src/HOL/Library/Multiset.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 58247 98d0f85d247f child 58881 b9556a055632 permissions -rw-r--r--
simpler proof
```     1 (*  Title:      HOL/Library/Multiset.thy
```
```     2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
```
```     3     Author:     Andrei Popescu, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* (Finite) multisets *}
```
```     7
```
```     8 theory Multiset
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 subsection {* The type of multisets *}
```
```    13
```
```    14 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
```
```    15
```
```    16 typedef 'a multiset = "multiset :: ('a => nat) set"
```
```    17   morphisms count Abs_multiset
```
```    18   unfolding multiset_def
```
```    19 proof
```
```    20   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
```
```    21 qed
```
```    22
```
```    23 setup_lifting type_definition_multiset
```
```    24
```
```    25 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
```
```    26   "a :# M == 0 < count M a"
```
```    27
```
```    28 notation (xsymbols)
```
```    29   Melem (infix "\<in>#" 50)
```
```    30
```
```    31 lemma multiset_eq_iff:
```
```    32   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
```
```    33   by (simp only: count_inject [symmetric] fun_eq_iff)
```
```    34
```
```    35 lemma multiset_eqI:
```
```    36   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
```
```    37   using multiset_eq_iff by auto
```
```    38
```
```    39 text {*
```
```    40  \medskip Preservation of the representing set @{term multiset}.
```
```    41 *}
```
```    42
```
```    43 lemma const0_in_multiset:
```
```    44   "(\<lambda>a. 0) \<in> multiset"
```
```    45   by (simp add: multiset_def)
```
```    46
```
```    47 lemma only1_in_multiset:
```
```    48   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
```
```    49   by (simp add: multiset_def)
```
```    50
```
```    51 lemma union_preserves_multiset:
```
```    52   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
```
```    53   by (simp add: multiset_def)
```
```    54
```
```    55 lemma diff_preserves_multiset:
```
```    56   assumes "M \<in> multiset"
```
```    57   shows "(\<lambda>a. M a - N a) \<in> multiset"
```
```    58 proof -
```
```    59   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
```
```    60     by auto
```
```    61   with assms show ?thesis
```
```    62     by (auto simp add: multiset_def intro: finite_subset)
```
```    63 qed
```
```    64
```
```    65 lemma filter_preserves_multiset:
```
```    66   assumes "M \<in> multiset"
```
```    67   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
```
```    68 proof -
```
```    69   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
```
```    70     by auto
```
```    71   with assms show ?thesis
```
```    72     by (auto simp add: multiset_def intro: finite_subset)
```
```    73 qed
```
```    74
```
```    75 lemmas in_multiset = const0_in_multiset only1_in_multiset
```
```    76   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
```
```    77
```
```    78
```
```    79 subsection {* Representing multisets *}
```
```    80
```
```    81 text {* Multiset enumeration *}
```
```    82
```
```    83 instantiation multiset :: (type) cancel_comm_monoid_add
```
```    84 begin
```
```    85
```
```    86 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
```
```    87 by (rule const0_in_multiset)
```
```    88
```
```    89 abbreviation Mempty :: "'a multiset" ("{#}") where
```
```    90   "Mempty \<equiv> 0"
```
```    91
```
```    92 lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
```
```    93 by (rule union_preserves_multiset)
```
```    94
```
```    95 instance
```
```    96 by default (transfer, simp add: fun_eq_iff)+
```
```    97
```
```    98 end
```
```    99
```
```   100 lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
```
```   101 by (rule only1_in_multiset)
```
```   102
```
```   103 syntax
```
```   104   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
```
```   105 translations
```
```   106   "{#x, xs#}" == "{#x#} + {#xs#}"
```
```   107   "{#x#}" == "CONST single x"
```
```   108
```
```   109 lemma count_empty [simp]: "count {#} a = 0"
```
```   110   by (simp add: zero_multiset.rep_eq)
```
```   111
```
```   112 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   113   by (simp add: single.rep_eq)
```
```   114
```
```   115
```
```   116 subsection {* Basic operations *}
```
```   117
```
```   118 subsubsection {* Union *}
```
```   119
```
```   120 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   121   by (simp add: plus_multiset.rep_eq)
```
```   122
```
```   123
```
```   124 subsubsection {* Difference *}
```
```   125
```
```   126 instantiation multiset :: (type) comm_monoid_diff
```
```   127 begin
```
```   128
```
```   129 lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
```
```   130 by (rule diff_preserves_multiset)
```
```   131
```
```   132 instance
```
```   133 by default (transfer, simp add: fun_eq_iff)+
```
```   134
```
```   135 end
```
```   136
```
```   137 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   138   by (simp add: minus_multiset.rep_eq)
```
```   139
```
```   140 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   141   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
```
```   142
```
```   143 lemma diff_cancel[simp]: "A - A = {#}"
```
```   144   by (fact Groups.diff_cancel)
```
```   145
```
```   146 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
```
```   147   by (fact add_diff_cancel_right')
```
```   148
```
```   149 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
```
```   150   by (fact add_diff_cancel_left')
```
```   151
```
```   152 lemma diff_right_commute:
```
```   153   "(M::'a multiset) - N - Q = M - Q - N"
```
```   154   by (fact diff_right_commute)
```
```   155
```
```   156 lemma diff_add:
```
```   157   "(M::'a multiset) - (N + Q) = M - N - Q"
```
```   158   by (rule sym) (fact diff_diff_add)
```
```   159
```
```   160 lemma insert_DiffM:
```
```   161   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
```
```   162   by (clarsimp simp: multiset_eq_iff)
```
```   163
```
```   164 lemma insert_DiffM2 [simp]:
```
```   165   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
```
```   166   by (clarsimp simp: multiset_eq_iff)
```
```   167
```
```   168 lemma diff_union_swap:
```
```   169   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
```
```   170   by (auto simp add: multiset_eq_iff)
```
```   171
```
```   172 lemma diff_union_single_conv:
```
```   173   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
```
```   174   by (simp add: multiset_eq_iff)
```
```   175
```
```   176
```
```   177 subsubsection {* Equality of multisets *}
```
```   178
```
```   179 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
```
```   180   by (simp add: multiset_eq_iff)
```
```   181
```
```   182 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
```
```   183   by (auto simp add: multiset_eq_iff)
```
```   184
```
```   185 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
```
```   186   by (auto simp add: multiset_eq_iff)
```
```   187
```
```   188 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
```
```   189   by (auto simp add: multiset_eq_iff)
```
```   190
```
```   191 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
```
```   192   by (auto simp add: multiset_eq_iff)
```
```   193
```
```   194 lemma diff_single_trivial:
```
```   195   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
```
```   196   by (auto simp add: multiset_eq_iff)
```
```   197
```
```   198 lemma diff_single_eq_union:
```
```   199   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
```
```   200   by auto
```
```   201
```
```   202 lemma union_single_eq_diff:
```
```   203   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
```
```   204   by (auto dest: sym)
```
```   205
```
```   206 lemma union_single_eq_member:
```
```   207   "M + {#x#} = N \<Longrightarrow> x \<in># N"
```
```   208   by auto
```
```   209
```
```   210 lemma union_is_single:
```
```   211   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
```
```   212 proof
```
```   213   assume ?rhs then show ?lhs by auto
```
```   214 next
```
```   215   assume ?lhs then show ?rhs
```
```   216     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
```
```   217 qed
```
```   218
```
```   219 lemma single_is_union:
```
```   220   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
```
```   221   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
```
```   222
```
```   223 lemma add_eq_conv_diff:
```
```   224   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
```
```   225 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
```
```   226 proof
```
```   227   assume ?rhs then show ?lhs
```
```   228   by (auto simp add: add.assoc add.commute [of "{#b#}"])
```
```   229     (drule sym, simp add: add.assoc [symmetric])
```
```   230 next
```
```   231   assume ?lhs
```
```   232   show ?rhs
```
```   233   proof (cases "a = b")
```
```   234     case True with `?lhs` show ?thesis by simp
```
```   235   next
```
```   236     case False
```
```   237     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
```
```   238     with False have "a \<in># N" by auto
```
```   239     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
```
```   240     moreover note False
```
```   241     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
```
```   242   qed
```
```   243 qed
```
```   244
```
```   245 lemma insert_noteq_member:
```
```   246   assumes BC: "B + {#b#} = C + {#c#}"
```
```   247    and bnotc: "b \<noteq> c"
```
```   248   shows "c \<in># B"
```
```   249 proof -
```
```   250   have "c \<in># C + {#c#}" by simp
```
```   251   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
```
```   252   then have "c \<in># B + {#b#}" using BC by simp
```
```   253   then show "c \<in># B" using nc by simp
```
```   254 qed
```
```   255
```
```   256 lemma add_eq_conv_ex:
```
```   257   "(M + {#a#} = N + {#b#}) =
```
```   258     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
```
```   259   by (auto simp add: add_eq_conv_diff)
```
```   260
```
```   261 lemma multi_member_split:
```
```   262   "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
```
```   263   by (rule_tac x = "M - {#x#}" in exI, simp)
```
```   264
```
```   265 lemma multiset_add_sub_el_shuffle:
```
```   266   assumes "c \<in># B" and "b \<noteq> c"
```
```   267   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
```
```   268 proof -
```
```   269   from `c \<in># B` obtain A where B: "B = A + {#c#}"
```
```   270     by (blast dest: multi_member_split)
```
```   271   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
```
```   272   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
```
```   273     by (simp add: ac_simps)
```
```   274   then show ?thesis using B by simp
```
```   275 qed
```
```   276
```
```   277
```
```   278 subsubsection {* Pointwise ordering induced by count *}
```
```   279
```
```   280 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
```
```   281 begin
```
```   282
```
```   283 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)" .
