src/HOL/Library/Multiset.thy
 author haftmann Fri Jul 04 20:18:47 2014 +0200 (2014-07-04) changeset 57512 cc97b347b301 parent 57492 74bf65a1910a child 57514 bdc2c6b40bf2 permissions -rw-r--r--
reduced name variants for assoc and commute on plus and mult
```     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
```
```   266 subsubsection {* Pointwise ordering induced by count *}
```
```   267
```
```   268 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
```
```   269 begin
```
```   270
```
```   271 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)" .
```
```   272
```
```   273 lemmas mset_le_def = less_eq_multiset_def
```
```   274
```
```   275 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
```
```   276   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
```
```   277
```
```   278 instance
```
```   279   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
```
```   280
```
```   281 end
```
```   282
```
```   283 lemma mset_less_eqI:
```
```   284   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
```
```   285   by (simp add: mset_le_def)
```
```   286
```
```   287 lemma mset_le_exists_conv:
```
```   288   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
```
```   289 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
```
```   290 apply (auto intro: multiset_eq_iff [THEN iffD2])
```
```   291 done
```
```   292
```
```   293 instance multiset :: (type) ordered_cancel_comm_monoid_diff
```
```   294   by default (simp, fact mset_le_exists_conv)
```
```   295
```
```   296 lemma mset_le_mono_add_right_cancel [simp]:
```
```   297   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
```
```   298   by (fact add_le_cancel_right)
```
```   299
```
```   300 lemma mset_le_mono_add_left_cancel [simp]:
```
```   301   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
```
```   302   by (fact add_le_cancel_left)
```
```   303
```
```   304 lemma mset_le_mono_add:
```
```   305   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
```
```   306   by (fact add_mono)
```
```   307
```
```   308 lemma mset_le_add_left [simp]:
```
```   309   "(A::'a multiset) \<le> A + B"
```
```   310   unfolding mset_le_def by auto
```
```   311
```
```   312 lemma mset_le_add_right [simp]:
```
```   313   "B \<le> (A::'a multiset) + B"
```
```   314   unfolding mset_le_def by auto
```
```   315
```
```   316 lemma mset_le_single:
```
```   317   "a :# B \<Longrightarrow> {#a#} \<le> B"
```
```   318   by (simp add: mset_le_def)
```
```   319
```
```   320 lemma multiset_diff_union_assoc:
```
```   321   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
```
```   322   by (simp add: multiset_eq_iff mset_le_def)
```
```   323
```
```   324 lemma mset_le_multiset_union_diff_commute:
```
```   325   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
```
```   326 by (simp add: multiset_eq_iff mset_le_def)
```
```   327
```
```   328 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
```
```   329 by(simp add: mset_le_def)
```
```   330
```
```   331 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   332 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   333 apply (erule_tac x=x in allE)
```
```   334 apply auto
```
```   335 done
```
```   336
```
```   337 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   338 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   339 apply (erule_tac x = x in allE)
```
```   340 apply auto
```
```   341 done
```
```   342
```
```   343 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
```
```   344 apply (rule conjI)
```
```   345  apply (simp add: mset_lessD)
```
```   346 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   347 apply safe
```
```   348  apply (erule_tac x = a in allE)
```
```   349  apply (auto split: split_if_asm)
```
```   350 done
```
```   351
```
```   352 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
```
```   353 apply (rule conjI)
```
```   354  apply (simp add: mset_leD)
```
```   355 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
```
```   356 done
```
```   357
```
```   358 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
```
```   359   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
```
```   360
```
```   361 lemma empty_le[simp]: "{#} \<le> A"
```
```   362   unfolding mset_le_exists_conv by auto
```
```   363
```
```   364 lemma le_empty[simp]: "(M \<le> {#}) = (M = {#})"
```
```   365   unfolding mset_le_exists_conv by auto
```
```   366
```
```   367 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
```
```   368   by (auto simp: mset_le_def mset_less_def)
```
```   369
```
```   370 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
```
```   371   by simp
```
```   372
```
```   373 lemma mset_less_add_bothsides:
```
```   374   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
```
```   375   by (fact add_less_imp_less_right)
```
```   376
```
```   377 lemma mset_less_empty_nonempty:
```
```   378   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
```
```   379   by (auto simp: mset_le_def mset_less_def)
```
```   380
```
```   381 lemma mset_less_diff_self:
```
```   382   "c \<in># B \<Longrightarrow> B - {#c#} < B"
```
```   383   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
```
```   384
```
```   385
```
```   386 subsubsection {* Intersection *}
```
```   387
```
```   388 instantiation multiset :: (type) semilattice_inf
```
```   389 begin
```
```   390
```
```   391 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   392   multiset_inter_def: "inf_multiset A B = A - (A - B)"
```
```   393
```
```   394 instance
```
```   395 proof -
```
```   396   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
```
```   397   show "OFCLASS('a multiset, semilattice_inf_class)"
```
```   398     by default (auto simp add: multiset_inter_def mset_le_def aux)
```
```   399 qed
```
```   400
```
```   401 end
```
```   402
```
```   403 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
```
```   404   "multiset_inter \<equiv> inf"
```
```   405
```
```   406 lemma multiset_inter_count [simp]:
```
```   407   "count (A #\<inter> B) x = min (count A x) (count B x)"
```
```   408   by (simp add: multiset_inter_def)
```
```   409
```
```   410 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
```
```   411   by (rule multiset_eqI) auto
```
```   412
```
```   413 lemma multiset_union_diff_commute:
```
```   414   assumes "B #\<inter> C = {#}"
```
```   415   shows "A + B - C = A - C + B"
```
```   416 proof (rule multiset_eqI)
```
```   417   fix x
```
```   418   from assms have "min (count B x) (count C x) = 0"
```
```   419     by (auto simp add: multiset_eq_iff)
```
```   420   then have "count B x = 0 \<or> count C x = 0"
```
```   421     by auto
```
```   422   then show "count (A + B - C) x = count (A - C + B) x"
```
```   423     by auto
```
```   424 qed
```
```   425
```
```   426 lemma empty_inter [simp]:
```
```   427   "{#} #\<inter> M = {#}"
```
```   428   by (simp add: multiset_eq_iff)
```
```   429
```
```   430 lemma inter_empty [simp]:
```
```   431   "M #\<inter> {#} = {#}"
```
```   432   by (simp add: multiset_eq_iff)
```
```   433
```
```   434 lemma inter_add_left1:
```
```   435   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
```
```   436   by (simp add: multiset_eq_iff)
```
```   437
```
```   438 lemma inter_add_left2:
```
```   439   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
```
```   440   by (simp add: multiset_eq_iff)
```
```   441
```
```   442 lemma inter_add_right1:
```
```   443   "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
```
```   444   by (simp add: multiset_eq_iff)
```
```   445
```
```   446 lemma inter_add_right2:
```
```   447   "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
```
```   448   by (simp add: multiset_eq_iff)
```
```   449
```
```   450
```
```   451 subsubsection {* Bounded union *}
```
```   452
```
```   453 instantiation multiset :: (type) semilattice_sup
```
```   454 begin
```
```   455
```
```   456 definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   457   "sup_multiset A B = A + (B - A)"
```
```   458
```
```   459 instance
```
```   460 proof -
```
```   461   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" by arith
```
```   462   show "OFCLASS('a multiset, semilattice_sup_class)"
```
```   463     by default (auto simp add: sup_multiset_def mset_le_def aux)
```
```   464 qed
```
```   465
```
```   466 end
```
```   467
```
```   468 abbreviation sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<union>" 70) where
```
```   469   "sup_multiset \<equiv> sup"
```
```   470
```
```   471 lemma sup_multiset_count [simp]:
```
```   472   "count (A #\<union> B) x = max (count A x) (count B x)"
```
```   473   by (simp add: sup_multiset_def)
```
```   474
```
```   475 lemma empty_sup [simp]:
```
```   476   "{#} #\<union> M = M"
```
```   477   by (simp add: multiset_eq_iff)
```
```   478
```
```   479 lemma sup_empty [simp]:
```
```   480   "M #\<union> {#} = M"
```
```   481   by (simp add: multiset_eq_iff)
```
```   482
```
```   483 lemma sup_add_left1:
```
```   484   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
```
```   485   by (simp add: multiset_eq_iff)
```
```   486
```
```   487 lemma sup_add_left2:
```
```   488   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
```
```   489   by (simp add: multiset_eq_iff)
```
```   490
```
```   491 lemma sup_add_right1:
```
```   492   "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
```
```   493   by (simp add: multiset_eq_iff)
```
```   494
```
```   495 lemma sup_add_right2:
```
```   496   "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
```
```   497   by (simp add: multiset_eq_iff)
```
```   498
```
```   499
```
```   500 subsubsection {* Filter (with comprehension syntax) *}
```
```   501
```
```   502 text {* Multiset comprehension *}
```
```   503
```
```   504 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"
```
```   505 by (rule filter_preserves_multiset)
```
```   506
```
```   507 hide_const (open) filter
```
```   508
```
```   509 lemma count_filter [simp]:
```
```   510   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
```
```   511   by (simp add: filter.rep_eq)
```
```   512
```
```   513 lemma filter_empty [simp]:
```
```   514   "Multiset.filter P {#} = {#}"
```
```   515   by (rule multiset_eqI) simp
```
```   516
```
```   517 lemma filter_single [simp]:
```
```   518   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
```
```   519   by (rule multiset_eqI) simp
```
```   520
```
```   521 lemma filter_union [simp]:
```
```   522   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
```
```   523   by (rule multiset_eqI) simp
```
```   524
```
```   525 lemma filter_diff [simp]:
```
```   526   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
```
```   527   by (rule multiset_eqI) simp
```
```   528
```
```   529 lemma filter_inter [simp]:
```
```   530   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
```
```   531   by (rule multiset_eqI) simp
```
```   532
```
```   533 syntax
```
```   534   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
```
```   535 syntax (xsymbol)
```
```   536   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
```
```   537 translations
```
```   538   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
```
```   539
```
```   540
```
```   541 subsubsection {* Set of elements *}
```
```   542
```
```   543 definition set_of :: "'a multiset => 'a set" where
```
```   544   "set_of M = {x. x :# M}"
```
```   545
```
```   546 lemma set_of_empty [simp]: "set_of {#} = {}"
```
```   547 by (simp add: set_of_def)
```
```   548
```
```   549 lemma set_of_single [simp]: "set_of {#b#} = {b}"
```
```   550 by (simp add: set_of_def)
```
```   551
```
```   552 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   553 by (auto simp add: set_of_def)
```
```   554
```
```   555 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   556 by (auto simp add: set_of_def multiset_eq_iff)
```
```   557
```
```   558 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   559 by (auto simp add: set_of_def)
```
```   560
```
```   561 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
```
```   562 by (auto simp add: set_of_def)
```
```   563
```
```   564 lemma finite_set_of [iff]: "finite (set_of M)"
```
```   565   using count [of M] by (simp add: multiset_def set_of_def)
```
```   566
```
```   567 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
```
```   568   unfolding set_of_def[symmetric] by simp
```
```   569
```
```   570 lemma set_of_mono: "A \<le> B \<Longrightarrow> set_of A \<subseteq> set_of B"
```
```   571   by (metis mset_leD subsetI mem_set_of_iff)
```
```   572
```
```   573 subsubsection {* Size *}
```
```   574
```
```   575 definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
```
```   576
```
```   577 lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
```
```   578   by (auto simp: wcount_def add_mult_distrib)
```
```   579
```
```   580 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
```
```   581   "size_multiset f M = setsum (wcount f M) (set_of M)"
```
```   582
```
```   583 lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
```
```   584
```
```   585 instantiation multiset :: (type) size begin
```
```   586 definition size_multiset where
```
```   587   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
```
```   588 instance ..
```
```   589 end
```
```   590
```
```   591 lemmas size_multiset_overloaded_eq =
```
```   592   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
```
```   593
```
```   594 lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
```
```   595 by (simp add: size_multiset_def)
```
```   596
```
```   597 lemma size_empty [simp]: "size {#} = 0"
```
```   598 by (simp add: size_multiset_overloaded_def)
```
```   599
```
```   600 lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
```
```   601 by (simp add: size_multiset_eq)
```
```   602
```
```   603 lemma size_single [simp]: "size {#b#} = 1"
```
```   604 by (simp add: size_multiset_overloaded_def)
```
```   605
```
```   606 lemma setsum_wcount_Int:
```
```   607   "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_of N) = setsum (wcount f N) A"
```
```   608 apply (induct rule: finite_induct)
```
```   609  apply simp
```
```   610 apply (simp add: Int_insert_left set_of_def wcount_def)
```
```   611 done
```
```   612
```
```   613 lemma size_multiset_union [simp]:
```
```   614   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
```
```   615 apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
```
```   616 apply (subst Int_commute)
```
```   617 apply (simp add: setsum_wcount_Int)
```
```   618 done
```
```   619
```
```   620 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   621 by (auto simp add: size_multiset_overloaded_def)
```
```   622
```
```   623 lemma size_multiset_eq_0_iff_empty [iff]: "(size_multiset f M = 0) = (M = {#})"
```
```   624 by (auto simp add: size_multiset_eq multiset_eq_iff)
```
```   625
```
```   626 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   627 by (auto simp add: size_multiset_overloaded_def)
```
```   628
```
```   629 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
```
```   630 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
```
```   631
```
```   632 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   633 apply (unfold size_multiset_overloaded_eq)
```
```   634 apply (drule setsum_SucD)
```
```   635 apply auto
```
```   636 done
```
```   637
```
```   638 lemma size_eq_Suc_imp_eq_union:
```
```   639   assumes "size M = Suc n"
```
```   640   shows "\<exists>a N. M = N + {#a#}"
```
```   641 proof -
```
```   642   from assms obtain a where "a \<in># M"
```
```   643     by (erule size_eq_Suc_imp_elem [THEN exE])
```
```   644   then have "M = M - {#a#} + {#a#}" by simp
```
```   645   then show ?thesis by blast
```
```   646 qed
```
```   647
```
```   648
```
```   649 subsection {* Induction and case splits *}
```
```   650
```
```   651 theorem multiset_induct [case_names empty add, induct type: multiset]:
```
```   652   assumes empty: "P {#}"
```
```   653   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
```
```   654   shows "P M"
```
```   655 proof (induct n \<equiv> "size M" arbitrary: M)
```
```   656   case 0 thus "P M" by (simp add: empty)
```
```   657 next
```
```   658   case (Suc k)
```
```   659   obtain N x where "M = N + {#x#}"
```
```   660     using `Suc k = size M` [symmetric]
```
```   661     using size_eq_Suc_imp_eq_union by fast
```
```   662   with Suc add show "P M" by simp
```
```   663 qed
```
```   664
```
```   665 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
```
```   666 by (induct M) auto
```
```   667
```
```   668 lemma multiset_cases [cases type]:
```
```   669   obtains (empty) "M = {#}"
```
```   670     | (add) N x where "M = N + {#x#}"
```
```   671   using assms by (induct M) simp_all
```
```   672
```
```   673 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
```
```   674 by (cases "B = {#}") (auto dest: multi_member_split)
```
```   675
```
```   676 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
```
```   677 apply (subst multiset_eq_iff)
```
```   678 apply auto
```
```   679 done
```
```   680
```
```   681 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
```
```   682 proof (induct A arbitrary: B)
```
```   683   case (empty M)
```
```   684   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
```
```   685   then obtain M' x where "M = M' + {#x#}"
```
```   686     by (blast dest: multi_nonempty_split)
```
```   687   then show ?case by simp
```
```   688 next
```
```   689   case (add S x T)
```
```   690   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
```
```   691   have SxsubT: "S + {#x#} < T" by fact
```
```   692   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
```
```   693   then obtain T' where T: "T = T' + {#x#}"
```
```   694     by (blast dest: multi_member_split)
```
```   695   then have "S < T'" using SxsubT
```
```   696     by (blast intro: mset_less_add_bothsides)
```
```   697   then have "size S < size T'" using IH by simp
```
```   698   then show ?case using T by simp
```
```   699 qed
```
```   700
```
```   701
```
```   702 subsubsection {* Strong induction and subset induction for multisets *}
```
```   703
```
```   704 text {* Well-foundedness of proper subset operator: *}
```
```   705
```
```   706 text {* proper multiset subset *}
```
```   707
```
```   708 definition
```
```   709   mset_less_rel :: "('a multiset * 'a multiset) set" where
```
```   710   "mset_less_rel = {(A,B). A < B}"
```
```   711
```
```   712 lemma multiset_add_sub_el_shuffle:
```
```   713   assumes "c \<in># B" and "b \<noteq> c"
```
```   714   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
```
```   715 proof -
```
```   716   from `c \<in># B` obtain A where B: "B = A + {#c#}"
```
```   717     by (blast dest: multi_member_split)
```
```   718   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
```
```   719   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
```
```   720     by (simp add: add_ac)
```
```   721   then show ?thesis using B by simp
```
```   722 qed
```
```   723
```
```   724 lemma wf_mset_less_rel: "wf mset_less_rel"
```
```   725 apply (unfold mset_less_rel_def)
```
```   726 apply (rule wf_measure [THEN wf_subset, where f1=size])
```
```   727 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
```
```   728 done
```
```   729
```
```   730 text {* The induction rules: *}
```
```   731
```
```   732 lemma full_multiset_induct [case_names less]:
```
```   733 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
```
```   734 shows "P B"
```
```   735 apply (rule wf_mset_less_rel [THEN wf_induct])
```
```   736 apply (rule ih, auto simp: mset_less_rel_def)
```
```   737 done
```
```   738
```
```   739 lemma multi_subset_induct [consumes 2, case_names empty add]:
```
```   740 assumes "F \<le> A"
```
```   741   and empty: "P {#}"
```
```   742   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
```
```   743 shows "P F"
```
```   744 proof -
```
```   745   from `F \<le> A`
```
```   746   show ?thesis
```
```   747   proof (induct F)
```
```   748     show "P {#}" by fact
```
```   749   next
```
```   750     fix x F
```
```   751     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
```
```   752     show "P (F + {#x#})"
```
```   753     proof (rule insert)
```
```   754       from i show "x \<in># A" by (auto dest: mset_le_insertD)
```
```   755       from i have "F \<le> A" by (auto dest: mset_le_insertD)
```
```   756       with P show "P F" .
