src/HOL/Library/Multiset.thy
 author Andreas Lochbihler Fri Sep 20 10:09:16 2013 +0200 (2013-09-20) changeset 53745 788730ab7da4 parent 52289 83ce5d2841e7 child 54295 45a5523d4a63 permissions -rw-r--r--
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```     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
```
```     9 begin
```
```    10
```
```    11 subsection {* The type of multisets *}
```
```    12
```
```    13 definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
```
```    14
```
```    15 typedef '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 setup_lifting type_definition_multiset
```
```    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) cancel_comm_monoid_add
```
```    83 begin
```
```    84
```
```    85 lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
```
```    86 by (rule const0_in_multiset)
```
```    87
```
```    88 abbreviation Mempty :: "'a multiset" ("{#}") where
```
```    89   "Mempty \<equiv> 0"
```
```    90
```
```    91 lift_definition plus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
```
```    92 by (rule union_preserves_multiset)
```
```    93
```
```    94 instance
```
```    95 by default (transfer, simp add: fun_eq_iff)+
```
```    96
```
```    97 end
```
```    98
```
```    99 lift_definition single :: "'a => 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
```
```   100 by (rule only1_in_multiset)
```
```   101
```
```   102 syntax
```
```   103   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
```
```   104 translations
```
```   105   "{#x, xs#}" == "{#x#} + {#xs#}"
```
```   106   "{#x#}" == "CONST single x"
```
```   107
```
```   108 lemma count_empty [simp]: "count {#} a = 0"
```
```   109   by (simp add: zero_multiset.rep_eq)
```
```   110
```
```   111 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   112   by (simp add: single.rep_eq)
```
```   113
```
```   114
```
```   115 subsection {* Basic operations *}
```
```   116
```
```   117 subsubsection {* Union *}
```
```   118
```
```   119 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   120   by (simp add: plus_multiset.rep_eq)
```
```   121
```
```   122
```
```   123 subsubsection {* Difference *}
```
```   124
```
```   125 instantiation multiset :: (type) comm_monoid_diff
```
```   126 begin
```
```   127
```
```   128 lift_definition minus_multiset :: "'a multiset => 'a multiset => 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
```
```   129 by (rule diff_preserves_multiset)
```
```   130
```
```   131 instance
```
```   132 by default (transfer, simp add: fun_eq_iff)+
```
```   133
```
```   134 end
```
```   135
```
```   136 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   137   by (simp add: minus_multiset.rep_eq)
```
```   138
```
```   139 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   140   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
```
```   141
```
```   142 lemma diff_cancel[simp]: "A - A = {#}"
```
```   143   by (fact Groups.diff_cancel)
```
```   144
```
```   145 lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
```
```   146   by (fact add_diff_cancel_right')
```
```   147
```
```   148 lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
```
```   149   by (fact add_diff_cancel_left')
```
```   150
```
```   151 lemma diff_right_commute:
```
```   152   "(M::'a multiset) - N - Q = M - Q - N"
```
```   153   by (fact diff_right_commute)
```
```   154
```
```   155 lemma diff_add:
```
```   156   "(M::'a multiset) - (N + Q) = M - N - Q"
```
```   157   by (rule sym) (fact diff_diff_add)
```
```   158
```
```   159 lemma insert_DiffM:
```
```   160   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
```
```   161   by (clarsimp simp: multiset_eq_iff)
```
```   162
```
```   163 lemma insert_DiffM2 [simp]:
```
```   164   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
```
```   165   by (clarsimp simp: multiset_eq_iff)
```
```   166
```
```   167 lemma diff_union_swap:
```
```   168   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
```
```   169   by (auto simp add: multiset_eq_iff)
```
```   170
```
```   171 lemma diff_union_single_conv:
```
```   172   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
```
```   173   by (simp add: multiset_eq_iff)
```
```   174
```
```   175
```
```   176 subsubsection {* Equality of multisets *}
```
```   177
```
```   178 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
```
```   179   by (simp add: multiset_eq_iff)
```
```   180
```
```   181 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
```
```   182   by (auto simp add: multiset_eq_iff)
```
```   183
```
```   184 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
```
```   185   by (auto simp add: multiset_eq_iff)
```
```   186
```
```   187 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
```
```   188   by (auto simp add: multiset_eq_iff)
```
```   189
```
```   190 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
```
```   191   by (auto simp add: multiset_eq_iff)
```
```   192
```
```   193 lemma diff_single_trivial:
```
```   194   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
```
```   195   by (auto simp add: multiset_eq_iff)
```
```   196
```
```   197 lemma diff_single_eq_union:
```
```   198   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
```
```   199   by auto
```
```   200
```
```   201 lemma union_single_eq_diff:
```
```   202   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
```
```   203   by (auto dest: sym)
```
```   204
```
```   205 lemma union_single_eq_member:
```
```   206   "M + {#x#} = N \<Longrightarrow> x \<in># N"
```
```   207   by auto
```
```   208
```
```   209 lemma union_is_single:
```
```   210   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
```
```   211 proof
```
```   212   assume ?rhs then show ?lhs by auto
```
```   213 next
```
```   214   assume ?lhs then show ?rhs
```
```   215     by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
```
```   216 qed
```
```   217
```
```   218 lemma single_is_union:
```
```   219   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
```
```   220   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
```
```   221
```
```   222 lemma add_eq_conv_diff:
```
```   223   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
```
```   224 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
```
```   225 proof
```
```   226   assume ?rhs then show ?lhs
```
```   227   by (auto simp add: add_assoc add_commute [of "{#b#}"])
```
```   228     (drule sym, simp add: add_assoc [symmetric])
```
```   229 next
```
```   230   assume ?lhs
```
```   231   show ?rhs
```
```   232   proof (cases "a = b")
```
```   233     case True with `?lhs` show ?thesis by simp
```
```   234   next
```
```   235     case False
```
```   236     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
```
```   237     with False have "a \<in># N" by auto
```
```   238     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
```
```   239     moreover note False
```
```   240     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
```
```   241   qed
```
```   242 qed
```
```   243
```
```   244 lemma insert_noteq_member:
```
```   245   assumes BC: "B + {#b#} = C + {#c#}"
```
```   246    and bnotc: "b \<noteq> c"
```
```   247   shows "c \<in># B"
```
```   248 proof -
```
```   249   have "c \<in># C + {#c#}" by simp
```
```   250   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
```
```   251   then have "c \<in># B + {#b#}" using BC by simp
```
```   252   then show "c \<in># B" using nc by simp
```
```   253 qed
```
```   254
```
```   255 lemma add_eq_conv_ex:
```
```   256   "(M + {#a#} = N + {#b#}) =
```
```   257     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
```
```   258   by (auto simp add: add_eq_conv_diff)
```
```   259
```
```   260 lemma multi_member_split:
```
```   261   "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
```
```   262   by (rule_tac x = "M - {#x#}" in exI, simp)
```
```   263
```
```   264
```
```   265 subsubsection {* Pointwise ordering induced by count *}
```
```   266
```
```   267 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
```
```   268 begin
```
```   269
```
```   270 lift_definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" is "\<lambda> A B. (\<forall>a. A a \<le> B a)"
```
```   271 by simp
```
```   272 lemmas mset_le_def = less_eq_multiset_def
```
```   273
```
```   274 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
```
```   275   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
```
```   276
```
```   277 instance
```
```   278   by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
```
```   279
```
```   280 end
```
```   281
```
```   282 lemma mset_less_eqI:
```
```   283   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
```
```   284   by (simp add: mset_le_def)
```
```   285
```
```   286 lemma mset_le_exists_conv:
```
```   287   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
```
```   288 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
```
```   289 apply (auto intro: multiset_eq_iff [THEN iffD2])
```
```   290 done
```
```   291
```
```   292 instance multiset :: (type) ordered_cancel_comm_monoid_diff
```
```   293   by default (simp, fact mset_le_exists_conv)
```
```   294
```
```   295 lemma mset_le_mono_add_right_cancel [simp]:
```
```   296   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
```
```   297   by (fact add_le_cancel_right)
```
```   298
```
```   299 lemma mset_le_mono_add_left_cancel [simp]:
```
```   300   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
```
```   301   by (fact add_le_cancel_left)
```
```   302
```
```   303 lemma mset_le_mono_add:
```
```   304   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
```
```   305   by (fact add_mono)
```
```   306
```
```   307 lemma mset_le_add_left [simp]:
```
```   308   "(A::'a multiset) \<le> A + B"
```
```   309   unfolding mset_le_def by auto
```
```   310
```
```   311 lemma mset_le_add_right [simp]:
```
```   312   "B \<le> (A::'a multiset) + B"
```
```   313   unfolding mset_le_def by auto
```
```   314
```
```   315 lemma mset_le_single:
```
```   316   "a :# B \<Longrightarrow> {#a#} \<le> B"