```
```   284
```
```   285 lemmas mset_le_def = less_eq_multiset_def
```
```   286
```
```   287 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
```
```   288   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
```
```   289
```
```   290 instance
```
```   291   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
```
```   292
```
```   293 end
```
```   294
```
```   295 lemma mset_less_eqI:
```
```   296   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
```
```   297   by (simp add: mset_le_def)
```
```   298
```
```   299 lemma mset_le_exists_conv:
```
```   300   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
```
```   301 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
```
```   302 apply (auto intro: multiset_eq_iff [THEN iffD2])
```
```   303 done
```
```   304
```
```   305 instance multiset :: (type) ordered_cancel_comm_monoid_diff
```
```   306   by default (simp, fact mset_le_exists_conv)
```
```   307
```
```   308 lemma mset_le_mono_add_right_cancel [simp]:
```
```   309   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
```
```   310   by (fact add_le_cancel_right)
```
```   311
```
```   312 lemma mset_le_mono_add_left_cancel [simp]:
```
```   313   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
```
```   314   by (fact add_le_cancel_left)
```
```   315
```
```   316 lemma mset_le_mono_add:
```
```   317   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
```
```   318   by (fact add_mono)
```
```   319
```
```   320 lemma mset_le_add_left [simp]:
```
```   321   "(A::'a multiset) \<le> A + B"
```
```   322   unfolding mset_le_def by auto
```
```   323
```
```   324 lemma mset_le_add_right [simp]:
```
```   325   "B \<le> (A::'a multiset) + B"
```
```   326   unfolding mset_le_def by auto
```
```   327
```
```   328 lemma mset_le_single:
```
```   329   "a :# B \<Longrightarrow> {#a#} \<le> B"
```
```   330   by (simp add: mset_le_def)
```
```   331
```
```   332 lemma multiset_diff_union_assoc:
```
```   333   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
```
```   334   by (simp add: multiset_eq_iff mset_le_def)
```
```   335
```
```   336 lemma mset_le_multiset_union_diff_commute:
```
```   337   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
```
```   338 by (simp add: multiset_eq_iff mset_le_def)
```
```   339
```
```   340 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
```
```   341 by(simp add: mset_le_def)
```
```   342
```
```   343 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   344 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   345 apply (erule_tac x=x in allE)
```
```   346 apply auto
```
```   347 done
```
```   348
```
```   349 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   350 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   351 apply (erule_tac x = x in allE)
```
```   352 apply auto
```
```   353 done
```
```   354
```
```   355 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
```
```   356 apply (rule conjI)
```
```   357  apply (simp add: mset_lessD)
```
```   358 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   359 apply safe
```
```   360  apply (erule_tac x = a in allE)
```
```   361  apply (auto split: split_if_asm)
```
```   362 done
```
```   363
```
```   364 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
```
```   365 apply (rule conjI)
```
```   366  apply (simp add: mset_leD)
```
```   367 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
```
```   368 done
```
```   369
```
```   370 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
```
```   371   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
```
```   372
```
```   373 lemma empty_le[simp]: "{#} \<le> A"
```
```   374   unfolding mset_le_exists_conv by auto
```
```   375
```
```   376 lemma le_empty[simp]: "(M \<le> {#}) = (M = {#})"
```
```   377   unfolding mset_le_exists_conv by auto
```
```   378
```
```   379 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
```
```   380   by (auto simp: mset_le_def mset_less_def)
```
```   381
```
```   382 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
```
```   383   by simp
```
```   384
```
```   385 lemma mset_less_add_bothsides:
```
```   386   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
```
```   387   by (fact add_less_imp_less_right)
```
```   388
```
```   389 lemma mset_less_empty_nonempty:
```
```   390   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
```
```   391   by (auto simp: mset_le_def mset_less_def)
```
```   392
```
```   393 lemma mset_less_diff_self:
```
```   394   "c \<in># B \<Longrightarrow> B - {#c#} < B"
```
```   395   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
```
```   396
```
```   397
```
```   398 subsubsection {* Intersection *}
```
```   399
```
```   400 instantiation multiset :: (type) semilattice_inf
```
```   401 begin
```
```   402
```
```   403 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   404   multiset_inter_def: "inf_multiset A B = A - (A - B)"
```
```   405
```
```   406 instance
```
```   407 proof -
```
```   408   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
```
```   409   show "OFCLASS('a multiset, semilattice_inf_class)"
```
```   410     by default (auto simp add: multiset_inter_def mset_le_def aux)
```
```   411 qed
```
```   412
```
```   413 end
```
```   414
```
```   415 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
```
```   416   "multiset_inter \<equiv> inf"
```
```   417
```
```   418 lemma multiset_inter_count [simp]:
```
```   419   "count (A #\<inter> B) x = min (count A x) (count B x)"
```
```   420   by (simp add: multiset_inter_def)
```
```   421
```
```   422 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
```
```   423   by (rule multiset_eqI) auto
```
```   424
```
```   425 lemma multiset_union_diff_commute:
```
```   426   assumes "B #\<inter> C = {#}"
```
```   427   shows "A + B - C = A - C + B"
```
```   428 proof (rule multiset_eqI)
```
```   429   fix x
```
```   430   from assms have "min (count B x) (count C x) = 0"
```
```   431     by (auto simp add: multiset_eq_iff)
```
```   432   then have "count B x = 0 \<or> count C x = 0"
```
```   433     by auto
```
```   434   then show "count (A + B - C) x = count (A - C + B) x"
```
```   435     by auto
```
```   436 qed
```
```   437
```
```   438 lemma empty_inter [simp]:
```
```   439   "{#} #\<inter> M = {#}"
```
```   440   by (simp add: multiset_eq_iff)
```
```   441
```
```   442 lemma inter_empty [simp]:
```
```   443   "M #\<inter> {#} = {#}"
```
```   444   by (simp add: multiset_eq_iff)
```
```   445
```
```   446 lemma inter_add_left1:
```
```   447   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
```
```   448   by (simp add: multiset_eq_iff)
```
```   449
```
```   450 lemma inter_add_left2:
```
```   451   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
```
```   452   by (simp add: multiset_eq_iff)
```
```   453
```
```   454 lemma inter_add_right1:
```
```   455   "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
```
```   456   by (simp add: multiset_eq_iff)
```
```   457
```
```   458 lemma inter_add_right2:
```
```   459   "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
```
```   460   by (simp add: multiset_eq_iff)
```
```   461
```
```   462
```
```   463 subsubsection {* Bounded union *}
```
```   464
```
```   465 instantiation multiset :: (type) semilattice_sup
```
```   466 begin
```
```   467
```
```   468 definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   469   "sup_multiset A B = A + (B - A)"
```
```   470
```
```   471 instance
```
```   472 proof -
```
```   473   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" by arith
```
```   474   show "OFCLASS('a multiset, semilattice_sup_class)"
```
```   475     by default (auto simp add: sup_multiset_def mset_le_def aux)
```
```   476 qed
```
```   477
```
```   478 end
```
```   479
```
```   480 abbreviation sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<union>" 70) where
```
```   481   "sup_multiset \<equiv> sup"
```
```   482
```
```   483 lemma sup_multiset_count [simp]:
```
```   484   "count (A #\<union> B) x = max (count A x) (count B x)"
```
```   485   by (simp add: sup_multiset_def)
```
```   486
```
```   487 lemma empty_sup [simp]:
```
```   488   "{#} #\<union> M = M"
```
```   489   by (simp add: multiset_eq_iff)
```
```   490
```
```   491 lemma sup_empty [simp]:
```
```   492   "M #\<union> {#} = M"
```
```   493   by (simp add: multiset_eq_iff)
```
```   494
```
```   495 lemma sup_add_left1:
```
```   496   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
```
```   497   by (simp add: multiset_eq_iff)
```
```   498
```
```   499 lemma sup_add_left2:
```
```   500   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
```
```   501   by (simp add: multiset_eq_iff)
```
```   502
```
```   503 lemma sup_add_right1:
```
```   504   "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
```
```   505   by (simp add: multiset_eq_iff)
```
```   506
```
```   507 lemma sup_add_right2:
```
```   508   "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
```
```   509   by (simp add: multiset_eq_iff)
```
```   510
```
```   511
```
```   512 subsubsection {* Filter (with comprehension syntax) *}
```
```   513
```
```   514 text {* Multiset comprehension *}
```
```   515
```
```   516 lift_definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
```
```   517 by (rule filter_preserves_multiset)
```
```   518
```
```   519 hide_const (open) filter
```
```   520
```
```   521 lemma count_filter [simp]:
```
```   522   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
```
```   523   by (simp add: filter.rep_eq)
```
```   524
```
```   525 lemma filter_empty [simp]:
```
```   526   "Multiset.filter P {#} = {#}"
```
```   527   by (rule multiset_eqI) simp
```
```   528
```
```   529 lemma filter_single [simp]:
```
```   530   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
```
```   531   by (rule multiset_eqI) simp
```
```   532
```
```   533 lemma filter_union [simp]:
```
```   534   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
```
```   535   by (rule multiset_eqI) simp
```
```   536
```
```   537 lemma filter_diff [simp]:
```
```   538   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
```
```   539   by (rule multiset_eqI) simp
```
```   540
```
```   541 lemma filter_inter [simp]:
```
```   542   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
```
```   543   by (rule multiset_eqI) simp
```
```   544
```
```   545 lemma multiset_filter_subset[simp]: "Multiset.filter f M \<le> M"
```
```   546   unfolding less_eq_multiset.rep_eq by auto
```
```   547
```
```   548 lemma multiset_filter_mono: assumes "A \<le> B"
```
```   549   shows "Multiset.filter f A \<le> Multiset.filter f B"
```
```   550 proof -
```
```   551   from assms[unfolded mset_le_exists_conv]
```
```   552   obtain C where B: "B = A + C" by auto
```
```   553   show ?thesis unfolding B by auto
```
```   554 qed
```
```   555
```
```   556 syntax
```
```   557   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
```
```   558 syntax (xsymbol)
```
```   559   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
```
```   560 translations
```
```   561   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
```
```   562
```
```   563
```
```   564 subsubsection {* Set of elements *}
```
```   565
```
```   566 definition set_of :: "'a multiset => 'a set" where
```
```   567   "set_of M = {x. x :# M}"
```
```   568
```
```   569 lemma set_of_empty [simp]: "set_of {#} = {}"
```
```   570 by (simp add: set_of_def)
```
```   571
```
```   572 lemma set_of_single [simp]: "set_of {#b#} = {b}"
```
```   573 by (simp add: set_of_def)
```
```   574
```
```   575 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   576 by (auto simp add: set_of_def)
```
```   577
```
```   578 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   579 by (auto simp add: set_of_def multiset_eq_iff)
```
```   580
```
```   581 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   582 by (auto simp add: set_of_def)
```
```   583
```
```   584 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
```
```   585 by (auto simp add: set_of_def)
```
```   586
```
```   587 lemma finite_set_of [iff]: "finite (set_of M)"
```
```   588   using count [of M] by (simp add: multiset_def set_of_def)
```
```   589
```
```   590 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
```
```   591   unfolding set_of_def[symmetric] by simp
```
```   592
```
```   593 lemma set_of_mono: "A \<le> B \<Longrightarrow> set_of A \<subseteq> set_of B"
```
```   594   by (metis mset_leD subsetI mem_set_of_iff)
```
```   595
```
```   596 subsubsection {* Size *}
```
```   597
```
```   598 definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
```
```   599
```
```   600 lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
```
```   601   by (auto simp: wcount_def add_mult_distrib)
```
```   602
```
```   603 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
```
```   604   "size_multiset f M = setsum (wcount f M) (set_of M)"
```
```   605
```
```   606 lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
```
```   607
```
```   608 instantiation multiset :: (type) size begin
```
```   609 definition size_multiset where
```
```   610   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
```
```   611 instance ..
```
```   612 end
```
```   613
```
```   614 lemmas size_multiset_overloaded_eq =
```
```   615   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
```
```   616
```
```   617 lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
```
```   618 by (simp add: size_multiset_def)
```
```   619
```
```   620 lemma size_empty [simp]: "size {#} = 0"
```
```   621 by (simp add: size_multiset_overloaded_def)
```
```   622
```
```   623 lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
```
```   624 by (simp add: size_multiset_eq)
```
```   625
```
```   626 lemma size_single [simp]: "size {#b#} = 1"
```
```   627 by (simp add: size_multiset_overloaded_def)
```
```   628
```
```   629 lemma setsum_wcount_Int:
```
```   630   "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_of N) = setsum (wcount f N) A"
```
```   631 apply (induct rule: finite_induct)
```
```   632  apply simp
```
```   633 apply (simp add: Int_insert_left set_of_def wcount_def)
```
```   634 done
```
```   635
```
```   636 lemma size_multiset_union [simp]:
```
```   637   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
```
```   638 apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
```
```   639 apply (subst Int_commute)
```
```   640 apply (simp add: setsum_wcount_Int)
```
```   641 done
```
```   642
```
```   643 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   644 by (auto simp add: size_multiset_overloaded_def)
```
```   645
```
```   646 lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
```
```   647 by (auto simp add: size_multiset_eq multiset_eq_iff)
```
```   648
```
```   649 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   650 by (auto simp add: size_multiset_overloaded_def)
```
```   651
```
```   652 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
```
```   653 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
```
```   654
```
```   655 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   656 apply (unfold size_multiset_overloaded_eq)
```
```   657 apply (drule setsum_SucD)
```
```   658 apply auto
```
```   659 done
```
```   660
```
```   661 lemma size_eq_Suc_imp_eq_union:
```
```   662   assumes "size M = Suc n"
```
```   663   shows "\<exists>a N. M = N + {#a#}"
```
```   664 proof -
```
```   665   from assms obtain a where "a \<in># M"
```
```   666     by (erule size_eq_Suc_imp_elem [THEN exE])
```
```   667   then have "M = M - {#a#} + {#a#}" by simp
```
```   668   then show ?thesis by blast
```
```   669 qed
```
```   670
```
```   671
```
```   672 subsection {* Induction and case splits *}
```
```   673
```
```   674 theorem multiset_induct [case_names empty add, induct type: multiset]:
```
```   675   assumes empty: "P {#}"
```
```   676   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
```
```   677   shows "P M"
```
```   678 proof (induct n \<equiv> "size M" arbitrary: M)
```
```   679   case 0 thus "P M" by (simp add: empty)
```
```   680 next
```
```   681   case (Suc k)
```
```   682   obtain N x where "M = N + {#x#}"
```
```   683     using `Suc k = size M` [symmetric]
```
```   684     using size_eq_Suc_imp_eq_union by fast
```
```   685   with Suc add show "P M" by simp
```
```   686 qed
```
```   687
```
```   688 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
```
```   689 by (induct M) auto
```
```   690
```
```   691 lemma multiset_cases [cases type]:
```
```   692   obtains (empty) "M = {#}"
```
```   693     | (add) N x where "M = N + {#x#}"
```
```   694   using assms by (induct M) simp_all
```
```   695
```
```   696 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
```
```   697 by (cases "B = {#}") (auto dest: multi_member_split)
```
```   698
```
```   699 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
```
```   700 apply (subst multiset_eq_iff)
```
```   701 apply auto
```
```   702 done
```
```   703
```
```   704 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
```
```   705 proof (induct A arbitrary: B)
```
```   706   case (empty M)
```
```   707   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
```
```   708   then obtain M' x where "M = M' + {#x#}"
```
```   709     by (blast dest: multi_nonempty_split)
```
```   710   then show ?case by simp
```
```   711 next
```
```   712   case (add S x T)
```
```   713   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
```
```   714   have SxsubT: "S + {#x#} < T" by fact
```
```   715   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
```
```   716   then obtain T' where T: "T = T' + {#x#}"
```
```   717     by (blast dest: multi_member_split)
```
```   718   then have "S < T'" using SxsubT
```
```   719     by (blast intro: mset_less_add_bothsides)
```
```   720   then have "size S < size T'" using IH by simp
```
```   721   then show ?case using T by simp
```
```   722 qed
```
```   723
```
```   724
```
```   725 subsubsection {* Strong induction and subset induction for multisets *}
```
```   726
```
```   727 text {* Well-foundedness of strict subset relation *}
```
```   728
```
```   729 lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M < N}"
```
```   730 apply (rule wf_measure [THEN wf_subset, where f1=size])
```
```   731 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
```
```   732 done
```
```   733
```
```   734 lemma full_multiset_induct [case_names less]:
```
```   735 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
```
```   736 shows "P B"
```
```   737 apply (rule wf_less_mset_rel [THEN wf_induct])
```
```   738 apply (rule ih, auto)
```
```   739 done
```
```   740
```
```   741 lemma multi_subset_induct [consumes 2, case_names empty add]:
```
```   742 assumes "F \<le> A"
```
```   743   and empty: "P {#}"
```
```   744   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
```
```   745 shows "P F"
```
```   746 proof -
```
```   747   from `F \<le> A`
```
```   748   show ?thesis
```
```   749   proof (induct F)
```
```   750     show "P {#}" by fact
```
```   751   next
```
```   752     fix x F
```
```   753     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
```
```   754     show "P (F + {#x#})"
```
```   755     proof (rule insert)
```
```   756       from i show "x \<in># A" by (auto dest: mset_le_insertD)
```
```   757       from i have "F \<le> A" by (auto dest: mset_le_insertD)
```
```   758       with P show "P F" .