```
```   757     qed
```
```   758   qed
```
```   759 qed
```
```   760
```
```   761
```
```   762 subsection {* The fold combinator *}
```
```   763
```
```   764 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
```
```   765 where
```
```   766   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
```
```   767
```
```   768 lemma fold_mset_empty [simp]:
```
```   769   "fold f s {#} = s"
```
```   770   by (simp add: fold_def)
```
```   771
```
```   772 context comp_fun_commute
```
```   773 begin
```
```   774
```
```   775 lemma fold_mset_insert:
```
```   776   "fold f s (M + {#x#}) = f x (fold f s M)"
```
```   777 proof -
```
```   778   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
```
```   779     by (fact comp_fun_commute_funpow)
```
```   780   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
```
```   781     by (fact comp_fun_commute_funpow)
```
```   782   show ?thesis
```
```   783   proof (cases "x \<in> set_of M")
```
```   784     case False
```
```   785     then have *: "count (M + {#x#}) x = 1" by simp
```
```   786     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
```
```   787       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
```
```   788       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   789     with False * show ?thesis
```
```   790       by (simp add: fold_def del: count_union)
```
```   791   next
```
```   792     case True
```
```   793     def N \<equiv> "set_of M - {x}"
```
```   794     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
```
```   795     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
```
```   796       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
```
```   797       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   798     with * show ?thesis by (simp add: fold_def del: count_union) simp
```
```   799   qed
```
```   800 qed
```
```   801
```
```   802 corollary fold_mset_single [simp]:
```
```   803   "fold f s {#x#} = f x s"
```
```   804 proof -
```
```   805   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
```
```   806   then show ?thesis by simp
```
```   807 qed
```
```   808
```
```   809 lemma fold_mset_fun_left_comm:
```
```   810   "f x (fold f s M) = fold f (f x s) M"
```
```   811   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
```
```   812
```
```   813 lemma fold_mset_union [simp]:
```
```   814   "fold f s (M + N) = fold f (fold f s M) N"
```
```   815 proof (induct M)
```
```   816   case empty then show ?case by simp
```
```   817 next
```
```   818   case (add M x)
```
```   819   have "M + {#x#} + N = (M + N) + {#x#}"
```
```   820     by (simp add: add_ac)
```
```   821   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
```
```   822 qed
```
```   823
```
```   824 lemma fold_mset_fusion:
```
```   825   assumes "comp_fun_commute g"
```
```   826   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")
```
```   827 proof -
```
```   828   interpret comp_fun_commute g by (fact assms)
```
```   829   show "PROP ?P" by (induct A) auto
```
```   830 qed
```
```   831
```
```   832 end
```
```   833
```
```   834 text {*
```
```   835   A note on code generation: When defining some function containing a
```
```   836   subterm @{term "fold F"}, code generation is not automatic. When
```
```   837   interpreting locale @{text left_commutative} with @{text F}, the
```
```   838   would be code thms for @{const fold} become thms like
```
```   839   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
```
```   840   contains defined symbols, i.e.\ is not a code thm. Hence a separate
```
```   841   constant with its own code thms needs to be introduced for @{text
```
```   842   F}. See the image operator below.
```
```   843 *}
```
```   844
```
```   845
```
```   846 subsection {* Image *}
```
```   847
```
```   848 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
```
```   849   "image_mset f = fold (plus o single o f) {#}"
```
```   850
```
```   851 lemma comp_fun_commute_mset_image:
```
```   852   "comp_fun_commute (plus o single o f)"
```
```   853 proof
```
```   854 qed (simp add: add_ac fun_eq_iff)
```
```   855
```
```   856 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
```
```   857   by (simp add: image_mset_def)
```
```   858
```
```   859 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
```
```   860 proof -
```
```   861   interpret comp_fun_commute "plus o single o f"
```
```   862     by (fact comp_fun_commute_mset_image)
```
```   863   show ?thesis by (simp add: image_mset_def)
```
```   864 qed
```
```   865
```
```   866 lemma image_mset_union [simp]:
```
```   867   "image_mset f (M + N) = image_mset f M + image_mset f N"
```
```   868 proof -
```
```   869   interpret comp_fun_commute "plus o single o f"
```
```   870     by (fact comp_fun_commute_mset_image)
```
```   871   show ?thesis by (induct N) (simp_all add: image_mset_def add_ac)
```
```   872 qed
```
```   873
```
```   874 corollary image_mset_insert:
```
```   875   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
```
```   876   by simp
```
```   877
```
```   878 lemma set_of_image_mset [simp]:
```
```   879   "set_of (image_mset f M) = image f (set_of M)"
```
```   880   by (induct M) simp_all
```
```   881
```
```   882 lemma size_image_mset [simp]:
```
```   883   "size (image_mset f M) = size M"
```
```   884   by (induct M) simp_all
```
```   885
```
```   886 lemma image_mset_is_empty_iff [simp]:
```
```   887   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
```
```   888   by (cases M) auto
```
```   889
```
```   890 syntax
```
```   891   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
```
```   892       ("({#_/. _ :# _#})")
```
```   893 translations
```
```   894   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
```
```   895
```
```   896 syntax
```
```   897   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
```
```   898       ("({#_/ | _ :# _./ _#})")
```
```   899 translations
```
```   900   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
```
```   901
```
```   902 text {*
```
```   903   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
```
```   904   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
```
```   905   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
```
```   906   @{term "{#x+x|x:#M. x<c#}"}.
```
```   907 *}
```
```   908
```
```   909 functor image_mset: image_mset
```
```   910 proof -
```
```   911   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
```
```   912   proof
```
```   913     fix A
```
```   914     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
```
```   915       by (induct A) simp_all
```
```   916   qed
```
```   917   show "image_mset id = id"
```
```   918   proof
```
```   919     fix A
```
```   920     show "image_mset id A = id A"
```
```   921       by (induct A) simp_all
```
```   922   qed
```
```   923 qed
```
```   924
```
```   925 declare image_mset.identity [simp]
```
```   926
```
```   927
```
```   928 subsection {* Further conversions *}
```
```   929
```
```   930 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
```
```   931   "multiset_of [] = {#}" |
```
```   932   "multiset_of (a # x) = multiset_of x + {# a #}"
```
```   933
```
```   934 lemma in_multiset_in_set:
```
```   935   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```   936   by (induct xs) simp_all
```
```   937
```
```   938 lemma count_multiset_of:
```
```   939   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```   940   by (induct xs) simp_all
```
```   941
```
```   942 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
```
```   943 by (induct x) auto
```
```   944
```
```   945 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
```
```   946 by (induct x) auto
```
```   947
```
```   948 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
```
```   949 by (induct x) auto
```
```   950
```
```   951 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
```
```   952 by (induct xs) auto
```
```   953
```
```   954 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
```
```   955   by (induct xs) simp_all
```
```   956
```
```   957 lemma multiset_of_append [simp]:
```
```   958   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
```
```   959   by (induct xs arbitrary: ys) (auto simp: add_ac)
```
```   960
```
```   961 lemma multiset_of_filter:
```
```   962   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
```
```   963   by (induct xs) simp_all
```
```   964
```
```   965 lemma multiset_of_rev [simp]:
```
```   966   "multiset_of (rev xs) = multiset_of xs"
```
```   967   by (induct xs) simp_all
```
```   968
```
```   969 lemma surj_multiset_of: "surj multiset_of"
```
```   970 apply (unfold surj_def)
```
```   971 apply (rule allI)
```
```   972 apply (rule_tac M = y in multiset_induct)
```
```   973  apply auto
```
```   974 apply (rule_tac x = "x # xa" in exI)
```
```   975 apply auto
```
```   976 done
```
```   977
```
```   978 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
```
```   979 by (induct x) auto
```
```   980
```
```   981 lemma distinct_count_atmost_1:
```
```   982   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
```
```   983 apply (induct x, simp, rule iffI, simp_all)
```
```   984 apply (rename_tac a b)
```
```   985 apply (rule conjI)
```
```   986 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
```
```   987 apply (erule_tac x = a in allE, simp, clarify)
```
```   988 apply (erule_tac x = aa in allE, simp)
```
```   989 done
```
```   990
```
```   991 lemma multiset_of_eq_setD:
```
```   992   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
```
```   993 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
```
```   994
```
```   995 lemma set_eq_iff_multiset_of_eq_distinct:
```
```   996   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
```
```   997     (set x = set y) = (multiset_of x = multiset_of y)"
```
```   998 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
```
```   999
```
```  1000 lemma set_eq_iff_multiset_of_remdups_eq:
```
```  1001    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
```
```  1002 apply (rule iffI)
```
```  1003 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
```
```  1004 apply (drule distinct_remdups [THEN distinct_remdups
```
```  1005       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
```
```  1006 apply simp
```
```  1007 done
```
```  1008
```
```  1009 lemma multiset_of_compl_union [simp]:
```
```  1010   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
```
```  1011   by (induct xs) (auto simp: add_ac)
```
```  1012
```
```  1013 lemma count_multiset_of_length_filter:
```
```  1014   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```  1015   by (induct xs) auto
```
```  1016
```
```  1017 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
```
```  1018 apply (induct ls arbitrary: i)
```
```  1019  apply simp
```
```  1020 apply (case_tac i)
```
```  1021  apply auto
```
```  1022 done
```
```  1023
```
```  1024 lemma multiset_of_remove1[simp]:
```
```  1025   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
```
```  1026 by (induct xs) (auto simp add: multiset_eq_iff)
```
```  1027
```
```  1028 lemma multiset_of_eq_length:
```
```  1029   assumes "multiset_of xs = multiset_of ys"
```
```  1030   shows "length xs = length ys"
```
```  1031   using assms by (metis size_multiset_of)
```
```  1032
```
```  1033 lemma multiset_of_eq_length_filter:
```
```  1034   assumes "multiset_of xs = multiset_of ys"
```
```  1035   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
```
```  1036   using assms by (metis count_multiset_of)
```
```  1037
```
```  1038 lemma fold_multiset_equiv:
```
```  1039   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"
```
```  1040     and equiv: "multiset_of xs = multiset_of ys"
```
```  1041   shows "List.fold f xs = List.fold f ys"
```
```  1042 using f equiv [symmetric]
```
```  1043 proof (induct xs arbitrary: ys)
```
```  1044   case Nil then show ?case by simp
```
```  1045 next
```
```  1046   case (Cons x xs)
```
```  1047   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
```
```  1048   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
```
```  1049     by (rule Cons.prems(1)) (simp_all add: *)
```
```  1050   moreover from * have "x \<in> set ys" by simp
```
```  1051   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
```
```  1052   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
```
```  1053   ultimately show ?case by simp
```
```  1054 qed
```
```  1055
```
```  1056 lemma multiset_of_insort [simp]:
```
```  1057   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
```
```  1058   by (induct xs) (simp_all add: ac_simps)
```
```  1059
```
```  1060 lemma in_multiset_of:
```
```  1061   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```  1062   by (induct xs) simp_all
```
```  1063
```
```  1064 lemma multiset_of_map:
```
```  1065   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
```
```  1066   by (induct xs) simp_all
```
```  1067
```
```  1068 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
```
```  1069 where
```
```  1070   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
```
```  1071
```
```  1072 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
```
```  1073 where
```
```  1074   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
```
```  1075 proof -
```
```  1076   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
```
```  1077   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
```
```  1078   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
```
```  1079 qed
```
```  1080
```
```  1081 lemma count_multiset_of_set [simp]:
```
```  1082   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
```
```  1083   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
```
```  1084   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
```
```  1085 proof -
```
```  1086   { fix A
```
```  1087     assume "x \<notin> A"
```
```  1088     have "count (multiset_of_set A) x = 0"
```
```  1089     proof (cases "finite A")
```
```  1090       case False then show ?thesis by simp
```
```  1091     next
```
```  1092       case True from True `x \<notin> A` show ?thesis by (induct A) auto
```
```  1093     qed
```
```  1094   } note * = this
```
```  1095   then show "PROP ?P" "PROP ?Q" "PROP ?R"
```
```  1096   by (auto elim!: Set.set_insert)
```
```  1097 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
```
```  1098
```
```  1099 context linorder
```
```  1100 begin
```
```  1101
```
```  1102 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
```
```  1103 where
```
```  1104   "sorted_list_of_multiset M = fold insort [] M"
```
```  1105
```
```  1106 lemma sorted_list_of_multiset_empty [simp]:
```
```  1107   "sorted_list_of_multiset {#} = []"
```
```  1108   by (simp add: sorted_list_of_multiset_def)
```
```  1109
```
```  1110 lemma sorted_list_of_multiset_singleton [simp]:
```
```  1111   "sorted_list_of_multiset {#x#} = [x]"
```
```  1112 proof -
```
```  1113   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1114   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1115 qed
```
```  1116
```
```  1117 lemma sorted_list_of_multiset_insert [simp]:
```
```  1118   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
```
```  1119 proof -
```
```  1120   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1121   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1122 qed
```
```  1123
```
```  1124 end
```
```  1125
```
```  1126 lemma multiset_of_sorted_list_of_multiset [simp]:
```
```  1127   "multiset_of (sorted_list_of_multiset M) = M"
```
```  1128   by (induct M) simp_all
```
```  1129
```
```  1130 lemma sorted_list_of_multiset_multiset_of [simp]:
```
```  1131   "sorted_list_of_multiset (multiset_of xs) = sort xs"
```
```  1132   by (induct xs) simp_all
```
```  1133
```
```  1134 lemma finite_set_of_multiset_of_set:
```
```  1135   assumes "finite A"
```
```  1136   shows "set_of (multiset_of_set A) = A"
```
```  1137   using assms by (induct A) simp_all
```
```  1138
```
```  1139 lemma infinite_set_of_multiset_of_set:
```
```  1140   assumes "\<not> finite A"
```
```  1141   shows "set_of (multiset_of_set A) = {}"
```
```  1142   using assms by simp
```
```  1143
```
```  1144 lemma set_sorted_list_of_multiset [simp]:
```
```  1145   "set (sorted_list_of_multiset M) = set_of M"
```
```  1146   by (induct M) (simp_all add: set_insort)
```
```  1147
```
```  1148 lemma sorted_list_of_multiset_of_set [simp]:
```
```  1149   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
```
```  1150   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
```
```  1151
```
```  1152
```
```  1153 subsection {* Big operators *}
```
```  1154
```
```  1155 no_notation times (infixl "*" 70)
```
```  1156 no_notation Groups.one ("1")
```
```  1157
```
```  1158 locale comm_monoid_mset = comm_monoid
```
```  1159 begin
```
```  1160
```
```  1161 definition F :: "'a multiset \<Rightarrow> 'a"
```
```  1162 where
```
```  1163   eq_fold: "F M = Multiset.fold f 1 M"
```
```  1164
```
```  1165 lemma empty [simp]:
```
```  1166   "F {#} = 1"
```
```  1167   by (simp add: eq_fold)
```
```  1168
```
```  1169 lemma singleton [simp]:
```
```  1170   "F {#x#} = x"
```
```  1171 proof -
```
```  1172   interpret comp_fun_commute
```
```  1173     by default (simp add: fun_eq_iff left_commute)
```
```  1174   show ?thesis by (simp add: eq_fold)
```
```  1175 qed
```
```  1176
```
```  1177 lemma union [simp]:
```
```  1178   "F (M + N) = F M * F N"
```
```  1179 proof -
```
```  1180   interpret comp_fun_commute f
```
```  1181     by default (simp add: fun_eq_iff left_commute)
```
```  1182   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
```
```  1183 qed
```
```  1184
```
```  1185 end
```
```  1186
```
```  1187 notation times (infixl "*" 70)
```
```  1188 notation Groups.one ("1")
```
```  1189
```
```  1190 context comm_monoid_add
```
```  1191 begin
```
```  1192
```
```  1193 definition msetsum :: "'a multiset \<Rightarrow> 'a"
```
```  1194 where
```
```  1195   "msetsum = comm_monoid_mset.F plus 0"
```
```  1196
```
```  1197 sublocale msetsum!: comm_monoid_mset plus 0
```
```  1198 where
```
```  1199   "comm_monoid_mset.F plus 0 = msetsum"
```
```  1200 proof -
```
```  1201   show "comm_monoid_mset plus 0" ..
