src/HOL/Library/Multiset.thy
 author haftmann Mon Jul 07 08:47:17 2008 +0200 (2008-07-07) changeset 27487 c8a6ce181805 parent 27368 9f90ac19e32b child 27611 2c01c0bdb385 permissions -rw-r--r--
absolute imports of HOL/*.thy theories
```     1 (*  Title:      HOL/Library/Multiset.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
```
```     4 *)
```
```     5
```
```     6 header {* Multisets *}
```
```     7
```
```     8 theory Multiset
```
```     9 imports Plain "~~/src/HOL/List"
```
```    10 begin
```
```    11
```
```    12 subsection {* The type of multisets *}
```
```    13
```
```    14 typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"
```
```    15 proof
```
```    16   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
```
```    17 qed
```
```    18
```
```    19 lemmas multiset_typedef [simp] =
```
```    20     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
```
```    21   and [simp] = Rep_multiset_inject [symmetric]
```
```    22
```
```    23 definition
```
```    24   Mempty :: "'a multiset"  ("{#}") where
```
```    25   "{#} = Abs_multiset (\<lambda>a. 0)"
```
```    26
```
```    27 definition
```
```    28   single :: "'a => 'a multiset" where
```
```    29   "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
```
```    30
```
```    31 declare
```
```    32   Mempty_def[code func del] single_def[code func del]
```
```    33
```
```    34 definition
```
```    35   count :: "'a multiset => 'a => nat" where
```
```    36   "count = Rep_multiset"
```
```    37
```
```    38 definition
```
```    39   MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
```
```    40   "MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
```
```    41
```
```    42 abbreviation
```
```    43   Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
```
```    44   "a :# M == 0 < count M a"
```
```    45
```
```    46 notation (xsymbols)
```
```    47   Melem (infix "\<in>#" 50)
```
```    48
```
```    49 syntax
```
```    50   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ :# _./ _#})")
```
```    51 translations
```
```    52   "{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"
```
```    53
```
```    54 definition
```
```    55   set_of :: "'a multiset => 'a set" where
```
```    56   "set_of M = {x. x :# M}"
```
```    57
```
```    58 instantiation multiset :: (type) "{plus, minus, zero, size}"
```
```    59 begin
```
```    60
```
```    61 definition
```
```    62   union_def[code func del]:
```
```    63   "M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
```
```    64
```
```    65 definition
```
```    66   diff_def: "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
```
```    67
```
```    68 definition
```
```    69   Zero_multiset_def [simp]: "0 = {#}"
```
```    70
```
```    71 definition
```
```    72   size_def[code func del]: "size M = setsum (count M) (set_of M)"
```
```    73
```
```    74 instance ..
```
```    75
```
```    76 end
```
```    77
```
```    78 definition
```
```    79   multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where
```
```    80   "multiset_inter A B = A - (A - B)"
```
```    81
```
```    82 text {* Multiset Enumeration *}
```
```    83 syntax
```
```    84   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
```
```    85 translations
```
```    86   "{#x, xs#}" == "{#x#} + {#xs#}"
```
```    87   "{#x#}" == "CONST single x"
```
```    88
```
```    89
```
```    90 text {*
```
```    91  \medskip Preservation of the representing set @{term multiset}.
```
```    92 *}
```
```    93
```
```    94 lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
```
```    95 by (simp add: multiset_def)
```
```    96
```
```    97 lemma only1_in_multiset: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
```
```    98 by (simp add: multiset_def)
```
```    99
```
```   100 lemma union_preserves_multiset:
```
```   101   "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
```
```   102 apply (simp add: multiset_def)
```
```   103 apply (drule (1) finite_UnI)
```
```   104 apply (simp del: finite_Un add: Un_def)
```
```   105 done
```
```   106
```
```   107 lemma diff_preserves_multiset:
```
```   108   "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
```
```   109 apply (simp add: multiset_def)
```
```   110 apply (rule finite_subset)
```
```   111  apply auto
```
```   112 done
```
```   113
```
```   114 lemma MCollect_preserves_multiset:
```
```   115   "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
```
```   116 apply (simp add: multiset_def)
```
```   117 apply (rule finite_subset, auto)
```
```   118 done
```
```   119
```
```   120 lemmas in_multiset = const0_in_multiset only1_in_multiset
```
```   121   union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
```
```   122
```
```   123
```
```   124 subsection {* Algebraic properties *}
```
```   125
```
```   126 subsubsection {* Union *}
```
```   127
```
```   128 lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
```
```   129 by (simp add: union_def Mempty_def in_multiset)
```
```   130
```
```   131 lemma union_commute: "M + N = N + (M::'a multiset)"
```
```   132 by (simp add: union_def add_ac in_multiset)
```
```   133
```
```   134 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```   135 by (simp add: union_def add_ac in_multiset)
```
```   136
```
```   137 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```   138 proof -
```
```   139   have "M + (N + K) = (N + K) + M" by (rule union_commute)
```
```   140   also have "\<dots> = N + (K + M)" by (rule union_assoc)
```
```   141   also have "K + M = M + K" by (rule union_commute)
```
```   142   finally show ?thesis .
