src/HOL/Library/Multiset.thy
 author paulson Mon Oct 22 11:54:22 2001 +0200 (2001-10-22) changeset 11868 56db9f3a6b3e parent 11701 3d51fbf81c17 child 12338 de0f4a63baa5 permissions -rw-r--r--
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
to their abstract counterparts, while other binary numerals work correctly.
```     1 (*  Title:      HOL/Library/Multiset.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, TU Muenchen
```
```     4     Author:     Markus Wenzel, TU Muenchen
```
```     5     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     6 *)
```
```     7
```
```     8 header {*
```
```     9  \title{Multisets}
```
```    10  \author{Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson}
```
```    11 *}
```
```    12
```
```    13 theory Multiset = Accessible_Part:
```
```    14
```
```    15 subsection {* The type of multisets *}
```
```    16
```
```    17 typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
```
```    18 proof
```
```    19   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
```
```    20 qed
```
```    21
```
```    22 lemmas multiset_typedef [simp] =
```
```    23     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
```
```    24   and [simp] = Rep_multiset_inject [symmetric]
```
```    25
```
```    26 constdefs
```
```    27   Mempty :: "'a multiset"    ("{#}")
```
```    28   "{#} == Abs_multiset (\<lambda>a. 0)"
```
```    29
```
```    30   single :: "'a => 'a multiset"    ("{#_#}")
```
```    31   "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
```
```    32
```
```    33   count :: "'a multiset => 'a => nat"
```
```    34   "count == Rep_multiset"
```
```    35
```
```    36   MCollect :: "'a multiset => ('a => bool) => 'a multiset"
```
```    37   "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
```
```    38
```
```    39 syntax
```
```    40   "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
```
```    41   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
```
```    42 translations
```
```    43   "a :# M" == "0 < count M a"
```
```    44   "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
```
```    45
```
```    46 constdefs
```
```    47   set_of :: "'a multiset => 'a set"
```
```    48   "set_of M == {x. x :# M}"
```
```    49
```
```    50 instance multiset :: ("term") plus ..
```
```    51 instance multiset :: ("term") minus ..
```
```    52 instance multiset :: ("term") zero ..
```
```    53
```
```    54 defs (overloaded)
```
```    55   union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
```
```    56   diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
```
```    57   Zero_multiset_def [simp]: "0 == {#}"
```
```    58   size_def: "size M == setsum (count M) (set_of M)"
```
```    59
```
```    60
```
```    61 text {*
```
```    62  \medskip Preservation of the representing set @{term multiset}.
```
```    63 *}
```
```    64
```
```    65 lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
```
```    66   apply (simp add: multiset_def)
```
```    67   done
```
```    68
```
```    69 lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
```
```    70   apply (simp add: multiset_def)
```
```    71   done
```
```    72
```
```    73 lemma union_preserves_multiset [simp]:
```
```    74     "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
```
```    75   apply (unfold multiset_def)
```
```    76   apply simp
```
```    77   apply (drule finite_UnI)
```
```    78    apply assumption
```
```    79   apply (simp del: finite_Un add: Un_def)
```
```    80   done
```
```    81
```
```    82 lemma diff_preserves_multiset [simp]:
```
```    83     "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
```
```    84   apply (unfold multiset_def)
```
```    85   apply simp
```
```    86   apply (rule finite_subset)
```
```    87    prefer 2
```
```    88    apply assumption
```
```    89   apply auto
```
```    90   done
```
```    91
```
```    92
```
```    93 subsection {* Algebraic properties of multisets *}
```
```    94
```
```    95 subsubsection {* Union *}
```
```    96
```
```    97 theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
```
```    98   apply (simp add: union_def Mempty_def)
```
```    99   done
```
```   100
```
```   101 theorem union_commute: "M + N = N + (M::'a multiset)"
```
```   102   apply (simp add: union_def add_ac)
```
```   103   done
```
```   104
```
```   105 theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
```
```   106   apply (simp add: union_def add_ac)
```
```   107   done
```
```   108
```
```   109 theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
```
```   110   apply (rule union_commute [THEN trans])
```
```   111   apply (rule union_assoc [THEN trans])
```
```   112   apply (rule union_commute [THEN arg_cong])
```
```   113   done
```
```   114
```
```   115 theorems union_ac = union_assoc union_commute union_lcomm
```
```   116
```