```
```  1202   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
```
```  1203 qed
```
```  1204
```
```  1205 lemma setsum_unfold_msetsum:
```
```  1206   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
```
```  1207   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1208
```
```  1209 abbreviation msetsum_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
```
```  1210 where
```
```  1211   "msetsum_image f M \<equiv> msetsum (image_mset f M)"
```
```  1212
```
```  1213 end
```
```  1214
```
```  1215 syntax
```
```  1216   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1217       ("(3SUM _:#_. _)" [0, 51, 10] 10)
```
```  1218
```
```  1219 syntax (xsymbols)
```
```  1220   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1221       ("(3\<Sum>_:#_. _)" [0, 51, 10] 10)
```
```  1222
```
```  1223 syntax (HTML output)
```
```  1224   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1225       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1226
```
```  1227 translations
```
```  1228   "SUM i :# A. b" == "CONST msetsum_image (\<lambda>i. b) A"
```
```  1229
```
```  1230 context comm_monoid_mult
```
```  1231 begin
```
```  1232
```
```  1233 definition msetprod :: "'a multiset \<Rightarrow> 'a"
```
```  1234 where
```
```  1235   "msetprod = comm_monoid_mset.F times 1"
```
```  1236
```
```  1237 sublocale msetprod!: comm_monoid_mset times 1
```
```  1238 where
```
```  1239   "comm_monoid_mset.F times 1 = msetprod"
```
```  1240 proof -
```
```  1241   show "comm_monoid_mset times 1" ..
```
```  1242   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
```
```  1243 qed
```
```  1244
```
```  1245 lemma msetprod_empty:
```
```  1246   "msetprod {#} = 1"
```
```  1247   by (fact msetprod.empty)
```
```  1248
```
```  1249 lemma msetprod_singleton:
```
```  1250   "msetprod {#x#} = x"
```
```  1251   by (fact msetprod.singleton)
```
```  1252
```
```  1253 lemma msetprod_Un:
```
```  1254   "msetprod (A + B) = msetprod A * msetprod B"
```
```  1255   by (fact msetprod.union)
```
```  1256
```
```  1257 lemma setprod_unfold_msetprod:
```
```  1258   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
```
```  1259   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1260
```
```  1261 lemma msetprod_multiplicity:
```
```  1262   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
```
```  1263   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
```
```  1264
```
```  1265 abbreviation msetprod_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
```
```  1266 where
```
```  1267   "msetprod_image f M \<equiv> msetprod (image_mset f M)"
```
```  1268
```
```  1269 end
```
```  1270
```
```  1271 syntax
```
```  1272   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1273       ("(3PROD _:#_. _)" [0, 51, 10] 10)
```
```  1274
```
```  1275 syntax (xsymbols)
```
```  1276   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1277       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1278
```
```  1279 syntax (HTML output)
```
```  1280   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1281       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1282
```
```  1283 translations
```
```  1284   "PROD i :# A. b" == "CONST msetprod_image (\<lambda>i. b) A"
```
```  1285
```
```  1286 lemma (in comm_semiring_1) dvd_msetprod:
```
```  1287   assumes "x \<in># A"
```
```  1288   shows "x dvd msetprod A"
```
```  1289 proof -
```
```  1290   from assms have "A = (A - {#x#}) + {#x#}" by simp
```
```  1291   then obtain B where "A = B + {#x#}" ..
```
```  1292   then show ?thesis by simp
```
```  1293 qed
```
```  1294
```
```  1295
```
```  1296 subsection {* Cardinality *}
```
```  1297
```
```  1298 definition mcard :: "'a multiset \<Rightarrow> nat"
```
```  1299 where
```
```  1300   "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
```
```  1301
```
```  1302 lemma mcard_empty [simp]:
```
```  1303   "mcard {#} = 0"
```
```  1304   by (simp add: mcard_def)
```
```  1305
```
```  1306 lemma mcard_singleton [simp]:
```
```  1307   "mcard {#a#} = Suc 0"
```
```  1308   by (simp add: mcard_def)
```
```  1309
```
```  1310 lemma mcard_plus [simp]:
```
```  1311   "mcard (M + N) = mcard M + mcard N"
```
```  1312   by (simp add: mcard_def)
```
```  1313
```
```  1314 lemma mcard_empty_iff [simp]:
```
```  1315   "mcard M = 0 \<longleftrightarrow> M = {#}"
```
```  1316   by (induct M) simp_all
```
```  1317
```
```  1318 lemma mcard_unfold_setsum:
```
```  1319   "mcard M = setsum (count M) (set_of M)"
```
```  1320 proof (induct M)
```
```  1321   case empty then show ?case by simp
```
```  1322 next
```
```  1323   case (add M x) then show ?case
```
```  1324     by (cases "x \<in> set_of M")
```
```  1325       (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
```
```  1326 qed
```
```  1327
```
```  1328 lemma size_eq_mcard:
```
```  1329   "size = mcard"
```
```  1330   by (simp add: fun_eq_iff size_multiset_overloaded_eq mcard_unfold_setsum)
```
```  1331
```
```  1332 lemma mcard_multiset_of:
```
```  1333   "mcard (multiset_of xs) = length xs"
```
```  1334   by (induct xs) simp_all
```
```  1335
```
```  1336
```
```  1337 subsection {* Alternative representations *}
```
```  1338
```
```  1339 subsubsection {* Lists *}
```
```  1340
```
```  1341 context linorder
```
```  1342 begin
```
```  1343
```
```  1344 lemma multiset_of_insort [simp]:
```
```  1345   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
```
```  1346   by (induct xs) (simp_all add: ac_simps)
```
```  1347
```
```  1348 lemma multiset_of_sort [simp]:
```
```  1349   "multiset_of (sort_key k xs) = multiset_of xs"
```
```  1350   by (induct xs) (simp_all add: ac_simps)
```
```  1351
```
```  1352 text {*
```
```  1353   This lemma shows which properties suffice to show that a function
```
```  1354   @{text "f"} with @{text "f xs = ys"} behaves like sort.
```
```  1355 *}
```
```  1356
```
```  1357 lemma properties_for_sort_key:
```
```  1358   assumes "multiset_of ys = multiset_of xs"
```
```  1359   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
```
```  1360   and "sorted (map f ys)"
```
```  1361   shows "sort_key f xs = ys"
```
```  1362 using assms
```
```  1363 proof (induct xs arbitrary: ys)
```
```  1364   case Nil then show ?case by simp
```
```  1365 next
```
```  1366   case (Cons x xs)
```
```  1367   from Cons.prems(2) have
```
```  1368     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
```
```  1369     by (simp add: filter_remove1)
```
```  1370   with Cons.prems have "sort_key f xs = remove1 x ys"
```
```  1371     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
```
```  1372   moreover from Cons.prems have "x \<in> set ys"
```
```  1373     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
```
```  1374   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
```
```  1375 qed
```
```  1376
```
```  1377 lemma properties_for_sort:
```
```  1378   assumes multiset: "multiset_of ys = multiset_of xs"
```
```  1379   and "sorted ys"
```
```  1380   shows "sort xs = ys"
```
```  1381 proof (rule properties_for_sort_key)
```
```  1382   from multiset show "multiset_of ys = multiset_of xs" .
```
```  1383   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
```
```  1384   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
```
```  1385     by (rule multiset_of_eq_length_filter)
```
```  1386   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
```
```  1387     by simp
```
```  1388   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
```
```  1389     by (simp add: replicate_length_filter)
```
```  1390 qed
```
```  1391
```
```  1392 lemma sort_key_by_quicksort:
```
```  1393   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
```
```  1394     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
```
```  1395     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
```
```  1396 proof (rule properties_for_sort_key)
```
```  1397   show "multiset_of ?rhs = multiset_of ?lhs"
```
```  1398     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
```
```  1399 next
```
```  1400   show "sorted (map f ?rhs)"
```
```  1401     by (auto simp add: sorted_append intro: sorted_map_same)
```
```  1402 next
```
```  1403   fix l
```
```  1404   assume "l \<in> set ?rhs"
```
```  1405   let ?pivot = "f (xs ! (length xs div 2))"
```
```  1406   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
```
```  1407   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
```
```  1408     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
```
```  1409   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
```
```  1410   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
```
```  1411   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
```
```  1412     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
```
```  1413   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
```
```  1414   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
```
```  1415   proof (cases "f l" ?pivot rule: linorder_cases)
```
```  1416     case less
```
```  1417     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
```
```  1418     with less show ?thesis
```
```  1419       by (simp add: filter_sort [symmetric] ** ***)
```
```  1420   next
```
```  1421     case equal then show ?thesis
```
```  1422       by (simp add: * less_le)
```
```  1423   next
```
```  1424     case greater
```
```  1425     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
```
```  1426     with greater show ?thesis
```
```  1427       by (simp add: filter_sort [symmetric] ** ***)
```
```  1428   qed
```
```  1429 qed
```
```  1430
```
```  1431 lemma sort_by_quicksort:
```
```  1432   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
```
```  1433     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
```
```  1434     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
```
```  1435   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
```
```  1436
```
```  1437 text {* A stable parametrized quicksort *}
```
```  1438
```
```  1439 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
```
```  1440   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
```
```  1441
```
```  1442 lemma part_code [code]:
```
```  1443   "part f pivot [] = ([], [], [])"
```
```  1444   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
```
```  1445      if x' < pivot then (x # lts, eqs, gts)
```
```  1446      else if x' > pivot then (lts, eqs, x # gts)
```
```  1447      else (lts, x # eqs, gts))"
```
```  1448   by (auto simp add: part_def Let_def split_def)
```
```  1449
```
```  1450 lemma sort_key_by_quicksort_code [code]:
```
```  1451   "sort_key f xs = (case xs of [] \<Rightarrow> []
```
```  1452     | [x] \<Rightarrow> xs
```
```  1453     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
```
```  1454     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1455        in sort_key f lts @ eqs @ sort_key f gts))"
```
```  1456 proof (cases xs)
```
```  1457   case Nil then show ?thesis by simp
```
```  1458 next
```
```  1459   case (Cons _ ys) note hyps = Cons show ?thesis
```
```  1460   proof (cases ys)
```
```  1461     case Nil with hyps show ?thesis by simp
```
```  1462   next
```
```  1463     case (Cons _ zs) note hyps = hyps Cons show ?thesis
```
```  1464     proof (cases zs)
```
```  1465       case Nil with hyps show ?thesis by auto
```
```  1466     next
```
```  1467       case Cons
```
```  1468       from sort_key_by_quicksort [of f xs]
```
```  1469       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1470         in sort_key f lts @ eqs @ sort_key f gts)"
```
```  1471       by (simp only: split_def Let_def part_def fst_conv snd_conv)
```
```  1472       with hyps Cons show ?thesis by (simp only: list.cases)
```
```  1473     qed
```
```  1474   qed
```
```  1475 qed
```
```  1476
```
```  1477 end
```
```  1478
```
```  1479 hide_const (open) part
```
```  1480
```
```  1481 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
```
```  1482   by (induct xs) (auto intro: order_trans)
```
```  1483
```
```  1484 lemma multiset_of_update:
```
```  1485   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
```
```  1486 proof (induct ls arbitrary: i)
```
```  1487   case Nil then show ?case by simp
```
```  1488 next
```
```  1489   case (Cons x xs)
```
```  1490   show ?case
```
```  1491   proof (cases i)
```
```  1492     case 0 then show ?thesis by simp
```
```  1493   next
```
```  1494     case (Suc i')
```
```  1495     with Cons show ?thesis
```
```  1496       apply simp
```
```  1497       apply (subst add.assoc)
```
```  1498       apply (subst add.commute [of "{#v#}" "{#x#}"])
```
```  1499       apply (subst add.assoc [symmetric])
```
```  1500       apply simp
```
```  1501       apply (rule mset_le_multiset_union_diff_commute)
```
```  1502       apply (simp add: mset_le_single nth_mem_multiset_of)
```
```  1503       done
```
```  1504   qed
```
```  1505 qed
```
```  1506
```
```  1507 lemma multiset_of_swap:
```
```  1508   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
```
```  1509     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
```
```  1510   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
```
```  1511
```
```  1512
```
```  1513 subsection {* The multiset order *}
```
```  1514
```
```  1515 subsubsection {* Well-foundedness *}
```
```  1516
```
```  1517 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1518   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```  1519       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```  1520
```
```  1521 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1522   "mult r = (mult1 r)\<^sup>+"
```
```  1523
```
```  1524 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```  1525 by (simp add: mult1_def)
```
```  1526
```
```  1527 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```  1528     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```  1529     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```  1530   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
```
```  1531 proof (unfold mult1_def)
```
```  1532   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```  1533   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```  1534   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```  1535
```
```  1536   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```  1537   then have "\<exists>a' M0' K.
```
```  1538       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```  1539   then show "?case1 \<or> ?case2"
```
```  1540   proof (elim exE conjE)
```
```  1541     fix a' M0' K
```
```  1542     assume N: "N = M0' + K" and r: "?r K a'"
```
```  1543     assume "M0 + {#a#} = M0' + {#a'#}"
```
```  1544     then have "M0 = M0' \<and> a = a' \<or>
```
```  1545         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```  1546       by (simp only: add_eq_conv_ex)
```
```  1547     then show ?thesis
```
```  1548     proof (elim disjE conjE exE)
```
```  1549       assume "M0 = M0'" "a = a'"
```
```  1550       with N r have "?r K a \<and> N = M0 + K" by simp
```
```  1551       then have ?case2 .. then show ?thesis ..
```
```  1552     next
```
```  1553       fix K'
```
```  1554       assume "M0' = K' + {#a#}"
```
```  1555       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
```
```  1556
```
```  1557       assume "M0 = K' + {#a'#}"
```
```  1558       with r have "?R (K' + K) M0" by blast
```
```  1559       with n have ?case1 by simp then show ?thesis ..
