src/HOL/Library/Multiset.thy
 author chaieb Mon Jun 11 11:06:04 2007 +0200 (2007-06-11) changeset 23315 df3a7e9ebadb parent 23281 e26ec695c9b3 child 23373 ead82c82da9e permissions -rw-r--r--
tuned Proof
```     1 (*  Title:      HOL/Library/Multiset.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
```
```     4 *)
```
```     5
```
```     6 header {* Multisets *}
```
```     7
```
```     8 theory Multiset
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 subsection {* The type of multisets *}
```
```    13
```
```    14 typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
```
```    15 proof
```
```    16   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
```
```    17 qed
```
```    18
```
```    19 lemmas multiset_typedef [simp] =
```
```    20     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
```
```    21   and [simp] = Rep_multiset_inject [symmetric]
```
```    22
```
```    23 definition
```
```    24   Mempty :: "'a multiset"  ("{#}") where
```
```    25   "{#} = Abs_multiset (\<lambda>a. 0)"
```
```    26
```
```    27 definition
```
```    28   single :: "'a => 'a multiset"  ("{#_#}") where
```
```    29   "{#a#} = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
```
```    30
```
```    31 definition
```
```    32   count :: "'a multiset => 'a => nat" where
```
```    33   "count = Rep_multiset"
```
```    34
```
```    35 definition
```
```    36   MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
```
```    37   "MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
```
```    38
```
```    39 abbreviation
```
```    40   Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
```
```    41   "a :# M == 0 < count M a"
```
```    42
```
```    43 syntax
```
```    44   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
```
```    45 translations
```
```    46   "{#x:M. P#}" == "CONST MCollect M (\<lambda>x. P)"
```
```    47
```
```    48 definition
```
```    49   set_of :: "'a multiset => 'a set" where
```
```    50   "set_of M = {x. x :# M}"
```
```    51
```
```    52 instance multiset :: (type) "{plus, minus, zero, size}"
```
```    53   union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
```
```    54   diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
```
```    55   Zero_multiset_def [simp]: "0 == {#}"
```
```    56   size_def: "size M == setsum (count M) (set_of M)" ..
```
```    57
```
```    58 definition
```
```    59   multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where
```
```    60   "multiset_inter A B = A - (A - B)"
```
```    61
```
```    62
```
```    63 text {*
```
```    64  \medskip Preservation of the representing set @{term multiset}.
```
```    65 *}
```
```    66
```
```    67 lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
```
```    68   by (simp add: multiset_def)
```
```    69
```
```    70 lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
```
```    71   by (simp add: multiset_def)
```
```    72
```
```    73 lemma union_preserves_multiset [simp]:
```
```    74     "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
```
```    75   apply (simp add: multiset_def)
```
```    76   apply (drule (1) finite_UnI)
```
```    77   apply (simp del: finite_Un add: Un_def)
```
```    78   done
```
```    79
```
```    80 lemma diff_preserves_multiset [simp]:
```
```    81     "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
```
```    82   apply (simp add: multiset_def)
```
```    83   apply (rule finite_subset)
```
```    84    apply auto
```
```    85   done
```
```    86
```
```    87
```
```    88 subsection {* Algebraic properties of multisets *}
```
```    89
```
```    90 subsubsection {* Union *}
```
```    91
```
```    92 lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
```
```    93   by (simp add: union_def Mempty_def)
```
```    94
```
```    95 lemma union_commute: "M + N = N + (M::'a multiset)"
```
```    96   by (simp add: union_def add_ac)
```
```    97
```
```    98 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```    99   by (simp add: union_def add_ac)
```
```   100
```
```   101 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```   102 proof -
```
```   103   have "M + (N + K) = (N + K) + M"
```
```   104     by (rule union_commute)
```
```   105   also have "\<dots> = N + (K + M)"
```
```   106     by (rule union_assoc)
```
```   107   also have "K + M = M + K"
```
```   108     by (rule union_commute)
```
```   109   finally show ?thesis .
