src/HOL/Library/Multiset.thy
 author wenzelm Sat May 01 22:01:57 2004 +0200 (2004-05-01) changeset 14691 e1eedc8cad37 parent 14445 4392cb82018b child 14706 71590b7733b7 permissions -rw-r--r--
tuned instance statements;
```     1 (*  Title:      HOL/Library/Multiset.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson
```
```     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
```
```     5 *)
```
```     6
```
```     7 header {*
```
```     8  \title{Multisets}
```
```     9  \author{Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson}
```
```    10 *}
```
```    11
```
```    12 theory Multiset = Accessible_Part:
```
```    13
```
```    14 subsection {* The type of multisets *}
```
```    15
```
```    16 typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
```
```    17 proof
```
```    18   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
```
```    19 qed
```
```    20
```
```    21 lemmas multiset_typedef [simp] =
```
```    22     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
```
```    23   and [simp] = Rep_multiset_inject [symmetric]
```
```    24
```
```    25 constdefs
```
```    26   Mempty :: "'a multiset"    ("{#}")
```
```    27   "{#} == Abs_multiset (\<lambda>a. 0)"
```
```    28
```
```    29   single :: "'a => 'a multiset"    ("{#_#}")
```
```    30   "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
```
```    31
```
```    32   count :: "'a multiset => 'a => nat"
```
```    33   "count == Rep_multiset"
```
```    34
```
```    35   MCollect :: "'a multiset => ('a => bool) => 'a multiset"
```
```    36   "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
```
```    37
```
```    38 syntax
```
```    39   "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
```
```    40   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
```
```    41 translations
```
```    42   "a :# M" == "0 < count M a"
```
```    43   "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
```
```    44
```
```    45 constdefs
```
```    46   set_of :: "'a multiset => 'a set"
```
```    47   "set_of M == {x. x :# M}"
```
```    48
```
```    49 instance multiset :: (type) "{plus, minus, zero}" ..
```
```    50
```
```    51 defs (overloaded)
```
```    52   union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
```
```    53   diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
```
```    54   Zero_multiset_def [simp]: "0 == {#}"
```
```    55   size_def: "size M == setsum (count M) (set_of M)"
```
```    56
```
```    57
```
```    58 text {*
```
```    59  \medskip Preservation of the representing set @{term multiset}.
```
```    60 *}
```
```    61
```
```    62 lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
```
```    63   apply (simp add: multiset_def)
```
```    64   done
```
```    65
```
```    66 lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
```
```    67   apply (simp add: multiset_def)
```
```    68   done
```
```    69
```
```    70 lemma union_preserves_multiset [simp]:
```
```    71     "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
```
```    72   apply (unfold multiset_def)
```
```    73   apply simp
```
```    74   apply (drule finite_UnI)
```
```    75    apply assumption
```
```    76   apply (simp del: finite_Un add: Un_def)
```
```    77   done
```
```    78
```
```    79 lemma diff_preserves_multiset [simp]:
```
```    80     "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
```
```    81   apply (unfold multiset_def)
```
```    82   apply simp
```
```    83   apply (rule finite_subset)
```
```    84    prefer 2
```
```    85    apply assumption
```
```    86   apply auto
```
```    87   done
```
```    88
```
```    89
```
```    90 subsection {* Algebraic properties of multisets *}
```
```    91
```
```    92 subsubsection {* Union *}
```
```    93
```
```    94 theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
```
```    95   apply (simp add: union_def Mempty_def)
```
```    96   done
```
```    97
```
```    98 theorem union_commute: "M + N = N + (M::'a multiset)"
```
```    99   apply (simp add: union_def add_ac)
```
```   100   done
```
```   101
```
```   102 theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```   103   apply (simp add: union_def add_ac)
```
```   104   done
```
```   105
```
```   106 theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```   107   apply (rule union_commute [THEN trans])
```
```   108   apply (rule union_assoc [THEN trans])
```
```   109   apply (rule union_commute [THEN arg_cong])
```
```   110   done
```
```   111
```
```   112 theorems union_ac = union_assoc union_commute union_lcomm
```
```   113
```
```   114 instance multiset :: (type) plus_ac0
```
```   115   apply intro_classes
```
```   116     apply (rule union_commute)
```
```   117    apply (rule union_assoc)
```
```   118   apply simp