```
```  1560     qed
```
```  1561   qed
```
```  1562 qed
```
```  1563
```
```  1564 lemma all_accessible: "wf r ==> \<forall>M. M \<in> Wellfounded.acc (mult1 r)"
```
```  1565 proof
```
```  1566   let ?R = "mult1 r"
```
```  1567   let ?W = "Wellfounded.acc ?R"
```
```  1568   {
```
```  1569     fix M M0 a
```
```  1570     assume M0: "M0 \<in> ?W"
```
```  1571       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1572       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```  1573     have "M0 + {#a#} \<in> ?W"
```
```  1574     proof (rule accI [of "M0 + {#a#}"])
```
```  1575       fix N
```
```  1576       assume "(N, M0 + {#a#}) \<in> ?R"
```
```  1577       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```  1578           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```  1579         by (rule less_add)
```
```  1580       then show "N \<in> ?W"
```
```  1581       proof (elim exE disjE conjE)
```
```  1582         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```  1583         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```  1584         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
```
```  1585         then show "N \<in> ?W" by (simp only: N)
```
```  1586       next
```
```  1587         fix K
```
```  1588         assume N: "N = M0 + K"
```
```  1589         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```  1590         then have "M0 + K \<in> ?W"
```
```  1591         proof (induct K)
```
```  1592           case empty
```
```  1593           from M0 show "M0 + {#} \<in> ?W" by simp
```
```  1594         next
```
```  1595           case (add K x)
```
```  1596           from add.prems have "(x, a) \<in> r" by simp
```
```  1597           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```  1598           moreover from add have "M0 + K \<in> ?W" by simp
```
```  1599           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
```
```  1600           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc)
```
```  1601         qed
```
```  1602         then show "N \<in> ?W" by (simp only: N)
```
```  1603       qed
```
```  1604     qed
```
```  1605   } note tedious_reasoning = this
```
```  1606
```
```  1607   assume wf: "wf r"
```
```  1608   fix M
```
```  1609   show "M \<in> ?W"
```
```  1610   proof (induct M)
```
```  1611     show "{#} \<in> ?W"
```
```  1612     proof (rule accI)
```
```  1613       fix b assume "(b, {#}) \<in> ?R"
```
```  1614       with not_less_empty show "b \<in> ?W" by contradiction
```
```  1615     qed
```
```  1616
```
```  1617     fix M a assume "M \<in> ?W"
```
```  1618     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1619     proof induct
```
```  1620       fix a
```
```  1621       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1622       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1623       proof
```
```  1624         fix M assume "M \<in> ?W"
```
```  1625         then show "M + {#a#} \<in> ?W"
```
```  1626           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
```
```  1627       qed
```
```  1628     qed
```
```  1629     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
```
```  1630   qed
```
```  1631 qed
```
```  1632
```
```  1633 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```  1634 by (rule acc_wfI) (rule all_accessible)
```
```  1635
```
```  1636 theorem wf_mult: "wf r ==> wf (mult r)"
```
```  1637 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
```
```  1638
```
```  1639
```
```  1640 subsubsection {* Closure-free presentation *}
```
```  1641
```
```  1642 text {* One direction. *}
```
```  1643
```
```  1644 lemma mult_implies_one_step:
```
```  1645   "trans r ==> (M, N) \<in> mult r ==>
```
```  1646     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```  1647     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```  1648 apply (unfold mult_def mult1_def set_of_def)
```
```  1649 apply (erule converse_trancl_induct, clarify)
```
```  1650  apply (rule_tac x = M0 in exI, simp, clarify)
```
```  1651 apply (case_tac "a :# K")
```
```  1652  apply (rule_tac x = I in exI)
```
```  1653  apply (simp (no_asm))
```
```  1654  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```  1655  apply (simp (no_asm_simp) add: add.assoc [symmetric])
```
```  1656  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="?S + ?T" in arg_cong)
```
```  1657  apply (simp add: diff_union_single_conv)
```
```  1658  apply (simp (no_asm_use) add: trans_def)
```
```  1659  apply blast
```
```  1660 apply (subgoal_tac "a :# I")
```
```  1661  apply (rule_tac x = "I - {#a#}" in exI)
```
```  1662  apply (rule_tac x = "J + {#a#}" in exI)
```
```  1663  apply (rule_tac x = "K + Ka" in exI)
```
```  1664  apply (rule conjI)
```
```  1665   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1666  apply (rule conjI)
```
```  1667   apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="?S + ?T" in arg_cong, simp)
```
```  1668   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1669  apply (simp (no_asm_use) add: trans_def)
```
```  1670  apply blast
```
```  1671 apply (subgoal_tac "a :# (M0 + {#a#})")
```
```  1672  apply simp
```
```  1673 apply (simp (no_asm))
```
```  1674 done
```
```  1675
```
```  1676 lemma one_step_implies_mult_aux:
```
```  1677   "trans r ==>
```
```  1678     \<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))
```
```  1679       --> (I + K, I + J) \<in> mult r"
```
```  1680 apply (induct_tac n, auto)
```
```  1681 apply (frule size_eq_Suc_imp_eq_union, clarify)
```
```  1682 apply (rename_tac "J'", simp)
```
```  1683 apply (erule notE, auto)
```
```  1684 apply (case_tac "J' = {#}")
```
```  1685  apply (simp add: mult_def)
```
```  1686  apply (rule r_into_trancl)
```
```  1687  apply (simp add: mult1_def set_of_def, blast)
```
```  1688 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```  1689 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```  1690 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```  1691 apply (erule ssubst)
```
```  1692 apply (simp add: Ball_def, auto)
```
```  1693 apply (subgoal_tac
```
```  1694   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
```
```  1695     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```  1696  prefer 2
```
```  1697  apply force
```
```  1698 apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def)
```
```  1699 apply (erule trancl_trans)
```
```  1700 apply (rule r_into_trancl)
```
```  1701 apply (simp add: mult1_def set_of_def)
```
```  1702 apply (rule_tac x = a in exI)
```
```  1703 apply (rule_tac x = "I + J'" in exI)
```
```  1704 apply (simp add: add_ac)
```
```  1705 done
```
```  1706
```
```  1707 lemma one_step_implies_mult:
```
```  1708   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```  1709     ==> (I + K, I + J) \<in> mult r"
```
```  1710 using one_step_implies_mult_aux by blast
```
```  1711
```
```  1712
```
```  1713 subsubsection {* Partial-order properties *}
```
```  1714
```
```  1715 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
```
```  1716   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
```
```  1717
```
```  1718 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
```
```  1719   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
```
```  1720
```
```  1721 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
```
```  1722 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
```
```  1723
```
```  1724 interpretation multiset_order: order le_multiset less_multiset
```
```  1725 proof -
```
```  1726   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
```
```  1727   proof
```
```  1728     fix M :: "'a multiset"
```
```  1729     assume "M \<subset># M"
```
```  1730     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
```
```  1731     have "trans {(x'::'a, x). x' < x}"
```
```  1732       by (rule transI) simp
```
```  1733     moreover note MM
```
```  1734     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
```
```  1735       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
```
```  1736       by (rule mult_implies_one_step)
```
```  1737     then obtain I J K where "M = I + J" and "M = I + K"
```
```  1738       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
```
```  1739     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
```
```  1740     have "finite (set_of K)" by simp
```
```  1741     moreover note aux2
```
```  1742     ultimately have "set_of K = {}"
```
```  1743       by (induct rule: finite_induct) (auto intro: order_less_trans)
```
```  1744     with aux1 show False by simp
```
```  1745   qed
```
```  1746   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
```
```  1747     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
```
```  1748   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
```
```  1749     by default (auto simp add: le_multiset_def irrefl dest: trans)
```
```  1750 qed
```
```  1751
```
```  1752 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
```
```  1753   by simp
```
```  1754
```
```  1755
```
```  1756 subsubsection {* Monotonicity of multiset union *}
```
```  1757
```
```  1758 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
```
```  1759 apply (unfold mult1_def)
```
```  1760 apply auto
```
```  1761 apply (rule_tac x = a in exI)
```
```  1762 apply (rule_tac x = "C + M0" in exI)
```
```  1763 apply (simp add: add.assoc)
```
```  1764 done
```
```  1765
```
```  1766 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
```
```  1767 apply (unfold less_multiset_def mult_def)
```
```  1768 apply (erule trancl_induct)
```
```  1769  apply (blast intro: mult1_union)
```
```  1770 apply (blast intro: mult1_union trancl_trans)
```
```  1771 done
```
```  1772
```
```  1773 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
```
```  1774 apply (subst add.commute [of B C])
```
```  1775 apply (subst add.commute [of D C])
```
```  1776 apply (erule union_less_mono2)
```
```  1777 done
```
```  1778
```
```  1779 lemma union_less_mono:
```
```  1780   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
```
```  1781   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
```
```  1782
```
```  1783 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
```
```  1784 proof
```
```  1785 qed (auto simp add: le_multiset_def intro: union_less_mono2)
```
```  1786
```
```  1787
```
```  1788 subsection {* Termination proofs with multiset orders *}
```
```  1789
```
```  1790 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
```
```  1791   and multi_member_this: "x \<in># {# x #} + XS"
```
```  1792   and multi_member_last: "x \<in># {# x #}"
```
```  1793   by auto
```
```  1794
```
```  1795 definition "ms_strict = mult pair_less"
```
```  1796 definition "ms_weak = ms_strict \<union> Id"
```
```  1797
```
```  1798 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
```
```  1799 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
```
```  1800 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
```
```  1801
```
```  1802 lemma smsI:
```
```  1803   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
```
```  1804   unfolding ms_strict_def
```
```  1805 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
```
```  1806
```
```  1807 lemma wmsI:
```
```  1808   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
```
```  1809   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
```
```  1810 unfolding ms_weak_def ms_strict_def
```
```  1811 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
```
```  1812
```
```  1813 inductive pw_leq
```
```  1814 where
```
```  1815   pw_leq_empty: "pw_leq {#} {#}"
```
```  1816 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
```
```  1817
```
```  1818 lemma pw_leq_lstep:
```
```  1819   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
```
```  1820 by (drule pw_leq_step) (rule pw_leq_empty, simp)
```
```  1821
```
```  1822 lemma pw_leq_split:
```
```  1823   assumes "pw_leq X Y"
```
```  1824   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 = {#}))"
```
```  1825   using assms
```
```  1826 proof (induct)
```
```  1827   case pw_leq_empty thus ?case by auto
```
```  1828 next
```
```  1829   case (pw_leq_step x y X Y)
```
```  1830   then obtain A B Z where
```
```  1831     [simp]: "X = A + Z" "Y = B + Z"
```
```  1832       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
```
```  1833     by auto
```
```  1834   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
```
```  1835     unfolding pair_leq_def by auto
```
```  1836   thus ?case
```
```  1837   proof
```
```  1838     assume [simp]: "x = y"
```
```  1839     have
```
```  1840       "{#x#} + X = A + ({#y#}+Z)
```
```  1841       \<and> {#y#} + Y = B + ({#y#}+Z)
```
```  1842       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
```
```  1843       by (auto simp: add_ac)
```
```  1844     thus ?case by (intro exI)
```
```  1845   next
```
```  1846     assume A: "(x, y) \<in> pair_less"
```
```  1847     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
```
```  1848     have "{#x#} + X = ?A' + Z"
```
```  1849       "{#y#} + Y = ?B' + Z"
```
```  1850       by (auto simp add: add_ac)
```
```  1851     moreover have
```
```  1852       "(set_of ?A', set_of ?B') \<in> max_strict"
```
```  1853       using 1 A unfolding max_strict_def
```
```  1854       by (auto elim!: max_ext.cases)
```
```  1855     ultimately show ?thesis by blast
```
```  1856   qed
```
```  1857 qed
```
```  1858
```
```  1859 lemma
```
```  1860   assumes pwleq: "pw_leq Z Z'"
```
```  1861   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
```
```  1862   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
```
```  1863   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
```
```  1864 proof -
```
```  1865   from pw_leq_split[OF pwleq]
```
```  1866   obtain A' B' Z''
```
```  1867     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
```
```  1868     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
```
```  1869     by blast
```
```  1870   {
```
```  1871     assume max: "(set_of A, set_of B) \<in> max_strict"
```
```  1872     from mx_or_empty
```
```  1873     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
```
```  1874     proof
```
```  1875       assume max': "(set_of A', set_of B') \<in> max_strict"
```
```  1876       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
```
```  1877         by (auto simp: max_strict_def intro: max_ext_additive)
```
```  1878       thus ?thesis by (rule smsI)
```
```  1879     next
```
```  1880       assume [simp]: "A' = {#} \<and> B' = {#}"
```
```  1881       show ?thesis by (rule smsI) (auto intro: max)
```
```  1882     qed
```
```  1883     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
```
```  1884     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
```
```  1885   }
```
```  1886   from mx_or_empty
```
```  1887   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
```
```  1888   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
```
```  1889 qed
```
```  1890
```
```  1891 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
```
```  1892 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
```
```  1893 and nonempty_single: "{# x #} \<noteq> {#}"
```
```  1894 by auto
```
```  1895
```
```  1896 setup {*
```
```  1897 let
```
```  1898   fun msetT T = Type (@{type_name multiset}, [T]);
```
```  1899
```
```  1900   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
```
```  1901     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
```
```  1902     | mk_mset T (x :: xs) =
```
```  1903           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
```
```  1904                 mk_mset T [x] \$ mk_mset T xs
```
```  1905
```
```  1906   fun mset_member_tac m i =
```
```  1907       (if m <= 0 then
```
```  1908            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
```
```  1909        else
```
```  1910            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
```
```  1911
```
```  1912   val mset_nonempty_tac =
```
```  1913       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
```
```  1914
```
```  1915   val regroup_munion_conv =
```
```  1916       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
```
```  1917         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))
```
```  1918
```
```  1919   fun unfold_pwleq_tac i =
```
```  1920     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
```
```  1921       ORELSE (rtac @{thm pw_leq_lstep} i)
```
```  1922       ORELSE (rtac @{thm pw_leq_empty} i)
```
```  1923
```
```  1924   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
```
```  1925                       @{thm Un_insert_left}, @{thm Un_empty_left}]
```
```  1926 in
```
```  1927   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
```
```  1928   {
```
```  1929     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
```
```  1930     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
```
```  1931     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
```
```  1932     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
```
```  1933     reduction_pair= @{thm ms_reduction_pair}
```
```  1934   })
```
```  1935 end
```
```  1936 *}
```
```  1937
```
```  1938
```
```  1939 subsection {* Legacy theorem bindings *}
```
```  1940
```
```  1941 lemmas multi_count_eq = multiset_eq_iff [symmetric]
```
```  1942
```
```  1943 lemma union_commute: "M + N = N + (M::'a multiset)"
```
```  1944   by (fact add.commute)
```
```  1945
```
```  1946 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```  1947   by (fact add.assoc)
```
```  1948
```
```  1949 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```  1950   by (fact add.left_commute)
```
```  1951
```
```  1952 lemmas union_ac = union_assoc union_commute union_lcomm
```
```  1953
```
```  1954 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
```
```  1955   by (fact add_right_cancel)
```
```  1956
```
```  1957 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
```
```  1958   by (fact add_left_cancel)
```
```  1959
```
```  1960 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
```
```  1961   by (fact add_imp_eq)
```
```  1962
```
```  1963 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
