src/HOL/Library/Multiset.thy
 author bulwahn Wed Mar 28 10:16:02 2012 +0200 (2012-03-28) changeset 47179 54b38de0620e parent 47177 2fa00264392a child 47198 cfd8ff62eab1 permissions -rw-r--r--
changing more definitions to quotient_definition
```     1 (*  Title:      HOL/Library/Multiset.thy
```
```     2     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
```
```     3 *)
```
```     4
```
```     5 header {* (Finite) multisets *}
```
```     6
```
```     7 theory Multiset
```
```     8 imports Main DAList
```
```     9 begin
```
```    10
```
```    11 subsection {* The type of multisets *}
```
```    12
```
```    13 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
```
```    14
```
```    15 typedef (open) 'a multiset = "multiset :: ('a => nat) set"
```
```    16   morphisms count Abs_multiset
```
```    17   unfolding multiset_def
```
```    18 proof
```
```    19   show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
```
```    20 qed
```
```    21
```
```    22 lemmas multiset_typedef = Abs_multiset_inverse count_inverse count
```
```    23
```
```    24 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
```
```    25   "a :# M == 0 < count M a"
```
```    26
```
```    27 notation (xsymbols)
```
```    28   Melem (infix "\<in>#" 50)
```
```    29
```
```    30 lemma multiset_eq_iff:
```
```    31   "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
```
```    32   by (simp only: count_inject [symmetric] fun_eq_iff)
```
```    33
```
```    34 lemma multiset_eqI:
```
```    35   "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
```
```    36   using multiset_eq_iff by auto
```
```    37
```
```    38 text {*
```
```    39  \medskip Preservation of the representing set @{term multiset}.
```
```    40 *}
```
```    41
```
```    42 lemma const0_in_multiset:
```
```    43   "(\<lambda>a. 0) \<in> multiset"
```
```    44   by (simp add: multiset_def)
```
```    45
```
```    46 lemma only1_in_multiset:
```
```    47   "(\<lambda>b. if b = a then n else 0) \<in> multiset"
```
```    48   by (simp add: multiset_def)
```
```    49
```
```    50 lemma union_preserves_multiset:
```
```    51   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
```
```    52   by (simp add: multiset_def)
```
```    53
```
```    54 lemma diff_preserves_multiset:
```
```    55   assumes "M \<in> multiset"
```
```    56   shows "(\<lambda>a. M a - N a) \<in> multiset"
```
```    57 proof -
```
```    58   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
```
```    59     by auto
```
```    60   with assms show ?thesis
```
```    61     by (auto simp add: multiset_def intro: finite_subset)
```
```    62 qed
```
```    63
```
```    64 lemma filter_preserves_multiset:
```
```    65   assumes "M \<in> multiset"
```
```    66   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
```
```    67 proof -
```
```    68   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
```
```    69     by auto
```
```    70   with assms show ?thesis
```
```    71     by (auto simp add: multiset_def intro: finite_subset)
```
```    72 qed
```
```    73
```
```    74 lemmas in_multiset = const0_in_multiset only1_in_multiset
```
```    75   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
```
```    76
```
```    77
```
```    78 subsection {* Representing multisets *}
```
```    79
```
```    80 text {* Multiset enumeration *}
```
```    81
```
```    82 instantiation multiset :: (type) "{zero, plus}"
```
```    83 begin
```
```    84
```
```    85 definition Mempty_def:
```
```    86   "0 = Abs_multiset (\<lambda>a. 0)"
```
```    87
```
```    88 abbreviation Mempty :: "'a multiset" ("{#}") where
```
```    89   "Mempty \<equiv> 0"
```
```    90
```
```    91 definition union_def:
```
```    92   "M + N = Abs_multiset (\<lambda>a. count M a + count N a)"
```
```    93
```
```    94 instance ..
```
```    95
```
```    96 end
```
```    97
```
```    98 definition single :: "'a => 'a multiset" where
```
```    99   "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
```
```   100
```
```   101 syntax
```
```   102   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
```
```   103 translations
```
```   104   "{#x, xs#}" == "{#x#} + {#xs#}"
```
```   105   "{#x#}" == "CONST single x"
```
```   106
```
```   107 lemma count_empty [simp]: "count {#} a = 0"
```
```   108   by (simp add: Mempty_def in_multiset multiset_typedef)
```
```   109
```
```   110 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   111   by (simp add: single_def in_multiset multiset_typedef)
```
```   112
```
```   113
```
```   114 subsection {* Basic operations *}
```
```   115
```
```   116 subsubsection {* Union *}
```
```   117
```
```   118 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   119   by (simp add: union_def in_multiset multiset_typedef)
```
```   120
```
```   121 instance multiset :: (type) cancel_comm_monoid_add
```
```   122   by default (simp_all add: multiset_eq_iff)
```
```   123
```
```   124
```
```   125 subsubsection {* Difference *}
```
```   126
```
```   127 instantiation multiset :: (type) minus
```
```   128 begin
```
```   129
```
```   130 definition diff_def:
```
```   131   "M - N = Abs_multiset (\<lambda>a. count M a - count N a)"
```
```   132
```
```   133 instance ..
```
```   134
```
```   135 end
```
```   136
```
```   137 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   138   by (simp add: diff_def in_multiset multiset_typedef)
```
```   139
```
```   140 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   141 by(simp add: multiset_eq_iff)
```
```   142
```
```   143 lemma diff_cancel[simp]: "A - A = {#}"
```
```   144 by (rule multiset_eqI) simp
```
```   145
```
```   146 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
```
```   147 by(simp add: multiset_eq_iff)
```
```   148
```
```   149 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
```
```   150 by(simp add: multiset_eq_iff)
```
```   151
```
```   152 lemma insert_DiffM:
```
```   153   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
```
```   154   by (clarsimp simp: multiset_eq_iff)
```
```   155
```
```   156 lemma insert_DiffM2 [simp]:
```
```   157   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
```
```   158   by (clarsimp simp: multiset_eq_iff)
```
```   159
```
```   160 lemma diff_right_commute:
```
```   161   "(M::'a multiset) - N - Q = M - Q - N"
```
```   162   by (auto simp add: multiset_eq_iff)
```
```   163
```
```   164 lemma diff_add:
```
```   165   "(M::'a multiset) - (N + Q) = M - N - Q"
```
```   166 by (simp add: 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
```
```   262 subsubsection {* Pointwise ordering induced by count *}
```
```   263
```
```   264 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
```
```   265 begin
```
```   266
```
```   267 definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
```
```   268   mset_le_def: "A \<le> B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
```
```   269
```
```   270 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
```
```   271   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
```
```   272
```
```   273 instance
```
```   274   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
```
```   275
```
```   276 end
```
```   277
```
```   278 lemma mset_less_eqI:
```
```   279   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
```
```   280   by (simp add: mset_le_def)
```
```   281
```
```   282 lemma mset_le_exists_conv:
```
```   283   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
```
```   284 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
```
```   285 apply (auto intro: multiset_eq_iff [THEN iffD2])
```
```   286 done
```
```   287
```
```   288 lemma mset_le_mono_add_right_cancel [simp]:
```
```   289   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
```
```   290   by (fact add_le_cancel_right)
```
```   291
```
```   292 lemma mset_le_mono_add_left_cancel [simp]:
```
```   293   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
```
```   294   by (fact add_le_cancel_left)
```
```   295
```
```   296 lemma mset_le_mono_add:
```
```   297   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
```
```   298   by (fact add_mono)
```
```   299
```
```   300 lemma mset_le_add_left [simp]:
```
```   301   "(A::'a multiset) \<le> A + B"
```
```   302   unfolding mset_le_def by auto
```
```   303
```
```   304 lemma mset_le_add_right [simp]:
```
```   305   "B \<le> (A::'a multiset) + B"
```
```   306   unfolding mset_le_def by auto
```
```   307
```
```   308 lemma mset_le_single:
```
```   309   "a :# B \<Longrightarrow> {#a#} \<le> B"
```
```   310   by (simp add: mset_le_def)
```
```   311
```
```   312 lemma multiset_diff_union_assoc:
```
```   313   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
```
```   314   by (simp add: multiset_eq_iff mset_le_def)
```
```   315
```
```   316 lemma mset_le_multiset_union_diff_commute:
```
```   317   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
```
```   318 by (simp add: multiset_eq_iff mset_le_def)
```
```   319
```
```   320 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
```
```   321 by(simp add: mset_le_def)
```
```   322
```
```   323 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   324 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   325 apply (erule_tac x=x in allE)
```
```   326 apply auto
```
```   327 done
```
```   328
```
```   329 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   330 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   331 apply (erule_tac x = x in allE)
```
```   332 apply auto
```
```   333 done
```
```   334
```
```   335 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
```
```   336 apply (rule conjI)
```
```   337  apply (simp add: mset_lessD)
```
```   338 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   339 apply safe
```
```   340  apply (erule_tac x = a in allE)
```
```   341  apply (auto split: split_if_asm)
```
```   342 done
```
```   343
```
```   344 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
```
```   345 apply (rule conjI)
```
```   346  apply (simp add: mset_leD)
```
```   347 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
```
```   348 done
```
