src/HOL/Library/Multiset.thy
 author nipkow Fri Apr 10 12:16:45 2015 +0200 (2015-04-10) changeset 59998 c54d36be22ef parent 59986 f38b94549dc8 child 59999 3fa68bacfa2b permissions -rw-r--r--
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
```     1 (*  Title:      HOL/Library/Multiset.thy
```
```     2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
```
```     3     Author:     Andrei Popescu, TU Muenchen
```
```     4     Author:     Jasmin Blanchette, Inria, LORIA, MPII
```
```     5     Author:     Dmitriy Traytel, TU Muenchen
```
```     6     Author:     Mathias Fleury, MPII
```
```     7 *)
```
```     8
```
```     9 section {* (Finite) multisets *}
```
```    10
```
```    11 theory Multiset
```
```    12 imports Main
```
```    13 begin
```
```    14
```
```    15 subsection {* The type of multisets *}
```
```    16
```
```    17 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
```
```    18
```
```    19 typedef 'a multiset = "multiset :: ('a => nat) set"
```
```    20   morphisms count Abs_multiset
```
```    21   unfolding multiset_def
```
```    22 proof
```
```    23   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
```
```    24 qed
```
```    25
```
```    26 setup_lifting type_definition_multiset
```
```    27
```
```    28 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
```
```    29   "a :# M == 0 < count M a"
```
```    30
```
```    31 notation (xsymbols)
```
```    32   Melem (infix "\<in>#" 50)
```
```    33
```
```    34 lemma multiset_eq_iff:
```
```    35   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
```
```    36   by (simp only: count_inject [symmetric] fun_eq_iff)
```
```    37
```
```    38 lemma multiset_eqI:
```
```    39   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
```
```    40   using multiset_eq_iff by auto
```
```    41
```
```    42 text {*
```
```    43  \medskip Preservation of the representing set @{term multiset}.
```
```    44 *}
```
```    45
```
```    46 lemma const0_in_multiset:
```
```    47   "(\<lambda>a. 0) \<in> multiset"
```
```    48   by (simp add: multiset_def)
```
```    49
```
```    50 lemma only1_in_multiset:
```
```    51   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
```
```    52   by (simp add: multiset_def)
```
```    53
```
```    54 lemma union_preserves_multiset:
```
```    55   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
```
```    56   by (simp add: multiset_def)
```
```    57
```
```    58 lemma diff_preserves_multiset:
```
```    59   assumes "M \<in> multiset"
```
```    60   shows "(\<lambda>a. M a - N a) \<in> multiset"
```
```    61 proof -
```
```    62   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
```
```    63     by auto
```
```    64   with assms show ?thesis
```
```    65     by (auto simp add: multiset_def intro: finite_subset)
```
```    66 qed
```
```    67
```
```    68 lemma filter_preserves_multiset:
```
```    69   assumes "M \<in> multiset"
```
```    70   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
```
```    71 proof -
```
```    72   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
```
```    73     by auto
```
```    74   with assms show ?thesis
```
```    75     by (auto simp add: multiset_def intro: finite_subset)
```
```    76 qed
```
```    77
```
```    78 lemmas in_multiset = const0_in_multiset only1_in_multiset
```
```    79   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
```
```    80
```
```    81
```
```    82 subsection {* Representing multisets *}
```
```    83
```
```    84 text {* Multiset enumeration *}
```
```    85
```
```    86 instantiation multiset :: (type) cancel_comm_monoid_add
```
```    87 begin
```
```    88
```
```    89 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
```
```    90 by (rule const0_in_multiset)
```
```    91
```
```    92 abbreviation Mempty :: "'a multiset" ("{#}") where
```
```    93   "Mempty \<equiv> 0"
```
```    94
```
```    95 lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
```
```    96 by (rule union_preserves_multiset)
```
```    97
```
```    98 lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
```
```    99 by (rule diff_preserves_multiset)
```
```   100
```
```   101 instance
```
```   102   by default (transfer, simp add: fun_eq_iff)+
```
```   103
```
```   104 end
```
```   105
```
```   106 lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
```
```   107 by (rule only1_in_multiset)
```
```   108
```
```   109 syntax
```
```   110   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
```
```   111 translations
```
```   112   "{#x, xs#}" == "{#x#} + {#xs#}"
```
```   113   "{#x#}" == "CONST single x"
```
```   114
```
```   115 lemma count_empty [simp]: "count {#} a = 0"
```
```   116   by (simp add: zero_multiset.rep_eq)
```
```   117
```
```   118 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   119   by (simp add: single.rep_eq)
```
```   120
```
```   121
```
```   122 subsection {* Basic operations *}
```
```   123
```
```   124 subsubsection {* Union *}
```
```   125
```
```   126 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   127   by (simp add: plus_multiset.rep_eq)
```
```   128
```
```   129
```
```   130 subsubsection {* Difference *}
```
```   131
```
```   132 instantiation multiset :: (type) comm_monoid_diff
```
```   133 begin
```
```   134
```
```   135 instance
```
```   136 by default (transfer, simp add: fun_eq_iff)+
```
```   137
```
```   138 end
```
```   139
```
```   140 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   141   by (simp add: minus_multiset.rep_eq)
```
```   142
```
```   143 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   144   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
```
```   145
```
```   146 lemma diff_cancel[simp]: "A - A = {#}"
```
```   147   by (fact Groups.diff_cancel)
```
```   148
```
```   149 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
```
```   150   by (fact add_diff_cancel_right')
```
```   151
```
```   152 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
```
```   153   by (fact add_diff_cancel_left')
```
```   154
```
```   155 lemma diff_right_commute:
```
```   156   "(M::'a multiset) - N - Q = M - Q - N"
```
```   157   by (fact diff_right_commute)
```
```   158
```
```   159 lemma diff_add:
```
```   160   "(M::'a multiset) - (N + Q) = M - N - Q"
```
```   161   by (rule sym) (fact diff_diff_add)
```
```   162
```
```   163 lemma insert_DiffM:
```
```   164   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
```
```   165   by (clarsimp simp: multiset_eq_iff)
```
```   166
```
```   167 lemma insert_DiffM2 [simp]:
```
```   168   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
```
```   169   by (clarsimp simp: multiset_eq_iff)
```
```   170
```
```   171 lemma diff_union_swap:
```
```   172   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
```
```   173   by (auto simp add: multiset_eq_iff)
```
```   174
```
```   175 lemma diff_union_single_conv:
```
```   176   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
```
```   177   by (simp add: multiset_eq_iff)
```
```   178
```
```   179
```
```   180 subsubsection {* Equality of multisets *}
```
```   181
```
```   182 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
```
```   183   by (simp add: multiset_eq_iff)
```
```   184
```
```   185 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
```
```   186   by (auto simp add: multiset_eq_iff)
```
```   187
```
```   188 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
```
```   189   by (auto simp add: multiset_eq_iff)
```
```   190
```
```   191 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
```
```   192   by (auto simp add: multiset_eq_iff)
```
```   193
```
```   194 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
```
```   195   by (auto simp add: multiset_eq_iff)
```
```   196
```
```   197 lemma diff_single_trivial:
```
```   198   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
```
```   199   by (auto simp add: multiset_eq_iff)
```
```   200
```
```   201 lemma diff_single_eq_union:
```
```   202   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
```
```   203   by auto
```
```   204
```
```   205 lemma union_single_eq_diff:
```
```   206   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
```
```   207   by (auto dest: sym)
```
```   208
```
```   209 lemma union_single_eq_member:
```
```   210   "M + {#x#} = N \<Longrightarrow> x \<in># N"
```
```   211   by auto
```
```   212
```
```   213 lemma union_is_single:
```
```   214   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
```
```   215 proof
```
```   216   assume ?rhs then show ?lhs by auto
```
```   217 next
```
```   218   assume ?lhs then show ?rhs
```
```   219     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
```
```   220 qed
```
```   221
```
```   222 lemma single_is_union:
```
```   223   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
```
```   224   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
```
```   225
```
```   226 lemma add_eq_conv_diff:
```
```   227   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
```
```   228 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
```
```   229 proof
```
```   230   assume ?rhs then show ?lhs
```
```   231   by (auto simp add: add.assoc add.commute [of "{#b#}"])
```
```   232     (drule sym, simp add: add.assoc [symmetric])
```
```   233 next
```
```   234   assume ?lhs
```
```   235   show ?rhs
```
```   236   proof (cases "a = b")
```
```   237     case True with `?lhs` show ?thesis by simp
```
```   238   next
```
```   239     case False
```
```   240     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
```
```   241     with False have "a \<in># N" by auto
```
```   242     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
```
```   243     moreover note False
```
```   244     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
```
```   245   qed
```
```   246 qed
```
```   247
```
```   248 lemma insert_noteq_member:
```
```   249   assumes BC: "B + {#b#} = C + {#c#}"
```
```   250    and bnotc: "b \<noteq> c"
```
```   251   shows "c \<in># B"
```
```   252 proof -
```
```   253   have "c \<in># C + {#c#}" by simp
```
```   254   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
```
```   255   then have "c \<in># B + {#b#}" using BC by simp
```
```   256   then show "c \<in># B" using nc by simp
```
```   257 qed
```
```   258
```
```   259 lemma add_eq_conv_ex:
```
```   260   "(M + {#a#} = N + {#b#}) =
```
```   261     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
```
```   262   by (auto simp add: add_eq_conv_diff)
```
```   263
```
```   264 lemma multi_member_split:
```
```   265   "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
```
```   266   by (rule_tac x = "M - {#x#}" in exI, simp)
```
```   267
```
```   268 lemma multiset_add_sub_el_shuffle:
```
```   269   assumes "c \<in># B" and "b \<noteq> c"
```
```   270   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
```
```   271 proof -
```
```   272   from `c \<in># B` obtain A where B: "B = A + {#c#}"
```
```   273     by (blast dest: multi_member_split)
```
```   274   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
```
```   275   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
```
```   276     by (simp add: ac_simps)
```
```   277   then show ?thesis using B by simp
```
```   278 qed
```
```   279
```
```   280
```
```   281 subsubsection {* Pointwise ordering induced by count *}
```
```   282
```
```   283 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
```
```   284 begin
```
```   285
```
```   286 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)" .