```
```  1964   by (fact order_less_trans)
```
```  1965
```
```  1966 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
```
```  1967   by (fact inf.commute)
```
```  1968
```
```  1969 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
```
```  1970   by (fact inf.assoc [symmetric])
```
```  1971
```
```  1972 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
```
```  1973   by (fact inf.left_commute)
```
```  1974
```
```  1975 lemmas multiset_inter_ac =
```
```  1976   multiset_inter_commute
```
```  1977   multiset_inter_assoc
```
```  1978   multiset_inter_left_commute
```
```  1979
```
```  1980 lemma mult_less_not_refl:
```
```  1981   "\<not> M \<subset># (M::'a::order multiset)"
```
```  1982   by (fact multiset_order.less_irrefl)
```
```  1983
```
```  1984 lemma mult_less_trans:
```
```  1985   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
```
```  1986   by (fact multiset_order.less_trans)
```
```  1987
```
```  1988 lemma mult_less_not_sym:
```
```  1989   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
```
```  1990   by (fact multiset_order.less_not_sym)
```
```  1991
```
```  1992 lemma mult_less_asym:
```
```  1993   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
```
```  1994   by (fact multiset_order.less_asym)
```
```  1995
```
```  1996 ML {*
```
```  1997 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
```
```  1998                       (Const _ \$ t') =
```
```  1999     let
```
```  2000       val (maybe_opt, ps) =
```
```  2001         Nitpick_Model.dest_plain_fun t' ||> op ~~
```
```  2002         ||> map (apsnd (snd o HOLogic.dest_number))
```
```  2003       fun elems_for t =
```
```  2004         case AList.lookup (op =) ps t of
```
```  2005           SOME n => replicate n t
```
```  2006         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
```
```  2007     in
```
```  2008       case maps elems_for (all_values elem_T) @
```
```  2009            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
```
```  2010             else []) of
```
```  2011         [] => Const (@{const_name zero_class.zero}, T)
```
```  2012       | ts => foldl1 (fn (t1, t2) =>
```
```  2013                          Const (@{const_name plus_class.plus}, T --> T --> T)
```
```  2014                          \$ t1 \$ t2)
```
```  2015                      (map (curry (op \$) (Const (@{const_name single},
```
```  2016                                                 elem_T --> T))) ts)
```
```  2017     end
```
```  2018   | multiset_postproc _ _ _ _ t = t
```
```  2019 *}
```
```  2020
```
```  2021 declaration {*
```
```  2022 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
```
```  2023     multiset_postproc
```
```  2024 *}
```
```  2025
```
```  2026 hide_const (open) fold
```
```  2027
```
```  2028
```
```  2029 subsection {* Naive implementation using lists *}
```
```  2030
```
```  2031 code_datatype multiset_of
```
```  2032
```
```  2033 lemma [code]:
```
```  2034   "{#} = multiset_of []"
```
```  2035   by simp
```
```  2036
```
```  2037 lemma [code]:
```
```  2038   "{#x#} = multiset_of [x]"
```
```  2039   by simp
```
```  2040
```
```  2041 lemma union_code [code]:
```
```  2042   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
```
```  2043   by simp
```
```  2044
```
```  2045 lemma [code]:
```
```  2046   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
```
```  2047   by (simp add: multiset_of_map)
```
```  2048
```
```  2049 lemma [code]:
```
```  2050   "Multiset.filter f (multiset_of xs) = multiset_of (filter f xs)"
```
```  2051   by (simp add: multiset_of_filter)
```
```  2052
```
```  2053 lemma [code]:
```
```  2054   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
```
```  2055   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
```
```  2056
```
```  2057 lemma [code]:
```
```  2058   "multiset_of xs #\<inter> multiset_of ys =
```
```  2059     multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2060       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
```
```  2061 proof -
```
```  2062   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2063     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
```
```  2064       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
```
```  2065     by (induct xs arbitrary: ys)
```
```  2066       (auto simp add: mem_set_multiset_eq inter_add_right1 inter_add_right2 ac_simps)
```
```  2067   then show ?thesis by simp
```
```  2068 qed
```
```  2069
```
```  2070 lemma [code]:
```
```  2071   "multiset_of xs #\<union> multiset_of ys =
```
```  2072     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
```
```  2073 proof -
```
```  2074   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
```
```  2075       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
```
```  2076     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
```
```  2077   then show ?thesis by simp
```
```  2078 qed
```
```  2079
```
```  2080 lemma [code_unfold]:
```
```  2081   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```  2082   by (simp add: in_multiset_of)
```
```  2083
```
```  2084 lemma [code]:
```
```  2085   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
```
```  2086 proof -
```
```  2087   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
```
```  2088     by (induct xs) simp_all
```
```  2089   then show ?thesis by simp
```
```  2090 qed
```
```  2091
```
```  2092 lemma [code]:
```
```  2093   "set_of (multiset_of xs) = set xs"
```
```  2094   by simp
```
```  2095
```
```  2096 lemma [code]:
```
```  2097   "sorted_list_of_multiset (multiset_of xs) = sort xs"
```
```  2098   by (induct xs) simp_all
```
```  2099
```
```  2100 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
```
```  2101   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
```
```  2102   apply (cases "finite A")
```
```  2103   apply simp_all
```
```  2104   apply (induct A rule: finite_induct)
```
```  2105   apply (simp_all add: union_commute)
```
```  2106   done
```
```  2107
```
```  2108 lemma [code]:
```
```  2109   "mcard (multiset_of xs) = length xs"
```
```  2110   by (simp add: mcard_multiset_of)
```
```  2111
```
```  2112 fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
```
```  2113   "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
```
```  2114 | "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
```
```  2115      None \<Rightarrow> None
```
```  2116    | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"
```
```  2117
```
```  2118 lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> multiset_of xs \<le> multiset_of ys) \<and>
```
```  2119   (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> multiset_of xs < multiset_of ys) \<and>
```
```  2120   (ms_lesseq_impl xs ys = Some False \<longrightarrow> multiset_of xs = multiset_of ys)"
```
```  2121 proof (induct xs arbitrary: ys)
```
```  2122   case (Nil ys)
```
```  2123   show ?case by (auto simp: mset_less_empty_nonempty)
```
```  2124 next
```
```  2125   case (Cons x xs ys)
```
```  2126   show ?case
```
```  2127   proof (cases "List.extract (op = x) ys")
```
```  2128     case None
```
```  2129     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
```
```  2130     {
```
```  2131       assume "multiset_of (x # xs) \<le> multiset_of ys"
```
```  2132       from set_of_mono[OF this] x have False by simp
```
```  2133     } note nle = this
```
```  2134     moreover
```
```  2135     {
```
```  2136       assume "multiset_of (x # xs) < multiset_of ys"
```
```  2137       hence "multiset_of (x # xs) \<le> multiset_of ys" by auto
```
```  2138       from nle[OF this] have False .
```
```  2139     }
```
```  2140     ultimately show ?thesis using None by auto
```
```  2141   next
```
```  2142     case (Some res)
```
```  2143     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
```
```  2144     note Some = Some[unfolded res]
```
```  2145     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
```
```  2146     hence id: "multiset_of ys = multiset_of (ys1 @ ys2) + {#x#}"
```
```  2147       by (auto simp: ac_simps)
```
```  2148     show ?thesis unfolding ms_lesseq_impl.simps
```
```  2149       unfolding Some option.simps split
```
```  2150       unfolding id
```
```  2151       using Cons[of "ys1 @ ys2"]
```
```  2152       unfolding mset_le_def mset_less_def by auto
```
```  2153   qed
```
```  2154 qed
```
```  2155
```
```  2156 lemma [code]: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
```
```  2157   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2158
```
```  2159 lemma [code]: "multiset_of xs < multiset_of ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
```
```  2160   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2161
```
```  2162 instantiation multiset :: (equal) equal
```
```  2163 begin
```
```  2164
```
```  2165 definition
```
```  2166   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
```
```  2167 lemma [code]: "HOL.equal (multiset_of xs) (multiset_of ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
```
```  2168   unfolding equal_multiset_def
```
```  2169   using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)
```
```  2170
```
```  2171 instance
```
```  2172   by default (simp add: equal_multiset_def)
```
```  2173 end
```
```  2174
```
```  2175 lemma [code]:
```
```  2176   "msetsum (multiset_of xs) = listsum xs"
```
```  2177   by (induct xs) (simp_all add: add.commute)
```
```  2178
```
```  2179 lemma [code]:
```
```  2180   "msetprod (multiset_of xs) = fold times xs 1"
```
```  2181 proof -
```
```  2182   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
```
```  2183     by (induct xs) (simp_all add: mult.assoc)
```
```  2184   then show ?thesis by simp
```
```  2185 qed
```
```  2186
```
```  2187 lemma [code]:
```
```  2188   "size = mcard"
```
```  2189   by (fact size_eq_mcard)
```
```  2190
```
```  2191 text {*
```
```  2192   Exercise for the casual reader: add implementations for @{const le_multiset}
```
```  2193   and @{const less_multiset} (multiset order).
```
```  2194 *}
```
```  2195
```
```  2196 text {* Quickcheck generators *}
```
```  2197
```
```  2198 definition (in term_syntax)
```
```  2199   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
```
```  2200     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
```
```  2201   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
```
```  2202
```
```  2203 notation fcomp (infixl "\<circ>>" 60)
```
```  2204 notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2205
```
```  2206 instantiation multiset :: (random) random
```
```  2207 begin
```
```  2208
```
```  2209 definition
```
```  2210   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
```
```  2211
```
```  2212 instance ..
```
```  2213
```
```  2214 end
```
```  2215
```
```  2216 no_notation fcomp (infixl "\<circ>>" 60)
```
```  2217 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2218
```
```  2219 instantiation multiset :: (full_exhaustive) full_exhaustive
```
```  2220 begin
```
```  2221
```
```  2222 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
```
```  2223 where
```
```  2224   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
```
```  2225
```
```  2226 instance ..
```
```  2227
```
```  2228 end
```
```  2229
```
```  2230 hide_const (open) msetify
```
```  2231
```
```  2232
```
```  2233 subsection {* BNF setup *}
```
```  2234
```
```  2235 lemma setsum_gt_0_iff:
```
```  2236 fixes f :: "'a \<Rightarrow> nat" assumes "finite A"
```
```  2237 shows "setsum f A > 0 \<longleftrightarrow> (\<exists> a \<in> A. f a > 0)"
```
```  2238 (is "?L \<longleftrightarrow> ?R")
```
```  2239 proof-
```
```  2240   have "?L \<longleftrightarrow> \<not> setsum f A = 0" by fast
```
```  2241   also have "... \<longleftrightarrow> (\<exists> a \<in> A. f a \<noteq> 0)" using assms by simp
```
```  2242   also have "... \<longleftrightarrow> ?R" by simp
```
```  2243   finally show ?thesis .
```
```  2244 qed
```
```  2245
```
```  2246 lift_definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" is
```
```  2247   "\<lambda>h f b. setsum f {a. h a = b \<and> f a > 0} :: nat"
```
```  2248 unfolding multiset_def proof safe
```
```  2249   fix h :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat"
```
```  2250   assume fin: "finite {a. 0 < f a}"  (is "finite ?A")
```
```  2251   show "finite {b. 0 < setsum f {a. h a = b \<and> 0 < f a}}"
```
```  2252   (is "finite {b. 0 < setsum f (?As b)}")
```
```  2253   proof- let ?B = "{b. 0 < setsum f (?As b)}"
```
```  2254     have "\<And> b. finite (?As b)" using fin by simp
```
```  2255     hence B: "?B = {b. ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
```
```  2256     hence "?B \<subseteq> h ` ?A" by auto
```
```  2257     thus ?thesis using finite_surj[OF fin] by auto
```
```  2258   qed
```
```  2259 qed
```
```  2260
```
```  2261 lemma mmap_id0: "mmap id = id"
```
```  2262 proof (intro ext multiset_eqI)
```
```  2263   fix f a show "count (mmap id f) a = count (id f) a"
```
```  2264   proof (cases "count f a = 0")
```
```  2265     case False
```
```  2266     hence 1: "{aa. aa = a \<and> aa \<in># f} = {a}" by auto
```
```  2267     thus ?thesis by transfer auto
```
```  2268   qed (transfer, simp)
```
```  2269 qed
```
```  2270
```
```  2271 lemma inj_on_setsum_inv:
```
```  2272 assumes 1: "(0::nat) < setsum (count f) {a. h a = b' \<and> a \<in># f}" (is "0 < setsum (count f) ?A'")
```
```  2273 and     2: "{a. h a = b \<and> a \<in># f} = {a. h a = b' \<and> a \<in># f}" (is "?A = ?A'")
```
```  2274 shows "b = b'"
```
```  2275 using assms by (auto simp add: setsum_gt_0_iff)
```
```  2276
```
```  2277 lemma mmap_comp:
```
```  2278 fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
```
```  2279 shows "mmap (h2 o h1) = mmap h2 o mmap h1"
```
```  2280 proof (intro ext multiset_eqI)
```
```  2281   fix f :: "'a multiset" fix c :: 'c
```
```  2282   let ?A = "{a. h2 (h1 a) = c \<and> a \<in># f}"
```
```  2283   let ?As = "\<lambda> b. {a. h1 a = b \<and> a \<in># f}"
```
```  2284   let ?B = "{b. h2 b = c \<and> 0 < setsum (count f) (?As b)}"
```
```  2285   have 0: "{?As b | b.  b \<in> ?B} = ?As ` ?B" by auto
```
```  2286   have "\<And> b. finite (?As b)" by transfer (simp add: multiset_def)
```
```  2287   hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
```
```  2288   hence A: "?A = \<Union> {?As b | b.  b \<in> ?B}" by auto
```
```  2289   have "setsum (count f) ?A = setsum (setsum (count f)) {?As b | b.  b \<in> ?B}"
```
```  2290     unfolding A by transfer (intro setsum.Union_disjoint [simplified], auto simp: multiset_def setsum.Union_disjoint)
```
```  2291   also have "... = setsum (setsum (count f)) (?As ` ?B)" unfolding 0 ..
```
```  2292   also have "... = setsum (setsum (count f) o ?As) ?B"
```
```  2293     by (intro setsum.reindex) (auto simp add: setsum_gt_0_iff inj_on_def)
```
```  2294   also have "... = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" unfolding comp_def ..
```
```  2295   finally have "setsum (count f) ?A = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" .
```
```  2296   thus "count (mmap (h2 \<circ> h1) f) c = count ((mmap h2 \<circ> mmap h1) f) c"
```
```  2297     by transfer (unfold comp_apply, blast)
```
```  2298 qed
```
```  2299
```
```  2300 lemma mmap_cong:
```
```  2301 assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
```
```  2302 shows "mmap f M = mmap g M"
```
```  2303 using assms by transfer (auto intro!: setsum.cong)
```
```  2304
```
```  2305 context
```
```  2306 begin
```
```  2307 interpretation lifting_syntax .
```
```  2308
```
```  2309 lemma set_of_transfer[transfer_rule]: "(pcr_multiset op = ===> op =) (\<lambda>f. {a. 0 < f a}) set_of"
```
```  2310   unfolding set_of_def pcr_multiset_def cr_multiset_def rel_fun_def by auto
```
```  2311
```
```  2312 end
```
```  2313
```
```  2314 lemma set_of_mmap: "set_of o mmap h = image h o set_of"
```
```  2315 proof (rule ext, unfold comp_apply)
```
```  2316   fix M show "set_of (mmap h M) = h ` set_of M"
```
```  2317     by transfer (auto simp add: multiset_def setsum_gt_0_iff)
```
```  2318 qed
```
```  2319
```
```  2320 lemma multiset_of_surj:
```
```  2321   "multiset_of ` {as. set as \<subseteq> A} = {M. set_of M \<subseteq> A}"
```
```  2322 proof safe
```
```  2323   fix M assume M: "set_of M \<subseteq> A"
```
```  2324   obtain as where eq: "M = multiset_of as" using surj_multiset_of unfolding surj_def by auto
```
```  2325   hence "set as \<subseteq> A" using M by auto
```
```  2326   thus "M \<in> multiset_of ` {as. set as \<subseteq> A}" using eq by auto
```
```  2327 next
```
```  2328   show "\<And>x xa xb. \<lbrakk>set xa \<subseteq> A; xb \<in> set_of (multiset_of xa)\<rbrakk> \<Longrightarrow> xb \<in> A"
```
```  2329   by (erule set_mp) (unfold set_of_multiset_of)
```
```  2330 qed
```
```  2331
```
```  2332 lemma card_of_set_of:
```
```  2333 "(card_of {M. set_of M \<subseteq> A}, card_of {as. set as \<subseteq> A}) \<in> ordLeq"
```
```  2334 apply(rule surj_imp_ordLeq[of _ multiset_of]) using multiset_of_surj by auto
```
```  2335
```
```  2336 lemma nat_sum_induct:
```
```  2337 assumes "\<And>n1 n2. (\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow> phi m1 m2) \<Longrightarrow> phi n1 n2"
```
```  2338 shows "phi (n1::nat) (n2::nat)"
```
```  2339 proof-
```
```  2340   let ?chi = "\<lambda> n1n2 :: nat * nat. phi (fst n1n2) (snd n1n2)"
```
```  2341   have "?chi (n1,n2)"
```
```  2342   apply(induct rule: measure_induct[of "\<lambda> n1n2. fst n1n2 + snd n1n2" ?chi])
```
```  2343   using assms by (metis fstI sndI)
```
```  2344   thus ?thesis by simp
```
```  2345 qed
```
```  2346
```
```  2347 lemma matrix_count:
```
```  2348 fixes ct1 ct2 :: "nat \<Rightarrow> nat"
```
```  2349 assumes "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
```
```  2350 shows
```
```  2351 "\<exists> ct. (\<forall> i1 \<le> n1. setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ct1 i1) \<and>
```
```  2352        (\<forall> i2 \<le> n2. setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ct2 i2)"
```
```  2353 (is "?phi ct1 ct2 n1 n2")
```
```  2354 proof-
```
```  2355   have "\<forall> ct1 ct2 :: nat \<Rightarrow> nat.