```   349
```
```   350 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
```
```   351   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
```
```   352
```
```   353 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
```
```   354   by (auto simp: mset_le_def mset_less_def)
```
```   355
```
```   356 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
```
```   357   by simp
```
```   358
```
```   359 lemma mset_less_add_bothsides:
```
```   360   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
```
```   361   by (fact add_less_imp_less_right)
```
```   362
```
```   363 lemma mset_less_empty_nonempty:
```
```   364   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
```
```   365   by (auto simp: mset_le_def mset_less_def)
```
```   366
```
```   367 lemma mset_less_diff_self:
```
```   368   "c \<in># B \<Longrightarrow> B - {#c#} < B"
```
```   369   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
```
```   370
```
```   371
```
```   372 subsubsection {* Intersection *}
```
```   373
```
```   374 instantiation multiset :: (type) semilattice_inf
```
```   375 begin
```
```   376
```
```   377 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   378   multiset_inter_def: "inf_multiset A B = A - (A - B)"
```
```   379
```
```   380 instance
```
```   381 proof -
```
```   382   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
```
```   383   show "OFCLASS('a multiset, semilattice_inf_class)"
```
```   384     by default (auto simp add: multiset_inter_def mset_le_def aux)
```
```   385 qed
```
```   386
```
```   387 end
```
```   388
```
```   389 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
```
```   390   "multiset_inter \<equiv> inf"
```
```   391
```
```   392 lemma multiset_inter_count [simp]:
```
```   393   "count (A #\<inter> B) x = min (count A x) (count B x)"
```
```   394   by (simp add: multiset_inter_def multiset_typedef)
```
```   395
```
```   396 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
```
```   397   by (rule multiset_eqI) auto
```
```   398
```
```   399 lemma multiset_union_diff_commute:
```
```   400   assumes "B #\<inter> C = {#}"
```
```   401   shows "A + B - C = A - C + B"
```
```   402 proof (rule multiset_eqI)
```
```   403   fix x
```
```   404   from assms have "min (count B x) (count C x) = 0"
```
```   405     by (auto simp add: multiset_eq_iff)
```
```   406   then have "count B x = 0 \<or> count C x = 0"
```
```   407     by auto
```
```   408   then show "count (A + B - C) x = count (A - C + B) x"
```
```   409     by auto
```
```   410 qed
```
```   411
```
```   412
```
```   413 subsubsection {* Filter (with comprehension syntax) *}
```
```   414
```
```   415 text {* Multiset comprehension *}
```
```   416
```
```   417 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   418   "filter P M = Abs_multiset (\<lambda>x. if P x then count M x else 0)"
```
```   419
```
```   420 hide_const (open) filter
```
```   421
```
```   422 lemma count_filter [simp]:
```
```   423   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
```
```   424   by (simp add: filter_def in_multiset multiset_typedef)
```
```   425
```
```   426 lemma filter_empty [simp]:
```
```   427   "Multiset.filter P {#} = {#}"
```
```   428   by (rule multiset_eqI) simp
```
```   429
```
```   430 lemma filter_single [simp]:
```
```   431   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
```
```   432   by (rule multiset_eqI) simp
```
```   433
```
```   434 lemma filter_union [simp]:
```
```   435   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
```
```   436   by (rule multiset_eqI) simp
```
```   437
```
```   438 lemma filter_diff [simp]:
```
```   439   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
```
```   440   by (rule multiset_eqI) simp
```
```   441
```
```   442 lemma filter_inter [simp]:
```
```   443   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
```
```   444   by (rule multiset_eqI) simp
```
```   445
```
```   446 syntax
```
```   447   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
```
```   448 syntax (xsymbol)
```
```   449   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
```
```   450 translations
```
```   451   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
```
```   452
```
```   453
```
```   454 subsubsection {* Set of elements *}
```
```   455
```
```   456 definition set_of :: "'a multiset => 'a set" where
```
```   457   "set_of M = {x. x :# M}"
```
```   458
```
```   459 lemma set_of_empty [simp]: "set_of {#} = {}"
```
```   460 by (simp add: set_of_def)
```
```   461
```
```   462 lemma set_of_single [simp]: "set_of {#b#} = {b}"
```
```   463 by (simp add: set_of_def)
```
```   464
```
```   465 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   466 by (auto simp add: set_of_def)
```
```   467
```
```   468 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   469 by (auto simp add: set_of_def multiset_eq_iff)
```
```   470
```
```   471 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   472 by (auto simp add: set_of_def)
```
```   473
```
```   474 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
```
```   475 by (auto simp add: set_of_def)
```
```   476
```
```   477 lemma finite_set_of [iff]: "finite (set_of M)"
```
```   478   using count [of M] by (simp add: multiset_def set_of_def)
```
```   479
```
```   480 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
```
```   481   unfolding set_of_def[symmetric] by simp
```
```   482
```
```   483 subsubsection {* Size *}
```
```   484
```
```   485 instantiation multiset :: (type) size
```
```   486 begin
```
```   487
```
```   488 definition size_def:
```
```   489   "size M = setsum (count M) (set_of M)"
```
```   490
```
```   491 instance ..
```
```   492
```
```   493 end
```
```   494
```
```   495 lemma size_empty [simp]: "size {#} = 0"
```
```   496 by (simp add: size_def)
```
```   497
```
```   498 lemma size_single [simp]: "size {#b#} = 1"
```
```   499 by (simp add: size_def)
```
```   500
```
```   501 lemma setsum_count_Int:
```
```   502   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
```
```   503 apply (induct rule: finite_induct)
```
```   504  apply simp
```
```   505 apply (simp add: Int_insert_left set_of_def)
```
```   506 done
```
```   507
```
```   508 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   509 apply (unfold size_def)
```
```   510 apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
```
```   511  prefer 2
```
```   512  apply (rule ext, simp)
```
```   513 apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
```
```   514 apply (subst Int_commute)
```
```   515 apply (simp (no_asm_simp) add: setsum_count_Int)
```
```   516 done
```
```   517
```
```   518 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   519 by (auto simp add: size_def multiset_eq_iff)
```
```   520
```
```   521 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
```
```   522 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
```
```   523
```
```   524 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   525 apply (unfold size_def)
```
```   526 apply (drule setsum_SucD)
```
```   527 apply auto
```
```   528 done
```
```   529
```
```   530 lemma size_eq_Suc_imp_eq_union:
```
```   531   assumes "size M = Suc n"
```
```   532   shows "\<exists>a N. M = N + {#a#}"
```
```   533 proof -
```
```   534   from assms obtain a where "a \<in># M"
```
```   535     by (erule size_eq_Suc_imp_elem [THEN exE])
```
```   536   then have "M = M - {#a#} + {#a#}" by simp
```
```   537   then show ?thesis by blast
```
```   538 qed
```
```   539
```
```   540
```
```   541 subsection {* Induction and case splits *}
```
```   542
```
```   543 lemma setsum_decr:
```
```   544   "finite F ==> (0::nat) < f a ==>
```
```   545     setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
```
```   546 apply (induct rule: finite_induct)
```
```   547  apply auto
```
```   548 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   549 done
```
```   550
```
```   551 lemma rep_multiset_induct_aux:
```
```   552 assumes 1: "P (\<lambda>a. (0::nat))"
```
```   553   and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
```
```   554 shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
```
```   555 apply (unfold multiset_def)
```
```   556 apply (induct_tac n, simp, clarify)
```
```   557  apply (subgoal_tac "f = (\<lambda>a.0)")
```
```   558   apply simp
```
```   559   apply (rule 1)
```
```   560  apply (rule ext, force, clarify)
```
```   561 apply (frule setsum_SucD, clarify)
```
```   562 apply (rename_tac a)
```
```   563 apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
```
```   564  prefer 2
```
```   565  apply (rule finite_subset)
```
```   566   prefer 2
```
```   567   apply assumption
```
```   568  apply simp
```
```   569  apply blast
```
```   570 apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
```
```   571  prefer 2
```
```   572  apply (rule ext)
```
```   573  apply (simp (no_asm_simp))
```
```   574  apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
```
```   575 apply (erule allE, erule impE, erule_tac  mp, blast)
```
```   576 apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
```
```   577 apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
```
```   578  prefer 2
```
```   579  apply blast
```
```   580 apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
```
```   581  prefer 2
```
```   582  apply blast
```
```   583 apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
```
```   584 done
```
```   585
```
```   586 theorem rep_multiset_induct:
```
```   587   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
```
```   588     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
```
```   589 using rep_multiset_induct_aux by blast
```
```   590
```
```   591 theorem multiset_induct [case_names empty add, induct type: multiset]:
```
```   592 assumes empty: "P {#}"
```
```   593   and add: "!!M x. P M ==> P (M + {#x#})"
```
```   594 shows "P M"
```
```   595 proof -
```
```   596   note defns = union_def single_def Mempty_def
```
```   597   note add' = add [unfolded defns, simplified]
```
```   598   have aux: "\<And>a::'a. count (Abs_multiset (\<lambda>b. if b = a then 1 else 0)) =
