src/HOL/Library/Multiset.thy
author wenzelm
Sat, 08 Apr 2006 22:51:06 +0200
changeset 19363 667b5ea637dd
parent 19086 1b3780be6cc2
child 19564 d3e2f532459a
permissions -rw-r--r--
refined 'abbreviation';
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     1
(*  Title:      HOL/Library/Multiset.thy
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     2
    ID:         $Id$
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
     3
    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     4
*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     5
14706
71590b7733b7 tuned document;
wenzelm
parents: 14691
diff changeset
     6
header {* Multisets *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     7
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15072
diff changeset
     8
theory Multiset
15140
322485b816ac import -> imports
nipkow
parents: 15131
diff changeset
     9
imports Accessible_Part
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15072
diff changeset
    10
begin
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    11
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    12
subsection {* The type of multisets *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    13
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    14
typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    15
proof
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    16
  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    17
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    18
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    19
lemmas multiset_typedef [simp] =
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
    20
    Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
    21
  and [simp] = Rep_multiset_inject [symmetric]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    22
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    23
definition
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    24
  Mempty :: "'a multiset"    ("{#}")
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    25
  "{#} = Abs_multiset (\<lambda>a. 0)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    26
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    27
  single :: "'a => 'a multiset"    ("{#_#}")
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    28
  "{#a#} = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    29
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    30
  count :: "'a multiset => 'a => nat"
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    31
  "count = Rep_multiset"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    32
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    33
  MCollect :: "'a multiset => ('a => bool) => 'a multiset"
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    34
  "MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    35
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19086
diff changeset
    36
abbreviation
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    37
  Melem :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19086
diff changeset
    38
  "a :# M == 0 < count M a"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    39
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    40
syntax
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    41
  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    42
translations
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    43
  "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    44
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    45
definition
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    46
  set_of :: "'a multiset => 'a set"
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    47
  "set_of M = {x. x :# M}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    48
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14445
diff changeset
    49
instance multiset :: (type) "{plus, minus, zero}" ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    50
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    51
defs (overloaded)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    52
  union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    53
  diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
    54
  Zero_multiset_def [simp]: "0 == {#}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    55
  size_def: "size M == setsum (count M) (set_of M)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    56
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    57
definition
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    58
  multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70)
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    59
  "multiset_inter A B = A - (A - B)"
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
    60
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    61
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    62
text {*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    63
 \medskip Preservation of the representing set @{term multiset}.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    64
*}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    65
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    66
lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    67
  by (simp add: multiset_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    68
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
    69
lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    70
  by (simp add: multiset_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    71
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    72
lemma union_preserves_multiset [simp]:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    73
    "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    74
  apply (simp add: multiset_def)
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    75
  apply (drule (1) finite_UnI)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    76
  apply (simp del: finite_Un add: Un_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    77
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    78
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    79
lemma diff_preserves_multiset [simp]:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    80
    "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    81
  apply (simp add: multiset_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    82
  apply (rule finite_subset)
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    83
   apply auto
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    84
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    85
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    86
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    87
subsection {* Algebraic properties of multisets *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    88
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    89
subsubsection {* Union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    90
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    91
lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    92
  by (simp add: union_def Mempty_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    93
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    94
lemma union_commute: "M + N = N + (M::'a multiset)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    95
  by (simp add: union_def add_ac)
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    96
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    97
lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
    98
  by (simp add: union_def add_ac)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    99
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   100
lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   101
proof -
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   102
  have "M + (N + K) = (N + K) + M"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   103
    by (rule union_commute)
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   104
  also have "\<dots> = N + (K + M)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   105
    by (rule union_assoc)
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   106
  also have "K + M = M + K"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   107
    by (rule union_commute)
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   108
  finally show ?thesis .