```
```   110 qed
```
```   111
```
```   112 lemmas union_ac = union_assoc union_commute union_lcomm
```
```   113
```
```   114 instance multiset :: (type) comm_monoid_add
```
```   115 proof
```
```   116   fix a b c :: "'a multiset"
```
```   117   show "(a + b) + c = a + (b + c)" by (rule union_assoc)
```
```   118   show "a + b = b + a" by (rule union_commute)
```
```   119   show "0 + a = a" by simp
```
```   120 qed
```
```   121
```
```   122
```
```   123 subsubsection {* Difference *}
```
```   124
```
```   125 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   126   by (simp add: Mempty_def diff_def)
```
```   127
```
```   128 lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
```
```   129   by (simp add: union_def diff_def)
```
```   130
```
```   131
```
```   132 subsubsection {* Count of elements *}
```
```   133
```
```   134 lemma count_empty [simp]: "count {#} a = 0"
```
```   135   by (simp add: count_def Mempty_def)
```
```   136
```
```   137 lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   138   by (simp add: count_def single_def)
```
```   139
```
```   140 lemma count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   141   by (simp add: count_def union_def)
```
```   142
```
```   143 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   144   by (simp add: count_def diff_def)
```
```   145
```
```   146
```
```   147 subsubsection {* Set of elements *}
```
```   148
```
```   149 lemma set_of_empty [simp]: "set_of {#} = {}"
```
```   150   by (simp add: set_of_def)
```
```   151
```
```   152 lemma set_of_single [simp]: "set_of {#b#} = {b}"
```
```   153   by (simp add: set_of_def)
```
```   154
```
```   155 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   156   by (auto simp add: set_of_def)
```
```   157
```
```   158 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   159   by (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
```
```   160
```
```   161 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   162   by (auto simp add: set_of_def)
```
```   163
```
```   164
```
```   165 subsubsection {* Size *}
```
```   166
```
```   167 lemma size_empty [simp]: "size {#} = 0"
```
```   168   by (simp add: size_def)
```
```   169
```
```   170 lemma size_single [simp]: "size {#b#} = 1"
```
```   171   by (simp add: size_def)
```
```   172
```
```   173 lemma finite_set_of [iff]: "finite (set_of M)"
```
```   174   using Rep_multiset [of M]
```
```   175   by (simp add: multiset_def set_of_def count_def)
```
```   176
```
```   177 lemma setsum_count_Int:
```
```   178     "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
```
```   179   apply (induct rule: finite_induct)
```
```   180    apply simp
```
```   181   apply (simp add: Int_insert_left set_of_def)
```
```   182   done
```
```   183
```
```   184 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   185   apply (unfold size_def)
```
```   186   apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
```
```   187    prefer 2
```
```   188    apply (rule ext, simp)
```
```   189   apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
```
```   190   apply (subst Int_commute)
```
```   191   apply (simp (no_asm_simp) add: setsum_count_Int)
```
```   192   done
```
```   193
```
```   194 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   195   apply (unfold size_def Mempty_def count_def, auto)
```
```   196   apply (simp add: set_of_def count_def expand_fun_eq)
```
```   197   done
```
```   198
```
```   199 lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   200   apply (unfold size_def)
```
```   201   apply (drule setsum_SucD, auto)
```
```   202   done
```
```   203
```
```   204
```
```   205 subsubsection {* Equality of multisets *}
```
```   206
```
```   207 lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
```
```   208   by (simp add: count_def expand_fun_eq)
```
```   209
```
```   210 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
```
```   211   by (simp add: single_def Mempty_def expand_fun_eq)
```
```   212
```
```   213 lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
```
```   214   by (auto simp add: single_def expand_fun_eq)
```
```   215
```
```   216 lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
```
```   217   by (auto simp add: union_def Mempty_def expand_fun_eq)
```
```   218
```
```   219 lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
```
```   220   by (auto simp add: union_def Mempty_def expand_fun_eq)
```
```   221
```
```   222 lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
```
```   223   by (simp add: union_def expand_fun_eq)