```
```   287
```
```   288 lemmas mset_le_def = less_eq_multiset_def
```
```   289
```
```   290 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
```
```   291   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
```
```   292
```
```   293 instance
```
```   294   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
```
```   295
```
```   296 end
```
```   297
```
```   298 abbreviation less_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
```
```   299   "A <# B \<equiv> A < B"
```
```   300 abbreviation (xsymbols) subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subset>#" 50) where
```
```   301   "A \<subset># B \<equiv> A < B"
```
```   302
```
```   303 abbreviation less_eq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
```
```   304   "A <=# B \<equiv> A \<le> B"
```
```   305 abbreviation (xsymbols) leq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<le>#" 50) where
```
```   306   "A \<le># B \<equiv> A \<le> B"
```
```   307 abbreviation (xsymbols) subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subseteq>#" 50) where
```
```   308   "A \<subseteq># B \<equiv> A \<le> B"
```
```   309
```
```   310 lemma mset_less_eqI:
```
```   311   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
```
```   312   by (simp add: mset_le_def)
```
```   313
```
```   314 lemma mset_le_exists_conv:
```
```   315   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
```
```   316 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
```
```   317 apply (auto intro: multiset_eq_iff [THEN iffD2])
```
```   318 done
```
```   319
```
```   320 instance multiset :: (type) ordered_cancel_comm_monoid_diff
```
```   321   by default (simp, fact mset_le_exists_conv)
```
```   322
```
```   323 lemma mset_le_mono_add_right_cancel [simp]:
```
```   324   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
```
```   325   by (fact add_le_cancel_right)
```
```   326
```
```   327 lemma mset_le_mono_add_left_cancel [simp]:
```
```   328   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
```
```   329   by (fact add_le_cancel_left)
```
```   330
```
```   331 lemma mset_le_mono_add:
```
```   332   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
```
```   333   by (fact add_mono)
```
```   334
```
```   335 lemma mset_le_add_left [simp]:
```
```   336   "(A::'a multiset) \<le> A + B"
```
```   337   unfolding mset_le_def by auto
```
```   338
```
```   339 lemma mset_le_add_right [simp]:
```
```   340   "B \<le> (A::'a multiset) + B"
```
```   341   unfolding mset_le_def by auto
```
```   342
```
```   343 lemma mset_le_single:
```
```   344   "a :# B \<Longrightarrow> {#a#} \<le> B"
```
```   345   by (simp add: mset_le_def)
```
```   346
```
```   347 lemma multiset_diff_union_assoc:
```
```   348   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
```
```   349   by (simp add: multiset_eq_iff mset_le_def)
```
```   350
```
```   351 lemma mset_le_multiset_union_diff_commute:
```
```   352   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
```
```   353 by (simp add: multiset_eq_iff mset_le_def)
```
```   354
```
```   355 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
```
```   356 by(simp add: mset_le_def)
```
```   357
```
```   358 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   359 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   360 apply (erule_tac x=x in allE)
```
```   361 apply auto
```
```   362 done
```
```   363
```
```   364 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   365 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   366 apply (erule_tac x = x in allE)
```
```   367 apply auto
```
```   368 done
```
```   369
```
```   370 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
```
```   371 apply (rule conjI)
```
```   372  apply (simp add: mset_lessD)
```
```   373 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   374 apply safe
```
```   375  apply (erule_tac x = a in allE)
```
```   376  apply (auto split: split_if_asm)
```
```   377 done
```
```   378
```
```   379 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
```
```   380 apply (rule conjI)
```
```   381  apply (simp add: mset_leD)
```
```   382 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
```
```   383 done
```
```   384
```
```   385 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
```
```   386   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
```
```   387
```
```   388 lemma empty_le[simp]: "{#} \<le> A"
```
```   389   unfolding mset_le_exists_conv by auto
```
```   390
```
```   391 lemma le_empty[simp]: "(M \<le> {#}) = (M = {#})"
```
```   392   unfolding mset_le_exists_conv by auto
```
```   393
```
```   394 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
```
```   395   by (auto simp: mset_le_def mset_less_def)
```
```   396
```
```   397 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
```
```   398   by simp
```
```   399
```
```   400 lemma mset_less_add_bothsides: "N + {#x#} < M + {#x#} \<Longrightarrow> N < M"
```
```   401   by (fact add_less_imp_less_right)
```
```   402
```
```   403 lemma mset_less_empty_nonempty:
```
```   404   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
```
```   405   by (auto simp: mset_le_def mset_less_def)
```
```   406
```
```   407 lemma mset_less_diff_self:
```
```   408   "c \<in># B \<Longrightarrow> B - {#c#} < B"
```
```   409   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
```
```   410
```
```   411
```
```   412 subsubsection {* Intersection *}
```
```   413
```
```   414 instantiation multiset :: (type) semilattice_inf
```
```   415 begin
```
```   416
```
```   417 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   418   multiset_inter_def: "inf_multiset A B = A - (A - B)"
```
```   419
```
```   420 instance
```
```   421 proof -
```
```   422   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
```
```   423   show "OFCLASS('a multiset, semilattice_inf_class)"
```
```   424     by default (auto simp add: multiset_inter_def mset_le_def aux)
```
```   425 qed
```
```   426
```
```   427 end
```
```   428
```
```   429 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
```
```   430   "multiset_inter \<equiv> inf"
```
```   431
```
```   432 lemma multiset_inter_count [simp]:
```
```   433   "count (A #\<inter> B) x = min (count A x) (count B x)"
```
```   434   by (simp add: multiset_inter_def)
```
```   435
```
```   436 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
```
```   437   by (rule multiset_eqI) auto
```
```   438
```
```   439 lemma multiset_union_diff_commute:
```
```   440   assumes "B #\<inter> C = {#}"
```
```   441   shows "A + B - C = A - C + B"
```
```   442 proof (rule multiset_eqI)
```
```   443   fix x
```
```   444   from assms have "min (count B x) (count C x) = 0"
```
```   445     by (auto simp add: multiset_eq_iff)
```
```   446   then have "count B x = 0 \<or> count C x = 0"
```
```   447     by auto
```
```   448   then show "count (A + B - C) x = count (A - C + B) x"
```
```   449     by auto
```
```   450 qed
```
```   451
```
```   452 lemma empty_inter [simp]:
```
```   453   "{#} #\<inter> M = {#}"
```
```   454   by (simp add: multiset_eq_iff)
```
```   455
```
```   456 lemma inter_empty [simp]:
```
```   457   "M #\<inter> {#} = {#}"
```
```   458   by (simp add: multiset_eq_iff)
```
```   459
```
```   460 lemma inter_add_left1:
```
```   461   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
```
```   462   by (simp add: multiset_eq_iff)
```
```   463
```
```   464 lemma inter_add_left2:
```
```   465   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
```
```   466   by (simp add: multiset_eq_iff)
```
```   467
```
```   468 lemma inter_add_right1:
```
```   469   "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
```
```   470   by (simp add: multiset_eq_iff)
```
```   471
```
```   472 lemma inter_add_right2:
```
```   473   "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
```
```   474   by (simp add: multiset_eq_iff)
```
```   475
```
```   476
```
```   477 subsubsection {* Bounded union *}
```
```   478
```
```   479 instantiation multiset :: (type) semilattice_sup
```
```   480 begin
```
```   481
```
```   482 definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   483   "sup_multiset A B = A + (B - A)"
```
```   484
```
```   485 instance
```
```   486 proof -
```
```   487   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" by arith
```
```   488   show "OFCLASS('a multiset, semilattice_sup_class)"
```
```   489     by default (auto simp add: sup_multiset_def mset_le_def aux)
```
```   490 qed
```
```   491
```
```   492 end
```
```   493
```
```   494 abbreviation sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<union>" 70) where
```
```   495   "sup_multiset \<equiv> sup"
```
```   496
```
```   497 lemma sup_multiset_count [simp]:
```
```   498   "count (A #\<union> B) x = max (count A x) (count B x)"
```
```   499   by (simp add: sup_multiset_def)
```
```   500
```
```   501 lemma empty_sup [simp]:
```
```   502   "{#} #\<union> M = M"
```
```   503   by (simp add: multiset_eq_iff)
```
```   504
```
```   505 lemma sup_empty [simp]:
```
```   506   "M #\<union> {#} = M"
```
```   507   by (simp add: multiset_eq_iff)
```
```   508
```
```   509 lemma sup_add_left1:
```
```   510   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
```
```   511   by (simp add: multiset_eq_iff)
```
```   512
```
```   513 lemma sup_add_left2:
```
```   514   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
```
```   515   by (simp add: multiset_eq_iff)
```
```   516
```
```   517 lemma sup_add_right1:
```
```   518   "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
```
```   519   by (simp add: multiset_eq_iff)
```
```   520
```
```   521 lemma sup_add_right2:
```
```   522   "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
```
```   523   by (simp add: multiset_eq_iff)
```
```   524
```
```   525
```
```   526 subsubsection {* Filter (with comprehension syntax) *}
```
```   527
```
```   528 text {* Multiset comprehension *}
```
```   529
```
```   530 lift_definition filter_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"
```
```   531 is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
```
```   532 by (rule filter_preserves_multiset)
```
```   533
```
```   534 lemma count_filter_mset [simp]:
```
```   535   "count (filter_mset P M) a = (if P a then count M a else 0)"
```
```   536   by (simp add: filter_mset.rep_eq)
```
```   537
```
```   538 lemma filter_empty_mset [simp]:
```
```   539   "filter_mset P {#} = {#}"
```
```   540   by (rule multiset_eqI) simp
```
```   541
```
```   542 lemma filter_single_mset [simp]:
```
```   543   "filter_mset P {#x#} = (if P x then {#x#} else {#})"
```
```   544   by (rule multiset_eqI) simp
```
```   545
```
```   546 lemma filter_union_mset [simp]:
```
```   547   "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
```
```   548   by (rule multiset_eqI) simp
```
```   549
```
```   550 lemma filter_diff_mset [simp]:
```
```   551   "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
```
```   552   by (rule multiset_eqI) simp
```
```   553
```
```   554 lemma filter_inter_mset [simp]:
```
```   555   "filter_mset P (M #\<inter> N) = filter_mset P M #\<inter> filter_mset P N"
```
```   556   by (rule multiset_eqI) simp
```
```   557
```
```   558 lemma multiset_filter_subset[simp]: "filter_mset f M \<le> M"
```
```   559   unfolding less_eq_multiset.rep_eq by auto
```
```   560
```
```   561 lemma multiset_filter_mono: assumes "A \<le> B"
```
```   562   shows "filter_mset f A \<le> filter_mset f B"
```
```   563 proof -
```
```   564   from assms[unfolded mset_le_exists_conv]
```
```   565   obtain C where B: "B = A + C" by auto
```
```   566   show ?thesis unfolding B by auto
```
```   567 qed
```
```   568
```
```   569 syntax
```
```   570   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
```
```   571 syntax (xsymbol)
```
```   572   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
```
```   573 translations
```
```   574   "{#x \<in># M. P#}" == "CONST filter_mset (\<lambda>x. P) M"
```
```   575
```
```   576
```
```   577 subsubsection {* Set of elements *}
```
```   578
```
```   579 definition set_of :: "'a multiset => 'a set" where
```
```   580   "set_of M = {x. x :# M}"
```
```   581
```
```   582 lemma set_of_empty [simp]: "set_of {#} = {}"
```
```   583 by (simp add: set_of_def)
```
```   584
```
```   585 lemma set_of_single [simp]: "set_of {#b#} = {b}"
```
```   586 by (simp add: set_of_def)
```
```   587
```
```   588 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   589 by (auto simp add: set_of_def)
```
```   590
```
```   591 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   592 by (auto simp add: set_of_def multiset_eq_iff)
```
```   593
```
```   594 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   595 by (auto simp add: set_of_def)
```
```   596
```
```   597 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
```
```   598 by (auto simp add: set_of_def)
```
```   599
```
```   600 lemma finite_set_of [iff]: "finite (set_of M)"
```
```   601   using count [of M] by (simp add: multiset_def set_of_def)
```
```   602
```
```   603 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
```
```   604   unfolding set_of_def[symmetric] by simp
```
```   605
```
```   606 lemma set_of_mono: "A \<le> B \<Longrightarrow> set_of A \<subseteq> set_of B"
```
```   607   by (metis mset_leD subsetI mem_set_of_iff)
```
```   608
```
```   609 lemma ball_set_of_iff: "(\<forall>x \<in> set_of M. P x) \<longleftrightarrow> (\<forall>x. x \<in># M \<longrightarrow> P x)"
```
```   610   by auto
```
```   611
```
```   612
```
```   613 subsubsection {* Size *}
```
```   614
```
```   615 definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
```
```   616
```
```   617 lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
```
```   618   by (auto simp: wcount_def add_mult_distrib)
```
```   619
```
```   620 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
```
```   621   "size_multiset f M = setsum (wcount f M) (set_of M)"
```
```   622
```
```   623 lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
```
```   624
```
```   625 instantiation multiset :: (type) size begin
```
```   626 definition size_multiset where
```
```   627   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
```
```   628 instance ..