```
```   119   done
```
```   120
```
```   121
```
```   122 subsubsection {* Difference *}
```
```   123
```
```   124 theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   125   apply (simp add: Mempty_def diff_def)
```
```   126   done
```
```   127
```
```   128 theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
```
```   129   apply (simp add: union_def diff_def)
```
```   130   done
```
```   131
```
```   132
```
```   133 subsubsection {* Count of elements *}
```
```   134
```
```   135 theorem count_empty [simp]: "count {#} a = 0"
```
```   136   apply (simp add: count_def Mempty_def)
```
```   137   done
```
```   138
```
```   139 theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   140   apply (simp add: count_def single_def)
```
```   141   done
```
```   142
```
```   143 theorem count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   144   apply (simp add: count_def union_def)
```
```   145   done
```
```   146
```
```   147 theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   148   apply (simp add: count_def diff_def)
```
```   149   done
```
```   150
```
```   151
```
```   152 subsubsection {* Set of elements *}
```
```   153
```
```   154 theorem set_of_empty [simp]: "set_of {#} = {}"
```
```   155   apply (simp add: set_of_def)
```
```   156   done
```
```   157
```
```   158 theorem set_of_single [simp]: "set_of {#b#} = {b}"
```
```   159   apply (simp add: set_of_def)
```
```   160   done
```
```   161
```
```   162 theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   163   apply (auto simp add: set_of_def)
```
```   164   done
```
```   165
```
```   166 theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   167   apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
```
```   168   done
```
```   169
```
```   170 theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   171   apply (auto simp add: set_of_def)
```
```   172   done
```
```   173
```
```   174
```
```   175 subsubsection {* Size *}
```
```   176
```
```   177 theorem size_empty [simp]: "size {#} = 0"
```
```   178   apply (simp add: size_def)
```
```   179   done
```
```   180
```
```   181 theorem size_single [simp]: "size {#b#} = 1"
```
```   182   apply (simp add: size_def)
```
```   183   done
```
```   184
```
```   185 theorem finite_set_of [iff]: "finite (set_of M)"
```
```   186   apply (cut_tac x = M in Rep_multiset)
```
```   187   apply (simp add: multiset_def set_of_def count_def)
```
```   188   done
```
```   189
```
```   190 theorem setsum_count_Int:
```
```   191     "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
```
```   192   apply (erule finite_induct)
```
```   193    apply simp
```
```   194   apply (simp add: Int_insert_left set_of_def)
```
```   195   done
```
```   196
```
```   197 theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   198   apply (unfold size_def)
```
```   199   apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
```
```   200    prefer 2
```
```   201    apply (rule ext)
```
```   202    apply simp
```
```   203   apply (simp (no_asm_simp) add: setsum_Un setsum_addf setsum_count_Int)
```
```   204   apply (subst Int_commute)
```
```   205   apply (simp (no_asm_simp) add: setsum_count_Int)
```
```   206   done
```
```   207
```
```   208 theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   209   apply (unfold size_def Mempty_def count_def)
```
```   210   apply auto
```
```   211   apply (simp add: set_of_def count_def expand_fun_eq)
```
```   212   done
```
```   213
```
```   214 theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   215   apply (unfold size_def)
```
```   216   apply (drule setsum_SucD)
```
```   217   apply auto
```
```   218   done
```
```   219
```
```   220
```
```   221 subsubsection {* Equality of multisets *}
```
```   222
```
```   223 theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
```
```   224   apply (simp add: count_def expand_fun_eq)
```
```   225   done
```
```   226
```
```   227 theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
```
```   228   apply (simp add: single_def Mempty_def expand_fun_eq)
```
```   229   done
```
```   230
```
```   231 theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
```
```   232   apply (auto simp add: single_def expand_fun_eq)
```
```   233   done
```
```   234
```
```   235 theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
```
```   236   apply (auto simp add: union_def Mempty_def expand_fun_eq)
```
```   237   done
```
```   238
```
```   239 theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
```
```   240   apply (auto simp add: union_def Mempty_def expand_fun_eq)
```
```   241   done
```
```   242
```
```   243 theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
```
```   244   apply (simp add: union_def expand_fun_eq)
```
```   245   done