```   117 instance multiset :: ("term") plus_ac0
```
```   118   apply intro_classes
```
```   119     apply (rule union_commute)
```
```   120    apply (rule union_assoc)
```
```   121   apply simp
```
```   122   done
```
```   123
```
```   124
```
```   125 subsubsection {* Difference *}
```
```   126
```
```   127 theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
```
```   128   apply (simp add: Mempty_def diff_def)
```
```   129   done
```
```   130
```
```   131 theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
```
```   132   apply (simp add: union_def diff_def)
```
```   133   done
```
```   134
```
```   135
```
```   136 subsubsection {* Count of elements *}
```
```   137
```
```   138 theorem count_empty [simp]: "count {#} a = 0"
```
```   139   apply (simp add: count_def Mempty_def)
```
```   140   done
```
```   141
```
```   142 theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
```
```   143   apply (simp add: count_def single_def)
```
```   144   done
```
```   145
```
```   146 theorem count_union [simp]: "count (M + N) a = count M a + count N a"
```
```   147   apply (simp add: count_def union_def)
```
```   148   done
```
```   149
```
```   150 theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
```
```   151   apply (simp add: count_def diff_def)
```
```   152   done
```
```   153
```
```   154
```
```   155 subsubsection {* Set of elements *}
```
```   156
```
```   157 theorem set_of_empty [simp]: "set_of {#} = {}"
```
```   158   apply (simp add: set_of_def)
```
```   159   done
```
```   160
```
```   161 theorem set_of_single [simp]: "set_of {#b#} = {b}"
```
```   162   apply (simp add: set_of_def)
```
```   163   done
```
```   164
```
```   165 theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
```
```   166   apply (auto simp add: set_of_def)
```
```   167   done
```
```   168
```
```   169 theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
```
```   170   apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
```
```   171   done
```
```   172
```
```   173 theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
```
```   174   apply (auto simp add: set_of_def)
```
```   175   done
```
```   176
```
```   177
```
```   178 subsubsection {* Size *}
```
```   179
```
```   180 theorem size_empty [simp]: "size {#} = 0"
```
```   181   apply (simp add: size_def)
```
```   182   done
```
```   183
```
```   184 theorem size_single [simp]: "size {#b#} = 1"
```
```   185   apply (simp add: size_def)
```
```   186   done
```
```   187
```
```   188 theorem finite_set_of [iff]: "finite (set_of M)"
```
```   189   apply (cut_tac x = M in Rep_multiset)
```
```   190   apply (simp add: multiset_def set_of_def count_def)
```
```   191   done
```
```   192
```
```   193 theorem setsum_count_Int:
```
```   194     "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
```
```   195   apply (erule finite_induct)
```
```   196    apply simp
```
```   197   apply (simp add: Int_insert_left set_of_def)
```
```   198   done
```
```   199
```
```   200 theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
```
```   201   apply (unfold size_def)
```
```   202   apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
```
```   203    prefer 2
```
```   204    apply (rule ext)
```
```   205    apply simp
```
```   206   apply (simp (no_asm_simp) add: setsum_Un setsum_addf setsum_count_Int)
```
```   207   apply (subst Int_commute)
```
```   208   apply (simp (no_asm_simp) add: setsum_count_Int)
```
```   209   done
```
```   210
```
```   211 theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
```
```   212   apply (unfold size_def Mempty_def count_def)
```
```   213   apply auto
```
```   214   apply (simp add: set_of_def count_def expand_fun_eq)
```
```   215   done
```
```   216
```
```   217 theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
```
```   218   apply (unfold size_def)
```
```   219   apply (drule setsum_SucD)
```
```   220   apply auto
```
```   221   done
```
```   222
```
```   223
```
```   224 subsubsection {* Equality of multisets *}
```
```   225
```
```   226 theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
```
```   227   apply (simp add: count_def expand_fun_eq)
```
```   228   done
```
```   229
```
```   230 theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
```
```   231   apply (simp add: single_def Mempty_def expand_fun_eq)
```
```   232   done
```
```   233
```
```   234 theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
```
```   235   apply (auto simp add: single_def expand_fun_eq)
```
```   236   done
```
```   237
```
```   238 theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
```
```   239   apply (auto simp add: union_def Mempty_def expand_fun_eq)
```
```   240   done
```
```   241
```
```   242 theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
```
```   243   apply (auto simp add: union_def Mempty_def expand_fun_eq)
```