```
```   143 qed
```
```   144
```
```   145 lemmas union_ac = union_assoc union_commute union_lcomm
```
```   146
```
```   147 instance multiset :: (type) comm_monoid_add
```
```   148 proof
```
```   149   fix a b c :: "'a multiset"
```
```   150   show "(a + b) + c = a + (b + c)" by (rule union_assoc)
```
```   151   show "a + b = b + a" by (rule union_commute)
```
```   152   show "0 + a = a" by simp
```
```   153 qed
```
```   154
```
```   155
```
```   156 subsubsection {* Difference *}
```
```   157
```
```   158 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   159 by (simp add: Mempty_def diff_def in_multiset)
```
```   160
```
```   161 lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
```
```   162 by (simp add: union_def diff_def in_multiset)
```
```   163
```
```   164 lemma diff_cancel: "A - A = {#}"
```
```   165 by (simp add: diff_def Mempty_def)
```
```   166
```
```   167
```
```   168 subsubsection {* Count of elements *}
```
```   169
```
```   170 lemma count_empty [simp]: "count {#} a = 0"
```
```   171 by (simp add: count_def Mempty_def in_multiset)
```
```   172
```
```   173 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   174 by (simp add: count_def single_def in_multiset)
```
```   175
```
```   176 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   177 by (simp add: count_def union_def in_multiset)
```
```   178
```
```   179 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   180 by (simp add: count_def diff_def in_multiset)
```
```   181
```
```   182 lemma count_MCollect [simp]:
```
```   183   "count {# x:#M. P x #} a = (if P a then count M a else 0)"
```
```   184 by (simp add: count_def MCollect_def in_multiset)
```
```   185
```
```   186
```
```   187 subsubsection {* Set of elements *}
```
```   188
```
```   189 lemma set_of_empty [simp]: "set_of {#} = {}"
```
```   190 by (simp add: set_of_def)
```
```   191
```
```   192 lemma set_of_single [simp]: "set_of {#b#} = {b}"
```
```   193 by (simp add: set_of_def)
```
```   194
```
```   195 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   196 by (auto simp add: set_of_def)
```
```   197
```
```   198 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   199 by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])
```
```   200
```
```   201 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   202 by (auto simp add: set_of_def)
```
```   203
```
```   204 lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
```
```   205 by (auto simp add: set_of_def)
```
```   206
```
```   207
```
```   208 subsubsection {* Size *}
```
```   209
```
```   210 lemma size_empty [simp,code func]: "size {#} = 0"
```
```   211 by (simp add: size_def)
```
```   212
```
```   213 lemma size_single [simp,code func]: "size {#b#} = 1"
```
```   214 by (simp add: size_def)
```
```   215
```
```   216 lemma finite_set_of [iff]: "finite (set_of M)"
```
```   217 using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)
```
```   218
```
```   219 lemma setsum_count_Int:
```
```   220   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
```
```   221 apply (induct rule: finite_induct)
```
```   222  apply simp
```
```   223 apply (simp add: Int_insert_left set_of_def)
```
```   224 done
```
```   225
```
```   226 lemma size_union[simp,code func]: "size (M + N::'a multiset) = size M + size N"
```
```   227 apply (unfold size_def)
```
```   228 apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
```
```   229  prefer 2
```
```   230  apply (rule ext, simp)
```
```   231 apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
```
```   232 apply (subst Int_commute)
```
```   233 apply (simp (no_asm_simp) add: setsum_count_Int)
```
```   234 done
```
```   235
```
```   236 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   237 apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)
```
```   238 apply (simp add: set_of_def count_def in_multiset expand_fun_eq)
```
```   239 done
```
```   240
```
```   241 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
```
```   242 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
```
```   243
```
```   244 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   245 apply (unfold size_def)
```
```   246 apply (drule setsum_SucD)
```
```   247 apply auto
```
```   248 done
```
```   249
```
```   250
```
```   251 subsubsection {* Equality of multisets *}
```
```   252
```
```   253 lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
```
```   254 by (simp add: count_def expand_fun_eq)
```
```   255
```
```   256 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
```
```   257 by (simp add: single_def Mempty_def in_multiset expand_fun_eq)
```
```   258
```
```   259 lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
```
```   260 by (auto simp add: single_def in_multiset expand_fun_eq)
```
```   261
```
```   262 lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
```
```   263 by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
```
```   264
```
```   265 lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
```
```   266 by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
```
```   267
```
```   268 lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
```
```   269 by (simp add: union_def in_multiset expand_fun_eq)
```
```   270
```
```   271 lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
```
```   272 by (simp add: union_def in_multiset expand_fun_eq)
```
```   273
```
```   274 lemma union_is_single:
```
```   275   "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
```
```   276 apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)
```
```   277 apply blast
```
```   278 done
```
```   279
```
```   280 lemma single_is_union:
```
```   281   "({#a#} = M + N) \<longleftrightarrow> ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
```
```   282 apply (unfold Mempty_def single_def union_def)
```
```   283 apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq)
```
```   284 apply (blast dest: sym)
```
```   285 done
```
```   286
```
```   287 lemma add_eq_conv_diff:
```
```   288   "(M + {#a#} = N + {#b#}) =
```
```   289    (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
```
```   290 using [[simproc del: neq]]
```
```   291 apply (unfold single_def union_def diff_def)
```
```   292 apply (simp (no_asm) add: in_multiset expand_fun_eq)
```
```   293 apply (rule conjI, force, safe, simp_all)
```
```   294 apply (simp add: eq_sym_conv)
```
```   295 done
```
```   296
```
```   297 declare Rep_multiset_inject [symmetric, simp del]
```
```   298
```
```   299 instance multiset :: (type) cancel_ab_semigroup_add
```
```   300 proof
```
```   301   fix a b c :: "'a multiset"
```
```   302   show "a + b = a + c \<Longrightarrow> b = c" by simp
```
```   303 qed
```
```   304
```
```   305 lemma insert_DiffM:
```
```   306   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
```
```   307 by (clarsimp simp: multiset_eq_conv_count_eq)
```
```   308
```
```   309 lemma insert_DiffM2[simp]:
```
```   310   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
```
```   311 by (clarsimp simp: multiset_eq_conv_count_eq)
```
```   312
```
```   313 lemma multi_union_self_other_eq:
```
```   314   "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
```
```   315 by (induct A arbitrary: X Y) auto
```
```   316
```
```   317 lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"
```
```   318 by (metis single_not_empty union_empty union_left_cancel)
```
```   319
```
```   320 lemma insert_noteq_member:
```
```   321   assumes BC: "B + {#b#} = C + {#c#}"
```
```   322    and bnotc: "b \<noteq> c"
```
```   323   shows "c \<in># B"
```
```   324 proof -
```
```   325   have "c \<in># C + {#c#}" by simp
```
```   326   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
```
```   327   then have "c \<in># B + {#b#}" using BC by simp
```
```   328   then show "c \<in># B" using nc by simp
```
```   329 qed
```
```   330
```
```   331
```
```   332 lemma add_eq_conv_ex:
```
```   333   "(M + {#a#} = N + {#b#}) =
```
```   334     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
```
```   335 by (auto simp add: add_eq_conv_diff)
```
```   336
```
```   337
```
```   338 lemma empty_multiset_count:
```
```   339   "(\<forall>x. count A x = 0) = (A = {#})"
```
```   340 by (metis count_empty multiset_eq_conv_count_eq)
```
```   341
```
```   342
```
```   343 subsubsection {* Intersection *}
```
```   344
```
```   345 lemma multiset_inter_count:
```
```   346   "count (A #\<inter> B) x = min (count A x) (count B x)"
```
```   347 by (simp add: multiset_inter_def min_def)
```
```   348
```
```   349 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
```