```
```   629 end
```
```   630
```
```   631 lemmas size_multiset_overloaded_eq =
```
```   632   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
```
```   633
```
```   634 lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
```
```   635 by (simp add: size_multiset_def)
```
```   636
```
```   637 lemma size_empty [simp]: "size {#} = 0"
```
```   638 by (simp add: size_multiset_overloaded_def)
```
```   639
```
```   640 lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
```
```   641 by (simp add: size_multiset_eq)
```
```   642
```
```   643 lemma size_single [simp]: "size {#b#} = 1"
```
```   644 by (simp add: size_multiset_overloaded_def)
```
```   645
```
```   646 lemma setsum_wcount_Int:
```
```   647   "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_of N) = setsum (wcount f N) A"
```
```   648 apply (induct rule: finite_induct)
```
```   649  apply simp
```
```   650 apply (simp add: Int_insert_left set_of_def wcount_def)
```
```   651 done
```
```   652
```
```   653 lemma size_multiset_union [simp]:
```
```   654   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
```
```   655 apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
```
```   656 apply (subst Int_commute)
```
```   657 apply (simp add: setsum_wcount_Int)
```
```   658 done
```
```   659
```
```   660 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   661 by (auto simp add: size_multiset_overloaded_def)
```
```   662
```
```   663 lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
```
```   664 by (auto simp add: size_multiset_eq multiset_eq_iff)
```
```   665
```
```   666 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   667 by (auto simp add: size_multiset_overloaded_def)
```
```   668
```
```   669 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
```
```   670 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
```
```   671
```
```   672 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   673 apply (unfold size_multiset_overloaded_eq)
```
```   674 apply (drule setsum_SucD)
```
```   675 apply auto
```
```   676 done
```
```   677
```
```   678 lemma size_eq_Suc_imp_eq_union:
```
```   679   assumes "size M = Suc n"
```
```   680   shows "\<exists>a N. M = N + {#a#}"
```
```   681 proof -
```
```   682   from assms obtain a where "a \<in># M"
```
```   683     by (erule size_eq_Suc_imp_elem [THEN exE])
```
```   684   then have "M = M - {#a#} + {#a#}" by simp
```
```   685   then show ?thesis by blast
```
```   686 qed
```
```   687
```
```   688 lemma size_mset_mono: assumes "A \<le> B"
```
```   689   shows "size A \<le> size(B::_ multiset)"
```
```   690 proof -
```
```   691   from assms[unfolded mset_le_exists_conv]
```
```   692   obtain C where B: "B = A + C" by auto
```
```   693   show ?thesis unfolding B by (induct C, auto)
```
```   694 qed
```
```   695
```
```   696 lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) \<le> size M"
```
```   697 by (rule size_mset_mono[OF multiset_filter_subset])
```
```   698
```
```   699 lemma size_Diff_submset:
```
```   700   "M \<le> M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)"
```
```   701 by (metis add_diff_cancel_left' size_union mset_le_exists_conv)
```
```   702
```
```   703 subsection {* Induction and case splits *}
```
```   704
```
```   705 theorem multiset_induct [case_names empty add, induct type: multiset]:
```
```   706   assumes empty: "P {#}"
```
```   707   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
```
```   708   shows "P M"
```
```   709 proof (induct n \<equiv> "size M" arbitrary: M)
```
```   710   case 0 thus "P M" by (simp add: empty)
```
```   711 next
```
```   712   case (Suc k)
```
```   713   obtain N x where "M = N + {#x#}"
```
```   714     using `Suc k = size M` [symmetric]
```
```   715     using size_eq_Suc_imp_eq_union by fast
```
```   716   with Suc add show "P M" by simp
```
```   717 qed
```
```   718
```
```   719 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
```
```   720 by (induct M) auto
```
```   721
```
```   722 lemma multiset_cases [cases type]:
```
```   723   obtains (empty) "M = {#}"
```
```   724     | (add) N x where "M = N + {#x#}"
```
```   725   using assms by (induct M) simp_all
```
```   726
```
```   727 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
```
```   728 by (cases "B = {#}") (auto dest: multi_member_split)
```
```   729
```
```   730 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
```
```   731 apply (subst multiset_eq_iff)
```
```   732 apply auto
```
```   733 done
```
```   734
```
```   735 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
```
```   736 proof (induct A arbitrary: B)
```
```   737   case (empty M)
```
```   738   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
```
```   739   then obtain M' x where "M = M' + {#x#}"
```
```   740     by (blast dest: multi_nonempty_split)
```
```   741   then show ?case by simp
```
```   742 next
```
```   743   case (add S x T)
```
```   744   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
```
```   745   have SxsubT: "S + {#x#} < T" by fact
```
```   746   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
```
```   747   then obtain T' where T: "T = T' + {#x#}"
```
```   748     by (blast dest: multi_member_split)
```
```   749   then have "S < T'" using SxsubT
```
```   750     by (blast intro: mset_less_add_bothsides)
```
```   751   then have "size S < size T'" using IH by simp
```
```   752   then show ?case using T by simp
```
```   753 qed
```
```   754
```
```   755
```
```   756 lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}"
```
```   757 by (cases M) auto
```
```   758
```
```   759 subsubsection {* Strong induction and subset induction for multisets *}
```
```   760
```
```   761 text {* Well-foundedness of strict subset relation *}
```
```   762
```
```   763 lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M < N}"
```
```   764 apply (rule wf_measure [THEN wf_subset, where f1=size])
```
```   765 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
```
```   766 done
```
```   767
```
```   768 lemma full_multiset_induct [case_names less]:
```
```   769 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
```
```   770 shows "P B"
```
```   771 apply (rule wf_less_mset_rel [THEN wf_induct])
```
```   772 apply (rule ih, auto)
```
```   773 done
```
```   774
```
```   775 lemma multi_subset_induct [consumes 2, case_names empty add]:
```
```   776 assumes "F \<le> A"
```
```   777   and empty: "P {#}"
```
```   778   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
```
```   779 shows "P F"
```
```   780 proof -
```
```   781   from `F \<le> A`
```
```   782   show ?thesis
```
```   783   proof (induct F)
```
```   784     show "P {#}" by fact
```
```   785   next
```
```   786     fix x F
```
```   787     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
```
```   788     show "P (F + {#x#})"
```
```   789     proof (rule insert)
```
```   790       from i show "x \<in># A" by (auto dest: mset_le_insertD)
```
```   791       from i have "F \<le> A" by (auto dest: mset_le_insertD)
```
```   792       with P show "P F" .
```
```   793     qed
```
```   794   qed
```
```   795 qed
```
```   796
```
```   797
```
```   798 subsection {* The fold combinator *}
```
```   799
```
```   800 definition fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
```
```   801 where
```
```   802   "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
```
```   803
```
```   804 lemma fold_mset_empty [simp]:
```
```   805   "fold_mset f s {#} = s"
```
```   806   by (simp add: fold_mset_def)
```
```   807
```
```   808 context comp_fun_commute
```
```   809 begin
```
```   810
```
```   811 lemma fold_mset_insert:
```
```   812   "fold_mset f s (M + {#x#}) = f x (fold_mset f s M)"
```
```   813 proof -
```
```   814   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
```
```   815     by (fact comp_fun_commute_funpow)
```
```   816   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
```
```   817     by (fact comp_fun_commute_funpow)
```
```   818   show ?thesis
```
```   819   proof (cases "x \<in> set_of M")
```
```   820     case False
```
```   821     then have *: "count (M + {#x#}) x = 1" by simp
```
```   822     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
```
```   823       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
```
```   824       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   825     with False * show ?thesis
```
```   826       by (simp add: fold_mset_def del: count_union)
```
```   827   next
```
```   828     case True
```
```   829     def N \<equiv> "set_of M - {x}"
```
```   830     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
```
```   831     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
```
```   832       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
```
```   833       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   834     with * show ?thesis by (simp add: fold_mset_def del: count_union) simp
```
```   835   qed
```
```   836 qed
```
```   837
```
```   838 corollary fold_mset_single [simp]:
```
```   839   "fold_mset f s {#x#} = f x s"
```
```   840 proof -
```
```   841   have "fold_mset f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
```
```   842   then show ?thesis by simp
```
```   843 qed
```
```   844
```
```   845 lemma fold_mset_fun_left_comm:
```
```   846   "f x (fold_mset f s M) = fold_mset f (f x s) M"
```
```   847   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
```
```   848
```
```   849 lemma fold_mset_union [simp]:
```
```   850   "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
```
```   851 proof (induct M)
```
```   852   case empty then show ?case by simp
```
```   853 next
```
```   854   case (add M x)
```
```   855   have "M + {#x#} + N = (M + N) + {#x#}"
```
```   856     by (simp add: ac_simps)
```
```   857   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
```
```   858 qed
```
```   859
```
```   860 lemma fold_mset_fusion:
```
```   861   assumes "comp_fun_commute g"
```
```   862   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")
```
```   863 proof -
```
```   864   interpret comp_fun_commute g by (fact assms)
```
```   865   show "PROP ?P" by (induct A) auto
```
```   866 qed
```
```   867
```
```   868 end
```
```   869
```
```   870 text {*
```
```   871   A note on code generation: When defining some function containing a
```
```   872   subterm @{term "fold_mset F"}, code generation is not automatic. When
```
```   873   interpreting locale @{text left_commutative} with @{text F}, the
```
```   874   would be code thms for @{const fold_mset} become thms like
```
```   875   @{term "fold_mset F z {#} = z"} where @{text F} is not a pattern but
```
```   876   contains defined symbols, i.e.\ is not a code thm. Hence a separate
```
```   877   constant with its own code thms needs to be introduced for @{text
```
```   878   F}. See the image operator below.
```
```   879 *}
```
```   880
```
```   881
```
```   882 subsection {* Image *}
```
```   883
```
```   884 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
```
```   885   "image_mset f = fold_mset (plus o single o f) {#}"
```
```   886
```
```   887 lemma comp_fun_commute_mset_image:
```
```   888   "comp_fun_commute (plus o single o f)"
```
```   889 proof
```
```   890 qed (simp add: ac_simps fun_eq_iff)
```
```   891
```
```   892 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
```
```   893   by (simp add: image_mset_def)
```
```   894
```
```   895 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
```
```   896 proof -
```
```   897   interpret comp_fun_commute "plus o single o f"
```
```   898     by (fact comp_fun_commute_mset_image)
```
```   899   show ?thesis by (simp add: image_mset_def)
```
```   900 qed
```
```   901
```
```   902 lemma image_mset_union [simp]:
```
```   903   "image_mset f (M + N) = image_mset f M + image_mset f N"
```
```   904 proof -
```
```   905   interpret comp_fun_commute "plus o single o f"
```
```   906     by (fact comp_fun_commute_mset_image)
```
```   907   show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
```
```   908 qed
```
```   909
```
```   910 corollary image_mset_insert:
```
```   911   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
```
```   912   by simp
```
```   913
```
```   914 lemma set_of_image_mset [simp]:
```
```   915   "set_of (image_mset f M) = image f (set_of M)"
```
```   916   by (induct M) simp_all
```
```   917
```
```   918 lemma size_image_mset [simp]:
```
```   919   "size (image_mset f M) = size M"
```
```   920   by (induct M) simp_all
```
```   921
```
```   922 lemma image_mset_is_empty_iff [simp]:
```
```   923   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
```
```   924   by (cases M) auto
```
```   925
```
```   926 syntax
```
```   927   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
```
```   928       ("({#_/. _ :# _#})")
```
```   929 translations
```
```   930   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
```
```   931
```
```   932 syntax (xsymbols)
```
```   933   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
```
```   934       ("({#_/. _ \<in># _#})")
```
```   935 translations
```
```   936   "{#e. x \<in># M#}" == "CONST image_mset (\<lambda>x. e) M"
```
```   937
```
```   938 syntax
```
```   939   "_comprehension3_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
```
```   940       ("({#_/ | _ :# _./ _#})")
```
```   941 translations
```
```   942   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
```
```   943
```
```   944 syntax
```
```   945   "_comprehension4_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
```
```   946       ("({#_/ | _ \<in># _./ _#})")
```
```   947 translations
```
```   948   "{#e | x\<in>#M. P#}" => "{#e. x \<in># {# x\<in>#M. P#}#}"
```
```   949
```
```   950 text {*
```
```   951   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
```
```   952   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
```
```   953   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
```
```   954   @{term "{#x+x|x:#M. x<c#}"}.