```
```   317   by (simp add: mset_le_def)
```
```   318
```
```   319 lemma multiset_diff_union_assoc:
```
```   320   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
```
```   321   by (simp add: multiset_eq_iff mset_le_def)
```
```   322
```
```   323 lemma mset_le_multiset_union_diff_commute:
```
```   324   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
```
```   325 by (simp add: multiset_eq_iff mset_le_def)
```
```   326
```
```   327 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
```
```   328 by(simp add: mset_le_def)
```
```   329
```
```   330 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   331 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   332 apply (erule_tac x=x in allE)
```
```   333 apply auto
```
```   334 done
```
```   335
```
```   336 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
```
```   337 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   338 apply (erule_tac x = x in allE)
```
```   339 apply auto
```
```   340 done
```
```   341
```
```   342 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
```
```   343 apply (rule conjI)
```
```   344  apply (simp add: mset_lessD)
```
```   345 apply (clarsimp simp: mset_le_def mset_less_def)
```
```   346 apply safe
```
```   347  apply (erule_tac x = a in allE)
```
```   348  apply (auto split: split_if_asm)
```
```   349 done
```
```   350
```
```   351 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
```
```   352 apply (rule conjI)
```
```   353  apply (simp add: mset_leD)
```
```   354 apply (force simp: mset_le_def mset_less_def split: split_if_asm)
```
```   355 done
```
```   356
```
```   357 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
```
```   358   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
```
```   359
```
```   360 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
```
```   361   by (auto simp: mset_le_def mset_less_def)
```
```   362
```
```   363 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
```
```   364   by simp
```
```   365
```
```   366 lemma mset_less_add_bothsides:
```
```   367   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
```
```   368   by (fact add_less_imp_less_right)
```
```   369
```
```   370 lemma mset_less_empty_nonempty:
```
```   371   "{#} < S \<longleftrightarrow> S \<noteq> {#}"
```
```   372   by (auto simp: mset_le_def mset_less_def)
```
```   373
```
```   374 lemma mset_less_diff_self:
```
```   375   "c \<in># B \<Longrightarrow> B - {#c#} < B"
```
```   376   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
```
```   377
```
```   378
```
```   379 subsubsection {* Intersection *}
```
```   380
```
```   381 instantiation multiset :: (type) semilattice_inf
```
```   382 begin
```
```   383
```
```   384 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   385   multiset_inter_def: "inf_multiset A B = A - (A - B)"
```
```   386
```
```   387 instance
```
```   388 proof -
```
```   389   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
```
```   390   show "OFCLASS('a multiset, semilattice_inf_class)"
```
```   391     by default (auto simp add: multiset_inter_def mset_le_def aux)
```
```   392 qed
```
```   393
```
```   394 end
```
```   395
```
```   396 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
```
```   397   "multiset_inter \<equiv> inf"
```
```   398
```
```   399 lemma multiset_inter_count [simp]:
```
```   400   "count (A #\<inter> B) x = min (count A x) (count B x)"
```
```   401   by (simp add: multiset_inter_def)
```
```   402
```
```   403 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
```
```   404   by (rule multiset_eqI) auto
```
```   405
```
```   406 lemma multiset_union_diff_commute:
```
```   407   assumes "B #\<inter> C = {#}"
```
```   408   shows "A + B - C = A - C + B"
```
```   409 proof (rule multiset_eqI)
```
```   410   fix x
```
```   411   from assms have "min (count B x) (count C x) = 0"
```
```   412     by (auto simp add: multiset_eq_iff)
```
```   413   then have "count B x = 0 \<or> count C x = 0"
```
```   414     by auto
```
```   415   then show "count (A + B - C) x = count (A - C + B) x"
```
```   416     by auto
```
```   417 qed
```
```   418
```
```   419 lemma empty_inter [simp]:
```
```   420   "{#} #\<inter> M = {#}"
```
```   421   by (simp add: multiset_eq_iff)
```
```   422
```
```   423 lemma inter_empty [simp]:
```
```   424   "M #\<inter> {#} = {#}"
```
```   425   by (simp add: multiset_eq_iff)
```
```   426
```
```   427 lemma inter_add_left1:
```
```   428   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
```
```   429   by (simp add: multiset_eq_iff)
```
```   430
```
```   431 lemma inter_add_left2:
```
```   432   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
```
```   433   by (simp add: multiset_eq_iff)
```
```   434
```
```   435 lemma inter_add_right1:
```
```   436   "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
```
```   437   by (simp add: multiset_eq_iff)
```
```   438
```
```   439 lemma inter_add_right2:
```
```   440   "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
```
```   441   by (simp add: multiset_eq_iff)
```
```   442
```
```   443
```
```   444 subsubsection {* Bounded union *}
```
```   445
```
```   446 instantiation multiset :: (type) semilattice_sup
```
```   447 begin
```
```   448
```
```   449 definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
```
```   450   "sup_multiset A B = A + (B - A)"
```
```   451
```
```   452 instance
```
```   453 proof -
```
```   454   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" by arith
```
```   455   show "OFCLASS('a multiset, semilattice_sup_class)"
```
```   456     by default (auto simp add: sup_multiset_def mset_le_def aux)
```
```   457 qed
```
```   458
```
```   459 end
```
```   460
```
```   461 abbreviation sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<union>" 70) where
```
```   462   "sup_multiset \<equiv> sup"
```
```   463
```
```   464 lemma sup_multiset_count [simp]:
```
```   465   "count (A #\<union> B) x = max (count A x) (count B x)"
```
```   466   by (simp add: sup_multiset_def)
```
```   467
```
```   468 lemma empty_sup [simp]:
```
```   469   "{#} #\<union> M = M"
```
```   470   by (simp add: multiset_eq_iff)
```
```   471
```
```   472 lemma sup_empty [simp]:
```
```   473   "M #\<union> {#} = M"
```
```   474   by (simp add: multiset_eq_iff)
```
```   475
```
```   476 lemma sup_add_left1:
```
```   477   "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
```
```   478   by (simp add: multiset_eq_iff)
```
```   479
```
```   480 lemma sup_add_left2:
```
```   481   "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
```
```   482   by (simp add: multiset_eq_iff)
```
```   483
```
```   484 lemma sup_add_right1:
```
```   485   "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
```
```   486   by (simp add: multiset_eq_iff)
```
```   487
```
```   488 lemma sup_add_right2:
```
```   489   "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
```
```   490   by (simp add: multiset_eq_iff)
```
```   491
```
```   492
```
```   493 subsubsection {* Filter (with comprehension syntax) *}
```
```   494
```
```   495 text {* Multiset comprehension *}
```
```   496
```
```   497 lift_definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
```
```   498 by (rule filter_preserves_multiset)
```
```   499
```
```   500 hide_const (open) filter
```
```   501
```
```   502 lemma count_filter [simp]:
```
```   503   "count (Multiset.filter P M) a = (if P a then count M a else 0)"
```
```   504   by (simp add: filter.rep_eq)
```
```   505
```
```   506 lemma filter_empty [simp]:
```
```   507   "Multiset.filter P {#} = {#}"
```
```   508   by (rule multiset_eqI) simp
```
```   509
```
```   510 lemma filter_single [simp]:
```
```   511   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
```
```   512   by (rule multiset_eqI) simp
```
```   513
```
```   514 lemma filter_union [simp]:
```
```   515   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
```
```   516   by (rule multiset_eqI) simp
```
```   517
```
```   518 lemma filter_diff [simp]:
```
```   519   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
```
```   520   by (rule multiset_eqI) simp
```
```   521
```
```   522 lemma filter_inter [simp]:
```
```   523   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
```
```   524   by (rule multiset_eqI) simp
```
```   525
```
```   526 syntax
```
```   527   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
```
```   528 syntax (xsymbol)
```
```   529   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
```
```   530 translations
```
```   531   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
```
```   532
```
```   533
```
```   534 subsubsection {* Set of elements *}
```
```   535
```
```   536 definition set_of :: "'a multiset => 'a set" where
```
```   537   "set_of M = {x. x :# M}"
```
```   538
```
```   539 lemma set_of_empty [simp]: "set_of {#} = {}"
```
```   540 by (simp add: set_of_def)
```
```   541
```
```   542 lemma set_of_single [simp]: "set_of {#b#} = {b}"
```
```   543 by (simp add: set_of_def)
```
```   544
```
```   545 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   546 by (auto simp add: set_of_def)
```
```   547
```
```   548 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   549 by (auto simp add: set_of_def multiset_eq_iff)
```
```   550
```
```   551 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   552 by (auto simp add: set_of_def)
```
```   553
```
```   554 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
```
```   555 by (auto simp add: set_of_def)
```
```   556
```
```   557 lemma finite_set_of [iff]: "finite (set_of M)"
```
```   558   using count [of M] by (simp add: multiset_def set_of_def)
```
```   559
```
```   560 lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
```
```   561   unfolding set_of_def[symmetric] by simp
```
```   562
```
```   563 subsubsection {* Size *}
```
```   564
```
```   565 instantiation multiset :: (type) size
```
```   566 begin
```
```   567
```
```   568 definition size_def:
```
```   569   "size M = setsum (count M) (set_of M)"
```
```   570
```
```   571 instance ..