```
```   599     (\<lambda>b. if b = a then 1 else 0)" by (simp add: Abs_multiset_inverse in_multiset)
```
```   600   show ?thesis
```
```   601     apply (rule count_inverse [THEN subst])
```
```   602     apply (rule count [THEN rep_multiset_induct])
```
```   603      apply (rule empty [unfolded defns])
```
```   604     apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
```
```   605      prefer 2
```
```   606      apply (simp add: fun_eq_iff)
```
```   607     apply (erule ssubst)
```
```   608     apply (erule Abs_multiset_inverse [THEN subst])
```
```   609     apply (drule add')
```
```   610     apply (simp add: aux)
```
```   611     done
```
```   612 qed
```
```   613
```
```   614 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
```
```   615 by (induct M) auto
```
```   616
```
```   617 lemma multiset_cases [cases type, case_names empty add]:
```
```   618 assumes em:  "M = {#} \<Longrightarrow> P"
```
```   619 assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
```
```   620 shows "P"
```
```   621 proof (cases "M = {#}")
```
```   622   assume "M = {#}" then show ?thesis using em by simp
```
```   623 next
```
```   624   assume "M \<noteq> {#}"
```
```   625   then obtain M' m where "M = M' + {#m#}"
```
```   626     by (blast dest: multi_nonempty_split)
```
```   627   then show ?thesis using add by simp
```
```   628 qed
```
```   629
```
```   630 lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
```
```   631 apply (cases M)
```
```   632  apply simp
```
```   633 apply (rule_tac x="M - {#x#}" in exI, simp)
```
```   634 done
```
```   635
```
```   636 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
```
```   637 by (cases "B = {#}") (auto dest: multi_member_split)
```
```   638
```
```   639 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
```
```   640 apply (subst multiset_eq_iff)
```
```   641 apply auto
```
```   642 done
```
```   643
```
```   644 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
```
```   645 proof (induct A arbitrary: B)
```
```   646   case (empty M)
```
```   647   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
```
```   648   then obtain M' x where "M = M' + {#x#}"
```
```   649     by (blast dest: multi_nonempty_split)
```
```   650   then show ?case by simp
```
```   651 next
```
```   652   case (add S x T)
```
```   653   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
```
```   654   have SxsubT: "S + {#x#} < T" by fact
```
```   655   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
```
```   656   then obtain T' where T: "T = T' + {#x#}"
```
```   657     by (blast dest: multi_member_split)
```
```   658   then have "S < T'" using SxsubT
```
```   659     by (blast intro: mset_less_add_bothsides)
```
```   660   then have "size S < size T'" using IH by simp
```
```   661   then show ?case using T by simp
```
```   662 qed
```
```   663
```
```   664
```
```   665 subsubsection {* Strong induction and subset induction for multisets *}
```
```   666
```
```   667 text {* Well-foundedness of proper subset operator: *}
```
```   668
```
```   669 text {* proper multiset subset *}
```
```   670
```
```   671 definition
```
```   672   mset_less_rel :: "('a multiset * 'a multiset) set" where
```
```   673   "mset_less_rel = {(A,B). A < B}"
```
```   674
```
```   675 lemma multiset_add_sub_el_shuffle:
```
```   676   assumes "c \<in># B" and "b \<noteq> c"
```
```   677   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
```
```   678 proof -
```
```   679   from `c \<in># B` obtain A where B: "B = A + {#c#}"
```
```   680     by (blast dest: multi_member_split)
```
```   681   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
```
```   682   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
```
```   683     by (simp add: add_ac)
```
```   684   then show ?thesis using B by simp
```
```   685 qed
```
```   686
```
```   687 lemma wf_mset_less_rel: "wf mset_less_rel"
```
```   688 apply (unfold mset_less_rel_def)
```
```   689 apply (rule wf_measure [THEN wf_subset, where f1=size])
```
```   690 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
```
```   691 done
```
```   692
```
```   693 text {* The induction rules: *}
```
```   694
```
```   695 lemma full_multiset_induct [case_names less]:
```
```   696 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
```
```   697 shows "P B"
```
```   698 apply (rule wf_mset_less_rel [THEN wf_induct])
```
```   699 apply (rule ih, auto simp: mset_less_rel_def)
```
```   700 done
```
```   701
```
```   702 lemma multi_subset_induct [consumes 2, case_names empty add]:
```
```   703 assumes "F \<le> A"
```
```   704   and empty: "P {#}"
```
```   705   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
```
```   706 shows "P F"
```
```   707 proof -
```
```   708   from `F \<le> A`
```
```   709   show ?thesis
```
```   710   proof (induct F)
```
```   711     show "P {#}" by fact
```
```   712   next
```
```   713     fix x F
```
```   714     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
```
```   715     show "P (F + {#x#})"
```
```   716     proof (rule insert)
```
```   717       from i show "x \<in># A" by (auto dest: mset_le_insertD)
```
```   718       from i have "F \<le> A" by (auto dest: mset_le_insertD)
```
```   719       with P show "P F" .
```
```   720     qed
```
```   721   qed
```
```   722 qed
```
```   723
```
```   724
```
```   725 subsection {* Alternative representations *}
```
```   726
```
```   727 subsubsection {* Lists *}
```
```   728
```
```   729 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
```
```   730   "multiset_of [] = {#}" |
```
```   731   "multiset_of (a # x) = multiset_of x + {# a #}"
```
```   732
```
```   733 lemma in_multiset_in_set:
```
```   734   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```   735   by (induct xs) simp_all
```
```   736
```
```   737 lemma count_multiset_of:
```
```   738   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```   739   by (induct xs) simp_all
```
```   740
```
```   741 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
```
```   742 by (induct x) auto
```
```   743
```
```   744 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
```
```   745 by (induct x) auto
```
```   746
```
```   747 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
```
```   748 by (induct x) auto
```
```   749
```
```   750 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
```
```   751 by (induct xs) auto
```
```   752
```
```   753 lemma multiset_of_append [simp]:
```
```   754   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
```
```   755   by (induct xs arbitrary: ys) (auto simp: add_ac)
```
```   756
```
```   757 lemma multiset_of_filter:
```
```   758   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
```
```   759   by (induct xs) simp_all
```
```   760
```
```   761 lemma multiset_of_rev [simp]:
```
```   762   "multiset_of (rev xs) = multiset_of xs"
```
```   763   by (induct xs) simp_all
```
```   764
```
```   765 lemma surj_multiset_of: "surj multiset_of"
```
```   766 apply (unfold surj_def)
```
```   767 apply (rule allI)
```
```   768 apply (rule_tac M = y in multiset_induct)
```
```   769  apply auto
```
```   770 apply (rule_tac x = "x # xa" in exI)
```
```   771 apply auto
```
```   772 done
```
```   773
```
```   774 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
```
```   775 by (induct x) auto
```
```   776
```
```   777 lemma distinct_count_atmost_1:
```
```   778   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
```
```   779 apply (induct x, simp, rule iffI, simp_all)
```
```   780 apply (rule conjI)
```
```   781 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
```
```   782 apply (erule_tac x = a in allE, simp, clarify)
```
```   783 apply (erule_tac x = aa in allE, simp)
```
```   784 done
```
```   785
```
```   786 lemma multiset_of_eq_setD:
```
```   787   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
```
```   788 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
```
```   789
```
```   790 lemma set_eq_iff_multiset_of_eq_distinct:
```
```   791   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
```
```   792     (set x = set y) = (multiset_of x = multiset_of y)"
```
```   793 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
```
```   794
```
```   795 lemma set_eq_iff_multiset_of_remdups_eq:
```
```   796    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
```
```   797 apply (rule iffI)
```
```   798 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
```
```   799 apply (drule distinct_remdups [THEN distinct_remdups
```
```   800       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
```
```   801 apply simp
```
```   802 done
```
```   803
```
```   804 lemma multiset_of_compl_union [simp]:
```
```   805   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
```
```   806   by (induct xs) (auto simp: add_ac)
```
```   807
```
```   808 lemma count_multiset_of_length_filter:
```
```   809   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```   810   by (induct xs) auto
```
```   811
```
```   812 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
```
```   813 apply (induct ls arbitrary: i)
```
```   814  apply simp
```
```   815 apply (case_tac i)
```
```   816  apply auto
```
```   817 done
```
```   818
```
```   819 lemma multiset_of_remove1[simp]:
```
```   820   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
```
```   821 by (induct xs) (auto simp add: multiset_eq_iff)
```
```   822
```
```   823 lemma multiset_of_eq_length:
```
```   824   assumes "multiset_of xs = multiset_of ys"
```
```   825   shows "length xs = length ys"
```
```   826 using assms
```
```   827 proof (induct xs arbitrary: ys)
```
```   828   case Nil then show ?case by simp
```
```   829 next
```
```   830   case (Cons x xs)
```
```   831   then have "x \<in># multiset_of ys" by (simp add: union_single_eq_member)
```
```   832   then have "x \<in> set ys" by (simp add: in_multiset_in_set)
```
```   833   from Cons.prems [symmetric] have "multiset_of xs = multiset_of (remove1 x ys)"
```
```   834     by simp
```
```   835   with Cons.hyps have "length xs = length (remove1 x ys)" .
