src/HOL/Library/Multiset.thy
author kleing
Thu Apr 28 09:21:15 2005 +0200 (2005-04-28)
changeset 15867 5c63b6c2f4a5
parent 15630 cc3776f004e2
child 15869 3aca7f05cd12
permissions -rw-r--r--
some more lemmas about multiset_of
wenzelm@10249
     1
(*  Title:      HOL/Library/Multiset.thy
wenzelm@10249
     2
    ID:         $Id$
paulson@15072
     3
    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
wenzelm@10249
     4
*)
wenzelm@10249
     5
wenzelm@14706
     6
header {* Multisets *}
wenzelm@10249
     7
nipkow@15131
     8
theory Multiset
nipkow@15140
     9
imports Accessible_Part
nipkow@15131
    10
begin
wenzelm@10249
    11
wenzelm@10249
    12
subsection {* The type of multisets *}
wenzelm@10249
    13
wenzelm@10249
    14
typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
wenzelm@10249
    15
proof
nipkow@11464
    16
  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
wenzelm@10249
    17
qed
wenzelm@10249
    18
wenzelm@10249
    19
lemmas multiset_typedef [simp] =
wenzelm@10277
    20
    Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
wenzelm@10277
    21
  and [simp] = Rep_multiset_inject [symmetric]
wenzelm@10249
    22
wenzelm@10249
    23
constdefs
wenzelm@10249
    24
  Mempty :: "'a multiset"    ("{#}")
nipkow@11464
    25
  "{#} == Abs_multiset (\<lambda>a. 0)"
wenzelm@10249
    26
wenzelm@10249
    27
  single :: "'a => 'a multiset"    ("{#_#}")
wenzelm@11701
    28
  "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
wenzelm@10249
    29
wenzelm@10249
    30
  count :: "'a multiset => 'a => nat"
wenzelm@10249
    31
  "count == Rep_multiset"
wenzelm@10249
    32
wenzelm@10249
    33
  MCollect :: "'a multiset => ('a => bool) => 'a multiset"
nipkow@11464
    34
  "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
wenzelm@10249
    35
wenzelm@10249
    36
syntax
wenzelm@10249
    37
  "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
wenzelm@10249
    38
  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
wenzelm@10249
    39
translations
wenzelm@10249
    40
  "a :# M" == "0 < count M a"
nipkow@11464
    41
  "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
wenzelm@10249
    42
wenzelm@10249
    43
constdefs
wenzelm@10249
    44
  set_of :: "'a multiset => 'a set"
wenzelm@10249
    45
  "set_of M == {x. x :# M}"
wenzelm@10249
    46
wenzelm@14691
    47
instance multiset :: (type) "{plus, minus, zero}" ..
wenzelm@10249
    48
wenzelm@10249
    49
defs (overloaded)
nipkow@11464
    50
  union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
nipkow@11464
    51
  diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
wenzelm@11701
    52
  Zero_multiset_def [simp]: "0 == {#}"
wenzelm@10249
    53
  size_def: "size M == setsum (count M) (set_of M)"
wenzelm@10249
    54
wenzelm@10249
    55
wenzelm@10249
    56
text {*
wenzelm@10249
    57
 \medskip Preservation of the representing set @{term multiset}.
wenzelm@10249
    58
*}
wenzelm@10249
    59
nipkow@11464
    60
lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
paulson@15072
    61
by (simp add: multiset_def)
wenzelm@10249
    62
wenzelm@11701
    63
lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
paulson@15072
    64
by (simp add: multiset_def)
wenzelm@10249
    65
wenzelm@10249
    66
lemma union_preserves_multiset [simp]:
nipkow@11464
    67
    "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
paulson@15072
    68
  apply (unfold multiset_def, simp)
paulson@15072
    69
  apply (drule finite_UnI, assumption)
wenzelm@10249
    70
  apply (simp del: finite_Un add: Un_def)
wenzelm@10249
    71
  done
wenzelm@10249
    72
wenzelm@10249
    73
lemma diff_preserves_multiset [simp]:
nipkow@11464
    74
    "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
paulson@15072
    75
  apply (unfold multiset_def, simp)
wenzelm@10249
    76
  apply (rule finite_subset)
wenzelm@10249
    77
   prefer 2
wenzelm@10249
    78
   apply assumption
wenzelm@10249
    79
  apply auto
wenzelm@10249
    80
  done
wenzelm@10249
    81
wenzelm@10249
    82
wenzelm@10249
    83
subsection {* Algebraic properties of multisets *}
wenzelm@10249
    84
wenzelm@10249
    85
subsubsection {* Union *}
wenzelm@10249
    86
nipkow@11464
    87
theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
paulson@15072
    88
by (simp add: union_def Mempty_def)
wenzelm@10249
    89
wenzelm@10249
    90
theorem union_commute: "M + N = N + (M::'a multiset)"
paulson@15072
    91
by (simp add: union_def add_ac)
wenzelm@10249
    92
wenzelm@10249
    93
theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
paulson@15072
    94
by (simp add: union_def add_ac)
wenzelm@10249
    95
wenzelm@10249
    96
theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
wenzelm@10249
    97
  apply (rule union_commute [THEN trans])
wenzelm@10249
    98
  apply (rule union_assoc [THEN trans])
wenzelm@10249
    99
  apply (rule union_commute [THEN arg_cong])
wenzelm@10249
   100
  done
wenzelm@10249
   101
wenzelm@10249
   102
theorems union_ac = union_assoc union_commute union_lcomm
wenzelm@10249
   103
obua@14738
   104
instance multiset :: (type) comm_monoid_add
obua@14722
   105
proof 
obua@14722
   106