```   350 by (simp add: multiset_eq_conv_count_eq multiset_inter_count
```
```   351     min_max.inf_commute)
```
```   352
```
```   353 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
```
```   354 by (simp add: multiset_eq_conv_count_eq multiset_inter_count
```
```   355     min_max.inf_assoc)
```
```   356
```
```   357 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
```
```   358 by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
```
```   359
```
```   360 lemmas multiset_inter_ac =
```
```   361   multiset_inter_commute
```
```   362   multiset_inter_assoc
```
```   363   multiset_inter_left_commute
```
```   364
```
```   365 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
```
```   366 by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
```
```   367
```
```   368 lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
```
```   369 apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
```
```   370     split: split_if_asm)
```
```   371 apply clarsimp
```
```   372 apply (erule_tac x = a in allE)
```
```   373 apply auto
```
```   374 done
```
```   375
```
```   376
```
```   377 subsubsection {* Comprehension (filter) *}
```
```   378
```
```   379 lemma MCollect_empty[simp, code func]: "MCollect {#} P = {#}"
```
```   380 by (simp add: MCollect_def Mempty_def Abs_multiset_inject
```
```   381     in_multiset expand_fun_eq)
```
```   382
```
```   383 lemma MCollect_single[simp, code func]:
```
```   384   "MCollect {#x#} P = (if P x then {#x#} else {#})"
```
```   385 by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
```
```   386     in_multiset expand_fun_eq)
```
```   387
```
```   388 lemma MCollect_union[simp, code func]:
```
```   389   "MCollect (M+N) f = MCollect M f + MCollect N f"
```
```   390 by (simp add: MCollect_def union_def Abs_multiset_inject
```
```   391     in_multiset expand_fun_eq)
```
```   392
```
```   393
```
```   394 subsection {* Induction and case splits *}
```
```   395
```
```   396 lemma setsum_decr:
```
```   397   "finite F ==> (0::nat) < f a ==>
```
```   398     setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
```
```   399 apply (induct rule: finite_induct)
```
```   400  apply auto
```
```   401 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   402 done
```
```   403
```
```   404 lemma rep_multiset_induct_aux:
```
```   405 assumes 1: "P (\<lambda>a. (0::nat))"
```
```   406   and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
```
```   407 shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
```
```   408 apply (unfold multiset_def)
```
```   409 apply (induct_tac n, simp, clarify)
```
```   410  apply (subgoal_tac "f = (\<lambda>a.0)")
```
```   411   apply simp
```
```   412   apply (rule 1)
```
```   413  apply (rule ext, force, clarify)
```
```   414 apply (frule setsum_SucD, clarify)
```
```   415 apply (rename_tac a)
```
```   416 apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
```
```   417  prefer 2
```
```   418  apply (rule finite_subset)
```
```   419   prefer 2
```
```   420   apply assumption
```
```   421  apply simp
```
```   422  apply blast
```
```   423 apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
```
```   424  prefer 2
```
```   425  apply (rule ext)
```
```   426  apply (simp (no_asm_simp))
```
```   427  apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
```
```   428 apply (erule allE, erule impE, erule_tac [2] mp, blast)
```
```   429 apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
```
```   430 apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
```
```   431  prefer 2
```
```   432  apply blast
```
```   433 apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
```
```   434  prefer 2
```
```   435  apply blast
```
```   436 apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
```
```   437 done
```
```   438
```
```   439 theorem rep_multiset_induct:
```
```   440   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
```
```   441     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
```
```   442 using rep_multiset_induct_aux by blast
```
```   443
```
```   444 theorem multiset_induct [case_names empty add, induct type: multiset]:
```
```   445 assumes empty: "P {#}"
```
```   446   and add: "!!M x. P M ==> P (M + {#x#})"
```
```   447 shows "P M"
```
```   448 proof -
```
```   449   note defns = union_def single_def Mempty_def
```
```   450   show ?thesis
```
```   451     apply (rule Rep_multiset_inverse [THEN subst])
```
```   452     apply (rule Rep_multiset [THEN rep_multiset_induct])
```
```   453      apply (rule empty [unfolded defns])
```
```   454     apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
```
```   455      prefer 2
```
```   456      apply (simp add: expand_fun_eq)
```
```   457     apply (erule ssubst)
```
```   458     apply (erule Abs_multiset_inverse [THEN subst])
```
```   459     apply (drule add [unfolded defns, simplified])
```
```   460     apply(simp add:in_multiset)
```
```   461     done
```
```   462 qed
```
```   463
```
```   464 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
```
```   465 by (induct M) auto
```
```   466
```
```   467 lemma multiset_cases [cases type, case_names empty add]:
```
```   468 assumes em:  "M = {#} \<Longrightarrow> P"
```
```   469 assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
```
```   470 shows "P"
```
```   471 proof (cases "M = {#}")
```
```   472   assume "M = {#}" then show ?thesis using em by simp
```
```   473 next
```
```   474   assume "M \<noteq> {#}"
```
```   475   then obtain M' m where "M = M' + {#m#}"
```
```   476     by (blast dest: multi_nonempty_split)
```
```   477   then show ?thesis using add by simp
```
```   478 qed
```
```   479
```
```   480 lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
```
```   481 apply (cases M)
```
```   482  apply simp
```
```   483 apply (rule_tac x="M - {#x#}" in exI, simp)
```
```   484 done
```
```   485
```
```   486 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
```
```   487 apply (subst multiset_eq_conv_count_eq)
```
```   488 apply auto
```
```   489 done
```
```   490
```
```   491 declare multiset_typedef [simp del]
```
```   492
```
```   493 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
```
```   494 by (cases "B = {#}") (auto dest: multi_member_split)
```
```   495
```
```   496
```
```   497 subsection {* Orderings *}
```
```   498
```
```   499 subsubsection {* Well-foundedness *}
```
```   500
```
```   501 definition
```
```   502   mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```   503   [code func del]:"mult1 r =
```
```   504     {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```   505       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```   506
```
```   507 definition
```
```   508   mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```   509   "mult r = (mult1 r)\<^sup>+"
```
```   510
```
```   511 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```   512 by (simp add: mult1_def)
```
```   513
```
```   514 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```   515     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```   516     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```   517   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
```
```   518 proof (unfold mult1_def)
```
```   519   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```   520   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```   521   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```   522
```
```   523   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```   524   then have "\<exists>a' M0' K.
```
```   525       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```   526   then show "?case1 \<or> ?case2"
```
```   527   proof (elim exE conjE)
```
```   528     fix a' M0' K
```
```   529     assume N: "N = M0' + K" and r: "?r K a'"
```
```   530     assume "M0 + {#a#} = M0' + {#a'#}"
```
```   531     then have "M0 = M0' \<and> a = a' \<or>
```
```   532         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```   533       by (simp only: add_eq_conv_ex)
```
```   534     then show ?thesis
```
```   535     proof (elim disjE conjE exE)
```
```   536       assume "M0 = M0'" "a = a'"
```
```   537       with N r have "?r K a \<and> N = M0 + K" by simp
```
```   538       then have ?case2 .. then show ?thesis ..
```
```   539     next
```
```   540       fix K'
```
```   541       assume "M0' = K' + {#a#}"
```
```   542       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
```
```   543
```
```   544       assume "M0 = K' + {#a'#}"
```
```   545       with r have "?R (K' + K) M0" by blast
```
```   546       with n have ?case1 by simp then show ?thesis ..