```
```   955 *}
```
```   956
```
```   957 lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_of M"
```
```   958   by (metis mem_set_of_iff set_of_image_mset)
```
```   959
```
```   960 functor image_mset: image_mset
```
```   961 proof -
```
```   962   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
```
```   963   proof
```
```   964     fix A
```
```   965     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
```
```   966       by (induct A) simp_all
```
```   967   qed
```
```   968   show "image_mset id = id"
```
```   969   proof
```
```   970     fix A
```
```   971     show "image_mset id A = id A"
```
```   972       by (induct A) simp_all
```
```   973   qed
```
```   974 qed
```
```   975
```
```   976 declare
```
```   977   image_mset.id [simp]
```
```   978   image_mset.identity [simp]
```
```   979
```
```   980 lemma image_mset_id[simp]: "image_mset id x = x"
```
```   981   unfolding id_def by auto
```
```   982
```
```   983 lemma image_mset_cong: "(\<And>x. x \<in># M \<Longrightarrow> f x = g x) \<Longrightarrow> {#f x. x \<in># M#} = {#g x. x \<in># M#}"
```
```   984   by (induct M) auto
```
```   985
```
```   986 lemma image_mset_cong_pair:
```
```   987   "(\<forall>x y. (x, y) \<in># M \<longrightarrow> f x y = g x y) \<Longrightarrow> {#f x y. (x, y) \<in># M#} = {#g x y. (x, y) \<in># M#}"
```
```   988   by (metis image_mset_cong split_cong)
```
```   989
```
```   990
```
```   991 subsection {* Further conversions *}
```
```   992
```
```   993 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
```
```   994   "multiset_of [] = {#}" |
```
```   995   "multiset_of (a # x) = multiset_of x + {# a #}"
```
```   996
```
```   997 lemma in_multiset_in_set:
```
```   998   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```   999   by (induct xs) simp_all
```
```  1000
```
```  1001 lemma count_multiset_of:
```
```  1002   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```  1003   by (induct xs) simp_all
```
```  1004
```
```  1005 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
```
```  1006   by (induct x) auto
```
```  1007
```
```  1008 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
```
```  1009 by (induct x) auto
```
```  1010
```
```  1011 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
```
```  1012 by (induct x) auto
```
```  1013
```
```  1014 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
```
```  1015 by (induct xs) auto
```
```  1016
```
```  1017 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
```
```  1018   by (induct xs) simp_all
```
```  1019
```
```  1020 lemma multiset_of_append [simp]:
```
```  1021   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
```
```  1022   by (induct xs arbitrary: ys) (auto simp: ac_simps)
```
```  1023
```
```  1024 lemma multiset_of_filter:
```
```  1025   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
```
```  1026   by (induct xs) simp_all
```
```  1027
```
```  1028 lemma multiset_of_rev [simp]:
```
```  1029   "multiset_of (rev xs) = multiset_of xs"
```
```  1030   by (induct xs) simp_all
```
```  1031
```
```  1032 lemma surj_multiset_of: "surj multiset_of"
```
```  1033 apply (unfold surj_def)
```
```  1034 apply (rule allI)
```
```  1035 apply (rule_tac M = y in multiset_induct)
```
```  1036  apply auto
```
```  1037 apply (rule_tac x = "x # xa" in exI)
```
```  1038 apply auto
```
```  1039 done
```
```  1040
```
```  1041 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
```
```  1042 by (induct x) auto
```
```  1043
```
```  1044 lemma distinct_count_atmost_1:
```
```  1045   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
```
```  1046 apply (induct x, simp, rule iffI, simp_all)
```
```  1047 apply (rename_tac a b)
```
```  1048 apply (rule conjI)
```
```  1049 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
```
```  1050 apply (erule_tac x = a in allE, simp, clarify)
```
```  1051 apply (erule_tac x = aa in allE, simp)
```
```  1052 done
```
```  1053
```
```  1054 lemma multiset_of_eq_setD:
```
```  1055   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
```
```  1056 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
```
```  1057
```
```  1058 lemma set_eq_iff_multiset_of_eq_distinct:
```
```  1059   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
```
```  1060     (set x = set y) = (multiset_of x = multiset_of y)"
```
```  1061 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
```
```  1062
```
```  1063 lemma set_eq_iff_multiset_of_remdups_eq:
```
```  1064    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
```
```  1065 apply (rule iffI)
```
```  1066 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
```
```  1067 apply (drule distinct_remdups [THEN distinct_remdups
```
```  1068       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
```
```  1069 apply simp
```
```  1070 done
```
```  1071
```
```  1072 lemma multiset_of_compl_union [simp]:
```
```  1073   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
```
```  1074   by (induct xs) (auto simp: ac_simps)
```
```  1075
```
```  1076 lemma count_multiset_of_length_filter:
```
```  1077   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```  1078   by (induct xs) auto
```
```  1079
```
```  1080 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
```
```  1081 apply (induct ls arbitrary: i)
```
```  1082  apply simp
```
```  1083 apply (case_tac i)
```
```  1084  apply auto
```
```  1085 done
```
```  1086
```
```  1087 lemma multiset_of_remove1[simp]:
```
```  1088   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
```
```  1089 by (induct xs) (auto simp add: multiset_eq_iff)
```
```  1090
```
```  1091 lemma multiset_of_eq_length:
```
```  1092   assumes "multiset_of xs = multiset_of ys"
```
```  1093   shows "length xs = length ys"
```
```  1094   using assms by (metis size_multiset_of)
```
```  1095
```
```  1096 lemma multiset_of_eq_length_filter:
```
```  1097   assumes "multiset_of xs = multiset_of ys"
```
```  1098   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
```
```  1099   using assms by (metis count_multiset_of)
```
```  1100
```
```  1101 lemma fold_multiset_equiv:
```
```  1102   assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
```
```  1103     and equiv: "multiset_of xs = multiset_of ys"
```
```  1104   shows "List.fold f xs = List.fold f ys"
```
```  1105 using f equiv [symmetric]
```
```  1106 proof (induct xs arbitrary: ys)
```
```  1107   case Nil then show ?case by simp
```
```  1108 next
```
```  1109   case (Cons x xs)
```
```  1110   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
```
```  1111   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
```
```  1112     by (rule Cons.prems(1)) (simp_all add: *)
```
```  1113   moreover from * have "x \<in> set ys" by simp
```
```  1114   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
```
```  1115   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
```
```  1116   ultimately show ?case by simp
```
```  1117 qed
```
```  1118
```
```  1119 lemma multiset_of_insort [simp]:
```
```  1120   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
```
```  1121   by (induct xs) (simp_all add: ac_simps)
```
```  1122
```
```  1123 lemma multiset_of_map:
```
```  1124   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
```
```  1125   by (induct xs) simp_all
```
```  1126
```
```  1127 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
```
```  1128 where
```
```  1129   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
```
```  1130
```
```  1131 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
```
```  1132 where
```
```  1133   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
```
```  1134 proof -
```
```  1135   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
```
```  1136   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
```
```  1137   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
```
```  1138 qed
```
```  1139
```
```  1140 lemma count_multiset_of_set [simp]:
```
```  1141   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
```
```  1142   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
```
```  1143   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
```
```  1144 proof -
```
```  1145   { fix A
```
```  1146     assume "x \<notin> A"
```
```  1147     have "count (multiset_of_set A) x = 0"
```
```  1148     proof (cases "finite A")
```
```  1149       case False then show ?thesis by simp
```
```  1150     next
```
```  1151       case True from True `x \<notin> A` show ?thesis by (induct A) auto
```
```  1152     qed
```
```  1153   } note * = this
```
```  1154   then show "PROP ?P" "PROP ?Q" "PROP ?R"
```
```  1155   by (auto elim!: Set.set_insert)
```
```  1156 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
```
```  1157
```
```  1158 lemma elem_multiset_of_set[simp, intro]: "finite A \<Longrightarrow> x \<in># multiset_of_set A \<longleftrightarrow> x \<in> A"
```
```  1159   by (induct A rule: finite_induct) simp_all
```
```  1160
```
```  1161 context linorder
```
```  1162 begin
```
```  1163
```
```  1164 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
```
```  1165 where
```
```  1166   "sorted_list_of_multiset M = fold_mset insort [] M"
```
```  1167
```
```  1168 lemma sorted_list_of_multiset_empty [simp]:
```
```  1169   "sorted_list_of_multiset {#} = []"
```
```  1170   by (simp add: sorted_list_of_multiset_def)
```
```  1171
```
```  1172 lemma sorted_list_of_multiset_singleton [simp]:
```
```  1173   "sorted_list_of_multiset {#x#} = [x]"
```
```  1174 proof -
```
```  1175   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1176   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1177 qed
```
```  1178
```
```  1179 lemma sorted_list_of_multiset_insert [simp]:
```
```  1180   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
```
```  1181 proof -
```
```  1182   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1183   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1184 qed
```
```  1185
```
```  1186 end
```
```  1187
```
```  1188 lemma multiset_of_sorted_list_of_multiset [simp]:
```
```  1189   "multiset_of (sorted_list_of_multiset M) = M"
```
```  1190   by (induct M) simp_all
```
```  1191
```
```  1192 lemma sorted_list_of_multiset_multiset_of [simp]:
```
```  1193   "sorted_list_of_multiset (multiset_of xs) = sort xs"
```
```  1194   by (induct xs) simp_all
```
```  1195
```
```  1196 lemma finite_set_of_multiset_of_set:
```
```  1197   assumes "finite A"
```
```  1198   shows "set_of (multiset_of_set A) = A"
```
```  1199   using assms by (induct A) simp_all
```
```  1200
```
```  1201 lemma infinite_set_of_multiset_of_set:
```
```  1202   assumes "\<not> finite A"
```
```  1203   shows "set_of (multiset_of_set A) = {}"
```
```  1204   using assms by simp
```
```  1205
```
```  1206 lemma set_sorted_list_of_multiset [simp]:
```
```  1207   "set (sorted_list_of_multiset M) = set_of M"
```
```  1208   by (induct M) (simp_all add: set_insort)
```
```  1209
```
```  1210 lemma sorted_list_of_multiset_of_set [simp]:
```
```  1211   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
```
```  1212   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
```
```  1213
```
```  1214
```
```  1215 subsection {* Big operators *}
```
```  1216
```
```  1217 no_notation times (infixl "*" 70)
```
```  1218 no_notation Groups.one ("1")
```
```  1219
```
```  1220 locale comm_monoid_mset = comm_monoid
```
```  1221 begin
```
```  1222
```
```  1223 definition F :: "'a multiset \<Rightarrow> 'a"
```
```  1224 where
```
```  1225   eq_fold: "F M = fold_mset f 1 M"
```
```  1226
```
```  1227 lemma empty [simp]:
```
```  1228   "F {#} = 1"
```
```  1229   by (simp add: eq_fold)
```
```  1230
```
```  1231 lemma singleton [simp]:
```
```  1232   "F {#x#} = x"
```
```  1233 proof -
```
```  1234   interpret comp_fun_commute
```
```  1235     by default (simp add: fun_eq_iff left_commute)
```
```  1236   show ?thesis by (simp add: eq_fold)
```
```  1237 qed
```
```  1238
```
```  1239 lemma union [simp]:
```
```  1240   "F (M + N) = F M * F N"
```
```  1241 proof -
```
```  1242   interpret comp_fun_commute f
```
```  1243     by default (simp add: fun_eq_iff left_commute)
```
```  1244   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
```
```  1245 qed
```
```  1246
```
```  1247 end
```
```  1248
```
```  1249 lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + \<Colon> 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
```
```  1250   by default (simp add: add_ac comp_def)
```
```  1251
```
```  1252 declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp]
```
```  1253
```
```  1254 lemma in_mset_fold_plus_iff[iff]: "x \<in># fold_mset (op +) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)"
```
```  1255   by (induct NN) auto
```
```  1256
```
```  1257 notation times (infixl "*" 70)
```
```  1258 notation Groups.one ("1")
```
```  1259
```
```  1260 context comm_monoid_add
```
```  1261 begin
```
```  1262
```
```  1263 definition msetsum :: "'a multiset \<Rightarrow> 'a"
```
```  1264 where
```
```  1265   "msetsum = comm_monoid_mset.F plus 0"
```
```  1266
```
```  1267 sublocale msetsum!: comm_monoid_mset plus 0
```
```  1268 where
```
```  1269   "comm_monoid_mset.F plus 0 = msetsum"
```
```  1270 proof -
```
```  1271   show "comm_monoid_mset plus 0" ..
```
```  1272   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
```
```  1273 qed
```
```  1274
```
```  1275 lemma setsum_unfold_msetsum:
```
```  1276   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
```
```  1277   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1278
```
```  1279 end
```
```  1280
```
```  1281 lemma msetsum_diff:
```
```  1282   fixes M N :: "('a \<Colon> ordered_cancel_comm_monoid_diff) multiset"
```
```  1283   shows "N \<le> M \<Longrightarrow> msetsum (M - N) = msetsum M - msetsum N"
```
```  1284   by (metis add_diff_cancel_left' msetsum.union ordered_cancel_comm_monoid_diff_class.add_diff_inverse)
```
```  1285
```
```  1286 lemma size_eq_msetsum: "size M = msetsum (image_mset (\<lambda>_. 1) M)"
```
```  1287 proof (induct M)
```
```  1288   case empty then show ?case by simp
```
```  1289 next
```
```  1290   case (add M x) then show ?case
```
```  1291     by (cases "x \<in> set_of M")
```
```  1292       (simp_all del: mem_set_of_iff add: size_multiset_overloaded_eq setsum.distrib setsum.delta' insert_absorb, simp)
```
```  1293 qed
```
```  1294
```
```  1295
```
```  1296 abbreviation Union_mset :: "'a multiset multiset \<Rightarrow> 'a multiset" where
```
```  1297   "Union_mset MM \<equiv> msetsum MM"
```
```  1298
```
```  1299 notation (xsymbols) Union_mset ("\<Union>#_" [900] 900)
```
```  1300
```
```  1301 lemma set_of_Union_mset[simp]: "set_of (\<Union># MM) = (\<Union>M \<in> set_of MM. set_of M)"
```
```  1302   by (induct MM) auto
```
```  1303
```
```  1304 lemma in_Union_mset_iff[iff]: "x \<in># \<Union># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
```
```  1305   by (induct MM) auto
```
```  1306
```
```  1307 syntax
```
```  1308   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1309       ("(3SUM _:#_. _)" [0, 51, 10] 10)
```
```  1310
```
```  1311 syntax (xsymbols)
```
```  1312   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1313       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1314
```
```  1315 syntax (HTML output)
```
```  1316   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1317       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1318
```
```  1319 translations
```
```  1320   "SUM i :# A. b" == "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"
```
```  1321
```
```  1322 context comm_monoid_mult
```
```  1323 begin
```
```  1324
```
```  1325 definition msetprod :: "'a multiset \<Rightarrow> 'a"
```
```  1326 where
```
```  1327   "msetprod = comm_monoid_mset.F times 1"
```
```  1328
```
```  1329 sublocale msetprod!: comm_monoid_mset times 1
```
```  1330 where
```
```  1331   "comm_monoid_mset.F times 1 = msetprod"
```
```  1332 proof -
```
```  1333   show "comm_monoid_mset times 1" ..
```
```  1334   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
```
```  1335 qed
```
```  1336
```
```  1337 lemma msetprod_empty:
```
```  1338   "msetprod {#} = 1"
```
```  1339   by (fact msetprod.empty)
```
```  1340
```
```  1341 lemma msetprod_singleton:
```
```  1342   "msetprod {#x#} = x"
```
```  1343   by (fact msetprod.singleton)
```
```  1344
```
```  1345 lemma msetprod_Un:
```
```  1346   "msetprod (A + B) = msetprod A * msetprod B"
```
```  1347   by (fact msetprod.union)
```
```  1348
```
```  1349 lemma setprod_unfold_msetprod:
```
```  1350   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
```
```  1351   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1352
```
```  1353 lemma msetprod_multiplicity:
```
```  1354   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
```
```  1355   by (simp add: fold_mset_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
```
```  1356
```
```  1357 end
```
```  1358
```
```  1359 syntax
```
```  1360   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1361       ("(3PROD _:#_. _)" [0, 51, 10] 10)
```
```  1362
```
```  1363 syntax (xsymbols)
```
```  1364   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1365       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1366
```
```  1367 syntax (HTML output)
```
```  1368   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1369       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1370
```
```  1371 translations
```
```  1372   "PROD i :# A. b" == "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"
```
```  1373
```
```  1374 lemma (in comm_semiring_1) dvd_msetprod:
```
```  1375   assumes "x \<in># A"
```
```  1376   shows "x dvd msetprod A"
```
```  1377 proof -
```
```  1378   from assms have "A = (A - {#x#}) + {#x#}" by simp
```
```  1379   then obtain B where "A = B + {#x#}" ..