  fix a b c :: "'a multiset"
obua@14722
   107
  show "(a + b) + c = a + (b + c)" by (rule union_assoc)
obua@14722
   108
  show "a + b = b + a" by (rule union_commute)
obua@14722
   109
  show "0 + a = a" by simp
obua@14722
   110
qed
wenzelm@10277
   111
wenzelm@10249
   112
wenzelm@10249
   113
subsubsection {* Difference *}
wenzelm@10249
   114
nipkow@11464
   115
theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
paulson@15072
   116
by (simp add: Mempty_def diff_def)
wenzelm@10249
   117
wenzelm@10249
   118
theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
paulson@15072
   119
by (simp add: union_def diff_def)
wenzelm@10249
   120
wenzelm@10249
   121
wenzelm@10249
   122
subsubsection {* Count of elements *}
wenzelm@10249
   123
wenzelm@10249
   124
theorem count_empty [simp]: "count {#} a = 0"
paulson@15072
   125
by (simp add: count_def Mempty_def)
wenzelm@10249
   126
wenzelm@11701
   127
theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
paulson@15072
   128
by (simp add: count_def single_def)
wenzelm@10249
   129
wenzelm@10249
   130
theorem count_union [simp]: "count (M + N) a = count M a + count N a"
paulson@15072
   131
by (simp add: count_def union_def)
wenzelm@10249
   132
wenzelm@10249
   133
theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
paulson@15072
   134
by (simp add: count_def diff_def)
wenzelm@10249
   135
wenzelm@10249
   136
wenzelm@10249
   137
subsubsection {* Set of elements *}
wenzelm@10249
   138
wenzelm@10249
   139
theorem set_of_empty [simp]: "set_of {#} = {}"
paulson@15072
   140
by (simp add: set_of_def)
wenzelm@10249
   141
wenzelm@10249
   142
theorem set_of_single [simp]: "set_of {#b#} = {b}"
paulson@15072
   143
by (simp add: set_of_def)
wenzelm@10249
   144
nipkow@11464
   145
theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
paulson@15072
   146
by (auto simp add: set_of_def)
wenzelm@10249
   147
wenzelm@10249
   148
theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
paulson@15072
   149
by (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
wenzelm@10249
   150
nipkow@11464
   151
theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
paulson@15072
   152
by (auto simp add: set_of_def)
wenzelm@10249
   153
wenzelm@10249
   154
wenzelm@10249
   155
subsubsection {* Size *}
wenzelm@10249
   156
wenzelm@10249
   157
theorem size_empty [simp]: "size {#} = 0"
paulson@15072
   158
by (simp add: size_def)
wenzelm@10249
   159
wenzelm@10249
   160
theorem size_single [simp]: "size {#b#} = 1"
paulson@15072
   161
by (simp add: size_def)
wenzelm@10249
   162
wenzelm@10249
   163
theorem finite_set_of [iff]: "finite (set_of M)"
wenzelm@10249
   164
  apply (cut_tac x = M in Rep_multiset)
wenzelm@10249
   165
  apply (simp add: multiset_def set_of_def count_def)
wenzelm@10249
   166
  done
wenzelm@10249
   167
wenzelm@10249
   168
theorem setsum_count_Int:
nipkow@11464
   169
    "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
paulson@15072
   170
  apply (erule finite_induct, simp)
wenzelm@10249
   171
  apply (simp add: Int_insert_left set_of_def)
wenzelm@10249
   172
  done
wenzelm@10249
   173
wenzelm@10249
   174
theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
wenzelm@10249
   175
  apply (unfold size_def)
nipkow@11464
   176
  apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
wenzelm@10249
   177
   prefer 2
paulson@15072
   178
   apply (rule ext, simp)
nipkow@15402
   179
  apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
wenzelm@10249
   180
  apply (subst Int_commute)
wenzelm@10249
   181
  apply (simp (no_asm_simp) add: setsum_count_Int)
wenzelm@10249
   182
  done
wenzelm@10249
   183
wenzelm@10249
   184
theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
paulson@15072
   185
  apply (unfold size_def Mempty_def count_def, auto)
wenzelm@10249
   186
  apply (simp add: set_of_def count_def expand_fun_eq)
wenzelm@10249
   187
  done
wenzelm@10249
   188
nipkow@11464
   189
theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
wenzelm@10249
   190
  apply (unfold size_def)
paulson@15072
   191
  apply (drule setsum_SucD, auto)
wenzelm@10249
   192
  done
wenzelm@10249
   193
wenzelm@10249
   194
wenzelm@10249
   195
subsubsection {* Equality of multisets *}
wenzelm@10249
   196
nipkow@11464
   197
theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
paulson@15072
   198
by (simp add: count_def expand_fun_eq)
wenzelm@10249
   199
nipkow@11464
   200
theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
paulson@15072
   201
by (simp add: single_def Mempty_def expand_fun_eq)
wenzelm@10249
   202
wenzelm@10249
   203
theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
paulson@15072
   204
by (auto simp add: single_def expand_fun_eq)
wenzelm@10249
   205
nipkow@11464
   206
theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
paulson@15072
   207
by (auto simp add: union_def Mempty_def expand_fun_eq)
wenzelm@10249
   208
nipkow@11464
   209
theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
paulson@15072
   210
by (auto simp add: union_def Mempty_def expand_fun_eq)
wenzelm@10249
   211
wenzelm@10249
   212
theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