```
```   836   with `x \<in> set ys` show ?case
```
```   837     by (auto simp add: length_remove1 dest: length_pos_if_in_set)
```
```   838 qed
```
```   839
```
```   840 lemma multiset_of_eq_length_filter:
```
```   841   assumes "multiset_of xs = multiset_of ys"
```
```   842   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
```
```   843 proof (cases "z \<in># multiset_of xs")
```
```   844   case False
```
```   845   moreover have "\<not> z \<in># multiset_of ys" using assms False by simp
```
```   846   ultimately show ?thesis by (simp add: count_multiset_of_length_filter)
```
```   847 next
```
```   848   case True
```
```   849   moreover have "z \<in># multiset_of ys" using assms True by simp
```
```   850   show ?thesis using assms
```
```   851   proof (induct xs arbitrary: ys)
```
```   852     case Nil then show ?case by simp
```
```   853   next
```
```   854     case (Cons x xs)
```
```   855     from `multiset_of (x # xs) = multiset_of ys` [symmetric]
```
```   856       have *: "multiset_of xs = multiset_of (remove1 x ys)"
```
```   857       and "x \<in> set ys"
```
```   858       by (auto simp add: mem_set_multiset_eq)
```
```   859     from * have "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) (remove1 x ys))" by (rule Cons.hyps)
```
```   860     moreover from `x \<in> set ys` have "length (filter (\<lambda>y. x = y) ys) > 0" by (simp add: filter_empty_conv)
```
```   861     ultimately show ?case using `x \<in> set ys`
```
```   862       by (simp add: filter_remove1) (auto simp add: length_remove1)
```
```   863   qed
```
```   864 qed
```
```   865
```
```   866 lemma fold_multiset_equiv:
```
```   867   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"
```
```   868     and equiv: "multiset_of xs = multiset_of ys"
```
```   869   shows "fold f xs = fold f ys"
```
```   870 using f equiv [symmetric]
```
```   871 proof (induct xs arbitrary: ys)
```
```   872   case Nil then show ?case by simp
```
```   873 next
```
```   874   case (Cons x xs)
```
```   875   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
```
```   876   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
```
```   877     by (rule Cons.prems(1)) (simp_all add: *)
```
```   878   moreover from * have "x \<in> set ys" by simp
```
```   879   ultimately have "fold f ys = fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
```
```   880   moreover from Cons.prems have "fold f xs = fold f (remove1 x ys)" by (auto intro: Cons.hyps)
```
```   881   ultimately show ?case by simp
```
```   882 qed
```
```   883
```
```   884 context linorder
```
```   885 begin
```
```   886
```
```   887 lemma multiset_of_insort [simp]:
```
```   888   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
```
```   889   by (induct xs) (simp_all add: ac_simps)
```
```   890
```
```   891 lemma multiset_of_sort [simp]:
```
```   892   "multiset_of (sort_key k xs) = multiset_of xs"
```
```   893   by (induct xs) (simp_all add: ac_simps)
```
```   894
```
```   895 text {*
```
```   896   This lemma shows which properties suffice to show that a function
```
```   897   @{text "f"} with @{text "f xs = ys"} behaves like sort.
```
```   898 *}
```
```   899
```
```   900 lemma properties_for_sort_key:
```
```   901   assumes "multiset_of ys = multiset_of xs"
```
```   902   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
```
```   903   and "sorted (map f ys)"
```
```   904   shows "sort_key f xs = ys"
```
```   905 using assms
```
```   906 proof (induct xs arbitrary: ys)
```
```   907   case Nil then show ?case by simp
```
```   908 next
```
```   909   case (Cons x xs)
```
```   910   from Cons.prems(2) have
```
```   911     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
```
```   912     by (simp add: filter_remove1)
```
```   913   with Cons.prems have "sort_key f xs = remove1 x ys"
```
```   914     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
```
```   915   moreover from Cons.prems have "x \<in> set ys"
```
```   916     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
```
```   917   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
```
```   918 qed
```
```   919
```
```   920 lemma properties_for_sort:
```
```   921   assumes multiset: "multiset_of ys = multiset_of xs"
```
```   922   and "sorted ys"
```
```   923   shows "sort xs = ys"
```
```   924 proof (rule properties_for_sort_key)
```
```   925   from multiset show "multiset_of ys = multiset_of xs" .
```
```   926   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
```
```   927   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
```
```   928     by (rule multiset_of_eq_length_filter)
```
```   929   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
```
```   930     by simp
```
```   931   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
```
```   932     by (simp add: replicate_length_filter)
```
```   933 qed
```
```   934
```
```   935 lemma sort_key_by_quicksort:
```
```   936   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
```
```   937     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
```
```   938     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
```
```   939 proof (rule properties_for_sort_key)
```
```   940   show "multiset_of ?rhs = multiset_of ?lhs"
```
```   941     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
```
```   942 next
```
```   943   show "sorted (map f ?rhs)"
```
```   944     by (auto simp add: sorted_append intro: sorted_map_same)
```
```   945 next
```
```   946   fix l
```
```   947   assume "l \<in> set ?rhs"
```
```   948   let ?pivot = "f (xs ! (length xs div 2))"
```
```   949   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
```
```   950   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
```
```   951     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
```
```   952   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
```
```   953   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
```
```   954   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
```
```   955     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
```
```   956   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
```
```   957   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
```
```   958   proof (cases "f l" ?pivot rule: linorder_cases)
```
```   959     case less
```
```   960     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
```
```   961     with less show ?thesis
```
```   962       by (simp add: filter_sort [symmetric] ** ***)
```
```   963   next
```
```   964     case equal then show ?thesis
```
```   965       by (simp add: * less_le)
```
```   966   next
```
```   967     case greater
```
```   968     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
```
```   969     with greater show ?thesis
```
```   970       by (simp add: filter_sort [symmetric] ** ***)
```
```   971   qed
```
```   972 qed
```
```   973
```
```   974 lemma sort_by_quicksort:
```
```   975   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
```
```   976     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
```
```   977     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
```
```   978   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
```
```   979
```
```   980 text {* A stable parametrized quicksort *}
```
```   981
```
```   982 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
```
```   983   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
```
```   984
```
```   985 lemma part_code [code]:
```
```   986   "part f pivot [] = ([], [], [])"
```
```   987   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
```
```   988      if x' < pivot then (x # lts, eqs, gts)
```
```   989      else if x' > pivot then (lts, eqs, x # gts)
```
```   990      else (lts, x # eqs, gts))"
```
```   991   by (auto simp add: part_def Let_def split_def)
```
```   992
```
```   993 lemma sort_key_by_quicksort_code [code]:
```
```   994   "sort_key f xs = (case xs of [] \<Rightarrow> []
```
```   995     | [x] \<Rightarrow> xs
```
```   996     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
```
```   997     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```   998        in sort_key f lts @ eqs @ sort_key f gts))"
```
```   999 proof (cases xs)
```
```  1000   case Nil then show ?thesis by simp
```
```  1001 next
```
```  1002   case (Cons _ ys) note hyps = Cons show ?thesis
```
```  1003   proof (cases ys)
```
```  1004     case Nil with hyps show ?thesis by simp
```
```  1005   next
```
```  1006     case (Cons _ zs) note hyps = hyps Cons show ?thesis
```
```  1007     proof (cases zs)
```
```  1008       case Nil with hyps show ?thesis by auto
```
```  1009     next
```
```  1010       case Cons
```
```  1011       from sort_key_by_quicksort [of f xs]
```
```  1012       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1013         in sort_key f lts @ eqs @ sort_key f gts)"
```
```  1014       by (simp only: split_def Let_def part_def fst_conv snd_conv)
```
```  1015       with hyps Cons show ?thesis by (simp only: list.cases)
```
```  1016     qed
```
```  1017   qed
```
```  1018 qed
```
```  1019
```
```  1020 end
```
```  1021
```
```  1022 hide_const (open) part
```
```  1023
```
```  1024 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
```
```  1025   by (induct xs) (auto intro: order_trans)
```
```  1026
```
```  1027 lemma multiset_of_update:
```
```  1028   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
```
```  1029 proof (induct ls arbitrary: i)
```
```  1030   case Nil then show ?case by simp
```
```  1031 next
```
```  1032   case (Cons x xs)
```
```  1033   show ?case
```
```  1034   proof (cases i)
```
```  1035     case 0 then show ?thesis by simp
```
```  1036   next
```
```  1037     case (Suc i')
```
```  1038     with Cons show ?thesis
```
```  1039       apply simp
```
```  1040       apply (subst add_assoc)
```
```  1041       apply (subst add_commute [of "{#v#}" "{#x#}"])
```
```  1042       apply (subst add_assoc [symmetric])
```
```  1043       apply simp
```
```  1044       apply (rule mset_le_multiset_union_diff_commute)
```
```  1045       apply (simp add: mset_le_single nth_mem_multiset_of)
```
```  1046       done
```
```  1047   qed
```
```  1048 qed
```
```  1049
```
```  1050 lemma multiset_of_swap:
```
```  1051   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
```
```  1052     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
```
```  1053   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
```
```  1054
```
```  1055
```
```  1056 subsubsection {* Association lists -- including code generation *}
```
```  1057
```
```  1058 text {* Preliminaries *}
```
```  1059
```
```  1060 text {* Raw operations on lists *}
```
```  1061
```
```  1062 definition join_raw :: "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
```
```  1063 where
```
```  1064   "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
```
```  1065
```
```  1066 lemma join_raw_Nil [simp]:
```
```  1067   "join_raw f xs [] = xs"
```
```  1068 by (simp add: join_raw_def)
```
```  1069
```
```  1070 lemma join_raw_Cons [simp]:
```
```  1071   "join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
```
```  1072 by (simp add: join_raw_def)
```
```  1073
```
```  1074 lemma map_of_join_raw:
```
```  1075   assumes "distinct (map fst ys)"
```
```  1076   shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v => (case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
