset.fold (\<lambda>a n. (if n = 0 then (\<lambda>m. m) else insert a)) {} ms"
```
```   420   unfolding set_mset_fold
```
```   421 proof (rule comp_fun_commute.DAList_Multiset_fold, unfold_locales, (auto simp: ac_simps)[1])
```
```   422   fix a n x
```
```   423   show "(if n = 0 then \<lambda>m. m else insert a) x = (insert a ^^ n) x" (is "?l n = ?r n")
```
```   424   proof (cases n)
```
```   425     case 0
```
```   426     then show ?thesis by simp
```
```   427   next
```
```   428     case (Suc m)
```
```   429     then have "?l n = insert a x" by simp
```
```   430     moreover have "?r n = insert a x" unfolding Suc by (induct m) auto
```
```   431     ultimately show ?thesis by auto
```
```   432   qed
```
```   433 qed
```
```   434
```
```   435
```
```   436 instantiation multiset :: (exhaustive) exhaustive
```
```   437 begin
```
```   438
```
```   439 definition exhaustive_multiset ::
```
```   440   "('a multiset \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
```
```   441   where "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (\<lambda>xs. f (Bag xs)) i"
```
```   442
```
```   443 instance ..
```
```   444
```
```   445 end
```
```   446
```
```   447 end
```
```   448
```