```
```   572
```
```   573 end
```
```   574
```
```   575 lemma size_empty [simp]: "size {#} = 0"
```
```   576 by (simp add: size_def)
```
```   577
```
```   578 lemma size_single [simp]: "size {#b#} = 1"
```
```   579 by (simp add: size_def)
```
```   580
```
```   581 lemma setsum_count_Int:
```
```   582   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
```
```   583 apply (induct rule: finite_induct)
```
```   584  apply simp
```
```   585 apply (simp add: Int_insert_left set_of_def)
```
```   586 done
```
```   587
```
```   588 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   589 apply (unfold size_def)
```
```   590 apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
```
```   591  prefer 2
```
```   592  apply (rule ext, simp)
```
```   593 apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
```
```   594 apply (subst Int_commute)
```
```   595 apply (simp (no_asm_simp) add: setsum_count_Int)
```
```   596 done
```
```   597
```
```   598 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   599 by (auto simp add: size_def multiset_eq_iff)
```
```   600
```
```   601 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
```
```   602 by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
```
```   603
```
```   604 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   605 apply (unfold size_def)
```
```   606 apply (drule setsum_SucD)
```
```   607 apply auto
```
```   608 done
```
```   609
```
```   610 lemma size_eq_Suc_imp_eq_union:
```
```   611   assumes "size M = Suc n"
```
```   612   shows "\<exists>a N. M = N + {#a#}"
```
```   613 proof -
```
```   614   from assms obtain a where "a \<in># M"
```
```   615     by (erule size_eq_Suc_imp_elem [THEN exE])
```
```   616   then have "M = M - {#a#} + {#a#}" by simp
```
```   617   then show ?thesis by blast
```
```   618 qed
```
```   619
```
```   620
```
```   621 subsection {* Induction and case splits *}
```
```   622
```
```   623 theorem multiset_induct [case_names empty add, induct type: multiset]:
```
```   624   assumes empty: "P {#}"
```
```   625   assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
```
```   626   shows "P M"
```
```   627 proof (induct n \<equiv> "size M" arbitrary: M)
```
```   628   case 0 thus "P M" by (simp add: empty)
```
```   629 next
```
```   630   case (Suc k)
```
```   631   obtain N x where "M = N + {#x#}"
```
```   632     using `Suc k = size M` [symmetric]
```
```   633     using size_eq_Suc_imp_eq_union by fast
```
```   634   with Suc add show "P M" by simp
```
```   635 qed
```
```   636
```
```   637 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
```
```   638 by (induct M) auto
```
```   639
```
```   640 lemma multiset_cases [cases type, case_names empty add]:
```
```   641 assumes em:  "M = {#} \<Longrightarrow> P"
```
```   642 assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
```
```   643 shows "P"
```
```   644 using assms by (induct M) simp_all
```
```   645
```
```   646 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
```
```   647 by (cases "B = {#}") (auto dest: multi_member_split)
```
```   648
```
```   649 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
```
```   650 apply (subst multiset_eq_iff)
```
```   651 apply auto
```
```   652 done
```
```   653
```
```   654 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
```
```   655 proof (induct A arbitrary: B)
```
```   656   case (empty M)
```
```   657   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
```
```   658   then obtain M' x where "M = M' + {#x#}"
```
```   659     by (blast dest: multi_nonempty_split)
```
```   660   then show ?case by simp
```
```   661 next
```
```   662   case (add S x T)
```
```   663   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
```
```   664   have SxsubT: "S + {#x#} < T" by fact
```
```   665   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
```
```   666   then obtain T' where T: "T = T' + {#x#}"
```
```   667     by (blast dest: multi_member_split)
```
```   668   then have "S < T'" using SxsubT
```
```   669     by (blast intro: mset_less_add_bothsides)
```
```   670   then have "size S < size T'" using IH by simp
```
```   671   then show ?case using T by simp
```
```   672 qed
```
```   673
```
```   674
```
```   675 subsubsection {* Strong induction and subset induction for multisets *}
```
```   676
```
```   677 text {* Well-foundedness of proper subset operator: *}
```
```   678
```
```   679 text {* proper multiset subset *}
```
```   680
```
```   681 definition
```
```   682   mset_less_rel :: "('a multiset * 'a multiset) set" where
```
```   683   "mset_less_rel = {(A,B). A < B}"
```
```   684
```
```   685 lemma multiset_add_sub_el_shuffle:
```
```   686   assumes "c \<in># B" and "b \<noteq> c"
```
```   687   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
```
```   688 proof -
```
```   689   from `c \<in># B` obtain A where B: "B = A + {#c#}"
```
```   690     by (blast dest: multi_member_split)
```
```   691   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
```
```   692   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
```
```   693     by (simp add: add_ac)
```
```   694   then show ?thesis using B by simp
```
```   695 qed
```
```   696
```
```   697 lemma wf_mset_less_rel: "wf mset_less_rel"
```
```   698 apply (unfold mset_less_rel_def)
```
```   699 apply (rule wf_measure [THEN wf_subset, where f1=size])
```
```   700 apply (clarsimp simp: measure_def inv_image_def mset_less_size)
```
```   701 done
```
```   702
```
```   703 text {* The induction rules: *}
```
```   704
```
```   705 lemma full_multiset_induct [case_names less]:
```
```   706 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
```
```   707 shows "P B"
```
```   708 apply (rule wf_mset_less_rel [THEN wf_induct])
```
```   709 apply (rule ih, auto simp: mset_less_rel_def)
```
```   710 done
```
```   711
```
```   712 lemma multi_subset_induct [consumes 2, case_names empty add]:
```
```   713 assumes "F \<le> A"
```
```   714   and empty: "P {#}"
```
```   715   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
```
```   716 shows "P F"
```
```   717 proof -
```
```   718   from `F \<le> A`
```
```   719   show ?thesis
```
```   720   proof (induct F)
```
```   721     show "P {#}" by fact
```
```   722   next
```
```   723     fix x F
```
```   724     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
```
```   725     show "P (F + {#x#})"
```
```   726     proof (rule insert)
```
```   727       from i show "x \<in># A" by (auto dest: mset_le_insertD)
```
```   728       from i have "F \<le> A" by (auto dest: mset_le_insertD)
```
```   729       with P show "P F" .
```
```   730     qed
```
```   731   qed
```
```   732 qed
```
```   733
```
```   734
```
```   735 subsection {* The fold combinator *}
```
```   736
```
```   737 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
```
```   738 where
```
```   739   "fold f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_of M)"
```
```   740
```
```   741 lemma fold_mset_empty [simp]:
```
```   742   "fold f s {#} = s"
```
```   743   by (simp add: fold_def)
```
```   744
```
```   745 context comp_fun_commute
```
```   746 begin
```
```   747
```
```   748 lemma fold_mset_insert:
```
```   749   "fold f s (M + {#x#}) = f x (fold f s M)"
```
```   750 proof -
```
```   751   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
```
```   752     by (fact comp_fun_commute_funpow)
```
```   753   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
```
```   754     by (fact comp_fun_commute_funpow)
```
```   755   show ?thesis
```
```   756   proof (cases "x \<in> set_of M")
```
```   757     case False
```
```   758     then have *: "count (M + {#x#}) x = 1" by simp
```
```   759     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_of M) =
```
```   760       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_of M)"
```
```   761       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   762     with False * show ?thesis
```
```   763       by (simp add: fold_def del: count_union)
```
```   764   next
```
```   765     case True
```
```   766     def N \<equiv> "set_of M - {x}"
```
```   767     from N_def True have *: "set_of M = insert x N" "x \<notin> N" "finite N" by auto
```
```   768     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
```
```   769       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
```
```   770       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
```
```   771     with * show ?thesis by (simp add: fold_def del: count_union) simp
```
```   772   qed
```
```   773 qed
```
```   774
```
```   775 corollary fold_mset_single [simp]:
```
```   776   "fold f s {#x#} = f x s"
```
```   777 proof -
```
```   778   have "fold f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
```
```   779   then show ?thesis by simp
```
```   780 qed
```
```   781
```
```   782 lemma fold_mset_fun_left_comm:
```
```   783   "f x (fold f s M) = fold f (f x s) M"
```
```   784   by (induct M) (simp_all add: fold_mset_insert fun_left_comm)
```
```   785
```
```   786 lemma fold_mset_union [simp]:
```
```   787   "fold f s (M + N) = fold f (fold f s M) N"
```
```   788 proof (induct M)
```
```   789   case empty then show ?case by simp
```
```   790 next
```
```   791   case (add M x)
```
```   792   have "M + {#x#} + N = (M + N) + {#x#}"
```
```   793     by (simp add: add_ac)
```
```   794   with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
```
```   795 qed
```
```   796
```
```   797 lemma fold_mset_fusion:
```
```   798   assumes "comp_fun_commute g"
```
```   799   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold g w A) = fold f (h w) A" (is "PROP ?P")
```
```   800 proof -
```
```   801   interpret comp_fun_commute g by (fact assms)
```
```   802   show "PROP ?P" by (induct A) auto
```
```   803 qed
```
```   804
```
```   805 end
```
```   806
```
```   807 text {*
```
```   808   A note on code generation: When defining some function containing a
```
```   809   subterm @{term "fold F"}, code generation is not automatic. When
```
```   810   interpreting locale @{text left_commutative} with @{text F}, the
```
```   811   would be code thms for @{const fold} become thms like
```
```   812   @{term "fold F z {#} = z"} where @{text F} is not a pattern but
```
```   813   contains defined symbols, i.e.\ is not a code thm. Hence a separate
```
```   814   constant with its own code thms needs to be introduced for @{text
```
```   815   F}. See the image operator below.
```
```   816 *}
```
```   817
```
```   818
```
```   819 subsection {* Image *}
```
```   820
```
```   821 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
```
```   822   "image_mset f = fold (plus o single o f) {#}"
```
```   823
```
```   824 lemma comp_fun_commute_mset_image:
```
```   825   "comp_fun_commute (plus o single o f)"
```
```   826 proof
```
```   827 qed (simp add: add_ac fun_eq_iff)
```
```   828
```
```   829 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
```
```   830   by (simp add: image_mset_def)
```
```   831
```
```   832 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
```
```   833 proof -
```
```   834   interpret comp_fun_commute "plus o single o f"
```
```   835     by (fact comp_fun_commute_mset_image)
```
```   836   show ?thesis by (simp add: image_mset_def)
```
```   837 qed
```
```   838
```
```   839 lemma image_mset_union [simp]:
```
```   840   "image_mset f (M + N) = image_mset f M + image_mset f N"
```
```   841 proof -
```
```   842   interpret comp_fun_commute "plus o single o f"
```
```   843     by (fact comp_fun_commute_mset_image)
```
```   844   show ?thesis by (induct N) (simp_all add: image_mset_def add_ac)
```
```   845 qed
```
```   846
```
```   847 corollary image_mset_insert:
```
```   848   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
```
```   849   by simp
```
```   850
```
```   851 lemma set_of_image_mset [simp]:
```
```   852   "set_of (image_mset f M) = image f (set_of M)"
```
```   853   by (induct M) simp_all
```
```   854
```
```   855 lemma size_image_mset [simp]:
```
```   856   "size (image_mset f M) = size M"
```
```   857   by (induct M) simp_all
```
```   858
```
```   859 lemma image_mset_is_empty_iff [simp]:
```
```   860   "image_mset f M = {#} \<longleftrightarrow> M = {#}"
```
```   861   by (cases M) auto
```
```   862
```
```   863 syntax
```
```   864   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
```
```   865       ("({#_/. _ :# _#})")
```
```   866 translations
```
```   867   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
```
```   868
```
```   869 syntax
```
```   870   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
```
```   871       ("({#_/ | _ :# _./ _#})")
```
```   872 translations
```
```   873   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
```
```   874
```
```   875 text {*
```
```   876   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
```
```   877   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
```
```   878   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
```
```   879   @{term "{#x+x|x:#M. x<c#}"}.