```
```  2356         setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"
```
```  2357   proof(induct rule: nat_sum_induct[of
```
```  2358 "\<lambda> n1 n2. \<forall> ct1 ct2 :: nat \<Rightarrow> nat.
```
```  2359      setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"],
```
```  2360       clarify)
```
```  2361   fix n1 n2 :: nat and ct1 ct2 :: "nat \<Rightarrow> nat"
```
```  2362   assume IH: "\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow>
```
```  2363                 \<forall> dt1 dt2 :: nat \<Rightarrow> nat.
```
```  2364                 setsum dt1 {..<Suc m1} = setsum dt2 {..<Suc m2} \<longrightarrow> ?phi dt1 dt2 m1 m2"
```
```  2365   and ss: "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
```
```  2366   show "?phi ct1 ct2 n1 n2"
```
```  2367   proof(cases n1)
```
```  2368     case 0 note n1 = 0
```
```  2369     show ?thesis
```
```  2370     proof(cases n2)
```
```  2371       case 0 note n2 = 0
```
```  2372       let ?ct = "\<lambda> i1 i2. ct2 0"
```
```  2373       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by simp
```
```  2374     next
```
```  2375       case (Suc m2) note n2 = Suc
```
```  2376       let ?ct = "\<lambda> i1 i2. ct2 i2"
```
```  2377       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
```
```  2378     qed
```
```  2379   next
```
```  2380     case (Suc m1) note n1 = Suc
```
```  2381     show ?thesis
```
```  2382     proof(cases n2)
```
```  2383       case 0 note n2 = 0
```
```  2384       let ?ct = "\<lambda> i1 i2. ct1 i1"
```
```  2385       show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
```
```  2386     next
```
```  2387       case (Suc m2) note n2 = Suc
```
```  2388       show ?thesis
```
```  2389       proof(cases "ct1 n1 \<le> ct2 n2")
```
```  2390         case True
```
```  2391         def dt2 \<equiv> "\<lambda> i2. if i2 = n2 then ct2 i2 - ct1 n1 else ct2 i2"
```
```  2392         have "setsum ct1 {..<Suc m1} = setsum dt2 {..<Suc n2}"
```
```  2393         unfolding dt2_def using ss n1 True by auto
```
```  2394         hence "?phi ct1 dt2 m1 n2" using IH[of m1 n2] n1 by simp
```
```  2395         then obtain dt where
```
```  2396         1: "\<And> i1. i1 \<le> m1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc n2} = ct1 i1" and
```
```  2397         2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc m1} = dt2 i2" by auto
```
```  2398         let ?ct = "\<lambda> i1 i2. if i1 = n1 then (if i2 = n2 then ct1 n1 else 0)
```
```  2399                                        else dt i1 i2"
```
```  2400         show ?thesis apply(rule exI[of _ ?ct])
```
```  2401         using n1 n2 1 2 True unfolding dt2_def by simp
```
```  2402       next
```
```  2403         case False
```
```  2404         hence False: "ct2 n2 < ct1 n1" by simp
```
```  2405         def dt1 \<equiv> "\<lambda> i1. if i1 = n1 then ct1 i1 - ct2 n2 else ct1 i1"
```
```  2406         have "setsum dt1 {..<Suc n1} = setsum ct2 {..<Suc m2}"
```
```  2407         unfolding dt1_def using ss n2 False by auto
```
```  2408         hence "?phi dt1 ct2 n1 m2" using IH[of n1 m2] n2 by simp
```
```  2409         then obtain dt where
```
```  2410         1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc m2} = dt1 i1" and
```
```  2411         2: "\<And> i2. i2 \<le> m2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc n1} = ct2 i2" by force
```
```  2412         let ?ct = "\<lambda> i1 i2. if i2 = n2 then (if i1 = n1 then ct2 n2 else 0)
```
```  2413                                        else dt i1 i2"
```
```  2414         show ?thesis apply(rule exI[of _ ?ct])
```
```  2415         using n1 n2 1 2 False unfolding dt1_def by simp
```
```  2416       qed
```
```  2417     qed
```
```  2418   qed
```
```  2419   qed
```
```  2420   thus ?thesis using assms by auto
```
```  2421 qed
```
```  2422
```
```  2423 definition
```
```  2424 "inj2 u B1 B2 \<equiv>
```
```  2425  \<forall> b1 b1' b2 b2'. {b1,b1'} \<subseteq> B1 \<and> {b2,b2'} \<subseteq> B2 \<and> u b1 b2 = u b1' b2'
```
```  2426                   \<longrightarrow> b1 = b1' \<and> b2 = b2'"
```
```  2427
```
```  2428 lemma matrix_setsum_finite:
```
```  2429 assumes B1: "B1 \<noteq> {}" "finite B1" and B2: "B2 \<noteq> {}" "finite B2" and u: "inj2 u B1 B2"
```
```  2430 and ss: "setsum N1 B1 = setsum N2 B2"
```
```  2431 shows "\<exists> M :: 'a \<Rightarrow> nat.
```
```  2432             (\<forall> b1 \<in> B1. setsum (\<lambda> b2. M (u b1 b2)) B2 = N1 b1) \<and>
```
```  2433             (\<forall> b2 \<in> B2. setsum (\<lambda> b1. M (u b1 b2)) B1 = N2 b2)"
```
```  2434 proof-
```
```  2435   obtain n1 where "card B1 = Suc n1" using B1 by (metis card_insert finite.simps)
```
```  2436   then obtain e1 where e1: "bij_betw e1 {..<Suc n1} B1"
```
```  2437   using ex_bij_betw_finite_nat[OF B1(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
```
```  2438   hence e1_inj: "inj_on e1 {..<Suc n1}" and e1_surj: "e1 ` {..<Suc n1} = B1"
```
```  2439   unfolding bij_betw_def by auto
```
```  2440   def f1 \<equiv> "inv_into {..<Suc n1} e1"
```
```  2441   have f1: "bij_betw f1 B1 {..<Suc n1}"
```
```  2442   and f1e1[simp]: "\<And> i1. i1 < Suc n1 \<Longrightarrow> f1 (e1 i1) = i1"
```
```  2443   and e1f1[simp]: "\<And> b1. b1 \<in> B1 \<Longrightarrow> e1 (f1 b1) = b1" unfolding f1_def
```
```  2444   apply (metis bij_betw_inv_into e1, metis bij_betw_inv_into_left e1 lessThan_iff)
```
```  2445   by (metis e1_surj f_inv_into_f)
```
```  2446   (*  *)
```
```  2447   obtain n2 where "card B2 = Suc n2" using B2 by (metis card_insert finite.simps)
```
```  2448   then obtain e2 where e2: "bij_betw e2 {..<Suc n2} B2"
```
```  2449   using ex_bij_betw_finite_nat[OF B2(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
```
```  2450   hence e2_inj: "inj_on e2 {..<Suc n2}" and e2_surj: "e2 ` {..<Suc n2} = B2"
```
```  2451   unfolding bij_betw_def by auto
```
```  2452   def f2 \<equiv> "inv_into {..<Suc n2} e2"
```
```  2453   have f2: "bij_betw f2 B2 {..<Suc n2}"
```
```  2454   and f2e2[simp]: "\<And> i2. i2 < Suc n2 \<Longrightarrow> f2 (e2 i2) = i2"
```
```  2455   and e2f2[simp]: "\<And> b2. b2 \<in> B2 \<Longrightarrow> e2 (f2 b2) = b2" unfolding f2_def
```
```  2456   apply (metis bij_betw_inv_into e2, metis bij_betw_inv_into_left e2 lessThan_iff)
```
```  2457   by (metis e2_surj f_inv_into_f)
```
```  2458   (*  *)
```
```  2459   let ?ct1 = "N1 o e1"  let ?ct2 = "N2 o e2"
```
```  2460   have ss: "setsum ?ct1 {..<Suc n1} = setsum ?ct2 {..<Suc n2}"
```
```  2461   unfolding setsum.reindex[OF e1_inj, symmetric] setsum.reindex[OF e2_inj, symmetric]
```
```  2462   e1_surj e2_surj using ss .
```
```  2463   obtain ct where
```
```  2464   ct1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ?ct1 i1" and
```
```  2465   ct2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ?ct2 i2"
```
```  2466   using matrix_count[OF ss] by blast
```
```  2467   (*  *)
```
```  2468   def A \<equiv> "{u b1 b2 | b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"
```
```  2469   have "\<forall> a \<in> A. \<exists> b1b2 \<in> B1 <*> B2. u (fst b1b2) (snd b1b2) = a"
```
```  2470   unfolding A_def Ball_def mem_Collect_eq by auto
```
```  2471   then obtain h1h2 where h12:
```
```  2472   "\<And>a. a \<in> A \<Longrightarrow> u (fst (h1h2 a)) (snd (h1h2 a)) = a \<and> h1h2 a \<in> B1 <*> B2" by metis
```
```  2473   def h1 \<equiv> "fst o h1h2"  def h2 \<equiv> "snd o h1h2"
```
```  2474   have h12[simp]: "\<And>a. a \<in> A \<Longrightarrow> u (h1 a) (h2 a) = a"
```
```  2475                   "\<And> a. a \<in> A \<Longrightarrow> h1 a \<in> B1"  "\<And> a. a \<in> A \<Longrightarrow> h2 a \<in> B2"
```
```  2476   using h12 unfolding h1_def h2_def by force+
```
```  2477   {fix b1 b2 assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2"
```
```  2478    hence inA: "u b1 b2 \<in> A" unfolding A_def by auto
```
```  2479    hence "u b1 b2 = u (h1 (u b1 b2)) (h2 (u b1 b2))" by auto
```
```  2480    moreover have "h1 (u b1 b2) \<in> B1" "h2 (u b1 b2) \<in> B2" using inA by auto
```
```  2481    ultimately have "h1 (u b1 b2) = b1 \<and> h2 (u b1 b2) = b2"
```
```  2482    using u b1 b2 unfolding inj2_def by fastforce
```
```  2483   }
```
```  2484   hence h1[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h1 (u b1 b2) = b1" and
```
```  2485         h2[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h2 (u b1 b2) = b2" by auto
```
```  2486   def M \<equiv> "\<lambda> a. ct (f1 (h1 a)) (f2 (h2 a))"
```
```  2487   show ?thesis
```
```  2488   apply(rule exI[of _ M]) proof safe
```
```  2489     fix b1 assume b1: "b1 \<in> B1"
```
```  2490     hence f1b1: "f1 b1 \<le> n1" using f1 unfolding bij_betw_def
```
```  2491     by (metis image_eqI lessThan_iff less_Suc_eq_le)
```
```  2492     have "(\<Sum>b2\<in>B2. M (u b1 b2)) = (\<Sum>i2<Suc n2. ct (f1 b1) (f2 (e2 i2)))"
```
```  2493     unfolding e2_surj[symmetric] setsum.reindex[OF e2_inj]
```
```  2494     unfolding M_def comp_def apply(intro setsum.cong) apply force
```
```  2495     by (metis e2_surj b1 h1 h2 imageI)
```
```  2496     also have "... = N1 b1" using b1 ct1[OF f1b1] by simp
```
```  2497     finally show "(\<Sum>b2\<in>B2. M (u b1 b2)) = N1 b1" .
```
```  2498   next
```
```  2499     fix b2 assume b2: "b2 \<in> B2"
```
```  2500     hence f2b2: "f2 b2 \<le> n2" using f2 unfolding bij_betw_def
```
```  2501     by (metis image_eqI lessThan_iff less_Suc_eq_le)
```
```  2502     have "(\<Sum>b1\<in>B1. M (u b1 b2)) = (\<Sum>i1<Suc n1. ct (f1 (e1 i1)) (f2 b2))"
```
```  2503     unfolding e1_surj[symmetric] setsum.reindex[OF e1_inj]
```
```  2504     unfolding M_def comp_def apply(intro setsum.cong) apply force
```
```  2505     by (metis e1_surj b2 h1 h2 imageI)
```
```  2506     also have "... = N2 b2" using b2 ct2[OF f2b2] by simp
```
```  2507     finally show "(\<Sum>b1\<in>B1. M (u b1 b2)) = N2 b2" .