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   109
qed
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   110
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   111
lemmas union_ac = union_assoc union_commute union_lcomm
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   112
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14722
diff changeset
   113
instance multiset :: (type) comm_monoid_add
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   114
proof
14722
8e739a6eaf11 replaced apply-style proof for instance Multiset :: plus_ac0 by recommended Isar proof style
obua
parents: 14706
diff changeset
   115
  fix a b c :: "'a multiset"
8e739a6eaf11 replaced apply-style proof for instance Multiset :: plus_ac0 by recommended Isar proof style
obua
parents: 14706
diff changeset
   116
  show "(a + b) + c = a + (b + c)" by (rule union_assoc)
8e739a6eaf11 replaced apply-style proof for instance Multiset :: plus_ac0 by recommended Isar proof style
obua
parents: 14706
diff changeset
   117
  show "a + b = b + a" by (rule union_commute)
8e739a6eaf11 replaced apply-style proof for instance Multiset :: plus_ac0 by recommended Isar proof style
obua
parents: 14706
diff changeset
   118
  show "0 + a = a" by simp
8e739a6eaf11 replaced apply-style proof for instance Multiset :: plus_ac0 by recommended Isar proof style
obua
parents: 14706
diff changeset
   119
qed
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   120
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   121
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   122
subsubsection {* Difference *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   123
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   124
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   125
  by (simp add: Mempty_def diff_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   126
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   127
lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   128
  by (simp add: union_def diff_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   129
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   130
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   131
subsubsection {* Count of elements *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   132
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   133
lemma count_empty [simp]: "count {#} a = 0"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   134
  by (simp add: count_def Mempty_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   135
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   136
lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   137
  by (simp add: count_def single_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   138
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   139
lemma count_union [simp]: "count (M + N) a = count M a + count N a"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   140
  by (simp add: count_def union_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   141
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   142
lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   143
  by (simp add: count_def diff_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   144
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   145
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   146
subsubsection {* Set of elements *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   147
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   148
lemma set_of_empty [simp]: "set_of {#} = {}"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   149
  by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   150
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   151
lemma set_of_single [simp]: "set_of {#b#} = {b}"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   152
  by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   153
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   154
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   155
  by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   156
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   157
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   158
  by (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   159
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   160
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   161
  by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   162
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   163
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   164
subsubsection {* Size *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   165
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   166
lemma size_empty [simp]: "size {#} = 0"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   167
  by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   168
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   169
lemma size_single [simp]: "size {#b#} = 1"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   170
  by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   171
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   172
lemma finite_set_of [iff]: "finite (set_of M)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   173
  using Rep_multiset [of M]
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   174
  by (simp add: multiset_def set_of_def count_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   175
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   176
lemma setsum_count_Int:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   177
    "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   178
  apply (induct rule: finite_induct)
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   179
   apply simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   180
  apply (simp add: Int_insert_left set_of_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   181
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   182
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   183
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   184
  apply (unfold size_def)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   185
  apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   186
   prefer 2
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   187
   apply (rule ext, simp)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15316
diff changeset
   188
  apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   189
  apply (subst Int_commute)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   190
  apply (simp (no_asm_simp) add: setsum_count_Int)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   191
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   192
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   193
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   194
  apply (unfold size_def Mempty_def count_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   195
  apply (simp add: set_of_def count_def expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   196
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   197
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   198
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   199
  apply (unfold size_def)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   200
  apply (drule setsum_SucD, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   201
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   202
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   203
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   204
subsubsection {* Equality of multisets *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   205
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   206
lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   207
  by (simp add: count_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   208
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   209
lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   210
  by (simp add: single_def Mempty_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   211
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   212
lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   213
  by (auto simp add: single_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   214
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   215
lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   216
  by (auto simp add: union_def Mempty_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   217
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   218
lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   219
  by (auto simp add: union_def Mempty_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   220
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   221
lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   222
  by (simp add: union_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   223
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   224
lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   225
  by (simp add: union_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   226
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   227
lemma union_is_single:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   228
    "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   229
  apply (simp add: Mempty_def single_def union_def add_is_1 expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   230
  apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   231
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   232
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   233
lemma single_is_union:
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   234
     "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   235
  apply (unfold Mempty_def single_def union_def)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   236
  apply (simp add: add_is_1 one_is_add expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   237
  apply (blast dest: sym)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   238
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   239
17778
93d7e524417a changes due to new neq_simproc in simpdata.ML
nipkow
parents: 17200
diff changeset
   240
ML"reset use_neq_simproc"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   241
lemma add_eq_conv_diff:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   242
  "(M + {#a#} = N + {#b#}) =
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   243
   (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   244
  apply (unfold single_def union_def diff_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   245
  apply (simp (no_asm) add: expand_fun_eq)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   246
  apply (rule conjI, force, safe, simp_all)
13601
fd3e3d6b37b2 Adapted to new simplifier.