```
```   246
```
```   247 theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
```
```   248   apply (simp add: union_def expand_fun_eq)
```
```   249   done
```
```   250
```
```   251 theorem union_is_single:
```
```   252     "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
```
```   253   apply (unfold Mempty_def single_def union_def)
```
```   254   apply (simp add: add_is_1 expand_fun_eq)
```
```   255   apply blast
```
```   256   done
```
```   257
```
```   258 theorem single_is_union:
```
```   259   "({#a#} = M + N) =
```
```   260     ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
```
```   261   apply (unfold Mempty_def single_def union_def)
```
```   262   apply (simp add: add_is_1 one_is_add expand_fun_eq)
```
```   263   apply (blast dest: sym)
```
```   264   done
```
```   265
```
```   266 theorem add_eq_conv_diff:
```
```   267   "(M + {#a#} = N + {#b#}) =
```
```   268     (M = N \<and> a = b \<or>
```
```   269       M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
```
```   270   apply (unfold single_def union_def diff_def)
```
```   271   apply (simp (no_asm) add: expand_fun_eq)
```
```   272   apply (rule conjI)
```
```   273    apply force
```
```   274   apply safe
```
```   275   apply simp_all
```
```   276   apply (simp add: eq_sym_conv)
```
```   277   done
```
```   278
```
```   279 (*
```
```   280 val prems = Goal
```
```   281  "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
```
```   282 by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
```
```   283      measure_induct 1);
```
```   284 by (Clarify_tac 1);
```
```   285 by (resolve_tac prems 1);
```
```   286  by (assume_tac 1);
```
```   287 by (Clarify_tac 1);
```
```   288 by (subgoal_tac "finite G" 1);
```
```   289  by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
```
```   290 by (etac allE 1);
```
```   291 by (etac impE 1);
```
```   292  by (Blast_tac 2);
```
```   293 by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
```
```   294 no_qed();
```
```   295 val lemma = result();
```
```   296
```
```   297 val prems = Goal
```
```   298  "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
```
```   299 by (rtac (lemma RS mp) 1);
```
```   300 by (REPEAT(ares_tac prems 1));
```
```   301 qed "finite_psubset_induct";
```
```   302
```
```   303 Better: use wf_finite_psubset in WF_Rel
```
```   304 *)
```
```   305
```
```   306
```
```   307 subsection {* Induction over multisets *}
```
```   308
```
```   309 lemma setsum_decr:
```
```   310   "finite F ==> (0::nat) < f a ==>
```
```   311     setsum (f (a := f a - 1)) F = (if a \<in> F then setsum f F - 1 else setsum f F)"
```
```   312   apply (erule finite_induct)
```
```   313    apply auto
```
```   314   apply (drule_tac a = a in mk_disjoint_insert)
```
```   315   apply auto
```
```   316   done
```
```   317
```
```   318 lemma rep_multiset_induct_aux:
```
```   319   "P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
```
```   320     ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
```
```   321 proof -
```
```   322   case rule_context
```
```   323   note premises = this [unfolded multiset_def]
```
```   324   show ?thesis
```
```   325     apply (unfold multiset_def)
```
```   326     apply (induct_tac n)
```
```   327      apply simp
```
```   328      apply clarify
```
```   329      apply (subgoal_tac "f = (\<lambda>a.0)")
```
```   330       apply simp
```
```   331       apply (rule premises)
```
```   332      apply (rule ext)
```
```   333      apply force
```
```   334     apply clarify
```
```   335     apply (frule setsum_SucD)
```
```   336     apply clarify
```
```   337     apply (rename_tac a)
```
```   338     apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
```
```   339      prefer 2
```
```   340      apply (rule finite_subset)
```
```   341       prefer 2
```
```   342       apply assumption
```
```   343      apply simp
```
```   344      apply blast
```
```   345     apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
```
```   346      prefer 2
```
```   347      apply (rule ext)
```
```   348      apply (simp (no_asm_simp))
```
```   349      apply (erule ssubst, rule premises)
```
```   350      apply blast
```
```   351     apply (erule allE, erule impE, erule_tac  mp)
```
```   352      apply blast
```
```   353     apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
```
```   354     apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
```
```   355      prefer 2
```
```   356      apply blast
```
```   357     apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
```
```   358      prefer 2
```
```   359      apply blast
```
```   360     apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
```
```   361     done
```
```   362 qed
```
```   363
```
```   364 theorem rep_multiset_induct:
```
```   365   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
```
```   366     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
```
```   367   apply (insert rep_multiset_induct_aux)
```
```   368   apply blast
```
```   369   done
```
```   370
```
```   371 theorem multiset_induct [induct type: multiset]:
```
```   372   "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
```
```   373 proof -
```
```   374   note defns = union_def single_def Mempty_def
```
```   375   assume prem1 [unfolded defns]: "P {#}"
```
```   376   assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
```
```   377   show ?thesis
```
```   378     apply (rule Rep_multiset_inverse [THEN subst])
```
```   379     apply (rule Rep_multiset [THEN rep_multiset_induct])
```
```   380      apply (rule prem1)
```
```   381     apply (subgoal_tac "f (b := f b + 1) = (\<lambda>a. f a + (if a = b then 1 else 0))")
```
```   382      prefer 2
```
```   383      apply (simp add: expand_fun_eq)
```
```   384     apply (erule ssubst)
```
```   385     apply (erule Abs_multiset_inverse [THEN subst])
```
```   386     apply (erule prem2 [simplified])
```
```   387     done
```
```   388 qed
```
```   389
```
```   390
```
```   391 lemma MCollect_preserves_multiset:
```
```   392     "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
```
```   393   apply (simp add: multiset_def)
```
```   394   apply (rule finite_subset)
```
```   395    apply auto
```
```   396   done
```
```   397
```
```   398 theorem count_MCollect [simp]:
```
```   399     "count {# x:M. P x #} a = (if P a then count M a else 0)"
```
```   400   apply (unfold count_def MCollect_def)
```
```   401   apply (simp add: MCollect_preserves_multiset)
```
```   402   done
```
```   403
```
```   404 theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
```
```   405   apply (auto simp add: set_of_def)
```
```   406   done
```
```   407
```
```   408 theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
```
```   409   apply (subst multiset_eq_conv_count_eq)
```
```   410   apply auto
```
```   411   done
```
```   412
```
```   413 declare Rep_multiset_inject [symmetric, simp del]
```
```   414 declare multiset_typedef [simp del]
```
```   415
```
```   416 theorem add_eq_conv_ex:
```
```   417   "(M + {#a#} = N + {#b#}) =
```
```   418     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
```
```   419   apply (auto simp add: add_eq_conv_diff)
```
```   420   done
```
```   421
```
```   422
```
```   423 subsection {* Multiset orderings *}
```
```   424
```
```   425 subsubsection {* Well-foundedness *}
```
```   426
```
```   427 constdefs
```
```   428   mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
```
```   429   "mult1 r ==
```
```   430     {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```   431       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```   432
```
```   433   mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
```
```   434   "mult r == (mult1 r)\<^sup>+"
```
```   435
```
```   436 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```   437   by (simp add: mult1_def)
```
```   438
```
```   439 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```   440     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```   441     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```   442   (concl is "?case1 (mult1 r) \<or> ?case2")
```
```   443 proof (unfold mult1_def)
```
```   444   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```   445   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```   446   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```   447
```
```   448   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```   449   hence "\<exists>a' M0' K.
```
```   450       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```   451   thus "?case1 \<or> ?case2"
```
```   452   proof (elim exE conjE)
```
```   453     fix a' M0' K
```
```   454     assume N: "N = M0' + K" and r: "?r K a'"
```
```   455     assume "M0 + {#a#} = M0' + {#a'#}"
```
```   456     hence "M0 = M0' \<and> a = a' \<or>
```
```   457         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```   458       by (simp only: add_eq_conv_ex)
```
```   459     thus ?thesis
```
```   460     proof (elim disjE conjE exE)
```
```   461       assume "M0 = M0'" "a = a'"
```
```   462       with N r have "?r K a \<and> N = M0 + K" by simp
```
```   463       hence ?case2 .. thus ?thesis ..
```
```   464     next
```
```   465       fix K'
```
```   466       assume "M0' = K' + {#a#}"
```
```   467       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
```
```   468
```
```   469       assume "M0 = K' + {#a'#}"
```
```   470       with r have "?R (K' + K) M0" by blast
```
```   471       with n have ?case1 by simp thus ?thesis ..