```
```   547     qed
```
```   548   qed
```
```   549 qed
```
```   550
```
```   551 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
```
```   552 proof
```
```   553   let ?R = "mult1 r"
```
```   554   let ?W = "acc ?R"
```
```   555   {
```
```   556     fix M M0 a
```
```   557     assume M0: "M0 \<in> ?W"
```
```   558       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```   559       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```   560     have "M0 + {#a#} \<in> ?W"
```
```   561     proof (rule accI [of "M0 + {#a#}"])
```
```   562       fix N
```
```   563       assume "(N, M0 + {#a#}) \<in> ?R"
```
```   564       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```   565           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```   566         by (rule less_add)
```
```   567       then show "N \<in> ?W"
```
```   568       proof (elim exE disjE conjE)
```
```   569         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```   570         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```   571         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
```
```   572         then show "N \<in> ?W" by (simp only: N)
```
```   573       next
```
```   574         fix K
```
```   575         assume N: "N = M0 + K"
```
```   576         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```   577         then have "M0 + K \<in> ?W"
```
```   578         proof (induct K)
```
```   579           case empty
```
```   580           from M0 show "M0 + {#} \<in> ?W" by simp
```
```   581         next
```
```   582           case (add K x)
```
```   583           from add.prems have "(x, a) \<in> r" by simp
```
```   584           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```   585           moreover from add have "M0 + K \<in> ?W" by simp
```
```   586           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
```
```   587           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
```
```   588         qed
```
```   589         then show "N \<in> ?W" by (simp only: N)
```
```   590       qed
```
```   591     qed
```
```   592   } note tedious_reasoning = this
```
```   593
```
```   594   assume wf: "wf r"
```
```   595   fix M
```
```   596   show "M \<in> ?W"
```
```   597   proof (induct M)
```
```   598     show "{#} \<in> ?W"
```
```   599     proof (rule accI)
```
```   600       fix b assume "(b, {#}) \<in> ?R"
```
```   601       with not_less_empty show "b \<in> ?W" by contradiction
```
```   602     qed
```
```   603
```
```   604     fix M a assume "M \<in> ?W"
```
```   605     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```   606     proof induct
```
```   607       fix a
```
```   608       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```   609       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```   610       proof
```
```   611         fix M assume "M \<in> ?W"
```
```   612         then show "M + {#a#} \<in> ?W"
```
```   613           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
```
```   614       qed
```
```   615     qed
```
```   616     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
```
```   617   qed
```
```   618 qed
```
```   619
```
```   620 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```   621 by (rule acc_wfI) (rule all_accessible)
```
```   622
```
```   623 theorem wf_mult: "wf r ==> wf (mult r)"
```
```   624 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
```
```   625
```
```   626
```
```   627 subsubsection {* Closure-free presentation *}
```
```   628
```
```   629 (*Badly needed: a linear arithmetic procedure for multisets*)
```
```   630
```
```   631 lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
```
```   632 by (simp add: multiset_eq_conv_count_eq)
```
```   633
```
```   634 text {* One direction. *}
```
```   635
```
```   636 lemma mult_implies_one_step:
```
```   637   "trans r ==> (M, N) \<in> mult r ==>
```
```   638     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```   639     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```   640 apply (unfold mult_def mult1_def set_of_def)
```
```   641 apply (erule converse_trancl_induct, clarify)
```
```   642  apply (rule_tac x = M0 in exI, simp, clarify)
```
```   643 apply (case_tac "a :# K")
```
```   644  apply (rule_tac x = I in exI)
```
```   645  apply (simp (no_asm))
```
```   646  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```   647  apply (simp (no_asm_simp) add: union_assoc [symmetric])
```
```   648  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```   649  apply (simp add: diff_union_single_conv)
```
```   650  apply (simp (no_asm_use) add: trans_def)
```
```   651  apply blast
```
```   652 apply (subgoal_tac "a :# I")
```
```   653  apply (rule_tac x = "I - {#a#}" in exI)
```
```   654  apply (rule_tac x = "J + {#a#}" in exI)
```
```   655  apply (rule_tac x = "K + Ka" in exI)
```
```   656  apply (rule conjI)
```
```   657   apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   658  apply (rule conjI)
```
```   659   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
```
```   660   apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   661  apply (simp (no_asm_use) add: trans_def)
```
```   662  apply blast
```
```   663 apply (subgoal_tac "a :# (M0 + {#a#})")
```
```   664  apply simp
```
```   665 apply (simp (no_asm))
```
```   666 done
```
```   667
```
```   668 lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
```
```   669 by (simp add: multiset_eq_conv_count_eq)
```
```   670
```
```   671 lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
```
```   672 apply (erule size_eq_Suc_imp_elem [THEN exE])
```
```   673 apply (drule elem_imp_eq_diff_union, auto)
```
```   674 done
```
```   675
```
```   676 lemma one_step_implies_mult_aux:
```
```   677   "trans r ==>
```
```   678     \<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))
```
```   679       --> (I + K, I + J) \<in> mult r"
```
```   680 apply (induct_tac n, auto)
```
```   681 apply (frule size_eq_Suc_imp_eq_union, clarify)
```
```   682 apply (rename_tac "J'", simp)
```
```   683 apply (erule notE, auto)
```
```   684 apply (case_tac "J' = {#}")
```
```   685  apply (simp add: mult_def)
```
```   686  apply (rule r_into_trancl)
```
```   687  apply (simp add: mult1_def set_of_def, blast)
```
```   688 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```   689 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```   690 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```   691 apply (erule ssubst)
```
```   692 apply (simp add: Ball_def, auto)
```
```   693 apply (subgoal_tac
```
```   694   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
```
```   695     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```   696  prefer 2
```
```   697  apply force
```
```   698 apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
```
```   699 apply (erule trancl_trans)
```
```   700 apply (rule r_into_trancl)
```
```   701 apply (simp add: mult1_def set_of_def)
```
```   702 apply (rule_tac x = a in exI)
```
```   703 apply (rule_tac x = "I + J'" in exI)
```
```   704 apply (simp add: union_ac)
```
```   705 done
```
```   706
```
```   707 lemma one_step_implies_mult:
```
```   708   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```   709     ==> (I + K, I + J) \<in> mult r"
```
```   710 using one_step_implies_mult_aux by blast
```
```   711
```
```   712
```
```   713 subsubsection {* Partial-order properties *}
```
```   714
```
```   715 instantiation multiset :: (order) order
```
```   716 begin
```
```   717
```
```   718 definition
```
```   719   less_multiset_def [code func del]: "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
```
```   720
```
```   721 definition
```
```   722   le_multiset_def [code func del]: "M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"
```
```   723
```
```   724 lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
```
```   725 unfolding trans_def by (blast intro: order_less_trans)
```
```   726
```
```   727 text {*
```
```   728  \medskip Irreflexivity.