```
```  1077 using assms
```
```  1078 apply (induct ys)
```
```  1079 apply (auto simp add: map_of_map_default split: option.split)
```
```  1080 apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
```
```  1081 by (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
```
```  1082
```
```  1083 lemma distinct_join_raw:
```
```  1084   assumes "distinct (map fst xs)"
```
```  1085   shows "distinct (map fst (join_raw f xs ys))"
```
```  1086 using assms
```
```  1087 proof (induct ys)
```
```  1088   case (Cons y ys)
```
```  1089   thus ?case by (cases y) (simp add: distinct_map_default)
```
```  1090 qed auto
```
```  1091
```
```  1092 definition
```
```  1093   "subtract_entries_raw xs ys = foldr (%(k, v). AList.map_entry k (%v'. v' - v)) ys xs"
```
```  1094
```
```  1095 lemma map_of_subtract_entries_raw:
```
```  1096   "distinct (map fst ys) ==> map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None => None | Some v => (case map_of ys x of None => Some v | Some v' => Some (v - v')))"
```
```  1097 unfolding subtract_entries_raw_def
```
```  1098 apply (induct ys)
```
```  1099 apply auto
```
```  1100 apply (simp split: option.split)
```
```  1101 apply (simp add: map_of_map_entry)
```
```  1102 apply (auto split: option.split)
```
```  1103 apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
```
```  1104 by (metis map_of_eq_None_iff option.simps(4) option.simps(5))
```
```  1105
```
```  1106 lemma distinct_subtract_entries_raw:
```
```  1107   assumes "distinct (map fst xs)"
```
```  1108   shows "distinct (map fst (subtract_entries_raw xs ys))"
```
```  1109 using assms
```
```  1110 unfolding subtract_entries_raw_def by (induct ys) (auto simp add: distinct_map_entry)
```
```  1111
```
```  1112 text {* Operations on alists with distinct keys *}
```
```  1113
```
```  1114 quotient_definition join :: "('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
```
```  1115 where
```
```  1116   "join" is "join_raw :: ('a \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"
```
```  1117 by (simp add: distinct_join_raw)
```
```  1118
```
```  1119 quotient_definition subtract_entries :: "('a, ('b :: minus)) alist \<Rightarrow> ('a, 'b) alist \<Rightarrow> ('a, 'b) alist"
```
```  1120 where
```
```  1121   "subtract_entries" is "subtract_entries_raw :: ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"
```
```  1122 by (simp add: distinct_subtract_entries_raw)
```
```  1123
```
```  1124 text {* Implementing multisets by means of association lists *}
```
```  1125
```
```  1126 definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
```
```  1127   "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
```
```  1128
```
```  1129 lemma count_of_multiset:
```
```  1130   "count_of xs \<in> multiset"
```
```  1131 proof -
```
```  1132   let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
```
```  1133   have "?A \<subseteq> dom (map_of xs)"
```
```  1134   proof
```
```  1135     fix x
```
```  1136     assume "x \<in> ?A"
```
```  1137     then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
```
```  1138     then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
```
```  1139     then show "x \<in> dom (map_of xs)" by auto
```
```  1140   qed
```
```  1141   with finite_dom_map_of [of xs] have "finite ?A"
```
```  1142     by (auto intro: finite_subset)
```
```  1143   then show ?thesis
```
```  1144     by (simp add: count_of_def fun_eq_iff multiset_def)
```
```  1145 qed
```
```  1146
```
```  1147 lemma count_simps [simp]:
```
```  1148   "count_of [] = (\<lambda>_. 0)"
```
```  1149   "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
```
```  1150   by (simp_all add: count_of_def fun_eq_iff)
```
```  1151
```
```  1152 lemma count_of_empty:
```
```  1153   "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
```
```  1154   by (induct xs) (simp_all add: count_of_def)
```
```  1155
```
```  1156 lemma count_of_filter:
```
```  1157   "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
```
```  1158   by (induct xs) auto
```
```  1159
```
```  1160 lemma count_of_map_default [simp]:
```
```  1161   "count_of (map_default x b (%x. x + b) xs) y = (if x = y then count_of xs x + b else count_of xs y)"
```
```  1162 unfolding count_of_def by (simp add: map_of_map_default split: option.split)
```
```  1163
```
```  1164 lemma count_of_join_raw:
```
```  1165   "distinct (map fst ys) ==> count_of xs x + count_of ys x = count_of (join_raw (%x (x, y). x + y) xs ys) x"
```
```  1166 unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
```
```  1167
```
```  1168 lemma count_of_subtract_entries_raw:
```
```  1169   "distinct (map fst ys) ==> count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
```
```  1170 unfolding count_of_def by (simp add: map_of_subtract_entries_raw split: option.split)
```
```  1171
```
```  1172 text {* Code equations for multiset operations *}
```
```  1173
```
```  1174 definition Bag :: "('a, nat) alist \<Rightarrow> 'a multiset" where
```
```  1175   "Bag xs = Abs_multiset (count_of (DAList.impl_of xs))"
```
```  1176
```
```  1177 code_datatype Bag
```
```  1178
```
```  1179 lemma count_Bag [simp, code]:
```
```  1180   "count (Bag xs) = count_of (DAList.impl_of xs)"
```
```  1181   by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
```
```  1182
```
```  1183 lemma Mempty_Bag [code]:
```
```  1184   "{#} = Bag (DAList.empty)"
```
```  1185   by (simp add: multiset_eq_iff alist.Alist_inverse DAList.empty_def)
```
```  1186
```
```  1187 lemma single_Bag [code]:
```
```  1188   "{#x#} = Bag (DAList.update x 1 DAList.empty)"
```
```  1189   by (simp add: multiset_eq_iff alist.Alist_inverse update_code_eqn empty_code_eqn)
```
```  1190
```
```  1191 lemma union_Bag [code]:
```
```  1192   "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
```
```  1193 by (rule multiset_eqI) (simp add: count_of_join_raw alist.Alist_inverse distinct_join_raw join_def)
```
```  1194
```
```  1195 lemma minus_Bag [code]:
```
```  1196   "Bag xs - Bag ys = Bag (subtract_entries xs ys)"
```
```  1197 by (rule multiset_eqI)
```
```  1198   (simp add: count_of_subtract_entries_raw alist.Alist_inverse distinct_subtract_entries_raw subtract_entries_def)
```
```  1199
```
```  1200 lemma filter_Bag [code]:
```
```  1201   "Multiset.filter P (Bag xs) = Bag (DAList.filter (P \<circ> fst) xs)"
```
```  1202 by (rule multiset_eqI) (simp add: count_of_filter filter_code_eqn)
```
```  1203
```
```  1204 lemma mset_less_eq_Bag [code]:
```
```  1205   "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set (DAList.impl_of xs). count_of (DAList.impl_of xs) x \<le> count A x)"
```
```  1206     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1207 proof
```
```  1208   assume ?lhs then show ?rhs
```
```  1209     by (auto simp add: mset_le_def)
```
```  1210 next
```
```  1211   assume ?rhs
```
```  1212   show ?lhs
```
```  1213   proof (rule mset_less_eqI)
```
```  1214     fix x
```
```  1215     from `?rhs` have "count_of (DAList.impl_of xs) x \<le> count A x"
```
```  1216       by (cases "x \<in> fst ` set (DAList.impl_of xs)") (auto simp add: count_of_empty)
```
```  1217     then show "count (Bag xs) x \<le> count A x"
```
```  1218       by (simp add: mset_le_def)
```
```  1219   qed
```
```  1220 qed
```
```  1221
```
```  1222 instantiation multiset :: (equal) equal
```
```  1223 begin
```
```  1224
```
```  1225 definition
```
```  1226   [code]: "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"
```
```  1227
```
```  1228 instance
```
```  1229   by default (simp add: equal_multiset_def eq_iff)
```
```  1230
```
```  1231 end
```
```  1232
```
```  1233 text {* Quickcheck generators *}
```
```  1234
```
```  1235 definition (in term_syntax)
```
```  1236   bagify :: "('a\<Colon>typerep, nat) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)
```
```  1237     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
```
```  1238   [code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs"
```
```  1239
```
```  1240 notation fcomp (infixl "\<circ>>" 60)
```
```  1241 notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  1242
```
```  1243 instantiation multiset :: (random) random
```
```  1244 begin
```
```  1245
```
```  1246 definition
```
```  1247   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (bagify xs))"
```
```  1248
```
```  1249 instance ..
```
```  1250
```
```  1251 end
```
```  1252
```
```  1253 no_notation fcomp (infixl "\<circ>>" 60)
```
```  1254 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  1255
```
```  1256 instantiation multiset :: (exhaustive) exhaustive
```
```  1257 begin
```
```  1258
```
```  1259 definition exhaustive_multiset :: "('a multiset => (bool * term list) option) => code_numeral => (bool * term list) option"
```
```  1260 where
```
```  1261   "exhaustive_multiset f i = Quickcheck_Exhaustive.exhaustive (%xs. f (Bag xs)) i"
```
```  1262
```
```  1263 instance ..
```
```  1264
```
```  1265 end
```
```  1266
```
```  1267 instantiation multiset :: (full_exhaustive) full_exhaustive
```
```  1268 begin
```
```  1269
```
```  1270 definition full_exhaustive_multiset :: "('a multiset * (unit => term) => (bool * term list) option) => code_numeral => (bool * term list) option"
```
```  1271 where
```
```  1272   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (%xs. f (bagify xs)) i"
```
```  1273
```
```  1274 instance ..
```
```  1275
```
```  1276 end
```
```  1277
```
```  1278 hide_const (open) bagify
```
```  1279
```
```  1280
```
```  1281 subsection {* The multiset order *}
```
```  1282
```
```  1283 subsubsection {* Well-foundedness *}
```
```  1284
```
```  1285 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1286   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```  1287       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```  1288
```
```  1289 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1290   "mult r = (mult1 r)\<^sup>+"
```
```  1291
```
```  1292 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```  1293 by (simp add: mult1_def)
```
```  1294
```
```  1295 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```  1296     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```  1297     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```  1298   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
```
```  1299 proof (unfold mult1_def)
```
```  1300   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```  1301   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```  1302   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```  1303
```
```  1304   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```  1305   then have "\<exists>a' M0' K.
```
```  1306       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```  1307   then show "?case1 \<or> ?case2"
```
```  1308   proof (elim exE conjE)
```
```  1309     fix a' M0' K
```
```  1310     assume N: "N = M0' + K" and r: "?r K a'"
```
```  1311     assume "M0 + {#a#} = M0' + {#a'#}"
```
```  1312     then have "M0 = M0' \<and> a = a' \<or>
```
```  1313         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```  1314       by (simp only: add_eq_conv_ex)
```
```  1315     then show ?thesis
```
```  1316     proof (elim disjE conjE exE)
```
```  1317       assume "M0 = M0'" "a = a'"
```
```  1318       with N r have "?r K a \<and> N = M0 + K" by simp
```
```  1319       then have ?case2 .. then show ?thesis ..