```   244   done
```
```   245
```
```   246 theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
```
```   247   apply (simp add: union_def expand_fun_eq)
```
```   248   done
```
```   249
```
```   250 theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
```
```   251   apply (simp add: union_def expand_fun_eq)
```
```   252   done
```
```   253
```
```   254 theorem union_is_single:
```
```   255     "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
```
```   256   apply (unfold Mempty_def single_def union_def)
```
```   257   apply (simp add: add_is_1 expand_fun_eq)
```
```   258   apply blast
```
```   259   done
```
```   260
```
```   261 theorem single_is_union:
```
```   262   "({#a#} = M + N) =
```
```   263     ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
```
```   264   apply (unfold Mempty_def single_def union_def)
```
```   265   apply (simp add: add_is_1 one_is_add expand_fun_eq)
```
```   266   apply (blast dest: sym)
```
```   267   done
```
```   268
```
```   269 theorem add_eq_conv_diff:
```
```   270   "(M + {#a#} = N + {#b#}) =
```
```   271     (M = N \<and> a = b \<or>
```
```   272       M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
```
```   273   apply (unfold single_def union_def diff_def)
```
```   274   apply (simp (no_asm) add: expand_fun_eq)
```
```   275   apply (rule conjI)
```
```   276    apply force
```
```   277   apply safe
```
```   278   apply (simp_all add: eq_sym_conv)
```
```   279   done
```
```   280
```
```   281 (*
```
```   282 val prems = Goal
```
```   283  "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
```
```   284 by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
```
```   285      measure_induct 1);
```
```   286 by (Clarify_tac 1);
```
```   287 by (resolve_tac prems 1);
```
```   288  by (assume_tac 1);
```
```   289 by (Clarify_tac 1);
```
```   290 by (subgoal_tac "finite G" 1);
```
```   291  by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
```
```   292 by (etac allE 1);
```
```   293 by (etac impE 1);
```
```   294  by (Blast_tac 2);
```
```   295 by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
```
```   296 no_qed();
```
```   297 val lemma = result();
```
```   298
```
```   299 val prems = Goal
```
```   300  "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
```
```   301 by (rtac (lemma RS mp) 1);
```
```   302 by (REPEAT(ares_tac prems 1));
```
```   303 qed "finite_psubset_induct";
```
```   304
```
```   305 Better: use wf_finite_psubset in WF_Rel
```
```   306 *)
```
```   307
```
```   308
```
```   309 subsection {* Induction over multisets *}
```
```   310
```
```   311 lemma setsum_decr:
```
```   312   "finite F ==> (0::nat) < f a ==>
```
```   313     setsum (f (a := f a - 1)) F = (if a \<in> F then setsum f F - 1 else setsum f F)"
```
```   314   apply (erule finite_induct)
```
```   315    apply auto
```
```   316   apply (drule_tac a = a in mk_disjoint_insert)
```
```   317   apply auto
```
```   318   done
```
```   319
```
```   320 lemma rep_multiset_induct_aux:
```
```   321   "P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
```
```   322     ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
```
```   323 proof -
```
```   324   case rule_context
```
```   325   note premises = this [unfolded multiset_def]
```
```   326   show ?thesis
```
```   327     apply (unfold multiset_def)
```
```   328     apply (induct_tac n)
```
```   329      apply simp
```
```   330      apply clarify
```
```   331      apply (subgoal_tac "f = (\<lambda>a.0)")
```
```   332       apply simp
```
```   333       apply (rule premises)
```
```   334      apply (rule ext)
```
```   335      apply force
```
```   336     apply clarify
```
```   337     apply (frule setsum_SucD)
```
```   338     apply clarify
```
```   339     apply (rename_tac a)
```
```   340     apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
```
```   341      prefer 2
```
```   342      apply (rule finite_subset)
```
```   343       prefer 2
```
```   344       apply assumption
```
```   345      apply simp
```
```   346      apply blast
```
```   347     apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
```
```   348      prefer 2
```
```   349      apply (rule ext)
```
```   350      apply (simp (no_asm_simp))
```
```   351      apply (erule ssubst, rule premises)
```
```   352      apply blast
```
```   353     apply (erule allE, erule impE, erule_tac [2] mp)
```
```   354      apply blast
```
```   355     apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
```
```   356     apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
```
```   357      prefer 2
```
```   358      apply blast
```
```   359     apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
```
```   360      prefer 2
```
```   361      apply blast
```
```   362     apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
```
```   363     done
```