```
```   729 *}
```
```   730
```
```   731 lemma mult_irrefl_aux:
```
```   732   "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
```
```   733 by (induct rule: finite_induct) (auto intro: order_less_trans)
```
```   734
```
```   735 lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
```
```   736 apply (unfold less_multiset_def, auto)
```
```   737 apply (drule trans_base_order [THEN mult_implies_one_step], auto)
```
```   738 apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
```
```   739 apply (simp add: set_of_eq_empty_iff)
```
```   740 done
```
```   741
```
```   742 lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
```
```   743 using insert mult_less_not_refl by fast
```
```   744
```
```   745
```
```   746 text {* Transitivity. *}
```
```   747
```
```   748 theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
```
```   749 unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
```
```   750
```
```   751 text {* Asymmetry. *}
```
```   752
```
```   753 theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
```
```   754 apply auto
```
```   755 apply (rule mult_less_not_refl [THEN notE])
```
```   756 apply (erule mult_less_trans, assumption)
```
```   757 done
```
```   758
```
```   759 theorem mult_less_asym:
```
```   760   "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
```
```   761 using mult_less_not_sym by blast
```
```   762
```
```   763 theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
```
```   764 unfolding le_multiset_def by auto
```
```   765
```
```   766 text {* Anti-symmetry. *}
```
```   767
```
```   768 theorem mult_le_antisym:
```
```   769   "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
```
```   770 unfolding le_multiset_def by (blast dest: mult_less_not_sym)
```
```   771
```
```   772 text {* Transitivity. *}
```
```   773
```
```   774 theorem mult_le_trans:
```
```   775   "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
```
```   776 unfolding le_multiset_def by (blast intro: mult_less_trans)
```
```   777
```
```   778 theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
```
```   779 unfolding le_multiset_def by auto
```
```   780
```
```   781 instance
```
```   782 apply intro_classes
```
```   783    apply (rule mult_less_le)
```
```   784   apply (rule mult_le_refl)
```
```   785  apply (erule mult_le_trans, assumption)
```
```   786 apply (erule mult_le_antisym, assumption)
```
```   787 done
```
```   788
```
```   789 end
```
```   790
```
```   791
```
```   792 subsubsection {* Monotonicity of multiset union *}
```
```   793
```
```   794 lemma mult1_union:
```
```   795   "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
```
```   796 apply (unfold mult1_def)
```
```   797 apply auto
```
```   798 apply (rule_tac x = a in exI)
```
```   799 apply (rule_tac x = "C + M0" in exI)
```
```   800 apply (simp add: union_assoc)
```
```   801 done
```
```   802
```
```   803 lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
```
```   804 apply (unfold less_multiset_def mult_def)
```
```   805 apply (erule trancl_induct)
```
```   806  apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
```
```   807 apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
```
```   808 done
```
```   809
```
```   810 lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
```
```   811 apply (subst union_commute [of B C])
```
```   812 apply (subst union_commute [of D C])
```
```   813 apply (erule union_less_mono2)
```
```   814 done
```
```   815
```
```   816 lemma union_less_mono:
```
```   817   "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
```
```   818 by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
```
```   819
```
```   820 lemma union_le_mono:
```
```   821   "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
```
```   822 unfolding le_multiset_def
```
```   823 by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
```
```   824
```
```   825 lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
```
```   826 apply (unfold le_multiset_def less_multiset_def)
```
```   827 apply (case_tac "M = {#}")
```
```   828  prefer 2
```
```   829  apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
```
```   830   prefer 2
```
```   831   apply (rule one_step_implies_mult)
```
```   832     apply (simp only: trans_def)
```
```   833     apply auto
```
```   834 done
```
```   835
```
```   836 lemma union_upper1: "A <= A + (B::'a::order multiset)"
```
```   837 proof -
```
```   838   have "A + {#} <= A + B" by (blast intro: union_le_mono)
```
```   839   then show ?thesis by simp
```
```   840 qed
```
```   841
```
```   842 lemma union_upper2: "B <= A + (B::'a::order multiset)"
```
```   843 by (subst union_commute) (rule union_upper1)
```
```   844
```
```   845 instance multiset :: (order) pordered_ab_semigroup_add
```
```   846 apply intro_classes
```
```   847 apply (erule union_le_mono[OF mult_le_refl])
```
```   848 done
```
```   849
```
```   850
```
```   851 subsection {* Link with lists *}
```
```   852
```
```   853 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
```
```   854   "multiset_of [] = {#}" |
```
```   855   "multiset_of (a # x) = multiset_of x + {# a #}"
```
```   856
```
```   857 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
```
```   858 by (induct x) auto
```
```   859
```
```   860 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
```
```   861 by (induct x) auto
```
```   862
```
```   863 lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
```
```   864 by (induct x) auto
```
```   865
```
```   866 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
```
```   867 by (induct xs) auto
```
```   868
```
```   869 lemma multiset_of_append [simp]:
```
```   870   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
```
```   871 by (induct xs arbitrary: ys) (auto simp: union_ac)
```
```   872
```
```   873 lemma surj_multiset_of: "surj multiset_of"
```
```   874 apply (unfold surj_def)
```
```   875 apply (rule allI)
```
```   876 apply (rule_tac M = y in multiset_induct)
```
```   877  apply auto
```
```   878 apply (rule_tac x = "x # xa" in exI)
```
```   879 apply auto
```
```   880 done
```
```   881
```
```   882 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
```
```   883 by (induct x) auto
```
```   884
```
```   885 lemma distinct_count_atmost_1:
```
```   886   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
```
```   887 apply (induct x, simp, rule iffI, simp_all)
```
```   888 apply (rule conjI)
```
```   889 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
```
```   890 apply (erule_tac x = a in allE, simp, clarify)
```
```   891 apply (erule_tac x = aa in allE, simp)
```
```   892 done
```
```   893
```
```   894 lemma multiset_of_eq_setD:
```
```   895   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
```
```   896 by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
```
```   897
```
```   898 lemma set_eq_iff_multiset_of_eq_distinct:
```
```   899   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
```
```   900     (set x = set y) = (multiset_of x = multiset_of y)"
```
```   901 by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
```
```   902
```
```   903 lemma set_eq_iff_multiset_of_remdups_eq:
```
```   904    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
```
```   905 apply (rule iffI)
```
```   906 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
```
```   907 apply (drule distinct_remdups [THEN distinct_remdups
```
```   908       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
```
```   909 apply simp
```
```   910 done
```
```   911
```
```   912 lemma multiset_of_compl_union [simp]:
```
```   913   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
```
```   914 by (induct xs) (auto simp: union_ac)
```
```   915
```
```   916 lemma count_filter:
```
```   917   "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
```
```   918 by (induct xs) auto
```
```   919
```
```   920 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
```
```   921 apply (induct ls arbitrary: i)
```
```   922  apply simp
```
```   923 apply (case_tac i)
```
```   924  apply auto
```
```   925 done
```
```   926
```
```   927 lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
```
```   928 by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
```
```   929
```
```   930 lemma multiset_of_eq_length:
```
```   931 assumes "multiset_of xs = multiset_of ys"
```
```   932 shows "length xs = length ys"
```
```   933 using assms
```
```   934 proof (induct arbitrary: ys rule: length_induct)
```
```   935   case (1 xs ys)
```
```   936   show ?case
```
```   937   proof (cases xs)
```
```   938     case Nil with "1.prems" show ?thesis by simp
```
```   939   next
```
```   940     case (Cons x xs')
```
```   941     note xCons = Cons
```
```   942     show ?thesis
```
```   943     proof (cases ys)
```
```   944       case Nil
```
```   945       with "1.prems" Cons show ?thesis by simp
```
```   946     next
```
```   947       case (Cons y ys')
```
```   948       have x_in_ys: "x = y \<or> x \<in> set ys'"
```
```   949       proof (cases "x = y")
```
```   950 	case True then show ?thesis ..
```
```   951       next
```
```   952 	case False
```
```   953 	from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
```
```   954 	with False show ?thesis by (simp add: mem_set_multiset_eq)
```
```   955       qed
```
```   956       from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
```
```   957 	(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
```
```   958       from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
```
```   959 	apply -
```
```   960 	apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
```
```   961 	apply fastsimp
```
```   962 	done
```
```   963       with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
```
```   964       from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
```
```   965       with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
```
```   966     qed
```
```   967   qed
```
```   968 qed
```
```   969
```
```   970 text {*
```
```   971   This lemma shows which properties suffice to show that a function
```
```   972   @{text "f"} with @{text "f xs = ys"} behaves like sort.