berghofe
parents: 13596
diff changeset
   247
  apply (simp add: eq_sym_conv)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   248
  done
17778
93d7e524417a changes due to new neq_simproc in simpdata.ML
nipkow
parents: 17200
diff changeset
   249
ML"set use_neq_simproc"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   250
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   251
declare Rep_multiset_inject [symmetric, simp del]
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   252
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   253
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   254
subsubsection {* Intersection *}
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   255
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   256
lemma multiset_inter_count:
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   257
    "count (A #\<inter> B) x = min (count A x) (count B x)"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   258
  by (simp add: multiset_inter_def min_def)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   259
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   260
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   261
  by (simp add: multiset_eq_conv_count_eq multiset_inter_count
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   262
    min_max.below_inf.inf_commute)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   263
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   264
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   265
  by (simp add: multiset_eq_conv_count_eq multiset_inter_count
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   266
    min_max.below_inf.inf_assoc)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   267
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   268
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   269
  by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   270
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   271
lemmas multiset_inter_ac =
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   272
  multiset_inter_commute
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   273
  multiset_inter_assoc
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   274
  multiset_inter_left_commute
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   275
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   276
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   277
  apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   278
    split: split_if_asm)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   279
  apply clarsimp
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   280
  apply (erule_tac x = a in allE)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   281
  apply auto
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   282
  done
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   283
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   284
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   285
subsection {* Induction over multisets *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   286
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   287
lemma setsum_decr:
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   288
  "finite F ==> (0::nat) < f a ==>
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   289
    setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   290
  apply (induct rule: finite_induct)
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   291
   apply auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   292
  apply (drule_tac a = a in mk_disjoint_insert, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   293
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   294
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   295
lemma rep_multiset_induct_aux:
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   296
  assumes 1: "P (\<lambda>a. (0::nat))"
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   297
    and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   298
  shows "\<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   299
  apply (unfold multiset_def)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   300
  apply (induct_tac n, simp, clarify)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   301
   apply (subgoal_tac "f = (\<lambda>a.0)")
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   302
    apply simp
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   303
    apply (rule 1)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   304
   apply (rule ext, force, clarify)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   305
  apply (frule setsum_SucD, clarify)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   306
  apply (rename_tac a)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   307
  apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   308
   prefer 2
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   309
   apply (rule finite_subset)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   310
    prefer 2
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   311
    apply assumption
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   312
   apply simp
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   313
   apply blast
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   314
  apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   315
   prefer 2
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   316
   apply (rule ext)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   317
   apply (simp (no_asm_simp))
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   318
   apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   319
  apply (erule allE, erule impE, erule_tac [2] mp, blast)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   320
  apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   321
  apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   322
   prefer 2
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   323
   apply blast
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   324
  apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   325
   prefer 2
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   326
   apply blast
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   327
  apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   328
  done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   329
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   330
theorem rep_multiset_induct:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   331
  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   332
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   333
  using rep_multiset_induct_aux by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   334
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   335
theorem multiset_induct [case_names empty add, induct type: multiset]:
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   336
  assumes empty: "P {#}"
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   337
    and add: "!!M x. P M ==> P (M + {#x#})"
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   338
  shows "P M"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   339
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   340
  note defns = union_def single_def Mempty_def
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   341
  show ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   342
    apply (rule Rep_multiset_inverse [THEN subst])
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   343
    apply (rule Rep_multiset [THEN rep_multiset_induct])
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   344
     apply (rule empty [unfolded defns])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   345
    apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   346
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   347