```
```   472     qed
```
```   473   qed
```
```   474 qed
```
```   475
```
```   476 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
```
```   477 proof
```
```   478   let ?R = "mult1 r"
```
```   479   let ?W = "acc ?R"
```
```   480   {
```
```   481     fix M M0 a
```
```   482     assume M0: "M0 \<in> ?W"
```
```   483       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```   484       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```   485     have "M0 + {#a#} \<in> ?W"
```
```   486     proof (rule accI [of "M0 + {#a#}"])
```
```   487       fix N
```
```   488       assume "(N, M0 + {#a#}) \<in> ?R"
```
```   489       hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```   490           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```   491         by (rule less_add)
```
```   492       thus "N \<in> ?W"
```
```   493       proof (elim exE disjE conjE)
```
```   494         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```   495         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```   496         hence "M + {#a#} \<in> ?W" ..
```
```   497         thus "N \<in> ?W" by (simp only: N)
```
```   498       next
```
```   499         fix K
```
```   500         assume N: "N = M0 + K"
```
```   501         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```   502         have "?this --> M0 + K \<in> ?W" (is "?P K")
```
```   503         proof (induct K)
```
```   504           from M0 have "M0 + {#} \<in> ?W" by simp
```
```   505           thus "?P {#}" ..
```
```   506
```
```   507           fix K x assume hyp: "?P K"
```
```   508           show "?P (K + {#x#})"
```
```   509           proof
```
```   510             assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
```
```   511             hence "(x, a) \<in> r" by simp
```
```   512             with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```   513
```
```   514             from a hyp have "M0 + K \<in> ?W" by simp
```
```   515             with b have "(M0 + K) + {#x#} \<in> ?W" ..
```
```   516             thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
```
```   517           qed
```
```   518         qed
```
```   519         hence "M0 + K \<in> ?W" ..
```
```   520         thus "N \<in> ?W" by (simp only: N)
```
```   521       qed
```
```   522     qed
```
```   523   } note tedious_reasoning = this
```
```   524
```
```   525   assume wf: "wf r"
```
```   526   fix M
```
```   527   show "M \<in> ?W"
```
```   528   proof (induct M)
```
```   529     show "{#} \<in> ?W"
```
```   530     proof (rule accI)
```
```   531       fix b assume "(b, {#}) \<in> ?R"
```
```   532       with not_less_empty show "b \<in> ?W" by contradiction
```
```   533     qed
```
```   534
```
```   535     fix M a assume "M \<in> ?W"
```
```   536     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```   537     proof induct
```
```   538       fix a
```
```   539       assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```   540       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```   541       proof
```
```   542         fix M assume "M \<in> ?W"
```
```   543         thus "M + {#a#} \<in> ?W"
```
```   544           by (rule acc_induct) (rule tedious_reasoning)
```
```   545       qed
```
```   546     qed
```
```   547     thus "M + {#a#} \<in> ?W" ..