```
```   759     qed
```
```   760   qed
```
```   761 qed
```
```   762
```
```   763
```
```   764 subsection {* The fold combinator *}
```
```   765
```
```   766 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
```
```   767 where
```
```   768   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
```
```   769
```
```   770 lemma fold_mset_empty [simp]:
```
```   771   "fold f s {#} = s"
```
```   772   by (simp add: fold_def)
```
```   773
```
```   774 context comp_fun_commute
```
```   775 begin
```
```   776
```
```   777 lemma fold_mset_insert:
```
```   778   "fold f s (M + {#x#}) = f x (fold f s M)"
```
```   779 proof -
```
```   780   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
```
```   781     by (fact comp_fun_commute_funpow)
```
```   782   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
```
```   783     by (fact comp_fun_commute_funpow)
```
```   784   show ?thesis
```
```   785   proof (cases "x \<in> set_of M")
```
```   786     case False
```
```   787     then have *: "count (M + {#x#}) x = 1" by simp
```
```   788     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
```
```   789       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
```
```   790       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   791     with False * show ?thesis
```
```   792       by (simp add: fold_def del: count_union)
```
```   793   next
```
```   794     case True
```
```   795     def N \<equiv> "set_of M - {x}"
```
```   796     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
```
```   797     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
```
```   798       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
```
```   799       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   800     with * show ?thesis by (simp add: fold_def del: count_union) simp
```
```   801   qed
```
```   802 qed
```
```   803
```
```   804 corollary fold_mset_single [simp]:
```
```   805   "fold f s {#x#} = f x s"
```
```   806 proof -
```
```   807   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
```
```   808   then show ?thesis by simp
```
```   809 qed
```
```   810
```
```   811 lemma fold_mset_fun_left_comm:
```
```   812   "f x (fold f s M) = fold f (f x s) M"
```
```   813   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
```
```   814
```
```   815 lemma fold_mset_union [simp]:
```
```   816   "fold f s (M + N) = fold f (fold f s M) N"
```
```   817 proof (induct M)
```
```   818   case empty then show ?case by simp
```
```   819 next
```
```   820   case (add M x)
```
```   821   have "M + {#x#} + N = (M + N) + {#x#}"
```
```   822     by (simp add: ac_simps)
```
```   823   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
```
```   824 qed
```
```   825
```
```   826 lemma fold_mset_fusion:
```
```   827   assumes "comp_fun_commute g"
```
```   828   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold g w A) = fold f (h w) A" (is "PROP ?P")
```
```   829 proof -
```
```   830   interpret comp_fun_commute g by (fact assms)
```
```   831   show "PROP ?P" by (induct A) auto
```
```   832 qed
```
```   833
```
```   834 end
```
```   835
```
```   836 text {*
```
```   837   A note on code generation: When defining some function containing a
```
```   838   subterm @{term "fold F"}, code generation is not automatic. When
```
```   839   interpreting locale @{text left_commutative} with @{text F}, the
```
```   840   would be code thms for @{const fold} become thms like
```
```   841   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
```
```   842   contains defined symbols, i.e.\ is not a code thm. Hence a separate
```
```   843   constant with its own code thms needs to be introduced for @{text
```
```   844   F}. See the image operator below.
```
```   845 *}
```
```   846
```
```   847
```
```   848 subsection {* Image *}
```
```   849
```
```   850 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
```
```   851   "image_mset f = fold (plus o single o f) {#}"
```
```   852
```
```   853 lemma comp_fun_commute_mset_image:
```
```   854   "comp_fun_commute (plus o single o f)"
```
```   855 proof
```
```   856 qed (simp add: ac_simps fun_eq_iff)
```
```   857
```
```   858 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
```
```   859   by (simp add: image_mset_def)
```
```   860
```
```   861 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
```
```   862 proof -
```
```   863   interpret comp_fun_commute "plus o single o f"
```
```   864     by (fact comp_fun_commute_mset_image)
```
```   865   show ?thesis by (simp add: image_mset_def)
```
```   866 qed
```
```   867
```
```   868 lemma image_mset_union [simp]:
```
```   869   "image_mset f (M + N) = image_mset f M + image_mset f N"
```
```   870 proof -
```
```   871   interpret comp_fun_commute "plus o single o f"
```
```   872     by (fact comp_fun_commute_mset_image)
```
```   873   show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
```
```   874 qed
```
```   875
```
```   876 corollary image_mset_insert:
```
```   877   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
```
```   878   by simp
```
```   879
```
```   880 lemma set_of_image_mset [simp]:
```
```   881   "set_of (image_mset f M) = image f (set_of M)"
```
```   882   by (induct M) simp_all
```
```   883
```
```   884 lemma size_image_mset [simp]:
```
```   885   "size (image_mset f M) = size M"
```
```   886   by (induct M) simp_all
```
```   887
```
```   888 lemma image_mset_is_empty_iff [simp]:
```
```   889   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
```
```   890   by (cases M) auto
```
```   891
```
```   892 syntax
```
```   893   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
```
```   894       ("({#_/. _ :# _#})")
```
```   895 translations
```
```   896   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
```
```   897
```
```   898 syntax
```
```   899   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
```
```   900       ("({#_/ | _ :# _./ _#})")
```
```   901 translations
```
```   902   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
```
```   903
```
```   904 text {*
```
```   905   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
```
```   906   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
```
```   907   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
```
```   908   @{term "{#x+x|x:#M. x<c#}"}.
```
```   909 *}
```
```   910
```
```   911 functor image_mset: image_mset
```
```   912 proof -
```
```   913   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
```
```   914   proof
```
```   915     fix A
```
```   916     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
```
```   917       by (induct A) simp_all
```
```   918   qed
```
```   919   show "image_mset id = id"
```
```   920   proof
```
```   921     fix A
```
```   922     show "image_mset id A = id A"
```
```   923       by (induct A) simp_all
```
```   924   qed
```
```   925 qed
```
```   926
```
```   927 declare image_mset.identity [simp]
```
```   928
```
```   929
```
```   930 subsection {* Further conversions *}
```
```   931
```
```   932 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
```
```   933   "multiset_of [] = {#}" |
```
```   934   "multiset_of (a # x) = multiset_of x + {# a #}"
```
```   935
```
```   936 lemma in_multiset_in_set:
```
```   937   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```   938   by (induct xs) simp_all
```
```   939
```
```   940 lemma count_multiset_of:
```
```   941   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```   942   by (induct xs) simp_all
```
```   943
```
```   944 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
```
```   945 by (induct x) auto
```
```   946
```
```   947 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
```
```   948 by (induct x) auto
```
```   949
```
```   950 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
```
```   951 by (induct x) auto
```
```   952
```
```   953 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
```
```   954 by (induct xs) auto
```
```   955
```
```   956 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
```
```   957   by (induct xs) simp_all
```
```   958
```
```   959 lemma multiset_of_append [simp]:
```
```   960   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
```
```   961   by (induct xs arbitrary: ys) (auto simp: ac_simps)
```
```   962
```
```   963 lemma multiset_of_filter:
```
```   964   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
```
```   965   by (induct xs) simp_all
```
```   966
```
```   967 lemma multiset_of_rev [simp]:
```
```   968   "multiset_of (rev xs) = multiset_of xs"
```
```   969   by (induct xs) simp_all
```
```   970
```
```   971 lemma surj_multiset_of: "surj multiset_of"
```
```   972 apply (unfold surj_def)
```
```   973 apply (rule allI)
```
```   974 apply (rule_tac M = y in multiset_induct)
```
```   975  apply auto
```
```   976 apply (rule_tac x = "x # xa" in exI)
```
```   977 apply auto
```
```   978 done
```
```   979
```
```   980 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
```
```   981 by (induct x) auto
```
```   982
```
```   983 lemma distinct_count_atmost_1:
```
```   984   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
```
```   985 apply (induct x, simp, rule iffI, simp_all)
```
```   986 apply (rename_tac a b)
```
```   987 apply (rule conjI)
```
```   988 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
```
```   989 apply (erule_tac x = a in allE, simp, clarify)
```
```   990 apply (erule_tac x = aa in allE, simp)
```
```   991 done
```
```   992
```
```   993 lemma multiset_of_eq_setD:
```
```   994   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
```
```   995 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
```
```   996
```
```   997 lemma set_eq_iff_multiset_of_eq_distinct:
```
```   998   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
```
```   999     (set x = set y) = (multiset_of x = multiset_of y)"
```
```  1000 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
```
```  1001
```
```  1002 lemma set_eq_iff_multiset_of_remdups_eq:
```
```  1003    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
```
```  1004 apply (rule iffI)
```
```  1005 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
```
```  1006 apply (drule distinct_remdups [THEN distinct_remdups
```
```  1007       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
```
```  1008 apply simp
```
```  1009 done
```
```  1010
```
```  1011 lemma multiset_of_compl_union [simp]:
```
```  1012   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
```
```  1013   by (induct xs) (auto simp: ac_simps)
```
```  1014
```
```  1015 lemma count_multiset_of_length_filter:
```
```  1016   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```  1017   by (induct xs) auto
```
```  1018
```
```  1019 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
```
```  1020 apply (induct ls arbitrary: i)
```
```  1021  apply simp
```
```  1022 apply (case_tac i)
```
```  1023  apply auto
```
```  1024 done
```
```  1025
```
```  1026 lemma multiset_of_remove1[simp]:
```
```  1027   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
```
```  1028 by (induct xs) (auto simp add: multiset_eq_iff)
```
```  1029
```
```  1030 lemma multiset_of_eq_length:
```
```  1031   assumes "multiset_of xs = multiset_of ys"
```
```  1032   shows "length xs = length ys"
```
```  1033   using assms by (metis size_multiset_of)
```
```  1034
```
```  1035 lemma multiset_of_eq_length_filter:
```
```  1036   assumes "multiset_of xs = multiset_of ys"
```
```  1037   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
```
```  1038   using assms by (metis count_multiset_of)
```
```  1039
```
```  1040 lemma fold_multiset_equiv:
```
```  1041   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"
```
```  1042     and equiv: "multiset_of xs = multiset_of ys"
```
```  1043   shows "List.fold f xs = List.fold f ys"
```
```  1044 using f equiv [symmetric]
```
```  1045 proof (induct xs arbitrary: ys)
```
```  1046   case Nil then show ?case by simp
```
```  1047 next
```
```  1048   case (Cons x xs)
```
```  1049   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
```