     apply (simp add: expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   348
    apply (erule ssubst)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   349
    apply (erule Abs_multiset_inverse [THEN subst])
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   350
    apply (erule add [unfolded defns, simplified])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   351
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   352
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   353
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   354
lemma MCollect_preserves_multiset:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   355
    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   356
  apply (simp add: multiset_def)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   357
  apply (rule finite_subset, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   358
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   359
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   360
lemma count_MCollect [simp]:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   361
    "count {# x:M. P x #} a = (if P a then count M a else 0)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   362
  by (simp add: count_def MCollect_def MCollect_preserves_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   363
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   364
lemma set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   365
  by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   366
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   367
lemma multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   368
  by (subst multiset_eq_conv_count_eq, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   369
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   370
lemma add_eq_conv_ex:
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   371
  "(M + {#a#} = N + {#b#}) =
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   372
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   373
  by (auto simp add: add_eq_conv_diff)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   374
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   375
declare multiset_typedef [simp del]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   376
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   377
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   378
subsection {* Multiset orderings *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   379
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   380
subsubsection {* Well-foundedness *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   381
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   382
definition
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   383
  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   384
  "mult1 r =
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   385
    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   386
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   387
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   388
  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   389
  "mult r = (mult1 r)\<^sup>+"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   390
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   391
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   392
  by (simp add: mult1_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   393
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   394
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   395
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   396
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   397
  (concl is "?case1 (mult1 r) \<or> ?case2")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   398
proof (unfold mult1_def)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   399
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   400
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   401
  let ?case1 = "?case1 {(N, M). ?R N M}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   402
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   403
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   404
  then have "\<exists>a' M0' K.
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   405
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   406
  then show "?case1 \<or> ?case2"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   407
  proof (elim exE conjE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   408
    fix a' M0' K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   409
    assume N: "N = M0' + K" and r: "?r K a'"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   410
    assume "M0 + {#a#} = M0' + {#a'#}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   411
    then have "M0 = M0' \<and> a = a' \<or>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   412
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   413
      by (simp only: add_eq_conv_ex)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   414
    then show ?thesis
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   415
    proof (elim disjE conjE exE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   416
      assume "M0 = M0'" "a = a'"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   417
      with N r have "?r K a \<and> N = M0 + K" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   418
      then have ?case2 .. then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   419
    next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   420
      fix K'
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   421
      assume "M0' = K' + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   422
      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   423
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   424
      assume "M0 = K' + {#a'#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   425
      with r have "?R (K' + K) M0" by blast
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   426
      with n have ?case1 by simp then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   427
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   428
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   429
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   430
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   431
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   432
proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   433
  let ?R = "mult1 r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   434
  let ?W = "acc ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   435
  {
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   436
    fix M M0 a
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   437
    assume M0: "M0 \<in> ?W"
12399
wenzelm
parents: 12338
diff changeset
   438
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   439
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   440
    have "M0 + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   441
    proof (rule accI [of "M0 + {#a#}"])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   442
      fix N
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   443
      assume "(N, M0 + {#a#}) \<in> ?R"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   444
      then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   445
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   446
        by (rule less_add)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   447
      then show "N \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   448
      proof (elim exE disjE conjE)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   449
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   450
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   451
        then have "M + {#a#} \<in> ?W" ..