```
```  1380   then show ?thesis by simp
```
```  1381 qed
```
```  1382
```
```  1383
```
```  1384 subsection {* Replicate operation *}
```
```  1385
```
```  1386 definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
```
```  1387   "replicate_mset n x = ((op + {#x#}) ^^ n) {#}"
```
```  1388
```
```  1389 lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}"
```
```  1390   unfolding replicate_mset_def by simp
```
```  1391
```
```  1392 lemma replicate_mset_Suc[simp]: "replicate_mset (Suc n) x = replicate_mset n x + {#x#}"
```
```  1393   unfolding replicate_mset_def by (induct n) (auto intro: add.commute)
```
```  1394
```
```  1395 lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y"
```
```  1396   unfolding replicate_mset_def by (induct n) simp_all
```
```  1397
```
```  1398 lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)"
```
```  1399   unfolding replicate_mset_def by (induct n) simp_all
```
```  1400
```
```  1401 lemma set_of_replicate_mset_subset[simp]: "set_of (replicate_mset n x) = (if n = 0 then {} else {x})"
```
```  1402   by (auto split: if_splits)
```
```  1403
```
```  1404 lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n"
```
```  1405   by (induct n, simp_all)
```
```  1406
```
```  1407 lemma count_le_replicate_mset_le: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<le> M"
```
```  1408   by (auto simp add: assms mset_less_eqI) (metis count_replicate_mset less_eq_multiset.rep_eq)
```
```  1409
```
```  1410 lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
```
```  1411   by (induct D) simp_all
```
```  1412
```
```  1413
```
```  1414 subsection {* Alternative representations *}
```
```  1415
```
```  1416 subsubsection {* Lists *}
```
```  1417
```
```  1418 context linorder
```
```  1419 begin
```
```  1420
```
```  1421 lemma multiset_of_insort [simp]:
```
```  1422   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
```
```  1423   by (induct xs) (simp_all add: ac_simps)
```
```  1424
```
```  1425 lemma multiset_of_sort [simp]:
```
```  1426   "multiset_of (sort_key k xs) = multiset_of xs"
```
```  1427   by (induct xs) (simp_all add: ac_simps)
```
```  1428
```
```  1429 text {*
```
```  1430   This lemma shows which properties suffice to show that a function
```
```  1431   @{text "f"} with @{text "f xs = ys"} behaves like sort.
```
```  1432 *}
```
```  1433
```
```  1434 lemma properties_for_sort_key:
```
```  1435   assumes "multiset_of ys = multiset_of xs"
```
```  1436   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
```
```  1437   and "sorted (map f ys)"
```
```  1438   shows "sort_key f xs = ys"
```
```  1439 using assms
```
```  1440 proof (induct xs arbitrary: ys)
```
```  1441   case Nil then show ?case by simp
```
```  1442 next
```
```  1443   case (Cons x xs)
```
```  1444   from Cons.prems(2) have
```
```  1445     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
```
```  1446     by (simp add: filter_remove1)
```
```  1447   with Cons.prems have "sort_key f xs = remove1 x ys"
```
```  1448     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
```
```  1449   moreover from Cons.prems have "x \<in> set ys"
```
```  1450     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
```
```  1451   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
```
```  1452 qed
```
```  1453
```
```  1454 lemma properties_for_sort:
```
```  1455   assumes multiset: "multiset_of ys = multiset_of xs"
```
```  1456   and "sorted ys"
```
```  1457   shows "sort xs = ys"
```
```  1458 proof (rule properties_for_sort_key)
```
```  1459   from multiset show "multiset_of ys = multiset_of xs" .
```
```  1460   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
```
```  1461   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
```
```  1462     by (rule multiset_of_eq_length_filter)
```
```  1463   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
```
```  1464     by simp
```
```  1465   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
```
```  1466     by (simp add: replicate_length_filter)
```
```  1467 qed
```
```  1468
```
```  1469 lemma sort_key_by_quicksort:
```
```  1470   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
```
```  1471     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
```
```  1472     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
```
```  1473 proof (rule properties_for_sort_key)
```
```  1474   show "multiset_of ?rhs = multiset_of ?lhs"
```
```  1475     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
```
```  1476 next
```
```  1477   show "sorted (map f ?rhs)"
```
```  1478     by (auto simp add: sorted_append intro: sorted_map_same)
```
```  1479 next
```
```  1480   fix l
```
```  1481   assume "l \<in> set ?rhs"
```
```  1482   let ?pivot = "f (xs ! (length xs div 2))"
```
```  1483   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
```
```  1484   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
```
```  1485     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
```
```  1486   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
```
```  1487   have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
```
```  1488   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
```
```  1489     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
```
```  1490   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
```
```  1491   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
```
```  1492   proof (cases "f l" ?pivot rule: linorder_cases)
```
```  1493     case less
```
```  1494     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
```
```  1495     with less show ?thesis
```
```  1496       by (simp add: filter_sort [symmetric] ** ***)
```
```  1497   next
```
```  1498     case equal then show ?thesis
```
```  1499       by (simp add: * less_le)
```
```  1500   next
```
```  1501     case greater
```
```  1502     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
```
```  1503     with greater show ?thesis
```
```  1504       by (simp add: filter_sort [symmetric] ** ***)
```
```  1505   qed
```
```  1506 qed
```
```  1507
```
```  1508 lemma sort_by_quicksort:
```
```  1509   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
```
```  1510     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
```
```  1511     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
```
```  1512   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
```
```  1513
```
```  1514 text {* A stable parametrized quicksort *}
```
```  1515
```
```  1516 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
```
```  1517   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
```
```  1518
```
```  1519 lemma part_code [code]:
```
```  1520   "part f pivot [] = ([], [], [])"
```
```  1521   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
```
```  1522      if x' < pivot then (x # lts, eqs, gts)
```
```  1523      else if x' > pivot then (lts, eqs, x # gts)
```
```  1524      else (lts, x # eqs, gts))"
```
```  1525   by (auto simp add: part_def Let_def split_def)
```
```  1526
```
```  1527 lemma sort_key_by_quicksort_code [code]:
```
```  1528   "sort_key f xs = (case xs of [] \<Rightarrow> []
```
```  1529     | [x] \<Rightarrow> xs
```
```  1530     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
```
```  1531     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1532        in sort_key f lts @ eqs @ sort_key f gts))"
```
```  1533 proof (cases xs)
```
```  1534   case Nil then show ?thesis by simp
```
```  1535 next
```
```  1536   case (Cons _ ys) note hyps = Cons show ?thesis
```
```  1537   proof (cases ys)
```
```  1538     case Nil with hyps show ?thesis by simp
```
```  1539   next
```
```  1540     case (Cons _ zs) note hyps = hyps Cons show ?thesis
```
```  1541     proof (cases zs)
```
```  1542       case Nil with hyps show ?thesis by auto
```
```  1543     next
```
```  1544       case Cons
```
```  1545       from sort_key_by_quicksort [of f xs]
```
```  1546       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1547         in sort_key f lts @ eqs @ sort_key f gts)"
```
```  1548       by (simp only: split_def Let_def part_def fst_conv snd_conv)
```
```  1549       with hyps Cons show ?thesis by (simp only: list.cases)
```
```  1550     qed
```
```  1551   qed
```
```  1552 qed
```
```  1553
```
```  1554 end
```
```  1555
```
```  1556 hide_const (open) part
```
```  1557
```
```  1558 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
```
```  1559   by (induct xs) (auto intro: order_trans)
```
```  1560
```
```  1561 lemma multiset_of_update:
```
```  1562   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
```
```  1563 proof (induct ls arbitrary: i)
```
```  1564   case Nil then show ?case by simp
```
```  1565 next
```
```  1566   case (Cons x xs)
```
```  1567   show ?case
```
```  1568   proof (cases i)
```
```  1569     case 0 then show ?thesis by simp
```
```  1570   next
```
```  1571     case (Suc i')
```
```  1572     with Cons show ?thesis
```
```  1573       apply simp
```
```  1574       apply (subst add.assoc)
```
```  1575       apply (subst add.commute [of "{#v#}" "{#x#}"])
```
```  1576       apply (subst add.assoc [symmetric])
```
```  1577       apply simp
```
```  1578       apply (rule mset_le_multiset_union_diff_commute)
```
```  1579       apply (simp add: mset_le_single nth_mem_multiset_of)
```
```  1580       done
```
```  1581   qed
```
```  1582 qed
```
```  1583
```
```  1584 lemma multiset_of_swap:
```
```  1585   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
```
```  1586     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
```
```  1587   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
```
```  1588
```
```  1589
```
```  1590 subsection {* The multiset order *}
```
```  1591
```
```  1592 subsubsection {* Well-foundedness *}
```
```  1593
```
```  1594 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1595   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```  1596       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```  1597
```
```  1598 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1599   "mult r = (mult1 r)\<^sup>+"
```
```  1600
```
```  1601 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```  1602 by (simp add: mult1_def)
```
```  1603
```
```  1604 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```  1605     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```  1606     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```  1607   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
```
```  1608 proof (unfold mult1_def)
```
```  1609   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```  1610   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```  1611   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```  1612
```
```  1613   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```  1614   then have "\<exists>a' M0' K.
```
```  1615       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```  1616   then show "?case1 \<or> ?case2"
```
```  1617   proof (elim exE conjE)
```
```  1618     fix a' M0' K
```
```  1619     assume N: "N = M0' + K" and r: "?r K a'"
```
```  1620     assume "M0 + {#a#} = M0' + {#a'#}"
```
```  1621     then have "M0 = M0' \<and> a = a' \<or>
```
```  1622         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```  1623       by (simp only: add_eq_conv_ex)
```
```  1624     then show ?thesis
```
```  1625     proof (elim disjE conjE exE)
```
```  1626       assume "M0 = M0'" "a = a'"
```
```  1627       with N r have "?r K a \<and> N = M0 + K" by simp
```
```  1628       then have ?case2 .. then show ?thesis ..
```
```  1629     next
```
```  1630       fix K'
```
```  1631       assume "M0' = K' + {#a#}"
```
```  1632       with N have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
```
```  1633
```
```  1634       assume "M0 = K' + {#a'#}"
```
```  1635       with r have "?R (K' + K) M0" by blast
```
```  1636       with n have ?case1 by simp then show ?thesis ..