```
```  1320     next
```
```  1321       fix K'
```
```  1322       assume "M0' = K' + {#a#}"
```
```  1323       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
```
```  1324
```
```  1325       assume "M0 = K' + {#a'#}"
```
```  1326       with r have "?R (K' + K) M0" by blast
```
```  1327       with n have ?case1 by simp then show ?thesis ..
```
```  1328     qed
```
```  1329   qed
```
```  1330 qed
```
```  1331
```
```  1332 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
```
```  1333 proof
```
```  1334   let ?R = "mult1 r"
```
```  1335   let ?W = "acc ?R"
```
```  1336   {
```
```  1337     fix M M0 a
```
```  1338     assume M0: "M0 \<in> ?W"
```
```  1339       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1340       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```  1341     have "M0 + {#a#} \<in> ?W"
```
```  1342     proof (rule accI [of "M0 + {#a#}"])
```
```  1343       fix N
```
```  1344       assume "(N, M0 + {#a#}) \<in> ?R"
```
```  1345       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```  1346           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```  1347         by (rule less_add)
```
```  1348       then show "N \<in> ?W"
```
```  1349       proof (elim exE disjE conjE)
```
```  1350         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```  1351         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```  1352         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
```
```  1353         then show "N \<in> ?W" by (simp only: N)
```
```  1354       next
```
```  1355         fix K
```
```  1356         assume N: "N = M0 + K"
```
```  1357         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```  1358         then have "M0 + K \<in> ?W"
```
```  1359         proof (induct K)
```
```  1360           case empty
```
```  1361           from M0 show "M0 + {#} \<in> ?W" by simp
```
```  1362         next
```
```  1363           case (add K x)
```
```  1364           from add.prems have "(x, a) \<in> r" by simp
```
```  1365           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```  1366           moreover from add have "M0 + K \<in> ?W" by simp
```
```  1367           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
```
```  1368           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
```
```  1369         qed
```
```  1370         then show "N \<in> ?W" by (simp only: N)
```
```  1371       qed
```
```  1372     qed
```
```  1373   } note tedious_reasoning = this
```
```  1374
```
```  1375   assume wf: "wf r"
```
```  1376   fix M
```
```  1377   show "M \<in> ?W"
```
```  1378   proof (induct M)
```
```  1379     show "{#} \<in> ?W"
```
```  1380     proof (rule accI)
```
```  1381       fix b assume "(b, {#}) \<in> ?R"
```
```  1382       with not_less_empty show "b \<in> ?W" by contradiction
```
```  1383     qed
```
```  1384
```
```  1385     fix M a assume "M \<in> ?W"
```
```  1386     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1387     proof induct
```
```  1388       fix a
```
```  1389       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1390       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1391       proof
```
```  1392         fix M assume "M \<in> ?W"
```
```  1393         then show "M + {#a#} \<in> ?W"
```
```  1394           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
```
```  1395       qed
```
```  1396     qed
```
```  1397     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
```
```  1398   qed
```
```  1399 qed
```
```  1400
```
```  1401 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```  1402 by (rule acc_wfI) (rule all_accessible)
```
```  1403
```
```  1404 theorem wf_mult: "wf r ==> wf (mult r)"
```
```  1405 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
```
```  1406
```
```  1407
```
```  1408 subsubsection {* Closure-free presentation *}
```
```  1409
```
```  1410 text {* One direction. *}
```
```  1411
```
```  1412 lemma mult_implies_one_step:
```
```  1413   "trans r ==> (M, N) \<in> mult r ==>
```
```  1414     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```  1415     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```  1416 apply (unfold mult_def mult1_def set_of_def)
```
```  1417 apply (erule converse_trancl_induct, clarify)
```
```  1418  apply (rule_tac x = M0 in exI, simp, clarify)
```
```  1419 apply (case_tac "a :# K")
```
```  1420  apply (rule_tac x = I in exI)
```
```  1421  apply (simp (no_asm))
```
```  1422  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```  1423  apply (simp (no_asm_simp) add: add_assoc [symmetric])
```
```  1424  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```  1425  apply (simp add: diff_union_single_conv)
```
```  1426  apply (simp (no_asm_use) add: trans_def)
```
```  1427  apply blast
```
```  1428 apply (subgoal_tac "a :# I")
```
```  1429  apply (rule_tac x = "I - {#a#}" in exI)
```
```  1430  apply (rule_tac x = "J + {#a#}" in exI)
```
```  1431  apply (rule_tac x = "K + Ka" in exI)
```
```  1432  apply (rule conjI)
```
```  1433   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1434  apply (rule conjI)
```
```  1435   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
```
```  1436   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1437  apply (simp (no_asm_use) add: trans_def)
```
```  1438  apply blast
```
```  1439 apply (subgoal_tac "a :# (M0 + {#a#})")
```
```  1440  apply simp
```
```  1441 apply (simp (no_asm))
```
```  1442 done
```
```  1443
```
```  1444 lemma one_step_implies_mult_aux:
```
```  1445   "trans r ==>
```
```  1446     \<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))
```
```  1447       --> (I + K, I + J) \<in> mult r"
```
```  1448 apply (induct_tac n, auto)
```
```  1449 apply (frule size_eq_Suc_imp_eq_union, clarify)
```
```  1450 apply (rename_tac "J'", simp)
```
```  1451 apply (erule notE, auto)
```
```  1452 apply (case_tac "J' = {#}")
```
```  1453  apply (simp add: mult_def)
```
```  1454  apply (rule r_into_trancl)
```
```  1455  apply (simp add: mult1_def set_of_def, blast)
```
```  1456 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```  1457 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```  1458 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```  1459 apply (erule ssubst)
```
```  1460 apply (simp add: Ball_def, auto)
```
```  1461 apply (subgoal_tac
```
```  1462   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
```
```  1463     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```  1464  prefer 2
```
```  1465  apply force
```
```  1466 apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
```
```  1467 apply (erule trancl_trans)
```
```  1468 apply (rule r_into_trancl)
```
```  1469 apply (simp add: mult1_def set_of_def)
```
```  1470 apply (rule_tac x = a in exI)
```
```  1471 apply (rule_tac x = "I + J'" in exI)
```
```  1472 apply (simp add: add_ac)
```
```  1473 done
```
```  1474
```
```  1475 lemma one_step_implies_mult:
```
```  1476   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```  1477     ==> (I + K, I + J) \<in> mult r"
```
```  1478 using one_step_implies_mult_aux by blast
```
```  1479
```
```  1480
```
```  1481 subsubsection {* Partial-order properties *}
```
```  1482
```
```  1483 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
```
```  1484   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
```
```  1485
```
```  1486 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
```
```  1487   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
```
```  1488
```
```  1489 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
```
```  1490 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
```
```  1491
```
```  1492 interpretation multiset_order: order le_multiset less_multiset
```
```  1493 proof -
```
```  1494   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
```
```  1495   proof
```
```  1496     fix M :: "'a multiset"
```
```  1497     assume "M \<subset># M"
```
```  1498     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
```
```  1499     have "trans {(x'::'a, x). x' < x}"
```
```  1500       by (rule transI) simp
```
```  1501     moreover note MM
```
```  1502     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
```
```  1503       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
```
```  1504       by (rule mult_implies_one_step)
```
```  1505     then obtain I J K where "M = I + J" and "M = I + K"
```
```  1506       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
```
```  1507     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
```
```  1508     have "finite (set_of K)" by simp
```
```  1509     moreover note aux2
```
```  1510     ultimately have "set_of K = {}"
```
```  1511       by (induct rule: finite_induct) (auto intro: order_less_trans)
```
```  1512     with aux1 show False by simp
```
```  1513   qed
```
```  1514   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
```
```  1515     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
```
```  1516   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
```
```  1517     by default (auto simp add: le_multiset_def irrefl dest: trans)
```
```  1518 qed
```
```  1519
```
```  1520 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
```
```  1521   by simp
```
```  1522
```
```  1523
```
```  1524 subsubsection {* Monotonicity of multiset union *}
```
```  1525
```
```  1526 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
```
```  1527 apply (unfold mult1_def)
```
```  1528 apply auto
```
```  1529 apply (rule_tac x = a in exI)
```
```  1530 apply (rule_tac x = "C + M0" in exI)
```
```  1531 apply (simp add: add_assoc)
```
```  1532 done
```
```  1533
```
```  1534 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
```
```  1535 apply (unfold less_multiset_def mult_def)
```
```  1536 apply (erule trancl_induct)
```
```  1537  apply (blast intro: mult1_union)
```
```  1538 apply (blast intro: mult1_union trancl_trans)
```
```  1539 done
```
```  1540
```
```  1541 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
```
```  1542 apply (subst add_commute [of B C])
```
```  1543 apply (subst add_commute [of D C])
```
```  1544 apply (erule union_less_mono2)
```
```  1545 done
```
```  1546
```
```  1547 lemma union_less_mono:
```
```  1548   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
```
```  1549   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
```
```  1550
```
```  1551 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
```
```  1552 proof
```
```  1553 qed (auto simp add: le_multiset_def intro: union_less_mono2)
```
```  1554
```
```  1555
```
```  1556 subsection {* The fold combinator *}
```
```  1557
```
```  1558 text {*
```
```  1559   The intended behaviour is
```
```  1560   @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
```
```  1561   if @{text f} is associative-commutative.