```   364 qed
```
```   365
```
```   366 theorem rep_multiset_induct:
```
```   367   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
```
```   368     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
```
```   369   apply (insert rep_multiset_induct_aux)
```
```   370   apply blast
```
```   371   done
```
```   372
```
```   373 theorem multiset_induct [induct type: multiset]:
```
```   374   "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
```
```   375 proof -
```
```   376   note defns = union_def single_def Mempty_def
```
```   377   assume prem1 [unfolded defns]: "P {#}"
```
```   378   assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
```
```   379   show ?thesis
```
```   380     apply (rule Rep_multiset_inverse [THEN subst])
```
```   381     apply (rule Rep_multiset [THEN rep_multiset_induct])
```
```   382      apply (rule prem1)
```
```   383     apply (subgoal_tac "f (b := f b + 1) = (\<lambda>a. f a + (if a = b then 1 else 0))")
```
```   384      prefer 2
```
```   385      apply (simp add: expand_fun_eq)
```
```   386     apply (erule ssubst)
```
```   387     apply (erule Abs_multiset_inverse [THEN subst])
```
```   388     apply (erule prem2 [simplified])
```
```   389     done
```
```   390 qed
```
```   391
```
```   392
```
```   393 lemma MCollect_preserves_multiset:
```
```   394     "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
```
```   395   apply (simp add: multiset_def)
```
```   396   apply (rule finite_subset)
```
```   397    apply auto
```
```   398   done
```
```   399
```
```   400 theorem count_MCollect [simp]:
```
```   401     "count {# x:M. P x #} a = (if P a then count M a else 0)"
```
```   402   apply (unfold count_def MCollect_def)
```
```   403   apply (simp add: MCollect_preserves_multiset)
```
```   404   done
```
```   405
```
```   406 theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
```
```   407   apply (auto simp add: set_of_def)
```
```   408   done
```
```   409
```
```   410 theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
```
```   411   apply (subst multiset_eq_conv_count_eq)
```
```   412   apply auto
```
```   413   done
```
```   414
```
```   415 declare Rep_multiset_inject [symmetric, simp del]
```
```   416 declare multiset_typedef [simp del]
```
```   417
```
```   418 theorem add_eq_conv_ex:
```
```   419   "(M + {#a#} = N + {#b#}) =
```
```   420     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
```
```   421   apply (auto simp add: add_eq_conv_diff)
```
```   422   done
```
```   423
```
```   424
```
```   425 subsection {* Multiset orderings *}
```
```   426
```
```   427 subsubsection {* Well-foundedness *}
```
```   428
```
```   429 constdefs
```
```   430   mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
```
```   431   "mult1 r ==
```
```   432     {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
```
```   433       (\<forall>b. b :# K --> (b, a) \<in> r)}"
```
```   434
```
```   435   mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
```
```   436   "mult r == (mult1 r)\<^sup>+"
```
```   437
```
```   438 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
```
```   439   by (simp add: mult1_def)
```
```   440
```
```   441 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
```
```   442     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
```
```   443     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
```
```   444   (concl is "?case1 (mult1 r) \<or> ?case2")
```
```   445 proof (unfold mult1_def)
```
```   446   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
```
```   447   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
```
```   448   let ?case1 = "?case1 {(N, M). ?R N M}"
```
```   449
```
```   450   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
```
```   451   hence "\<exists>a' M0' K.
```
```   452       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
```
```   453   thus "?case1 \<or> ?case2"
```
```   454   proof (elim exE conjE)
```
```   455     fix a' M0' K
```
```   456     assume N: "N = M0' + K" and r: "?r K a'"
```
```   457     assume "M0 + {#a#} = M0' + {#a'#}"
```
```   458     hence "M0 = M0' \<and> a = a' \<or>
```
```   459         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
```
```   460       by (simp only: add_eq_conv_ex)
```
```   461     thus ?thesis
```
```   462     proof (elim disjE conjE exE)
```
```   463       assume "M0 = M0'" "a = a'"
```
```   464       with N r have "?r K a \<and> N = M0 + K" by simp
```
```   465       hence ?case2 .. thus ?thesis ..
```
```   466     next
```
```   467       fix K'
```
```   468       assume "M0' = K' + {#a#}"
```
```   469       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
```
```   470
```
```   471       assume "M0 = K' + {#a'#}"
```
```   472       with r have "?R (K' + K) M0" by blast
```
```   473       with n have ?case1 by simp thus ?thesis ..