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   452
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   453
      next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   454
        fix K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   455
        assume N: "N = M0 + K"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   456
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   457
        then have "M0 + K \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   458
        proof (induct K)
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   459
          case empty
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   460
          from M0 show "M0 + {#} \<in> ?W" by simp
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   461
        next
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   462
          case (add K x)
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   463
          from add.prems have "(x, a) \<in> r" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   464
          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   465
          moreover from add have "M0 + K \<in> ?W" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   466
          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   467
          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   468
        qed
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   469
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   470
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   471
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   472
  } note tedious_reasoning = this
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   473
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   474
  assume wf: "wf r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   475
  fix M
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   476
  show "M \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   477
  proof (induct M)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   478
    show "{#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   479
    proof (rule accI)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   480
      fix b assume "(b, {#}) \<in> ?R"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   481
      with not_less_empty show "b \<in> ?W" by contradiction
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   482
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   483
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   484
    fix M a assume "M \<in> ?W"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   485
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   486
    proof induct
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   487
      fix a
12399
wenzelm
parents: 12338
diff changeset
   488
      assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   489
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   490
      proof
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   491
        fix M assume "M \<in> ?W"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   492
        then show "M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   493
          by (rule acc_induct) (rule tedious_reasoning)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   494
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   495
    qed
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   496
    then show "M + {#a#} \<in> ?W" ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   497
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   498
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   499
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   500
theorem wf_mult1: "wf r ==> wf (mult1 r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   501
  by (rule acc_wfI, rule all_accessible)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   502
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   503
theorem wf_mult: "wf r ==> wf (mult r)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   504
  by (unfold mult_def, rule wf_trancl, rule wf_mult1)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   505
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   506
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   507
subsubsection {* Closure-free presentation *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   508
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   509
(*Badly needed: a linear arithmetic procedure for multisets*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   510
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   511
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   512
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   513
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   514
text {* One direction. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   515
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   516
lemma mult_implies_one_step:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   517
  "trans r ==> (M, N) \<in> mult r ==>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   518
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   519
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   520
  apply (unfold mult_def mult1_def set_of_def)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   521
  apply (erule converse_trancl_induct, clarify)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   522
   apply (rule_tac x = M0 in exI, simp, clarify)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   523
  apply (case_tac "a :# K")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   524
   apply (rule_tac x = I in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   525
   apply (simp (no_asm))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   526
   apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   527
   apply (simp (no_asm_simp) add: union_assoc [symmetric])
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   528
   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   529
   apply (simp add: diff_union_single_conv)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   530
   apply (simp (no_asm_use) add: trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   531
   apply blast
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   532
  apply (subgoal_tac "a :# I")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   533
   apply (rule_tac x = "I - {#a#}" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   534
   apply (rule_tac x = "J + {#a#}" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   535
   apply (rule_tac x = "K + Ka" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   536
   apply (rule conjI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   537
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   538
   apply (rule conjI)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   539
    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   540
    apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   541
   apply (simp (no_asm_use) add: trans_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   542
   apply blast
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   543
  apply (subgoal_tac "a :# (M0 + {#a#})")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   544
   apply simp
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   545
  apply (simp (no_asm))
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   546
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   547
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   548
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   549
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   550
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   551
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   552
  apply (erule size_eq_Suc_imp_elem [THEN exE])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   553
  apply (drule elem_imp_eq_diff_union, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   554
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   555
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   556
lemma one_step_implies_mult_aux:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   557
  "trans r ==>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   558
    \<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))
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   559
      --> (I + K, I + J) \<in> mult r"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   560
  apply (induct_tac n, auto)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   561
  apply (frule size_eq_Suc_imp_eq_union, clarify)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   562
  apply (rename_tac "J'", simp)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   563
  apply (erule notE, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   564
  apply (case_tac "J' = {#}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   565
   apply (simp add: mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   566
   apply (rule r_into_trancl)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   567
   apply (simp add: mult1_def set_of_def, blast)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   568
  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   569
  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   570
  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   571
  apply (erule ssubst)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   572
  apply (simp add: Ball_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   573
  apply (subgoal_tac
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   574
    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   575
      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   576
   prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   577
   apply force
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   578
  apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   579
  apply (erule trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   580
  apply (rule r_into_trancl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   581