```
```  1562 *}
```
```  1563
```
```  1564 text {*
```
```  1565   The graph of @{text "fold_mset"}, @{text "z"}: the start element,
```
```  1566   @{text "f"}: folding function, @{text "A"}: the multiset, @{text
```
```  1567   "y"}: the result.
```
```  1568 *}
```
```  1569 inductive
```
```  1570   fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool"
```
```  1571   for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1572   and z :: 'b
```
```  1573 where
```
```  1574   emptyI [intro]:  "fold_msetG f z {#} z"
```
```  1575 | insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
```
```  1576
```
```  1577 inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
```
```  1578 inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y"
```
```  1579
```
```  1580 definition
```
```  1581   fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
```
```  1582   "fold_mset f z A = (THE x. fold_msetG f z A x)"
```
```  1583
```
```  1584 lemma Diff1_fold_msetG:
```
```  1585   "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
```
```  1586 apply (frule_tac x = x in fold_msetG.insertI)
```
```  1587 apply auto
```
```  1588 done
```
```  1589
```
```  1590 lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
```
```  1591 apply (induct A)
```
```  1592  apply blast
```
```  1593 apply clarsimp
```
```  1594 apply (drule_tac x = x in fold_msetG.insertI)
```
```  1595 apply auto
```
```  1596 done
```
```  1597
```
```  1598 lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
```
```  1599 unfolding fold_mset_def by blast
```
```  1600
```
```  1601 context comp_fun_commute
```
```  1602 begin
```
```  1603
```
```  1604 lemma fold_msetG_determ:
```
```  1605   "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
```
```  1606 proof (induct arbitrary: x y z rule: full_multiset_induct)
```
```  1607   case (less M x\<^isub>1 x\<^isub>2 Z)
```
```  1608   have IH: "\<forall>A. A < M \<longrightarrow>
```
```  1609     (\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'
```
```  1610                \<longrightarrow> x' = x)" by fact
```
```  1611   have Mfoldx\<^isub>1: "fold_msetG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "fold_msetG f Z M x\<^isub>2" by fact+
```
```  1612   show ?case
```
```  1613   proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])
```
```  1614     assume "M = {#}" and "x\<^isub>1 = Z"
```
```  1615     then show ?case using Mfoldx\<^isub>2 by auto
```
```  1616   next
```
```  1617     fix B b u
```
```  1618     assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"
```
```  1619     then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto
```
```  1620     show ?case
```
```  1621     proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])
```
```  1622       assume "M = {#}" "x\<^isub>2 = Z"
```
```  1623       then show ?case using Mfoldx\<^isub>1 by auto
```
```  1624     next
```
```  1625       fix C c v
```
```  1626       assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"
```
```  1627       then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto
```
```  1628       then have CsubM: "C < M" by simp
```
```  1629       from MBb have BsubM: "B < M" by simp
```
```  1630       show ?case
```
```  1631       proof cases
```
```  1632         assume *: "b = c"
```
```  1633         then have "B = C" using MBb MCc by auto
```
```  1634         with * show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto
```
```  1635       next
```
```  1636         assume diff: "b \<noteq> c"
```
```  1637         let ?D = "B - {#c#}"
```
```  1638         have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
```
```  1639           by (auto intro: insert_noteq_member dest: sym)
```
```  1640         have "B - {#c#} < B" using cinB by (rule mset_less_diff_self)
```
```  1641         then have DsubM: "?D < M" using BsubM by (blast intro: order_less_trans)
```
```  1642         from MBb MCc have "B + {#b#} = C + {#c#}" by blast
```
```  1643         then have [simp]: "B + {#b#} - {#c#} = C"
```
```  1644           using MBb MCc binC cinB by auto
```
```  1645         have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
```
```  1646           using MBb MCc diff binC cinB
```
```  1647           by (auto simp: multiset_add_sub_el_shuffle)
```
```  1648         then obtain d where Dfoldd: "fold_msetG f Z ?D d"
```
```  1649           using fold_msetG_nonempty by iprover
```
```  1650         then have "fold_msetG f Z B (f c d)" using cinB
```
```  1651           by (rule Diff1_fold_msetG)
```
```  1652         then have "f c d = u" using IH BsubM Bu by blast
```
```  1653         moreover
```
```  1654         have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
```
```  1655           by (auto simp: multiset_add_sub_el_shuffle
```
```  1656             dest: fold_msetG.insertI [where x=b])
```
```  1657         then have "f b d = v" using IH CsubM Cv by blast
```
```  1658         ultimately show ?thesis using x\<^isub>1 x\<^isub>2
```
```  1659           by (auto simp: fun_left_comm)
```
```  1660       qed
```
```  1661     qed
```
```  1662   qed
```
```  1663 qed
```
```  1664
```
```  1665 lemma fold_mset_insert_aux:
```
```  1666   "(fold_msetG f z (A + {#x#}) v) =
```
```  1667     (\<exists>y. fold_msetG f z A y \<and> v = f x y)"
```
```  1668 apply (rule iffI)
```
```  1669  prefer 2
```
```  1670  apply blast
```
```  1671 apply (rule_tac A1=A and f1=f in fold_msetG_nonempty [THEN exE])
```
```  1672 apply (blast intro: fold_msetG_determ)
```
```  1673 done
```
```  1674
```
```  1675 lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
```
```  1676 unfolding fold_mset_def by (blast intro: fold_msetG_determ)
```
```  1677
```
```  1678 lemma fold_mset_insert:
```
```  1679   "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
```
```  1680 apply (simp add: fold_mset_def fold_mset_insert_aux)
```
```  1681 apply (rule the_equality)
```
```  1682  apply (auto cong add: conj_cong
```
```  1683      simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
```
```  1684 done
```
```  1685
```
```  1686 lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
```
```  1687 by (induct A) (auto simp: fold_mset_insert fun_left_comm [of x])
```
```  1688
```
```  1689 lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
```
```  1690 using fold_mset_insert [of z "{#}"] by simp
```
```  1691
```
```  1692 lemma fold_mset_union [simp]:
```
```  1693   "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
```
```  1694 proof (induct A)
```
```  1695   case empty then show ?case by simp
```
```  1696 next
```
```  1697   case (add A x)
```
```  1698   have "A + {#x#} + B = (A+B) + {#x#}" by (simp add: add_ac)
```
```  1699   then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))"
```
```  1700     by (simp add: fold_mset_insert)
```
```  1701   also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
```
```  1702     by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
```
```  1703   finally show ?case .
```
```  1704 qed
```
```  1705
```
```  1706 lemma fold_mset_fusion:
```
```  1707   assumes "comp_fun_commute g"
```
```  1708   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")
```
```  1709 proof -
```
```  1710   interpret comp_fun_commute g by (fact assms)
```
```  1711   show "PROP ?P" by (induct A) auto
```
```  1712 qed
```
```  1713
```
```  1714 lemma fold_mset_rec:
```
```  1715   assumes "a \<in># A"
```
```  1716   shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
```
```  1717 proof -
```
```  1718   from assms obtain A' where "A = A' + {#a#}"
```
```  1719     by (blast dest: multi_member_split)
```
```  1720   then show ?thesis by simp
```
```  1721 qed
```
```  1722
```
```  1723 end
```
```  1724
```
```  1725 text {*
```
```  1726   A note on code generation: When defining some function containing a
```
```  1727   subterm @{term"fold_mset F"}, code generation is not automatic. When
```
```  1728   interpreting locale @{text left_commutative} with @{text F}, the
```
```  1729   would be code thms for @{const fold_mset} become thms like
```
```  1730   @{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
```
```  1731   contains defined symbols, i.e.\ is not a code thm. Hence a separate
```
```  1732   constant with its own code thms needs to be introduced for @{text
```
```  1733   F}. See the image operator below.
```
```  1734 *}
```
```  1735
```
```  1736
```
```  1737 subsection {* Image *}
```
```  1738
```
```  1739 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
```
```  1740   "image_mset f = fold_mset (op + o single o f) {#}"
```
```  1741
```
```  1742 interpretation image_fun_commute: comp_fun_commute "op + o single o f" for f
```
```  1743 proof qed (simp add: add_ac fun_eq_iff)
```
```  1744
```
```  1745 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
```
```  1746 by (simp add: image_mset_def)
```
```  1747
```
```  1748 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
```
```  1749 by (simp add: image_mset_def)
```
```  1750
```
```  1751 lemma image_mset_insert:
```
```  1752   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
```
```  1753 by (simp add: image_mset_def add_ac)
```
```  1754
```
```  1755 lemma image_mset_union [simp]:
```
```  1756   "image_mset f (M+N) = image_mset f M + image_mset f N"
```
```  1757 apply (induct N)
```
```  1758  apply simp
```
```  1759 apply (simp add: add_assoc [symmetric] image_mset_insert)
```
```  1760 done
```
```  1761
```
```  1762 lemma size_image_mset [simp]: "size (image_mset f M) = size M"
```
```  1763 by (induct M) simp_all
```
```  1764
```
```  1765 lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
```
```  1766 by (cases M) auto
```
```  1767
```
```  1768 syntax
```
```  1769   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
```
```  1770       ("({#_/. _ :# _#})")
```
```  1771 translations
```
```  1772   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
```
```  1773
```
```  1774 syntax
```
```  1775   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
```
```  1776       ("({#_/ | _ :# _./ _#})")
```
```  1777 translations
```
```  1778   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
```
```  1779
```
```  1780 text {*
```
```  1781   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
```
```  1782   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
```
```  1783   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
```
```  1784   @{term "{#x+x|x:#M. x<c#}"}.