```
```  2508   qed
```
```  2509 qed
```
```  2510
```
```  2511 lemma supp_vimage_mmap: "set_of M \<subseteq> f -` (set_of (mmap f M))"
```
```  2512   by transfer (auto simp: multiset_def setsum_gt_0_iff)
```
```  2513
```
```  2514 lemma mmap_ge_0: "b \<in># mmap f M \<longleftrightarrow> (\<exists>a. a \<in># M \<and> f a = b)"
```
```  2515   by transfer (auto simp: multiset_def setsum_gt_0_iff)
```
```  2516
```
```  2517 lemma finite_twosets:
```
```  2518 assumes "finite B1" and "finite B2"
```
```  2519 shows "finite {u b1 b2 |b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"  (is "finite ?A")
```
```  2520 proof-
```
```  2521   have A: "?A = (\<lambda> b1b2. u (fst b1b2) (snd b1b2)) ` (B1 <*> B2)" by force
```
```  2522   show ?thesis unfolding A using finite_cartesian_product[OF assms] by auto
```
```  2523 qed
```
```  2524
```
```  2525 (* Weak pullbacks: *)
```
```  2526 definition wpull where
```
```  2527 "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow>
```
```  2528  (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow> (\<exists> a \<in> A. p1 a = b1 \<and> p2 a = b2))"
```
```  2529
```
```  2530 (* Weak pseudo-pullbacks *)
```
```  2531 definition wppull where
```
```  2532 "wppull A B1 B2 f1 f2 e1 e2 p1 p2 \<longleftrightarrow>
```
```  2533  (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow>
```
```  2534            (\<exists> a \<in> A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2))"
```
```  2535
```
```  2536
```
```  2537 (* The pullback of sets *)
```
```  2538 definition thePull where
```
```  2539 "thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
```
```  2540
```
```  2541 lemma wpull_thePull:
```
```  2542 "wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
```
```  2543 unfolding wpull_def thePull_def by auto
```
```  2544
```
```  2545 lemma wppull_thePull:
```
```  2546 assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
```
```  2547 shows
```
```  2548 "\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
```
```  2549    j a' \<in> A \<and>
```
```  2550    e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
```
```  2551 (is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
```
```  2552 proof(rule bchoice[of ?A' ?phi], default)
```
```  2553   fix a' assume a': "a' \<in> ?A'"
```
```  2554   hence "fst a' \<in> B1" unfolding thePull_def by auto
```
```  2555   moreover
```
```  2556   from a' have "snd a' \<in> B2" unfolding thePull_def by auto
```
```  2557   moreover have "f1 (fst a') = f2 (snd a')"
```
```  2558   using a' unfolding csquare_def thePull_def by auto
```
```  2559   ultimately show "\<exists> ja'. ?phi a' ja'"
```
```  2560   using assms unfolding wppull_def by blast
```
```  2561 qed
```
```  2562
```
```  2563 lemma wpull_wppull:
```
```  2564 assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
```
```  2565 1: "\<forall> a' \<in> A'. j a' \<in> A \<and> e1 (p1 (j a')) = e1 (p1' a') \<and> e2 (p2 (j a')) = e2 (p2' a')"
```
```  2566 shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
```
```  2567 unfolding wppull_def proof safe
```
```  2568   fix b1 b2
```
```  2569   assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
```
```  2570   then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
```
```  2571   using wp unfolding wpull_def by blast
```
```  2572   show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
```
```  2573   apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
```
```  2574 qed
```
```  2575
```
```  2576 lemma wppull_fstOp_sndOp:
```
```  2577 shows "wppull (Collect (split (P OO Q))) (Collect (split P)) (Collect (split Q))
```
```  2578   snd fst fst snd (BNF_Def.fstOp P Q) (BNF_Def.sndOp P Q)"
```
```  2579 using pick_middlep unfolding wppull_def fstOp_def sndOp_def relcompp.simps by auto
```
```  2580
```
```  2581 lemma wpull_mmap:
```
```  2582 fixes A :: "'a set" and B1 :: "'b1 set" and B2 :: "'b2 set"
```
```  2583 assumes wp: "wpull A B1 B2 f1 f2 p1 p2"
```
```  2584 shows
```
```  2585 "wpull {M. set_of M \<subseteq> A}
```
```  2586        {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
```
```  2587        (mmap f1) (mmap f2) (mmap p1) (mmap p2)"
```
```  2588 unfolding wpull_def proof (safe, unfold Bex_def mem_Collect_eq)
```
```  2589   fix N1 :: "'b1 multiset" and N2 :: "'b2 multiset"
```
```  2590   assume mmap': "mmap f1 N1 = mmap f2 N2"
```
```  2591   and N1[simp]: "set_of N1 \<subseteq> B1"
```
```  2592   and N2[simp]: "set_of N2 \<subseteq> B2"
```
```  2593   def P \<equiv> "mmap f1 N1"
```
```  2594   have P1: "P = mmap f1 N1" and P2: "P = mmap f2 N2" unfolding P_def using mmap' by auto
```
```  2595   note P = P1 P2
```
```  2596   have fin_N1[simp]: "finite (set_of N1)"
```
```  2597    and fin_N2[simp]: "finite (set_of N2)"
```
```  2598    and fin_P[simp]: "finite (set_of P)" by auto
```
```  2599
```
```  2600   def set1 \<equiv> "\<lambda> c. {b1 \<in> set_of N1. f1 b1 = c}"
```
```  2601   have set1[simp]: "\<And> c b1. b1 \<in> set1 c \<Longrightarrow> f1 b1 = c" unfolding set1_def by auto
```
```  2602   have fin_set1: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set1 c)"
```
```  2603     using N1(1) unfolding set1_def multiset_def by auto
```
```  2604   have set1_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<noteq> {}"
```
```  2605    unfolding set1_def set_of_def P mmap_ge_0 by auto
```
```  2606   have supp_N1_set1: "set_of N1 = (\<Union> c \<in> set_of P. set1 c)"
```
```  2607     using supp_vimage_mmap[of N1 f1] unfolding set1_def P1 by auto
```
```  2608   hence set1_inclN1: "\<And>c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> set_of N1" by auto
```
```  2609   hence set1_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> B1" using N1 by blast
```
```  2610   have set1_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set1 c \<inter> set1 c' = {}"
```
```  2611     unfolding set1_def by auto
```
```  2612   have setsum_set1: "\<And> c. setsum (count N1) (set1 c) = count P c"
```
```  2613     unfolding P1 set1_def by transfer (auto intro: setsum.cong)
```
```  2614
```
```  2615   def set2 \<equiv> "\<lambda> c. {b2 \<in> set_of N2. f2 b2 = c}"
```
```  2616   have set2[simp]: "\<And> c b2. b2 \<in> set2 c \<Longrightarrow> f2 b2 = c" unfolding set2_def by auto
```
```  2617   have fin_set2: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set2 c)"
```
```  2618   using N2(1) unfolding set2_def multiset_def by auto
```
```  2619   have set2_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<noteq> {}"
```
```  2620     unfolding set2_def P2 mmap_ge_0 set_of_def by auto
```
```  2621   have supp_N2_set2: "set_of N2 = (\<Union> c \<in> set_of P. set2 c)"
```
```  2622     using supp_vimage_mmap[of N2 f2] unfolding set2_def P2 by auto
```
```  2623   hence set2_inclN2: "\<And>c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> set_of N2" by auto
```
```  2624   hence set2_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> B2" using N2 by blast
```
```  2625   have set2_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set2 c \<inter> set2 c' = {}"
```
```  2626     unfolding set2_def by auto
```
```  2627   have setsum_set2: "\<And> c. setsum (count N2) (set2 c) = count P c"
```
```  2628     unfolding P2 set2_def by transfer (auto intro: setsum.cong)
```
```  2629
```
```  2630   have ss: "\<And> c. c \<in> set_of P \<Longrightarrow> setsum (count N1) (set1 c) = setsum (count N2) (set2 c)"
```
```  2631     unfolding setsum_set1 setsum_set2 ..
```
```  2632   have "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
```
```  2633           \<exists> a \<in> A. p1 a = fst b1b2 \<and> p2 a = snd b1b2"
```
```  2634     using wp set1_incl set2_incl unfolding wpull_def Ball_def mem_Collect_eq
```
```  2635     by simp (metis set1 set2 set_rev_mp)
```
```  2636   then obtain uu where uu:
```
```  2637   "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
```
```  2638      uu c b1b2 \<in> A \<and> p1 (uu c b1b2) = fst b1b2 \<and> p2 (uu c b1b2) = snd b1b2" by metis
```
```  2639   def u \<equiv> "\<lambda> c b1 b2. uu c (b1,b2)"
```
```  2640   have u[simp]:
```
```  2641   "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> A"
```
```  2642   "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p1 (u c b1 b2) = b1"
```
```  2643   "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p2 (u c b1 b2) = b2"
```
```  2644     using uu unfolding u_def by auto
```
```  2645   {fix c assume c: "c \<in> set_of P"
```
```  2646    have "inj2 (u c) (set1 c) (set2 c)" unfolding inj2_def proof clarify
```
```  2647      fix b1 b1' b2 b2'
```
```  2648      assume "{b1, b1'} \<subseteq> set1 c" "{b2, b2'} \<subseteq> set2 c" and 0: "u c b1 b2 = u c b1' b2'"
```
```  2649      hence "p1 (u c b1 b2) = b1 \<and> p2 (u c b1 b2) = b2 \<and>
```
```  2650             p1 (u c b1' b2') = b1' \<and> p2 (u c b1' b2') = b2'"
```
```  2651      using u(2)[OF c] u(3)[OF c] by simp metis
```
```  2652      thus "b1 = b1' \<and> b2 = b2'" using 0 by auto
```
```  2653    qed
```
```  2654   } note inj = this
```
```  2655   def sset \<equiv> "\<lambda> c. {u c b1 b2 | b1 b2. b1 \<in> set1 c \<and> b2 \<in> set2 c}"
```
```  2656   have fin_sset[simp]: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (sset c)" unfolding sset_def
```
```  2657     using fin_set1 fin_set2 finite_twosets by blast
```
```  2658   have sset_A: "\<And> c. c \<in> set_of P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
```
```  2659   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
```
```  2660    then obtain b1 b2 where b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
```
```  2661    and a: "a = u c b1 b2" unfolding sset_def by auto
```
```  2662    have "p1 a \<in> set1 c" and p2a: "p2 a \<in> set2 c"
```
```  2663    using ac a b1 b2 c u(2) u(3) by simp+
```
```  2664    hence "u c (p1 a) (p2 a) = a" unfolding a using b1 b2 inj[OF c]
```
```  2665    unfolding inj2_def by (metis c u(2) u(3))
```
```  2666   } note u_p12[simp] = this
```
```  2667   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
```
```  2668    hence "p1 a \<in> set1 c" unfolding sset_def by auto
```
```  2669   }note p1[simp] = this
```
```  2670   {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
```
```  2671    hence "p2 a \<in> set2 c" unfolding sset_def by auto
```
```  2672   }note p2[simp] = this
```
```  2673
```
```  2674   {fix c assume c: "c \<in> set_of P"
```
```  2675    hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = count N1 b1) \<and>
```
```  2676                (\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = count N2 b2)"
```
```  2677    unfolding sset_def
```
```  2678    using matrix_setsum_finite[OF set1_NE[OF c] fin_set1[OF c]
```
```  2679                                  set2_NE[OF c] fin_set2[OF c] inj[OF c] ss[OF c]] by auto
```
```  2680   }
```
```  2681   then obtain Ms where
```
```  2682   ss1: "\<And> c b1. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
```
```  2683                    setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = count N1 b1" and
```
```  2684   ss2: "\<And> c b2. \<lbrakk>c \<in> set_of P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
```
```  2685                    setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = count N2 b2"
```
```  2686   by metis
```
```  2687   def SET \<equiv> "\<Union> c \<in> set_of P. sset c"
```
```  2688   have fin_SET[simp]: "finite SET" unfolding SET_def apply(rule finite_UN_I) by auto
```
```  2689   have SET_A: "SET \<subseteq> A" unfolding SET_def using sset_A by blast
```
```  2690   have u_SET[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> SET"
```
```  2691     unfolding SET_def sset_def by blast
```
```  2692   {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
```
```  2693    then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
```
```  2694     unfolding SET_def by auto
```
```  2695    hence "p1 a \<in> set1 c'" unfolding sset_def by auto
```
```  2696    hence eq: "c = c'" using p1a c c' set1_disj by auto
```
```  2697    hence "a \<in> sset c" using ac' by simp
```
```  2698   } note p1_rev = this
```
```  2699   {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
```
```  2700    then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
```
```  2701    unfolding SET_def by auto
```
```  2702    hence "p2 a \<in> set2 c'" unfolding sset_def by auto
```
```  2703    hence eq: "c = c'" using p2a c c' set2_disj by auto
```
```  2704    hence "a \<in> sset c" using ac' by simp
```
```  2705   } note p2_rev = this
```
```  2706
```
```  2707   have "\<forall> a \<in> SET. \<exists> c \<in> set_of P. a \<in> sset c" unfolding SET_def by auto
```
```  2708   then obtain h where h: "\<forall> a \<in> SET. h a \<in> set_of P \<and> a \<in> sset (h a)" by metis
```
```  2709   have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
```
```  2710                       \<Longrightarrow> h (u c b1 b2) = c"
```
```  2711   by (metis h p2 set2 u(3) u_SET)
```
```  2712   have h_u1: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
```
```  2713                       \<Longrightarrow> h (u c b1 b2) = f1 b1"
```
```  2714   using h unfolding sset_def by auto
```
```  2715   have h_u2: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
```
```  2716                       \<Longrightarrow> h (u c b1 b2) = f2 b2"
```
```  2717   using h unfolding sset_def by auto
```
```  2718   def M \<equiv>
```
```  2719     "Abs_multiset (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0)"
```
```  2720   have "(\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) \<in> multiset"
```
```  2721     unfolding multiset_def by auto
```
```  2722   hence [transfer_rule]: "pcr_multiset op = (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) M"
```
```  2723     unfolding M_def pcr_multiset_def cr_multiset_def by (auto simp: Abs_multiset_inverse)
```
```  2724   have sM: "set_of M \<subseteq> SET" "set_of M \<subseteq> p1 -` (set_of N1)" "set_of M \<subseteq> p2 -` set_of N2"
```
```  2725     by (transfer, auto split: split_if_asm)+
```
```  2726   show "\<exists>M. set_of M \<subseteq> A \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
```
```  2727   proof(rule exI[of _ M], safe)
```
```  2728     fix a assume *: "a \<in> set_of M"
```
```  2729     from SET_A show "a \<in> A"
```
```  2730     proof (cases "a \<in> SET")
```
```  2731       case False thus ?thesis using * by transfer' auto
```
```  2732     qed blast
```
```  2733   next
```
```  2734     show "mmap p1 M = N1"
```
```  2735     proof(intro multiset_eqI)
```
```  2736       fix b1
```
```  2737       let ?K = "{a. p1 a = b1 \<and> a \<in># M}"
```
```  2738       have "setsum (count M) ?K = count N1 b1"
```
```  2739       proof(cases "b1 \<in> set_of N1")
```
```  2740         case False
```
```  2741         hence "?K = {}" using sM(2) by auto
```
```  2742         thus ?thesis using False by auto
```
```  2743       next
```
```  2744         case True
```
```  2745         def c \<equiv> "f1 b1"
```
```  2746         have c: "c \<in> set_of P" and b1: "b1 \<in> set1 c"
```
```  2747           unfolding set1_def c_def P1 using True by (auto simp: comp_eq_dest[OF set_of_mmap])
```
```  2748         with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p1 a = b1 \<and> a \<in> SET}"
```
```  2749           by transfer (force intro: setsum.mono_neutral_cong_left split: split_if_asm)
```
```  2750         also have "... = setsum (count M) ((\<lambda> b2. u c b1 b2) ` (set2 c))"
```
```  2751           apply(rule setsum.cong) using c b1 proof safe
```
```  2752           fix a assume p1a: "p1 a \<in> set1 c" and "c \<in> set_of P" and "a \<in> SET"
```
```  2753           hence ac: "a \<in> sset c" using p1_rev by auto
```
```  2754           hence "a = u c (p1 a) (p2 a)" using c by auto
```
```  2755           moreover have "p2 a \<in> set2 c" using ac c by auto
```
```  2756           ultimately show "a \<in> u c (p1 a) ` set2 c" by auto
```
```  2757         qed auto
```
```  2758         also have "... = setsum (\<lambda> b2. count M (u c b1 b2)) (set2 c)"
```
```  2759           unfolding comp_def[symmetric] apply(rule setsum.reindex)
```
```  2760           using inj unfolding inj_on_def inj2_def using b1 c u(3) by blast
```
```  2761         also have "... = count N1 b1" unfolding ss1[OF c b1, symmetric]
```
```  2762           apply(rule setsum.cong[OF refl]) apply (transfer fixing: Ms u c b1 set2)
```
```  2763           using True h_u[OF c b1] set2_def u(2,3)[OF c b1] u_SET[OF c b1]
```
```  2764             [[hypsubst_thin = true]]
```
```  2765           by fastforce
```
```  2766         finally show ?thesis .
```
```  2767       qed
```
```  2768       thus "count (mmap p1 M) b1 = count N1 b1" by transfer
```
```  2769     qed
```
```  2770   next
```
```  2771     show "mmap p2 M = N2"
```
```  2772     proof(intro multiset_eqI)
```
```  2773       fix b2
```
```  2774       let ?K = "{a. p2 a = b2 \<and> a \<in># M}"
```
```  2775       have "setsum (count M) ?K = count N2 b2"
```
```  2776       proof(cases "b2 \<in> set_of N2")
```
```  2777         case False
```
```  2778         hence "?K = {}" using sM(3) by auto
```
```  2779         thus ?thesis using False by auto
```
```  2780       next
```
```  2781         case True
```
```  2782         def c \<equiv> "f2 b2"
```
```  2783         have c: "c \<in> set_of P" and b2: "b2 \<in> set2 c"
```
```  2784           unfolding set2_def c_def P2 using True by (auto simp: comp_eq_dest[OF set_of_mmap])
```
```  2785         with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p2 a = b2 \<and> a \<in> SET}"
```
```  2786           by transfer (force intro: setsum.mono_neutral_cong_left split: split_if_asm)
```
```  2787         also have "... = setsum (count M) ((\<lambda> b1. u c b1 b2) ` (set1 c))"
```
```  2788           apply(rule setsum.cong) using c b2 proof safe
```
```  2789           fix a assume p2a: "p2 a \<in> set2 c" and "c \<in> set_of P" and "a \<in> SET"
```
```  2790           hence ac: "a \<in> sset c" using p2_rev by auto
```
```  2791           hence "a = u c (p1 a) (p2 a)" using c by auto
```
```  2792           moreover have "p1 a \<in> set1 c" using ac c by auto
```
```  2793           ultimately show "a \<in> (\<lambda>x. u c x (p2 a)) ` set1 c" by auto
```
```  2794         qed auto
```
```  2795         also have "... = setsum (count M o (\<lambda> b1. u c b1 b2)) (set1 c)"
```
```  2796           apply(rule setsum.reindex)
```
```  2797           using inj unfolding inj_on_def inj2_def using b2 c u(2) by blast
```
```  2798         also have "... = setsum (\<lambda> b1. count M (u c b1 b2)) (set1 c)" by simp
```
```  2799         also have "... = count N2 b2" unfolding ss2[OF c b2, symmetric] comp_def
```
```  2800           apply(rule setsum.cong[OF refl]) apply (transfer fixing: Ms u c b2 set1)
```
```  2801           using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2] set1_def
```
```  2802             [[hypsubst_thin = true]]
```
```  2803           by fastforce
```
```  2804         finally show ?thesis .