  apply (simp add: mult1_def set_of_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   582
  apply (rule_tac x = a in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   583
  apply (rule_tac x = "I + J'" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   584
  apply (simp add: union_ac)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   585
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   586
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   587
lemma one_step_implies_mult:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   588
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   589
    ==> (I + K, I + J) \<in> mult r"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   590
  apply (insert one_step_implies_mult_aux, blast)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   591
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   592
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   593
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   594
subsubsection {* Partial-order properties *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   595
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 11868
diff changeset
   596
instance multiset :: (type) ord ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   597
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   598
defs (overloaded)
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   599
  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   600
  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   601
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   602
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   603
  unfolding trans_def by (blast intro: order_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   604
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   605
text {*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   606
 \medskip Irreflexivity.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   607
*}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   608
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   609
lemma mult_irrefl_aux:
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   610
    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   611
  apply (induct rule: finite_induct)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   612
   apply (auto intro: order_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   613
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   614
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   615
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   616
  apply (unfold less_multiset_def, auto)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   617
  apply (drule trans_base_order [THEN mult_implies_one_step], auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   618
  apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   619
  apply (simp add: set_of_eq_empty_iff)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   620
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   621
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   622
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   623
by (insert mult_less_not_refl, fast)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   624
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   625
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   626
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   627
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   628
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   629
  apply (unfold less_multiset_def mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   630
  apply (blast intro: trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   631
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   632
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   633
text {* Asymmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   634
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   635
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   636
  apply auto
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   637
  apply (rule mult_less_not_refl [THEN notE])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   638
  apply (erule mult_less_trans, assumption)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   639
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   640
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   641
theorem mult_less_asym:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   642
    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   643
  by (insert mult_less_not_sym, blast)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   644
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   645
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   646
  unfolding le_multiset_def by auto
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   647
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   648
text {* Anti-symmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   649
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   650
theorem mult_le_antisym:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   651
    "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   652
  unfolding le_multiset_def by (blast dest: mult_less_not_sym)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   653
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   654
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   655
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   656
theorem mult_le_trans:
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   657
    "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   658
  unfolding le_multiset_def by (blast intro: mult_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   659
11655
923e4d0d36d5 tuned parentheses in relational expressions;
wenzelm
parents: 11549
diff changeset
   660
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   661
  unfolding le_multiset_def by auto
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   662
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   663
text {* Partial order. *}
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   664
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   665
instance multiset :: (order) order
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   666
  apply intro_classes
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   667
     apply (rule mult_le_refl)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   668
    apply (erule mult_le_trans, assumption)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   669
   apply (erule mult_le_antisym, assumption)
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   670
  apply (rule mult_less_le)
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   671
  done
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   672
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   673
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   674
subsubsection {* Monotonicity of multiset union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   675
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   676
lemma mult1_union:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   677
    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   678
  apply (unfold mult1_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   679
  apply (rule_tac x = a in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   680
  apply (rule_tac x = "C + M0" in exI)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   681
  apply (simp add: union_assoc)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   682
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   683
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   684
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   685
  apply (unfold less_multiset_def mult_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   686
  apply (erule trancl_induct)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   687
   apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   688
  apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   689
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   690
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   691
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   692
  apply (subst union_commute [of B C])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   693
  apply (subst union_commute [of D C])
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   694
  apply (erule union_less_mono2)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   695
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   696
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   697
lemma union_less_mono:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   698
    "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   699
  apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   700
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   701
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   702
lemma union_le_mono:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   703
    "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   704
  unfolding le_multiset_def
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   705
  by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   706
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   707
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   708
  apply (unfold le_multiset_def less_multiset_def)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   709
  apply (case_tac "M = {#}")
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   710
   prefer 2
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   711
   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   712
    prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   713
    apply (rule one_step_implies_mult)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   714
      apply (simp only: trans_def, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   715
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   716