```
```   224
```
```   225 lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
```
```   226   by (simp add: union_def expand_fun_eq)
```
```   227
```
```   228 lemma union_is_single:
```
```   229     "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
```
```   230   apply (simp add: Mempty_def single_def union_def add_is_1 expand_fun_eq)
```
```   231   apply blast
```
```   232   done
```
```   233
```
```   234 lemma single_is_union:
```
```   235      "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
```
```   236   apply (unfold Mempty_def single_def union_def)
```
```   237   apply (simp add: add_is_1 one_is_add expand_fun_eq)
```
```   238   apply (blast dest: sym)
```
```   239   done
```
```   240
```
```   241 ML"reset use_neq_simproc"
```
```   242 lemma add_eq_conv_diff:
```
```   243   "(M + {#a#} = N + {#b#}) =
```
```   244    (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
```
```   245   apply (unfold single_def union_def diff_def)
```
```   246   apply (simp (no_asm) add: expand_fun_eq)
```
```   247   apply (rule conjI, force, safe, simp_all)
```
```   248   apply (simp add: eq_sym_conv)
```
```   249   done
```
```   250 ML"set use_neq_simproc"
```
```   251
```
```   252 declare Rep_multiset_inject [symmetric, simp del]
```
```   253
```
```   254
```
```   255 subsubsection {* Intersection *}
```
```   256
```
```   257 lemma multiset_inter_count:
```
```   258     "count (A #\<inter> B) x = min (count A x) (count B x)"
```
```   259   by (simp add: multiset_inter_def min_def)
```
```   260
```
```   261 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
```
```   262   by (simp add: multiset_eq_conv_count_eq multiset_inter_count
```
```   263     min_max.inf_commute)
```
```   264
```
```   265 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
```
```   266   by (simp add: multiset_eq_conv_count_eq multiset_inter_count
```
```   267     min_max.inf_assoc)
```
```   268
```
```   269 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
```
```   270   by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
```
```   271
```
```   272 lemmas multiset_inter_ac =
```
```   273   multiset_inter_commute
```
```   274   multiset_inter_assoc
```
```   275   multiset_inter_left_commute
```
```   276
```
```   277 lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
```
```   278   apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
```
```   279     split: split_if_asm)
```
```   280   apply clarsimp
```
```   281   apply (erule_tac x = a in allE)
```
```   282   apply auto
```
```   283   done
```
```   284
```
```   285
```
```   286 subsection {* Induction over multisets *}
```
```   287
```
```   288 lemma setsum_decr:
```
```   289   "finite F ==> (0::nat) < f a ==>
```
```   290     setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
```
```   291   apply (induct rule: finite_induct)
```
```   292    apply auto
```
```   293   apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   294   done
```
```   295
```
```   296 lemma rep_multiset_induct_aux:
```
```   297   assumes 1: "P (\<lambda>a. (0::nat))"
```
```   298     and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
```
```   299   shows "\<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
```
```   300   apply (unfold multiset_def)
```
```   301   apply (induct_tac n, simp, clarify)
```
```   302    apply (subgoal_tac "f = (\<lambda>a.0)")
```
```   303     apply simp
```
```   304     apply (rule 1)
```
```   305    apply (rule ext, force, clarify)
```
```   306   apply (frule setsum_SucD, clarify)
```
```   307   apply (rename_tac a)
```
```   308   apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
```
```   309    prefer 2
```
```   310    apply (rule finite_subset)
```
```   311     prefer 2
```
```   312     apply assumption
```
```   313    apply simp
```
```   314    apply blast
```
```   315   apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
```
```   316    prefer 2
```
```   317    apply (rule ext)
```
```   318    apply (simp (no_asm_simp))
```
```   319    apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
```
```   320   apply (erule allE, erule impE, erule_tac [2] mp, blast)
```
```   321   apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
```
```   322   apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
```
```   323    prefer 2
```
```   324    apply blast
```
```   325   apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
```
```   326    prefer 2
```
```   327    apply blast
```
```   328   apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
```