```
```  2805       qed
```
```  2806       thus "count (mmap p2 M) b2 = count N2 b2" by transfer
```
```  2807     qed
```
```  2808   qed
```
```  2809 qed
```
```  2810
```
```  2811 lemma set_of_bd: "(card_of (set_of x), natLeq) \<in> ordLeq"
```
```  2812   by transfer
```
```  2813     (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
```
```  2814
```
```  2815 lemma wppull_mmap:
```
```  2816   assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
```
```  2817   shows "wppull {M. set_of M \<subseteq> A} {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
```
```  2818     (mmap f1) (mmap f2) (mmap e1) (mmap e2) (mmap p1) (mmap p2)"
```
```  2819 proof -
```
```  2820   from assms obtain j where j: "\<forall>a'\<in>thePull B1 B2 f1 f2.
```
```  2821     j a' \<in> A \<and> e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
```
```  2822     by (blast dest: wppull_thePull)
```
```  2823   then show ?thesis
```
```  2824     by (intro wpull_wppull[OF wpull_mmap[OF wpull_thePull], of _ _ _ _ "mmap j"])
```
```  2825       (auto simp: comp_eq_dest_lhs[OF mmap_comp[symmetric]] comp_eq_dest[OF set_of_mmap]
```
```  2826         intro!: mmap_cong simp del: mem_set_of_iff simp: mem_set_of_iff[symmetric])
```
```  2827 qed
```
```  2828
```
```  2829 bnf "'a multiset"
```
```  2830   map: mmap
```
```  2831   sets: set_of
```
```  2832   bd: natLeq
```
```  2833   wits: "{#}"
```
```  2834 by (auto simp add: mmap_id0 mmap_comp set_of_mmap natLeq_card_order natLeq_cinfinite set_of_bd
```
```  2835   Grp_def relcompp.simps intro: mmap_cong)
```
```  2836   (metis wppull_mmap[OF wppull_fstOp_sndOp, unfolded wppull_def
```
```  2837     o_eq_dest_lhs[OF mmap_comp[symmetric]] fstOp_def sndOp_def comp_def, simplified])
```
```  2838
```
```  2839 inductive rel_multiset' where
```
```  2840   Zero[intro]: "rel_multiset' R {#} {#}"
```
```  2841 | Plus[intro]: "\<lbrakk>R a b; rel_multiset' R M N\<rbrakk> \<Longrightarrow> rel_multiset' R (M + {#a#}) (N + {#b#})"
```
```  2842
```
```  2843 lemma map_multiset_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"
```
```  2844 by (metis image_is_empty multiset.set_map set_of_eq_empty_iff)
```
```  2845
```
```  2846 lemma map_multiset_Zero[simp]: "mmap f {#} = {#}" by simp
```
```  2847
```
```  2848 lemma rel_multiset_Zero: "rel_multiset R {#} {#}"
```
```  2849 unfolding rel_multiset_def Grp_def by auto
```
```  2850
```
```  2851 declare multiset.count[simp]
```
```  2852 declare Abs_multiset_inverse[simp]
```
```  2853 declare multiset.count_inverse[simp]
```
```  2854 declare union_preserves_multiset[simp]
```
```  2855
```
```  2856 lemma map_multiset_Plus[simp]: "mmap f (M1 + M2) = mmap f M1 + mmap f M2"
```
```  2857 proof (intro multiset_eqI, transfer fixing: f)
```
```  2858   fix x :: 'a and M1 M2 :: "'b \<Rightarrow> nat"
```
```  2859   assume "M1 \<in> multiset" "M2 \<in> multiset"
```
```  2860   hence "setsum M1 {a. f a = x \<and> 0 < M1 a} = setsum M1 {a. f a = x \<and> 0 < M1 a + M2 a}"
```
```  2861         "setsum M2 {a. f a = x \<and> 0 < M2 a} = setsum M2 {a. f a = x \<and> 0 < M1 a + M2 a}"
```
```  2862     by (auto simp: multiset_def intro!: setsum.mono_neutral_cong_left)
```
```  2863   then show "(\<Sum>a | f a = x \<and> 0 < M1 a + M2 a. M1 a + M2 a) =
```
```  2864        setsum M1 {a. f a = x \<and> 0 < M1 a} +
```
```  2865        setsum M2 {a. f a = x \<and> 0 < M2 a}"
```
```  2866     by (auto simp: setsum.distrib[symmetric])
```
```  2867 qed
```
```  2868
```
```  2869 lemma map_multiset_single[simp]: "mmap f {#a#} = {#f a#}"
```
```  2870   by transfer auto
```
```  2871
```
```  2872 lemma rel_multiset_Plus:
```
```  2873 assumes ab: "R a b" and MN: "rel_multiset R M N"
```
```  2874 shows "rel_multiset R (M + {#a#}) (N + {#b#})"
```
```  2875 proof-
```
```  2876   {fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
```
```  2877    hence "\<exists>ya. mmap fst y + {#a#} = mmap fst ya \<and>
```
```  2878                mmap snd y + {#b#} = mmap snd ya \<and>
```
```  2879                set_of ya \<subseteq> {(x, y). R x y}"
```
```  2880    apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
```
```  2881   }
```
```  2882   thus ?thesis
```
```  2883   using assms
```
```  2884   unfolding rel_multiset_def Grp_def by force
```
```  2885 qed
```
```  2886
```
```  2887 lemma rel_multiset'_imp_rel_multiset:
```
```  2888 "rel_multiset' R M N \<Longrightarrow> rel_multiset R M N"
```
```  2889 apply(induct rule: rel_multiset'.induct)
```
```  2890 using rel_multiset_Zero rel_multiset_Plus by auto
```
```  2891
```
```  2892 lemma mcard_mmap[simp]: "mcard (mmap f M) = mcard M"
```
```  2893 proof -
```
```  2894   def A \<equiv> "\<lambda> b. {a. f a = b \<and> a \<in># M}"
```
```  2895   let ?B = "{b. 0 < setsum (count M) (A b)}"
```
```  2896   have "{b. \<exists>a. f a = b \<and> a \<in># M} \<subseteq> f ` {a. a \<in># M}" by auto
```
```  2897   moreover have "finite (f ` {a. a \<in># M})" apply(rule finite_imageI)
```
```  2898   using finite_Collect_mem .
```
```  2899   ultimately have fin: "finite {b. \<exists>a. f a = b \<and> a \<in># M}" by(rule finite_subset)
```
```  2900   have i: "inj_on A ?B" unfolding inj_on_def A_def apply clarsimp
```
```  2901     by (metis (lifting, full_types) mem_Collect_eq neq0_conv setsum.neutral)
```
```  2902   have 0: "\<And> b. 0 < setsum (count M) (A b) \<longleftrightarrow> (\<exists> a \<in> A b. count M a > 0)"
```
```  2903   apply safe
```
```  2904     apply (metis less_not_refl setsum_gt_0_iff setsum.infinite)
```
```  2905     by (metis A_def finite_Collect_conjI finite_Collect_mem setsum_gt_0_iff)
```
```  2906   hence AB: "A ` ?B = {A b | b. \<exists> a \<in> A b. count M a > 0}" by auto
```
```  2907
```
```  2908   have "setsum (\<lambda> x. setsum (count M) (A x)) ?B = setsum (setsum (count M) o A) ?B"
```
```  2909   unfolding comp_def ..
```
```  2910   also have "... = (\<Sum>x\<in> A ` ?B. setsum (count M) x)"
```
```  2911   unfolding setsum.reindex [OF i, symmetric] ..
```
```  2912   also have "... = setsum (count M) (\<Union>x\<in>A ` {b. 0 < setsum (count M) (A b)}. x)"
```
```  2913   (is "_ = setsum (count M) ?J")
```
```  2914   apply(rule setsum.UNION_disjoint[symmetric])
```
```  2915   using 0 fin unfolding A_def by auto
```
```  2916   also have "?J = {a. a \<in># M}" unfolding AB unfolding A_def by auto
```
```  2917   finally have "setsum (\<lambda> x. setsum (count M) (A x)) ?B =
```
```  2918                 setsum (count M) {a. a \<in># M}" .
```
```  2919   then show ?thesis unfolding mcard_unfold_setsum A_def by transfer
```
```  2920 qed
```
```  2921
```
```  2922 lemma rel_multiset_mcard:
```
```  2923 assumes "rel_multiset R M N"
```
```  2924 shows "mcard M = mcard N"
```
```  2925 using assms unfolding rel_multiset_def Grp_def by auto
```
```  2926
```
```  2927 lemma multiset_induct2[case_names empty addL addR]:
```
```  2928 assumes empty: "P {#} {#}"
```
```  2929 and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
```
```  2930 and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
```
```  2931 shows "P M N"
```
```  2932 apply(induct N rule: multiset_induct)
```
```  2933   apply(induct M rule: multiset_induct, rule empty, erule addL)
```
```  2934   apply(induct M rule: multiset_induct, erule addR, erule addR)
```
```  2935 done
```
```  2936
```
```  2937 lemma multiset_induct2_mcard[consumes 1, case_names empty add]:
```
```  2938 assumes c: "mcard M = mcard N"
```
```  2939 and empty: "P {#} {#}"
```
```  2940 and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
```
```  2941 shows "P M N"
```
```  2942 using c proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
```
```  2943   case (less M)  show ?case
```
```  2944   proof(cases "M = {#}")
```
```  2945     case True hence "N = {#}" using less.prems by auto
```
```  2946     thus ?thesis using True empty by auto
```
```  2947   next
```
```  2948     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
```
```  2949     have "N \<noteq> {#}" using False less.prems by auto
```
```  2950     then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
```
```  2951     have "mcard M1 = mcard N1" using less.prems unfolding M N by auto
```
```  2952     thus ?thesis using M N less.hyps add by auto
```
```  2953   qed
```
```  2954 qed
```
```  2955
```
```  2956 lemma msed_map_invL:
```
```  2957 assumes "mmap f (M + {#a#}) = N"
```
```  2958 shows "\<exists> N1. N = N1 + {#f a#} \<and> mmap f M = N1"
```
```  2959 proof-
```
```  2960   have "f a \<in># N"
```
```  2961   using assms multiset.set_map[of f "M + {#a#}"] by auto
```
```  2962   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
```
```  2963   have "mmap f M = N1" using assms unfolding N by simp
```
```  2964   thus ?thesis using N by blast
```
```  2965 qed
```
```  2966
```
```  2967 lemma msed_map_invR:
```
```  2968 assumes "mmap f M = N + {#b#}"
```
```  2969 shows "\<exists> M1 a. M = M1 + {#a#} \<and> f a = b \<and> mmap f M1 = N"
```
```  2970 proof-
```
```  2971   obtain a where a: "a \<in># M" and fa: "f a = b"
```
```  2972   using multiset.set_map[of f M] unfolding assms
```
```  2973   by (metis image_iff mem_set_of_iff union_single_eq_member)
```
```  2974   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
```
```  2975   have "mmap f M1 = N" using assms unfolding M fa[symmetric] by simp
```
```  2976   thus ?thesis using M fa by blast
```
```  2977 qed
```
```  2978
```
```  2979 lemma msed_rel_invL:
```
```  2980 assumes "rel_multiset R (M + {#a#}) N"
```
```  2981 shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_multiset R M N1"
```
```  2982 proof-
```
```  2983   obtain K where KM: "mmap fst K = M + {#a#}"
```
```  2984   and KN: "mmap snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
```
```  2985   using assms
```
```  2986   unfolding rel_multiset_def Grp_def by auto
```
```  2987   obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
```
```  2988   and K1M: "mmap fst K1 = M" using msed_map_invR[OF KM] by auto
```
```  2989   obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "mmap snd K1 = N1"
```
```  2990   using msed_map_invL[OF KN[unfolded K]] by auto
```
```  2991   have Rab: "R a (snd ab)" using sK a unfolding K by auto
```
```  2992   have "rel_multiset R M N1" using sK K1M K1N1
```
```  2993   unfolding K rel_multiset_def Grp_def by auto
```
```  2994   thus ?thesis using N Rab by auto
```
```  2995 qed
```
```  2996
```
```  2997 lemma msed_rel_invR:
```
```  2998 assumes "rel_multiset R M (N + {#b#})"
```
```  2999 shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_multiset R M1 N"
```
```  3000 proof-
```
```  3001   obtain K where KN: "mmap snd K = N + {#b#}"
```
```  3002   and KM: "mmap fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
```
```  3003   using assms
```
```  3004   unfolding rel_multiset_def Grp_def by auto
```
```  3005   obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
```
```  3006   and K1N: "mmap snd K1 = N" using msed_map_invR[OF KN] by auto
```
```  3007   obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "mmap fst K1 = M1"
```
```  3008   using msed_map_invL[OF KM[unfolded K]] by auto
```
```  3009   have Rab: "R (fst ab) b" using sK b unfolding K by auto
```
```  3010   have "rel_multiset R M1 N" using sK K1N K1M1
```
```  3011   unfolding K rel_multiset_def Grp_def by auto
```
```  3012   thus ?thesis using M Rab by auto
```
```  3013 qed
```
```  3014
```
```  3015 lemma rel_multiset_imp_rel_multiset':
```
```  3016 assumes "rel_multiset R M N"
```
```  3017 shows "rel_multiset' R M N"
```
```  3018 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of mcard])
```
```  3019   case (less M)
```
```  3020   have c: "mcard M = mcard N" using rel_multiset_mcard[OF less.prems] .
```
```  3021   show ?case
```
```  3022   proof(cases "M = {#}")
```
```  3023     case True hence "N = {#}" using c by simp
```
```  3024     thus ?thesis using True rel_multiset'.Zero by auto
```
```  3025   next
```
```  3026     case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
```
```  3027     obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_multiset R M1 N1"
```
```  3028     using msed_rel_invL[OF less.prems[unfolded M]] by auto
```
```  3029     have "rel_multiset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
```
```  3030     thus ?thesis using rel_multiset'.Plus[of R a b, OF R] unfolding M N by simp
```
```  3031   qed
```
```  3032 qed
```
```  3033
```
```  3034 lemma rel_multiset_rel_multiset':
```
```  3035 "rel_multiset R M N = rel_multiset' R M N"
```
```  3036 using  rel_multiset_imp_rel_multiset' rel_multiset'_imp_rel_multiset by auto
```
```  3037
```
```  3038 (* The main end product for rel_multiset: inductive characterization *)
```
```  3039 theorems rel_multiset_induct[case_names empty add, induct pred: rel_multiset] =
```
```  3040          rel_multiset'.induct[unfolded rel_multiset_rel_multiset'[symmetric]]
```
```  3041
```
```  3042
```
```  3043 subsection {* Size setup *}
```
```  3044
```
```  3045 lemma multiset_size_o_map: "size_multiset g \<circ> mmap f = size_multiset (g \<circ> f)"
```
```  3046 apply (rule ext)
```
```  3047 apply (unfold o_apply)
```
```  3048 apply (induct_tac x)
```
```  3049 apply auto
```
```  3050 done
```
```  3051
```
```  3052 setup {*
```
```  3053 BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
```
```  3054   @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
```
```  3055     size_union}
```
```  3056   @{thms multiset_size_o_map}
```
```  3057 *}
```
```  3058
```
```  3059 hide_const (open) wcount
```
```  3060
```
```  3061 end
```