```
```   474     qed
```
```   475   qed
```
```   476 qed
```
```   477
```
```   478 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
```
```   479 proof
```
```   480   let ?R = "mult1 r"
```
```   481   let ?W = "acc ?R"
```
```   482   {
```
```   483     fix M M0 a
```
```   484     assume M0: "M0 \<in> ?W"
```
```   485       and wf_hyp: "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```   486       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
```
```   487     have "M0 + {#a#} \<in> ?W"
```
```   488     proof (rule accI [of "M0 + {#a#}"])
```
```   489       fix N
```
```   490       assume "(N, M0 + {#a#}) \<in> ?R"
```
```   491       hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
```
```   492           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
```
```   493         by (rule less_add)
```
```   494       thus "N \<in> ?W"
```
```   495       proof (elim exE disjE conjE)
```
```   496         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
```
```   497         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
```
```   498         hence "M + {#a#} \<in> ?W" ..
```
```   499         thus "N \<in> ?W" by (simp only: N)
```
```   500       next
```
```   501         fix K
```
```   502         assume N: "N = M0 + K"
```
```   503         assume "\<forall>b. b :# K --> (b, a) \<in> r"
```
```   504         have "?this --> M0 + K \<in> ?W" (is "?P K")
```
```   505         proof (induct K)
```
```   506           from M0 have "M0 + {#} \<in> ?W" by simp
```
```   507           thus "?P {#}" ..
```
```   508
```
```   509           fix K x assume hyp: "?P K"
```
```   510           show "?P (K + {#x#})"
```
```   511           proof
```
```   512             assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
```
```   513             hence "(x, a) \<in> r" by simp
```
```   514             with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
```
```   515
```
```   516             from a hyp have "M0 + K \<in> ?W" by simp
```
```   517             with b have "(M0 + K) + {#x#} \<in> ?W" ..
```
```   518             thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
```
```   519           qed
```
```   520         qed
```
```   521         hence "M0 + K \<in> ?W" ..
```
```   522         thus "N \<in> ?W" by (simp only: N)
```
```   523       qed
```
```   524     qed
```
```   525   } note tedious_reasoning = this
```
```   526
```
```   527   assume wf: "wf r"
```
```   528   fix M
```
```   529   show "M \<in> ?W"
```
```   530   proof (induct M)
```
```   531     show "{#} \<in> ?W"
```
```   532     proof (rule accI)
```
```   533       fix b assume "(b, {#}) \<in> ?R"
```
```   534       with not_less_empty show "b \<in> ?W" by contradiction
```
```   535     qed
```
```   536
```
```   537     fix M a assume "M \<in> ?W"
```
```   538     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```   539     proof induct
```
```   540       fix a
```
```   541       assume "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
```
```   542       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
```
```   543       proof
```
```   544         fix M assume "M \<in> ?W"
```
```   545         thus "M + {#a#} \<in> ?W"
```
```   546           by (rule acc_induct) (rule tedious_reasoning)
```
```   547       qed
```
```   548     qed
```
```   549     thus "M + {#a#} \<in> ?W" ..