```  1050   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
```
```  1051     by (rule Cons.prems(1)) (simp_all add: *)
```
```  1052   moreover from * have "x \<in> set ys" by simp
```
```  1053   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
```
```  1054   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
```
```  1055   ultimately show ?case by simp
```
```  1056 qed
```
```  1057
```
```  1058 lemma multiset_of_insort [simp]:
```
```  1059   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
```
```  1060   by (induct xs) (simp_all add: ac_simps)
```
```  1061
```
```  1062 lemma in_multiset_of:
```
```  1063   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```  1064   by (induct xs) simp_all
```
```  1065
```
```  1066 lemma multiset_of_map:
```
```  1067   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
```
```  1068   by (induct xs) simp_all
```
```  1069
```
```  1070 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
```
```  1071 where
```
```  1072   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
```
```  1073
```
```  1074 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
```
```  1075 where
```
```  1076   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
```
```  1077 proof -
```
```  1078   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
```
```  1079   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
```
```  1080   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
```
```  1081 qed
```
```  1082
```
```  1083 lemma count_multiset_of_set [simp]:
```
```  1084   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
```
```  1085   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
```
```  1086   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
```
```  1087 proof -
```
```  1088   { fix A
```
```  1089     assume "x \<notin> A"
```
```  1090     have "count (multiset_of_set A) x = 0"
```
```  1091     proof (cases "finite A")
```
```  1092       case False then show ?thesis by simp
```
```  1093     next
```
```  1094       case True from True `x \<notin> A` show ?thesis by (induct A) auto
```
```  1095     qed
```
```  1096   } note * = this
```
```  1097   then show "PROP ?P" "PROP ?Q" "PROP ?R"
```
```  1098   by (auto elim!: Set.set_insert)
```
```  1099 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
```
```  1100
```
```  1101 context linorder
```
```  1102 begin
```
```  1103
```
```  1104 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
```
```  1105 where
```
```  1106   "sorted_list_of_multiset M = fold insort [] M"
```
```  1107
```
```  1108 lemma sorted_list_of_multiset_empty [simp]:
```
```  1109   "sorted_list_of_multiset {#} = []"
```
```  1110   by (simp add: sorted_list_of_multiset_def)
```
```  1111
```
```  1112 lemma sorted_list_of_multiset_singleton [simp]:
```
```  1113   "sorted_list_of_multiset {#x#} = [x]"
```
```  1114 proof -
```
```  1115   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1116   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1117 qed
```
```  1118
```
```  1119 lemma sorted_list_of_multiset_insert [simp]:
```
```  1120   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
```
```  1121 proof -
```
```  1122   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1123   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1124 qed
```
```  1125
```
```  1126 end
```
```  1127
```
```  1128 lemma multiset_of_sorted_list_of_multiset [simp]:
```
```  1129   "multiset_of (sorted_list_of_multiset M) = M"
```
```  1130   by (induct M) simp_all
```
```  1131
```
```  1132 lemma sorted_list_of_multiset_multiset_of [simp]:
```
```  1133   "sorted_list_of_multiset (multiset_of xs) = sort xs"
```
```  1134   by (induct xs) simp_all
```
```  1135
```
```  1136 lemma finite_set_of_multiset_of_set:
```
```  1137   assumes "finite A"
```
```  1138   shows "set_of (multiset_of_set A) = A"
```
```  1139   using assms by (induct A) simp_all
```
```  1140
```
```  1141 lemma infinite_set_of_multiset_of_set:
```
```  1142   assumes "\<not> finite A"
```
```  1143   shows "set_of (multiset_of_set A) = {}"
```
```  1144   using assms by simp
```
```  1145
```
```  1146 lemma set_sorted_list_of_multiset [simp]:
```
```  1147   "set (sorted_list_of_multiset M) = set_of M"
```
```  1148   by (induct M) (simp_all add: set_insort)
```
```  1149
```
```  1150 lemma sorted_list_of_multiset_of_set [simp]:
```
```  1151   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
```
```  1152   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
```
```  1153
```
```  1154
```
```  1155 subsection {* Big operators *}
```
```  1156
```
```  1157 no_notation times (infixl "*" 70)
```
```  1158 no_notation Groups.one ("1")
```
```  1159
```
```  1160 locale comm_monoid_mset = comm_monoid
```
```  1161 begin
```
```  1162
```
```  1163 definition F :: "'a multiset \<Rightarrow> 'a"
```
```  1164 where
```
```  1165   eq_fold: "F M = Multiset.fold f 1 M"
```
```  1166
```
```  1167 lemma empty [simp]:
```
```  1168   "F {#} = 1"
```
```  1169   by (simp add: eq_fold)
```
```  1170
```
```  1171 lemma singleton [simp]:
```
```  1172   "F {#x#} = x"
```
```  1173 proof -
```
```  1174   interpret comp_fun_commute
```
```  1175     by default (simp add: fun_eq_iff left_commute)
```
```  1176   show ?thesis by (simp add: eq_fold)
```
```  1177 qed
```
```  1178
```
```  1179 lemma union [simp]:
```
```  1180   "F (M + N) = F M * F N"
```
```  1181 proof -
```
```  1182   interpret comp_fun_commute f
```
```  1183     by default (simp add: fun_eq_iff left_commute)
```
```  1184   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
```
```  1185 qed
```
```  1186
```
```  1187 end
```
```  1188
```
```  1189 notation times (infixl "*" 70)
```
```  1190 notation Groups.one ("1")
```
```  1191
```
```  1192 context comm_monoid_add
```
```  1193 begin
```
```  1194
```
```  1195 definition msetsum :: "'a multiset \<Rightarrow> 'a"
```
```  1196 where
```
```  1197   "msetsum = comm_monoid_mset.F plus 0"
```
```  1198
```
```  1199 sublocale msetsum!: comm_monoid_mset plus 0
```
```  1200 where
```
```  1201   "comm_monoid_mset.F plus 0 = msetsum"
```
```  1202 proof -
```
```  1203   show "comm_monoid_mset plus 0" ..
```
```  1204   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
```
```  1205 qed
```
```  1206
```
```  1207 lemma setsum_unfold_msetsum:
```
```  1208   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
```
```  1209   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1210
```
```  1211 end
```
```  1212
```
```  1213 syntax
```
```  1214   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1215       ("(3SUM _:#_. _)" [0, 51, 10] 10)
```
```  1216
```
```  1217 syntax (xsymbols)
```
```  1218   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1219       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1220
```
```  1221 syntax (HTML output)
```
```  1222   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1223       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1224
```
```  1225 translations
```
```  1226   "SUM i :# A. b" == "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"
```
```  1227
```
```  1228 context comm_monoid_mult
```
```  1229 begin
```
```  1230
```
```  1231 definition msetprod :: "'a multiset \<Rightarrow> 'a"
```
```  1232 where
```
```  1233   "msetprod = comm_monoid_mset.F times 1"
```
```  1234
```
```  1235 sublocale msetprod!: comm_monoid_mset times 1
```
```  1236 where
```
```  1237   "comm_monoid_mset.F times 1 = msetprod"
```
```  1238 proof -
```
```  1239   show "comm_monoid_mset times 1" ..
```
```  1240   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
```
```  1241 qed
```
```  1242
```
```  1243 lemma msetprod_empty:
```
```  1244   "msetprod {#} = 1"
```
```  1245   by (fact msetprod.empty)
```
```  1246
```
```  1247 lemma msetprod_singleton:
```
```  1248   "msetprod {#x#} = x"
```
```  1249   by (fact msetprod.singleton)
```
```  1250
```
```  1251 lemma msetprod_Un:
```
```  1252   "msetprod (A + B) = msetprod A * msetprod B"
```
```  1253   by (fact msetprod.union)
```
```  1254
```
```  1255 lemma setprod_unfold_msetprod:
```
```  1256   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
```
```  1257   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1258
```
```  1259 lemma msetprod_multiplicity:
```
```  1260   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
```
```  1261   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
```
```  1262
```
```  1263 end
```
```  1264
```
```  1265 syntax
```
```  1266   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1267       ("(3PROD _:#_. _)" [0, 51, 10] 10)
```
```  1268
```
```  1269 syntax (xsymbols)
```
```  1270   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1271       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1272
```
```  1273 syntax (HTML output)
```
```  1274   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1275       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1276
```
```  1277 translations
```
```  1278   "PROD i :# A. b" == "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"
```
```  1279
```
```  1280 lemma (in comm_semiring_1) dvd_msetprod:
```
```  1281   assumes "x \<in># A"
```
```  1282   shows "x dvd msetprod A"
```
```  1283 proof -
```
```  1284   from assms have "A = (A - {#x#}) + {#x#}" by simp
```
```  1285   then obtain B where "A = B + {#x#}" ..
```
```  1286   then show ?thesis by simp
```
```  1287 qed
```
```  1288
```
```  1289
```
```  1290 subsection {* Cardinality *}
```
```  1291
```
```  1292 definition mcard :: "'a multiset \<Rightarrow> nat"
```
```  1293 where
```
```  1294   "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
```
```  1295
```
```  1296 lemma mcard_empty [simp]:
```
```  1297   "mcard {#} = 0"
```
```  1298   by (simp add: mcard_def)
```
```  1299
```
```  1300 lemma mcard_singleton [simp]:
```
```  1301   "mcard {#a#} = Suc 0"
```
```  1302   by (simp add: mcard_def)
```
```  1303
```
```  1304 lemma mcard_plus [simp]:
```
```  1305   "mcard (M + N) = mcard M + mcard N"
```
```  1306   by (simp add: mcard_def)
```
```  1307
```
```  1308 lemma mcard_empty_iff [simp]:
```
```  1309   "mcard M = 0 \<longleftrightarrow> M = {#}"
```
```  1310   by (induct M) simp_all
```
```  1311
```
```  1312 lemma mcard_unfold_setsum:
```
```  1313   "mcard M = setsum (count M) (set_of M)"
```
```  1314 proof (induct M)
```
```  1315   case empty then show ?case by simp
```
```  1316 next
```
```  1317   case (add M x) then show ?case
```
```  1318     by (cases "x \<in> set_of M")
```
```  1319       (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
```
```  1320 qed
```
```  1321
```
```  1322 lemma size_eq_mcard:
```
```  1323   "size = mcard"
```
```  1324   by (simp add: fun_eq_iff size_multiset_overloaded_eq mcard_unfold_setsum)
```
```  1325
```
```  1326 lemma mcard_multiset_of:
```
```  1327   "mcard (multiset_of xs) = length xs"
```
```  1328   by (induct xs) simp_all
```
```  1329
```
```  1330 lemma mcard_mono: assumes "A \<le> B"
```
```  1331   shows "mcard A \<le> mcard B"
```
```  1332 proof -
```
```  1333   from assms[unfolded mset_le_exists_conv]
```
```  1334   obtain C where B: "B = A + C" by auto
```
```  1335   show ?thesis unfolding B by (induct C, auto)
```
```  1336 qed
```
```  1337
```
```  1338 lemma mcard_filter_lesseq[simp]: "mcard (Multiset.filter f M) \<le> mcard M"
```
```  1339   by (rule mcard_mono[OF multiset_filter_subset])
```
```  1340
```
```  1341
```
```  1342 subsection {* Alternative representations *}
```
```  1343
```
```  1344 subsubsection {* Lists *}
```
```  1345
```
```  1346 context linorder
```
```  1347 begin
```
```  1348
```
```  1349 lemma multiset_of_insort [simp]:
```
```  1350   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
```
```  1351   by (induct xs) (simp_all add: ac_simps)
```
```  1352
```
```  1353 lemma multiset_of_sort [simp]:
```
```  1354   "multiset_of (sort_key k xs) = multiset_of xs"
```
```  1355   by (induct xs) (simp_all add: ac_simps)
```
```  1356
```
```  1357 text {*
```
```  1358   This lemma shows which properties suffice to show that a function
```
```  1359   @{text "f"} with @{text "f xs = ys"} behaves like sort.