```
```   548   qed
```
```   549 qed
```
```   550
```
```   551 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```   552   by (rule acc_wfI, rule all_accessible)
```
```   553
```
```   554 theorem wf_mult: "wf r ==> wf (mult r)"
```
```   555   by (unfold mult_def, rule wf_trancl, rule wf_mult1)
```
```   556
```
```   557
```
```   558 subsubsection {* Closure-free presentation *}
```
```   559
```
```   560 (*Badly needed: a linear arithmetic procedure for multisets*)
```
```   561
```
```   562 lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
```
```   563   apply (simp add: multiset_eq_conv_count_eq)
```
```   564   done
```
```   565
```
```   566 text {* One direction. *}
```
```   567
```
```   568 lemma mult_implies_one_step:
```
```   569   "trans r ==> (M, N) \<in> mult r ==>
```
```   570     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```   571     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```   572   apply (unfold mult_def mult1_def set_of_def)
```
```   573   apply (erule converse_trancl_induct)
```
```   574   apply clarify
```
```   575    apply (rule_tac x = M0 in exI)
```
```   576    apply simp
```
```   577   apply clarify
```
```   578   apply (case_tac "a :# K")
```
```   579    apply (rule_tac x = I in exI)
```
```   580    apply (simp (no_asm))
```
```   581    apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```   582    apply (simp (no_asm_simp) add: union_assoc [symmetric])
```
```   583    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```   584    apply (simp add: diff_union_single_conv)
```
```   585    apply (simp (no_asm_use) add: trans_def)
```
```   586    apply blast
```
```   587   apply (subgoal_tac "a :# I")
```
```   588    apply (rule_tac x = "I - {#a#}" in exI)
```
```   589    apply (rule_tac x = "J + {#a#}" in exI)
```
```   590    apply (rule_tac x = "K + Ka" in exI)
```
```   591    apply (rule conjI)
```
```   592     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   593    apply (rule conjI)
```
```   594     apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```   595     apply simp
```
```   596     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   597    apply (simp (no_asm_use) add: trans_def)
```
```   598    apply blast
```
```   599   apply (subgoal_tac "a :# (M0 + {#a#})")
```
```   600    apply simp
```
```   601   apply (simp (no_asm))
```
```   602   done
```
```   603
```
```   604 lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
```
```   605   apply (simp add: multiset_eq_conv_count_eq)
```
```   606   done
```
```   607
```
```   608 lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
```
```   609   apply (erule size_eq_Suc_imp_elem [THEN exE])
```
```   610   apply (drule elem_imp_eq_diff_union)
```
```   611   apply auto
```
```   612   done
```
```   613
```
```   614 lemma one_step_implies_mult_aux:
```
```   615   "trans r ==>
```
```   616     \<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))
```
```   617       --> (I + K, I + J) \<in> mult r"
```
```   618   apply (induct_tac n)
```
```   619    apply auto
```
```   620   apply (frule size_eq_Suc_imp_eq_union)
```
```   621   apply clarify
```
```   622   apply (rename_tac "J'")
```
```   623   apply simp
```
```   624   apply (erule notE)
```
```   625    apply auto
```
```   626   apply (case_tac "J' = {#}")
```
```   627    apply (simp add: mult_def)
```
```   628    apply (rule r_into_trancl)
```
```   629    apply (simp add: mult1_def set_of_def)
```
```   630    apply blast
```
```   631   txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```   632   apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```   633   apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```   634   apply (erule ssubst)
```
```   635   apply (simp add: Ball_def)
```
```   636   apply auto
```
```   637   apply (subgoal_tac
```
```   638     "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
```
```   639       (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```   640    prefer 2
```
```   641    apply force
```
```   642   apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
```
```   643   apply (erule trancl_trans)
```
```   644   apply (rule r_into_trancl)
```
```   645   apply (simp add: mult1_def set_of_def)
```
```   646   apply (rule_tac x = a in exI)
```
```   647   apply (rule_tac x = "I + J'" in exI)
```
```   648   apply (simp add: union_ac)
```
```   649   done
```
```   650
```
```   651 theorem one_step_implies_mult:
```
```   652   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```   653     ==> (I + K, I + J) \<in> mult r"
```
```   654   apply (insert one_step_implies_mult_aux)
```
```   655   apply blast
```
```   656   done
```
```   657
```
```   658
```
```   659 subsubsection {* Partial-order properties *}
```
```   660
```
```   661 instance multiset :: (type) ord ..
```
```   662
```
```   663 defs (overloaded)
```
```   664   less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
```
```   665   le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
```
```   666
```
```   667 lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
```
```   668   apply (unfold trans_def)
```
```   669   apply (blast intro: order_less_trans)
```
```   670   done
```
```   671
```
```   672 text {*
```
```   673  \medskip Irreflexivity.