```
```   973 *}
```
```   974 lemma properties_for_sort:
```
```   975   "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
```
```   976 proof (induct xs arbitrary: ys)
```
```   977   case Nil then show ?case by simp
```
```   978 next
```
```   979   case (Cons x xs)
```
```   980   then have "x \<in> set ys"
```
```   981     by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
```
```   982   with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
```
```   983     by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
```
```   984 qed
```
```   985
```
```   986
```
```   987 subsection {* Pointwise ordering induced by count *}
```
```   988
```
```   989 definition
```
```   990   mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
```
```   991   [code func del]: "(A \<le># B) = (\<forall>a. count A a \<le> count B a)"
```
```   992
```
```   993 definition
```
```   994   mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
```
```   995   [code func del]: "(A <# B) = (A \<le># B \<and> A \<noteq> B)"
```
```   996
```
```   997 notation mset_le  (infix "\<subseteq>#" 50)
```
```   998 notation mset_less  (infix "\<subset>#" 50)
```
```   999
```
```  1000 lemma mset_le_refl[simp]: "A \<le># A"
```
```  1001 unfolding mset_le_def by auto
```
```  1002
```
```  1003 lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
```
```  1004 unfolding mset_le_def by (fast intro: order_trans)
```
```  1005
```
```  1006 lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
```
```  1007 apply (unfold mset_le_def)
```
```  1008 apply (rule multiset_eq_conv_count_eq [THEN iffD2])
```
```  1009 apply (blast intro: order_antisym)
```
```  1010 done
```
```  1011
```
```  1012 lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
```
```  1013 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
```
```  1014 apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
```
```  1015 done
```
```  1016
```
```  1017 lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
```
```  1018 unfolding mset_le_def by auto
```
```  1019
```
```  1020 lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
```
```  1021 unfolding mset_le_def by auto
```
```  1022
```
```  1023 lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
```
```  1024 apply (unfold mset_le_def)
```
```  1025 apply auto
```
```  1026 apply (erule_tac x = a in allE)+
```
```  1027 apply auto
```
```  1028 done
```
```  1029
```
```  1030 lemma mset_le_add_left[simp]: "A \<le># A + B"
```
```  1031 unfolding mset_le_def by auto
```
```  1032
```
```  1033 lemma mset_le_add_right[simp]: "B \<le># A + B"
```
```  1034 unfolding mset_le_def by auto
```
```  1035
```
```  1036 lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
```
```  1037 by (simp add: mset_le_def)
```
```  1038
```
```  1039 lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
```
```  1040 by (simp add: multiset_eq_conv_count_eq mset_le_def)
```
```  1041
```
```  1042 lemma mset_le_multiset_union_diff_commute:
```
```  1043 assumes "B \<le># A"
```
```  1044 shows "A - B + C = A + C - B"
```
```  1045 proof -
```
```  1046   from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
```
```  1047   from this obtain D where "A = B + D" ..
```
```  1048   then show ?thesis
```
```  1049     apply simp
```
```  1050     apply (subst union_commute)
```
```  1051     apply (subst multiset_diff_union_assoc)
```
```  1052     apply simp
```
```  1053     apply (simp add: diff_cancel)
```
```  1054     apply (subst union_assoc)
```
```  1055     apply (subst union_commute[of "B" _])
```
```  1056     apply (subst multiset_diff_union_assoc)
```
```  1057     apply simp
```
```  1058     apply (simp add: diff_cancel)
```
```  1059     done
```
```  1060 qed
```
```  1061
```
```  1062 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
```
```  1063 apply (induct xs)
```
```  1064  apply auto
```
```  1065 apply (rule mset_le_trans)
```
```  1066  apply auto
```
```  1067 done
```
```  1068
```
```  1069 lemma multiset_of_update:
```
```  1070   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
```
```  1071 proof (induct ls arbitrary: i)
```
```  1072   case Nil then show ?case by simp
```
```  1073 next
```
```  1074   case (Cons x xs)
```
```  1075   show ?case
```
```  1076   proof (cases i)
```
```  1077     case 0 then show ?thesis by simp
```
```  1078   next
```
```  1079     case (Suc i')
```
```  1080     with Cons show ?thesis
```
```  1081       apply simp
```
```  1082       apply (subst union_assoc)
```
```  1083       apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
```
```  1084       apply (subst union_assoc [symmetric])
```
```  1085       apply simp
```
```  1086       apply (rule mset_le_multiset_union_diff_commute)
```
```  1087       apply (simp add: mset_le_single nth_mem_multiset_of)
```
```  1088       done
```
```  1089   qed
```
```  1090 qed
```
```  1091
```
```  1092 lemma multiset_of_swap:
```
```  1093   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
```
```  1094     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
```
```  1095 apply (case_tac "i = j")
```
```  1096  apply simp
```
```  1097 apply (simp add: multiset_of_update)
```
```  1098 apply (subst elem_imp_eq_diff_union[symmetric])
```
```  1099  apply (simp add: nth_mem_multiset_of)
```
```  1100 apply simp
```
```  1101 done
```
```  1102
```
```  1103 interpretation mset_order: order ["op \<le>#" "op <#"]
```
```  1104 by (auto intro: order.intro mset_le_refl mset_le_antisym
```
```  1105     mset_le_trans simp: mset_less_def)
```
```  1106
```
```  1107 interpretation mset_order_cancel_semigroup:
```
```  1108     pordered_cancel_ab_semigroup_add ["op +" "op \<le>#" "op <#"]
```
```  1109 by unfold_locales (erule mset_le_mono_add [OF mset_le_refl])
```
```  1110
```
```  1111 interpretation mset_order_semigroup_cancel:
```
```  1112     pordered_ab_semigroup_add_imp_le ["op +" "op \<le>#" "op <#"]
```
```  1113 by (unfold_locales) simp
```
```  1114
```
```  1115
```
```  1116 lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```  1117 apply (clarsimp simp: mset_le_def mset_less_def)
```
```  1118 apply (erule_tac x=x in allE)
```
```  1119 apply auto
```
```  1120 done
```
```  1121
```
```  1122 lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```  1123 apply (clarsimp simp: mset_le_def mset_less_def)
```
```  1124 apply (erule_tac x = x in allE)
```
```  1125 apply auto
```
```  1126 done
```
```  1127
```
```  1128 lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
```
```  1129 apply (rule conjI)
```
```  1130  apply (simp add: mset_lessD)
```
```  1131 apply (clarsimp simp: mset_le_def mset_less_def)
```
```  1132 apply safe
```
```  1133  apply (erule_tac x = a in allE)
```
```  1134  apply (auto split: split_if_asm)
```
```  1135 done
```
```  1136
```
```  1137 lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
```
```  1138 apply (rule conjI)
```
```  1139  apply (simp add: mset_leD)
```
```  1140 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
```
```  1141 done
```
```  1142
```
```  1143 lemma mset_less_of_empty[simp]: "A \<subset># {#} = False"
```
```  1144 by (induct A) (auto simp: mset_le_def mset_less_def)
```
```  1145
```
```  1146 lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
```
```  1147 by (auto simp: mset_le_def mset_less_def)
```
```  1148
```
```  1149 lemma multi_psub_self[simp]: "A \<subset># A = False"
```
```  1150 by (auto simp: mset_le_def mset_less_def)
```
```  1151
```
```  1152 lemma mset_less_add_bothsides:
```
```  1153   "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
```
```  1154 by (auto simp: mset_le_def mset_less_def)
```
```  1155
```
```  1156 lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
```
```  1157 by (auto simp: mset_le_def mset_less_def)
```
```  1158
```