```
```  1360 *}
```
```  1361
```
```  1362 lemma properties_for_sort_key:
```
```  1363   assumes "multiset_of ys = multiset_of xs"
```
```  1364   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
```
```  1365   and "sorted (map f ys)"
```
```  1366   shows "sort_key f xs = ys"
```
```  1367 using assms
```
```  1368 proof (induct xs arbitrary: ys)
```
```  1369   case Nil then show ?case by simp
```
```  1370 next
```
```  1371   case (Cons x xs)
```
```  1372   from Cons.prems(2) have
```
```  1373     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
```
```  1374     by (simp add: filter_remove1)
```
```  1375   with Cons.prems have "sort_key f xs = remove1 x ys"
```
```  1376     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
```
```  1377   moreover from Cons.prems have "x \<in> set ys"
```
```  1378     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
```
```  1379   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
```
```  1380 qed
```
```  1381
```
```  1382 lemma properties_for_sort:
```
```  1383   assumes multiset: "multiset_of ys = multiset_of xs"
```
```  1384   and "sorted ys"
```
```  1385   shows "sort xs = ys"
```
```  1386 proof (rule properties_for_sort_key)
```
```  1387   from multiset show "multiset_of ys = multiset_of xs" .
```
```  1388   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
```
```  1389   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
```
```  1390     by (rule multiset_of_eq_length_filter)
```
```  1391   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
```
```  1392     by simp
```
```  1393   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
```
```  1394     by (simp add: replicate_length_filter)
```
```  1395 qed
```
```  1396
```
```  1397 lemma sort_key_by_quicksort:
```
```  1398   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
```
```  1399     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
```
```  1400     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
```
```  1401 proof (rule properties_for_sort_key)
```
```  1402   show "multiset_of ?rhs = multiset_of ?lhs"
```
```  1403     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
```
```  1404 next
```
```  1405   show "sorted (map f ?rhs)"
```
```  1406     by (auto simp add: sorted_append intro: sorted_map_same)
```
```  1407 next
```
```  1408   fix l
```
```  1409   assume "l \<in> set ?rhs"
```
```  1410   let ?pivot = "f (xs ! (length xs div 2))"
```
```  1411   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
```
```  1412   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
```
```  1413     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
```
```  1414   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
```
```  1415   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
```
```  1416   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
```
```  1417     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
```
```  1418   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
```
```  1419   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
```
```  1420   proof (cases "f l" ?pivot rule: linorder_cases)
```
```  1421     case less
```
```  1422     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
```
```  1423     with less show ?thesis
```
```  1424       by (simp add: filter_sort [symmetric] ** ***)
```
```  1425   next
```
```  1426     case equal then show ?thesis
```
```  1427       by (simp add: * less_le)
```
```  1428   next
```
```  1429     case greater
```
```  1430     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
```
```  1431     with greater show ?thesis
```
```  1432       by (simp add: filter_sort [symmetric] ** ***)
```
```  1433   qed
```
```  1434 qed
```
```  1435
```
```  1436 lemma sort_by_quicksort:
```
```  1437   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
```
```  1438     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
```
```  1439     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
```
```  1440   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
```
```  1441
```
```  1442 text {* A stable parametrized quicksort *}
```
```  1443
```
```  1444 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
```
```  1445   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
```
```  1446
```
```  1447 lemma part_code [code]:
```
```  1448   "part f pivot [] = ([], [], [])"
```
```  1449   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
```
```  1450      if x' < pivot then (x # lts, eqs, gts)
```
```  1451      else if x' > pivot then (lts, eqs, x # gts)
```
```  1452      else (lts, x # eqs, gts))"
```
```  1453   by (auto simp add: part_def Let_def split_def)
```
```  1454
```
```  1455 lemma sort_key_by_quicksort_code [code]:
```
```  1456   "sort_key f xs = (case xs of [] \<Rightarrow> []
```
```  1457     | [x] \<Rightarrow> xs
```
```  1458     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
```
```  1459     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1460        in sort_key f lts @ eqs @ sort_key f gts))"
```
```  1461 proof (cases xs)
```
```  1462   case Nil then show ?thesis by simp
```
```  1463 next
```
```  1464   case (Cons _ ys) note hyps = Cons show ?thesis
```
```  1465   proof (cases ys)
```
```  1466     case Nil with hyps show ?thesis by simp
```
```  1467   next
```
```  1468     case (Cons _ zs) note hyps = hyps Cons show ?thesis
```
```  1469     proof (cases zs)
```
```  1470       case Nil with hyps show ?thesis by auto
```
```  1471     next
```
```  1472       case Cons
```
```  1473       from sort_key_by_quicksort [of f xs]
```
```  1474       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1475         in sort_key f lts @ eqs @ sort_key f gts)"
```
```  1476       by (simp only: split_def Let_def part_def fst_conv snd_conv)
```
```  1477       with hyps Cons show ?thesis by (simp only: list.cases)
```
```  1478     qed
```
```  1479   qed
```
```  1480 qed
```
```  1481
```
```  1482 end
```
```  1483
```
```  1484 hide_const (open) part
```
```  1485
```
```  1486 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
```
```  1487   by (induct xs) (auto intro: order_trans)
```
```  1488
```
```  1489 lemma multiset_of_update:
```
```  1490   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
```
```  1491 proof (induct ls arbitrary: i)
```
```  1492   case Nil then show ?case by simp
```
```  1493 next
```
```  1494   case (Cons x xs)
```
```  1495   show ?case
```
```  1496   proof (cases i)
```
```  1497     case 0 then show ?thesis by simp
```
```  1498   next
```
```  1499     case (Suc i')
```
```  1500     with Cons show ?thesis
```
```  1501       apply simp
```
```  1502       apply (subst add.assoc)
```
```  1503       apply (subst add.commute [of "{#v#}" "{#x#}"])
```
```  1504       apply (subst add.assoc [symmetric])
```
```  1505       apply simp
```
```  1506       apply (rule mset_le_multiset_union_diff_commute)
```
```  1507       apply (simp add: mset_le_single nth_mem_multiset_of)
```
```  1508       done
```
```  1509   qed
```
```  1510 qed
```
```  1511
```
```  1512 lemma multiset_of_swap:
```
```  1513   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
```
```  1514     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
```
```  1515   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
```
```  1516
```
```  1517
```
```  1518 subsection {* The multiset order *}
```
```  1519
```
```  1520 subsubsection {* Well-foundedness *}
```
```  1521
```
```  1522 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1523   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```  1524       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```  1525
```
```  1526 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1527   "mult r = (mult1 r)\<^sup>+"
```
```  1528
```
```  1529 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```  1530 by (simp add: mult1_def)
```
```  1531
```
```  1532 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```  1533     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```  1534     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```  1535   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
```
```  1536 proof (unfold mult1_def)
```
```  1537   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```  1538   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```  1539   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```  1540
```
```  1541   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```  1542   then have "\<exists>a' M0' K.
```
```  1543       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```  1544   then show "?case1 \<or> ?case2"
```
```  1545   proof (elim exE conjE)
```
```  1546     fix a' M0' K
```
```  1547     assume N: "N = M0' + K" and r: "?r K a'"
```
```  1548     assume "M0 + {#a#} = M0' + {#a'#}"
```
```  1549     then have "M0 = M0' \<and> a = a' \<or>
```
```  1550         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```  1551       by (simp only: add_eq_conv_ex)
```
```  1552     then show ?thesis
```
```  1553     proof (elim disjE conjE exE)
```
```  1554       assume "M0 = M0'" "a = a'"
```
```  1555       with N r have "?r K a \<and> N = M0 + K" by simp
```
```  1556       then have ?case2 .. then show ?thesis ..
```
```  1557     next
```
```  1558       fix K'
```
```  1559       assume "M0' = K' + {#a#}"
```
```  1560       with N have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
```
```  1561
```
```  1562       assume "M0 = K' + {#a'#}"
```
```  1563       with r have "?R (K' + K) M0" by blast
```
```  1564       with n have ?case1 by simp then show ?thesis ..
```
```  1565     qed
```
```  1566   qed
```
```  1567 qed
```
```  1568
```
```  1569 lemma all_accessible: "wf r ==> \<forall>M. M \<in> Wellfounded.acc (mult1 r)"
```
```  1570 proof
```
```  1571   let ?R = "mult1 r"
```
```  1572   let ?W = "Wellfounded.acc ?R"
```
```  1573   {
```
```  1574     fix M M0 a
```
```  1575     assume M0: "M0 \<in> ?W"
```
```  1576       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1577       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```  1578     have "M0 + {#a#} \<in> ?W"
```
```  1579     proof (rule accI [of "M0 + {#a#}"])
```
```  1580       fix N
```
```  1581       assume "(N, M0 + {#a#}) \<in> ?R"
```
```  1582       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```  1583           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```  1584         by (rule less_add)
```
```  1585       then show "N \<in> ?W"
```
```  1586       proof (elim exE disjE conjE)
```
```  1587         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```  1588         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```  1589         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
```
```  1590         then show "N \<in> ?W" by (simp only: N)
```
```  1591       next
```
```  1592         fix K
```
```  1593         assume N: "N = M0 + K"
```
```  1594         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```  1595         then have "M0 + K \<in> ?W"
```
```  1596         proof (induct K)
```
```  1597           case empty
```
```  1598           from M0 show "M0 + {#} \<in> ?W" by simp
```
```  1599         next
```
```  1600           case (add K x)
```
```  1601           from add.prems have "(x, a) \<in> r" by simp
```
```  1602           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```  1603           moreover from add have "M0 + K \<in> ?W" by simp
```
```  1604           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
```
```  1605           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
```
```  1606         qed
```
```  1607         then show "N \<in> ?W" by (simp only: N)
```
```  1608       qed
```
```  1609     qed
```
```  1610   } note tedious_reasoning = this
```
```  1611
```
```  1612   assume wf: "wf r"
```
```  1613   fix M
```
```  1614   show "M \<in> ?W"
```
```  1615   proof (induct M)
```
```  1616     show "{#} \<in> ?W"
```
```  1617     proof (rule accI)
```
```  1618       fix b assume "(b, {#}) \<in> ?R"
```
```  1619       with not_less_empty show "b \<in> ?W" by contradiction
```
```  1620     qed
```
```  1621
```
```  1622     fix M a assume "M \<in> ?W"
```
```  1623     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1624     proof induct
```
```  1625       fix a
```
```  1626       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1627       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1628       proof
```
```  1629         fix M assume "M \<in> ?W"
```
```  1630         then show "M + {#a#} \<in> ?W"
```
```  1631           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
```
```  1632       qed
```
```  1633     qed
```
```  1634     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
```
```  1635   qed
```
```  1636 qed
```
```  1637
```
```  1638 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```  1639 by (rule acc_wfI) (rule all_accessible)
```
```  1640
```
```  1641 theorem wf_mult: "wf r ==> wf (mult r)"
```
```  1642 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
```
```  1643
```
```  1644
```
```  1645 subsubsection {* Closure-free presentation *}
```
```  1646
```
```  1647 text {* One direction. *}
```
```  1648
```
```  1649 lemma mult_implies_one_step:
```
```  1650   "trans r ==> (M, N) \<in> mult r ==>
```
```  1651     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```  1652     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```  1653 apply (unfold mult_def mult1_def set_of_def)
```
```  1654 apply (erule converse_trancl_induct, clarify)
```
```  1655  apply (rule_tac x = M0 in exI, simp, clarify)
```
```  1656 apply (case_tac "a :# K")
```
```  1657  apply (rule_tac x = I in exI)
```
```  1658  apply (simp (no_asm))
```
```  1659  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```  1660  apply (simp (no_asm_simp) add: add.assoc [symmetric])