```
```   550   qed
```
```   551 qed
```
```   552
```
```   553 theorem wf_mult1: "wf r ==> wf (mult1 r)"
```
```   554   by (rule acc_wfI, rule all_accessible)
```
```   555
```
```   556 theorem wf_mult: "wf r ==> wf (mult r)"
```
```   557   by (unfold mult_def, rule wf_trancl, rule wf_mult1)
```
```   558
```
```   559
```
```   560 subsubsection {* Closure-free presentation *}
```
```   561
```
```   562 (*Badly needed: a linear arithmetic procedure for multisets*)
```
```   563
```
```   564 lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
```
```   565   apply (simp add: multiset_eq_conv_count_eq)
```
```   566   done
```
```   567
```
```   568 text {* One direction. *}
```
```   569
```
```   570 lemma mult_implies_one_step:
```
```   571   "trans r ==> (M, N) \<in> mult r ==>
```
```   572     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
```
```   573     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
```
```   574   apply (unfold mult_def mult1_def set_of_def)
```
```   575   apply (erule converse_trancl_induct)
```
```   576   apply clarify
```
```   577    apply (rule_tac x = M0 in exI)
```
```   578    apply simp
```
```   579   apply clarify
```
```   580   apply (case_tac "a :# K")
```
```   581    apply (rule_tac x = I in exI)
```
```   582    apply (simp (no_asm))
```
```   583    apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
```
```   584    apply (simp (no_asm_simp) add: union_assoc [symmetric])
```
```   585    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```   586    apply (simp add: diff_union_single_conv)
```
```   587    apply (simp (no_asm_use) add: trans_def)
```
```   588    apply blast
```
```   589   apply (subgoal_tac "a :# I")
```
```   590    apply (rule_tac x = "I - {#a#}" in exI)
```
```   591    apply (rule_tac x = "J + {#a#}" in exI)
```
```   592    apply (rule_tac x = "K + Ka" in exI)
```
```   593    apply (rule conjI)
```
```   594     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   595    apply (rule conjI)
```
```   596     apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
```
```   597     apply simp
```
```   598     apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
```
```   599    apply (simp (no_asm_use) add: trans_def)
```
```   600    apply blast
```
```   601   apply (subgoal_tac "a :# (M0 + {#a#})")
```
```   602    apply simp
```
```   603   apply (simp (no_asm))
```
```   604   done
```
```   605
```
```   606 lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
```
```   607   apply (simp add: multiset_eq_conv_count_eq)
```
```   608   done
```
```   609
```
```   610 lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
```
```   611   apply (erule size_eq_Suc_imp_elem [THEN exE])
```
```   612   apply (drule elem_imp_eq_diff_union)
```
```   613   apply auto
```
```   614   done
```
```   615
```
```   616 lemma one_step_implies_mult_aux:
```
```   617   "trans r ==>
```
```   618     \<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))
```
```   619       --> (I + K, I + J) \<in> mult r"
```
```   620   apply (induct_tac n)
```
```   621    apply auto
```
```   622   apply (frule size_eq_Suc_imp_eq_union)
```
```   623   apply clarify
```
```   624   apply (rename_tac "J'")
```
```   625   apply simp
```
```   626   apply (erule notE)
```
```   627    apply auto
```
```   628   apply (case_tac "J' = {#}")
```
```   629    apply (simp add: mult_def)
```
```   630    apply (rule r_into_trancl)
```
```   631    apply (simp add: mult1_def set_of_def)
```
```   632    apply blast
```
```   633   txt {* Now we know @{term "J' \<noteq> {#}"}. *}
```
```   634   apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
```
```   635   apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
```
```   636   apply (erule ssubst)
```
```   637   apply (simp add: Ball_def)
```
```   638   apply auto
```
```   639   apply (subgoal_tac
```
```   640     "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
```
```   641       (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
```
```   642    prefer 2
```
```   643    apply force
```
```   644   apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
```
```   645   apply (erule trancl_trans)
```
```   646   apply (rule r_into_trancl)
```
```   647   apply (simp add: mult1_def set_of_def)
```
```   648   apply (rule_tac x = a in exI)
```
```   649   apply (rule_tac x = "I + J'" in exI)
```
```   650   apply (simp add: union_ac)
```
```   651   done
```
```   652
```
```   653 theorem one_step_implies_mult:
```
```   654   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
```
```   655     ==> (I + K, I + J) \<in> mult r"
```
```   656   apply (insert one_step_implies_mult_aux)
```
```   657   apply blast
```
```   658   done
```
```   659
```
```   660
```
```   661 subsubsection {* Partial-order properties *}
```
```   662
```
```   663 instance multiset :: ("term") ord ..
```
```   664
```
```   665 defs (overloaded)
```
```   666   less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
```
```   667   le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
```
```   668
```
```   669 lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
```
```   670   apply (unfold trans_def)
```
```   671   apply (blast intro: order_less_trans)
```
```   672   done
```
```   673
```
```   674 text {*
```
```   675  \medskip Irreflexivity.