```
```  1637     qed
```
```  1638   qed
```
```  1639 qed
```
```  1640
```
```  1641 lemma all_accessible: "wf r ==> \<forall>M. M \<in> Wellfounded.acc (mult1 r)"
```
```  1642 proof
```
```  1643   let ?R = "mult1 r"
```
```  1644   let ?W = "Wellfounded.acc ?R"
```
```  1645   {
```
```  1646     fix M M0 a
```
```  1647     assume M0: "M0 \<in> ?W"
```
```  1648       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1649       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```  1650     have "M0 + {#a#} \<in> ?W"
```
```  1651     proof (rule accI [of "M0 + {#a#}"])
```
```  1652       fix N
```
```  1653       assume "(N, M0 + {#a#}) \<in> ?R"
```
```  1654       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```  1655           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```  1656         by (rule less_add)
```
```  1657       then show "N \<in> ?W"
```
```  1658       proof (elim exE disjE conjE)
```
```  1659         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```  1660         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```  1661         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
```
```  1662         then show "N \<in> ?W" by (simp only: N)
```
```  1663       next
```
```  1664         fix K
```
```  1665         assume N: "N = M0 + K"
```
```  1666         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```  1667         then have "M0 + K \<in> ?W"
```
```  1668         proof (induct K)
```
```  1669           case empty
```
```  1670           from M0 show "M0 + {#} \<in> ?W" by simp
```
```  1671         next
```
```  1672           case (add K x)
```
```  1673           from add.prems have "(x, a) \<in> r" by simp
```
```  1674           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```  1675           moreover from add have "M0 + K \<in> ?W" by simp
```
```  1676           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
```
```  1677           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
```
```  1678         qed
```
```  1679         then show "N \<in> ?W" by (simp only: N)
```
```  1680       qed
```
```  1681     qed
```
```  1682   } note tedious_reasoning = this
```
```  1683
```
```  1684   assume wf: "wf r"
```
```  1685   fix M
```
```  1686   show "M \<in> ?W"
```
```  1687   proof (induct M)
```
```  1688     show "{#} \<in> ?W"
```
```  1689     proof (rule accI)
```
```  1690       fix b assume "(b, {#}) \<in> ?R"
```
```  1691       with not_less_empty show "b \<in> ?W" by contradiction
```
```  1692     qed
```
```  1693
```
```  1694     fix M a assume "M \<in> ?W"
```
```  1695     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1696     proof induct
```
```  1697       fix a
```
```  1698       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1699       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1700       proof
```
```  1701         fix M assume "M \<in> ?W"
```
```  1702         then show "M + {#a#} \<in> ?W"
```
```  1703           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
```
```  1704       qed
```
```  1705     qed
```
```  1706     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
```
```  1707   qed
```
```  1708 qed
```
```  1709
```
```  1710 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```  1711 by (rule acc_wfI) (rule all_accessible)
```
```  1712
```
```  1713 theorem wf_mult: "wf r ==> wf (mult r)"
```
```  1714 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
```
```  1715
```
```  1716
```
```  1717 subsubsection {* Closure-free presentation *}
```
```  1718
```
```  1719 text {* One direction. *}
```
```  1720
```
```  1721 lemma mult_implies_one_step:
```
```  1722   "trans r ==> (M, N) \<in> mult r ==>
```
```  1723     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```  1724     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```  1725 apply (unfold mult_def mult1_def set_of_def)
```
```  1726 apply (erule converse_trancl_induct, clarify)
```
```  1727  apply (rule_tac x = M0 in exI, simp, clarify)
```
```  1728 apply (case_tac "a :# K")
```
```  1729  apply (rule_tac x = I in exI)
```
```  1730  apply (simp (no_asm))
```
```  1731  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```  1732  apply (simp (no_asm_simp) add: add.assoc [symmetric])
```
```  1733  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong)
```
```  1734  apply (simp add: diff_union_single_conv)
```
```  1735  apply (simp (no_asm_use) add: trans_def)
```
```  1736  apply blast
```
```  1737 apply (subgoal_tac "a :# I")
```
```  1738  apply (rule_tac x = "I - {#a#}" in exI)
```
```  1739  apply (rule_tac x = "J + {#a#}" in exI)
```
```  1740  apply (rule_tac x = "K + Ka" in exI)
```
```  1741  apply (rule conjI)
```
```  1742   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1743  apply (rule conjI)
```
```  1744   apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
```
```  1745   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1746  apply (simp (no_asm_use) add: trans_def)
```
```  1747  apply blast
```
```  1748 apply (subgoal_tac "a :# (M0 + {#a#})")
```
```  1749  apply simp
```
```  1750 apply (simp (no_asm))
```
```  1751 done
```
```  1752
```
```  1753 lemma one_step_implies_mult_aux:
```
```  1754   "trans r ==>
```
```  1755     \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
```
```  1756       --> (I + K, I + J) \<in> mult r"
```
```  1757 apply (induct_tac n, auto)
```
```  1758 apply (frule size_eq_Suc_imp_eq_union, clarify)
```
```  1759 apply (rename_tac "J'", simp)
```
```  1760 apply (erule notE, auto)
```
```  1761 apply (case_tac "J' = {#}")
```
```  1762  apply (simp add: mult_def)
```
```  1763  apply (rule r_into_trancl)
```
```  1764  apply (simp add: mult1_def set_of_def, blast)
```
```  1765 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```  1766 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```  1767 apply (erule_tac P = "\<forall>k \<in> set_of K. P k" for P in rev_mp)
```
```  1768 apply (erule ssubst)
```
```  1769 apply (simp add: Ball_def, auto)
```
```  1770 apply (subgoal_tac
```
```  1771   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
```
```  1772     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```  1773  prefer 2
```
```  1774  apply force
```
```  1775 apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
```
```  1776 apply (erule trancl_trans)
```
```  1777 apply (rule r_into_trancl)
```
```  1778 apply (simp add: mult1_def set_of_def)
```
```  1779 apply (rule_tac x = a in exI)
```
```  1780 apply (rule_tac x = "I + J'" in exI)
```
```  1781 apply (simp add: ac_simps)
```
```  1782 done
```
```  1783
```
```  1784 lemma one_step_implies_mult:
```
```  1785   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```  1786     ==> (I + K, I + J) \<in> mult r"
```
```  1787 using one_step_implies_mult_aux by blast
```
```  1788
```
```  1789
```
```  1790 subsubsection {* Partial-order properties *}
```
```  1791
```
```  1792 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "#<#" 50) where
```
```  1793   "M' #<# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
```
```  1794
```
```  1795 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "#<=#" 50) where
```
```  1796   "M' #<=# M \<longleftrightarrow> M' #<# M \<or> M' = M"
```
```  1797
```
```  1798 notation (xsymbols) less_multiset (infix "#\<subset>#" 50)
```
```  1799 notation (xsymbols) le_multiset (infix "#\<subseteq>#" 50)
```
```  1800
```
```  1801 interpretation multiset_order: order le_multiset less_multiset
```
```  1802 proof -
```
```  1803   have irrefl: "\<And>M :: 'a multiset. \<not> M #\<subset># M"
```
```  1804   proof
```
```  1805     fix M :: "'a multiset"
```
```  1806     assume "M #\<subset># M"
```
```  1807     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
```
```  1808     have "trans {(x'::'a, x). x' < x}"
```
```  1809       by (rule transI) simp
```
```  1810     moreover note MM
```
```  1811     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
```
```  1812       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
```
```  1813       by (rule mult_implies_one_step)
```
```  1814     then obtain I J K where "M = I + J" and "M = I + K"
```
```  1815       and "J \<noteq> {#}" and "(\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})" by blast
```
```  1816     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
```
```  1817     have "finite (set_of K)" by simp
```
```  1818     moreover note aux2
```
```  1819     ultimately have "set_of K = {}"
```
```  1820       by (induct rule: finite_induct) (auto intro: order_less_trans)
```
```  1821     with aux1 show False by simp
```
```  1822   qed
```
```  1823   have trans: "\<And>K M N :: 'a multiset. K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># N"
```
```  1824     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
```
```  1825   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
```
```  1826     by default (auto simp add: le_multiset_def irrefl dest: trans)
```
```  1827 qed
```
```  1828
```
```  1829 lemma mult_less_irrefl [elim!]: "M #\<subset># (M::'a::order multiset) ==> R"
```
```  1830   by simp
```
```  1831
```
```  1832
```
```  1833 subsubsection {* Monotonicity of multiset union *}
```
```  1834
```
```  1835 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
```
```  1836 apply (unfold mult1_def)
```
```  1837 apply auto
```
```  1838 apply (rule_tac x = a in exI)
```
```  1839 apply (rule_tac x = "C + M0" in exI)
```
```  1840 apply (simp add: add.assoc)
```
```  1841 done
```
```  1842
```
```  1843 lemma union_less_mono2: "B #\<subset># D ==> C + B #\<subset># C + (D::'a::order multiset)"
```
```  1844 apply (unfold less_multiset_def mult_def)
```
```  1845 apply (erule trancl_induct)
```
```  1846  apply (blast intro: mult1_union)
```
```  1847 apply (blast intro: mult1_union trancl_trans)
```
```  1848 done
```
```  1849
```
```  1850 lemma union_less_mono1: "B #\<subset># D ==> B + C #\<subset># D + (C::'a::order multiset)"
```
```  1851 apply (subst add.commute [of B C])
```
```  1852 apply (subst add.commute [of D C])
```
```  1853 apply (erule union_less_mono2)
```
```  1854 done
```
```  1855
```
```  1856 lemma union_less_mono:
```
```  1857   "A #\<subset># C ==> B #\<subset># D ==> A + B #\<subset># C + (D::'a::order multiset)"
```
```  1858   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
```
```  1859
```
```  1860 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
```
```  1861 proof
```
```  1862 qed (auto simp add: le_multiset_def intro: union_less_mono2)
```
```  1863
```
```  1864
```
```  1865 subsubsection {* Termination proofs with multiset orders *}
```
```  1866
```
```  1867 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
```
```  1868   and multi_member_this: "x \<in># {# x #} + XS"
```
```  1869   and multi_member_last: "x \<in># {# x #}"
```
```  1870   by auto
```
```  1871
```
```  1872 definition "ms_strict = mult pair_less"
```
```  1873 definition "ms_weak = ms_strict \<union> Id"
```
```  1874
```
```  1875 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
```
```  1876 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
```
```  1877 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
```
```  1878
```
```  1879 lemma smsI:
```
```  1880   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
```
```  1881   unfolding ms_strict_def
```
```  1882 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
```
```  1883
```
```  1884 lemma wmsI:
```
```  1885   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
```
```  1886   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
```
```  1887 unfolding ms_weak_def ms_strict_def
```
```  1888 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
```
```  1889
```
```  1890 inductive pw_leq
```
```  1891 where
```
```  1892   pw_leq_empty: "pw_leq {#} {#}"
```
```  1893 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
```
```  1894
```
```  1895 lemma pw_leq_lstep:
```
```  1896   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
```
```  1897 by (drule pw_leq_step) (rule pw_leq_empty, simp)
```
```  1898
```
```  1899 lemma pw_leq_split:
```
```  1900   assumes "pw_leq X Y"
```
```  1901   shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
```
```  1902   using assms
```
```  1903 proof (induct)
```
```  1904   case pw_leq_empty thus ?case by auto
```
```  1905 next
```
```  1906   case (pw_leq_step x y X Y)
```
```  1907   then obtain A B Z where
```
```  1908     [simp]: "X = A + Z" "Y = B + Z"
```
```  1909       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
```
```  1910     by auto
```
```  1911   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
```
```  1912     unfolding pair_leq_def by auto
```
```  1913   thus ?case
```
```  1914   proof
```
```  1915     assume [simp]: "x = y"
```
```  1916     have
```
```  1917       "{#x#} + X = A + ({#y#}+Z)
```
```  1918       \<and> {#y#} + Y = B + ({#y#}+Z)
```
```  1919       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
```
```  1920       by (auto simp: ac_simps)
```
```  1921     thus ?case by (intro exI)
```
```  1922   next
```
```  1923     assume A: "(x, y) \<in> pair_less"
```
```  1924     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
```
```  1925     have "{#x#} + X = ?A' + Z"
```
```  1926       "{#y#} + Y = ?B' + Z"
```
```  1927       by (auto simp add: ac_simps)
```
```  1928     moreover have
```
```  1929       "(set_of ?A', set_of ?B') \<in> max_strict"
```
```  1930       using 1 A unfolding max_strict_def
```
```  1931       by (auto elim!: max_ext.cases)
```
```  1932     ultimately show ?thesis by blast
```
```  1933   qed
```
```  1934 qed
```
```  1935
```
```  1936 lemma
```
```  1937   assumes pwleq: "pw_leq Z Z'"
```
```  1938   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
```
```  1939   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
```
```  1940   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
```
```  1941 proof -
```
```  1942   from pw_leq_split[OF pwleq]
```
```  1943   obtain A' B' Z''
```
```  1944     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
```
```  1945     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
```
```  1946     by blast
```
```  1947   {
```
```  1948     assume max: "(set_of A, set_of B) \<in> max_strict"
```
```  1949     from mx_or_empty
```
```  1950     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
```
```  1951     proof
```
```  1952       assume max': "(set_of A', set_of B') \<in> max_strict"
```
```  1953       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
```