```   329   done
```
```   330
```
```   331 theorem rep_multiset_induct:
```
```   332   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
```
```   333     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
```
```   334   using rep_multiset_induct_aux by blast
```
```   335
```
```   336 theorem multiset_induct [case_names empty add, induct type: multiset]:
```
```   337   assumes empty: "P {#}"
```
```   338     and add: "!!M x. P M ==> P (M + {#x#})"
```
```   339   shows "P M"
```
```   340 proof -
```
```   341   note defns = union_def single_def Mempty_def
```
```   342   show ?thesis
```
```   343     apply (rule Rep_multiset_inverse [THEN subst])
```
```   344     apply (rule Rep_multiset [THEN rep_multiset_induct])
```
```   345      apply (rule empty [unfolded defns])
```
```   346     apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
```
```   347      prefer 2
```
```   348      apply (simp add: expand_fun_eq)
```
```   349     apply (erule ssubst)
```
```   350     apply (erule Abs_multiset_inverse [THEN subst])
```
```   351     apply (erule add [unfolded defns, simplified])
```
```   352     done
```
```   353 qed
```
```   354
```
```   355 lemma MCollect_preserves_multiset:
```
```   356     "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
```
```   357   apply (simp add: multiset_def)
```
```   358   apply (rule finite_subset, auto)
```
```   359   done
```
```   360
```
```   361 lemma count_MCollect [simp]:
```
```   362     "count {# x:M. P x #} a = (if P a then count M a else 0)"
```
```   363   by (simp add: count_def MCollect_def MCollect_preserves_multiset)
```
```   364
```
```   365 lemma set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
```
```   366   by (auto simp add: set_of_def)
```
```   367
```
```   368 lemma multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
```
```   369   by (subst multiset_eq_conv_count_eq, auto)
```
```   370
```
```   371 lemma add_eq_conv_ex:
```
```   372   "(M + {#a#} = N + {#b#}) =
```
```   373     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
```
```   374   by (auto simp add: add_eq_conv_diff)
```
```   375
```
```   376 declare multiset_typedef [simp del]
```
```   377
```
```   378
```
```   379 subsection {* Multiset orderings *}
```
```   380
```
```   381 subsubsection {* Well-foundedness *}
```
```   382
```
```   383 definition
```
```   384   mult1 :: "('a => 'a => bool) => 'a multiset => 'a multiset => bool" where
```
```   385   "mult1 r =
```
```   386     (%N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```   387       (\<forall>b. b :# K --> r b a))"
```
```   388
```
```   389 definition
```
```   390   mult :: "('a => 'a => bool) => 'a multiset => 'a multiset => bool" where
```
```   391   "mult r = (mult1 r)\<^sup>+\<^sup>+"
```
```   392
```
```   393 lemma not_less_empty [iff]: "\<not> mult1 r M {#}"
```
```   394   by (simp add: mult1_def)
```
```   395
```
```   396 lemma less_add: "mult1 r N (M0 + {#a#})==>
```
```   397     (\<exists>M. mult1 r M M0 \<and> N = M + {#a#}) \<or>
```
```   398     (\<exists>K. (\<forall>b. b :# K --> r b a) \<and> N = M0 + K)"
```
```   399   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
```
```   400 proof (unfold mult1_def)
```
```   401   let ?r = "\<lambda>K a. \<forall>b. b :# K --> r b a"
```
```   402   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```   403   let ?case1 = "?case1 ?R"
```
```   404
```
```   405   assume "?R N (M0 + {#a#})"
```
```   406   then have "\<exists>a' M0' K.
```
```   407       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```   408   then show "?case1 \<or> ?case2"
```
```   409   proof (elim exE conjE)
```
```   410     fix a' M0' K
```
```   411     assume N: "N = M0' + K" and r: "?r K a'"
```
```   412     assume "M0 + {#a#} = M0' + {#a'#}"
```
```   413     then have "M0 = M0' \<and> a = a' \<or>
```
```   414         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```   415       by (simp only: add_eq_conv_ex)
```
```   416     then show ?thesis
```
```   417     proof (elim disjE conjE exE)
```
```   418       assume "M0 = M0'" "a = a'"
```
```   419       with N r have "?r K a \<and> N = M0 + K" by simp
```
```   420       then have ?case2 .. then show ?thesis ..
```
```   421     next
```
```   422       fix K'
```
```   423       assume "M0' = K' + {#a#}"
```
```   424       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
```
```   425
```
```   426       assume "M0 = K' + {#a'#}"
```
```   427       with r have "?R (K' + K) M0" by blast
```
```   428       with n have ?case1 by simp then show ?thesis ..