```
```  1661  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="?S + ?T" in arg_cong)
```
```  1662  apply (simp add: diff_union_single_conv)
```
```  1663  apply (simp (no_asm_use) add: trans_def)
```
```  1664  apply blast
```
```  1665 apply (subgoal_tac "a :# I")
```
```  1666  apply (rule_tac x = "I - {#a#}" in exI)
```
```  1667  apply (rule_tac x = "J + {#a#}" in exI)
```
```  1668  apply (rule_tac x = "K + Ka" in exI)
```
```  1669  apply (rule conjI)
```
```  1670   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1671  apply (rule conjI)
```
```  1672   apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="?S + ?T" in arg_cong, simp)
```
```  1673   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1674  apply (simp (no_asm_use) add: trans_def)
```
```  1675  apply blast
```
```  1676 apply (subgoal_tac "a :# (M0 + {#a#})")
```
```  1677  apply simp
```
```  1678 apply (simp (no_asm))
```
```  1679 done
```
```  1680
```
```  1681 lemma one_step_implies_mult_aux:
```
```  1682   "trans r ==>
```
```  1683     \<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))
```
```  1684       --> (I + K, I + J) \<in> mult r"
```
```  1685 apply (induct_tac n, auto)
```
```  1686 apply (frule size_eq_Suc_imp_eq_union, clarify)
```
```  1687 apply (rename_tac "J'", simp)
```
```  1688 apply (erule notE, auto)
```
```  1689 apply (case_tac "J' = {#}")
```
```  1690  apply (simp add: mult_def)
```
```  1691  apply (rule r_into_trancl)
```
```  1692  apply (simp add: mult1_def set_of_def, blast)
```
```  1693 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```  1694 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```  1695 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```  1696 apply (erule ssubst)
```
```  1697 apply (simp add: Ball_def, auto)
```
```  1698 apply (subgoal_tac
```
```  1699   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
```
```  1700     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```  1701  prefer 2
```
```  1702  apply force
```
```  1703 apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
```
```  1704 apply (erule trancl_trans)
```
```  1705 apply (rule r_into_trancl)
```
```  1706 apply (simp add: mult1_def set_of_def)
```
```  1707 apply (rule_tac x = a in exI)
```
```  1708 apply (rule_tac x = "I + J'" in exI)
```
```  1709 apply (simp add: ac_simps)
```
```  1710 done
```
```  1711
```
```  1712 lemma one_step_implies_mult:
```
```  1713   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```  1714     ==> (I + K, I + J) \<in> mult r"
```
```  1715 using one_step_implies_mult_aux by blast
```
```  1716
```
```  1717
```
```  1718 subsubsection {* Partial-order properties *}
```
```  1719
```
```  1720 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
```
```  1721   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
```
```  1722
```
```  1723 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
```
```  1724   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
```
```  1725
```
```  1726 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
```
```  1727 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
```
```  1728
```
```  1729 interpretation multiset_order: order le_multiset less_multiset
```
```  1730 proof -
```
```  1731   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
```
```  1732   proof
```
```  1733     fix M :: "'a multiset"
```
```  1734     assume "M \<subset># M"
```
```  1735     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
```
```  1736     have "trans {(x'::'a, x). x' < x}"
```
```  1737       by (rule transI) simp
```
```  1738     moreover note MM
```
```  1739     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
```
```  1740       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
```
```  1741       by (rule mult_implies_one_step)
```
```  1742     then obtain I J K where "M = I + J" and "M = I + K"
```
```  1743       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
```
```  1744     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
```
```  1745     have "finite (set_of K)" by simp
```
```  1746     moreover note aux2
```
```  1747     ultimately have "set_of K = {}"
```
```  1748       by (induct rule: finite_induct) (auto intro: order_less_trans)
```
```  1749     with aux1 show False by simp
```
```  1750   qed
```
```  1751   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
```
```  1752     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
```
```  1753   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
```
```  1754     by default (auto simp add: le_multiset_def irrefl dest: trans)
```
```  1755 qed
```
```  1756
```
```  1757 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
```
```  1758   by simp
```
```  1759
```
```  1760
```
```  1761 subsubsection {* Monotonicity of multiset union *}
```
```  1762
```
```  1763 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
```
```  1764 apply (unfold mult1_def)
```
```  1765 apply auto
```
```  1766 apply (rule_tac x = a in exI)
```
```  1767 apply (rule_tac x = "C + M0" in exI)
```
```  1768 apply (simp add: add.assoc)
```
```  1769 done
```
```  1770
```
```  1771 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
```
```  1772 apply (unfold less_multiset_def mult_def)
```
```  1773 apply (erule trancl_induct)
```
```  1774  apply (blast intro: mult1_union)
```
```  1775 apply (blast intro: mult1_union trancl_trans)
```
```  1776 done
```
```  1777
```
```  1778 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
```
```  1779 apply (subst add.commute [of B C])
```
```  1780 apply (subst add.commute [of D C])
```
```  1781 apply (erule union_less_mono2)
```
```  1782 done
```
```  1783
```
```  1784 lemma union_less_mono:
```
```  1785   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
```
```  1786   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
```
```  1787
```
```  1788 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
```
```  1789 proof
```
```  1790 qed (auto simp add: le_multiset_def intro: union_less_mono2)
```
```  1791
```
```  1792
```
```  1793 subsection {* Termination proofs with multiset orders *}
```
```  1794
```
```  1795 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
```
```  1796   and multi_member_this: "x \<in># {# x #} + XS"
```
```  1797   and multi_member_last: "x \<in># {# x #}"
```
```  1798   by auto
```
```  1799
```
```  1800 definition "ms_strict = mult pair_less"
```
```  1801 definition "ms_weak = ms_strict \<union> Id"
```
```  1802
```
```  1803 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
```
```  1804 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
```
```  1805 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
```
```  1806
```
```  1807 lemma smsI:
```
```  1808   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
```
```  1809   unfolding ms_strict_def
```
```  1810 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
```
```  1811
```
```  1812 lemma wmsI:
```
```  1813   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
```
```  1814   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
```
```  1815 unfolding ms_weak_def ms_strict_def
```
```  1816 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
```
```  1817
```
```  1818 inductive pw_leq
```
```  1819 where
```
```  1820   pw_leq_empty: "pw_leq {#} {#}"
```
```  1821 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
```
```  1822
```
```  1823 lemma pw_leq_lstep:
```
```  1824   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
```
```  1825 by (drule pw_leq_step) (rule pw_leq_empty, simp)
```
```  1826
```
```  1827 lemma pw_leq_split:
```
```  1828   assumes "pw_leq X Y"
```
```  1829   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 = {#}))"
```
```  1830   using assms
```
```  1831 proof (induct)
```
```  1832   case pw_leq_empty thus ?case by auto
```
```  1833 next
```
```  1834   case (pw_leq_step x y X Y)
```
```  1835   then obtain A B Z where
```
```  1836     [simp]: "X = A + Z" "Y = B + Z"
```
```  1837       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
```
```  1838     by auto
```
```  1839   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
```
```  1840     unfolding pair_leq_def by auto
```
```  1841   thus ?case
```
```  1842   proof
```
```  1843     assume [simp]: "x = y"
```
```  1844     have
```
```  1845       "{#x#} + X = A + ({#y#}+Z)
```
```  1846       \<and> {#y#} + Y = B + ({#y#}+Z)
```
```  1847       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
```
```  1848       by (auto simp: ac_simps)
```
```  1849     thus ?case by (intro exI)
```
```  1850   next
```
```  1851     assume A: "(x, y) \<in> pair_less"
```
```  1852     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
```
```  1853     have "{#x#} + X = ?A' + Z"
```
```  1854       "{#y#} + Y = ?B' + Z"
```
```  1855       by (auto simp add: ac_simps)
```
```  1856     moreover have
```
```  1857       "(set_of ?A', set_of ?B') \<in> max_strict"
```
```  1858       using 1 A unfolding max_strict_def
```
```  1859       by (auto elim!: max_ext.cases)
```
```  1860     ultimately show ?thesis by blast
```
```  1861   qed
```
```  1862 qed
```
```  1863
```
```  1864 lemma
```
```  1865   assumes pwleq: "pw_leq Z Z'"
```
```  1866   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
```
```  1867   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
```
```  1868   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
```
```  1869 proof -
```
```  1870   from pw_leq_split[OF pwleq]
```
```  1871   obtain A' B' Z''
```
```  1872     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
```
```  1873     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
```
```  1874     by blast
```
```  1875   {
```
```  1876     assume max: "(set_of A, set_of B) \<in> max_strict"
```
```  1877     from mx_or_empty
```
```  1878     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
```
```  1879     proof
```
```  1880       assume max': "(set_of A', set_of B') \<in> max_strict"
```
```  1881       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
```
```  1882         by (auto simp: max_strict_def intro: max_ext_additive)
```
```  1883       thus ?thesis by (rule smsI)
```
```  1884     next
```
```  1885       assume [simp]: "A' = {#} \<and> B' = {#}"
```
```  1886       show ?thesis by (rule smsI) (auto intro: max)
```
```  1887     qed
```
```  1888     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:ac_simps)
```
```  1889     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
```
```  1890   }
```
```  1891   from mx_or_empty
```
```  1892   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
```
```  1893   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
```
```  1894 qed
```
```  1895
```
```  1896 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
```
```  1897 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
```
```  1898 and nonempty_single: "{# x #} \<noteq> {#}"
```
```  1899 by auto
```
```  1900
```
```  1901 setup {*
```
```  1902 let
```
```  1903   fun msetT T = Type (@{type_name multiset}, [T]);
```
```  1904
```
```  1905   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
```
```  1906     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
```
```  1907     | mk_mset T (x :: xs) =
```
```  1908           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
```
```  1909                 mk_mset T [x] \$ mk_mset T xs
```
```  1910
```
```  1911   fun mset_member_tac m i =
```
```  1912       (if m <= 0 then
```
```  1913            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
```
```  1914        else
```
```  1915            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
```
```  1916
```
```  1917   val mset_nonempty_tac =
```
```  1918       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
```
```  1919
```
```  1920   val regroup_munion_conv =
```
```  1921       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
```
```  1922         (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
```
```  1923
```
```  1924   fun unfold_pwleq_tac i =
```
```  1925     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
```
```  1926       ORELSE (rtac @{thm pw_leq_lstep} i)
```
```  1927       ORELSE (rtac @{thm pw_leq_empty} i)
```
```  1928
```
```  1929   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
```
```  1930                       @{thm Un_insert_left}, @{thm Un_empty_left}]
```
```  1931 in
```
```  1932   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
```
```  1933   {
```
```  1934     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
```
```  1935     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
```
```  1936     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
```
```  1937     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
```
```  1938     reduction_pair= @{thm ms_reduction_pair}
```
```  1939   })
```
```  1940 end
```
```  1941 *}
```
```  1942
```
```  1943
```
```  1944 subsection {* Legacy theorem bindings *}
```
```  1945
```
```  1946 lemmas multi_count_eq = multiset_eq_iff [symmetric]
```
```  1947
```
```  1948 lemma union_commute: "M + N = N + (M::'a multiset)"
```
```  1949   by (fact add.commute)
```
```  1950
```
```  1951 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```  1952   by (fact add.assoc)
```
```  1953
```
```  1954 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```  1955   by (fact add.left_commute)
```
```  1956
```
```  1957 lemmas union_ac = union_assoc union_commute union_lcomm
```
```  1958
```
```  1959 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
```
```  1960   by (fact add_right_cancel)
```
```  1961
```
```  1962 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
```
```  1963   by (fact add_left_cancel)
```
```  1964
```
```  1965 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
```