```
```   880 *}
```
```   881
```
```   882 enriched_type image_mset: image_mset
```
```   883 proof -
```
```   884   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
```
```   885   proof
```
```   886     fix A
```
```   887     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
```
```   888       by (induct A) simp_all
```
```   889   qed
```
```   890   show "image_mset id = id"
```
```   891   proof
```
```   892     fix A
```
```   893     show "image_mset id A = id A"
```
```   894       by (induct A) simp_all
```
```   895   qed
```
```   896 qed
```
```   897
```
```   898 declare image_mset.identity [simp]
```
```   899
```
```   900
```
```   901 subsection {* Further conversions *}
```
```   902
```
```   903 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
```
```   904   "multiset_of [] = {#}" |
```
```   905   "multiset_of (a # x) = multiset_of x + {# a #}"
```
```   906
```
```   907 lemma in_multiset_in_set:
```
```   908   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```   909   by (induct xs) simp_all
```
```   910
```
```   911 lemma count_multiset_of:
```
```   912   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```   913   by (induct xs) simp_all
```
```   914
```
```   915 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
```
```   916 by (induct x) auto
```
```   917
```
```   918 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
```
```   919 by (induct x) auto
```
```   920
```
```   921 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
```
```   922 by (induct x) auto
```
```   923
```
```   924 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
```
```   925 by (induct xs) auto
```
```   926
```
```   927 lemma size_multiset_of [simp]: "size (multiset_of xs) = length xs"
```
```   928   by (induct xs) simp_all
```
```   929
```
```   930 lemma multiset_of_append [simp]:
```
```   931   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
```
```   932   by (induct xs arbitrary: ys) (auto simp: add_ac)
```
```   933
```
```   934 lemma multiset_of_filter:
```
```   935   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
```
```   936   by (induct xs) simp_all
```
```   937
```
```   938 lemma multiset_of_rev [simp]:
```
```   939   "multiset_of (rev xs) = multiset_of xs"
```
```   940   by (induct xs) simp_all
```
```   941
```
```   942 lemma surj_multiset_of: "surj multiset_of"
```
```   943 apply (unfold surj_def)
```
```   944 apply (rule allI)
```
```   945 apply (rule_tac M = y in multiset_induct)
```
```   946  apply auto
```
```   947 apply (rule_tac x = "x # xa" in exI)
```
```   948 apply auto
```
```   949 done
```
```   950
```
```   951 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
```
```   952 by (induct x) auto
```
```   953
```
```   954 lemma distinct_count_atmost_1:
```
```   955   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
```
```   956 apply (induct x, simp, rule iffI, simp_all)
```
```   957 apply (rule conjI)
```
```   958 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
```
```   959 apply (erule_tac x = a in allE, simp, clarify)
```
```   960 apply (erule_tac x = aa in allE, simp)
```
```   961 done
```
```   962
```
```   963 lemma multiset_of_eq_setD:
```
```   964   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
```
```   965 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
```
```   966
```
```   967 lemma set_eq_iff_multiset_of_eq_distinct:
```
```   968   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
```
```   969     (set x = set y) = (multiset_of x = multiset_of y)"
```
```   970 by (auto simp: multiset_eq_iff distinct_count_atmost_1)
```
```   971
```
```   972 lemma set_eq_iff_multiset_of_remdups_eq:
```
```   973    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
```
```   974 apply (rule iffI)
```
```   975 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
```
```   976 apply (drule distinct_remdups [THEN distinct_remdups
```
```   977       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
```
```   978 apply simp
```
```   979 done
```
```   980
```
```   981 lemma multiset_of_compl_union [simp]:
```
```   982   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
```
```   983   by (induct xs) (auto simp: add_ac)
```
```   984
```
```   985 lemma count_multiset_of_length_filter:
```
```   986   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
```
```   987   by (induct xs) auto
```
```   988
```
```   989 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
```
```   990 apply (induct ls arbitrary: i)
```
```   991  apply simp
```
```   992 apply (case_tac i)
```
```   993  apply auto
```
```   994 done
```
```   995
```
```   996 lemma multiset_of_remove1[simp]:
```
```   997   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
```
```   998 by (induct xs) (auto simp add: multiset_eq_iff)
```
```   999
```
```  1000 lemma multiset_of_eq_length:
```
```  1001   assumes "multiset_of xs = multiset_of ys"
```
```  1002   shows "length xs = length ys"
```
```  1003   using assms by (metis size_multiset_of)
```
```  1004
```
```  1005 lemma multiset_of_eq_length_filter:
```
```  1006   assumes "multiset_of xs = multiset_of ys"
```
```  1007   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
```
```  1008   using assms by (metis count_multiset_of)
```
```  1009
```
```  1010 lemma fold_multiset_equiv:
```
```  1011   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"
```
```  1012     and equiv: "multiset_of xs = multiset_of ys"
```
```  1013   shows "List.fold f xs = List.fold f ys"
```
```  1014 using f equiv [symmetric]
```
```  1015 proof (induct xs arbitrary: ys)
```
```  1016   case Nil then show ?case by simp
```
```  1017 next
```
```  1018   case (Cons x xs)
```
```  1019   then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
```
```  1020   have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
```
```  1021     by (rule Cons.prems(1)) (simp_all add: *)
```
```  1022   moreover from * have "x \<in> set ys" by simp
```
```  1023   ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
```
```  1024   moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" by (auto intro: Cons.hyps)
```
```  1025   ultimately show ?case by simp
```
```  1026 qed
```
```  1027
```
```  1028 lemma multiset_of_insort [simp]:
```
```  1029   "multiset_of (insort x xs) = multiset_of xs + {#x#}"
```
```  1030   by (induct xs) (simp_all add: ac_simps)
```
```  1031
```
```  1032 lemma in_multiset_of:
```
```  1033   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```  1034   by (induct xs) simp_all
```
```  1035
```
```  1036 lemma multiset_of_map:
```
```  1037   "multiset_of (map f xs) = image_mset f (multiset_of xs)"
```
```  1038   by (induct xs) simp_all
```
```  1039
```
```  1040 definition multiset_of_set :: "'a set \<Rightarrow> 'a multiset"
```
```  1041 where
```
```  1042   "multiset_of_set = folding.F (\<lambda>x M. {#x#} + M) {#}"
```
```  1043
```
```  1044 interpretation multiset_of_set!: folding "\<lambda>x M. {#x#} + M" "{#}"
```
```  1045 where
```
```  1046   "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set"
```
```  1047 proof -
```
```  1048   interpret comp_fun_commute "\<lambda>x M. {#x#} + M" by default (simp add: fun_eq_iff ac_simps)
```
```  1049   show "folding (\<lambda>x M. {#x#} + M)" by default (fact comp_fun_commute)
```
```  1050   from multiset_of_set_def show "folding.F (\<lambda>x M. {#x#} + M) {#} = multiset_of_set" ..
```
```  1051 qed
```
```  1052
```
```  1053 lemma count_multiset_of_set [simp]:
```
```  1054   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (multiset_of_set A) x = 1" (is "PROP ?P")
```
```  1055   "\<not> finite A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?Q")
```
```  1056   "x \<notin> A \<Longrightarrow> count (multiset_of_set A) x = 0" (is "PROP ?R")
```
```  1057 proof -
```
```  1058   { fix A
```
```  1059     assume "x \<notin> A"
```
```  1060     have "count (multiset_of_set A) x = 0"
```
```  1061     proof (cases "finite A")
```
```  1062       case False then show ?thesis by simp
```
```  1063     next
```
```  1064       case True from True `x \<notin> A` show ?thesis by (induct A) auto
```
```  1065     qed
```
```  1066   } note * = this
```
```  1067   then show "PROP ?P" "PROP ?Q" "PROP ?R"
```
```  1068   by (auto elim!: Set.set_insert)
```
```  1069 qed -- {* TODO: maybe define @{const multiset_of_set} also in terms of @{const Abs_multiset} *}
```
```  1070
```
```  1071 context linorder
```
```  1072 begin
```
```  1073
```
```  1074 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
```
```  1075 where
```
```  1076   "sorted_list_of_multiset M = fold insort [] M"
```
```  1077
```
```  1078 lemma sorted_list_of_multiset_empty [simp]:
```
```  1079   "sorted_list_of_multiset {#} = []"
```
```  1080   by (simp add: sorted_list_of_multiset_def)
```
```  1081
```
```  1082 lemma sorted_list_of_multiset_singleton [simp]:
```
```  1083   "sorted_list_of_multiset {#x#} = [x]"
```
```  1084 proof -
```
```  1085   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1086   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1087 qed
```
```  1088
```
```  1089 lemma sorted_list_of_multiset_insert [simp]:
```
```  1090   "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
```
```  1091 proof -
```
```  1092   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
```
```  1093   show ?thesis by (simp add: sorted_list_of_multiset_def)
```
```  1094 qed
```
```  1095
```
```  1096 end
```
```  1097
```
```  1098 lemma multiset_of_sorted_list_of_multiset [simp]:
```
```  1099   "multiset_of (sorted_list_of_multiset M) = M"
```
```  1100   by (induct M) simp_all
```
```  1101
```
```  1102 lemma sorted_list_of_multiset_multiset_of [simp]:
```
```  1103   "sorted_list_of_multiset (multiset_of xs) = sort xs"
```
```  1104   by (induct xs) simp_all
```
```  1105
```
```  1106 lemma finite_set_of_multiset_of_set:
```
```  1107   assumes "finite A"
```
```  1108   shows "set_of (multiset_of_set A) = A"
```
```  1109   using assms by (induct A) simp_all
```
```  1110
```
```  1111 lemma infinite_set_of_multiset_of_set:
```
```  1112   assumes "\<not> finite A"
```
```  1113   shows "set_of (multiset_of_set A) = {}"
```
```  1114   using assms by simp
```
```  1115
```
```  1116 lemma set_sorted_list_of_multiset [simp]:
```
```  1117   "set (sorted_list_of_multiset M) = set_of M"
```
```  1118   by (induct M) (simp_all add: set_insort)
```
```  1119
```
```  1120 lemma sorted_list_of_multiset_of_set [simp]:
```
```  1121   "sorted_list_of_multiset (multiset_of_set A) = sorted_list_of_set A"
```
```  1122   by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)
```
```  1123
```
```  1124
```
```  1125 subsection {* Big operators *}
```
```  1126
```
```  1127 no_notation times (infixl "*" 70)
```
```  1128 no_notation Groups.one ("1")
```
```  1129
```
```  1130 locale comm_monoid_mset = comm_monoid
```
```  1131 begin
```
```  1132
```
```  1133 definition F :: "'a multiset \<Rightarrow> 'a"
```
```  1134 where
```
```  1135   eq_fold: "F M = Multiset.fold f 1 M"
```
```  1136
```
```  1137 lemma empty [simp]:
```
```  1138   "F {#} = 1"
```
```  1139   by (simp add: eq_fold)
```
```  1140
```
```  1141 lemma singleton [simp]:
```
```  1142   "F {#x#} = x"
```
```  1143 proof -
```
```  1144   interpret comp_fun_commute
```
```  1145     by default (simp add: fun_eq_iff left_commute)
```
```  1146   show ?thesis by (simp add: eq_fold)
```
```  1147 qed
```
```  1148
```
```  1149 lemma union [simp]:
```
```  1150   "F (M + N) = F M * F N"
```
```  1151 proof -
```
```  1152   interpret comp_fun_commute f
```
```  1153     by default (simp add: fun_eq_iff left_commute)
```
```  1154   show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
```
```  1155 qed
```
```  1156
```
```  1157 end
```
```  1158
```
```  1159 notation times (infixl "*" 70)
```
```  1160 notation Groups.one ("1")
```
```  1161
```
```  1162 definition (in comm_monoid_add) msetsum :: "'a multiset \<Rightarrow> 'a"
```
```  1163 where
```
```  1164   "msetsum = comm_monoid_mset.F plus 0"
```
```  1165
```
```  1166 definition (in comm_monoid_mult) msetprod :: "'a multiset \<Rightarrow> 'a"
```
```  1167 where
```
```  1168   "msetprod = comm_monoid_mset.F times 1"
```
```  1169
```
```  1170 sublocale comm_monoid_add < msetsum!: comm_monoid_mset plus 0
```
```  1171 where
```
```  1172   "comm_monoid_mset.F plus 0 = msetsum"
```
```  1173 proof -
```
```  1174   show "comm_monoid_mset plus 0" ..