paulson@15072
   213
by (simp add: union_def expand_fun_eq)
wenzelm@10249
   214
wenzelm@10249
   215
theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
paulson@15072
   216
by (simp add: union_def expand_fun_eq)
wenzelm@10249
   217
wenzelm@10249
   218
theorem union_is_single:
nipkow@11464
   219
    "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
paulson@15072
   220
  apply (simp add: Mempty_def single_def union_def add_is_1 expand_fun_eq)
wenzelm@10249
   221
  apply blast
wenzelm@10249
   222
  done
wenzelm@10249
   223
wenzelm@10249
   224
theorem single_is_union:
paulson@15072
   225
     "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
wenzelm@10249
   226
  apply (unfold Mempty_def single_def union_def)
nipkow@11464
   227
  apply (simp add: add_is_1 one_is_add expand_fun_eq)
wenzelm@10249
   228
  apply (blast dest: sym)
wenzelm@10249
   229
  done
wenzelm@10249
   230
wenzelm@10249
   231
theorem add_eq_conv_diff:
wenzelm@10249
   232
  "(M + {#a#} = N + {#b#}) =
paulson@15072
   233
   (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
wenzelm@10249
   234
  apply (unfold single_def union_def diff_def)
wenzelm@10249
   235
  apply (simp (no_asm) add: expand_fun_eq)
paulson@15072
   236
  apply (rule conjI, force, safe, simp_all)
berghofe@13601
   237
  apply (simp add: eq_sym_conv)
wenzelm@10249
   238
  done
wenzelm@10249
   239
wenzelm@10249
   240
(*
wenzelm@10249
   241
val prems = Goal
wenzelm@10249
   242
 "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
nipkow@11464
   243
by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
wenzelm@10249
   244
     measure_induct 1);
paulson@15072
   245
by (Clarify_tac 1)
paulson@15072
   246
by (resolve_tac prems 1)
paulson@15072
   247
 by (assume_tac 1)
paulson@15072
   248
by (Clarify_tac 1)
paulson@15072
   249
by (subgoal_tac "finite G" 1)
wenzelm@10249
   250
 by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
paulson@15072
   251
by (etac allE 1)
paulson@15072
   252
by (etac impE 1)
paulson@15072
   253
 by (Blast_tac 2)
wenzelm@10249
   254
by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
wenzelm@10249
   255
no_qed();
wenzelm@10249
   256
val lemma = result();
wenzelm@10249
   257
wenzelm@10249
   258
val prems = Goal
wenzelm@10249
   259
 "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
wenzelm@10249
   260
by (rtac (lemma RS mp) 1);
wenzelm@10249
   261
by (REPEAT(ares_tac prems 1));
wenzelm@10249
   262
qed "finite_psubset_induct";
wenzelm@10249
   263
wenzelm@10249
   264
Better: use wf_finite_psubset in WF_Rel
wenzelm@10249
   265
*)
wenzelm@10249
   266
wenzelm@10249
   267
wenzelm@10249
   268
subsection {* Induction over multisets *}
wenzelm@10249
   269
wenzelm@10249
   270
lemma setsum_decr:
wenzelm@11701
   271
  "finite F ==> (0::nat) < f a ==>
paulson@15072
   272
    setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
paulson@15072
   273
  apply (erule finite_induct, auto)
paulson@15072
   274
  apply (drule_tac a = a in mk_disjoint_insert, auto)
wenzelm@10249
   275
  done
wenzelm@10249
   276
wenzelm@10313
   277
lemma rep_multiset_induct_aux:
wenzelm@11701
   278
  "P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
nipkow@11464
   279
    ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
wenzelm@10249
   280
proof -
wenzelm@11549
   281
  case rule_context
wenzelm@11549
   282
  note premises = this [unfolded multiset_def]
wenzelm@10249
   283
  show ?thesis
wenzelm@10249
   284
    apply (unfold multiset_def)
paulson@15072
   285
    apply (induct_tac n, simp, clarify)
nipkow@11464
   286
     apply (subgoal_tac "f = (\<lambda>a.0)")
wenzelm@10249
   287
      apply simp
wenzelm@11549
   288
      apply (rule premises)
paulson@15072
   289
     apply (rule ext, force, clarify)
paulson@15072
   290
    apply (frule setsum_SucD, clarify)
wenzelm@10249
   291
    apply (rename_tac a)
wenzelm@11701
   292
    apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
wenzelm@10249
   293
     prefer 2
wenzelm@10249
   294
     apply (rule finite_subset)
wenzelm@10249
   295
      prefer 2
wenzelm@10249
   296
      apply assumption
wenzelm@10249
   297
     apply simp
wenzelm@10249
   298
     apply blast
wenzelm@11701
   299
    apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
wenzelm@10249
   300
     prefer 2
wenzelm@10249
   301
     apply (rule ext)
wenzelm@10249
   302
     apply (simp (no_asm_simp))
paulson@15072
   303
     apply (erule ssubst, rule premises, blast)
paulson@15072
   304
    apply (erule allE, erule impE, erule_tac [2] mp, blast)
wenzelm@11701
   305
    apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow@11464
   306
    apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
wenzelm@10249
   307
     prefer 2
wenzelm@10249
   308
     apply blast
nipkow@11464
   309
    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
wenzelm@10249
   310
     prefer 2
wenzelm@10249
   311
     apply blast
nipkow@15316
   312
    apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
wenzelm@10249
   313
    done
wenzelm@10249
   314
qed
wenzelm@10249
   315
wenzelm@10313
   316
theorem rep_multiset_induct:
nipkow@11464
   317
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
wenzelm@11701
   318
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
paulson@15072
   319
  by (insert rep_multiset_induct_aux, blast)
wenzelm@10249
   320
wenzelm@10249
   321
theorem multiset_induct [induct type: multiset]:
wenzelm@10249
   322
  "P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
wenzelm@10249
   323
proof -
wenzelm@10249
   324
  note defns = union_def single_def Mempty_def
wenzelm@10249
   325
  assume prem1 [unfolded defns]: "P {#}"
wenzelm@10249
   326
  assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
wenzelm@10249
   327
  show ?thesis
wenzelm@10249
   328
    apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313
   329
    apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@10249
   330
     apply (rule prem1)
paulson@15072
   331
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
wenzelm@10249
   332
     prefer 2
wenzelm@10249
   333
     apply (simp add: expand_fun_eq)
wenzelm@10249
   334
    apply (erule ssubst)
wenzelm@10249
   335
    apply (erule Abs_multiset_inverse [THEN subst])
wenzelm@10249
   336
    apply (erule prem2 [simplified])
wenzelm@10249
   337
    done
wenzelm@10249
   338
qed
wenzelm@10249
   339
wenzelm@10249
   340
wenzelm@10249
   341
lemma MCollect_preserves_multiset:
nipkow@11464
   342
    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
wenzelm@10249
   343
  apply (simp add: multiset_def)
paulson@15072
   344
  apply (rule finite_subset, auto)
wenzelm@10249
   345
  done
wenzelm@10249
   346
wenzelm@10249
   347
theorem count_MCollect [simp]:
wenzelm@10249
   348
    "count {# x:M. P x #} a = (if P a then count M a else 0)"
paulson@15072
   349
  by (simp add: count_def MCollect_def MCollect_preserves_multiset)
wenzelm@10249
   350
nipkow@11464
   351
theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
paulson@15072
   352
by (auto simp add: set_of_def)
wenzelm@10249
   353
nipkow@11464
   354
theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
paulson@15072
   355
by (subst multiset_eq_conv_count_eq, auto)
wenzelm@10249
   356
wenzelm@10277
   357
declare Rep_multiset_inject [symmetric, simp del]
wenzelm@10249
   358
declare multiset_typedef [simp del]
wenzelm@10249
   359
wenzelm@10249
   360
theorem add_eq_conv_ex:
paulson@15072
   361
      "(M + {#a#} = N + {#b#}) =
paulson@15072
   362
       (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
paulson@15072
   363
  by (auto simp add: add_eq_conv_diff)
wenzelm@10249
   364
wenzelm@10249
   365
wenzelm@10249
   366
subsection {* Multiset orderings *}
wenzelm@10249
   367
wenzelm@10249
   368
subsubsection {* Well-foundedness *}
wenzelm@10249
   369
wenzelm@10249
   370
constdefs
nipkow@11464
   371
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10249
   372
  "mult1 r ==
nipkow@11464
   373
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
nipkow@11464
   374
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249
   375
nipkow@11464
   376
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
wenzelm@10392
   377
  "mult r == (mult1 r)\<^sup>+"
wenzelm@10249
   378
nipkow@11464
   379
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
wenzelm@10277
   380
  by (simp add: mult1_def)
wenzelm@10249
   381
nipkow@11464
   382
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
nipkow@11464
   383
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
nipkow@11464
   384
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
nipkow@11464
   385
  (concl is "?case1 (mult1 r) \<or> ?case2")
wenzelm@10249
   386
proof (unfold mult1_def)
nipkow@11464
   387
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   388
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
wenzelm@10249
   389
  let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249
   390
nipkow@11464
   391
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
nipkow@11464
   392
  hence "\<exists>a' M0' K.
nipkow@11464
   393
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
nipkow@11464
   394
  thus "?case1 \<or> ?case2"
wenzelm@10249
   395
  proof (elim exE conjE)
wenzelm@10249
   396
    fix a' M0' K
wenzelm@10249
   397
    assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249
   398
    assume "M0 + {#a#} = M0' + {#a'#}"
nipkow@11464
   399
    hence "M0 = M0' \<and> a = a' \<or>
nipkow@11464
   400
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249
   401
      by (simp only: add_eq_conv_ex)
wenzelm@10249
   402
    thus ?thesis
wenzelm@10249
   403
    proof (elim disjE conjE exE)
wenzelm@10249
   404
      assume "M0 = M0'" "a = a'"
nipkow@11464
   405
      with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@10249
   406
      hence ?case2 .. thus ?thesis ..
wenzelm@10249
   407
    next
wenzelm@10249
   408
      fix K'
wenzelm@10249
   409
      assume "M0' = K' + {#a#}"
wenzelm@10249
   410
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249
   411
wenzelm@10249
   412
      assume "M0 = K' + {#a'#}"
wenzelm@10249
   413
      with r have "?R (K' + K) M0" by blast
wenzelm@10249
   414
      with n have ?case1 by simp thus ?thesis ..