```  1159 lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
```
```  1160 proof (induct A arbitrary: B)
```
```  1161   case (empty M)
```
```  1162   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
```
```  1163   then obtain M' x where "M = M' + {#x#}"
```
```  1164     by (blast dest: multi_nonempty_split)
```
```  1165   then show ?case by simp
```
```  1166 next
```
```  1167   case (add S x T)
```
```  1168   have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
```
```  1169   have SxsubT: "S + {#x#} \<subset># T" by fact
```
```  1170   then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
```
```  1171   then obtain T' where T: "T = T' + {#x#}"
```
```  1172     by (blast dest: multi_member_split)
```
```  1173   then have "S \<subset># T'" using SxsubT
```
```  1174     by (blast intro: mset_less_add_bothsides)
```
```  1175   then have "size S < size T'" using IH by simp
```
```  1176   then show ?case using T by simp
```
```  1177 qed
```
```  1178
```
```  1179 lemmas mset_less_trans = mset_order.less_eq_less.less_trans
```
```  1180
```
```  1181 lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
```
```  1182 by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
```
```  1183
```
```  1184
```
```  1185 subsection {* Strong induction and subset induction for multisets *}
```
```  1186
```
```  1187 text {* Well-foundedness of proper subset operator: *}
```
```  1188
```
```  1189 text {* proper multiset subset *}
```
```  1190 definition
```
```  1191   mset_less_rel :: "('a multiset * 'a multiset) set" where
```
```  1192   "mset_less_rel = {(A,B). A \<subset># B}"
```
```  1193
```
```  1194 lemma multiset_add_sub_el_shuffle:
```
```  1195   assumes "c \<in># B" and "b \<noteq> c"
```
```  1196   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
```
```  1197 proof -
```
```  1198   from `c \<in># B` obtain A where B: "B = A + {#c#}"
```
```  1199     by (blast dest: multi_member_split)
```
```  1200   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
```
```  1201   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
```
```  1202     by (simp add: union_ac)
```
```  1203   then show ?thesis using B by simp
```
```  1204 qed
```
```  1205
```
```  1206 lemma wf_mset_less_rel: "wf mset_less_rel"
```
```  1207 apply (unfold mset_less_rel_def)
```
```  1208 apply (rule wf_measure [THEN wf_subset, where f1=size])
```
```  1209 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
```
```  1210 done
```
```  1211
```
```  1212 text {* The induction rules: *}
```
```  1213
```
```  1214 lemma full_multiset_induct [case_names less]:
```
```  1215 assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
```
```  1216 shows "P B"
```
```  1217 apply (rule wf_mset_less_rel [THEN wf_induct])
```
```  1218 apply (rule ih, auto simp: mset_less_rel_def)
```
```  1219 done
```
```  1220
```
```  1221 lemma multi_subset_induct [consumes 2, case_names empty add]:
```
```  1222 assumes "F \<subseteq># A"
```
```  1223   and empty: "P {#}"
```
```  1224   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
```
```  1225 shows "P F"
```
```  1226 proof -
```
```  1227   from `F \<subseteq># A`
```
```  1228   show ?thesis
```
```  1229   proof (induct F)
```
```  1230     show "P {#}" by fact
```
```  1231   next
```
```  1232     fix x F
```
```  1233     assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
```
```  1234     show "P (F + {#x#})"
```
```  1235     proof (rule insert)
```
```  1236       from i show "x \<in># A" by (auto dest: mset_le_insertD)
```
```  1237       from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
```
```  1238       with P show "P F" .
```
```  1239     qed
```
```  1240   qed
```
```  1241 qed
```
```  1242
```
```  1243 text{* A consequence: Extensionality. *}
```
```  1244
```
```  1245 lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
```
```  1246 apply (rule iffI)
```
```  1247  prefer 2
```
```  1248  apply clarsimp
```
```  1249 apply (induct A arbitrary: B rule: full_multiset_induct)
```
```  1250 apply (rename_tac C)
```
```  1251 apply (case_tac B rule: multiset_cases)
```
```  1252  apply (simp add: empty_multiset_count)
```
```  1253 apply simp
```
```  1254 apply (case_tac "x \<in># C")
```
```  1255  apply (force dest: multi_member_split)
```
```  1256 apply (erule_tac x = x in allE)
```
```  1257 apply simp
```
```  1258 done
```
```  1259
```
```  1260 lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
```
```  1261
```
```  1262
```
```  1263 subsection {* The fold combinator *}
```
```  1264
```
```  1265 text {*
```
```  1266   The intended behaviour is
```
```  1267   @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
```
```  1268   if @{text f} is associative-commutative.
```
```  1269 *}
```
```  1270
```
```  1271 text {*
```
```  1272   The graph of @{text "fold_mset"}, @{text "z"}: the start element,
```
```  1273   @{text "f"}: folding function, @{text "A"}: the multiset, @{text
```
```  1274   "y"}: the result.
```
```  1275 *}
```
```  1276 inductive
```
```  1277   fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool"
```
```  1278   for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1279   and z :: 'b
```
```  1280 where
```
```  1281   emptyI [intro]:  "fold_msetG f z {#} z"
```
```  1282 | insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
```
```  1283
```
```  1284 inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
```
```  1285 inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y"
```
```  1286
```
```  1287 definition
```
```  1288   fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
```
```  1289   "fold_mset f z A = (THE x. fold_msetG f z A x)"
```
```  1290
```
```  1291 lemma Diff1_fold_msetG:
```
```  1292   "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
```
```  1293 apply (frule_tac x = x in fold_msetG.insertI)
```
```  1294 apply auto
```
```  1295 done
```
```  1296
```
```  1297 lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
```
```  1298 apply (induct A)
```
```  1299  apply blast
```
```  1300 apply clarsimp
```
```  1301 apply (drule_tac x = x in fold_msetG.insertI)
```
```  1302 apply auto
```
```  1303 done
```
```  1304
```
```  1305 lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
```
```  1306 unfolding fold_mset_def by blast
```
```  1307
```
```  1308 locale left_commutative =
```
```  1309 fixes f :: "'a => 'b => 'b"
```
```  1310 assumes left_commute: "f x (f y z) = f y (f x z)"
```
```  1311 begin
```
```  1312
```
```  1313 lemma fold_msetG_determ:
```
```  1314   "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
```
```  1315 proof (induct arbitrary: x y z rule: full_multiset_induct)
```
```  1316   case (less M x\<^isub>1 x\<^isub>2 Z)
```
```  1317   have IH: "\<forall>A. A \<subset># M \<longrightarrow>
```
```  1318     (\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'
```
```  1319                \<longrightarrow> x' = x)" by fact
```
```  1320   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+
```
```  1321   show ?case
```
```  1322   proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])
```
```  1323     assume "M = {#}" and "x\<^isub>1 = Z"
```
```  1324     then show ?case using Mfoldx\<^isub>2 by auto
```
```  1325   next
```
```  1326     fix B b u
```
```  1327     assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"
```
```  1328     then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto
```
```  1329     show ?case
```
```  1330     proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])
```
```  1331       assume "M = {#}" "x\<^isub>2 = Z"
```
```  1332       then show ?case using Mfoldx\<^isub>1 by auto
```
```  1333     next
```
```  1334       fix C c v
```
```  1335       assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"
```