```  1966   by (fact add_imp_eq)
```
```  1967
```
```  1968 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
```
```  1969   by (fact order_less_trans)
```
```  1970
```
```  1971 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
```
```  1972   by (fact inf.commute)
```
```  1973
```
```  1974 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
```
```  1975   by (fact inf.assoc [symmetric])
```
```  1976
```
```  1977 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
```
```  1978   by (fact inf.left_commute)
```
```  1979
```
```  1980 lemmas multiset_inter_ac =
```
```  1981   multiset_inter_commute
```
```  1982   multiset_inter_assoc
```
```  1983   multiset_inter_left_commute
```
```  1984
```
```  1985 lemma mult_less_not_refl:
```
```  1986   "\<not> M \<subset># (M::'a::order multiset)"
```
```  1987   by (fact multiset_order.less_irrefl)
```
```  1988
```
```  1989 lemma mult_less_trans:
```
```  1990   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
```
```  1991   by (fact multiset_order.less_trans)
```
```  1992
```
```  1993 lemma mult_less_not_sym:
```
```  1994   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
```
```  1995   by (fact multiset_order.less_not_sym)
```
```  1996
```
```  1997 lemma mult_less_asym:
```
```  1998   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
```
```  1999   by (fact multiset_order.less_asym)
```
```  2000
```
```  2001 ML {*
```
```  2002 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
```
```  2003                       (Const _ \$ t') =
```
```  2004     let
```
```  2005       val (maybe_opt, ps) =
```
```  2006         Nitpick_Model.dest_plain_fun t' ||> op ~~
```
```  2007         ||> map (apsnd (snd o HOLogic.dest_number))
```
```  2008       fun elems_for t =
```
```  2009         case AList.lookup (op =) ps t of
```
```  2010           SOME n => replicate n t
```
```  2011         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
```
```  2012     in
```
```  2013       case maps elems_for (all_values elem_T) @
```
```  2014            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
```
```  2015             else []) of
```
```  2016         [] => Const (@{const_name zero_class.zero}, T)
```
```  2017       | ts => foldl1 (fn (t1, t2) =>
```
```  2018                          Const (@{const_name plus_class.plus}, T --> T --> T)
```
```  2019                          \$ t1 \$ t2)
```
```  2020                      (map (curry (op \$) (Const (@{const_name single},
```
```  2021                                                 elem_T --> T))) ts)
```
```  2022     end
```
```  2023   | multiset_postproc _ _ _ _ t = t
```
```  2024 *}
```
```  2025
```
```  2026 declaration {*
```
```  2027 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
```
```  2028     multiset_postproc
```
```  2029 *}
```
```  2030
```
```  2031 hide_const (open) fold
```
```  2032
```
```  2033
```
```  2034 subsection {* Naive implementation using lists *}
```
```  2035
```
```  2036 code_datatype multiset_of
```
```  2037
```
```  2038 lemma [code]:
```
```  2039   "{#} = multiset_of []"
```
```  2040   by simp
```
```  2041
```
```  2042 lemma [code]:
```
```  2043   "{#x#} = multiset_of [x]"
```
```  2044   by simp
```
```  2045
```
```  2046 lemma union_code [code]:
```
```  2047   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
```
```  2048   by simp
```
```  2049
```
```  2050 lemma [code]:
```
```  2051   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
```
```  2052   by (simp add: multiset_of_map)
```
```  2053
```
```  2054 lemma [code]:
```
```  2055   "Multiset.filter f (multiset_of xs) = multiset_of (filter f xs)"
```
```  2056   by (simp add: multiset_of_filter)
```
```  2057
```
```  2058 lemma [code]:
```
```  2059   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
```
```  2060   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
```
```  2061
```
```  2062 lemma [code]:
```
```  2063   "multiset_of xs #\<inter> multiset_of ys =
```
```  2064     multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2065       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
```
```  2066 proof -
```
```  2067   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2068     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
```
```  2069       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
```
```  2070     by (induct xs arbitrary: ys)
```
```  2071       (auto simp add: mem_set_multiset_eq inter_add_right1 inter_add_right2 ac_simps)
```
```  2072   then show ?thesis by simp
```
```  2073 qed
```
```  2074
```
```  2075 lemma [code]:
```
```  2076   "multiset_of xs #\<union> multiset_of ys =
```
```  2077     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
```
```  2078 proof -
```
```  2079   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
```
```  2080       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
```
```  2081     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
```
```  2082   then show ?thesis by simp
```
```  2083 qed
```
```  2084
```
```  2085 lemma [code_unfold]:
```
```  2086   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```  2087   by (simp add: in_multiset_of)
```
```  2088
```
```  2089 lemma [code]:
```
```  2090   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
```
```  2091 proof -
```
```  2092   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
```
```  2093     by (induct xs) simp_all
```
```  2094   then show ?thesis by simp
```
```  2095 qed
```
```  2096
```
```  2097 lemma [code]:
```
```  2098   "set_of (multiset_of xs) = set xs"
```
```  2099   by simp
```
```  2100
```
```  2101 lemma [code]:
```
```  2102   "sorted_list_of_multiset (multiset_of xs) = sort xs"
```
```  2103   by (induct xs) simp_all
```
```  2104
```
```  2105 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
```
```  2106   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
```
```  2107   apply (cases "finite A")
```
```  2108   apply simp_all
```
```  2109   apply (induct A rule: finite_induct)
```
```  2110   apply (simp_all add: union_commute)
```
```  2111   done
```
```  2112
```
```  2113 lemma [code]:
```
```  2114   "mcard (multiset_of xs) = length xs"
```
```  2115   by (simp add: mcard_multiset_of)
```
```  2116
```
```  2117 fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
```
```  2118   "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
```
```  2119 | "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
```
```  2120      None \<Rightarrow> None
```
```  2121    | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
```
```  2122
```
```  2123 lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> multiset_of xs \<le> multiset_of ys) \<and>
```
```  2124   (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> multiset_of xs < multiset_of ys) \<and>
```
```  2125   (ms_lesseq_impl xs ys = Some False \<longrightarrow> multiset_of xs = multiset_of ys)"
```
```  2126 proof (induct xs arbitrary: ys)
```
```  2127   case (Nil ys)
```
```  2128   show ?case by (auto simp: mset_less_empty_nonempty)
```
```  2129 next
```
```  2130   case (Cons x xs ys)
```
```  2131   show ?case
```
```  2132   proof (cases "List.extract (op = x) ys")
```
```  2133     case None
```
```  2134     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
```
```  2135     {
```
```  2136       assume "multiset_of (x # xs) \<le> multiset_of ys"
```
```  2137       from set_of_mono[OF this] x have False by simp
```
```  2138     } note nle = this
```
```  2139     moreover
```
```  2140     {
```
```  2141       assume "multiset_of (x # xs) < multiset_of ys"
```
```  2142       hence "multiset_of (x # xs) \<le> multiset_of ys" by auto
```
```  2143       from nle[OF this] have False .
```
```  2144     }
```
```  2145     ultimately show ?thesis using None by auto
```
```  2146   next
```
```  2147     case (Some res)
```
```  2148     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
```
```  2149     note Some = Some[unfolded res]
```
```  2150     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
```
```  2151     hence id: "multiset_of ys = multiset_of (ys1 @ ys2) + {#x#}"
```
```  2152       by (auto simp: ac_simps)
```
```  2153     show ?thesis unfolding ms_lesseq_impl.simps
```
```  2154       unfolding Some option.simps split
```
```  2155       unfolding id
```
```  2156       using Cons[of "ys1 @ ys2"]
```
```  2157       unfolding mset_le_def mset_less_def by auto
```
```  2158   qed
```
```  2159 qed
```
```  2160
```
```  2161 lemma [code]: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
```
```  2162   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2163
```
```  2164 lemma [code]: "multiset_of xs < multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
```
```  2165   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2166
```
```  2167 instantiation multiset :: (equal) equal
```
```  2168 begin
```
```  2169
```
```  2170 definition
```
```  2171   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
```
```  2172 lemma [code]: "HOL.equal (multiset_of xs) (multiset_of ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
```
```  2173   unfolding equal_multiset_def
```
```  2174   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2175
```
```  2176 instance
```
```  2177   by default (simp add: equal_multiset_def)
```
```  2178 end
```
```  2179
```
```  2180 lemma [code]:
```
```  2181   "msetsum (multiset_of xs) = listsum xs"
```
```  2182   by (induct xs) (simp_all add: add.commute)
```
```  2183
```
```  2184 lemma [code]:
```
```  2185   "msetprod (multiset_of xs) = fold times xs 1"
```
```  2186 proof -
```
```  2187   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
```
```  2188     by (induct xs) (simp_all add: mult.assoc)
```
```  2189   then show ?thesis by simp
```
```  2190 qed
```
```  2191
```
```  2192 lemma [code]:
```
```  2193   "size = mcard"
```
```  2194   by (fact size_eq_mcard)
```
```  2195
```
```  2196 text {*
```
```  2197   Exercise for the casual reader: add implementations for @{const le_multiset}
```
```  2198   and @{const less_multiset} (multiset order).
```
```  2199 *}
```
```  2200
```
```  2201 text {* Quickcheck generators *}
```
```  2202
```
```  2203 definition (in term_syntax)
```
```  2204   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
```
```  2205     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
```
```  2206   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
```
```  2207
```
```  2208 notation fcomp (infixl "\<circ>>" 60)
```
```  2209 notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2210
```
```  2211 instantiation multiset :: (random) random
```
```  2212 begin
```
```  2213
```
```  2214 definition
```
```  2215   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
```
```  2216
```
```  2217 instance ..
```
```  2218
```
```  2219 end
```
```  2220
```
```  2221 no_notation fcomp (infixl "\<circ>>" 60)
```
```  2222 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2223
```
```  2224 instantiation multiset :: (full_exhaustive) full_exhaustive
```
```  2225 begin
```
```  2226
```
```  2227 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
```
```  2228 where
```
```  2229   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
```
```  2230
```
```  2231 instance ..
```
```  2232
```
```  2233 end
```
```  2234
```
```  2235 hide_const (open) msetify
```
```  2236
```
```  2237
```
```  2238 subsection {* BNF setup *}
```
```  2239
```
```  2240 definition rel_mset where
```
```  2241   "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. multiset_of xs = X \<and> multiset_of ys = Y \<and> list_all2 R xs ys)"
```
```  2242
```
```  2243 lemma multiset_of_zip_take_Cons_drop_twice:
```
```  2244   assumes "length xs = length ys" "j \<le> length xs"
```
```  2245   shows "multiset_of (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
```
```  2246     multiset_of (zip xs ys) + {#(x, y)#}"
```
```  2247 using assms
```
```  2248 proof (induct xs ys arbitrary: x y j rule: list_induct2)
```
```  2249   case Nil
```
```  2250   thus ?case
```
```  2251     by simp
```
```  2252 next
```
```  2253   case (Cons x xs y ys)
```
```  2254   thus ?case
```
```  2255   proof (cases "j = 0")
```
```  2256     case True
```
```  2257     thus ?thesis
```
```  2258       by simp
```
```  2259   next
```
```  2260     case False
```
```  2261     then obtain k where k: "j = Suc k"
```
```  2262       by (case_tac j) simp
```
```  2263     hence "k \<le> length xs"
```
```  2264       using Cons.prems by auto
```
```  2265     hence "multiset_of (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
```
```  2266       multiset_of (zip xs ys) + {#(x, y)#}"
```
```  2267       by (rule Cons.hyps(2))
```
```  2268     thus ?thesis
```
```  2269       unfolding k by (auto simp: add.commute union_lcomm)
```
```  2270   qed
```
```  2271 qed
```
```  2272
```
```  2273 lemma ex_multiset_of_zip_left:
```
```  2274   assumes "length xs = length ys" "multiset_of xs' = multiset_of xs"
```
```  2275   shows "\<exists>ys'. length ys' = length xs' \<and> multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
```
```  2276 using assms
```
```  2277 proof (induct xs ys arbitrary: xs' rule: list_induct2)
```
```  2278   case Nil
```
```  2279   thus ?case
```
```  2280     by auto
```
```  2281 next
```
```  2282   case (Cons x xs y ys xs')
```
```  2283   obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
```
```  2284     by (metis Cons.prems in_set_conv_nth list.set_intros(1) multiset_of_eq_setD)
```
```  2285
```
```  2286   def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
```
```  2287   have "multiset_of xs' = {#x#} + multiset_of xsa"
```
```  2288     unfolding xsa_def using j_len nth_j
```
```  2289     by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
```
```  2290       multiset_of.simps(2) union_code union_commute)
```