```
```  1175   from msetsum_def show "comm_monoid_mset.F plus 0 = msetsum" ..
```
```  1176 qed
```
```  1177
```
```  1178 context comm_monoid_add
```
```  1179 begin
```
```  1180
```
```  1181 lemma setsum_unfold_msetsum:
```
```  1182   "setsum f A = msetsum (image_mset f (multiset_of_set A))"
```
```  1183   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1184
```
```  1185 abbreviation msetsum_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
```
```  1186 where
```
```  1187   "msetsum_image f M \<equiv> msetsum (image_mset f M)"
```
```  1188
```
```  1189 end
```
```  1190
```
```  1191 syntax
```
```  1192   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1193       ("(3SUM _:#_. _)" [0, 51, 10] 10)
```
```  1194
```
```  1195 syntax (xsymbols)
```
```  1196   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1197       ("(3\<Sum>_:#_. _)" [0, 51, 10] 10)
```
```  1198
```
```  1199 syntax (HTML output)
```
```  1200   "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"
```
```  1201       ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1202
```
```  1203 translations
```
```  1204   "SUM i :# A. b" == "CONST msetsum_image (\<lambda>i. b) A"
```
```  1205
```
```  1206 sublocale comm_monoid_mult < msetprod!: comm_monoid_mset times 1
```
```  1207 where
```
```  1208   "comm_monoid_mset.F times 1 = msetprod"
```
```  1209 proof -
```
```  1210   show "comm_monoid_mset times 1" ..
```
```  1211   from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" ..
```
```  1212 qed
```
```  1213
```
```  1214 context comm_monoid_mult
```
```  1215 begin
```
```  1216
```
```  1217 lemma msetprod_empty:
```
```  1218   "msetprod {#} = 1"
```
```  1219   by (fact msetprod.empty)
```
```  1220
```
```  1221 lemma msetprod_singleton:
```
```  1222   "msetprod {#x#} = x"
```
```  1223   by (fact msetprod.singleton)
```
```  1224
```
```  1225 lemma msetprod_Un:
```
```  1226   "msetprod (A + B) = msetprod A * msetprod B"
```
```  1227   by (fact msetprod.union)
```
```  1228
```
```  1229 lemma setprod_unfold_msetprod:
```
```  1230   "setprod f A = msetprod (image_mset f (multiset_of_set A))"
```
```  1231   by (cases "finite A") (induct A rule: finite_induct, simp_all)
```
```  1232
```
```  1233 lemma msetprod_multiplicity:
```
```  1234   "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
```
```  1235   by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
```
```  1236
```
```  1237 abbreviation msetprod_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
```
```  1238 where
```
```  1239   "msetprod_image f M \<equiv> msetprod (image_mset f M)"
```
```  1240
```
```  1241 end
```
```  1242
```
```  1243 syntax
```
```  1244   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1245       ("(3PROD _:#_. _)" [0, 51, 10] 10)
```
```  1246
```
```  1247 syntax (xsymbols)
```
```  1248   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1249       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1250
```
```  1251 syntax (HTML output)
```
```  1252   "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"
```
```  1253       ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
```
```  1254
```
```  1255 translations
```
```  1256   "PROD i :# A. b" == "CONST msetprod_image (\<lambda>i. b) A"
```
```  1257
```
```  1258 lemma (in comm_semiring_1) dvd_msetprod:
```
```  1259   assumes "x \<in># A"
```
```  1260   shows "x dvd msetprod A"
```
```  1261 proof -
```
```  1262   from assms have "A = (A - {#x#}) + {#x#}" by simp
```
```  1263   then obtain B where "A = B + {#x#}" ..
```
```  1264   then show ?thesis by simp
```
```  1265 qed
```
```  1266
```
```  1267
```
```  1268 subsection {* Cardinality *}
```
```  1269
```
```  1270 definition mcard :: "'a multiset \<Rightarrow> nat"
```
```  1271 where
```
```  1272   "mcard = msetsum \<circ> image_mset (\<lambda>_. 1)"
```
```  1273
```
```  1274 lemma mcard_empty [simp]:
```
```  1275   "mcard {#} = 0"
```
```  1276   by (simp add: mcard_def)
```
```  1277
```
```  1278 lemma mcard_singleton [simp]:
```
```  1279   "mcard {#a#} = Suc 0"
```
```  1280   by (simp add: mcard_def)
```
```  1281
```
```  1282 lemma mcard_plus [simp]:
```
```  1283   "mcard (M + N) = mcard M + mcard N"
```
```  1284   by (simp add: mcard_def)
```
```  1285
```
```  1286 lemma mcard_empty_iff [simp]:
```
```  1287   "mcard M = 0 \<longleftrightarrow> M = {#}"
```
```  1288   by (induct M) simp_all
```
```  1289
```
```  1290 lemma mcard_unfold_setsum:
```
```  1291   "mcard M = setsum (count M) (set_of M)"
```
```  1292 proof (induct M)
```
```  1293   case empty then show ?case by simp
```
```  1294 next
```
```  1295   case (add M x) then show ?case
```
```  1296     by (cases "x \<in> set_of M")
```
```  1297       (simp_all del: mem_set_of_iff add: setsum.distrib setsum.delta' insert_absorb, simp)
```
```  1298 qed
```
```  1299
```
```  1300 lemma size_eq_mcard:
```
```  1301   "size = mcard"
```
```  1302   by (simp add: fun_eq_iff size_def mcard_unfold_setsum)
```
```  1303
```
```  1304 lemma mcard_multiset_of:
```
```  1305   "mcard (multiset_of xs) = length xs"
```
```  1306   by (induct xs) simp_all
```
```  1307
```
```  1308
```
```  1309 subsection {* Alternative representations *}
```
```  1310
```
```  1311 subsubsection {* Lists *}
```
```  1312
```
```  1313 context linorder
```
```  1314 begin
```
```  1315
```
```  1316 lemma multiset_of_insort [simp]:
```
```  1317   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
```
```  1318   by (induct xs) (simp_all add: ac_simps)
```
```  1319
```
```  1320 lemma multiset_of_sort [simp]:
```
```  1321   "multiset_of (sort_key k xs) = multiset_of xs"
```
```  1322   by (induct xs) (simp_all add: ac_simps)
```
```  1323
```
```  1324 text {*
```
```  1325   This lemma shows which properties suffice to show that a function
```
```  1326   @{text "f"} with @{text "f xs = ys"} behaves like sort.
```
```  1327 *}
```
```  1328
```
```  1329 lemma properties_for_sort_key:
```
```  1330   assumes "multiset_of ys = multiset_of xs"
```
```  1331   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
```
```  1332   and "sorted (map f ys)"
```
```  1333   shows "sort_key f xs = ys"
```
```  1334 using assms
```
```  1335 proof (induct xs arbitrary: ys)
```
```  1336   case Nil then show ?case by simp
```
```  1337 next
```
```  1338   case (Cons x xs)
```
```  1339   from Cons.prems(2) have
```
```  1340     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
```
```  1341     by (simp add: filter_remove1)
```
```  1342   with Cons.prems have "sort_key f xs = remove1 x ys"
```
```  1343     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
```
```  1344   moreover from Cons.prems have "x \<in> set ys"
```
```  1345     by (auto simp add: mem_set_multiset_eq intro!: ccontr)
```
```  1346   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
```
```  1347 qed
```
```  1348
```
```  1349 lemma properties_for_sort:
```
```  1350   assumes multiset: "multiset_of ys = multiset_of xs"
```
```  1351   and "sorted ys"
```
```  1352   shows "sort xs = ys"
```
```  1353 proof (rule properties_for_sort_key)
```
```  1354   from multiset show "multiset_of ys = multiset_of xs" .