```
```   429     qed
```
```   430   qed
```
```   431 qed
```
```   432
```
```   433 lemma all_accessible: "wfP r ==> \<forall>M. acc (mult1 r) M"
```
```   434 proof
```
```   435   let ?R = "mult1 r"
```
```   436   let ?W = "acc ?R"
```
```   437   {
```
```   438     fix M M0 a
```
```   439     assume M0: "?W M0"
```
```   440       and wf_hyp: "!!b. r b a ==> \<forall>M \<triangleright> ?W. ?W (M + {#b#})"
```
```   441       and acc_hyp: "\<forall>M. ?R M M0 --> ?W (M + {#a#})"
```
```   442     have "?W (M0 + {#a#})"
```
```   443     proof (rule accI [of _ "M0 + {#a#}"])
```
```   444       fix N
```
```   445       assume "?R N (M0 + {#a#})"
```
```   446       then have "((\<exists>M. ?R M M0 \<and> N = M + {#a#}) \<or>
```
```   447           (\<exists>K. (\<forall>b. b :# K --> r b a) \<and> N = M0 + K))"
```
```   448         by (rule less_add)
```
```   449       then show "?W N"
```
```   450       proof (elim exE disjE conjE)
```
```   451         fix M assume "?R M M0" and N: "N = M + {#a#}"
```
```   452         from acc_hyp have "?R M M0 --> ?W (M + {#a#})" ..
```
```   453         then have "?W (M + {#a#})" ..
```
```   454         then show "?W N" by (simp only: N)
```
```   455       next
```
```   456         fix K
```
```   457         assume N: "N = M0 + K"
```
```   458         assume "\<forall>b. b :# K --> r b a"
```
```   459         then have "?W (M0 + K)"
```
```   460         proof (induct K)
```
```   461           case empty
```
```   462           from M0 show "?W (M0 + {#})" by simp
```
```   463         next
```
```   464           case (add K x)
```
```   465           from add.prems have "r x a" by simp
```
```   466           with wf_hyp have "\<forall>M \<triangleright> ?W. ?W (M + {#x#})" by blast
```
```   467           moreover from add have "?W (M0 + K)" by simp
```
```   468           ultimately have "?W ((M0 + K) + {#x#})" ..
```
```   469           then show "?W (M0 + (K + {#x#}))" by (simp only: union_assoc)
```
```   470         qed
```
```   471         then show "?W N" by (simp only: N)
```
```   472       qed
```
```   473     qed
```
```   474   } note tedious_reasoning = this
```
```   475
```
```   476   assume wf: "wfP r"
```
```   477   fix M
```
```   478   show "?W M"
```
```   479   proof (induct M)
```
```   480     show "?W {#}"
```
```   481     proof (rule accI)
```
```   482       fix b assume "?R b {#}"
```
```   483       with not_less_empty show "?W b" by contradiction
```
```   484     qed
```
```   485
```
```   486     fix M a assume "?W M"
```
```   487     from wf have "\<forall>M \<triangleright> ?W. ?W (M + {#a#})"
```
```   488     proof induct
```
```   489       fix a
```
```   490       assume "!!b. r b a ==> \<forall>M \<triangleright> ?W. ?W (M + {#b#})"
```
```   491       show "\<forall>M \<triangleright> ?W. ?W (M + {#a#})"
```
```   492       proof
```
```   493         fix M assume "?W M"
```
```   494         then show "?W (M + {#a#})"
```
```   495           by (rule acc_induct) (rule tedious_reasoning)
```
```   496       qed
```
```   497     qed
```
```   498     then show "?W (M + {#a#})" ..