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   717
lemma union_upper1: "A <= A + (B::'a::order multiset)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   718
proof -
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   719
  have "A + {#} <= A + B" by (blast intro: union_le_mono)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   720
  then show ?thesis by simp
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   721
qed
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   722
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   723
lemma union_upper2: "B <= A + (B::'a::order multiset)"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   724
  by (subst union_commute) (rule union_upper1)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   725
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   726
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   727
subsection {* Link with lists *}
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   728
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   729
consts
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   730
  multiset_of :: "'a list \<Rightarrow> 'a multiset"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   731
primrec
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   732
  "multiset_of [] = {#}"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   733
  "multiset_of (a # x) = multiset_of x + {# a #}"
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   734
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   735
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   736
  by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   737
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   738
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   739
  by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   740
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   741
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   742
  by (induct x) auto
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   743
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   744
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   745
  by (induct xs) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   746
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   747
lemma multiset_of_append [simp]:
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   748
    "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   749
  by (induct xs fixing: ys) (auto simp: union_ac)
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   750
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   751
lemma surj_multiset_of: "surj multiset_of"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   752
  apply (unfold surj_def, rule allI)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   753
  apply (rule_tac M=y in multiset_induct, auto)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   754
  apply (rule_tac x = "x # xa" in exI, auto)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   755
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   756
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   757
lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   758
  by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   759
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   760
lemma distinct_count_atmost_1:
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   761
   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   762
   apply (induct x, simp, rule iffI, simp_all)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   763
   apply (rule conjI)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   764
   apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   765
   apply (erule_tac x=a in allE, simp, clarify)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   766
   apply (erule_tac x=aa in allE, simp)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   767
   done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   768
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   769
lemma multiset_of_eq_setD:
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   770
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   771
  by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   772
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   773
lemma set_eq_iff_multiset_of_eq_distinct:
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   774
  "\<lbrakk>distinct x; distinct y\<rbrakk>
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   775
   \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   776
  by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   777
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   778
lemma set_eq_iff_multiset_of_remdups_eq:
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   779
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   780
  apply (rule iffI)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   781
  apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   782
  apply (drule distinct_remdups[THEN distinct_remdups
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   783
                      [THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   784
  apply simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   785
  done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   786
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   787
lemma multiset_of_compl_union [simp]:
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   788
    "multiset_of [x\<in>xs. P x] + multiset_of [x\<in>xs. \<not>P x] = multiset_of xs"
15630
cc3776f004e2 fixed typo (multiset_append)
kleing
parents: 15402
diff changeset
   789
  by (induct xs) (auto simp: union_ac)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   790
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   791
lemma count_filter:
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   792
    "count (multiset_of xs) x = length [y \<in> xs. y = x]"
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   793
  by (induct xs) auto
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   794
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   795
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   796
subsection {* Pointwise ordering induced by count *}
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   797
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   798
definition
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   799
  mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool"   ("_ \<le># _"  [50,51] 50)
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
   800
  "(xs \<le># ys) = (\<forall>a. count xs a \<le> count ys a)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   801
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   802
lemma mset_le_refl[simp]: "xs \<le># xs"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   803
  unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   804
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   805
lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   806
  unfolding mset_le_def by (fast intro: order_trans)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   807
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   808
lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   809
  apply (unfold mset_le_def)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   810
  apply (rule multiset_eq_conv_count_eq[THEN iffD2])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   811
  apply (blast intro: order_antisym)
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   812
  done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   813
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   814
lemma mset_le_exists_conv:
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   815
  "(xs \<le># ys) = (\<exists>zs. ys = xs + zs)"
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   816
  apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   817
  apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   818
  done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   819
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   820
lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   821
  unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   822
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   823
lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   824
  unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   825
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   826
lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws"
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   827
  apply (unfold mset_le_def)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   828
  apply auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   829
  apply (erule_tac x=a in allE)+
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   830
  apply auto
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   831
  done
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   832
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   833
lemma mset_le_add_left[simp]: "xs \<le># xs + ys"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   834
  unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   835
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   836
lemma mset_le_add_right[simp]: "ys \<le># xs + ys"
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   837
  unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   838
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   839
lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x"
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   840
  apply (induct x)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   841
   apply auto
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   842
  apply (rule mset_le_trans)
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   843
   apply auto
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   844
  done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   845
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   846
end