```  2291   hence ms_x: "multiset_of xsa = multiset_of xs"
```
```  2292     by (metis Cons.prems add.commute add_right_imp_eq multiset_of.simps(2))
```
```  2293   then obtain ysa where
```
```  2294     len_a: "length ysa = length xsa" and ms_a: "multiset_of (zip xsa ysa) = multiset_of (zip xs ys)"
```
```  2295     using Cons.hyps(2) by blast
```
```  2296
```
```  2297   def ys' \<equiv> "take j ysa @ y # drop j ysa"
```
```  2298   have xs': "xs' = take j xsa @ x # drop j xsa"
```
```  2299     using ms_x j_len nth_j Cons.prems xsa_def
```
```  2300     by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
```
```  2301       length_drop mcard_multiset_of)
```
```  2302   have j_len': "j \<le> length xsa"
```
```  2303     using j_len xs' xsa_def
```
```  2304     by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
```
```  2305   have "length ys' = length xs'"
```
```  2306     unfolding ys'_def using Cons.prems len_a ms_x
```
```  2307     by (metis add_Suc_right append_take_drop_id length_Cons length_append multiset_of_eq_length)
```
```  2308   moreover have "multiset_of (zip xs' ys') = multiset_of (zip (x # xs) (y # ys))"
```
```  2309     unfolding xs' ys'_def
```
```  2310     by (rule trans[OF multiset_of_zip_take_Cons_drop_twice])
```
```  2311       (auto simp: len_a ms_a j_len' add.commute)
```
```  2312   ultimately show ?case
```
```  2313     by blast
```
```  2314 qed
```
```  2315
```
```  2316 lemma list_all2_reorder_left_invariance:
```
```  2317   assumes rel: "list_all2 R xs ys" and ms_x: "multiset_of xs' = multiset_of xs"
```
```  2318   shows "\<exists>ys'. list_all2 R xs' ys' \<and> multiset_of ys' = multiset_of ys"
```
```  2319 proof -
```
```  2320   have len: "length xs = length ys"
```
```  2321     using rel list_all2_conv_all_nth by auto
```
```  2322   obtain ys' where
```
```  2323     len': "length xs' = length ys'" and ms_xy: "multiset_of (zip xs' ys') = multiset_of (zip xs ys)"
```
```  2324     using len ms_x by (metis ex_multiset_of_zip_left)
```
```  2325   have "list_all2 R xs' ys'"
```
```  2326     using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: multiset_of_eq_setD)
```
```  2327   moreover have "multiset_of ys' = multiset_of ys"
```
```  2328     using len len' ms_xy map_snd_zip multiset_of_map by metis
```
```  2329   ultimately show ?thesis
```
```  2330     by blast
```
```  2331 qed
```
```  2332
```
```  2333 lemma ex_multiset_of: "\<exists>xs. multiset_of xs = X"
```
```  2334   by (induct X) (simp, metis multiset_of.simps(2))
```
```  2335
```
```  2336 bnf "'a multiset"
```
```  2337   map: image_mset
```
```  2338   sets: set_of
```
```  2339   bd: natLeq
```
```  2340   wits: "{#}"
```
```  2341   rel: rel_mset
```
```  2342 proof -
```
```  2343   show "image_mset id = id"
```
```  2344     by (rule image_mset.id)
```
```  2345 next
```
```  2346   show "\<And>f g. image_mset (g \<circ> f) = image_mset g \<circ> image_mset f"
```
```  2347     unfolding comp_def by (rule ext) (simp add: image_mset.compositionality comp_def)
```
```  2348 next
```
```  2349   fix X :: "'a multiset"
```
```  2350   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"
```
```  2351     by (induct X, (simp (no_asm))+,
```
```  2352       metis One_nat_def Un_iff count_single mem_set_of_iff set_of_union zero_less_Suc)
```
```  2353 next
```
```  2354   show "\<And>f. set_of \<circ> image_mset f = op ` f \<circ> set_of"
```
```  2355     by auto
```
```  2356 next
```
```  2357   show "card_order natLeq"
```
```  2358     by (rule natLeq_card_order)
```
```  2359 next
```
```  2360   show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
```
```  2361     by (rule natLeq_cinfinite)
```
```  2362 next
```
```  2363   show "\<And>X. ordLeq3 (card_of (set_of X)) natLeq"
```
```  2364     by transfer
```
```  2365       (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
```
```  2366 next
```
```  2367   show "\<And>R S. rel_mset R OO rel_mset S \<le> rel_mset (R OO S)"
```
```  2368     unfolding rel_mset_def[abs_def] OO_def
```
```  2369     apply clarify
```
```  2370     apply (rename_tac X Z Y xs ys' ys zs)
```
```  2371     apply (drule_tac xs = ys' and ys = zs and xs' = ys in list_all2_reorder_left_invariance)
```
```  2372     by (auto intro: list_all2_trans)
```
```  2373 next
```
```  2374   show "\<And>R. rel_mset R =
```
```  2375     (BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset fst))\<inverse>\<inverse> OO
```
```  2376     BNF_Def.Grp {x. set_of x \<subseteq> {(x, y). R x y}} (image_mset snd)"
```
```  2377     unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
```
```  2378     apply (rule ext)+
```
```  2379     apply auto
```
```  2380      apply (rule_tac x = "multiset_of (zip xs ys)" in exI)
```
```  2381      apply auto
```
```  2382         apply (metis list_all2_lengthD map_fst_zip multiset_of_map)
```
```  2383        apply (auto simp: list_all2_iff)
```
```  2384       apply (metis list_all2_lengthD map_snd_zip multiset_of_map)
```
```  2385      apply (auto simp: list_all2_iff)
```
```  2386     apply (rename_tac XY)
```
```  2387     apply (cut_tac X = XY in ex_multiset_of)
```
```  2388     apply (erule exE)
```
```  2389     apply (rename_tac xys)
```
```  2390     apply (rule_tac x = "map fst xys" in exI)
```
```  2391     apply (auto simp: multiset_of_map)
```
```  2392     apply (rule_tac x = "map snd xys" in exI)
```
```  2393     by (auto simp: multiset_of_map list_all2I subset_eq zip_map_fst_snd)
```
```  2394 next
```
```  2395   show "\<And>z. z \<in> set_of {#} \<Longrightarrow> False"
```
```  2396     by auto
```
```  2397 qed
```
```  2398
```
```  2399 inductive rel_mset' where
```
```  2400   Zero[intro]: "rel_mset' R {#} {#}"
```
```  2401 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"
```
```  2402
```
```  2403 lemma rel_mset_Zero: "rel_mset R {#} {#}"
```
```  2404 unfolding rel_mset_def Grp_def by auto
```
```  2405
```
```  2406 declare multiset.count[simp]
```
```  2407 declare Abs_multiset_inverse[simp]
```
```  2408 declare multiset.count_inverse[simp]
```
```  2409 declare union_preserves_multiset[simp]
```
```  2410
```
```  2411 lemma rel_mset_Plus:
```
```  2412 assumes ab: "R a b" and MN: "rel_mset R M N"
```
```  2413 shows "rel_mset R (M + {#a#}) (N + {#b#})"
```
```  2414 proof-
```
```  2415   {fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
```
```  2416    hence "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
```
```  2417                image_mset snd y + {#b#} = image_mset snd ya \<and>
```
```  2418                set_of ya \<subseteq> {(x, y). R x y}"
```
```  2419    apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
```
```  2420   }
```
```  2421   thus ?thesis
```
```  2422   using assms
```
```  2423   unfolding multiset.rel_compp_Grp Grp_def by blast
```
```  2424 qed
```
```  2425
```
```  2426 lemma rel_mset'_imp_rel_mset:
```
```  2427 "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
```
```  2428 apply(induct rule: rel_mset'.induct)
```
```  2429 using rel_mset_Zero rel_mset_Plus by auto
```
```  2430
```
```  2431 lemma mcard_image_mset[simp]: "mcard (image_mset f M) = mcard M"
```
```  2432   unfolding size_eq_mcard[symmetric] by (rule size_image_mset)
```
```  2433
```
```  2434 lemma rel_mset_mcard:
```
```  2435   assumes "rel_mset R M N"
```
```  2436   shows "mcard M = mcard N"
```
```  2437 using assms unfolding multiset.rel_compp_Grp Grp_def by auto
```
```  2438
```
```  2439 lemma multiset_induct2[case_names empty addL addR]:
```
```  2440 assumes empty: "P {#} {#}"
```
```  2441 and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
```
```  2442 and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
```
```  2443 shows "P M N"
```
```  2444 apply(induct N rule: multiset_induct)
```
```  2445   apply(induct M rule: multiset_induct, rule empty, erule addL)
```
```  2446   apply(induct M rule: multiset_induct, erule addR, erule addR)
```
```  2447 done
```
```  2448
```
```  2449 lemma multiset_induct2_mcard[consumes 1, case_names empty add]:
```
```  2450 assumes c: "mcard M = mcard N"
```
```  2451 and empty: "P {#} {#}"
```
```  2452 and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
```
```  2453 shows "P M N"
```
```  2454 using c proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
```
```  2455   case (less M)  show ?case
```
```  2456   proof(cases "M = {#}")
```
```  2457     case True hence "N = {#}" using less.prems by auto
```
```  2458     thus ?thesis using True empty by auto
```
```  2459   next
```
```  2460     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
```
```  2461     have "N \<noteq> {#}" using False less.prems by auto
```
```  2462     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
```
```  2463     have "mcard M1 = mcard N1" using less.prems unfolding M N by auto
```
```  2464     thus ?thesis using M N less.hyps add by auto
```
```  2465   qed
```
```  2466 qed
```
```  2467
```
```  2468 lemma msed_map_invL:
```
```  2469 assumes "image_mset f (M + {#a#}) = N"
```
```  2470 shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1"
```
```  2471 proof-
```
```  2472   have "f a \<in># N"
```
```  2473   using assms multiset.set_map[of f "M + {#a#}"] by auto
```
```  2474   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
```
```  2475   have "image_mset f M = N1" using assms unfolding N by simp
```
```  2476   thus ?thesis using N by blast
```
```  2477 qed
```
```  2478
```
```  2479 lemma msed_map_invR:
```
```  2480 assumes "image_mset f M = N + {#b#}"
```
```  2481 shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N"
```
```  2482 proof-
```
```  2483   obtain a where a: "a \<in># M" and fa: "f a = b"
```
```  2484   using multiset.set_map[of f M] unfolding assms
```
```  2485   by (metis image_iff mem_set_of_iff union_single_eq_member)
```
```  2486   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
```
```  2487   have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
```
```  2488   thus ?thesis using M fa by blast
```
```  2489 qed
```
```  2490
```
```  2491 lemma msed_rel_invL:
```
```  2492 assumes "rel_mset R (M + {#a#}) N"
```
```  2493 shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1"
```
```  2494 proof-
```
```  2495   obtain K where KM: "image_mset fst K = M + {#a#}"
```
```  2496   and KN: "image_mset snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
```
```  2497   using assms
```
```  2498   unfolding multiset.rel_compp_Grp Grp_def by auto
```
```  2499   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
```
```  2500   and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
```
```  2501   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1"
```
```  2502   using msed_map_invL[OF KN[unfolded K]] by auto
```
```  2503   have Rab: "R a (snd ab)" using sK a unfolding K by auto
```
```  2504   have "rel_mset R M N1" using sK K1M K1N1
```
```  2505   unfolding K multiset.rel_compp_Grp Grp_def by auto
```
```  2506   thus ?thesis using N Rab by auto
```
```  2507 qed
```
```  2508
```
```  2509 lemma msed_rel_invR:
```
```  2510 assumes "rel_mset R M (N + {#b#})"
```
```  2511 shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N"
```
```  2512 proof-
```
```  2513   obtain K where KN: "image_mset snd K = N + {#b#}"
```
```  2514   and KM: "image_mset fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
```
```  2515   using assms
```
```  2516   unfolding multiset.rel_compp_Grp Grp_def by auto
```
```  2517   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
```
```  2518   and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
```
```  2519   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1"
```
```  2520   using msed_map_invL[OF KM[unfolded K]] by auto
```
```  2521   have Rab: "R (fst ab) b" using sK b unfolding K by auto
```
```  2522   have "rel_mset R M1 N" using sK K1N K1M1
```
```  2523   unfolding K multiset.rel_compp_Grp Grp_def by auto
```
```  2524   thus ?thesis using M Rab by auto
```
```  2525 qed
```
```  2526
```
```  2527 lemma rel_mset_imp_rel_mset':
```
```  2528 assumes "rel_mset R M N"
```
```  2529 shows "rel_mset' R M N"
```
```  2530 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
```
```  2531   case (less M)
```
```  2532   have c: "mcard M = mcard N" using rel_mset_mcard[OF less.prems] .
```
```  2533   show ?case
```
```  2534   proof(cases "M = {#}")
```
```  2535     case True hence "N = {#}" using c by simp
```
```  2536     thus ?thesis using True rel_mset'.Zero by auto
```
```  2537   next
```
```  2538     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
```
```  2539     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1"
```
```  2540     using msed_rel_invL[OF less.prems[unfolded M]] by auto
```
```  2541     have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
```
```  2542     thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
```
```  2543   qed
```
```  2544 qed
```
```  2545
```
```  2546 lemma rel_mset_rel_mset':
```
```  2547 "rel_mset R M N = rel_mset' R M N"
```
```  2548 using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
```
```  2549
```
```  2550 (* The main end product for rel_mset: inductive characterization *)
```
```  2551 theorems rel_mset_induct[case_names empty add, induct pred: rel_mset] =
```
```  2552          rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
```
```  2553
```
```  2554
```
```  2555 subsection {* Size setup *}
```
```  2556
```
```  2557 lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
```
```  2558   unfolding o_apply by (rule ext) (induct_tac, auto)
```
```  2559
```
```  2560 setup {*
```
```  2561 BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
```
```  2562   @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
```
```  2563     size_union}
```
```  2564   @{thms multiset_size_o_map}
```
```  2565 *}
```
```  2566
```
```  2567 hide_const (open) wcount
```
```  2568
```
```  2569 end
```