```
```   499   qed
```
```   500 qed
```
```   501
```
```   502 theorem wf_mult1: "wfP r ==> wfP (mult1 r)"
```
```   503   by (rule acc_wfI, rule all_accessible)
```
```   504
```
```   505 theorem wf_mult: "wfP r ==> wfP (mult r)"
```
```   506   by (unfold mult_def, rule wfP_trancl, rule wf_mult1)
```
```   507
```
```   508
```
```   509 subsubsection {* Closure-free presentation *}
```
```   510
```
```   511 (*Badly needed: a linear arithmetic procedure for multisets*)
```
```   512
```
```   513 lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
```
```   514 by (simp add: multiset_eq_conv_count_eq)
```
```   515
```
```   516 text {* One direction. *}
```
```   517
```
```   518 lemma mult_implies_one_step:
```
```   519   "transP r ==> mult r M N ==>
```
```   520     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```   521     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. r k j)"
```
```   522   apply (unfold mult_def mult1_def set_of_def)
```
```   523   apply (erule converse_trancl_induct', clarify)
```
```   524    apply (rule_tac x = M0 in exI, simp, clarify)
```
```   525   apply (case_tac "a :# Ka")
```
```   526    apply (rule_tac x = I in exI)
```
```   527    apply (simp (no_asm))
```
```   528    apply (rule_tac x = "(Ka - {#a#}) + K" in exI)
```
```   529    apply (simp (no_asm_simp) add: union_assoc [symmetric])
```
```   530    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```   531    apply (simp add: diff_union_single_conv)
```
```   532    apply (simp (no_asm_use) add: trans_def)
```
```   533    apply blast
```
```   534   apply (subgoal_tac "a :# I")
```
```   535    apply (rule_tac x = "I - {#a#}" in exI)
```
```   536    apply (rule_tac x = "J + {#a#}" in exI)
```
```   537    apply (rule_tac x = "K + Ka" in exI)
```
```   538    apply (rule conjI)
```
```   539     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   540    apply (rule conjI)
```
```   541     apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
```
```   542     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   543    apply (simp (no_asm_use) add: trans_def)
```
```   544    apply blast
```
```   545   apply (subgoal_tac "a :# (M0 + {#a#})")
```
```   546    apply simp
```
```   547   apply (simp (no_asm))
```
```   548   done
```
```   549
```
```   550 lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
```
```   551 by (simp add: multiset_eq_conv_count_eq)
```
```   552
```
```   553 lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
```
```   554   apply (erule size_eq_Suc_imp_elem [THEN exE])
```
```   555   apply (drule elem_imp_eq_diff_union, auto)
```
```   556   done
```
```   557
```
```   558 lemma one_step_implies_mult_aux:
```
```   559   "\<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. r k j))
```
```   560     --> mult r (I + K) (I + J)"
```
```   561   apply (induct_tac n, auto)
```
```   562   apply (frule size_eq_Suc_imp_eq_union, clarify)
```
```   563   apply (rename_tac "J'", simp)
```
```   564   apply (erule notE, auto)
```
```   565   apply (case_tac "J' = {#}")
```
```   566    apply (simp add: mult_def)
```
```   567    apply (rule trancl.r_into_trancl)
```
```   568    apply (simp add: mult1_def set_of_def, blast)
```
```   569   txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```   570   apply (cut_tac M = K and P = "\<lambda>x. r x a" in multiset_partition)
```
```   571   apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```   572   apply (erule ssubst)
```
```   573   apply (simp add: Ball_def, auto)
```
```   574   apply (subgoal_tac
```
```   575     "mult r ((I + {# x : K. r x a #}) + {# x : K. \<not> r x a #})
```
```   576       ((I + {# x : K. r x a #}) + J')")
```
```   577    prefer 2
```
```   578    apply force
```
```   579   apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
```
```   580   apply (erule trancl_trans')
```
```   581   apply (rule trancl.r_into_trancl)
```
```   582   apply (simp add: mult1_def set_of_def)
```
```   583   apply (rule_tac x = a in exI)
```
```   584   apply (rule_tac x = "I + J'" in exI)
```
```   585   apply (simp add: union_ac)
```
```   586   done
```
```   587
```
```   588 lemma one_step_implies_mult:
```
```   589   "J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. r k j
```
```   590     ==> mult r (I + K) (I + J)"
```
```   591   apply (insert one_step_implies_mult_aux, blast)
```
```   592   done
```
```   593
```
```   594
```
```   595 subsubsection {* Partial-order properties *}
```
```   596
```
```   597 instance multiset :: (type) ord ..
```
```   598
```
```   599 defs (overloaded)
```
```   600   less_multiset_def: "op < == mult op <"
```
```   601   le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
```
```   602
```
```   603 lemma trans_base_order: "transP (op < :: 'a::order => 'a => bool)"
```
```   604   unfolding trans_def by (blast intro: order_less_trans)
```
```   605
```
```   606 text {*
```
```   607  \medskip Irreflexivity.