```
```  1355   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
```
```  1356   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
```
```  1357     by (rule multiset_of_eq_length_filter)
```
```  1358   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
```
```  1359     by simp
```
```  1360   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
```
```  1361     by (simp add: replicate_length_filter)
```
```  1362 qed
```
```  1363
```
```  1364 lemma sort_key_by_quicksort:
```
```  1365   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
```
```  1366     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
```
```  1367     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
```
```  1368 proof (rule properties_for_sort_key)
```
```  1369   show "multiset_of ?rhs = multiset_of ?lhs"
```
```  1370     by (rule multiset_eqI) (auto simp add: multiset_of_filter)
```
```  1371 next
```
```  1372   show "sorted (map f ?rhs)"
```
```  1373     by (auto simp add: sorted_append intro: sorted_map_same)
```
```  1374 next
```
```  1375   fix l
```
```  1376   assume "l \<in> set ?rhs"
```
```  1377   let ?pivot = "f (xs ! (length xs div 2))"
```
```  1378   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
```
```  1379   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
```
```  1380     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
```
```  1381   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
```
```  1382   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
```
```  1383   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
```
```  1384     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
```
```  1385   note *** = this [of "op <"] this [of "op >"] this [of "op ="]
```
```  1386   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
```
```  1387   proof (cases "f l" ?pivot rule: linorder_cases)
```
```  1388     case less
```
```  1389     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
```
```  1390     with less show ?thesis
```
```  1391       by (simp add: filter_sort [symmetric] ** ***)
```
```  1392   next
```
```  1393     case equal then show ?thesis
```
```  1394       by (simp add: * less_le)
```
```  1395   next
```
```  1396     case greater
```
```  1397     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
```
```  1398     with greater show ?thesis
```
```  1399       by (simp add: filter_sort [symmetric] ** ***)
```
```  1400   qed
```
```  1401 qed
```
```  1402
```
```  1403 lemma sort_by_quicksort:
```
```  1404   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
```
```  1405     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
```
```  1406     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
```
```  1407   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
```
```  1408
```
```  1409 text {* A stable parametrized quicksort *}
```
```  1410
```
```  1411 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
```
```  1412   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
```
```  1413
```
```  1414 lemma part_code [code]:
```
```  1415   "part f pivot [] = ([], [], [])"
```
```  1416   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
```
```  1417      if x' < pivot then (x # lts, eqs, gts)
```
```  1418      else if x' > pivot then (lts, eqs, x # gts)
```
```  1419      else (lts, x # eqs, gts))"
```
```  1420   by (auto simp add: part_def Let_def split_def)
```
```  1421
```
```  1422 lemma sort_key_by_quicksort_code [code]:
```
```  1423   "sort_key f xs = (case xs of [] \<Rightarrow> []
```
```  1424     | [x] \<Rightarrow> xs
```
```  1425     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
```
```  1426     | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1427        in sort_key f lts @ eqs @ sort_key f gts))"
```
```  1428 proof (cases xs)
```
```  1429   case Nil then show ?thesis by simp
```
```  1430 next
```
```  1431   case (Cons _ ys) note hyps = Cons show ?thesis
```
```  1432   proof (cases ys)
```
```  1433     case Nil with hyps show ?thesis by simp
```
```  1434   next
```
```  1435     case (Cons _ zs) note hyps = hyps Cons show ?thesis
```
```  1436     proof (cases zs)
```
```  1437       case Nil with hyps show ?thesis by auto
```
```  1438     next
```
```  1439       case Cons
```
```  1440       from sort_key_by_quicksort [of f xs]
```
```  1441       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
```
```  1442         in sort_key f lts @ eqs @ sort_key f gts)"
```
```  1443       by (simp only: split_def Let_def part_def fst_conv snd_conv)
```
```  1444       with hyps Cons show ?thesis by (simp only: list.cases)
```
```  1445     qed
```
```  1446   qed
```
```  1447 qed
```
```  1448
```
```  1449 end
```
```  1450
```
```  1451 hide_const (open) part
```
```  1452
```
```  1453 lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
```
```  1454   by (induct xs) (auto intro: order_trans)
```
```  1455
```
```  1456 lemma multiset_of_update:
```
```  1457   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
```
```  1458 proof (induct ls arbitrary: i)
```
```  1459   case Nil then show ?case by simp
```
```  1460 next
```
```  1461   case (Cons x xs)
```
```  1462   show ?case
```
```  1463   proof (cases i)
```
```  1464     case 0 then show ?thesis by simp
```
```  1465   next
```
```  1466     case (Suc i')
```
```  1467     with Cons show ?thesis
```
```  1468       apply simp
```
```  1469       apply (subst add_assoc)
```
```  1470       apply (subst add_commute [of "{#v#}" "{#x#}"])
```
```  1471       apply (subst add_assoc [symmetric])
```
```  1472       apply simp
```
```  1473       apply (rule mset_le_multiset_union_diff_commute)
```
```  1474       apply (simp add: mset_le_single nth_mem_multiset_of)
```
```  1475       done
```
```  1476   qed
```
```  1477 qed
```
```  1478
```
```  1479 lemma multiset_of_swap:
```
```  1480   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
```
```  1481     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
```
```  1482   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
```
```  1483
```
```  1484
```
```  1485 subsection {* The multiset order *}
```
```  1486
```
```  1487 subsubsection {* Well-foundedness *}
```
```  1488
```
```  1489 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1490   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```  1491       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```  1492
```
```  1493 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
```
```  1494   "mult r = (mult1 r)\<^sup>+"
```
```  1495
```
```  1496 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```  1497 by (simp add: mult1_def)
```
```  1498
```
```  1499 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```  1500     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```  1501     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```  1502   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
```
```  1503 proof (unfold mult1_def)
```
```  1504   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```  1505   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```  1506   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```  1507
```
```  1508   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```  1509   then have "\<exists>a' M0' K.
```
```  1510       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```  1511   then show "?case1 \<or> ?case2"
```
```  1512   proof (elim exE conjE)
```
```  1513     fix a' M0' K
```
```  1514     assume N: "N = M0' + K" and r: "?r K a'"
```
```  1515     assume "M0 + {#a#} = M0' + {#a'#}"
```
```  1516     then have "M0 = M0' \<and> a = a' \<or>
```
```  1517         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```  1518       by (simp only: add_eq_conv_ex)
```
```  1519     then show ?thesis
```
```  1520     proof (elim disjE conjE exE)
```
```  1521       assume "M0 = M0'" "a = a'"
```
```  1522       with N r have "?r K a \<and> N = M0 + K" by simp
```
```  1523       then have ?case2 .. then show ?thesis ..
```
```  1524     next
```
```  1525       fix K'
```
```  1526       assume "M0' = K' + {#a#}"
```
```  1527       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
```
```  1528
```
```  1529       assume "M0 = K' + {#a'#}"
```
```  1530       with r have "?R (K' + K) M0" by blast
```
```  1531       with n have ?case1 by simp then show ?thesis ..
```
```  1532     qed
```
```  1533   qed
```
```  1534 qed
```
```  1535
```
```  1536 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
```
```  1537 proof
```
```  1538   let ?R = "mult1 r"
```
```  1539   let ?W = "acc ?R"
```
```  1540   {
```
```  1541     fix M M0 a
```
```  1542     assume M0: "M0 \<in> ?W"
```
```  1543       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1544       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```  1545     have "M0 + {#a#} \<in> ?W"
```
```  1546     proof (rule accI [of "M0 + {#a#}"])
```
```  1547       fix N
```
```  1548       assume "(N, M0 + {#a#}) \<in> ?R"
```
```  1549       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```  1550           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```  1551         by (rule less_add)
```
```  1552       then show "N \<in> ?W"
```
```  1553       proof (elim exE disjE conjE)
```
```  1554         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```  1555         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```  1556         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
```
```  1557         then show "N \<in> ?W" by (simp only: N)
```
```  1558       next
```
```  1559         fix K
```
```  1560         assume N: "N = M0 + K"
```
```  1561         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```  1562         then have "M0 + K \<in> ?W"
```
```  1563         proof (induct K)
```
```  1564           case empty
```
```  1565           from M0 show "M0 + {#} \<in> ?W" by simp
```
```  1566         next
```
```  1567           case (add K x)
```
```  1568           from add.prems have "(x, a) \<in> r" by simp
```
```  1569           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```  1570           moreover from add have "M0 + K \<in> ?W" by simp
```
```  1571           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
```
```  1572           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
```
```  1573         qed
```
```  1574         then show "N \<in> ?W" by (simp only: N)
```
```  1575       qed
```
```  1576     qed
```
```  1577   } note tedious_reasoning = this
```
```  1578
```
```  1579   assume wf: "wf r"
```
```  1580   fix M
```
```  1581   show "M \<in> ?W"
```
```  1582   proof (induct M)
```
```  1583     show "{#} \<in> ?W"
```
```  1584     proof (rule accI)
```
```  1585       fix b assume "(b, {#}) \<in> ?R"
```
```  1586       with not_less_empty show "b \<in> ?W" by contradiction
```
```  1587     qed
```
```  1588
```
```  1589     fix M a assume "M \<in> ?W"
```
```  1590     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1591     proof induct
```
```  1592       fix a
```
```  1593       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```  1594       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```  1595       proof
```
```  1596         fix M assume "M \<in> ?W"
```
```  1597         then show "M + {#a#} \<in> ?W"
```
```  1598           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
```
```  1599       qed
```
```  1600     qed
```
```  1601     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
```
```  1602   qed
```
```  1603 qed
```
```  1604
```
```  1605 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```  1606 by (rule acc_wfI) (rule all_accessible)
```
```  1607
```
```  1608 theorem wf_mult: "wf r ==> wf (mult r)"
```
```  1609 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
```
```  1610
```
```  1611
```
```  1612 subsubsection {* Closure-free presentation *}