```  1954         by (auto simp: max_strict_def intro: max_ext_additive)
```
```  1955       thus ?thesis by (rule smsI)
```
```  1956     next
```
```  1957       assume [simp]: "A' = {#} \<and> B' = {#}"
```
```  1958       show ?thesis by (rule smsI) (auto intro: max)
```
```  1959     qed
```
```  1960     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:ac_simps)
```
```  1961     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
```
```  1962   }
```
```  1963   from mx_or_empty
```
```  1964   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
```
```  1965   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
```
```  1966 qed
```
```  1967
```
```  1968 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
```
```  1969 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
```
```  1970 and nonempty_single: "{# x #} \<noteq> {#}"
```
```  1971 by auto
```
```  1972
```
```  1973 setup {*
```
```  1974 let
```
```  1975   fun msetT T = Type (@{type_name multiset}, [T]);
```
```  1976
```
```  1977   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
```
```  1978     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
```
```  1979     | mk_mset T (x :: xs) =
```
```  1980           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
```
```  1981                 mk_mset T [x] \$ mk_mset T xs
```
```  1982
```
```  1983   fun mset_member_tac m i =
```
```  1984       (if m <= 0 then
```
```  1985            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
```
```  1986        else
```
```  1987            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
```
```  1988
```
```  1989   val mset_nonempty_tac =
```
```  1990       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
```
```  1991
```
```  1992   fun regroup_munion_conv ctxt =
```
```  1993     Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus}
```
```  1994       (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
```
```  1995
```
```  1996   fun unfold_pwleq_tac i =
```
```  1997     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
```
```  1998       ORELSE (rtac @{thm pw_leq_lstep} i)
```
```  1999       ORELSE (rtac @{thm pw_leq_empty} i)
```
```  2000
```
```  2001   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
```
```  2002                       @{thm Un_insert_left}, @{thm Un_empty_left}]
```
```  2003 in
```
```  2004   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
```
```  2005   {
```
```  2006     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
```
```  2007     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
```
```  2008     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
```
```  2009     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
```
```  2010     reduction_pair= @{thm ms_reduction_pair}
```
```  2011   })
```
```  2012 end
```
```  2013 *}
```
```  2014
```
```  2015
```
```  2016 subsection {* Legacy theorem bindings *}
```
```  2017
```
```  2018 lemmas multi_count_eq = multiset_eq_iff [symmetric]
```
```  2019
```
```  2020 lemma union_commute: "M + N = N + (M::'a multiset)"
```
```  2021   by (fact add.commute)
```
```  2022
```
```  2023 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```  2024   by (fact add.assoc)
```
```  2025
```
```  2026 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```  2027   by (fact add.left_commute)
```
```  2028
```
```  2029 lemmas union_ac = union_assoc union_commute union_lcomm
```
```  2030
```
```  2031 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
```
```  2032   by (fact add_right_cancel)
```
```  2033
```
```  2034 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
```
```  2035   by (fact add_left_cancel)
```
```  2036
```
```  2037 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
```
```  2038   by (fact add_left_imp_eq)
```
```  2039
```
```  2040 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
```
```  2041   by (fact order_less_trans)
```
```  2042
```
```  2043 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
```
```  2044   by (fact inf.commute)
```
```  2045
```
```  2046 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
```
```  2047   by (fact inf.assoc [symmetric])
```
```  2048
```
```  2049 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
```
```  2050   by (fact inf.left_commute)
```
```  2051
```
```  2052 lemmas multiset_inter_ac =
```
```  2053   multiset_inter_commute
```
```  2054   multiset_inter_assoc
```
```  2055   multiset_inter_left_commute
```
```  2056
```
```  2057 lemma mult_less_not_refl:
```
```  2058   "\<not> M #\<subset># (M::'a::order multiset)"
```
```  2059   by (fact multiset_order.less_irrefl)
```
```  2060
```
```  2061 lemma mult_less_trans:
```
```  2062   "K #\<subset># M ==> M #\<subset># N ==> K #\<subset># (N::'a::order multiset)"
```
```  2063   by (fact multiset_order.less_trans)
```
```  2064
```
```  2065 lemma mult_less_not_sym:
```
```  2066   "M #\<subset># N ==> \<not> N #\<subset># (M::'a::order multiset)"
```
```  2067   by (fact multiset_order.less_not_sym)
```
```  2068
```
```  2069 lemma mult_less_asym:
```
```  2070   "M #\<subset># N ==> (\<not> P ==> N #\<subset># (M::'a::order multiset)) ==> P"
```
```  2071   by (fact multiset_order.less_asym)
```
```  2072
```
```  2073 ML {*
```
```  2074 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
```
```  2075                       (Const _ \$ t') =
```
```  2076     let
```
```  2077       val (maybe_opt, ps) =
```
```  2078         Nitpick_Model.dest_plain_fun t' ||> op ~~
```
```  2079         ||> map (apsnd (snd o HOLogic.dest_number))
```
```  2080       fun elems_for t =
```
```  2081         case AList.lookup (op =) ps t of
```
```  2082           SOME n => replicate n t
```
```  2083         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
```
```  2084     in
```
```  2085       case maps elems_for (all_values elem_T) @
```
```  2086            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
```
```  2087             else []) of
```
```  2088         [] => Const (@{const_name zero_class.zero}, T)
```
```  2089       | ts => foldl1 (fn (t1, t2) =>
```
```  2090                          Const (@{const_name plus_class.plus}, T --> T --> T)
```
```  2091                          \$ t1 \$ t2)
```
```  2092                      (map (curry (op \$) (Const (@{const_name single},
```
```  2093                                                 elem_T --> T))) ts)
```
```  2094     end
```
```  2095   | multiset_postproc _ _ _ _ t = t
```
```  2096 *}
```
```  2097
```
```  2098 declaration {*
```
```  2099 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
```
```  2100     multiset_postproc
```
```  2101 *}
```
```  2102
```
```  2103
```
```  2104 subsection {* Naive implementation using lists *}
```
```  2105
```
```  2106 code_datatype multiset_of
```
```  2107
```
```  2108 lemma [code]:
```
```  2109   "{#} = multiset_of []"
```
```  2110   by simp
```
```  2111
```
```  2112 lemma [code]:
```
```  2113   "{#x#} = multiset_of [x]"
```
```  2114   by simp
```
```  2115
```
```  2116 lemma union_code [code]:
```
```  2117   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
```
```  2118   by simp
```
```  2119
```
```  2120 lemma [code]:
```
```  2121   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
```
```  2122   by (simp add: multiset_of_map)
```
```  2123
```
```  2124 lemma [code]:
```
```  2125   "filter_mset f (multiset_of xs) = multiset_of (filter f xs)"
```
```  2126   by (simp add: multiset_of_filter)
```
```  2127
```
```  2128 lemma [code]:
```
```  2129   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
```
```  2130   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
```
```  2131
```
```  2132 lemma [code]:
```
```  2133   "multiset_of xs #\<inter> multiset_of ys =
```
```  2134     multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2135       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
```
```  2136 proof -
```
```  2137   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2138     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
```
```  2139       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
```
```  2140     by (induct xs arbitrary: ys)
```
```  2141       (auto simp add: mem_set_multiset_eq inter_add_right1 inter_add_right2 ac_simps)
```
```  2142   then show ?thesis by simp
```
```  2143 qed
```
```  2144
```
```  2145 lemma [code]:
```
```  2146   "multiset_of xs #\<union> multiset_of ys =
```
```  2147     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
```
```  2148 proof -
```
```  2149   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
```
```  2150       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
```
```  2151     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
```
```  2152   then show ?thesis by simp
```
```  2153 qed
```
```  2154
```
```  2155 declare in_multiset_in_set [code_unfold]
```
```  2156
```
```  2157 lemma [code]:
```
```  2158   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
```
```  2159 proof -
```
```  2160   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
```
```  2161     by (induct xs) simp_all
```
```  2162   then show ?thesis by simp
```
```  2163 qed
```
```  2164
```
```  2165 declare set_of_multiset_of [code]
```
```  2166
```
```  2167 declare sorted_list_of_multiset_multiset_of [code]
```
```  2168
```
```  2169 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
```
```  2170   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
```
```  2171   apply (cases "finite A")
```
```  2172   apply simp_all
```
```  2173   apply (induct A rule: finite_induct)
```
```  2174   apply (simp_all add: add.commute)
```
```  2175   done
```
```  2176
```
```  2177 declare size_multiset_of [code]
```
```  2178
```
```  2179 fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
```
```  2180   "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
```
```  2181 | "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
```
```  2182      None \<Rightarrow> None
```
```  2183    | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
```
```  2184
```
```  2185 lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> multiset_of xs \<le> multiset_of ys) \<and>
```
```  2186   (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> multiset_of xs < multiset_of ys) \<and>
```
```  2187   (ms_lesseq_impl xs ys = Some False \<longrightarrow> multiset_of xs = multiset_of ys)"
```
```  2188 proof (induct xs arbitrary: ys)
```
```  2189   case (Nil ys)
```
```  2190   show ?case by (auto simp: mset_less_empty_nonempty)
```
```  2191 next
```
```  2192   case (Cons x xs ys)
```
```  2193   show ?case
```
```  2194   proof (cases "List.extract (op = x) ys")
```
```  2195     case None
```
```  2196     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
```
```  2197     {
```
```  2198       assume "multiset_of (x # xs) \<le> multiset_of ys"
```
```  2199       from set_of_mono[OF this] x have False by simp
```
```  2200     } note nle = this
```
```  2201     moreover
```
```  2202     {
```
```  2203       assume "multiset_of (x # xs) < multiset_of ys"
```
```  2204       hence "multiset_of (x # xs) \<le> multiset_of ys" by auto
```
```  2205       from nle[OF this] have False .
```
```  2206     }
```
```  2207     ultimately show ?thesis using None by auto
```
```  2208   next
```
```  2209     case (Some res)
```
```  2210     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
```
```  2211     note Some = Some[unfolded res]
```
```  2212     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
```
```  2213     hence id: "multiset_of ys = multiset_of (ys1 @ ys2) + {#x#}"
```
```  2214       by (auto simp: ac_simps)
```
```  2215     show ?thesis unfolding ms_lesseq_impl.simps
```
```  2216       unfolding Some option.simps split
```
```  2217       unfolding id
```
```  2218       using Cons[of "ys1 @ ys2"]
```
```  2219       unfolding mset_le_def mset_less_def by auto
```
```  2220   qed
```
```  2221 qed
```
```  2222
```
```  2223 lemma [code]: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
```
```  2224   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2225
```
```  2226 lemma [code]: "multiset_of xs < multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
```
```  2227   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2228
```
```  2229 instantiation multiset :: (equal) equal
```
```  2230 begin
```
```  2231
```
```  2232 definition
```
```  2233   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
```
```  2234 lemma [code]: "HOL.equal (multiset_of xs) (multiset_of ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
```
```  2235   unfolding equal_multiset_def
```
```  2236   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2237
```
```  2238 instance
```
```  2239   by default (simp add: equal_multiset_def)
```
```  2240 end
```
```  2241
```
```  2242 lemma [code]:
```
```  2243   "msetsum (multiset_of xs) = listsum xs"
```
```  2244   by (induct xs) (simp_all add: add.commute)
```
```  2245
```
```  2246 lemma [code]:
```
```  2247   "msetprod (multiset_of xs) = fold times xs 1"
```
```  2248 proof -
```
```  2249   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
```
```  2250     by (induct xs) (simp_all add: mult.assoc)
```
```  2251   then show ?thesis by simp
```
```  2252 qed
```
```  2253
```
```  2254 text {*
```
```  2255   Exercise for the casual reader: add implementations for @{const le_multiset}
```
```  2256   and @{const less_multiset} (multiset order).
```
```  2257 *}
```
```  2258
```
```  2259 text {* Quickcheck generators *}
```
```  2260
```
```  2261 definition (in term_syntax)
```
```  2262   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
```
```  2263     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
```
```  2264   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
```
```  2265
```
```  2266 notation fcomp (infixl "\<circ>>" 60)
```
```  2267 notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2268
```
```  2269 instantiation multiset :: (random) random
```
```  2270 begin
```
```  2271
```
```  2272 definition
```
```  2273   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
```
```  2274
```
```  2275 instance ..
```
```  2276
```
```  2277 end
```
```  2278
```
```  2279 no_notation fcomp (infixl "\<circ>>" 60)
```
```  2280 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2281
```
```  2282 instantiation multiset :: (full_exhaustive) full_exhaustive
```
```  2283 begin
```
```  2284
```
```  2285 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
```
```  2286 where
```
```  2287   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
```
```  2288
```
```  2289 instance ..