```
```   608 *}
```
```   609
```
```   610 lemma mult_irrefl_aux:
```
```   611     "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
```
```   612   apply (induct rule: finite_induct)
```
```   613    apply (auto intro: order_less_trans)
```
```   614   done
```
```   615
```
```   616 lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
```
```   617   apply (unfold less_multiset_def, auto)
```
```   618   apply (drule trans_base_order [THEN mult_implies_one_step], auto)
```
```   619   apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
```
```   620   apply (simp add: set_of_eq_empty_iff)
```
```   621   done
```
```   622
```
```   623 lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
```
```   624 by (insert mult_less_not_refl, fast)
```
```   625
```
```   626
```
```   627 text {* Transitivity. *}
```
```   628
```
```   629 theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
```
```   630   apply (unfold less_multiset_def mult_def)
```
```   631   apply (blast intro: trancl_trans')
```
```   632   done
```
```   633
```
```   634 text {* Asymmetry. *}
```
```   635
```
```   636 theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
```
```   637   apply auto
```
```   638   apply (rule mult_less_not_refl [THEN notE])
```
```   639   apply (erule mult_less_trans, assumption)
```
```   640   done
```
```   641
```
```   642 theorem mult_less_asym:
```
```   643     "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
```
```   644   by (insert mult_less_not_sym, blast)
```
```   645
```
```   646 theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
```
```   647   unfolding le_multiset_def by auto
```
```   648
```
```   649 text {* Anti-symmetry. *}
```
```   650
```
```   651 theorem mult_le_antisym:
```
```   652     "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
```
```   653   unfolding le_multiset_def by (blast dest: mult_less_not_sym)
```
```   654
```
```   655 text {* Transitivity. *}
```
```   656
```
```   657 theorem mult_le_trans:
```
```   658     "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
```
```   659   unfolding le_multiset_def by (blast intro: mult_less_trans)
```
```   660
```
```   661 theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
```
```   662   unfolding le_multiset_def by auto
```
```   663
```
```   664 text {* Partial order. *}
```
```   665
```
```   666 instance multiset :: (order) order
```
```   667   apply intro_classes
```
```   668     apply (rule mult_less_le)
```
```   669     apply (rule mult_le_refl)
```
```   670     apply (erule mult_le_trans, assumption)
```
```   671     apply (erule mult_le_antisym, assumption)
```
```   672   done
```
```   673
```
```   674
```
```   675 subsubsection {* Monotonicity of multiset union *}
```
```   676
```
```   677 lemma mult1_union:
```
```   678     "mult1 r B D ==> mult1 r (C + B) (C + D)"
```
```   679   apply (unfold mult1_def, auto)
```
```   680   apply (rule_tac x = a in exI)
```
```   681   apply (rule_tac x = "C + M0" in exI)
```
```   682   apply (simp add: union_assoc)
```
```   683   done
```
```   684
```
```   685 lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
```
```   686   apply (unfold less_multiset_def mult_def)
```
```   687   apply (erule trancl_induct')
```
```   688    apply (blast intro: mult1_union)
```
```   689   apply (blast intro: mult1_union trancl.r_into_trancl trancl_trans')
```
```   690   done
```
```   691
```
```   692 lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
```
```   693   apply (subst union_commute [of B C])
```
```   694   apply (subst union_commute [of D C])
```
```   695   apply (erule union_less_mono2)
```
```   696   done
```
```   697
```
```   698 lemma union_less_mono:
```
```   699     "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
```
```   700   apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
```
```   701   done
```
```   702
```
```   703 lemma union_le_mono:
```
```   704     "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
```
```   705   unfolding le_multiset_def
```
```   706   by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
```
```   707
```
```   708 lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
```
```   709   apply (unfold le_multiset_def less_multiset_def)
```
```   710   apply (case_tac "M = {#}")
```
```   711    prefer 2
```
```   712    apply (subgoal_tac "mult op < ({#} + {#}) ({#} + M)")
```
```   713     prefer 2
```
```   714     apply (rule one_step_implies_mult)
```
```   715       apply auto
```
```   716   done
```
```   717
```
```   718 lemma union_upper1: "A <= A + (B::'a::order multiset)"
```
```   719 proof -
```
```   720   have "A + {#} <= A + B" by (blast intro: union_le_mono)
```
```   721   then show ?thesis by simp
```
```   722 qed
```
```   723
```
```   724 lemma union_upper2: "B <= A + (B::'a::order multiset)"
```
```   725   by (subst union_commute) (rule union_upper1)
```
```   726
```
```   727
```
```   728 subsection {* Link with lists *}
```
```   729
```
```   730 consts
```