```
```  1613
```
```  1614 text {* One direction. *}
```
```  1615
```
```  1616 lemma mult_implies_one_step:
```
```  1617   "trans r ==> (M, N) \<in> mult r ==>
```
```  1618     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```  1619     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```  1620 apply (unfold mult_def mult1_def set_of_def)
```
```  1621 apply (erule converse_trancl_induct, clarify)
```
```  1622  apply (rule_tac x = M0 in exI, simp, clarify)
```
```  1623 apply (case_tac "a :# K")
```
```  1624  apply (rule_tac x = I in exI)
```
```  1625  apply (simp (no_asm))
```
```  1626  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```  1627  apply (simp (no_asm_simp) add: add_assoc [symmetric])
```
```  1628  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```  1629  apply (simp add: diff_union_single_conv)
```
```  1630  apply (simp (no_asm_use) add: trans_def)
```
```  1631  apply blast
```
```  1632 apply (subgoal_tac "a :# I")
```
```  1633  apply (rule_tac x = "I - {#a#}" in exI)
```
```  1634  apply (rule_tac x = "J + {#a#}" in exI)
```
```  1635  apply (rule_tac x = "K + Ka" in exI)
```
```  1636  apply (rule conjI)
```
```  1637   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1638  apply (rule conjI)
```
```  1639   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
```
```  1640   apply (simp add: multiset_eq_iff split: nat_diff_split)
```
```  1641  apply (simp (no_asm_use) add: trans_def)
```
```  1642  apply blast
```
```  1643 apply (subgoal_tac "a :# (M0 + {#a#})")
```
```  1644  apply simp
```
```  1645 apply (simp (no_asm))
```
```  1646 done
```
```  1647
```
```  1648 lemma one_step_implies_mult_aux:
```
```  1649   "trans r ==>
```
```  1650     \<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))
```
```  1651       --> (I + K, I + J) \<in> mult r"
```
```  1652 apply (induct_tac n, auto)
```
```  1653 apply (frule size_eq_Suc_imp_eq_union, clarify)
```
```  1654 apply (rename_tac "J'", simp)
```
```  1655 apply (erule notE, auto)
```
```  1656 apply (case_tac "J' = {#}")
```
```  1657  apply (simp add: mult_def)
```
```  1658  apply (rule r_into_trancl)
```
```  1659  apply (simp add: mult1_def set_of_def, blast)
```
```  1660 txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```  1661 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```  1662 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```  1663 apply (erule ssubst)
```
```  1664 apply (simp add: Ball_def, auto)
```
```  1665 apply (subgoal_tac
```
```  1666   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
```
```  1667     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```  1668  prefer 2
```
```  1669  apply force
```
```  1670 apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
```
```  1671 apply (erule trancl_trans)
```
```  1672 apply (rule r_into_trancl)
```
```  1673 apply (simp add: mult1_def set_of_def)
```
```  1674 apply (rule_tac x = a in exI)
```
```  1675 apply (rule_tac x = "I + J'" in exI)
```
```  1676 apply (simp add: add_ac)
```
```  1677 done
```
```  1678
```
```  1679 lemma one_step_implies_mult:
```
```  1680   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```  1681     ==> (I + K, I + J) \<in> mult r"
```
```  1682 using one_step_implies_mult_aux by blast
```
```  1683
```
```  1684
```
```  1685 subsubsection {* Partial-order properties *}
```
```  1686
```
```  1687 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where
```
```  1688   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
```
```  1689
```
```  1690 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where
```
```  1691   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"
```
```  1692
```
```  1693 notation (xsymbols) less_multiset (infix "\<subset>#" 50)
```
```  1694 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)
```
```  1695
```
```  1696 interpretation multiset_order: order le_multiset less_multiset
```
```  1697 proof -
```
```  1698   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"
```
```  1699   proof
```
```  1700     fix M :: "'a multiset"
```
```  1701     assume "M \<subset># M"
```
```  1702     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
```
```  1703     have "trans {(x'::'a, x). x' < x}"
```
```  1704       by (rule transI) simp
```
```  1705     moreover note MM
```
```  1706     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
```
```  1707       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"
```
```  1708       by (rule mult_implies_one_step)
```
```  1709     then obtain I J K where "M = I + J" and "M = I + K"
```
```  1710       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
```
```  1711     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto
```
```  1712     have "finite (set_of K)" by simp
```
```  1713     moreover note aux2
```
```  1714     ultimately have "set_of K = {}"
```
```  1715       by (induct rule: finite_induct) (auto intro: order_less_trans)
```
```  1716     with aux1 show False by simp
```
```  1717   qed
```
```  1718   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"
```
```  1719     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
```
```  1720   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
```
```  1721     by default (auto simp add: le_multiset_def irrefl dest: trans)
```
```  1722 qed
```
```  1723
```
```  1724 lemma mult_less_irrefl [elim!]: "M \<subset># (M::'a::order multiset) ==> R"
```
```  1725   by simp
```
```  1726
```
```  1727
```
```  1728 subsubsection {* Monotonicity of multiset union *}
```
```  1729
```
```  1730 lemma mult1_union: "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"
```
```  1731 apply (unfold mult1_def)
```
```  1732 apply auto
```
```  1733 apply (rule_tac x = a in exI)
```
```  1734 apply (rule_tac x = "C + M0" in exI)
```
```  1735 apply (simp add: add_assoc)
```
```  1736 done
```
```  1737
```
```  1738 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"
```
```  1739 apply (unfold less_multiset_def mult_def)
```
```  1740 apply (erule trancl_induct)
```
```  1741  apply (blast intro: mult1_union)
```
```  1742 apply (blast intro: mult1_union trancl_trans)
```
```  1743 done
```
```  1744
```
```  1745 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"
```
```  1746 apply (subst add_commute [of B C])
```
```  1747 apply (subst add_commute [of D C])
```
```  1748 apply (erule union_less_mono2)
```
```  1749 done
```
```  1750
```
```  1751 lemma union_less_mono:
```
```  1752   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"
```
```  1753   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)
```
```  1754
```
```  1755 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
```
```  1756 proof
```
```  1757 qed (auto simp add: le_multiset_def intro: union_less_mono2)
```
```  1758
```
```  1759
```
```  1760 subsection {* Termination proofs with multiset orders *}
```
```  1761
```
```  1762 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
```
```  1763   and multi_member_this: "x \<in># {# x #} + XS"
```
```  1764   and multi_member_last: "x \<in># {# x #}"
```
```  1765   by auto
```
```  1766
```
```  1767 definition "ms_strict = mult pair_less"
```
```  1768 definition "ms_weak = ms_strict \<union> Id"
```
```  1769
```
```  1770 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
```
```  1771 unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
```
```  1772 by (auto intro: wf_mult1 wf_trancl simp: mult_def)
```
```  1773
```
```  1774 lemma smsI:
```
```  1775   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
```
```  1776   unfolding ms_strict_def
```
```  1777 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
```
```  1778
```
```  1779 lemma wmsI:
```
```  1780   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
```
```  1781   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
```
```  1782 unfolding ms_weak_def ms_strict_def
```
```  1783 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
```
```  1784
```
```  1785 inductive pw_leq
```
```  1786 where
```
```  1787   pw_leq_empty: "pw_leq {#} {#}"
```
```  1788 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
```
```  1789
```
```  1790 lemma pw_leq_lstep:
```
```  1791   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
```
```  1792 by (drule pw_leq_step) (rule pw_leq_empty, simp)
```
```  1793
```
```  1794 lemma pw_leq_split:
```
```  1795   assumes "pw_leq X Y"
```
```  1796   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 = {#}))"
```
```  1797   using assms
```
```  1798 proof (induct)
```
```  1799   case pw_leq_empty thus ?case by auto
```
```  1800 next
```
```  1801   case (pw_leq_step x y X Y)
```
```  1802   then obtain A B Z where
```
```  1803     [simp]: "X = A + Z" "Y = B + Z"
```
```  1804       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
```
```  1805     by auto
```
```  1806   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less"
```
```  1807     unfolding pair_leq_def by auto
```
```  1808   thus ?case
```
```  1809   proof
```
```  1810     assume [simp]: "x = y"
```
```  1811     have
```
```  1812       "{#x#} + X = A + ({#y#}+Z)
```
```  1813       \<and> {#y#} + Y = B + ({#y#}+Z)
```
```  1814       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
```
```  1815       by (auto simp: add_ac)
```
```  1816     thus ?case by (intro exI)
```
```  1817   next
```
```  1818     assume A: "(x, y) \<in> pair_less"
```
```  1819     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
```
```  1820     have "{#x#} + X = ?A' + Z"
```
```  1821       "{#y#} + Y = ?B' + Z"
```
```  1822       by (auto simp add: add_ac)
```
```  1823     moreover have
```
```  1824       "(set_of ?A', set_of ?B') \<in> max_strict"
```
```  1825       using 1 A unfolding max_strict_def
```
```  1826       by (auto elim!: max_ext.cases)
```
```  1827     ultimately show ?thesis by blast
```
```  1828   qed
```
```  1829 qed
```
```  1830
```
```  1831 lemma
```
```  1832   assumes pwleq: "pw_leq Z Z'"
```
```  1833   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
```
```  1834   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
```
```  1835   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
```
```  1836 proof -
```
```  1837   from pw_leq_split[OF pwleq]
```
```  1838   obtain A' B' Z''
```
```  1839     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
```
```  1840     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
```
```  1841     by blast
```
```  1842   {
```
```  1843     assume max: "(set_of A, set_of B) \<in> max_strict"
```
```  1844     from mx_or_empty
```
```  1845     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
```
```  1846     proof
```
```  1847       assume max': "(set_of A', set_of B') \<in> max_strict"
```
```  1848       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
```
```  1849         by (auto simp: max_strict_def intro: max_ext_additive)
```
```  1850       thus ?thesis by (rule smsI)
```
```  1851     next
```
```  1852       assume [simp]: "A' = {#} \<and> B' = {#}"
```
```  1853       show ?thesis by (rule smsI) (auto intro: max)
```
```  1854     qed
```
```  1855     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
```
```  1856     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
```
```  1857   }
```
```  1858   from mx_or_empty
```
```  1859   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
```
```  1860   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
```
```  1861 qed
```
```  1862
```
```  1863 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
```
```  1864 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
```
```  1865 and nonempty_single: "{# x #} \<noteq> {#}"
```
```  1866 by auto
```
```  1867
```
```  1868 setup {*
```
```  1869 let
```
```  1870   fun msetT T = Type (@{type_name multiset}, [T]);
```
```  1871
```
```  1872   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