wenzelm@10249
   415
    qed
wenzelm@10249
   416
  qed
wenzelm@10249
   417
qed
wenzelm@10249
   418
nipkow@11464
   419
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249
   420
proof
wenzelm@10249
   421
  let ?R = "mult1 r"
wenzelm@10249
   422
  let ?W = "acc ?R"
wenzelm@10249
   423
  {
wenzelm@10249
   424
    fix M M0 a
nipkow@11464
   425
    assume M0: "M0 \<in> ?W"
wenzelm@12399
   426
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   427
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
nipkow@11464
   428
    have "M0 + {#a#} \<in> ?W"
wenzelm@10249
   429
    proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249
   430
      fix N
nipkow@11464
   431
      assume "(N, M0 + {#a#}) \<in> ?R"
nipkow@11464
   432
      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
nipkow@11464
   433
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249
   434
        by (rule less_add)
nipkow@11464
   435
      thus "N \<in> ?W"
wenzelm@10249
   436
      proof (elim exE disjE conjE)
nipkow@11464
   437
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
nipkow@11464
   438
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
nipkow@11464
   439
        hence "M + {#a#} \<in> ?W" ..
nipkow@11464
   440
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   441
      next
wenzelm@10249
   442
        fix K
wenzelm@10249
   443
        assume N: "N = M0 + K"
nipkow@11464
   444
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464
   445
        have "?this --> M0 + K \<in> ?W" (is "?P K")
wenzelm@10249
   446
        proof (induct K)
nipkow@11464
   447
          from M0 have "M0 + {#} \<in> ?W" by simp
wenzelm@10249
   448
          thus "?P {#}" ..
wenzelm@10249
   449
wenzelm@10249
   450
          fix K x assume hyp: "?P K"
wenzelm@10249
   451
          show "?P (K + {#x#})"
wenzelm@10249
   452
          proof
nipkow@11464
   453
            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
nipkow@11464
   454
            hence "(x, a) \<in> r" by simp
nipkow@11464
   455
            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
wenzelm@10249
   456
nipkow@11464
   457
            from a hyp have "M0 + K \<in> ?W" by simp
nipkow@11464
   458
            with b have "(M0 + K) + {#x#} \<in> ?W" ..
nipkow@11464
   459
            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249
   460
          qed
wenzelm@10249
   461
        qed
nipkow@11464
   462
        hence "M0 + K \<in> ?W" ..
nipkow@11464
   463
        thus "N \<in> ?W" by (simp only: N)
wenzelm@10249
   464
      qed
wenzelm@10249
   465
    qed
wenzelm@10249
   466
  } note tedious_reasoning = this
wenzelm@10249
   467
wenzelm@10249
   468
  assume wf: "wf r"
wenzelm@10249
   469
  fix M
nipkow@11464
   470
  show "M \<in> ?W"
wenzelm@10249
   471
  proof (induct M)
nipkow@11464
   472
    show "{#} \<in> ?W"
wenzelm@10249
   473
    proof (rule accI)
nipkow@11464
   474
      fix b assume "(b, {#}) \<in> ?R"
nipkow@11464
   475
      with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249
   476
    qed
wenzelm@10249
   477
nipkow@11464
   478
    fix M a assume "M \<in> ?W"
nipkow@11464
   479
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   480
    proof induct
wenzelm@10249
   481
      fix a
wenzelm@12399
   482
      assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
nipkow@11464
   483
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249
   484
      proof
nipkow@11464
   485
        fix M assume "M \<in> ?W"
nipkow@11464
   486
        thus "M + {#a#} \<in> ?W"
wenzelm@10249
   487
          by (rule acc_induct) (rule tedious_reasoning)
wenzelm@10249
   488
      qed
wenzelm@10249
   489
    qed
nipkow@11464
   490
    thus "M + {#a#} \<in> ?W" ..
wenzelm@10249
   491
  qed
wenzelm@10249
   492
qed
wenzelm@10249
   493
wenzelm@10249
   494
theorem wf_mult1: "wf r ==> wf (mult1 r)"
wenzelm@10249
   495
  by (rule acc_wfI, rule all_accessible)
wenzelm@10249
   496
wenzelm@10249
   497
theorem wf_mult: "wf r ==> wf (mult r)"
wenzelm@10249
   498
  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
wenzelm@10249
   499
wenzelm@10249
   500
wenzelm@10249
   501
subsubsection {* Closure-free presentation *}
wenzelm@10249
   502
wenzelm@10249
   503
(*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249
   504
wenzelm@10249
   505
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
paulson@15072
   506
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   507
wenzelm@10249
   508
text {* One direction. *}
wenzelm@10249
   509
wenzelm@10249
   510
lemma mult_implies_one_step:
nipkow@11464
   511
  "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464
   512
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
nipkow@11464
   513
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
wenzelm@10249
   514
  apply (unfold mult_def mult1_def set_of_def)
paulson@15072
   515
  apply (erule converse_trancl_induct, clarify)
paulson@15072
   516
   apply (rule_tac x = M0 in exI, simp, clarify)
wenzelm@10249
   517
  apply (case_tac "a :# K")
wenzelm@10249
   518
   apply (rule_tac x = I in exI)
wenzelm@10249
   519
   apply (simp (no_asm))
wenzelm@10249
   520
   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
wenzelm@10249
   521
   apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@11464
   522
   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
wenzelm@10249
   523