```
```  1785 *}
```
```  1786
```
```  1787 enriched_type image_mset: image_mset
```
```  1788 proof -
```
```  1789   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
```
```  1790   proof
```
```  1791     fix A
```
```  1792     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
```
```  1793       by (induct A) simp_all
```
```  1794   qed
```
```  1795   show "image_mset id = id"
```
```  1796   proof
```
```  1797     fix A
```
```  1798     show "image_mset id A = id A"
```
```  1799       by (induct A) simp_all
```
```  1800   qed
```
```  1801 qed
```
```  1802
```
```  1803
```
```  1804 subsection {* Termination proofs with multiset orders *}
```
```  1805
```
```  1806 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
```
```  1807   and multi_member_this: "x \<in># {# x #} + XS"
```
```  1808   and multi_member_last: "x \<in># {# x #}"
```
```  1809   by auto
```
```  1810
```
```  1811 definition "ms_strict = mult pair_less"
```
```  1812 definition "ms_weak = ms_strict \<union> Id"
```
```  1813
```
```  1814 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
```
```  1815 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
```
```  1816 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
```
```  1817
```
```  1818 lemma smsI:
```
```  1819   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
```
```  1820   unfolding ms_strict_def
```
```  1821 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
```
```  1822
```
```  1823 lemma wmsI:
```
```  1824   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
```
```  1825   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
```
```  1826 unfolding ms_weak_def ms_strict_def
```
```  1827 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
```
```  1828
```
```  1829 inductive pw_leq
```
```  1830 where
```
```  1831   pw_leq_empty: "pw_leq {#} {#}"
```
```  1832 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
```
```  1833
```
```  1834 lemma pw_leq_lstep:
```
```  1835   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
```
```  1836 by (drule pw_leq_step) (rule pw_leq_empty, simp)
```
```  1837
```
```  1838 lemma pw_leq_split:
```
```  1839   assumes "pw_leq X Y"
```
```  1840   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 = {#}))"
```
```  1841   using assms
```
```  1842 proof (induct)
```
```  1843   case pw_leq_empty thus ?case by auto
```
```  1844 next
```
```  1845   case (pw_leq_step x y X Y)
```
```  1846   then obtain A B Z where
```
```  1847     [simp]: "X = A + Z" "Y = B + Z"
```
```  1848       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
```
```  1849     by auto
```
```  1850   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
```
```  1851     unfolding pair_leq_def by auto
```
```  1852   thus ?case
```
```  1853   proof
```
```  1854     assume [simp]: "x = y"
```
```  1855     have
```
```  1856       "{#x#} + X = A + ({#y#}+Z)
```
```  1857       \<and> {#y#} + Y = B + ({#y#}+Z)
```
```  1858       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
```
```  1859       by (auto simp: add_ac)
```
```  1860     thus ?case by (intro exI)
```
```  1861   next
```
```  1862     assume A: "(x, y) \<in> pair_less"
```
```  1863     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
```
```  1864     have "{#x#} + X = ?A' + Z"
```
```  1865       "{#y#} + Y = ?B' + Z"
```
```  1866       by (auto simp add: add_ac)
```
```  1867     moreover have
```
```  1868       "(set_of ?A', set_of ?B') \<in> max_strict"
```
```  1869       using 1 A unfolding max_strict_def
```
```  1870       by (auto elim!: max_ext.cases)
```
```  1871     ultimately show ?thesis by blast
```
```  1872   qed
```
```  1873 qed
```
```  1874
```
```  1875 lemma
```
```  1876   assumes pwleq: "pw_leq Z Z'"
```
```  1877   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
```
```  1878   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
```
```  1879   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
```
```  1880 proof -
```
```  1881   from pw_leq_split[OF pwleq]
```
```  1882   obtain A' B' Z''
```
```  1883     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
```
```  1884     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
```
```  1885     by blast
```
```  1886   {
```
```  1887     assume max: "(set_of A, set_of B) \<in> max_strict"
```
```  1888     from mx_or_empty
```
```  1889     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
```
```  1890     proof
```
```  1891       assume max': "(set_of A', set_of B') \<in> max_strict"
```
```  1892       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
```
```  1893         by (auto simp: max_strict_def intro: max_ext_additive)
```
```  1894       thus ?thesis by (rule smsI)
```
```  1895     next
```
```  1896       assume [simp]: "A' = {#} \<and> B' = {#}"
```
```  1897       show ?thesis by (rule smsI) (auto intro: max)
```
```  1898     qed
```
```  1899     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
```
```  1900     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
```
```  1901   }
```
```  1902   from mx_or_empty
```
```  1903   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
```
```  1904   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
```
```  1905 qed
```
```  1906
```
```  1907 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
```
```  1908 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
```
```  1909 and nonempty_single: "{# x #} \<noteq> {#}"
```
```  1910 by auto
```
```  1911
```
```  1912 setup {*
```
```  1913 let
```
```  1914   fun msetT T = Type (@{type_name multiset}, [T]);
```
```  1915
```
```  1916   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
```
```  1917     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
```
```  1918     | mk_mset T (x :: xs) =
```
```  1919           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
```
```  1920                 mk_mset T [x] \$ mk_mset T xs
```
```  1921
```
```  1922   fun mset_member_tac m i =
```
```  1923       (if m <= 0 then
```
```  1924            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
```
```  1925        else
```
```  1926            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
```
```  1927
```
```  1928   val mset_nonempty_tac =
```
```  1929       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
```
```  1930
```
```  1931   val regroup_munion_conv =
```
```  1932       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
```
```  1933         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))
```
```  1934
```
```  1935   fun unfold_pwleq_tac i =
```
```  1936     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
```
```  1937       ORELSE (rtac @{thm pw_leq_lstep} i)
```
```  1938       ORELSE (rtac @{thm pw_leq_empty} i)
```
```  1939
```
```  1940   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
```
```  1941                       @{thm Un_insert_left}, @{thm Un_empty_left}]
```
```  1942 in
```
```  1943   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
```
```  1944   {
```
```  1945     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
```
```  1946     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
```
```  1947     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
```
```  1948     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
```
```  1949     reduction_pair= @{thm ms_reduction_pair}
```
```  1950   })
```
```  1951 end
```
```  1952 *}
```
```  1953
```
```  1954
```
```  1955 subsection {* Legacy theorem bindings *}
```
```  1956
```
```  1957 lemmas multi_count_eq = multiset_eq_iff [symmetric]
```
```  1958
```
```  1959 lemma union_commute: "M + N = N + (M::'a multiset)"
```
```  1960   by (fact add_commute)
```
```  1961
```
```  1962 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```  1963   by (fact add_assoc)
```
```  1964
```
```  1965 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```  1966   by (fact add_left_commute)
```
```  1967
```
```  1968 lemmas union_ac = union_assoc union_commute union_lcomm
```
```  1969
```
```  1970 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
```
```  1971   by (fact add_right_cancel)
```
```  1972
```
```  1973 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
```
```  1974   by (fact add_left_cancel)
```
```  1975
```
```  1976 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
```
```  1977   by (fact add_imp_eq)
```
```  1978
```
```  1979 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
```
```  1980   by (fact order_less_trans)
```
```  1981
```
```  1982 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
```
```  1983   by (fact inf.commute)
```
```  1984
```
```  1985 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
```
```  1986   by (fact inf.assoc [symmetric])
```
```  1987
```
```  1988 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
```
```  1989   by (fact inf.left_commute)
```
```  1990
```
```  1991 lemmas multiset_inter_ac =
```
```  1992   multiset_inter_commute
```
```  1993   multiset_inter_assoc
```
```  1994   multiset_inter_left_commute
```
```  1995
```
```  1996 lemma mult_less_not_refl:
```
```  1997   "\<not> M \<subset># (M::'a::order multiset)"
```
```  1998   by (fact multiset_order.less_irrefl)
```
```  1999
```
```  2000 lemma mult_less_trans:
```
```  2001   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
```
```  2002   by (fact multiset_order.less_trans)
```
```  2003
```
```  2004 lemma mult_less_not_sym:
```
```  2005   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
```
```  2006   by (fact multiset_order.less_not_sym)
```
```  2007
```
```  2008 lemma mult_less_asym:
```
```  2009   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
```
```  2010   by (fact multiset_order.less_asym)
```
```  2011
```
```  2012 ML {*
```
```  2013 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
```
```  2014                       (Const _ \$ t') =
```
```  2015     let
```
```  2016       val (maybe_opt, ps) =
```
```  2017         Nitpick_Model.dest_plain_fun t' ||> op ~~
```
```  2018         ||> map (apsnd (snd o HOLogic.dest_number))
```
```  2019       fun elems_for t =
```
```  2020         case AList.lookup (op =) ps t of
```
```  2021           SOME n => replicate n t
```
```  2022         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
```
```  2023     in
```
```  2024       case maps elems_for (all_values elem_T) @
```
```  2025            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
```
```  2026             else []) of
```
```  2027         [] => Const (@{const_name zero_class.zero}, T)
```
```  2028       | ts => foldl1 (fn (t1, t2) =>
```
```  2029                          Const (@{const_name plus_class.plus}, T --> T --> T)
```
```  2030                          \$ t1 \$ t2)
```
```  2031                      (map (curry (op \$) (Const (@{const_name single},
```
```  2032                                                 elem_T --> T))) ts)
```
```  2033     end
```
```  2034   | multiset_postproc _ _ _ _ t = t
```
```  2035 *}
```
```  2036
```
```  2037 declaration {*
```
```  2038 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
```
```  2039     multiset_postproc
```
```  2040 *}
```
```  2041
```
```  2042 end
```