```
```   674 *}
```
```   675
```
```   676 lemma mult_irrefl_aux:
```
```   677     "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
```
```   678   apply (erule finite_induct)
```
```   679    apply (auto intro: order_less_trans)
```
```   680   done
```
```   681
```
```   682 theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
```
```   683   apply (unfold less_multiset_def)
```
```   684   apply auto
```
```   685   apply (drule trans_base_order [THEN mult_implies_one_step])
```
```   686   apply auto
```
```   687   apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
```
```   688   apply (simp add: set_of_eq_empty_iff)
```
```   689   done
```
```   690
```
```   691 lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
```
```   692   apply (insert mult_less_not_refl)
```
```   693   apply fast
```
```   694   done
```
```   695
```
```   696
```
```   697 text {* Transitivity. *}
```
```   698
```
```   699 theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
```
```   700   apply (unfold less_multiset_def mult_def)
```
```   701   apply (blast intro: trancl_trans)
```
```   702   done
```
```   703
```
```   704 text {* Asymmetry. *}
```
```   705
```
```   706 theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
```
```   707   apply auto
```
```   708   apply (rule mult_less_not_refl [THEN notE])
```
```   709   apply (erule mult_less_trans)
```
```   710   apply assumption
```
```   711   done
```
```   712
```
```   713 theorem mult_less_asym:
```
```   714     "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
```
```   715   apply (insert mult_less_not_sym)
```
```   716   apply blast
```
```   717   done
```
```   718
```
```   719 theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
```
```   720   apply (unfold le_multiset_def)
```
```   721   apply auto
```
```   722   done
```
```   723
```
```   724 text {* Anti-symmetry. *}
```
```   725
```
```   726 theorem mult_le_antisym:
```
```   727     "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
```
```   728   apply (unfold le_multiset_def)
```
```   729   apply (blast dest: mult_less_not_sym)
```
```   730   done
```
```   731
```
```   732 text {* Transitivity. *}
```
```   733
```
```   734 theorem mult_le_trans:
```
```   735     "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
```
```   736   apply (unfold le_multiset_def)
```
```   737   apply (blast intro: mult_less_trans)
```
```   738   done
```
```   739
```
```   740 theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
```
```   741   apply (unfold le_multiset_def)
```
```   742   apply auto
```
```   743   done
```
```   744
```
```   745 text {* Partial order. *}
```
```   746
```
```   747 instance multiset :: (order) order
```
```   748   apply intro_classes
```
```   749      apply (rule mult_le_refl)
```
```   750     apply (erule mult_le_trans)
```
```   751     apply assumption
```
```   752    apply (erule mult_le_antisym)
```
```   753    apply assumption
```
```   754   apply (rule mult_less_le)
```
```   755   done
```
```   756
```
```   757
```
```   758 subsubsection {* Monotonicity of multiset union *}
```
```   759
```
```   760 theorem mult1_union:
```
```   761     "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
```
```   762   apply (unfold mult1_def)
```
```   763   apply auto
```
```   764   apply (rule_tac x = a in exI)
```
```   765   apply (rule_tac x = "C + M0" in exI)
```
```   766   apply (simp add: union_assoc)
```
```   767   done
```
```   768
```
```   769 lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
```
```   770   apply (unfold less_multiset_def mult_def)
```
```   771   apply (erule trancl_induct)
```
```   772    apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
```
```   773   apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
```
```   774   done
```
```   775
```
```   776 lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
```
```   777   apply (subst union_commute [of B C])
```
```   778   apply (subst union_commute [of D C])
```
```   779   apply (erule union_less_mono2)
```
```   780   done
```
```   781
```
```   782 theorem union_less_mono:
```
```   783     "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
```
```   784   apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
```
```   785   done
```
```   786
```
```   787 theorem union_le_mono:
```
```   788     "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
```
```   789   apply (unfold le_multiset_def)
```
```   790   apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
```
```   791   done
```
```   792
```
```   793 theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
```
```   794   apply (unfold le_multiset_def less_multiset_def)
```
```   795   apply (case_tac "M = {#}")
```
```   796    prefer 2
```
```   797    apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
```
```   798     prefer 2
```
```   799     apply (rule one_step_implies_mult)
```
```   800       apply (simp only: trans_def)
```
```   801       apply auto
```
```   802   done
```
```   803
```
```   804 theorem union_upper1: "A <= A + (B::'a::order multiset)"
```
```   805   apply (subgoal_tac "A + {#} <= A + B")
```
```   806    prefer 2
```
```   807    apply (rule union_le_mono)
```
```   808     apply auto
```
```   809   done
```
```   810
```
```   811 theorem union_upper2: "B <= A + (B::'a::order multiset)"
```
```   812   apply (subst union_commute, rule union_upper1)
```
```   813   done
```
```   814
```
```   815 end
```