   apply (simp add: diff_union_single_conv)
wenzelm@10249
   524
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   525
   apply blast
wenzelm@10249
   526
  apply (subgoal_tac "a :# I")
wenzelm@10249
   527
   apply (rule_tac x = "I - {#a#}" in exI)
wenzelm@10249
   528
   apply (rule_tac x = "J + {#a#}" in exI)
wenzelm@10249
   529
   apply (rule_tac x = "K + Ka" in exI)
wenzelm@10249
   530
   apply (rule conjI)
wenzelm@10249
   531
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   532
   apply (rule conjI)
paulson@15072
   533
    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
wenzelm@10249
   534
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
wenzelm@10249
   535
   apply (simp (no_asm_use) add: trans_def)
wenzelm@10249
   536
   apply blast
wenzelm@10277
   537
  apply (subgoal_tac "a :# (M0 + {#a#})")
wenzelm@10249
   538
   apply simp
wenzelm@10249
   539
  apply (simp (no_asm))
wenzelm@10249
   540
  done
wenzelm@10249
   541
wenzelm@10249
   542
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
paulson@15072
   543
by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249
   544
nipkow@11464
   545
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
wenzelm@10249
   546
  apply (erule size_eq_Suc_imp_elem [THEN exE])
paulson@15072
   547
  apply (drule elem_imp_eq_diff_union, auto)
wenzelm@10249
   548
  done
wenzelm@10249
   549
wenzelm@10249
   550
lemma one_step_implies_mult_aux:
wenzelm@10249
   551
  "trans r ==>
nipkow@11464
   552
    \<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))
nipkow@11464
   553
      --> (I + K, I + J) \<in> mult r"
paulson@15072
   554
  apply (induct_tac n, auto)
paulson@15072
   555
  apply (frule size_eq_Suc_imp_eq_union, clarify)
paulson@15072
   556
  apply (rename_tac "J'", simp)
paulson@15072
   557
  apply (erule notE, auto)
wenzelm@10249
   558
  apply (case_tac "J' = {#}")
wenzelm@10249
   559
   apply (simp add: mult_def)
wenzelm@10249
   560
   apply (rule r_into_trancl)
paulson@15072
   561
   apply (simp add: mult1_def set_of_def, blast)
nipkow@11464
   562
  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@11464
   563
  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@11464
   564
  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
wenzelm@10249
   565
  apply (erule ssubst)
paulson@15072
   566
  apply (simp add: Ball_def, auto)
wenzelm@10249
   567
  apply (subgoal_tac
nipkow@11464
   568
    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
nipkow@11464
   569
      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
wenzelm@10249
   570
   prefer 2
wenzelm@10249
   571
   apply force
wenzelm@10249
   572
  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
wenzelm@10249
   573
  apply (erule trancl_trans)
wenzelm@10249
   574
  apply (rule r_into_trancl)
wenzelm@10249
   575
  apply (simp add: mult1_def set_of_def)
wenzelm@10249
   576
  apply (rule_tac x = a in exI)
wenzelm@10249
   577
  apply (rule_tac x = "I + J'" in exI)
wenzelm@10249
   578
  apply (simp add: union_ac)
wenzelm@10249
   579
  done
wenzelm@10249
   580
wenzelm@10249
   581
theorem one_step_implies_mult:
nipkow@11464
   582
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
nipkow@11464
   583
    ==> (I + K, I + J) \<in> mult r"
paulson@15072
   584
  apply (insert one_step_implies_mult_aux, blast)
wenzelm@10249
   585
  done
wenzelm@10249
   586
wenzelm@10249
   587
wenzelm@10249
   588
subsubsection {* Partial-order properties *}
wenzelm@10249
   589
wenzelm@12338
   590
instance multiset :: (type) ord ..
wenzelm@10249
   591
wenzelm@10249
   592
defs (overloaded)
nipkow@11464
   593
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
nipkow@11464
   594
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249
   595
wenzelm@10249
   596
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
wenzelm@10249
   597
  apply (unfold trans_def)
wenzelm@10249
   598
  apply (blast intro: order_less_trans)
wenzelm@10249
   599
  done
wenzelm@10249
   600
wenzelm@10249
   601
text {*
wenzelm@10249
   602
 \medskip Irreflexivity.
wenzelm@10249
   603
*}
wenzelm@10249
   604
wenzelm@10249
   605
lemma mult_irrefl_aux:
nipkow@11464
   606
    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
wenzelm@10249
   607
  apply (erule finite_induct)
wenzelm@10249
   608
   apply (auto intro: order_less_trans)
wenzelm@10249
   609
  done
wenzelm@10249
   610
nipkow@11464
   611
theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
paulson@15072
   612
  apply (unfold less_multiset_def, auto)
paulson@15072
   613
  apply (drule trans_base_order [THEN mult_implies_one_step], auto)
wenzelm@10249
   614
  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
wenzelm@10249
   615
  apply (simp add: set_of_eq_empty_iff)
wenzelm@10249
   616
  done
wenzelm@10249
   617
wenzelm@10249
   618
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
paulson@15072
   619
by (insert mult_less_not_refl, fast)
wenzelm@10249
   620
wenzelm@10249
   621
wenzelm@10249
   622
text {* Transitivity. *}
wenzelm@10249
   623
wenzelm@10249
   624
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
wenzelm@10249
   625
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   626
  apply (blast intro: trancl_trans)
wenzelm@10249
   627
  done
wenzelm@10249
   628
wenzelm@10249
   629
text {* Asymmetry. *}
wenzelm@10249
   630
nipkow@11464
   631