```  1336       then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto
```
```  1337       then have CsubM: "C \<subset># M" by simp
```
```  1338       from MBb have BsubM: "B \<subset># M" by simp
```
```  1339       show ?case
```
```  1340       proof cases
```
```  1341         assume "b=c"
```
```  1342         then moreover have "B = C" using MBb MCc by auto
```
```  1343         ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto
```
```  1344       next
```
```  1345         assume diff: "b \<noteq> c"
```
```  1346         let ?D = "B - {#c#}"
```
```  1347         have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
```
```  1348           by (auto intro: insert_noteq_member dest: sym)
```
```  1349         have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self)
```
```  1350         then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans)
```
```  1351         from MBb MCc have "B + {#b#} = C + {#c#}" by blast
```
```  1352         then have [simp]: "B + {#b#} - {#c#} = C"
```
```  1353           using MBb MCc binC cinB by auto
```
```  1354         have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
```
```  1355           using MBb MCc diff binC cinB
```
```  1356           by (auto simp: multiset_add_sub_el_shuffle)
```
```  1357         then obtain d where Dfoldd: "fold_msetG f Z ?D d"
```
```  1358           using fold_msetG_nonempty by iprover
```
```  1359         then have "fold_msetG f Z B (f c d)" using cinB
```
```  1360           by (rule Diff1_fold_msetG)
```
```  1361         then have "f c d = u" using IH BsubM Bu by blast
```
```  1362         moreover
```
```  1363         have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
```
```  1364           by (auto simp: multiset_add_sub_el_shuffle
```
```  1365             dest: fold_msetG.insertI [where x=b])
```
```  1366         then have "f b d = v" using IH CsubM Cv by blast
```
```  1367         ultimately show ?thesis using x\<^isub>1 x\<^isub>2
```
```  1368           by (auto simp: left_commute)
```
```  1369       qed
```
```  1370     qed
```
```  1371   qed
```
```  1372 qed
```
```  1373
```
```  1374 lemma fold_mset_insert_aux:
```
```  1375   "(fold_msetG f z (A + {#x#}) v) =
```
```  1376     (\<exists>y. fold_msetG f z A y \<and> v = f x y)"
```
```  1377 apply (rule iffI)
```
```  1378  prefer 2
```
```  1379  apply blast
```
```  1380 apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])
```
```  1381 apply (blast intro: fold_msetG_determ)
```
```  1382 done
```
```  1383
```
```  1384 lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
```
```  1385 unfolding fold_mset_def by (blast intro: fold_msetG_determ)
```
```  1386
```
```  1387 lemma fold_mset_insert:
```
```  1388   "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
```
```  1389 apply (simp add: fold_mset_def fold_mset_insert_aux union_commute)
```
```  1390 apply (rule the_equality)
```
```  1391  apply (auto cong add: conj_cong
```
```  1392      simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
```
```  1393 done
```
```  1394
```
```  1395 lemma fold_mset_insert_idem:
```
```  1396   "fold_mset f z (A + {#a#}) = f a (fold_mset f z A)"
```
```  1397 apply (simp add: fold_mset_def fold_mset_insert_aux)
```
```  1398 apply (rule the_equality)
```
```  1399  apply (auto cong add: conj_cong
```
```  1400      simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
```
```  1401 done
```
```  1402
```
```  1403 lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
```
```  1404 by (induct A) (auto simp: fold_mset_insert left_commute [of x])
```
```  1405
```
```  1406 lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
```
```  1407 using fold_mset_insert [of z "{#}"] by simp
```
```  1408
```
```  1409 lemma fold_mset_union [simp]:
```
```  1410   "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
```
```  1411 proof (induct A)
```
```  1412   case empty then show ?case by simp
```
```  1413 next
```
```  1414   case (add A x)
```
```  1415   have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac)
```
```  1416   then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))"
```
```  1417     by (simp add: fold_mset_insert)
```
```  1418   also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
```
```  1419     by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
```
```  1420   finally show ?case .
```
```  1421 qed
```
```  1422
```
```  1423 lemma fold_mset_fusion:
```
```  1424   includes left_commutative g
```
```  1425   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"
```
```  1426 by (induct A) auto
```
```  1427
```
```  1428 lemma fold_mset_rec:
```
```  1429   assumes "a \<in># A"
```
```  1430   shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
```
```  1431 proof -
```
```  1432   from assms obtain A' where "A = A' + {#a#}"
```
```  1433     by (blast dest: multi_member_split)
```
```  1434   then show ?thesis by simp
```
```  1435 qed
```
```  1436
```
```  1437 end
```
```  1438
```
```  1439 text {*
```
```  1440   A note on code generation: When defining some function containing a
```
```  1441   subterm @{term"fold_mset F"}, code generation is not automatic. When
```
```  1442   interpreting locale @{text left_commutative} with @{text F}, the
```
```  1443   would be code thms for @{const fold_mset} become thms like
```
```  1444   @{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
```
```  1445   contains defined symbols, i.e.\ is not a code thm. Hence a separate
```
```  1446   constant with its own code thms needs to be introduced for @{text
```
```  1447   F}. See the image operator below.
```
```  1448 *}
```
```  1449
```
```  1450
```
```  1451 subsection {* Image *}
```
```  1452
```
```  1453 definition [code func del]: "image_mset f == fold_mset (op + o single o f) {#}"
```
```  1454
```
```  1455 interpretation image_left_comm: left_commutative ["op + o single o f"]
```
```  1456 by (unfold_locales) (simp add:union_ac)
```
```  1457
```
```  1458 lemma image_mset_empty [simp, code func]: "image_mset f {#} = {#}"
```
```  1459 by (simp add: image_mset_def)
```
```  1460
```
```  1461 lemma image_mset_single [simp, code func]: "image_mset f {#x#} = {#f x#}"
```
```  1462 by (simp add: image_mset_def)
```
```  1463
```
```  1464 lemma image_mset_insert:
```
```  1465   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
```
```  1466 by (simp add: image_mset_def add_ac)
```
```  1467
```
```  1468 lemma image_mset_union[simp, code func]:
```
```  1469   "image_mset f (M+N) = image_mset f M + image_mset f N"
```
```  1470 apply (induct N)
```
```  1471  apply simp
```
```  1472 apply (simp add: union_assoc [symmetric] image_mset_insert)
```
```  1473 done
```
```  1474
```
```  1475 lemma size_image_mset [simp]: "size (image_mset f M) = size M"
```
```  1476 by (induct M) simp_all
```
```  1477
```
```  1478 lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
```
```  1479 by (cases M) auto
```
```  1480
```
```  1481
```
```  1482 syntax
```
```  1483   comprehension1_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
```
```  1484       ("({#_/. _ :# _#})")
```
```  1485 translations
```
```  1486   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
```
```  1487
```
```  1488 syntax
```
```  1489   comprehension2_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
```
```  1490       ("({#_/ | _ :# _./ _#})")
```
```  1491 translations
```
```  1492   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
```
```  1493
```
```  1494 text {*
```
```  1495   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
```
```  1496   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
```
```  1497   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
```
```  1498   @{term "{#x+x|x:#M. x<c#}"}.
```
```  1499 *}
```
```  1500
```
```  1501 end
```