```
```  1873     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) \$ x
```
```  1874     | mk_mset T (x :: xs) =
```
```  1875           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
```
```  1876                 mk_mset T [x] \$ mk_mset T xs
```
```  1877
```
```  1878   fun mset_member_tac m i =
```
```  1879       (if m <= 0 then
```
```  1880            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
```
```  1881        else
```
```  1882            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
```
```  1883
```
```  1884   val mset_nonempty_tac =
```
```  1885       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
```
```  1886
```
```  1887   val regroup_munion_conv =
```
```  1888       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
```
```  1889         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))
```
```  1890
```
```  1891   fun unfold_pwleq_tac i =
```
```  1892     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
```
```  1893       ORELSE (rtac @{thm pw_leq_lstep} i)
```
```  1894       ORELSE (rtac @{thm pw_leq_empty} i)
```
```  1895
```
```  1896   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
```
```  1897                       @{thm Un_insert_left}, @{thm Un_empty_left}]
```
```  1898 in
```
```  1899   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
```
```  1900   {
```
```  1901     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
```
```  1902     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
```
```  1903     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
```
```  1904     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
```
```  1905     reduction_pair= @{thm ms_reduction_pair}
```
```  1906   })
```
```  1907 end
```
```  1908 *}
```
```  1909
```
```  1910
```
```  1911 subsection {* Legacy theorem bindings *}
```
```  1912
```
```  1913 lemmas multi_count_eq = multiset_eq_iff [symmetric]
```
```  1914
```
```  1915 lemma union_commute: "M + N = N + (M::'a multiset)"
```
```  1916   by (fact add_commute)
```
```  1917
```
```  1918 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```  1919   by (fact add_assoc)
```
```  1920
```
```  1921 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```  1922   by (fact add_left_commute)
```
```  1923
```
```  1924 lemmas union_ac = union_assoc union_commute union_lcomm
```
```  1925
```
```  1926 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
```
```  1927   by (fact add_right_cancel)
```
```  1928
```
```  1929 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
```
```  1930   by (fact add_left_cancel)
```
```  1931
```
```  1932 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
```
```  1933   by (fact add_imp_eq)
```
```  1934
```
```  1935 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"
```
```  1936   by (fact order_less_trans)
```
```  1937
```
```  1938 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
```
```  1939   by (fact inf.commute)
```
```  1940
```
```  1941 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
```
```  1942   by (fact inf.assoc [symmetric])
```
```  1943
```
```  1944 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
```
```  1945   by (fact inf.left_commute)
```
```  1946
```
```  1947 lemmas multiset_inter_ac =
```
```  1948   multiset_inter_commute
```
```  1949   multiset_inter_assoc
```
```  1950   multiset_inter_left_commute
```
```  1951
```
```  1952 lemma mult_less_not_refl:
```
```  1953   "\<not> M \<subset># (M::'a::order multiset)"
```
```  1954   by (fact multiset_order.less_irrefl)
```
```  1955
```
```  1956 lemma mult_less_trans:
```
```  1957   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"
```
```  1958   by (fact multiset_order.less_trans)
```
```  1959
```
```  1960 lemma mult_less_not_sym:
```
```  1961   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"
```
```  1962   by (fact multiset_order.less_not_sym)
```
```  1963
```
```  1964 lemma mult_less_asym:
```
```  1965   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"
```
```  1966   by (fact multiset_order.less_asym)
```
```  1967
```
```  1968 ML {*
```
```  1969 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
```
```  1970                       (Const _ \$ t') =
```
```  1971     let
```
```  1972       val (maybe_opt, ps) =
```
```  1973         Nitpick_Model.dest_plain_fun t' ||> op ~~
```
```  1974         ||> map (apsnd (snd o HOLogic.dest_number))
```
```  1975       fun elems_for t =
```
```  1976         case AList.lookup (op =) ps t of
```
```  1977           SOME n => replicate n t
```
```  1978         | NONE => [Const (maybe_name, elem_T --> elem_T) \$ t]
```
```  1979     in
```
```  1980       case maps elems_for (all_values elem_T) @
```
```  1981            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
```
```  1982             else []) of
```
```  1983         [] => Const (@{const_name zero_class.zero}, T)
```
```  1984       | ts => foldl1 (fn (t1, t2) =>
```
```  1985                          Const (@{const_name plus_class.plus}, T --> T --> T)
```
```  1986                          \$ t1 \$ t2)
```
```  1987                      (map (curry (op \$) (Const (@{const_name single},
```
```  1988                                                 elem_T --> T))) ts)
```
```  1989     end
```
```  1990   | multiset_postproc _ _ _ _ t = t
```
```  1991 *}
```
```  1992
```
```  1993 declaration {*
```
```  1994 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}
```
```  1995     multiset_postproc
```
```  1996 *}
```
```  1997
```
```  1998 hide_const (open) fold
```
```  1999
```
```  2000
```
```  2001 subsection {* Naive implementation using lists *}
```
```  2002
```
```  2003 code_datatype multiset_of
```
```  2004
```
```  2005 lemma [code]:
```
```  2006   "{#} = multiset_of []"
```
```  2007   by simp
```
```  2008
```
```  2009 lemma [code]:
```
```  2010   "{#x#} = multiset_of [x]"
```
```  2011   by simp
```
```  2012
```
```  2013 lemma union_code [code]:
```
```  2014   "multiset_of xs + multiset_of ys = multiset_of (xs @ ys)"
```
```  2015   by simp
```
```  2016
```
```  2017 lemma [code]:
```
```  2018   "image_mset f (multiset_of xs) = multiset_of (map f xs)"
```
```  2019   by (simp add: multiset_of_map)
```
```  2020
```
```  2021 lemma [code]:
```
```  2022   "Multiset.filter f (multiset_of xs) = multiset_of (filter f xs)"
```
```  2023   by (simp add: multiset_of_filter)
```
```  2024
```
```  2025 lemma [code]:
```
```  2026   "multiset_of xs - multiset_of ys = multiset_of (fold remove1 ys xs)"
```
```  2027   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)
```
```  2028
```
```  2029 lemma [code]:
```
```  2030   "multiset_of xs #\<inter> multiset_of ys =
```
```  2031     multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2032       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
```
```  2033 proof -
```
```  2034   have "\<And>zs. multiset_of (snd (fold (\<lambda>x (ys, zs).
```
```  2035     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
```
```  2036       (multiset_of xs #\<inter> multiset_of ys) + multiset_of zs"
```
```  2037     by (induct xs arbitrary: ys)
```
```  2038       (auto simp add: mem_set_multiset_eq inter_add_right1 inter_add_right2 ac_simps)
```
```  2039   then show ?thesis by simp
```
```  2040 qed
```
```  2041
```
```  2042 lemma [code]:
```
```  2043   "multiset_of xs #\<union> multiset_of ys =
```
```  2044     multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
```
```  2045 proof -
```
```  2046   have "\<And>zs. multiset_of (split append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
```
```  2047       (multiset_of xs #\<union> multiset_of ys) + multiset_of zs"
```
```  2048     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
```
```  2049   then show ?thesis by simp
```
```  2050 qed
```
```  2051
```
```  2052 lemma [code_unfold]:
```
```  2053   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
```
```  2054   by (simp add: in_multiset_of)
```
```  2055
```
```  2056 lemma [code]:
```
```  2057   "count (multiset_of xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
```
```  2058 proof -
```
```  2059   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (multiset_of xs) x + n"
```
```  2060     by (induct xs) simp_all
```
```  2061   then show ?thesis by simp
```
```  2062 qed
```
```  2063
```
```  2064 lemma [code]:
```
```  2065   "set_of (multiset_of xs) = set xs"
```
```  2066   by simp
```
```  2067
```
```  2068 lemma [code]:
```
```  2069   "sorted_list_of_multiset (multiset_of xs) = sort xs"
```
```  2070   by (induct xs) simp_all
```
```  2071
```
```  2072 lemma [code]: -- {* not very efficient, but representation-ignorant! *}
```
```  2073   "multiset_of_set A = multiset_of (sorted_list_of_set A)"
```
```  2074   apply (cases "finite A")
```
```  2075   apply simp_all
```
```  2076   apply (induct A rule: finite_induct)
```
```  2077   apply (simp_all add: union_commute)
```
```  2078   done
```
```  2079
```
```  2080 lemma [code]:
```
```  2081   "mcard (multiset_of xs) = length xs"
```
```  2082   by (simp add: mcard_multiset_of)
```
```  2083
```
```  2084 lemma [code]:
```
```  2085   "A \<le> B \<longleftrightarrow> A #\<inter> B = A"
```
```  2086   by (auto simp add: inf.order_iff)
```
```  2087
```
```  2088 instantiation multiset :: (equal) equal
```
```  2089 begin
```
```  2090
```
```  2091 definition
```
```  2092   [code]: "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"
```
```  2093
```
```  2094 instance
```
```  2095   by default (simp add: equal_multiset_def eq_iff)
```
```  2096
```
```  2097 end
```
```  2098
```
```  2099 lemma [code]:
```
```  2100   "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
```
```  2101   by auto
```
```  2102
```
```  2103 lemma [code]:
```
```  2104   "msetsum (multiset_of xs) = listsum xs"
```
```  2105   by (induct xs) (simp_all add: add.commute)
```
```  2106
```
```  2107 lemma [code]:
```
```  2108   "msetprod (multiset_of xs) = fold times xs 1"
```
```  2109 proof -
```
```  2110   have "\<And>x. fold times xs x = msetprod (multiset_of xs) * x"
```
```  2111     by (induct xs) (simp_all add: mult.assoc)
```
```  2112   then show ?thesis by simp
```
```  2113 qed
```
```  2114
```
```  2115 lemma [code]:
```
```  2116   "size = mcard"
```
```  2117   by (fact size_eq_mcard)
```
```  2118
```
```  2119 text {*
```
```  2120   Exercise for the casual reader: add implementations for @{const le_multiset}
```
```  2121   and @{const less_multiset} (multiset order).
```
```  2122 *}
```
```  2123
```
```  2124 text {* Quickcheck generators *}
```
```  2125
```
```  2126 definition (in term_syntax)
```
```  2127   msetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
```
```  2128     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
```
```  2129   [code_unfold]: "msetify xs = Code_Evaluation.valtermify multiset_of {\<cdot>} xs"
```
```  2130
```
```  2131 notation fcomp (infixl "\<circ>>" 60)
```
```  2132 notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2133
```
```  2134 instantiation multiset :: (random) random
```
```  2135 begin
```
```  2136
```
```  2137 definition
```
```  2138   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
```
```  2139
```
```  2140 instance ..
```
```  2141
```
```  2142 end
```
```  2143
```
```  2144 no_notation fcomp (infixl "\<circ>>" 60)
```
```  2145 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  2146
```
```  2147 instantiation multiset :: (full_exhaustive) full_exhaustive
```
```  2148 begin
```
```  2149
```
```  2150 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
```
```  2151 where
```
```  2152   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
```
```  2153
```
```  2154 instance ..
```
```  2155
```
```  2156 end
```
```  2157
```
```  2158 hide_const (open) msetify
```
```  2159
```
```  2160 end
```
```  2161
```