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
wenzelm@10249
   632
  apply auto
wenzelm@10249
   633
  apply (rule mult_less_not_refl [THEN notE])
paulson@15072
   634
  apply (erule mult_less_trans, assumption)
wenzelm@10249
   635
  done
wenzelm@10249
   636
wenzelm@10249
   637
theorem mult_less_asym:
nipkow@11464
   638
    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
paulson@15072
   639
  by (insert mult_less_not_sym, blast)
wenzelm@10249
   640
wenzelm@10249
   641
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
paulson@15072
   642
by (unfold le_multiset_def, auto)
wenzelm@10249
   643
wenzelm@10249
   644
text {* Anti-symmetry. *}
wenzelm@10249
   645
wenzelm@10249
   646
theorem mult_le_antisym:
wenzelm@10249
   647
    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
wenzelm@10249
   648
  apply (unfold le_multiset_def)
wenzelm@10249
   649
  apply (blast dest: mult_less_not_sym)
wenzelm@10249
   650
  done
wenzelm@10249
   651
wenzelm@10249
   652
text {* Transitivity. *}
wenzelm@10249
   653
wenzelm@10249
   654
theorem mult_le_trans:
wenzelm@10249
   655
    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
wenzelm@10249
   656
  apply (unfold le_multiset_def)
wenzelm@10249
   657
  apply (blast intro: mult_less_trans)
wenzelm@10249
   658
  done
wenzelm@10249
   659
wenzelm@11655
   660
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
paulson@15072
   661
by (unfold le_multiset_def, auto)
wenzelm@10249
   662
wenzelm@10277
   663
text {* Partial order. *}
wenzelm@10277
   664
wenzelm@10277
   665
instance multiset :: (order) order
wenzelm@10277
   666
  apply intro_classes
wenzelm@10277
   667
     apply (rule mult_le_refl)
paulson@15072
   668
    apply (erule mult_le_trans, assumption)
paulson@15072
   669
   apply (erule mult_le_antisym, assumption)
wenzelm@10277
   670
  apply (rule mult_less_le)
wenzelm@10277
   671
  done
wenzelm@10277
   672
wenzelm@10249
   673
wenzelm@10249
   674
subsubsection {* Monotonicity of multiset union *}
wenzelm@10249
   675
wenzelm@10249
   676
theorem mult1_union:
nipkow@11464
   677
    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
paulson@15072
   678
  apply (unfold mult1_def, auto)
wenzelm@10249
   679
  apply (rule_tac x = a in exI)
wenzelm@10249
   680
  apply (rule_tac x = "C + M0" in exI)
wenzelm@10249
   681
  apply (simp add: union_assoc)
wenzelm@10249
   682
  done
wenzelm@10249
   683
wenzelm@10249
   684
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
wenzelm@10249
   685
  apply (unfold less_multiset_def mult_def)
wenzelm@10249
   686
  apply (erule trancl_induct)
wenzelm@10249
   687
   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
wenzelm@10249
   688
  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
wenzelm@10249
   689
  done
wenzelm@10249
   690
wenzelm@10249
   691
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
wenzelm@10249
   692
  apply (subst union_commute [of B C])
wenzelm@10249
   693
  apply (subst union_commute [of D C])
wenzelm@10249
   694
  apply (erule union_less_mono2)
wenzelm@10249
   695
  done
wenzelm@10249
   696
wenzelm@10249
   697
theorem union_less_mono:
wenzelm@10249
   698
    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
wenzelm@10249
   699
  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249
   700
  done
wenzelm@10249
   701
wenzelm@10249
   702
theorem union_le_mono:
wenzelm@10249
   703
    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
wenzelm@10249
   704
  apply (unfold le_multiset_def)
wenzelm@10249
   705
  apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249
   706
  done
wenzelm@10249
   707
wenzelm@10249
   708
theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
wenzelm@10249
   709
  apply (unfold le_multiset_def less_multiset_def)
wenzelm@10249
   710
  apply (case_tac "M = {#}")
wenzelm@10249
   711
   prefer 2
nipkow@11464
   712
   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
wenzelm@10249
   713
    prefer 2
wenzelm@10249
   714
    apply (rule one_step_implies_mult)
paulson@15072
   715
      apply (simp only: trans_def, auto)
wenzelm@10249
   716
  done
wenzelm@10249
   717
wenzelm@10249
   718
theorem union_upper1: "A <= A + (B::'a::order multiset)"
paulson@15072
   719
proof -
paulson@15072
   720
  have "A + {#} <= A + B" by (blast intro: union_le_mono) 
paulson@15072
   721
  thus ?thesis by simp
paulson@15072
   722
qed
paulson@15072
   723
paulson@15072
   724
theorem union_upper2: "B <= A + (B::'a::order multiset)"
paulson@15072
   725
by (subst union_commute, rule union_upper1)
paulson@15072
   726
paulson@15072
   727
paulson@15072
   728
subsection {* Link with lists *} 
paulson@15072
   729
paulson@15072
   730
consts 
paulson@15072
   731
  multiset_of :: "'a list \<Rightarrow> 'a multiset"
paulson@15072
   732
primrec
paulson@15072
   733
  "multiset_of [] = {#}"
paulson@15072
   734
  "multiset_of (a # x) = multiset_of x + {# a #}"
paulson@15072
   735
paulson@15072
   736
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
paulson@15072
   737
  by (induct_tac x, auto) 
paulson@15072
   738
paulson@15072
   739
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
paulson@15072
   740
  by (induct_tac x, auto)
paulson@15072
   741
paulson@15072
   742
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
kleing@15867
   743
  by (induct_tac x, auto) 
kleing@15867
   744
kleing@15867
   745