```
```  2290
```
```  2291 end
```
```  2292
```
```  2293 hide_const (open) msetify
```
```  2294
```
```  2295
```
```  2296 subsection {* BNF setup *}
```
```  2297
```
```  2298 definition rel_mset where
```
```  2299   "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. multiset_of xs = X \<and> multiset_of ys = Y \<and> list_all2 R xs ys)"
```
```  2300
```
```  2301 lemma multiset_of_zip_take_Cons_drop_twice:
```
```  2302   assumes "length xs = length ys" "j \<le> length xs"
```
```  2303   shows "multiset_of (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
```
```  2304     multiset_of (zip xs ys) + {#(x, y)#}"
```
```  2305 using assms
```
```  2306 proof (induct xs ys arbitrary: x y j rule: list_induct2)
```
```  2307   case Nil
```
```  2308   thus ?case
```
```  2309     by simp
```
```  2310 next
```
```  2311   case (Cons x xs y ys)
```
```  2312   thus ?case
```
```  2313   proof (cases "j = 0")
```
```  2314     case True
```
```  2315     thus ?thesis
```
```  2316       by simp
```
```  2317   next
```
```  2318     case False
```
```  2319     then obtain k where k: "j = Suc k"
```
```  2320       by (case_tac j) simp
```
```  2321     hence "k \<le> length xs"
```
```  2322       using Cons.prems by auto
```
```  2323     hence "multiset_of (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
```
```  2324       multiset_of (zip xs ys) + {#(x, y)#}"
```
```  2325       by (rule Cons.hyps(2))
```
```  2326     thus ?thesis
```
```  2327       unfolding k by (auto simp: add.commute union_lcomm)
```
```  2328   qed
```
```  2329 qed
```
```  2330
```
```  2331 lemma ex_multiset_of_zip_left:
```
```  2332   assumes "length xs = length ys" "multiset_of xs' = multiset_of xs"
```
```  2333   shows "\<exists>ys'. length ys' = length xs' \<and> multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
```
```  2334 using assms
```
```  2335 proof (induct xs ys arbitrary: xs' rule: list_induct2)
```
```  2336   case Nil
```
```  2337   thus ?case
```
```  2338     by auto
```
```  2339 next
```
```  2340   case (Cons x xs y ys xs')
```
```  2341   obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
```
```  2342     by (metis Cons.prems in_set_conv_nth list.set_intros(1) multiset_of_eq_setD)
```
```  2343
```
```  2344   def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
```
```  2345   have "multiset_of xs' = {#x#} + multiset_of xsa"
```
```  2346     unfolding xsa_def using j_len nth_j
```
```  2347     by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
```
```  2348       multiset_of.simps(2) union_code add.commute)
```
```  2349   hence ms_x: "multiset_of xsa = multiset_of xs"
```
```  2350     by (metis Cons.prems add.commute add_right_imp_eq multiset_of.simps(2))
```
```  2351   then obtain ysa where
```
```  2352     len_a: "length ysa = length xsa" and ms_a: "multiset_of (zip xsa ysa) = multiset_of (zip xs ys)"
```
```  2353     using Cons.hyps(2) by blast
```
```  2354
```
```  2355   def ys' \<equiv> "take j ysa @ y # drop j ysa"
```
```  2356   have xs': "xs' = take j xsa @ x # drop j xsa"
```
```  2357     using ms_x j_len nth_j Cons.prems xsa_def
```
```  2358     by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
```
```  2359       length_drop size_multiset_of)
```
```  2360   have j_len': "j \<le> length xsa"
```
```  2361     using j_len xs' xsa_def
```
```  2362     by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
```
```  2363   have "length ys' = length xs'"
```
```  2364     unfolding ys'_def using Cons.prems len_a ms_x
```
```  2365     by (metis add_Suc_right append_take_drop_id length_Cons length_append multiset_of_eq_length)
```
```  2366   moreover have "multiset_of (zip xs' ys') = multiset_of (zip (x # xs) (y # ys))"
```
```  2367     unfolding xs' ys'_def
```
```  2368     by (rule trans[OF multiset_of_zip_take_Cons_drop_twice])
```
```  2369       (auto simp: len_a ms_a j_len' add.commute)
```
```  2370   ultimately show ?case
```
```  2371     by blast
```
```  2372 qed
```
```  2373
```
```  2374 lemma list_all2_reorder_left_invariance:
```
```  2375   assumes rel: "list_all2 R xs ys" and ms_x: "multiset_of xs' = multiset_of xs"
```
```  2376   shows "\<exists>ys'. list_all2 R xs' ys' \<and> multiset_of ys' = multiset_of ys"
```
```  2377 proof -
```
```  2378   have len: "length xs = length ys"
```
```  2379     using rel list_all2_conv_all_nth by auto
```
```  2380   obtain ys' where
```
```  2381     len': "length xs' = length ys'" and ms_xy: "multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
```
```  2382     using len ms_x by (metis ex_multiset_of_zip_left)
```
```  2383   have "list_all2 R xs' ys'"
```
```  2384     using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: multiset_of_eq_setD)
```
```  2385   moreover have "multiset_of ys' = multiset_of ys"
```
```  2386     using len len' ms_xy map_snd_zip multiset_of_map by metis
```
```  2387   ultimately show ?thesis
```
```  2388     by blast
```
```  2389 qed
```
```  2390
```
```  2391 lemma ex_multiset_of: "\<exists>xs. multiset_of xs = X"
```
```  2392   by (induct X) (simp, metis multiset_of.simps(2))
```
```  2393
```
```  2394 bnf "'a multiset"
```
```  2395   map: image_mset
```
```  2396   sets: set_of
```
```  2397   bd: natLeq
```
```  2398   wits: "{#}"
```
```  2399   rel: rel_mset
```
```  2400 proof -
```
```  2401   show "image_mset id = id"
```
```  2402     by (rule image_mset.id)
```
```  2403 next
```
```  2404   show "\<And>f g. image_mset (g \<circ> f) = image_mset g \<circ> image_mset f"
```
```  2405     unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality)
```
```  2406 next
```
```  2407   fix X :: "'a multiset"
```
```  2408   show "\<And>f g. (\<And>z. z \<in> set_of X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X"
```
```  2409     by (induct X, (simp (no_asm))+,
```
```  2410       metis One_nat_def Un_iff count_single mem_set_of_iff set_of_union zero_less_Suc)
```
```  2411 next
```
```  2412   show "\<And>f. set_of \<circ> image_mset f = op ` f \<circ> set_of"
```
```  2413     by auto
```
```  2414 next
```
```  2415   show "card_order natLeq"
```
```  2416     by (rule natLeq_card_order)
```
```  2417 next
```
```  2418   show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
```
```  2419     by (rule natLeq_cinfinite)
```
```  2420 next
```
```  2421   show "\<And>X. ordLeq3 (card_of (set_of X)) natLeq"
```
```  2422     by transfer
```
```  2423       (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
```
```  2424 next
```
```  2425   show "\<And>R S. rel_mset R OO rel_mset S \<le> rel_mset (R OO S)"
```
```  2426     unfolding rel_mset_def[abs_def] OO_def
```
```  2427     apply clarify
```
```  2428     apply (rename_tac X Z Y xs ys' ys zs)
```
```  2429     apply (drule_tac xs = ys' and ys = zs and xs' = ys in list_all2_reorder_left_invariance)
```
```  2430     by (auto intro: list_all2_trans)
```
```  2431 next
```
```  2432   show "\<And>R. rel_mset R =
```
```  2433     (BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
```
```  2434     BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset snd)"
```
```  2435     unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
```
```  2436     apply (rule ext)+
```
```  2437     apply auto
```
```  2438      apply (rule_tac x = "multiset_of (zip xs ys)" in exI)
```
```  2439      apply auto[1]
```
```  2440         apply (metis list_all2_lengthD map_fst_zip multiset_of_map)
```
```  2441        apply (auto simp: list_all2_iff)[1]
```
```  2442       apply (metis list_all2_lengthD map_snd_zip multiset_of_map)
```
```  2443      apply (auto simp: list_all2_iff)[1]
```
```  2444     apply (rename_tac XY)
```
```  2445     apply (cut_tac X = XY in ex_multiset_of)
```
```  2446     apply (erule exE)
```
```  2447     apply (rename_tac xys)
```
```  2448     apply (rule_tac x = "map fst xys" in exI)
```
```  2449     apply (auto simp: multiset_of_map)
```
```  2450     apply (rule_tac x = "map snd xys" in exI)
```
```  2451     by (auto simp: multiset_of_map list_all2I subset_eq zip_map_fst_snd)
```
```  2452 next
```
```  2453   show "\<And>z. z \<in> set_of {#} \<Longrightarrow> False"
```
```  2454     by auto
```
```  2455 qed
```
```  2456
```
```  2457 inductive rel_mset' where
```
```  2458   Zero[intro]: "rel_mset' R {#} {#}"
```
```  2459 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
```
```  2460
```
```  2461 lemma rel_mset_Zero: "rel_mset R {#} {#}"
```
```  2462 unfolding rel_mset_def Grp_def by auto
```
```  2463
```
```  2464 declare multiset.count[simp]
```
```  2465 declare Abs_multiset_inverse[simp]
```
```  2466 declare multiset.count_inverse[simp]
```
```  2467 declare union_preserves_multiset[simp]
```
```  2468
```
```  2469 lemma rel_mset_Plus:
```
```  2470 assumes ab: "R a b" and MN: "rel_mset R M N"
```
```  2471 shows "rel_mset R (M + {#a#}) (N + {#b#})"
```
```  2472 proof-
```
```  2473   {fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
```
```  2474    hence "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
```
```  2475                image_mset snd y + {#b#} = image_mset snd ya \<and>
```
```  2476                set_of ya \<subseteq> {(x, y). R x y}"
```
```  2477    apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
```
```  2478   }
```
```  2479   thus ?thesis
```
```  2480   using assms
```
```  2481   unfolding multiset.rel_compp_Grp Grp_def by blast
```
```  2482 qed
```
```  2483
```
```  2484 lemma rel_mset'_imp_rel_mset:
```
```  2485   "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
```
```  2486 apply(induct rule: rel_mset'.induct)
```
```  2487 using rel_mset_Zero rel_mset_Plus by auto
```
```  2488
```
```  2489 lemma rel_mset_size:
```
```  2490   "rel_mset R M N \<Longrightarrow> size M = size N"
```
```  2491 unfolding multiset.rel_compp_Grp Grp_def by auto
```
```  2492
```
```  2493 lemma multiset_induct2[case_names empty addL addR]:
```
```  2494 assumes empty: "P {#} {#}"
```
```  2495 and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
```
```  2496 and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
```
```  2497 shows "P M N"
```
```  2498 apply(induct N rule: multiset_induct)
```
```  2499   apply(induct M rule: multiset_induct, rule empty, erule addL)
```
```  2500   apply(induct M rule: multiset_induct, erule addR, erule addR)
```
```  2501 done
```
```  2502
```
```  2503 lemma multiset_induct2_size[consumes 1, case_names empty add]:
```
```  2504 assumes c: "size M = size N"
```
```  2505 and empty: "P {#} {#}"
```
```  2506 and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
```
```  2507 shows "P M N"
```
```  2508 using c proof(induct M arbitrary: N rule: measure_induct_rule[of size])
```
```  2509   case (less M)  show ?case
```
```  2510   proof(cases "M = {#}")
```
```  2511     case True hence "N = {#}" using less.prems by auto
```
```  2512     thus ?thesis using True empty by auto
```
```  2513   next
```
```  2514     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
```
```  2515     have "N \<noteq> {#}" using False less.prems by auto
```
```  2516     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
```
```  2517     have "size M1 = size N1" using less.prems unfolding M N by auto
```
```  2518     thus ?thesis using M N less.hyps add by auto
```
```  2519   qed
```
```  2520 qed
```
```  2521
```
```  2522 lemma msed_map_invL:
```
```  2523 assumes "image_mset f (M + {#a#}) = N"
```
```  2524 shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1"
```
```  2525 proof-
```
```  2526   have "f a \<in># N"
```
```  2527   using assms multiset.set_map[of f "M + {#a#}"] by auto
```
```  2528   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
```
```  2529   have "image_mset f M = N1" using assms unfolding N by simp
```
```  2530   thus ?thesis using N by blast
```
```  2531 qed
```
```  2532
```
```  2533 lemma msed_map_invR:
```
```  2534 assumes "image_mset f M = N + {#b#}"
```
```  2535 shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N"
```
```  2536 proof-
```
```  2537   obtain a where a: "a \<in># M" and fa: "f a = b"
```
```  2538   using multiset.set_map[of f M] unfolding assms
```
```  2539   by (metis image_iff mem_set_of_iff union_single_eq_member)
```
```  2540   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
```
```  2541   have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
```
```  2542   thus ?thesis using M fa by blast
```
```  2543 qed
```
```  2544
```
```  2545 lemma msed_rel_invL:
```
```  2546 assumes "rel_mset R (M + {#a#}) N"
```
```  2547 shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1"
```
```  2548 proof-
```
```  2549   obtain K where KM: "image_mset fst K = M + {#a#}"
```
```  2550   and KN: "image_mset snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
```
```  2551   using assms
```
```  2552   unfolding multiset.rel_compp_Grp Grp_def by auto
```
```  2553   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
```
```  2554   and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
```
```  2555   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1"
```
```  2556   using msed_map_invL[OF KN[unfolded K]] by auto
```
```  2557   have Rab: "R a (snd ab)" using sK a unfolding K by auto
```
```  2558   have "rel_mset R M N1" using sK K1M K1N1
```
```  2559   unfolding K multiset.rel_compp_Grp Grp_def by auto
```
```  2560   thus ?thesis using N Rab by auto
```
```  2561 qed
```
```  2562
```
```  2563 lemma msed_rel_invR:
```
```  2564 assumes "rel_mset R M (N + {#b#})"
```
```  2565 shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N"
```
```  2566 proof-
```
```  2567   obtain K where KN: "image_mset snd K = N + {#b#}"
```
```  2568   and KM: "image_mset fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
```
```  2569   using assms
```
```  2570   unfolding multiset.rel_compp_Grp Grp_def by auto
```
```  2571   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
```
```  2572   and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
```
```  2573   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1"
```
```  2574   using msed_map_invL[OF KM[unfolded K]] by auto
```
```  2575   have Rab: "R (fst ab) b" using sK b unfolding K by auto
```
```  2576   have "rel_mset R M1 N" using sK K1N K1M1
```
```  2577   unfolding K multiset.rel_compp_Grp Grp_def by auto
```
```  2578   thus ?thesis using M Rab by auto
```
```  2579 qed
```
```  2580
```
```  2581 lemma rel_mset_imp_rel_mset':
```
```  2582 assumes "rel_mset R M N"
```
```  2583 shows "rel_mset' R M N"
```
```  2584 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of size])
```
```  2585   case (less M)
```
```  2586   have c: "size M = size N" using rel_mset_size[OF less.prems] .
```
```  2587   show ?case
```
```  2588   proof(cases "M = {#}")
```
```  2589     case True hence "N = {#}" using c by simp
```
```  2590     thus ?thesis using True rel_mset'.Zero by auto
```
```  2591   next
```
```  2592     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
```
```  2593     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1"
```
```  2594     using msed_rel_invL[OF less.prems[unfolded M]] by auto
```
```  2595     have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
```
```  2596     thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
```
```  2597   qed
```
```  2598 qed
```
```  2599
```
```  2600 lemma rel_mset_rel_mset':
```
```  2601 "rel_mset R M N = rel_mset' R M N"
```
```  2602 using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
```
```  2603
```
```  2604 (* The main end product for rel_mset: inductive characterization *)
```
```  2605 theorems rel_mset_induct[case_names empty add, induct pred: rel_mset] =
```
```  2606          rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
```
```  2607
```
```  2608
```
```  2609 subsection {* Size setup *}
```
```  2610
```
```  2611 lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
```
```  2612   unfolding o_apply by (rule ext) (induct_tac, auto)
```
```  2613
```
```  2614 setup {*
```
```  2615 BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
```
```  2616   @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
```
```  2617     size_union}
```
```  2618   @{thms multiset_size_o_map}
```
```  2619 *}
```
```  2620
```
```  2621 hide_const (open) wcount
```
```  2622
```
```  2623 end
```