```
```   676 *}
```
```   677
```
```   678 lemma mult_irrefl_aux:
```
```   679     "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
```
```   680   apply (erule finite_induct)
```
```   681    apply (auto intro: order_less_trans)
```
```   682   done
```
```   683
```
```   684 theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
```
```   685   apply (unfold less_multiset_def)
```
```   686   apply auto
```
```   687   apply (drule trans_base_order [THEN mult_implies_one_step])
```
```   688   apply auto
```
```   689   apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
```
```   690   apply (simp add: set_of_eq_empty_iff)
```
```   691   done
```
```   692
```
```   693 lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
```
```   694   apply (insert mult_less_not_refl)
```
```   695   apply blast
```
```   696   done
```
```   697
```
```   698
```
```   699 text {* Transitivity. *}
```
```   700
```
```   701 theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
```
```   702   apply (unfold less_multiset_def mult_def)
```
```   703   apply (blast intro: trancl_trans)
```
```   704   done
```
```   705
```
```   706 text {* Asymmetry. *}
```
```   707
```
```   708 theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
```
```   709   apply auto
```
```   710   apply (rule mult_less_not_refl [THEN notE])
```
```   711   apply (erule mult_less_trans)
```
```   712   apply assumption
```
```   713   done
```
```   714
```
```   715 theorem mult_less_asym:
```
```   716     "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
```
```   717   apply (insert mult_less_not_sym)
```
```   718   apply blast
```
```   719   done
```
```   720
```
```   721 theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
```
```   722   apply (unfold le_multiset_def)
```
```   723   apply auto
```
```   724   done
```
```   725
```
```   726 text {* Anti-symmetry. *}
```
```   727
```
```   728 theorem mult_le_antisym:
```
```   729     "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
```
```   730   apply (unfold le_multiset_def)
```
```   731   apply (blast dest: mult_less_not_sym)
```
```   732   done
```
```   733
```
```   734 text {* Transitivity. *}
```
```   735
```
```   736 theorem mult_le_trans:
```
```   737     "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
```
```   738   apply (unfold le_multiset_def)
```
```   739   apply (blast intro: mult_less_trans)
```
```   740   done
```
```   741
```
```   742 theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
```
```   743   apply (unfold le_multiset_def)
```
```   744   apply auto
```
```   745   done
```
```   746
```
```   747 text {* Partial order. *}
```
```   748
```
```   749 instance multiset :: (order) order
```
```   750   apply intro_classes
```
```   751      apply (rule mult_le_refl)
```
```   752     apply (erule mult_le_trans)
```
```   753     apply assumption
```
```   754    apply (erule mult_le_antisym)
```
```   755    apply assumption
```
```   756   apply (rule mult_less_le)
```
```   757   done
```
```   758
```
```   759
```
```   760 subsubsection {* Monotonicity of multiset union *}
```
```   761
```
```   762 theorem mult1_union:
```
```   763     "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
```
```   764   apply (unfold mult1_def)
```
```   765   apply auto
```
```   766   apply (rule_tac x = a in exI)
```
```   767   apply (rule_tac x = "C + M0" in exI)
```
```   768   apply (simp add: union_assoc)
```
```   769   done
```
```   770
```
```   771 lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
```
```   772   apply (unfold less_multiset_def mult_def)
```
```   773   apply (erule trancl_induct)
```
```   774    apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
```
```   775   apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
```
```   776   done
```
```   777
```
```   778 lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
```
```   779   apply (subst union_commute [of B C])
```
```   780   apply (subst union_commute [of D C])
```
```   781   apply (erule union_less_mono2)
```
```   782   done
```
```   783
```
```   784 theorem union_less_mono:
```
```   785     "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
```
```   786   apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
```
```   787   done
```
```   788
```
```   789 theorem union_le_mono:
```
```   790     "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
```
```   791   apply (unfold le_multiset_def)
```
```   792   apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
```
```   793   done
```
```   794
```
```   795 theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
```
```   796   apply (unfold le_multiset_def less_multiset_def)
```
```   797   apply (case_tac "M = {#}")
```
```   798    prefer 2
```
```   799    apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
```
```   800     prefer 2
```
```   801     apply (rule one_step_implies_mult)
```
```   802       apply (simp only: trans_def)
```
```   803       apply auto
```
```   804   apply (blast intro: order_less_trans)
```
```   805   done
```
```   806
```
```   807 theorem union_upper1: "A <= A + (B::'a::order multiset)"
```
```   808   apply (subgoal_tac "A + {#} <= A + B")
```
```   809    prefer 2
```
```   810    apply (rule union_le_mono)
```
```   811     apply auto
```
```   812   done
```
```   813
```
```   814 theorem union_upper2: "B <= A + (B::'a::order multiset)"
```
```   815   apply (subst union_commute, rule union_upper1)
```
```   816   done
```
```   817
```
```   818 end
```