```   731   multiset_of :: "'a list \<Rightarrow> 'a multiset"
```
```   732 primrec
```
```   733   "multiset_of [] = {#}"
```
```   734   "multiset_of (a # x) = multiset_of x + {# a #}"
```
```   735
```
```   736 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
```
```   737   by (induct x) auto
```
```   738
```
```   739 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
```
```   740   by (induct x) auto
```
```   741
```
```   742 lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
```
```   743   by (induct x) auto
```
```   744
```
```   745 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
```
```   746   by (induct xs) auto
```
```   747
```
```   748 lemma multiset_of_append [simp]:
```
```   749     "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
```
```   750   by (induct xs arbitrary: ys) (auto simp: union_ac)
```
```   751
```
```   752 lemma surj_multiset_of: "surj multiset_of"
```
```   753   apply (unfold surj_def, rule allI)
```
```   754   apply (rule_tac M=y in multiset_induct, auto)
```
```   755   apply (rule_tac x = "x # xa" in exI, auto)
```
```   756   done
```
```   757
```
```   758 lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}"
```
```   759   by (induct x) auto
```
```   760
```
```   761 lemma distinct_count_atmost_1:
```
```   762    "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
```
```   763    apply (induct x, simp, rule iffI, simp_all)
```
```   764    apply (rule conjI)
```
```   765    apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
```
```   766    apply (erule_tac x=a in allE, simp, clarify)
```
```   767    apply (erule_tac x=aa in allE, simp)
```
```   768    done
```
```   769
```
```   770 lemma multiset_of_eq_setD:
```
```   771   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
```
```   772   by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
```
```   773
```
```   774 lemma set_eq_iff_multiset_of_eq_distinct:
```
```   775   "\<lbrakk>distinct x; distinct y\<rbrakk>
```
```   776    \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
```
```   777   by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
```
```   778
```
```   779 lemma set_eq_iff_multiset_of_remdups_eq:
```
```   780    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
```
```   781   apply (rule iffI)
```
```   782   apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
```
```   783   apply (drule distinct_remdups[THEN distinct_remdups
```
```   784                       [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]])
```
```   785   apply simp
```
```   786   done
```
```   787
```
```   788 lemma multiset_of_compl_union [simp]:
```
```   789     "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
```
```   790   by (induct xs) (auto simp: union_ac)
```
```   791
```
```   792 lemma count_filter:
```
```   793     "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
```
```   794   by (induct xs) auto
```
```   795
```
```   796
```
```   797 subsection {* Pointwise ordering induced by count *}
```
```   798
```
```   799 definition
```
```   800   mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool"  ("_ \<le># _"  [50,51] 50) where
```
```   801   "(xs \<le># ys) = (\<forall>a. count xs a \<le> count ys a)"
```
```   802
```
```   803 lemma mset_le_refl[simp]: "xs \<le># xs"
```
```   804   unfolding mset_le_def by auto
```
```   805
```
```   806 lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs"
```
```   807   unfolding mset_le_def by (fast intro: order_trans)
```
```   808
```
```   809 lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys"
```
```   810   apply (unfold mset_le_def)
```
```   811   apply (rule multiset_eq_conv_count_eq[THEN iffD2])
```
```   812   apply (blast intro: order_antisym)
```
```   813   done
```
```   814
```
```   815 lemma mset_le_exists_conv:
```
```   816   "(xs \<le># ys) = (\<exists>zs. ys = xs + zs)"
```
```   817   apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI)
```
```   818   apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
```
```   819   done
```
```   820
```
```   821 lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)"
```
```   822   unfolding mset_le_def by auto
```
```   823
```
```   824 lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)"
```
```   825   unfolding mset_le_def by auto
```
```   826
```
```   827 lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws"
```
```   828   apply (unfold mset_le_def)
```
```   829   apply auto
```
```   830   apply (erule_tac x=a in allE)+
```
```   831   apply auto
```
```   832   done
```
```   833
```
```   834 lemma mset_le_add_left[simp]: "xs \<le># xs + ys"
```
```   835   unfolding mset_le_def by auto
```
```   836
```
```   837 lemma mset_le_add_right[simp]: "ys \<le># xs + ys"
```
```   838   unfolding mset_le_def by auto
```
```   839
```
```   840 lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x"
```
```   841   apply (induct x)
```
```   842    apply auto
```
```   843   apply (rule mset_le_trans)
```
```   844    apply auto
```
```   845   done
```
```   846
```
```   847 end
```