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
kleing@15867
   746
  by (induct xs) auto
paulson@15072
   747
kleing@15630
   748
lemma multiset_of_append[simp]: 
paulson@15072
   749
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
paulson@15072
   750
  by (rule_tac x=ys in spec, induct_tac xs, auto simp: union_ac) 
paulson@15072
   751
paulson@15072
   752
lemma surj_multiset_of: "surj multiset_of"
paulson@15072
   753
  apply (unfold surj_def, rule allI) 
paulson@15072
   754
  apply (rule_tac M=y in multiset_induct, auto) 
paulson@15072
   755
  apply (rule_tac x = "x # xa" in exI, auto) 
wenzelm@10249
   756
  done
wenzelm@10249
   757
paulson@15072
   758
lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}"
paulson@15072
   759
  by (induct_tac x, auto)  
paulson@15072
   760
paulson@15072
   761
lemma distinct_count_atmost_1: 
paulson@15072
   762
   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
paulson@15072
   763
   apply ( induct_tac x, simp, rule iffI, simp_all)
paulson@15072
   764
   apply (rule conjI)  
paulson@15072
   765
   apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) 
paulson@15072
   766
   apply (erule_tac x=a in allE, simp, clarify)
paulson@15072
   767
   apply (erule_tac x=aa in allE, simp) 
paulson@15072
   768
   done
paulson@15072
   769
kleing@15867
   770
lemma multiset_of_eq_setD: 
kleing@15867
   771
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
kleing@15867
   772
  by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
kleing@15867
   773
paulson@15072
   774
lemma set_eq_iff_multiset_of_eq_distinct: 
paulson@15072
   775
  "\<lbrakk>distinct x; distinct y\<rbrakk> 
paulson@15072
   776
   \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
paulson@15072
   777
  by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) 
paulson@15072
   778
paulson@15072
   779
lemma set_eq_iff_multiset_of_remdups_eq: 
paulson@15072
   780
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
paulson@15072
   781
  apply (rule iffI) 
paulson@15072
   782
  apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) 
paulson@15072
   783
  apply (drule distinct_remdups[THEN distinct_remdups 
paulson@15072
   784
                      [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]]) 
paulson@15072
   785
  apply simp
wenzelm@10249
   786
  done
wenzelm@10249
   787
kleing@15630
   788
lemma multiset_of_compl_union[simp]:
kleing@15630
   789
 "multiset_of [x\<in>xs. P x] + multiset_of [x\<in>xs. \<not>P x] = multiset_of xs"
kleing@15630
   790
  by (induct xs) (auto simp: union_ac)
paulson@15072
   791
kleing@15867
   792
lemma count_filter: 
kleing@15867
   793
  "count (multiset_of xs) x = length [y \<in> xs. y = x]"
kleing@15867
   794
  by (induct xs, auto)
kleing@15867
   795
kleing@15867
   796
paulson@15072
   797
subsection {* Pointwise ordering induced by count *}
paulson@15072
   798
paulson@15072
   799
consts 
paulson@15072
   800
  mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool"
paulson@15072
   801
paulson@15072
   802
syntax 
paulson@15072
   803
  "_mset_le" :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"   ("_ \<le># _"  [50,51] 50) 
paulson@15072
   804
translations 
paulson@15072
   805
  "x \<le># y" == "mset_le x y"
paulson@15072
   806
paulson@15072
   807
defs 
paulson@15072
   808
  mset_le_def:   "xs \<le># ys  == (! a. count xs a \<le> count ys a)"
paulson@15072
   809
paulson@15072
   810
lemma mset_le_refl[simp]: "xs \<le># xs"
paulson@15072
   811
  by (unfold mset_le_def, auto) 
paulson@15072
   812
paulson@15072
   813
lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs"
paulson@15072
   814
  by (unfold mset_le_def, fast intro: order_trans) 
paulson@15072
   815
paulson@15072
   816
lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys"
paulson@15072
   817
  apply (unfold mset_le_def) 
paulson@15072
   818
  apply (rule multiset_eq_conv_count_eq[THEN iffD2]) 
paulson@15072
   819
  apply (blast intro: order_antisym)
paulson@15072
   820
  done
paulson@15072
   821
paulson@15072
   822
lemma mset_le_exists_conv: 
paulson@15072
   823
  "(xs \<le># ys) = (? zs. ys = xs + zs)"
paulson@15072
   824
  apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI) 
paulson@15072
   825
  apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
paulson@15072
   826
  done
paulson@15072
   827
paulson@15072
   828
lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)"
paulson@15072
   829
  by (unfold mset_le_def, auto) 
paulson@15072
   830
paulson@15072
   831
lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)"
paulson@15072
   832
  by (unfold mset_le_def, auto) 
paulson@15072
   833
paulson@15072
   834
lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws" 
paulson@15072
   835
  apply (unfold mset_le_def, auto) 
paulson@15072
   836
  apply (erule_tac x=a in allE)+
paulson@15072
   837
  apply auto
paulson@15072
   838
  done
paulson@15072
   839
paulson@15072
   840
lemma mset_le_add_left[simp]: "xs \<le># xs + ys"
paulson@15072
   841
  by (unfold mset_le_def, auto) 
paulson@15072
   842
paulson@15072
   843
lemma mset_le_add_right[simp]: "ys \<le># xs + ys"
paulson@15072
   844
  by (unfold mset_le_def, auto)
paulson@15072
   845
paulson@15072
   846
lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x"
paulson@15072
   847
  by (induct_tac x, auto, rule mset_le_trans, auto)
paulson@15072
   848
wenzelm@10249
   849
end